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\begin{document}
\title{Gluon mass and small-$x$ dynamics in hadrons}
\author{ Stanis{\l}aw D. G{\l}azek }
%\email{stglazek@fuw.edu.pl} 
\affiliation{ Institute of Theoretical Physics \\ 
Faculty of Physics, University of Warsaw \\  
ul. Pasteura 5, 02-093 Warsaw, Poland }
\date{\today, 20260306 14:38 pia Ekologiczna po Camp4} 
\begin{abstract}
Computational access to the logarithmically scale-dependent Hamiltonian eigenstate picture for hadrons in the space of virtual quark and gluon states in the canonical front form of QCD is impeded by the small-x divergences stronger than logarithmic. We propose introducing a gluon mass parameter and an auxiliary color-octet scalar field to regulate and break through this barrier, using the renormalization group procedure for effective particles (RGPEP) [1] in the limit of the gluon mass parameter going to zero at the end of a calculation. At that point the auxiliary field decouples from the finessed effective constituent dynamics, as required in the gauge theory. The same method yields cancellation of the quadratic ultraviolet transverse divergences in the self-interactions. We explain how this approach works in virtual quark and gluon scattering amplitudes and in Hamiltonian eigenvalue problems for bound states. We focus the discussion on the case of heavy quarkonia [2], where one obtains rotationally symmetric effective interactions at small interquark distances, akin to harmonic oscillator potential, and only globally colorless states having finite masses in the limit of the gluon mass going to zero. Previously [3], it was suggested that the QCD vacuum effects, such as in the QCD sum rules [4] and due to the instantons [5], can manifest themselves effectively through the front form counterterms to the small-x regularization singularities. We advance the hypothesis that the corresponding effective interaction terms can instead appear in the finite renormalization-group evolution when the scale parameter is lowered towards $\Lambda_{\rm QCD}$ in the RGPEP scheme.

[1] S.D. Głazek,  Eur. Phys. J. Spec. Top. (2026), https://doi.org/10.1140/epjs/s11734-026-02127-y, \cite{Glazek:2026xnp}

[2] K. Serafin et al., Phys.Rev.D 109, 016017 (2024), https://arxiv.org/abs/2310.00365, \cite{Serafin:2023pkf}

[3] K. G. Wilson et al., Phys.Rev.D 49, 6720 (1994), https://arxiv.org/abs/hep-th/9401153, \cite{Wilson:1994fk}

[4] M. Shifman, A. Vainshtein, I.  Zakharov, Nucl.Phys.B 147, 385 (1979), \cite{Shifman:1978bx,Shifman:1978by}

[5] E. Shuryak, I. Zahed, https://arxiv.org/abs/2601.15085, \cite{Shuryak:2026pqt}

Nordsieck small energy photons

Precise derivation of the logarithmically scale-dependent Hamiltonian eigenstate picture for hadrons 
in the space of virtual quark and gluon states of the canonical front form of QCD cannot be achieved 
without first  addressing the problem of small$x$ divergences stronger than logarithmic. We propose 
to facilitate cancellation of these strong divergences using a mass parameter for gluons and an auxiliary 
color-octet scalar field corresponding to the longitudinal polarization. The auxiliary field decouples from 
the hadronic constituent dynamics in the limit of the gluon mass parameter going to zero, as required 
in the gauge theory. The same method simultaneously facilitates cancellation of the quadratic ultraviolet 
transverse divergences in the self-interactions. After explaining how this approach works for virtual quark 
and gluon scattering amplitudes, we describe how it applies in the Hamiltonian eigenvalue problems for 
bound states. 

Saturation of gluons because small $p^+$ leads to large $s$?

Explanation of how Susskind's vacuum freezing occurs, by making small-$x$ gluons freeze into the bound-state
eigenvalue shape  

$E(P^\perp=0, P^+=M) = M$. In exact solution the eigenvalues do not depend on $s$ and 
therefore do not depend on $P^+$ and $P^\perp$ Precision of solving the RGPEP
equation and the associated eigenvalue problems is reflected by degree of independence 
of the eigenvalues from $P^+$.

van der Waals forces?

Instead of seeking vacuum effects in reaction to the cutoff $\epsilon^+$, like Wilson et al., 
it is proposed to work with $m_g$ and approach the vacuum issues for finite size $s$ of 
the effective quanta in the limit $m_g \to 0$, the parameter that can be used to describe 
confinement in the Fock space, including the quantum gluon string formation mechanism.
So, instead of studying vacuum effects through regularization, study them using the RGPEP
in the theory with $m_g \to 0$. The point is that the parameter $m_g$ regulates infrared
singularities and the RGPEP can be used to take care of the ultraviolet effects in the 
eigenvalue problems.

Small denominators are avoided in computation of $H_s$.

{\red Rephrase deeply: Advance the hypothesis that the vacuum-like effects develop in the renormalized small-$x$ dynamics 
and can be found using the RGPEP by matching the parton dynamics~\cite{Casher:1973vh,Kogut:1972di},
QCD sum rules~\cite{Shifman:1978bx,Shifman:1978by}, large momentum effective theory methods of 
lattice approach~\cite{Gao:2021dbh,Ji:2024oka},  and including vacuum effects in the parton 
picture~\cite{Shuryak:2026pqt}.}

Quarks and gluons with large $p^+$ are little evolved in the RGPEP. They closely resemble partons 
understood as quanta in canonical QCD. By comparison, quarks and gluons with small $p^+$, on the 
order of a hadron mass, are evolved a lot in the RGPEP towards the picture of the constituent quarks. 
This means that the quanta with large $p^+$, like in the parton model, interact nearly as in the canonical 
QCD, while the interaction vertices of the quanta of small $p^+$, such as in the rest frame of a hadron, 
are tempered by relatively narrow form factors, include terms resembling potentials and are expected 
to theoretically support the constituent quark models success in the classification of hadrons.

Technical improvements: boost invariant $f^r$, simple generator $[\cH_f, \cH_I]$, no need for counterterm 
$\delta(q_{gluon})$, confinement for $m_g \to 0$.  


\end{abstract}
\maketitle
\section{Verbatim}
\begin{verbatim}
\section{Introduction}
\label{intro}
\section{Bare Hamiltonian}                                                                
\label{bareH}  
        \subsection{FF Hamiltonian of QCD}                                                 
        \label{FFHQCD}
        \subsection{Gluon mass and auxiliary field}                                       
        \label{amendedH}
        \subsection{Quantum Hamiltonian}                                                  
        \label{QH}
\section{Regularization}                                                                     
\label{regularization} 
        \subsection{Operator labels and cutoff $\epsilon^+$}                                     
        \label{labels}
        \subsection{The weak-coupling issue of vacuum}                               
        \label{vacuum}
        \subsection{Regulating interaction terms}                                         
        \label{regf}
               \subsubsection{Terms linear in $g$ }                                              
               \label{linearing}
               \subsubsection{Terms quadratic in $g$ }                                         
               \label{linearing}
               \subsubsection{Fully regulated bare Hamiltonian}                              
               \label{fullyreg}
\section{Gluon mass in perturbative transition amplitudes}                                      
\label{transition}
        \subsection{Gluon severe small-$x$ singularity}                                
        \label{gluonsmallx}
        \subsection{Canceling out of $1/x^2$ in scattering amplitudes}           
        \label{xintransition}
        \subsection{Small $x$ in self-interactions}                                        
        \label{Sigmax}
\section{Bound-state eigenvalue problem}                                         
\label{BSeqs}
\section{Perturbative series for $H_s$}                                                 
\label{seriesHt}
\section{Homogeneous solutions for $H_s$ and small-$x$}                   
\label{homogeneoussol}
        \subsection{Example of the homogeneous small-$x$ divergence}
        \label{Uqbarq}
        \subsection{Homogenous counterterm computation}                 
        \label{homogeneousCT}
\section{Inhomogeneous solutions for $H_s$}
\label{inhomogeneoussol}
        \subsection{Weak-coupling expansion} 
        \label{2nd3rdorderHt}
\section{Quarkonium dynamics}
\label{Qdynamics}
        \subsection{Cancelling out of $1/x^2$ in eigenvalue problems}
        \label{xinboundstates}
\section{Eigenvalue problems for $m_g \to 0$, $g \to g_s$, $m_g \to m_G$} 
\label{mgto0}
        \subsection{Single-quark eigenvalue problem}
        \label{singleQ}
        \subsection{Heavy effective gluons}
        \label{heavyG}
        \subsection{Details of quarkonium eigenvalue problem}
        \label{DetailsEVforHLR}
        \subsection{Bound-state equation recapitulated}
        \label{BSrecap}
\section{Beyond perturbative expansion for $H_s$}
\label{beyondPT}
         \subsection{Successive approximations}
         \label{succapprox}
         \subsection{Operator cutoff}
         \label{operatorcutoff
\section{Conclusion}                                                                          
\label{concl}
\begin{appendix}
\section{Notation}
\label{notation}
\end{appendix}
\end{verbatim}
%{\red goto \ref{3311}}
\tableofcontents
%%%%%%%%%%%%
\section{Introduction}
\label{intro}
%%%%%%%%%%%%
For precise application to the physics of hadron constituents, the Hamiltonian of QCD 
needs to be constructed as an operator acting in the space of states of virtual quarks and 
gluons in a definite form of dynamics. Among the forms identified in~\cite{Dirac:1949cp}, 
the commonly known way of describing the evolution of physical systems using time $s$ 
of some observer, is called the instant form (IF). The name originates from the space-time 
hyperplane corresponding to one value of $s$, an instant. The less familiar form, in which 
an observer describes the system evolution from one value of $x^+=t+z$ to another, is 
called the front form (FF). The name comes from the fact that a space-time hyperplane 
corresponding to one value of $x^+$ is swept by the front of a plane wave of light moving 
against the $z$-axis. The choice of the FF of dynamics for describing how quarks and gluons 
form hadrons stems from the desire to describe the hadrons in any state of motion, such as 
being at rest in a laboratory or moving with a speed nearly equal to the speed of light after 
leaving an accelerator. In the IF, boosting hadrons is associated with complex dynamical 
effects because the quantum generators of boosts involve interactions. In the FF, there are 
3 independent Lorentz transformations that are free from interaction effects and the dynamics 
is invariant with respect to them. This feature suggests a simplification of the task of relating 
the observed parton structure of fast moving hadrons and their spectroscopic classification 
at rest in terms of states of 3 quarks or a quark-antiquark pair~\cite{ParticleDataGroup:2024cfk}. 
Besides offering a way to approach the quantum boost problem for bound states, the FF 
Hamiltonian of QCD naturally provides the eigenvalue equations determining how the 
physical hadrons, represented by the eigenstates, are built from the quanta of quark and 
gluon fields.

However, the canonical FF Hamiltonian of QCD is a singular operator. One needs to identify 
a computational path that leads from the canonical theory to the finite solutions of the 
bound-state eigenvalue problems. Asymptotic freedom suggests that the binding mechanism 
for hadronic constituents involves logarithms of their relative momenta. But to reliably compute 
the logarithmic effects in bound-state dynamics, one first needs to find a way to control the 
functions more singular than the logarithms. This paper proposes a way to do so. The difficulty 
to overcome is that the FF of the theory involves singular functions of two different kinds of 
variables. 

In particular, the constituent momenta measured along the $z$-axis greatly differ from the 
the momenta in directions perpendicular to that axis. Consequently, the FF Hamiltonians do 
not possess any explicit three-dimensional rotation symmetry that the IF Hamiltonians have. 
Instead of the three-dimensional momenta, say $\vec p$, one works with the transverse momenta 
of quanta, denoted by $p^\perp = (p^1,p^2)$, and the longitudinal momenta, denoted by $p^+
=p^0 + p^3$. The corresponding boost-invariant variables of two kinds are the quanta relative 
transverse momenta $k^\perp = p^\perp - x P^\perp$ and the ratios $x=p^+/P^+$, where $P$ 
denotes the total momentum of the quanta involved in a Hamiltonian interaction term. Divergences 
occur due to $k^\perp$ going to $\infty$ or 0 and due to $x$ going to 0 or 1. For example, there 
are singular functions that behave as $1/x^2$ or $1/x$. Reliable treatment of $\ln x$ when 
$x \to 0$ requires a Hamiltonian renormalization procedure for the divergences in $x$ and 
$k^\perp$~\cite{Wilson:1994fk}. The first step of any such procedure is to regulate the diverging 
terms.

Regularization of the Hamiltonian terms can be arranged by limiting the values of $k^\perp$ and 
$x$. Suppose one does it for the transverse and longitudinal coordinates separately. In that case, 
the ultraviolet divergences associated with the longitudinal and transverse directions pile up on 
each other, which leads to a complex renormalization procedure. One can correlate the regularization 
cutoffs on $k^\perp$ and $x$ by limiting the total invariant-mass squared, denoted by $\cM^2$, of 
the quanta involved in an interaction term. Each of the quanta contributes to $\cM^2$ via $P^+$ 
times 
\beq
k^- \es k^0 - k^3 \rs {k^{\perp 2}  + m^2 \over x P^+} \ ,
\eeq
with $m$ being a mass assigned to a quantum in the free part of the Hamiltonian. Limiting $\cM$ by 
a cutoff from above, say $\Lambda$, one simultaneously limits $|k^\perp|$ from above and prevents 
$x$ from approaching 0 because $m^2/x$ is limited. The role of a mass parameter as regulator is of 
interest because mass does not distinguish any direction in the space of four-momenta. The issue 
with gluons, however, is that they are massless, as befits gauge bosons. For them, 
\beq
\label{kperponly}
P^+ k^- \es {k^{\perp 2} \over x } \ .
\eeq
Small $x$ is not limited from below by the upper limit on the invariant mass $\cM$ when $k^\perp$ is 
correspondingly small. Therefore, the ultraviolet renormalization procedure is interfered with by the infrared 
issues of the theory. Namely, the small-$x$ region belongs to the ultraviolet regime when $k^\perp$ is 
sizable and it also contributes to the infrared regime when $k^\perp$ is small. This is where the gluon 
mass parameter, $m_g$, can be introduced as a regulator disentangling these regimes. Replacement of 
$k^{\perp 2}$ in Eq.~(\ref{kperponly}) by  $k^{\perp 2} + m_g^2$ makes small $x$ correspond to the 
ultraviolet regime even if $k^\perp$ vanishes. But a gluon mass term the canonical 
FF Hamiltonian of QCD for the regularization purposes, the resulting virtual quark and gluon transition 
amplitudes involve integrals of, in essence,
\beq
\label{severedivergences}
{m_g^2 \over k^{\perp 2} + m_g^2} \ {1 \over x^2} \ .
\eeq
The small-$x$ singularity is associated with a function of the transverse momentum. The singularity 
strengthens when $k^\perp$ decreases. We propose to counter this gluon mass effect by adding quanta 
of an auxiliary scalar octet field $\phi = \phi^a T^a$ to the dynamics. These quanta correspond to the 
longitudinally polarized gluons. In the limit $m_g \to 0$, the longitudinal quanta decouple from the theory 
when $m_g \to 0$. Such decoupling is know in case of the FF of QED~\cite{Soper:1971wn,Yan:1973qf}, 
but as far as the author knows has not been taken advantage of in the FF Hamiltonian of QCD as it is 
done in this paper.

Since the paper is focused on the use of gluon mass for the purpose of regularization and renormalization 
of the FF Hamiltonian of QCD, it does not report on or directly refer to most of the rich literature on the FF 
formulation of quantum field theory (QFT) and its applications. The literature can be traced 
following~\cite{Kogut:1972di,Yan:1973qg,Lepage:1980fj,Namyslowski:1985zq,Wilson:1994fk,Burkardt:1995ct,
Brodsky:1997de,Carbonell:1998rj,Brodsky:2014yha,Ji:2020ect}. 

In Sec.~\ref{bareH} . . . {\red Paper outline:} 
%%%%%%%%%%%%
\section{Bare Hamiltonian}                       
\label{bareH}  
%%%%%%%%%%%%
Canonical FF Hamiltonian of QCD is derived~\cite{Yan:1973qg,Lepage:1980fj} from the Lagrangian density
\beq
\label{LQCD}
\cL_{\rm QCD}
\es
\bar \psi \left( i / \hspace{-5pt}\partial - g / \hspace{-7pt}A - m \right) \psi 
- 
\2 {\rm ~Tr~} G_{\mu \nu} G^{\mu \nu } \ ,
\eeq
where the sum over quark flavors is assumed but not indicated. The derivation is recalled below for it is used 
later to explain our procedure for dealing with the small-$x$ divergences caused by gluons. 
%%%%%%%%%%%%%%%%%%%%%
\subsection{FF Hamiltonian of QCD}
\label{FFHQCD}
%%%%%%%%%%%%%%%%%%%%%
By minimizing action described by $\cL_{\rm QCD}$ of Eq.~(\ref{LQCD}), one obtains equations
\beq
\label{quarkEoM}
( i \f \partial - g \f \hspace{-2pt}A - m ) \psi \es 0 \ , \\
\label{gluonEoM}
\partial_\alpha G^{a\alpha \beta} 
+ 
ig [ A_\alpha , G^{\alpha \beta}]^a + J_\psi^{a \beta}
\es 0 \ ,
\eeq
where the quark current is $J_\psi^{a \beta} = -g \bar \psi 
T^a \gamma^\beta  \psi$. In the gauge $A^+=0$, 
Eq.~(\ref{quarkEoM}) relates the field $\psi_-= \Lambda_-\psi$ 
to the fields $\psi_+ = \Lambda_+ \psi$ and $A^\perp$, 
where $\Lambda_\pm = \2 \gamma^0 \gamma^\pm$. 
Namely,
\beq
\label{cpsi} 
\psi_- 
\es 
{ 1 \over i \partial^+ } [ \alpha^\perp (i \partial^\perp - g A^\perp) + \beta m ]  \psi_+ \ .
\eeq
Similarly, Eq.~(\ref{gluonEoM}) with $\beta=+$, provides the constraint 
\beq
\label{A-c}
A^- \es  {2 \partial^\perp \over \partial^+} A^\perp 
+ {2 \over \partial^{+2}} (J_A^+  + J_\psi^+) \ ,
\eeq
with $J_A^{a\mu} = - ig [  \partial^\mu A^\perp , A^\perp ]^a$. Inversion of $\partial^{+2}$ is a remote analog of 
inverting Laplasian in the Poisson equation to obtain the Coulomb potential in the IF of Hamiltonian dynamics. Generally, 
inversion of the derivative $\partial^+$ is understood here in terms of the inverse of momentum in the Fourier transform. 
Field modes constant in $x^-$ will not appear in the regulated theory, see below. When the coupling constant $g$ is set
to 0, the constraints simplify to  the constraint equations for free particles,
\beq
\label{psif-} 
\psi_{f-} 
\es 
{ 1 \over i \partial^+ } \left( \alpha^\perp i \partial^\perp + \beta m \right) \psi_+ \ , \\
\label{Af-}
A_f^- \es  {2 \partial^\perp \over \partial^+} A^\perp \ .
\eeq
One uses these equations to define the fields
\beq
\label{psi_f}
\psi_f \es \psi_+ + \psi_{f-} \ , \\
\label{A_f}
A_f \es \left(A^-_f, A^+=0, A^\perp \right) \ .
\eeq
Canonical formula for the FF Hamiltonian of QCD expressed in terms of the fields $\psi_f$ and $A_f$, is provided 
by the component $P^-$ of the total four-momentum carried by the quark and gluon fields. Thus, $P^-$  is an 
integral of Noether's energy-momentum tensor-density component $+-$ over the front,  
\beq
H_{\rm QCD} \es \int_F \  T_{\rm QCD}^{+ -} \ ,
\eeq
where $\int_F = \int d^4x \, \delta(x^+)$.
The Noether tensor is 
\beq
T_{\rm QCD}^{\mu \nu} 
\es
\sum_\chi
       { \partial \cL_{\rm QCD} \over \partial \partial_\mu \chi } \
       \partial^\nu \chi
-      
g^{\mu \nu}  \cL_{\rm QCD} \ .
\eeq
The sum extends over all fields $\chi$ in $\cL_{\rm QCD}$
with $A^+=0$. The Hamiltonian reads
\beq
\label{HrmQCD}
H_{\rm QCD} \es \int d^2x^\perp dx^- \ \cH_{\rm QCD} \ ,
\eeq
where the integrated density is, {\it cf.} \cite{Lepage:1982gd},
\beq
\label{cHcanQCD}
\cH_{\rm QCD} 
\es
\bar \psi_f { \gamma^+ (- \partial^{\perp 2} + m^2) \over 2i \partial^+ }  \psi_f
+
\2 A_{f \mu}^a  \partial^{\perp 2}  A_f^{a \mu}   
-
(J_{\psi f}^{a \mu}  + J_{A f}^{a \mu}) A_{f\mu}^a 
\np
\2 g^2 \bar \psi_f \, \f \hspace{-2pt} A_f 
{ \gamma^+ \over i \partial^+ } 
                          \, \f \hspace{-2pt} A_f \psi_f 
-
\4 g^2 [A_{f\mu}, A_{f_\nu}]^a [A_f^\mu, A_f^\nu]^a  
\nm
\2 (J_{\psi f}^{a+} + J_{A f}^{a+}) {1 \over \partial^{+2}} (J_{Af}^{a+} + J_{\psi f}^{a+}) \ ,
\eeq
with currents 
\beq
\label{quarkcurrent}
J_{\psi f}^{a \mu} \es -g \bar \psi_f T^a \gamma^\mu  \psi_f \ , \\
\label{gluoncurrent}
J_{A f}^{a\mu} \es ig [  \partial^\mu A_{f \nu}, A_f^\nu ]^a \ .
\eeq
These currents depend on the fields $A^\perp$ and $\psi_+$ only, since they include 
explicit solutions for $\psi_{f-}$ and $A_f^-$. They differ from the currents
\beq
\label{fullJpsi}
J_\psi^{a \mu} \es -g \bar \psi T^a \gamma^\mu  \psi 
%\rs
%-g (\bar \psi_f - { g \over i \partial^+ } \alpha^\perp A^\perp \psi_+)  
%T^a \gamma^\mu  
%(\psi_f - { g \over i \partial^+ } \alpha^\perp A^\perp \psi_+) 
\ , \\
\label{fullJA}
J_A^{a\mu}
\es
ig [ A_\alpha , G^{\alpha \mu}]^a
%\rs
%J_{Af}^{a\mu}
%+
%ig [ A_\alpha , \partial^\alpha A^\mu]^a 
%+
%(ig)^2 \left[ A_\alpha , {[}A^\alpha, A^\mu {]} \right]^a  \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gluon mass and auxiliary field} 
\label{amendedH}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The gluon mass term and the associated term for the auxiliary octet scalar field $\phi = \phi^a T^a$, are added to the density 
$\cH_{\rm QCD}$ of Eq.~(\ref{cHcanQCD}) to form the amended density,
\beq
\label{cHcanQCDmg}
\cH^{m_g}_{\rm QCD} 
\es
\cH_{\rm QCD} 
-
\2 m_g^2 A_{f \mu}^a   A_f^{a \mu}
+ 
\2 \phi^a (- \partial^{\perp 2} + m_g^2 ) \phi^a 
+
m_g \phi^a {1 \over \partial^+} (J_{Af}^{a+} + J_{\psi f}^{a+}) \ .
\eeq
The gluon mass parameter $m_g$ and field $\phi$ introduced in Eq.~(\ref{cHcanQCDmg}) suffice for the purpose
of regulating the gluon dynamics, see Sec.~\ref{regularization}. When $m_g$ is set to 0, the amended density 
becomes equal to the canonical $\cH_{\rm QCD}$ of Eq.~(\ref{cHcanQCD}) plus a density for a decoupled, free 
massless field $\phi$. The latter one can be ignored. Thus the limit $m_g \to 0$ recovers the canonical FF 
Hamiltonian of QCD.

In Eq.~(\ref{cHcanQCDmg}), the mass term for the gluon field $A$ involves only the constrained field $A_f$ of 
Eq.~(\ref{A_f}). The kinetic term for the field $\phi$ has the standard form for scalar bosons. The coupling of $\phi$ 
to the quark current is similar in structure, besides the color factors, to the couplings introduced in massive 
QED~\cite{Stueckelberg:1938zz,Soper:1971wn,Glazek:2020xpi}. The auxiliary field $\phi$ couples to the color 
current of gluons. However, it does not couple directly to itself, despite that it carries color. Our scheme differs 
from the generalized Stueckelberg formalism for massive Yang-Mills fields, in which additional constraints are 
imposed to eliminate terms inversely proportional to the gauge-boson mass parameter~\cite{Kunimasa:1967zza}. 
Such additional constraints are not needed here. 

The Fourier modes of the field $\phi$ with momentum $p$ couple to the color currents proportionally to 
$m_g/p^+$. This ratio diverges for $p^+ \to 0$. Precisely such coupling yields the cancellation of the severe 
small-$x$ divergences due to the assignment of mass $m_g$ to the transverse gluos, see Sec.~\ref{transition}.

The density $\cH^{m_g}_{\rm QCD}$ can also be derived from the Lagrangian density 
\beq
\label{LQCDmg}
\cL^{m_g}_{\rm QCD}
\es
\cL_{\rm QCD} +
\2 ( m_g A^a_\mu + \partial_\mu \phi^a )^2 \ ,
\eeq
proceeding in parallel to Sec.~\ref{FFHQCD} using gauge $A^+=0$. The term added to $\cL_{\rm QCD}$ in 
Eq.~(\ref{LQCDmg}) is not gauge invariant, as the regularization purpose it serves is not. Since the field $\phi$ 
is assumed very weakly coupled to gluons and since color is confined, any physical effects due to that field, if 
it existed, would be hard to detect. In principle, $\phi$ might contribute to the gravitational effects.

Quark fields $\psi$ obey the same equations in the case of $\cL^{m_g}_{\rm QCD}$ as in the case of 
$\cL_{\rm QCD}$, for the added term does not depend on the fermion fields. Therefore, the constraint 
on $\psi_-$ is the same as in QCD with $m_g=0$, Eq.~~(\ref{cpsi}). Keeping $A^+=0$, variation of $A^-$ 
yields the constraint 
\beq
\label{A-c1}
A^- \es  A_f^- 
+ {2 \over \partial^{+2}} (J_{Af}^+  + J_{\psi f}^+) + {2m_g \over \partial^+} \phi \ ,
\eeq
which differs from the canonical Eq.~(\ref{A-c}) by the last 
term only, given the identities 
$J_A^+ = J_{Af}^+$ and $J_\psi^+ = J_{\psi f}^+$ 
that follow from $\gamma^{+2}=0$. The field $\phi$ 
obeys
\beq
\Box \phi + m_g \partial_\mu  A^\mu \es 0 \ .
\eeq
The constraint Eq.~(\ref{A-c1}) implies
\beq
\label{constraintpartialA1}
\partial_\mu A^\mu \es  {1 \over \partial^+} (J_A^+ + J_\psi^+) + m_g \phi \ , \\
\label{phiJJ}
( \Box + m_g^2 ) \phi
\es
- {m_g \over \partial^+} (J_A^+ + J_\psi^+)   \ .
\eeq
The latter equation exhibits the so-called ``longitudinal'' or ``good'' currents~\cite{Fritzsch:1972jv} as the only 
sources of the field $\phi$. The amended Hamiltonian density $\cH_{\rm QCD}^{m_g}$ is obtained by evaluating 
$P^-$ that follows from the energy-momentum tensor density implied by $\cL^{m_g}_{\rm QCD}$.
%%%%%%%%%%%%%%%%
\subsection{Quantum Hamiltonian}
\label{QH}
%%%%%%%%%%%%%%%%
The Hamiltonian,
\beq
H_{\rm QCD}^{m_g} \es \int d^2x^\perp dx^- \ \cH_{\rm QCD}^{m_g} \ ,
\eeq
where the density $\cH_{\rm QCD}^{m_g}$ is a function of fields $\psi_f$, $A_f$, and $\phi$, given in 
Eq.~(\ref{cHcanQCDmg}), is turned into a quantum operator $\hat H_{\rm QCD}^{m_g}$ by replacing 
the fields at $x^+=0$ with operators, 
\beq
\label{barepsif}
\hat \psi_f
\es
\sum_{c=1}^3 \sum_{\sigma = 1}^2  \int [p]
\left[  u_{p\sigma} \chi_c \hat b_{p \sigma c} e^{-ipx}
      +    v_{p\sigma} \chi_c \hat d^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0} \ , \\
\label{bareAf}
\hat A_f^\mu
\es
\sum_{c=1}^8 
\sum_{\sigma =1}^2  \int [p]
\left[  \varepsilon^\mu_{p\sigma} T^c \hat a_{p\sigma c} e^{-ipx}
        +  \varepsilon^{\mu *}_{p\sigma} T^c \hat a^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0} \ , \\
\label{hatphi}
\hat \phi
\es
\sum_{c=1}^8 
\int [p]
\left[  -i T^c \hat a_{p3c} e^{-ipx}
        +  i T^c \hat a^\dagger_{p3c} e^{ipx}
\right]_{x^+=0} \ .
\eeq
In this notation, $c$ refers to colors of $SU(3)$ and $\sigma$ denotes spin projections on $z$-axis. 
Momentum integration measure reads $[p] = dp^+ d^2p^\perp /[2p^+ (2\pi)^3]$ with $p^+ > 0$, 
$p^\perp = (p^1,p^2)$, $px = p^-x^+/2 + p^+x^-/2 - p^\perp x^\perp$. The quark spinors, $u_{p\sigma}$, 
$v_{p\sigma}$, and gluon polarization vectors, $\varepsilon^\mu_{p\sigma}$, are given in~\cite{Lepage:1980fj}, 
App. A. Flavors of quarks are kept in mind. The creation and annihilation operators, $\hat b, \hat b^\dagger,
\hat d, \hat d^\dagger, a, a^\dagger$, will be commonly referred to as particle operators. They are assumed to 
satisfy the anti-commutation relations for fermions and commutation relations for bosons, 
\beq
\label{comrules}
\left\{ \hat b_{p \sigma c}, \hat b_{p' \sigma' c'}^\dagger \right\}
=
\left\{ \hat d_{p \sigma c}, \hat d_{p' \sigma' c'}^\dagger \right\}
=
\left[ \hat a_{p \sigma c}, \hat a_{p' \sigma' c'}^\dagger \right]
=
2p^+(2\pi)^3 \delta^3(p-p') \delta_{\sigma \sigma'} \delta_{cc'} \ .
\eeq
Other commutators or anti-commutators vanish. Fermion operators commute with the boson 
operators. By definition, the annihilation operators $b, d, a$ produce 0 when they act on the 
vacuum state, denoted by $|0\rangle$. Products of particle operators in all terms of the Hamiltonian 
$\hat H_{\rm QCD}^{m_g}$ are normal-ordered according to the pattern $a^\dagger b^\dagger 
d^\dagger d b a$. The normal-ordering is indicated by double dots. Thus, 
\beq
\label{EqbareH}
\hat H_{\rm QCD}^{m_g} \es \int d^2x^\perp dx^- \, 
: \cH_{\rm QCD}^{m_g}(\hat \psi_f, \hat A_f, \hat \phi): \ .
\eeq
This Hamiltonian is singular and requires regularization as a part of the renormalization procedure
described in next sections. The normal ordering will be always present but not indicated. Hats will 
be omitted to farther simplify the notation.
%%%%%%%%%%%%%%%%
\section{Regularization}                   
\label{regularization} 
%%%%%%%%%%%%%%%%
Regulating the Hamiltonian of Eq.~(\ref{EqbareH}) begins with limiting from below the range of momentum 
$p^+$ labeling creation and annihilation operators in the quantum fields in Eqs.~(\ref{barepsif}), (\ref{bareAf}) 
and (\ref{hatphi}). The lower bound on $p^+$ is denoted by $\epsilon^+ > 0$. This lower bound is set to be 
smaller than any $p^+$ in any physical process.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Particle momenta and cutoff $\epsilon^+$}
\label{labels}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To introduce the cutoff $\epsilon^+$, we define the integrals over the momenta that label creation and annihilation 
operators in Eqs.~(\ref{barepsif}), (\ref{bareAf}) and (\ref{hatphi}) as limits of discrete sums. The continuous notation 
is meant to correspond to the discrete momentum spacing that is too small to notice in physical applications. Care 
needs to be exercised because there are two limits to consider in discussion of physical processes, a large-volume limit 
and a long-time limit, {\it cf.}~\cite{Gell-Mann:1953dcn}. We introduce a large cuboid extending from $-2L$ to $2L$ in 
$x^-$ and from $-L$ to $L$ in $x^1$ and $x^2$. The cuboid is assumed large enough to easily contain all processes of 
physical interest. These processes are meant to develop over front ``time'' $x^+$ much shorter than $4L$. 

The discrete plane-wave momenta in the quark and gluon fields are defined using the periodic boundary conditions on 
the cuboid walls at $x^+=0$,
\beq
p_n = \epsilon \, n \ , \ n \rs (n^+,n^1,n^2) \ ,
\eeq
where $\epsilon = \pi/L$, $n^+$ is a natural number while $n^1$ and $n^2$ are integers. The number $n^+$ is positive 
because a free particle of positive mass and finite energy $E$ has $p^+= E + p^z > 0$ no matter how fast and in what 
direction the particle moves. The assumption is that, in the absence of interaction, the theory describes the energy of free 
particles, which we consider to be the field quanta, see {\it e.g.},~\cite{Weinberg:1995mt}, p. 297. Since the box can be 
arbitrarily large, $\epsilon$ can be arbitrarily small. 

Suppose one sets an ultraviolet cutoff $\Delta$ on the particle energy~\cite{Harindranath:1987ex,Bartnik:1988fu,Glazek:1986hs,
Glazek:1997sd},
\beq
E_n \es ( p_n^+ + p_n^-)/2 < \Delta \ ,
\eeq
where $p_n^-=(p_n^{\perp 2} + \mu^2)/p_n^+$. Such cutoff implies that
\beq
\Delta - \Delta_n& < & p_n^+ < \Delta + \Delta_n \ ,
\eeq
with $\Delta_n = \sqrt{\Delta^2-p_n^{\perp 2} - \mu^2 }$. Thus, $p_n^+ > \epsilon^+$,  where $\epsilon^+ = 
(\mu^2+ p^{\perp 2}_n)/(2\Delta) > 0$ for $\Delta \gg \mu^2 \gg \epsilon^2$. Thus, the energy cutoff implies 
that the momentum $p_n^+$ for particles with positive masses is separated by the positive gap $\epsilon^+$ 
from zero. In the infrared box limit, $L \to \infty$, and the ultraviolet cutoff limit, $\Delta \to \infty$, the question 
of which quantity is greater, $\epsilon$ or $\epsilon^+$, has an answer depending on the order of limits,
\beq 
{\epsilon^+ \over \epsilon}  \es {\mu L \over 2\pi} \, {\mu \over \Delta} \ . 
\eeq
This formula relates the ratio ${\epsilon^+ /\epsilon}$ to the ratio of the number of the particle Compton wavelengths 
that fit in the cube edge and the number of free particles at rest whose total energy equals the cutoff $\Delta$. 
We assume that the infrared limit $L \to \infty$ is taken first for a fixed mass $\mu$. In that case, $\epsilon^+ 
\gg \epsilon$. In the field expansions, the sum over momenta $p_n$ involves $p_n^+$ separated from 0 by the 
gap $\mu^2/(2\Delta)$. Therefore, the integrals in Eqs.~(\ref{barepsif}), (\ref{bareAf}), and (\ref{hatphi}), do not 
extend down to $p^+=0$. Consequently, the Hamiltonian $H_{\rm QCD}^{m_g}$ does not contain any terms that 
are made only of creation or only of  annihilation operators. Such terms would imply that a natural multiple of 
$\epsilon^+$ equals 0. 

The largest value of $p^+$ allowed in the operator labels in Eqs.~(\ref{barepsif}), (\ref{bareAf}) and (\ref{hatphi}), 
say $Q^+$, is assumed large enough to consider all physical processes of interest. Since every physical process is 
characterized by some conserved value of its total plus momentum, say $P^+$, and since the plus momentum 
of each participating field quantum is smaller than $P^+$, it is sufficient to assume that $Q^+$ is finite but 
greater than the total $P^+$ in any process of physical interest. 
%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The weak-coupling issue of vacuum}
\label{vacuum}
%%%%%%%%%%%%%%%%%%%%%%%
The Hamiltonian of Eq.~(\ref{EqbareH}) is normal-ordered and, according to Sec.~\ref{labels}, all its terms 
contain at least one annihilation operator. Such terms annihilate the bare vacuum state, denoted by $|0\rangle$. 
Therefore, the state $|0\rangle$ is an eigenstate of $H$ with an eigenvalue 0. It is also an eigenstate with 
eigenvalue 0 of the normal-ordered operators $P^+$ and $P^\perp$, which makes it a candidate for the 
theory vacuum state.

The issue is that the interactions can cause formation of Hamiltonian eigenstates with eigenvalues smaller 
than 0, for any finite eigenvalues of $P^+$ and $P^\perp$. But when the regulated interaction terms are 
made arbitrarily small, by making the bare coupling constant $g$ arbitrarily small, all eigenstates with positive 
$P^+$ must have eigenvalues $P^-$ greater than 0. It has to be so because all the quanta of the regulated 
theory are assigned masses squared greater than 0. Therefore, all quanta necessarily contribute only positive 
amounts to the total value of free $P^-$. The interaction energy is too small to cancel that free positive 
$P^-$ when $g$ tends to 0. Hadrons are then considered to be states obtained by acting with the creation 
operators on $|0\rangle$. 

The situation changes when the coupling constant is increased and the Hamiltonian develops negative eigenvalues 
due to the interactions. Such eigenvalues would invalidate the assumption that the eigenstate $|0\rangle$ of 
eigenvalue 0 is the theory's ground state. Therefore, we assume that no such strong binding effect occurs in QCD 
with physically justified values of the coupling constant and masses of the quanta. The issue becomes whether the 
squares of masses in the renormalized effective-particle FF kinetic energies, including the adjustment of the mass-squared 
counterterms to be discussed later, are large enough to prevent interactions from generating negative eigenvalues 
$P^-$ for bound states of quarks and gluons. The author does not know any example of such negative eigenvalues 
in any realistic theory. %, {\it cf.} Sec.~\ref{mgto0}. 
%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Regulating interaction terms}    
\label{regf}
%%%%%%%%%%%%%%%%%%%%%%%%%
Interaction terms in $H_{\rm QCD}^{m_g}$ do not converge as functions of the particle momenta. Such interactions 
cause arbitrarily large changes of momenta and produce diverging transition amplitudes. The pertinent measure of 
momentum is $p^-$, an eigenvalue of the free part of the Hamiltonian, obtained from $H_{\rm QCD}^{m_g}$ by 
setting the bare coupling constant $g$ to 0. The free part is denoted by $H_f$,
\beq
\label{hatHf}
H_f \es  
\sum_{\sigma = 1}^2 \int[p] \, {p^{\perp 2} + m^2 \over p^+} 
\left( b^\dagger_{p\sigma}b_{p\sigma} + d^\dagger_{p\sigma} d_{p\sigma} \right)
+
\sum_{\sigma = 1}^3 \int[p] \, {p^{\perp 2} + m_g^2 \over p^+} \, a^\dagger_{p\sigma} a_{p\sigma} \ .
\eeq
It results from the first two density terms on the right-hand side of Eq.~(\ref{cHcanQCD}) and from the terms second 
and third on the right-hand side of Eq.~(\ref{cHcanQCDmg}). Quanta of the auxiliary gluon field $\phi$ are accounted 
for by summing over 3 instead of only 2 polarizations. The interaction part of $H_{\rm QCD}^{m_g}$ is denoted by $H_I$,
\beq
\label{H0}
H_{\rm QCD}^{m_g} \es H_f + H_I \ .
\eeq
With all annihilation operators commonly denoted by $a$, the structure of $H_I$ reads
\beq
\label{HIcan}
H_I \es \sum_i H_i \ , \\
\label{initialHi}
H_i \es 
           \left[ \prod_{m =1 }^{c_i} \int [p_m]  a_{p_m}^\dagger \right] 
           \left[ \prod_{ n = 1}^{a_i} \int [q_n ]  a_{q_n }              \right] 
           \tilde \delta_{c.a} \ h_i(\bar p, \bar q) \ ,
\eeq
where $c_i$ and $a_i$ are the numbers of creation and annihilation operators in the term $i$, respectively. The coefficient
$h_i(\bar p, \bar q)$ depends on the set of created particles' quantum numbers, $\bar p$, which includes $p_m$ with 
$m=1, ...,c_i$, and the set of annihilated particles' quantum numbers, $\bar q$, which includes $q_n$ with $n=1, ...,a_i$. 
Canonical coefficients $h_i$ are proportional to the coupling constant $g$ or its square. The symbol $\tilde \delta_{c.a}$ 
stands for the $\delta$-function that secures conservation of  the total kinematic momentum of quanta involved in a term,
\beq
\label{deltac.a}
\tilde \delta_{c.a} 
\es 2 (2\pi)^3 \delta    \left(p_c^+ - p_a^- \right) \
                    \delta^2 \left(p_c^\perp - p_a^\perp \right) \ ,
\eeq
where the subscripts $c$ and $a$ refer to quanta created and annihilated by a term, respectively,
\beq
\label{PCPA}
p_c \es \sum_{m=1}^{c_i} p_m \ , \ p_a \rs \sum_{n=1}^{a_i} q_n \ .
\eeq
Our regularization scheme sets a limit on the magnitude of change of the particle momenta that the interaction terms
can cause. We first describe the regularization of the interaction terms that are linear in $g$ and then of the terms 
that are quadratic in $g$.
%%%%%%%%%%%%%%%%%
\subsubsection{ Terms linear in $g$ }
\label{linearing}
%%%%%%%%%%%%%%%%%
Regularization of terms linear in $g$ amounts to inserting in them regulating factors $f^r_{\bar p, \bar q}$, 
\beq
\label{Tif}
T(\bar p , \bar q) & \to & f^r_{\bar p, \bar q} \ T(\bar p , \bar q) \ ,
\eeq
where, using notation of Eq.~(\ref{PCPA}), 
\beq
\label{fr}
f^r_{\bar p, \bar q} \es e^{ - [ r \,  (p_c^-  - p_a^-) ]^2 }  
                            \rs e^{ - [ r \,  (\cM_c^2  - \cM_a^2) /p_c^+]^2 } \ ,
\eeq
with $\cM_c^2 = p_c^2$  and $\cM_a^2 = p_a^2$. The specific form of $f^r_{\bar p, \bar q}$ , Eq.~(\ref{fr}), 
is chosen to match the solutions of the homogeneous renormalization group equations for Hamiltonians,
described in Sec.~\ref{BSeqs}. 

The insertion of regulating factors in terms linear in $g$, is illustrated below using the three-gluon interaction 
term. The term originates from the density term $-J_{A f}^{a \mu} A_{f\mu}^a$ for transverse gluons in 
Eq.~(\ref{cHcanQCD}). Before regularization, the canonical term reads
\beq
\label{HA3k12}
H_{A^3} \es g \sum_{c_1, c_2, c_3 =1}^8 
                     \sum_{\sigma_1, \sigma_2, \sigma_3 =1}^2 \int [123] 
\, a_1^\dagger a_2^\dagger a_3 \ Y_{123} \, \tilde \delta_{12.3} + h.c. \ ,
\eeq
where 
\beq
\label{Y123}
Y_{123} \es i f^{c_1 c_2 c_3}
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right] \ .
\eeq
The symbols $\varepsilon$ denote the gluon polarization four-vectors. The three-gluon vertex function 
$Y_{123}$ grows proportionally to $k^\perp_{12}$, the relative transverse momentum of the created gluons 
1 and 2, $k_{12}^\perp = x_2 p_1^\perp - x_1 p_2^\perp$, $x_1 = p_1^+/p_3^+$, $x_2 = p_2^+/p_3^+ = 
1 - x_1 $, $k_{12}^+=0$. This growth leads to the ultraviolet transverse divergences in transition amplitudes. 
The vertex also grows as $1/x_1$ or $1/x_2$ when $x_1$ or $x_2$ tend to 0, which leads to the small-$x$ 
divergences. Additional complexity results from the ratios $k_{12}^\perp/x_1$ and $k_{12}^\perp/x_2$. For 
example, a term with an infrared $k^\perp$ may become ultraviolet when $x \to 0$. We use the gluon mass 
parameter $m_g$ to untangle the mixture of ultraviolet and infrared divergences. The regulated three-gluon 
interaction term reads
\beq
\label{regHA3}
H_{A^3}^r \es g \sum_{\sigma_1, \sigma_2, \sigma_3 =1}^2 \int [123] 
\, a_1^\dagger a_2^\dagger a_3 \ Y_{123} \, \tilde \delta_{12.3} \ f^r_{12,3} + h.c. \ ,
\eeq
where 
\beq
f^r_{12,3} \es e^{ -[ r \,  (\cM_{12}^2 - m_g^2)/p_3^+]^2 } \ .
\eeq
The invariant mass $\cM_{12}$ only depends on the relative motion 
variables of gluons 1 and 2 , $x=x_1$ and $k^\perp = k_{12}^\perp$,
\beq
\cM_{12}^2 \es {k^{\perp 2} + m_g^2 \over x(1-x) } \ .
\eeq
When the invariant mass is finite and $r \to 0$, the regulating factor $f^r_{12,3}$ tends to 1. When the invariant mass 
diverges and $r$ is fixed, the factor tends to 0. The regulating suppression of the vertex occurs when $\cM_{12}^2$ 
exceeds the magnitude of $p_3^+/r$ or, for $|k^\perp|$ smaller than $m_g$, when $x$ decreases below $r m_g^2/
p_3^+$. The presence of $m_g$ guarantees that $\cM_{12} \to \infty$ when $x \to 0$. Thus the small-$x$ gluon 
divergences are turned into ultraviolet divergences. Without $m_g$, the region of small $x$ would not be regulated 
by the factor $f^r_{12,3}$, for one could have $k^\perp \sim x^\alpha $ with $\alpha \ge 1/2$. The non-locality of 
vertices regulated using factors $f^r_{12,3}$ is analyzed in~\cite{Glazek:2010zr,Glazek:2010zza}.

All terms linear in $g$ in the Hamiltonian $\hat H_{\rm QCD}^{m_g}$ are regulated using factors $f^r_{\bar p,\bar q}$ 
as in the above example. Such terms contain two creation operators, one for a quantum of some mass $m_1$ and 
another one for a quantum of some mass $m_2$, and an annihilation operator for some quantum of mass $m_3$, in correspondence to  $a_1^\dagger$, $a_2^\dagger$ and $a_3$ in Eq.~(\ref{regHA3}). One uses the invariant mass of 
quanta 1 and 2, squared, 
\beq
\label{cM12}
\cM_{12}^2 \es {k_{12}^{\perp 2} + m_1^2 \over x_1}
+ {k_{12}^{\perp 2} + m_2^2 \over x_2} \ .
\eeq
The regulating factor is
\beq
\label{3fields}
f^r_{12,3} \es e^{ - [ r \, (\cM_{12}^2 - m_3^2)/p_3^+]^2} \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{ Terms quadratic in $g$ }
\label{quadraticing}
%%%%%%%%%%%%%%%%%%%%%%
Regularization of the canonical terms proportional to $g^2$ concerns products of four fields. There are two kinds
of them. One kind results from the constraint Eqs.~(\ref{cpsi}) and (\ref{A-c1}). These terms involve an inverse of 
$\partial^+$. They are traditionally called seagulls. The other kind originates from the square of $G^{a\mu \nu}$ 
in the Lagrangian density. Both kinds are treated in the same way, described below using the example of a seagull.

Constraint Eqs.~(\ref{A-c}) and (\ref{A-c1}) relate gauge field component $A^-$ to $1/\partial^{+2}$ acting on the 
color current $J$. The latter involves products of two fields. Therefore, when a color current is multiplied by the 
constrained $A^-$, the resulting density term is of the form $J (1/ \partial^{+2}) J$. Each $J$ can be though of as 
creating or annihilating a {\it gedanken} gluon that carries the momentum associated with $1/\partial^{+2}$. The 
imagined gluon together with the current $J$ form a vertex that is regulated by the factor $f^r_{12,3}$ in the same 
way as the three-gluon vertex is in Eq.~(\ref{regHA3}). In the fermion seagulls, the {\it gedanken} quantum is a 
fermion rather than a boson. In the Hamiltonian term due to the product of commutators, $g^2[A^i,A^j]^a [A^i,A^j]^a$, 
each commutator is treated as a current, even though the inverse of $\partial^+$ is absent.

The rgularization of terms proportional to $g^2$ is illustrated using the gluon seagull term
\beq
\label{HJJ}
H_{JJ} \es \int dx^- d^2x^\perp \ 
\2 (J_{\psi f}^{a+} + J_{A f}^{a+}) 
{-1 \over \partial^{+2}} 
(J_{Af}^{a+} + J_{\psi f}^{a+}) \ ,
\eeq
originating from the last density term in Eq.~(\ref{cHcanQCD}). It involves four terms, each of which is a product of 
four fields. All these products are regulated according to the same rule. It suffices to describe one product. We describe
in detail the regularization of the term that involves only quark and antiquark operators,
\beq
\label{HJJbeforeNO}
H_{J_\psi J_\psi} 
\es \int dx^- d^2x^\perp \ 
\2 J_{\psi f}^{a+} {1 \over (i \partial^+)^2} J_{\psi f}^{a+} \ .
\eeq
The current at $x^+=0$ is
\beq
J_{\psi f}^{a +} (x) \es -g
\sum_{c_1, c_2} \sum_{\sigma_1, \sigma_2} \ 
\chi_{c_1}^\dagger T^a \chi_{c_2}
\int[p_1p_2] 
\nt
\left[  
           \bar u_{p_1\sigma_1} b_{p_1\sigma_1 c_1}^\dagger  e^{ip_1x}
      +   \bar v_{p_1\sigma_1} d_{p_1\sigma_1 c_1}               e^{-ip_1x}
\right]
\nt
\gamma^+ \ 
\left[  
           u_{p_2 \sigma_2} b_{p_2 \sigma_2 c_2} e^{-ip_2x}
      +   v_{p_2 \sigma_2} d^\dagger_{p_2 \sigma_2 c_2} e^{ip_2x}
\right] \ .
\eeq
The spinor products are 
\beq
\label{uu1+2}
\bar u_1 \gamma^+ u_2 \es 
\bar v_1 \gamma^+ v_2 \rs 
2 \sqrt{p_1^+ p_2^+} \ \delta_{\sigma_1, \sigma_2} \ , \\
\label{uv1+2}
\bar u_1 \gamma^+ v_2 \es
\bar v_1 \gamma^+ u_2 \rs
- 2 \sqrt{p_1^+ p_2^+} \ \delta_{\sigma_1, - \sigma_2} \ .
\eeq
Hence the normal-ordered, regulated current is
\beq
\label{jquark1}
:J_{\psi f}^{a +}: \es -g
\sum_{c_1, c_2} \sum_{\sigma_1, \sigma_2} \ 
\chi_{c_1}^\dagger T^a \chi_{c_2} 
\int[p_1p_2]  \ 2 \sqrt{p_1^+ p_2^+} 
\
\left[ :\{ ~~ \}: \right]^r \ ,
\eeq
where the bracket, 
\beq
\label{jquark2}
\left[ :\{ ~~ \}: \right]^r
\es
            b_1^\dagger 
            b_2 e^{i(p_1-p_2)x}
            \left[  \theta_{1-2} f^r_{1,2(1-2)} + \theta_{2-1}f^r_{2,1(2-1)} \right]
            \delta_{\sigma_1 , \sigma_2}
\nm                    
            d^\dagger_2 
            d_1 e^{i(p_2-p_1)x}
            \left[ \theta_{2-1} f^r_{2,1(2-1)} + \theta_{1-2}f^r_{1,2(1-2)} \right]
            \delta_{\sigma_1 , \sigma_2}
\nm
            b_1^\dagger  
            d^\dagger_2 e^{i(p_1+p_2)x} f^r_{12,(1+2)}
            \delta_{-\sigma_1 , \sigma_2}
\nm
            d_1            
            b_2 e^{-i(p_1+p_2)x} f^r_{12,(1+2)}
            \delta_{-\sigma_1 , \sigma_2} \ ,
\eeq
carries the superscript $r$ to indicate the insertion of the regulating factors. The {\it gedanken} gluon is assigned 
the mass $m_g$ for evaluating its minus momentum. The Heaviside's $\theta$-function $\theta_{i-j} = 
\theta(p_i^+-p_j^+)$. Thus the regulated quark seagull is 
\beq
\label{HJJregulatedNO}
H_{J_\psi J_\psi}^r \es \int dx^- d^2x^\perp \ 
\left[ \2 J_{\psi f}^{a+} {1 \over (i \partial^+)^2} J_{\psi f}^{a+} \right]^r \\
\es
\2
\int dx^- d^2x^\perp \ 
:
\left[  : J_{\psi f}^{a+} : \right]^r 
{1 \over (i \partial^+)^2} 
\left[  : J_{\psi f}^{a+} : \right]^r 
: \ .
\label{HJJregulatedNO}
\eeq
The colon, a symbol for the normal-ordering, is dropped below. For the purpose of illustration of 
Eq.~(\ref{HJJregulatedNO}). We display details of the term that transforms a quark-antiquark pair 
into another such pair.
\beq
\label{seagull}
H_{J_\psi J_\psi}^{r \, q\bar q}
\es
\2 g^2
\sum_{1234} \  T^{\tilde c}_{12} T^{\tilde c}_{34} \int[1234]  \
\tilde \delta_{c.a} \, \left\{ ~~ \right\}  4 \sqrt{p_1^+ p_2^+ p_3^+ p_4^+}  \ ,
\eeq
where one sums over the color superscripts $\tilde c$ from 1 to 8, and
\beq
\label{currentfr}
\left\{ ~~ \right\} 
\es    
            b_1^\dagger  d^\dagger_4 d_3 b_2 \
            \theta_{1-2} f^r_{1,2(1-2)} \delta_{\sigma_1 , \sigma_2} 
           {- 2 \over (p_3^+-p_4^+)^2}
            \theta_{3-4}f^r_{3,4(3-4)}  \delta_{\sigma_3 , \sigma_4}
\np      
            b_1^\dagger  d^\dagger_4 d_3 b_2  \
            \theta_{2-1}f^r_{2,1(2-1)}  \delta_{\sigma_1 , \sigma_2}
           {- 2 \over (p_3^+-p_4^+)^2}                    
            \theta_{4-3} f^r_{4,3(4-3)} \delta_{\sigma_3 , \sigma_4} 
\np       
            b_1^\dagger d^\dagger_2 d_3 b_4  \
            f^r_{12,(1+2)} \delta_{-\sigma_1 , \sigma_2}
           {2 \over (p_3^++p_4^+)^2}  
            f^r_{34,(3+4)} \delta_{-\sigma_3 , \sigma_4} 
            \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Fully regulated bare Hamiltonian}                                              
\label{fullyreg}
%%%%%%%%%%%%%%%%%%%%%%%
Fully regulated QCD Hamiltonian with the gluon mass $m_g$ and field $\phi$, denoted by $H^{m_g \,  r}_{\rm QCD}$, reads
\beq
\label{EqbareHr}
H_{\rm QCD}^{m_g \, r} \es \int d^2x^\perp dx^- \, 
\cH_{\rm QCD}^{m_g \, r}(\psi_f, A_f, \phi) \ ,
\eeq
where
\beq
\label{cHcanQCDmgr}
\cH^{m_g \,  r}_{\rm QCD} 
\es
\cH_{\rm QCD}^r 
-
\2 m_g^2 A_{f \mu}^a   A_f^{a \mu}
\np  
\2 \phi^a (- \partial^{\perp 2} + m_g^2 ) \phi^a 
+
\left[
m_g \phi^a {1 \over \partial^+} (J_{Af}^{a+} + J_{\psi f}^{a+}) \right]^r \ ,
\eeq
and
\beq
\label{cHcanQCDr}
\cH_{\rm QCD}^r 
\es
\bar \psi_f { \gamma^+ (- \partial^{\perp 2} + m^2) \over 2i \partial^+ }  \psi_f
+
\2 A_{f \mu}^a  \partial^{\perp 2}  A_f^{a \mu}   
-
\left[ (J_{\psi f}^{a \mu}  + J_{A f}^{a \mu}) A_{f\mu}^a \right]^r
\np
\2 g^2 \left[ \bar \psi_f \, \f \hspace{-2pt} A_f 
{ \gamma^+ \over i \partial^+ } 
                          \, \f \hspace{-2pt} A_f \psi_f \right]^r
-
\4 g^2 \left[ [A_{f\mu}, A_{f_\nu}]^a [A_f^\mu, A_f^\nu]^a  \right]^r
\nm
\2 \left[ (J_{\psi f}^{a+} + J_{A f}^{a+}) {1 \over \partial^{+2}} (J_{Af}^{a+} + J_{\psi f}^{a+}) 
\right]^r \ .
\eeq
The regularization brackets $[~~~]^r$ embrace all interaction terms. The regularization brackets may be omitted 
to simplify notation. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Gluon mass and small $x$ in scattering amplitudes}  
\label{transition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Insertion of a mass term for gluons in the FF Hamiltonian of QCD leads to severe small-$x$ divergences due to inverse of 
$\partial^{+2}$, see, {\it e.g.,}~\cite{Wilson:1994fk}, Sec. IX A. In contrast, the Hamiltonian $H^{m_g \,  r}_{\rm QCD}$ 
of Eq.~(\ref{EqbareHr}), with density $\cH^{m_g \,  r}_{\rm QCD}$ given in Eq.~(\ref{cHcanQCDmgr}), leads to the cancellation 
of these singularities despite that $m_g > 0$. The reason is that $H^{m_g \,  r}_{\rm QCD}$ includes also a kinetic term for a 
color octet of scalar fields of the same mass and an interaction term that couples that octet field to the ``good'' color currents 
of quarks and transverse gluons, proportionally to the gluon mass. 
%%%%%%%%%%%%%%%%%%%%%%
\subsection{Severe small-$x$ singularity}  
\label{gluonsmallx}
%%%%%%%%%%%%%%%%%%%%%%
To identify the source of the gluon severe small-$x$ divergences, the current-gluon coupling term, {\it i.e.}, the Hamiltonian 
term resulting from the third density term on the right-hand side of Eq.~(\ref{cHcanQCDr}), is separated into its transverse 
and longitudinal parts,
\beq
- (J_{\psi f}^{a \mu}  + J_{A f}^{a \mu}) A_{f\mu}^a 
\es
(J_{\psi f}^{a j}  + J_{A f}^{a j}) A_{f}^{a j} 
-
\2 (J_{\psi f}^{a +}  + J_{A f}^{a +}) A_{f}^{a -} \ . 
\eeq
Only the longitudinal part involves the inverse of $\partial^+$. Using Eq.~(\ref{Af-}) for $A_f^{a-}$ and integrating by parts, the 
Hamiltonian density of Eq.~(\ref{cHcanQCDmgr}) for fields that vanish at large distances is transformed to the equivalent form
\beq
\label{HQCDmgphir}
\cH_{\rm QCD}^{m_g \, r}
\es
\bar \psi_f { \gamma^+ (- \partial^{\perp 2} + m^2) \over 2i \partial^+ }  \psi_f
+
\2 A_f^{a j}  (- \partial^{\perp 2} + m_g^2 )  A_f^{a j}   
\np
  (J_{\psi f}^{a j}  + J_{A f}^{a j}) A_f^{a j}
+
\2 g^2 
\bar \psi_f  A_f^j \gamma^j { \gamma^+ \over i \partial^+ } \gamma^k A_f^k \psi_f 
\nm
\4 g^2 [A_f^j, A_f^k]^a [A_f^j, A_f^k]^a  
-
\2 (J_{\psi f}^{a+} + J_{A f}^{a+}) {1 \over \partial^{+2}} (J_{Af}^{a+} + J_{\psi f}^{a+}) 
\np
\2 \phi^a (- \partial^{\perp 2} + m_g^2 ) \phi^a 
\np
\left( \partial^\perp A_f^{a\perp} + m_g \phi^a \right) 
{1 \over \partial^+} \left(J_{Af}^{a+} + J_{\psi f}^{a+}\right) \ .
\eeq
The superscripts $j$ and $k$ take only two values, 1 and 2, for the front transverse directions. The only terms leading 
to $1/\partial^{+2}$, or $1/x^2$ for gluons, are
\beq
\label{H+1}
\cH_{+1} 
\es
\left( \partial^\perp A_f^{a\perp} + m_g \phi^a \right) {1 \over \partial^+} \eta J_f^a \ , \\
\label{H+2}
\cH_{+2}
\es
- \2  \eta J_f^a {1 \over \partial^{+2}} \eta J_f^a \ ,
\eeq
where $\eta J_f^a = J_{\psi f}^{a+} + J_{A f}^{a+}$. The four-vector $\eta$ has components $\eta^-=2$, $\eta^+=0$, and
$\eta^\perp =0$. The term $\cH_{+1}$ is proportional to $g$ and the term $\cH_{+2}$ to $g^2$. Integration of these two 
densities over the front yields the Hamiltonian terms denoted by $H_{+1}$ and $H_{+2}$, respectively. Regularization factors 
in these terms are introduced according to the rules described in the previous section. The gluon severe small-$x$ singularities 
occur due to $H_{+2}$ and square of $H_{+1}$. It is shown in the next section that they cancel out in the transition amplitudes 
for quarks and gluons despite the presence of the mass parameter $m_g$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cancellation of $1/x^2$ in scattering amplitudes}
\label{xintransition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Scattering of quarks and gluons in the femtouniverse~\cite{Bjorken:1979hv} is described assuming that in the first 
approximation they propagate as free. Interactions are accounted for using an expansion in a series of  powers of 
the coupling constant $g$. Our discussion begins with the small-$x$ divergences in the second-order scattering 
matrix operator~\cite{Gell-Mann:1953dcn},
\beq
\label{T1}
T^{(2)} \es H_I^{(1)} {1 \over P^- - H_f + i\epsilon} H_I^{(1)} + H_I^{(2)} \ .
\eeq
Symbols $H_I^{(1)}$ and $H_I^{(2)}$ denote the Hamiltonian terms of order $g$ and $g^2$, respectively. $P^-$ denotes 
the initial-state eigenvalue of the free Hamiltonian $H_f$. 

Consider the quark-antiquark scattering in which a pair $q\bar q$ turns into $q'\bar q'$. Dropping the overall momentum 
conservation $\tilde \delta$-function, using $j_q$ and $j_{\bar q}$ to denote the quark currents, which are free from the 
gluon mall-$x$ singularities, denoting the four-momentum transfer from the quark to the antiquark by $k=q-q'=\bar q' - 
\bar q$, $k^+>0$, and omitting $i \epsilon$, one obtains the transition amplitude on-shell of total $P^-$ in the form 
\beq
\langle q'\bar q'|T^{(2)} |q\bar q\rangle \es j_{q \alpha} T^{\alpha \beta} j_{\bar q \beta} \ , \\
\label{exchangecurrentsfactor}
T^{\alpha \beta} \es
{1 \over k^+(P^- - H_f)} \sum_{\sigma = 1}^3 
\epsilon_{k\sigma}^\alpha \epsilon_{k\sigma}^{*\beta} 
+ {\eta^\alpha \eta^\beta \over k^{+2}} \ .
\eeq
The first term on the right-hand side of Eq.~(\ref{exchangecurrentsfactor}) comes from the exchange of gluons between 
the quarks. The second term, proportional to $\eta^\alpha \eta^\beta$, is contributed by the seagull term, or $H_{+2}$ 
corresponding to the density $\cH_{+2}$ in Eq.~(\ref{H+2}). Both, the exchange and seagull contributions diverge as 
$1/k^{+ 2}$ when $k^+ \to 0$. The sum over gluon polarizations extends from 1 to 3 because it includes the contribution 
of the longitudinal gluons described by the field operator $\hat \phi$ in Eq.~(\ref{hatphi}). 
\beq
\label{sumeps1}
\sum_{\sigma = 1}^3 \epsilon_{k\sigma}^\alpha 
                                \epsilon_{k\sigma}^{*\beta} 
\es
- g^{\alpha \beta}
+(k_0^\alpha \eta^\beta + \eta^\alpha k_0^\beta)/k^+
+ m_g^2 \eta^\alpha \eta^\beta/k^{+2} \ .
\eeq 
The first two terms above come from the transverse gluons, with $k_0^- = k^{\perp 2}/k^+$, and the third term comes 
from the longitudinal gluons. So, the amplitude is
\beq
\label{T11}
\langle q'\bar q'|T^{(2)} |q\bar q\rangle
\es j_{q \alpha} \left[ {-g^{\alpha \beta} + \Pi^{\alpha \beta}
\over k^2 - m_g^2} \right] j_{\bar q \beta} \ , 
\eeq
where
\beq
\label{sumeps2}
\Pi^{\alpha \beta} 
\es
(k_0^\alpha \eta^\beta + \eta^\alpha k_0^\beta)/k^+
+ 
m_g^2 \eta^\alpha \eta^\beta/k^{+2}
+ 
(k^2 - m_g^2) \, \eta^\alpha \eta^\beta / k^{+2} \ .
\eeq
The quark and anti-quark currents in Eq.~(\ref{T11}) are conserved, $k_\alpha \, j_q^\alpha = k_\alpha \, j_{\bar q}^\alpha = 0$.
The formula $k_0^\alpha = k^\alpha + (k_0^- - k^-)\eta^\alpha/2$ implies 
\beq
\label{polarPI}
\Pi^{\alpha \beta} \eqv \Pi \ \eta^\alpha \eta^\beta \ , \\
\label{Pitransition}
\Pi
\es
(k_0^- - k^-)/k^+
+
m_g^2 /k^{+2}
+ 
(k^2 - m_g^2) /k^{+2} \ .
\eeq
The key result is that $\Pi$ vanishes. The terms $\sim 1/x^2$ cancel out completely. The regularization factors $f^r$ do not 
interfere with the cancellation because they are the same in the gluon exchange and the seagull term. The resulting scattering 
amplitude has the co-variant form of Eq.~(\ref{T11}) with $\Pi^{\alpha \beta}=0$. Moreover, the amplitude is free from any 
small-$x$ singularity, not only the severe $1/x^2$. An alternative way of stating this result is that the diagrammatic rule (R8) 
of perturbative QCD in App. A of~\cite{Lepage:1980fj}, continues to be valid even though the gluons are assigned the mass 
$m_g$. 

In order to extend the above reasoning to the transition amplitudes between intermediate states that are off-shell of total 
$P^-$,  we first note that the severe gluon singularities appear in perturbative calculations due to the operator 
\beq
\label{Ts}
T_{\rm singular} \es H_{+1} {1 \over P^- - H_f + i\epsilon} H_{+1} + H_{+2} \ ,
\eeq
which sums the exchange of a gluon between two color currents and the seagull. The divergences occur in the coefficients 
of the tensor $\eta^\alpha \eta^\beta$, contracted with the quark or gluon currents. The sum over transverse polarizations 
of the exchanged gluon contributes $k^{\perp 2}/k^{+2}$, as dictated by $H_{+1}$. The intermediate quanta of field $\phi$ contribute $m_g^2/k^{+2}$, also dictated by $H_{+1}$. Since the eigenvalues $k^-$ of $H_f$ for the quanta of fields $A$ 
and $\phi$ are the same, $k_g^- = (k^{\perp 2} + m_g^2)/ k^+$, the most singular contribution to the transition amplitude 
due to $k^+ \to 0$, takes the form
\beq
\label{cancellation11}
\langle T_{\rm singular} \rangle \es - j_1 
\left[  { k^{\perp 2}/k^{+2}  + m_g^2 /k^{+ 2} \over k^+(P^- - Q^- - k_g^-)} 
+ {1 \over k^{+2}} \right] j_2 \,  \ ,
\eeq
where the second term in the bracket, $1/k^{+2}$ , comes from $H_{+2}$. The sum of eigenvalues of $H_f$ for other quanta in 
the same intermediate state, in which the exchanged gluons appear, is denoted by $Q^-$. Symbols $j_1$ and $j_2$ stand for the 
contractions of the quark or gluon currents with $\eta$. The three-momentum conservation $\tilde \delta$-function and $i\epsilon$ 
are omitted. The term $k^{\perp 2}/k^{+2}$ in the numerator in Eq.~(\ref{cancellation11}) is provided by the transverse gluons and 
the term $m_g^2 /k^{+ 2}$ by the longitudinal ones. The inverse of $k^{+2}$ would produce the severe singularity for $k^+ \to 0$ 
if the free Hamiltonian $H_f$ would include the mass $m_g$ for transverse gluons and the longitudinal gluons were absent. When their contribution is included,
\beq
\label{cancellation}
\langle T_{\rm singular} \rangle 
\es 
j_1 {  Q^- - P^- \over k^+ (P^- - Q^-) - (k^{\perp 2} + m_g^2)} \,  j_2  \  {1 \over k^+}\ .
\eeq
The divergence $\sim 1/k^{+2}$ for $k^+ \to 0$ cancels out despite that $m_g > 0$. At the same time, $m_g^2 > 0$ regulates 
the infrared divergence due to $k^\perp \to 0$ when $k^+ \to 0$. 

The same mechanism works to all orders of perturbation theory. Our demonstration begins with Eq.~(\ref{H+1}). Every numerator 
term associated with propagation of a transverse gluon, $(k^\perp/k^+)^2$, is accompanied by a term due to the propagation of 
a longitudinal gluon, $(m_g/k^+)^2$. The transverse-gluon exchange and the longitudinal-gluon exchange involve the same 
denominator factor with the same value of the gluon $k_g^-$. The exchange extending over one intermediate state always appears 
in addition to a contribution of the Hamiltonian term $H_{+2}$. Therefore, the cancellation in exchanges extending over one 
intermediate state occurs as described above. The cancellation holds no matter how many additional interactions precede or follow 
the exchange. When the gluon exchange extends over more than one intermediate state, the contribution of $H_{+2}$ is absent. 
However, in that case, the gluon $k_g^-$ appears in more than one denominator. The additional denominators provide additional 
powers of $k^+$ in the numerator, which eliminate the severe singularity $1/k^{+2}$. 
%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Small $x$ in self-interactions}
\label{Sigmax}
%%%%%%%%%%%%%%%%%%%%%%%
In the amplitudes discussed in Sec.~\ref{xintransition}, the severe small-$x$ divergences of the gluon exchange and seagull 
terms cancel each other. But in the quark and gluon self-interactions, schematically illustrated in Fig.~\ref{fig:self}, the 
normal-ordered seagulls do not contribute. 
\begin{figure}[ht!]
          \caption{Self-interaction of particle of type 1 via emission and absorption of particle of type 2.}
          \label{fig:self}
\begin{center}
         \includegraphics[width=0.4\textwidth]{SelfinteractionSchematic}
\end{center} 
\end{figure}
The second-order self-interaction sketched in Fig.~\ref{fig:self} requires a counterterm. Including the counterterm, the self-interaction 
reads
\beq
\label{selfinteraction1text}
\langle p' | p\rangle \, {\Sigma^{(2)} \over p^+} 
\es
\langle p' |  H_{+1} {1 \over P^- - Q^- - H_f } H_{+1} |  p\rangle 
+
\langle p' | p\rangle \, {C^{(2)} \over p^+} \ ,
\eeq
where the counterterm contribution is denoted by $C^{(2)}$. The factor $\langle p' | p\rangle $ accounts for the state normalization. 
We obtain
\beq
\label{selfinteraction1quark333full}
{\Sigma_q^{(2)} \over p^+}
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
  \left[ {N \over D} + 4\sigma_q^{(2)} \right]  \, f^{r 2}_{qg,p} \ , \\
\label{NumSigmaQfull}
N
\es 4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x) \ ,\\
\label{NumSigmaQfullD}
D
\es
x(1-x) p^+( P^- - Q^-  - p_q^- - p^-_g) \ ,
\eeq 
where $4\sigma_q^{(2)}$ stands for the integrand of the counterterm. $x$ is the fraction of the quark momentum $p^+$, carried 
by the emitted gluon, ranging between 0 and 1. $k^\perp = p_g^\perp - x p^\perp$. For $N$ colors of quarks, $C_F = (N^2-1)
(2N)$. The squared regularization-factor, $f^{r2}_{q g, q}$, comes from the double action of the Hamiltonian term $H_{+1}$. The 
severe small-$x$ singularities are contained in 
\beq
\label{selfinteraction1quark333}
{\Sigma_{q \, \rm sing}^{(2)} \over p^+}
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
\left[ {4p^-_g/x \over x ( P^- - Q^- - p_q^- - p^-_g)} + 4\sigma_q^{(2)} \right] 
\, 
f^{r 2}_{q g, q}
\ .
\eeq 
In our notation, $p_q$ denotes the free four-momentum of a quark of mass $m$, after emitting and before absorbing a gluon.
$p_g$ denotes the free four momentum of the gluon, of mass $m_g$. It is visible that to counter the singularity $\sim 1/x^2$, 
one needs the counterterm integrand 
\beq
\label{sigmaqint}
\sigma_q^{(2)} \es 1/x^2 \ .
\eeq
Inclusion of the less singular terms than $4p_g^-/x$, shows that the small-$x$ divergence $1/x^2$ and the ultraviolet quadratic 
transverse divergence in $\Sigma_q^{(2)}/ p^+$ are simultaneously countered by the counterterm whose integrand is 
\beq
\label{sigmaqintfull}
\sigma_q^{(2)} \es 1/x^2 + 1/[2(1-x)] \ .
\eeq
Similarly, the second-order transverse-gluon self-interaction reads, 
\beq
\label{SigmaTgluon}
{\Sigma_{g \, \rm sing}^{(2)} \over p^+}
\es {2 g^2 C_A \over p^+} \int {dx d^2k^\perp \over  2 (2\pi)^3 } 
\left[ {N \over D}  + \sigma_g^{(2)} \right] \, f^{r 2}_{g g,g} \ , \\
N \es 
(k^{\perp 2} + m_g^2) \ \left[ 1 + {1 \over x^2} + {1 \over (1-x)^2 } \right] 
-m_g^2 \left[ \2 + {1 \over x(1-x)} \right] \ , \\
D \es
x (1 - x)p^+(P^- - Q^-) - (k^{\perp 2} + m_g^2) \ .
\eeq
For $N$ colors $C_A=N$. It is visible that the counterterm integrand
\beq
\label{sigmag2}
\sigma_g^{(2)} \es 1 + {1 \over x^2} + {1 \over (1-x)^2 } \ ,
\eeq
removes the severe divergences of type $1/x^2$ for $x \to 0$. Inclusion of 1 in Eq.~(\ref{sigmag2}), secures simultaneous 
cancellation of the ultraviolet quadratic transverse divergence. The gluon self-interaction due to the intermediate quark 
pairs does not involve divergences $\sim 1/x^2$. The self-interaction of longitudinal gluons is free from such divergences
as well. The remaining singularities are only logarithmic, as promised. In orders higher than second, the one-particle irreducible 
self-interactions require more complex expressions for the counterterms. Self-interactions in the renormalized bound-state 
eigenvalue problems of the Hamiltonian $H^{m_g \,  r}_{\rm QCD}$ are discussed in the following sections.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hamiltonian eigenvalue problem }
\label{BSeqs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Eigenstates of the Hamiltonian $H^{m_g \,  r}_{\rm QCD}$ of Eq.~(\ref{EqbareHr}) represent hadrons of QCD in the limit 
of lifting the regularization, $r \to 0$ and $m_g \to 0$. The eigenstates are the combinations 
\beq
|h\rangle \es \sum_i \psi_i |i\rangle ,
\eeq 
where $|i\rangle$ are the basis states obtained by acting on the state $|0\rangle$ with products of creation operators 
introduced in Eqs.~(\ref{barepsif}), (\ref{bareAf}) and (\ref{hatphi}). To describe a hadron, one needs to compute the relevant 
eigenvalue of the matrix $H^{m_g \, r}_{ij} = \langle i |H^{m_g \,  r}_{\rm QCD}|j\rangle$ and evaluate the corresponding set
of coefficients $\psi_i$. Analytic calculations appear not feasible because of the matrix complexity. To use computers, one
needs to overcome several difficulties.

The Hamiltonian matrix has infinite size. One must replace it by some equivalent finite matrix. To begin with, one could limit 
the momenta and number of quarks and gluons in the basis states. However, setting such limits is not straightforward because 
the matrix elements $H^{m_g \, r}_{ij}$ diverge as functions of the momenta, {\it e.g.,} see Eq.~(\ref{HA3k12}) for $k^\perp \to 
\infty$ or $x \to 0$. The diverging matrix elements dominate the dynamics. They need to be dealt with first thing to gain a 
computational access to the finite quantities of physical interest~\cite{Wilson:1965zzb,Wilson:1970tp,Wilson:1994fk}, including 
the issue of explaining the relativistic quantum-binding mechanism for quarks and gluons. The singular interactions can change 
the number of quanta. Therefore, the states with quarks and gluons in fast relative motion are degenerate with states of less 
rapid but more numerous quanta. Traces of these complexities appear in QCD phenomenology. For example, applications of the 
parton model suggest that the gluon distribution in proton grows for small $x$ and requires some mechanism of saturation. 
Eventually, one needs a method of estimating the accuracy of a computation.

To overcome the difficulties mentioned above, the QCD bound-state eigenvalue problems are formulated not in terms 
of the canonical creation and annihilation operators introduced in Eqs.~(\ref{barepsif}), (\ref{bareAf}) and (\ref{hatphi}). 
Instead of $b$, $d$, and $a$, commonly denoted below by $q$, one uses operators $b_s$, $d_s$, and $a_s$, denoted 
by $q_s$. The latter are called the operators for effective particles. The parameter $s$ can be thought about as a size 
of the particles. The canonical operators $q$ correspond to the point-like quanta, $q = q_{s=0}$. When $s \sim 1/m$,
where $m$ is a hadron mass, the effective particles are meant to correspond to the constituent quarks and gluons. 
Evaluation of the QCD Hamiltonian in terms of operators $q_s$ is carried out using the renormalization group procedure 
for effective particles (RGPEP)~\cite{Glazek:2026xnp}. The procedure is designed to rewrite the canonical FF of QCD 
bound-state eigenvalue equations in an equivalent way using the effective particles, whereby the singular canonical 
interactions are replaced by the smoothed ones, which depend on the parameter $s$ and do not create the eigenstate 
components with unlimited numbers and momenta of the constituents. To describe how the gluon mass $m_g$ and 
field $\phi$ contribute to the effective bound-state dynamics, we need to recall the elements of the RGPEP.%\cite{Glazek:2026xnp}
%%%%%%%%%%%%%%%%%%%
\subsection{Elements of the RGPEP}
%%%%%%%%%%%%%%%%%%%
Quantum numbers of the phenomenological constituent quarks~\cite{ParticleDataGroup:2024cfk} are the same as 
those of the bare quarks in the canonical QCD, with the exception of the mass parameters. Therefore, the RGPEP for
QCD assumes that the effective particles are unitarily related to the bare particles, 
\beq
\label{qt}
q_s \es \cU_s^\dagger q \, \cU_s \ ,
\eeq
and the quantum numbers of $q_s$ and $q$ are the same. The Hamiltonian remains unchanged. Consequently, the 
products of operators $q_s$ in the Hamiltonian have different coefficients, say $c_s$, from the coefficients $c$ of 
the corresponding products of operators $q$, but
\beq
\label{cHctqt}
H(c_s,q_s) \es  H(c,q) \ .
\eeq
The renormalized expressions for the coefficients $c_s$, are obtained by solving a first-order differential equation in $s$, 
called here the RGPEP equation, see Eq.~(\ref{RGPEP1}) below. The initial condition at $s=0$ is provided by the canonical
coefficients, $c$, including the modifications resulting from the counterterms determined while solving the RGPEP equation,
according to the rules of the similarity renormalization group procedure (SRG)~\cite{Glazek:1993rc,Glazek:1994qc}. 
Computation of the coefficients $c_s$ is carried out using the operator
\beq
\cH \es H(c_s,q) \rs \cU_s H(c_s,q_s) \, \cU_s^\dagger \ ,
\eeq
whose structure implies
\beq
\label{RGPEP}
\cH' \es \left[ \cG , \cH \right] \ ,
\eeq
where prime denotes the derivative $d/ds^2$; the differentiation with respect to $s^2$ will be needed for dimensional 
reasons. The anti-Hermitian operator 
\beq
\label{GenaH}
\cG \es \cU' \cU^\dagger 
\eeq
is called the generator of the RGPEP transformation of the Hamiltonian. Whole class of generators can be 
considered~\cite{Glazek:1997gt,Glazek:2006mg}. Hamiltonian operators we discuss have the form $\cH = 
\cH_f + \cH_I$, where $\cH_f$ is given in Eq.~(\ref{hatHf}). The anti-Hermitian commutator $[ \cH_f, \cH_I]$ 
can be used as a generator of a unitary transformation,
\beq
\label{Gcommutator1}
\cG \es [ \cH_f, \cH_I] \ .
\eeq
Evolution in $s$ with such generator tends to bring the matrix of $\cH_I$ to a band diagonal form in the basis of 
space of states provided by the eigenstates of $\cH_f$. The width of the band decreases as $s$ increases. With this generator, 
Eq.~(\ref{RGPEP}) takes the double-commutator form, 
\beq
\label{RGPEP1}
\cH' \es \left[ [ \cH_f, \cH_I] , \cH \right] \ ,
\eeq
developed by Wegner for Hamiltonians in condensed matter physics~\cite{Wegner:1994fdg}, see also~\cite{Mielke:1998ag,
Wegner_2006,Kehrein:2006ti,PhysRevX.9.021037,Deift1983OrdinaryDE,doi:10.1137/1030090,BROCKETT199179}. The double 
commutator structure implies that $\cH$ fulfills the cluster decomposition principle~\cite{Weinberg:1995mt}, there are no 
disconnected interactions. Model studies of the RGPEP equations with various generators show a need for alterations of the 
generator $\cG$ of Eq.~(\ref{Gcommutator1}) when one seeks to achieve convergence of solutions for $\cH$ obtained using 
the weak-coupling expansion~\cite{Glazek:2002iw,Glazek:2003xr}. Here we are concerned with the lowest orders 
of the expansion and no need arises for altering Eqs.~(\ref{Gcommutator1}) and (\ref{RGPEP1}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Renormalized effective eigenvalue problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Assuming a solution for $\cH$ as a function of $s$ is available, the QCD Hamiltonian is obtained in the form
\beq
\label{Ht}
H_s \equiv H(c_s,q_s) \es  \cU_s^\dagger [ \cH = H(c_s,q_0) ] \, \cU_s \ .
\eeq
The eigenvalue problem for hadrons as bound states of effective quarks and gluons reads
\beq
\label{Htep}
H_s | \psi \rangle \es P^- | \psi \rangle \ , \\
\label{Hteppsi}
|\psi \rangle \es \sum_i \psi_i(s) |i\rangle_s \ ,
\eeq 
where $P^-$ is the eigenvalue. The basis states $|i\rangle_s$ are created by applying products of the creation operators 
$q_s$ to the state $|0\rangle$. If one solved the RGPEP equation for $H_s$ exactly and found exact eigenvalues and 
eigenstates of $H_s$, the coefficients $\psi_i(s)$ and basis states $|i\rangle_s$ would depend on $s$ but the eigenvalues 
and corresponding eigenstates $| \psi \rangle $ would not~\cite{Glazek:2021vnw}. Approximate calculations yield results 
varying with $s$. The magnitude of such variation indicates how large the theoretical error of an approximate calculation
may be. Another measure is provided by the accuracy of obeying symmetries, such as the rotation symmetry.   
%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weak-coupling expansion for $H_s$}
\label{seriesHt}
%%%%%%%%%%%%%%%%%%%%%%%%%
Given the initial condition at $s=0$ in the form of the regulated Hamiltonian $H^{m_g \,  r}_{\rm QCD}$ of Eq.~(\ref{EqbareHr}), 
the differential Eq.~(\ref{RGPEP1}) can be solved order-by-order in the expansion of the coefficients $c_s$ in series of powers of 
the coupling constant $g$. Such expansion is valid when the coupling constant is made so small that all computed interaction 
terms are small. But solutions for the coefficients $c_s$ may contain inverse powers or logarithms of the regularization parameter $r$. 
When one lifts the regularization, taking the limit $r \to 0$, such coefficients become infinite despite that $s$ is finite and $g$ small. Consequently, the corresponding matrix elements of $H_s$ between the effective basis states of finite momenta diverge and the 
eigenvalue problem is ill-defined. To formulate a soluble theory, the initial condition at $s=0$, given by $H^{m_g \,  r}_{\rm QCD}$ 
in Eq.~(\ref{EqbareHr}), needs to be modified by adding terms, called counterterms, determined by the condition that the divergences 
in $c_s$ are eliminated. Thus, the initial condition for solving the RGPEP equation is changed from $H^{m_g \,  r}_{\rm QCD}$ of Eq.~(\ref{EqbareHr}) to
\beq
\label{initcond}
H^{m_g \,  r }_{\rm QCD \, \cC}
\es 
\int dx^- d^2x^\perp \, \left( \cH^{m_g \,  r}_{\rm QCD} + \cC^{m_g \,  r}_{\rm QCD} \right) \ .
\eeq
The subscript $\cC$ indicates the inclusion of the counterterms, denoted by $\cC^{m_g \,  r}_{\rm QCD}$. Perturbative solution 
of the RGPEP equation for $H_s$, including the counterterms in the initial condition at $s=0$, becomes calculable order-by-order 
for very small coupling constants, though the radius of convergence is not known. To describe hadrons, one needs to extrapolate 
the weak-coupling expansion to the realistic values of the interaction strength~\cite{Wilson:1994fk}. The realistic magnitudes 
ought to match the running interaction strength fitted to high-energy data within the same scheme. Outlines of the entire procedure, 
using simple models, are available in~\cite{Glazek:2002iw,Glazek:2003xr,Glazek:2021vnw}.
%%%%%%%%%%%%%%%%%%%%%
\subsection{Solutions of lowest orders}
%%%%%%%%%%%%%%%%%%%%%
This section illustrates the weak-coupling approach to solving the RGPEP equation for terms up to the third order. 
The second-order formulas are used later on to discuss the bound-state eigenvalue problems in QCD of heavy quarks 
including the mass $m_g$ and field $\phi$. The third-order solution shows how the RGPEP computations proceed 
beyond the second order, {\it cf.}~\cite{Glazek:2012qj}. 

In the series expansion of the operator $\cH$ in powers of $g$,
\beq
\label{cHexpansion}
\cH \es \cH_f + \cH^{(1)} + \cH^{(2)} + \cH^{(3)} + O(g^4) \ ,
\eeq
where the terms $\cH^{(n)}$ are proportional to $g^n$. The operator $\cH^{(n)}$ obeys the equation obtained by equating 
coefficients of $g^n$ on both sides of Eq.~(\ref{RGPEP1}). For the terms of first 3 orders, besides $\cH^{(0)} = \cH_f$, 
the equations read
\beq
\label{h31'text}
\cH^{(1)'}
\es
%1%
   \left[ \left[ \cH_f , \cH^{(1)} \right] , \cH_f \right]  \ , \\
\label{h32'text}
\cH^{(2)'}
\es
%2%
   \left[ \left[ \cH_f ,\cH^{(2)}  \right] , \cH_f \right]  
+ \left[ \left[ \cH_f , \cH^{(1)} \right] , \cH^{(1)} \right]  \ , \\
\label{h33'text}
\cH^{(3)'}
\es
%3%
   \left[ \left[ \cH_f ,\cH^{(3)}  \right] , \cH_f \right] 
+\left[ \left[ \cH_f ,\cH^{(2)}  \right] , \cH^{(1)} \right]
+\left[ \left[ \cH_f , \cH^{(1)} \right] , \cH^{(2)} \right]   \ .
\eeq
In general, the derivative of $\cH^{(n)}$ only involves operators $\cH^{(k)}$ with $k < n$, which facilitates solving 
for $\cH$ order-by-order to all orders. 

The solution for terms of the 1st order reads
\beq
\label{cH1t}
\cH^{(1)}
\es
f_{LR} \, \cH^{(1)}_0 \ ,
\eeq
where the subscript 0 refers to the initial condition at $s=0$, and
\beq
\label{f1t}
f_{LR} \es e^{-(s \Delta_{LR})^2 } \ , \\
\label{f1t2}
\Delta_{LR} \es p_L^- - p_R^- \ .
\eeq
The subscript $L$ refers to the product of creation operators in a term in $\cH$, standing on its left-hand side, and the 
subscript $R$ refers to the product of annihilation operators in a term, standing on its right-hand side. Using the 
notation introduced in Eqs.~(\ref{initialHi}) - (\ref{fr}) for an interaction term $\cH_i$, action of $f_{LR}$ on any term in 
Eq.~(\ref{cH1t}), and on any other operator at any value of $s$, is defined by
\beq
\label{Tft}
f_{LR} \cH_i \es 
           \left[ \prod_{m =1 }^{c_i} \int [p_m]  a_{p_m}^\dagger \right] 
           \left[ \prod_{ n = 1}^{a_i} \int [q_n ]  a_{q_n }              \right] 
           \tilde \delta_{c.a} \ T_i(\bar p, \bar q) \, f^s_{\bar p,\bar q} \ .
\eeq
In words, the action of $f_{LR}$ on $\cH_i$ implies insertion of the factor $f^s_{\bar p,\bar q}$ in the integrand of $\cH_i$. 
{\red  Correct for the difference between $f^r$ and $f^s$: The first-order solution explains our earlier adoption of the factor 
$f^r_{\bar p,\bar q}$ as a regulator in Eqs.~(\ref{Tif}) and (\ref{fr}), since $p_L^- = p_c^-$ and $p_R^- = p_a^-$. Namely, the 
regularization factors match the 1st-order solution of the RGPEP equation with $s$ set to $r$. }

The solution for terms of the second order, obtained using solutions for the first-order terms, reads
\beq
\label{cH2t}
\cH^{(2)}
\es
f_{LR} \, \cH^{(2)}_0
+ 
(f_{LR} - f_{LI} f_{IR}) \
\Delta_{LIR} \
\cH^{(1)}_0 \, \cH^{(1)}_0 \ .
\eeq
The subscript $I$ refers to the labels of the annihilation operators in $f_{LI}\cH_0^{(1)}$, or to the labels of 
the creation operators in $f_{IR}\cH_0^{(1)}$.
\beq
\label{DeltaLIR}
\Delta_{LIR} \es { \Delta_{LI} - \Delta_{IR} 
\over 
\Delta_{LI}^2 + \Delta_{IR}^2 - \Delta_{LR}^2 } \ .
\eeq
When the denominator in $\Delta_{LIR}$ vanishes, the difference of the form factors, $f_{LR} - f_{LI} f_{IR}$, vanishes, too. 

The solution for the third-order terms reads
\beq
\label{RGPEP3solution3}
\cH^{(3)} \es 
( f_{LR} - f_{LI} f_{IR} ) 
\, \Delta_{LIR}  
\left[ \Delta_{IJR} \, \cH^{(1)}_0 \, \cH^{(1)}_0 \, \cH^{(1)}_0 + \cH^{(1)}_0 \, \cH^{(2)}_0  \right]
\np
( f_{LR} - f_{LJ} f_{JR} )        
\, \Delta_{LJR} 
\left[ \Delta_{LIJ} \, \cH^{(1)}_0 \, \cH^{(1)}_0 \, \cH^{(1)}_0 + \cH^{(2)}_0 \, \cH^{(1)}_0 \right]  
\nm
( f_{LR} - f_{LI} f_{IJ} f_{JR} )  \, \Delta_{LIJR} 
\,  \cH^{(1)}_0 \, \cH^{(1)}_0 \cH^{(1)}_0 \ ,  
\eeq
where
\beq
\Delta_{LIJR} 
\es
{  (\Delta_{LJ} - \Delta_{JR}) \Delta_{LIJ}
+ (\Delta_{LI} - \Delta_{IR}) \Delta_{IJR} 
\over 
    \Delta_{LI}^2 + \Delta_{IJ}^2 + \Delta_{JR}^2 - \Delta_{LR}^2 } \ .
\eeq
Subscripts $I$ and $J$ refer to the intermediate configurations of the particle operators involved in the interaction terms. 
Solutions for the Hamiltonian terms of orders higher than 3rd are obtained following the pattern shown by these examples.
%%%%%%%%%%%%%%%
\subsection{Counterterms}
\label{cC}
%%%%%%%%%%%%%%%
Computation of the counterterms in the initial Hamiltonian, $H^{m_g \,  r }_{\rm QCD \, \cC}$, is outlined above 
Eq.~(\ref{initcond}). It is illustrated below in the case of the quark self-interaction, which contributes to the quark 
bound-state dynamics. The self-interaction appears first in the second-order Hamiltonian in Eq.~(\ref{cH2t}). The 
right-hand side contains a product of two first-order terms, $\cH^{(1)}_0 \, \cH^{(1)}_0$. Consider the part of that 
product corresponding to Fig.~\ref{fig:self}, where a quark emits and absorbes gluons.  The part involves the product 
$b_1^\dagger a_3 b_2 \ b_{\tilde 2}^\dagger a_{\tilde 3}^\dagger b_{\tilde 1}$. Commuting $b_2$ to the right of 
$b_{\tilde 2}^\dagger$ and $a_3$ to the right of $a_{\tilde 3}^\dagger$ yields $\delta_{2 \tilde 2} \delta_{3 \tilde 3}$, 
according to the commutation rules specified in Eq.~(\ref{comrules}). Integration $\int [\tilde 1 \tilde 2 \tilde 3]$ 
results in the operator $\hat C=\int[1] \, c_1 \, \hat b_1^\dagger \hat b_1$, with the coefficient
\beq
\label{c1quark}
c_1 \es 
{ g^2  C_F \over p_1^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, \left( f_{LR} - f_{LI} f_{IR}  \right) {N \over D}  \, f^{r 2}_{qg,p} \ .
\eeq 
The numerator $N$ and denominator $D$ are given in Eqs.~(\ref{NumSigmaQfull}) and (\ref{NumSigmaQfullD}), 
{\it cf.} Eq.~(\ref{selfinteraction1quark333full}). Integration variables $x$ and $k^\perp$ are defined below Eq.~(\ref{selfinteraction1quark333}). The color factor $C_F$ equals 4/3 for $SU(3)$. The operators $\hat b_1^\dagger$ 
and $\hat b_1$ correspond to the same quark $p_1^-$. Therefore, Eq.~(\ref{f1t}) impllies $f_{LR}=1$. In contrast, the 
four-momenta $p_2$ and $p_3$, corresponding to the labels of the contracted quark and gluon operators 2 and 3, 
yield $f_{LI} = f_{IR} = \exp \left\{ - (s/p_1^+)^2 [m_1^2 - (p_2 + p_3)^2]^2\right\}$, which results in
\beq
f_{LI} f_{IR} \es \exp \left\{ - 2 (s D/p_1^+)^2 /[x(1-x)]^2 \right\} \ .
\eeq
This factor vanishes for small $x$ and large $k^\perp$. Therefore, the integrand involving $f_{LI} f_{IR}$ contributes a finite 
quantity to the coefficient $c_1(s>0)$ in the limit $r \to 0$. 

The only source of severe small-$x$ singularity in the coefficient $c_1$ is the term $4(k^{\perp 2} + m_g^2)/x^2$ in the 
numerator $N$, in the integrand with $f_{LR}=1$. This term is also singular as a function of $k^\perp \to \infty$. Both 
singularities are regulated by the factor $f^{r 2}_{qg,p}$. When the regularization is lifted by going to the limit $r \to 0$, 
the quark self-interaction diverges to negative infinity proportionally to $1/r$. But the solution for $\cH^{(2)}$ in Eq.~(\ref{cH2t}) 
contains also the term $f_{LR} \, \cH^{(2)}_0$, where $\cH^{(2)}_0$ is the initial condition for solving the RGPEP equation.
The initial condition can be set to contain a part canceling the divergent dependence of the self-interaction on $r$. It is visible 
that the required part amounts to the replacement of the integrand $f_{LR} N/D$ by $f_{LR}[ N/D+ 4 \sigma_q^{(2)} ]$, where 
$\sigma_q^{(2)}$ is given in Eq.~(\ref{sigmaqintfull}). This is how the RGPEP explains the guess made in the scattering theory. 
The resulting integrand,
\beq
f_{LR} [N/D+ 4 \sigma_q^{(2)}] \es - (4 m_1^2 + 2 m_g^2)/D \ ,
\eeq
integrates in the limit $r \to 0$ to a logarithmically divergent term,
\beq
\label{leftlog}
-\delta m^2_{1\ln} \es - {g^2  C_F \over  (2\pi)^2 } m_1^2 
\left[ \ln (\sqrt{2}\,r_p \, m_1^2) + \gamma_E/2 \right] \ .
\eeq
Consequently, the Hamiltonian $\cH_0^{(2)}$ is further supplied with a mass-squared counterterm $\delta m^2_{1 \ln}$. The 
result of including these quark self-interaction counterterms is that the quark masses appearing in the eigenvalue problems 
for effective Hamiltonian matrices, such as in Eq.~(\ref{HLR1pom1text}), are the ones introduced in $H_f$ in Eq.~(\ref{hatHf}). 
Similar analysis can be carried out for the gluon self-interaction counterterms.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Gluon mass and small $x$ in heavy quarkonia}
\label{Qdynamics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To discuss the cancellation of gluon singularities $\sim 1/x^2$ in the QCD eigenvalue problems with $m_g > 0$, we consider
the theory with only heavy quarks, {\it i.e.}, the theory in which $\Lambda_{\rm QCD}$ of the RGPEP scheme is set to values
much smaller than the quark masses, {\it cf.}~\cite{Serafin:2023pkf}. In such a theory, the running coupling constant for the 
parameter $s \lesssim 1/m$, where $m$ is the quark mass, can be made sufficiently small for using the weak-coupling expansion 
of Sec.~\ref{seriesHt} to approximately compute the Hamiltonian $H_s$ in Eq.~(\ref{Ht}). 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Eigenvalue problem for the dominant component}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
According to Eqs.~(\ref{Htep}) and \ref{Hteppsi}), the quarkonium eigenstate of the Hamiltonian $H_s$ has infinitely many components 
with various numbers of the effective quarks, antiquarks, and gluons, {\it cf.}~\cite{Glazek:2021vnw}. However, the interaction terms that change the number of effective particles include a small coupling constant and the vertex factors $f^s$, which implies that a quarkonium eigenstate of the lowest mass is dominated by its quark-antiquark component,  
\beq
\label{Pnt1}
|P \rangle \es \int[12] \, \psi_P(1,2) \, |12\rangle_s \ ,
\eeq
where $|12\rangle_s = b_{s1}^\dagger d_{s2}^\dagger |0\rangle$ and $P$ indicates that $|P\rangle$ is an 
eigenstate of the operators $\hat P^+$ and $\hat P^\perp$ with the eigenvalues $P^+$ and $P^\perp$, 
respectively. In other words, the matrix elements of the Hamiltonian $H_s$ between the effective quark-antiquark basis states are assumed to form the central part of the window matrix whose eigenstates approximately represent quarkonia in the theory, see Sec. 4.3 in~\cite{Glazek:2026xnp}. The assumption of dominance of the two-body component is successfully used in interpreting quarkonium data~\cite{ParticleDataGroup:2024cfk,Appelquist:1974zd,
Appelquist:1974yr,Eichten:1978tg,Eichten:1979ms,Wilson:1974sk,Bodwin:1994jh,Brambilla:2010cs}. In our approach, the two-body component can, as a first approximation, be described by solutions to the eigenvalue problem for the part of the operator $H_s$ whose form is
\beq
H_{Q\bar Q} \es \int[122'1'] \, {_s\langle 12|} H_s | 1'2' \rangle_s  \  b_{s1}^\dagger d_{s2}^\dagger d_{s2'} b_{s1'} \ . 
\eeq
The integration includes summing over the discrete quantum numbers. Without losing generality, the wave function is given the form
\beq
\psi_P(1,2) \es P^+ \tilde \delta_{12.P} \ \psi_{1 2}(x_1,k_{12}^\perp) \ ,
\eeq 
where $\tilde \delta_{12.P}$ constrains the quarks to carry together $P^+$ and $P^\perp$, {\it cf.} Eq.~(\ref{deltac.a}). The relative momentum variables $x_1$ and $k_{12}^\perp$ are defined as in the case of an interaction term in Eq.~(\ref{HA3k12}). The eigenvalue equation for the 
wave function reads
\beq
\label{EV2}
\int[1'2'] \, _s\langle 12| H_s | 1'2' \rangle_s \, \psi_P(1',2') \es E \, \psi_P(1,2) \ ,
\eeq
where $E(P^+,P^\perp) = (P^{\perp 2} + M^2)/P^+$ and $M$ denotes the quarkonium mass, $E(M,0^\perp)=M$. In the leading order of 
the weak-coupling expansion, $H_s = H_f + H_s^{(2)}$, where $H_s^{(2)}$ is obtained from $\cH^{(2)}$ of Eq.~(\ref{cH2t}) using Eq.~(\ref{Ht}). Hence, 
\beq
\label{2bodyMatrixHs}
_s\langle 12| H_s^{(2)} | 1'2' \rangle_s 
\es  
\langle 12| \left[ \cH_f + \cH^{(2)} \right] |1'2'\rangle \ ,
\eeq
which allows one to directly obtain the interaction matrix elements $_s\langle 12| H_s^{(2)} | 1'2' \rangle_s$ 
from Eq.~(\ref{cH2t}) for $\cH^{(2)}$. The direct relationship between matrix elements of $H_s$ and $\cH$
is a general feature of the RGPEP. Also, notice that there are no first-order terms in Eq.~(\ref{2bodyMatrixHs}), corresponding to $\cH^{(1)}$ in Eq.~(\ref{cHexpansion}), because such terms change the number of quanta.    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hamiltonian matrix for dominant component}
\label{Hin2bodySpace}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
According to Eq.~(\ref{cH2t}), the Hamiltonian matrix of Eq.~(\ref{2bodyMatrixHs}) reads   
\beq
\label{HLR1pom1text}
_s\langle 12| H_s | 1'2' \rangle_s
\es \langle 12|  \left\{  {\cH_f} + f_{LR} \, \cH_r^{(2)}  - f_{LI} f_{IR} \, \Delta_{LIR} \, \cH_0^{(1)}  \, \cH_0^{(1)} \right\}  | 1'2' \rangle \ , 
\eeq
where 
\beq
\label{cHr}
\cH_r^{(2)} \es \cH^{(2)}_0 + \Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0 \ .
\eeq
In Eq.~(\ref{HLR1pom1text}), the factor $f_{LI}f_{IR}$  suppresses all singularities generated by the product $\cH^{(1)}_0 \, 
\cH^{(1)}_0$. The factor $f_{LR}$ does not produce such suppression. Instead, the divergent quark self-interactions, resulting 
from $\Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0$, are canceled by the self-interaction counterterms introduced in $\cH^{(2)}_0$, 
as described in Sec.~\ref{cC}. But the gluon-exchange terms in $\Delta_{LIR} \ \cH^{(1)}_0 \, \cH^{(1)}_0$ in Eq.~(\ref{cHr}) 
still diverge like $1/x^2$, where $x$ is the fraction carried by the exchanged gluon of the total bound-state momentum 
$P^+$. Similar divergence is contributed by the gluon seagull term in $\cH^{(2)}_0 $.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cancellation of $1/x^2$ in bound states}
\label{xinboundstates}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We now describe how the singularities $\sim 1/x^2$ in terms $\Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0$ and $\cH^{(2)}_0$,
both contained in $\cH_r^{(2)}$, cancel each other in the bound-state eigenvalue problem for the dominant component. 
The cancellation occurs despite that $m_g$ is not zero. 
\begin{figure}[ht!]
          \caption{Notation for the matrix elements $\langle 12| f_{LR}\cH_r^{(2)} | 1'2' \rangle$ 
                        involving two orderings A and B of the gluon exchange and the seagull C.} 
          \label{fig:ABC}
\begin{center}
         \includegraphics[width=0.5\textwidth]{ExchangeNotation}
\end{center} 
\end{figure}
The matrix elements $\langle 12| f_{LR}\cH_r^{(2)} | 1'2' \rangle$ have the form  $g^2 \tilde \delta j_\mu h^{\mu \nu} \bar j_\nu$,
where $\tilde \delta$ secures conservation of the total momentum components $P^+$ and $P^\perp$, $j$ and $\bar j$ are the
quark and anti-quark currents, and  
\beq
\label{exchangefr}
h^{\mu \nu} \es
f_{LR} \left\{ \left[ - g^{\mu \nu} + \eta^\mu \eta^\nu (\rho_1 + \rho_2)/ (2q^{+2})\right] \2
\left( {1 \over \rho_1 } + {1 \over \rho_2} \right)  - {\eta^\mu \eta^\nu \over q^{+2}} \right\} f^r_{\overline 1,\underline 1 \rm g} f^r_{\overline 2, \underline 2 \rm g} \ .
\eeq
This formula is illustrated using diagrams in Fig.~\ref{fig:ABC}. We introduce $\overline 1$ and $\underline 1$





The term with $-g^{\mu \nu}$ provides the Coulomb potential term in the Hamiltonian for the 2-body component
in the limit $m_g \to 0$, with the kernel
\beq
- f_{LR} \ g^{\mu \nu} \  \2 \left( {1 \over \rho_1 } + {1 \over \rho_2} \right)  \ .
\eeq
This kernel is contracted with the quark and antiquark currents providing spin effects. The factor $f_{LR}$ deviates from 1 
in the non-relativistic limit only by small amounts discussed in~\cite{Glazek:2014ria}. In that limit, $\rho_1=\rho_2 =  
\vec q\,^2 + m_g^2 $. The terms with $\eta$ sum up to
\beq
f_{LR} \left[ (\rho_1 + \rho_2) \left( {1 \over \rho_1 } + {1 \over \rho_2} \right) -  4 \right]
\ { \eta^\mu \eta^\nu \over 4q^{+2}} 
\es
f_{LR} \ { (\rho_1 - \rho_2)^2  \over \rho_1 \rho_2} \ { \eta^\mu \eta^\nu \over 4q^{+2}}
\ .
\eeq
Using Eq.~(\ref{rho1-rho2}), one obtains
\beq
f_{LR} \ { (\rho_1 - \rho_2)^2  \over \rho_1 \rho_2} \ { \eta^\mu \eta^\nu \over 4q^{+2}}
\es
f_{LR} \ { (\cM_{12}^2 - \cM_{1'2'}^2 )^2  \over \rho_1 \rho_2} \ { \eta^\mu \eta^\nu \over 4 P_{12}^{+2}} \ ,
\eeq
which is a spin-independent interaction kernel integrated with the measure $d^3k_{1'2'}$ and the wave function $\psi_{1'2'}
(\vec k_{1'2'})$ in the 2-body eigenvalue equation. This term vanishes on shell of the free 2-body $P^-$ and hence has no 
counterpart in any classical approximation for the quarkonium dynamics. Its magnitude can be illustrated with the case of 
equal quark masses, in which $\cM_{12}^2 - \cM_{1'2'}^2 = 4(\vec k_{12}^2 - \vec k_{1'2'}^2)$. In the non-relativistic limit, 
$\rho_1 = \rho_2 = (\vec k_{12} - \vec k_{1'2'})^2$ for $m_g \to 0$. The term is suppressed by the factor $\sim K^2 c^2/(2m)^2$ 
in comparison with the Coulomb term, where $\vec K =\vec k_{12} + \vec k_{1'2'}$ and $c$ denotes the cosine of the angle 
between $\vec q$ and $\vec K$. Its quantitative role in the eigenvalue equation requires a study. For example, it contributes
to the quarkonium mass splittings between states with the same spins.

\beq
\label{HLR1pom1text2bodysummary}
_s\langle 12| H_s | 1'2' \rangle_s
\es \langle 12|  \left\{ {\cH_f} + f_{LR} \, \cH_r  - f_{LI} f_{IR} \,  \Delta_{LIR} \, \cH_0^{(1)} \cH_0^{(1)} \right\}  | 1'2' \rangle \ , \\
\label{cHr2bodysummary}
\cH_r \es \cH^{(2)}_0 + \Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0
\ .
\eeq
\begin{enumerate}
\item
$\cH_r$: small-$x$ divergences in the seagull and exchange, and the ultraviolet divergences in the quark self-interactions.
\item
$\cH^{(2)}_0$: the seagull, the seagull counterterm, the counterterm to gluon xchange, the quark self-interaction counterterms.
\item
$\Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0$ in $\cH_r$: the gluon exchange, the quark self-interactions.
\item
$ f_{LI} f_{IR} \, \Delta_{LIR} \, \cH_0^{(1)}  \cH_0^{(1)}$: the finite gluon exchange between the quarks 
and the finite quark self-interactions, depending on the gluon mass $m_g$. 
\end{enumerate}

The terms in $\cH_r$ are independent of the gluon mass $m_g \to 0$.

The terms in $f_{LI} f_{IR} \, \Delta_{LIR} \, \cH_0^{(1)}  \cH_0^{(1)}$ depend on $m_g \to 0$ in a useful way.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{\bf Contribution of $- f_{LI} f_{IR} \, \Delta_{LIR} \, \cH_0^{(1)} \cH_0^{(1)}$ to Eq.~(\ref{HLR1pom1text})}
\label{termsffHH}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The last term in the 2-body Hamiltonian matrix in Eq.~(\ref{HLR1pom1text}) is 
\beq
H_{\rm 2body} \es - f_{LI} f_{IR} \, \Delta_{LIR} \, \cH_0^{(1)}  \cH_0^{(1)} \ .
\eeq 
It includes the effective self-interaction of quarks and an exchange of an effective gluon between them. These terms are 
all finite because the vertex form factors prevent production of singular configurations from regular ones. But they do 
depend on the gluon mass $m_g$. The question is what happens in the limit $m_g \to 0$.

\vskip.1in \centerline{\bf The exchange terms}  \vskip.1in 

The exchange-term kernel 
has necessarily the form shared with the exchange term in Eq.~(\ref{exchangefr}). To interpret these interaction terms, I need the 
operator $- f_{LI} f_{IR} \, \Delta_{LIR} \, \cH_0^{(1)} \cH_0^{(1)}$. Using Eqs.~(\ref{cHcanQCD}) and (\ref{cHcanQCDmg}), I have 
\beq
\cH_0^{(1)} \es 
\int d^2x^\perp dx^- \ \left[ :  A_{f\mu}^a  (- J_{\psi f}^{a \mu})  %
+ i m_g \phi^a {1 \over i \partial^+} J_{\psi f}^{a+}  : \right]^r  \\
%
%\es \int d^2x^\perp dx^- \ \left[ :  A_{f\mu}^a \ g \bar \psi_f T^a \gamma^\mu  \psi_f
%- i m_g \phi^a {\eta_\mu \over i \partial^+} \,  g \bar \psi_f T^a \gamma^\mu \psi_f :
% \right]^r \\
%\es \int d^2x^\perp dx^- \ \left[ :  A_{f\mu}^a \ g \bar \psi_f T^a \gamma^\mu  \psi_f
%+ (i m_g {\eta_\mu \over i \partial^+} \phi^a )\,  g \bar \psi_f T^a \gamma^\mu \psi_f
%: \right]^r \\
%
\es \int d^2x^\perp dx^- \ \left[ :  \left( A_{f\mu}^a + i m_g {\eta_\mu \over i \partial^+} \phi^a \right) 
\,  g \bar \psi_f T^a \gamma^\mu \psi_f
: \right]^r
\eeq 
and according to Eqs.~(\ref{barepsif}), (\ref{bareAf}), and (\ref{hatphi}), I insert
\beq
\label{barepsifpom}
\hat \psi_f
\es
\sum_{c=1}^3 \sum_{\sigma = 1}^2  \int [p]
\left[  u_{p\sigma} \chi_c \hat b_{p \sigma c} e^{-ipx}
      +    v_{p\sigma} \chi_c \hat d^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0} \ , \\
\label{bareAfpom}
\hat A_f^\mu
\es
\sum_{c=1}^8 
\sum_{\sigma =1}^2  \int [p]
\left[  \varepsilon^\mu_{p\sigma} T^c \hat a_{p\sigma c} e^{-ipx}
        +  \varepsilon^{\mu *}_{p\sigma} T^c \hat a^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0}
\ ,
\\
\label{hatphipom}
i m_g {\eta_\mu \over i \partial^+} \hat \phi
\es
\sum_{c=1}^8 
\int [p] {\eta^\mu m_g  T^c \over p^+} \, 
\left[ \hat a_{p3c} e^{-ipx} + \hat a^\dagger_{p3c} e^{ipx} \right]_{x^+=0} \ .
\eeq
I can introduce the third polarization vector for the longitudinal gluons, by writing
\beq
\label{bareAfpom3}
\hat A_f^\mu
\es
\sum_{c=1}^8 
\sum_{\sigma =1}^3  \int [p]
\left[  \varepsilon^\mu_{p\sigma} T^c \hat a_{p\sigma c} e^{-ipx}
        +  \varepsilon^{\mu *}_{p\sigma} T^c \hat a^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0} \ , \  \epsilon_{p3}^\mu \rs m_g \eta^\mu/p^+ \ .
\eeq
The current is
\beq
- J_{\psi f}^{a \mu} \es g \bar \psi_f T^a \gamma^\mu \psi_f \\
\es
\int[pp']
\left[  \bar u_{p} \chi^\dagger_p b^\dagger_{p} e^{ipx} T^q \gamma^\mu u_{p'} \chi_{p'} b_{p'} e^{-ip'x}  
+       \bar v_{p} \chi^\dagger_p  d_p e^{-ipx} T^q \gamma^\mu  v_{p'} \chi_{p'} d^\dagger_{p'} e^{ip'x} \right] \ .
\eeq
The Hamiltonian term is
\beq
\cH_0^{(1)} \es g \int[pp'q] \ \tilde \delta_{c.a} \, f^t 
\left(  \bar u_{p} \gamma_\mu u_{p'} \ \chi^\dagger_p T^q \chi_{p'} \ b^\dagger_{p} b_{p'}  
-        \bar v_{p'} \gamma_\mu  v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} \ d^\dagger_{p} d_{p'}  \right)
\, \left(  \varepsilon^\mu_{q} a_{q} +  \varepsilon^{\mu *}_{q} a^\dagger_{q} \right) \ .
\eeq
Consider only real polarization vectors for gluons; I sum over their polarization anyway. 
\beq
\cH_0^{(1)} \es g \int[pp'q] \ \tilde \delta_{c.a} \, f^r 
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\ (   a_{q} + a^\dagger_{q} ) b^\dagger_{p} b_{p'}  
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\ (   a_{q} + a^\dagger_{q} ) d^\dagger_{p} d_{p'}  \right] \ .
\eeq
So,
\beq
H_{\rm 2body} \es - f_1^t f_2^t \, \Delta_{12g2'1'} \, \cH_0^{(1)} \cH_0^{(1)} 
\eeq
involves the product $\cH_0^{(1)} \cH_0^{(1)}$ reading
\beq
&& g \int[pp'q] \ \tilde \delta_{c.a} \, f^r
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\ (   a_{q} + a^\dagger_{q} ) b^\dagger_{p} b_{p'}  
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\ (   a_{q} + a^\dagger_{q} ) d^\dagger_{p} d_{p'}  \right] 
\nt
g \int[pp'q] \ \tilde \delta_{c.a} \, f^r
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\ (   a_{q} + a^\dagger_{q} ) b^\dagger_{p} b_{p'}  
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\ (   a_{q} + a^\dagger_{q} ) d^\dagger_{p} d_{p'}  \right] \ .
\eeq
The gluon operators need to be contracted and only terms with $b^\dagger b$, $d^\dagger d$, and 
$b^\dagger d^\dagger d b$ are kept. The products that contribute are
\beq
&& g \int[pp'q] \ \tilde \delta_{c.a} \, f^r
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\   b^\dagger_{p} b_{p'}  a_{q}
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\  d^\dagger_{p} d_{p'} a_{q} \right] 
\nt
g \int[pp'q] \ \tilde \delta_{c.a} \, f^r
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\ a^\dagger_{q} b^\dagger_{p} b_{p'}  
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\ a^\dagger_{q} d^\dagger_{p} d_{p'}  \right] \ .
\eeq
The exchange terms between a quark and antiquark come from 
\beq
&& g \int[pp'q] \ \tilde \delta_{c.a} \, f^r
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} \ b^\dagger_{p} b_{p'}  a_{q} \right] 
\
g \int[pp'q] \ \tilde \delta_{c.a} \, f^r
\left[  -  \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} \ a^\dagger_{q} d^\dagger_{p} d_{p'}  \right] 
\np
 g \int[pp'q] \ \tilde \delta_{c.a} \, f^r
\left[  -   \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} \  d^\dagger_{p} d_{p'} a_{q} \right] 
\
g \int[pp'q] \ \tilde \delta_{c.a} \, f^r
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} \ a^\dagger_{q} b^\dagger_{p} b_{p'}  \right] \ .
\eeq
In the notation 123, the exchange terms read
\beq
&& g \int[11'3] \ \tilde \delta_{1.1'3} \, f^r
\left[  \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} \ \chi^\dagger_1 T^3 \chi_{1'} \ 
b^\dagger_{1} b_{1'}  a_{3} \right] 
\
g \int[22'\tilde 3] \ \tilde \delta_{2\tilde 3.2'} \, f^r
\left[  -  \bar v_{2'} \gamma_\mu \varepsilon^\mu_{\tilde 3} v_{2} \ \chi^\dagger_{2'} T^{\tilde 3} \chi_{2} \ 
a^\dagger_{\tilde 3} d^\dagger_{2} d_{2'}  \right] 
\np
 g \int[22'3] \ \tilde \delta_{2.2'3} \, f^r
\left[  -   \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2} \ \chi^\dagger_{2'} T^3 \chi_{2} \  d^\dagger_{2} d_{2'} a_{3} \right] 
\
g \int[11'\tilde 3] \ \tilde \delta_{1\tilde 3.1'} \, f^r
\left[  \bar u_{1} \gamma_\mu \varepsilon^\mu_{\tilde 3} u_{1'} \ \chi^\dagger_1 T^{\tilde 3} \chi_{1'} \ 
a^\dagger_{\tilde 3} b^\dagger_{1} b_{1'}  \right] \ .
\eeq
Contraction of $\tilde 3$ with $3$ yields the exchange terms
\beq
\label{contractionexch}
&& - g^2 \int[11'3] \ \tilde \delta_{1.1'3} \, f^r
\ \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} \ \chi^\dagger_1 T^3 \chi_{1'} \
\int[22'] \ \tilde \delta_{23.2'} \, f^r
\  \bar v_{2'} \gamma_\nu \varepsilon^\nu_{3} v_{2} \ \chi^\dagger_{2'} T^{3} \chi_{2} 
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} 
\nm
 g^2 \int[22'3] \ \tilde \delta_{2.2'3} \, f^r
\ \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2} \ \chi^\dagger_{2'} T^3 \chi_{2} \
\int[11'] \ \tilde \delta_{13.1'} \, f^r
\ \bar u_{1} \gamma_\nu \varepsilon^\nu_{3} u_{1'} \ \chi^\dagger_1 T^{3} \chi_{1'} 
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} \ .
\eeq
Integration over 3 yields $\tilde \delta_{12.2'1'}/p_g^+$ and $\theta_{1-1'}$ in the first term and $\theta_{1'-1}$ in the second term.
\beq
&& - g^2 \sum_3 \int[122'1'] \ \tilde \delta_{12.1'2'} \ \theta_{1-1'}/p_g^+ \, f^r
\ \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} \ \chi^\dagger_1 T^3 \chi_{1'} \ f^r
\  \bar v_{2'} \gamma_\nu \varepsilon^\nu_{3} v_{2} \ \chi^\dagger_{2'} T^{3} \chi_{2} 
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} 
\nm
 g^2 \sum_3 \int[122'1'] \ \tilde \delta_{12.1'2'} \ \theta_{1'-1} /p_g^+ \, f^r
\ \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2} \ \chi^\dagger_{2'} T^3 \chi_{2} \
\int[11'] \ \tilde \delta_{13.1'} \, f^r
\ \bar u_{1} \gamma_\nu \varepsilon^\nu_{3} u_{1'} \ \chi^\dagger_1 T^{3} \chi_{1'} 
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} \ .
\eeq
Denoting
\beq
j_\mu \es \bar u_{1} \gamma_\mu u_{1'}  \ , \  \bar j_\nu  \rs \bar v_{2'} \gamma_\nu  v_{2} \ ,
\eeq
I obtain
\beq
H_{\rm 2body \ exch} \es - f_1^t f_2^t \, \Delta_{12q2'1'} \, \cH_0^{(1)} \cH_0^{(1)} \\
\es
g^2 f_1^t f_2^t \, \Delta_{12q2'1'} \ \sum_3 \int[122'1'] \ \tilde \delta_{12.1'2'} \ \theta_{1-1'}/p_g^+ \, f^r
\ \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} \ \chi^\dagger_1 T^3 \chi_{1'} \ f^r
\  \bar v_{2'} \gamma_\nu \varepsilon^\nu_{3} v_{2} \ \chi^\dagger_{2'} T^{3} \chi_{2} 
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} 
\np
 g^2 f_1^t f_2^t \, \Delta_{12q2'1'} \ \sum_3 \int[122'1'] \ \tilde \delta_{12.1'2'} \ \theta_{1'-1} /p_g^+ \, f^r
\ \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2} \ \chi^\dagger_{2'} T^3 \chi_{2} 
\, f^r
\ \bar u_{1} \gamma_\nu \varepsilon^\nu_{3} u_{1'} \ \chi^\dagger_1 T^{3} \chi_{1'} 
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} \nn
\eeq
Introducing the color factor and sum over gluon polarization, 
\beq
C \es \sum_3 \chi^\dagger_1 T^3 \chi_{1'} \ \chi^\dagger_{2'} T^3 \chi_{2} \ , \\
d^{\mu \nu} \es - g^{\mu\nu} + (p_g^\mu \eta^\nu + \eta^\mu p_g^\nu)/p_g^+ \ , \\
\eeq
and neglecting factors $f^r$ in the presence of the factors $f^t$, I have
\beq
H_{\rm 2body} \es - f_1^t f_2^t \, \Delta_{12g2'1'} \, \cH_0^{(1)} \cH_0^{(1)} \\
\es
g^2 f_1^t f_2^t \, \Delta_{12g2'1'} \int[122'1'] \ C \ \theta_{1-1'}/p_g^+ 
\ j_\mu d^{\mu\nu} \bar j_\nu \ \tilde \delta_{12.1'2'}
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} 
\np
 g^2 f_1^t f_2^t \, \Delta_{12g2'1'} \int[122'1'] \ C \ \theta_{1'-1} /p_g^+ 
\  j_\mu d^{\mu\nu} \bar j_\nu \ \tilde \delta_{12.1'2'} 
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} \ .
\eeq
The two terms can be written together as
\beq
H_{\rm 2body} \es - f_1^t f_2^t \, \Delta_{12q2'1'} \, \cH_0^{(1)} \cH_0^{(1)} \\
\es
g^2 C \int[122'1'] \, f_{\underline 1 g, \overline 1}^t f_{\overline 2,\underline 2 g}^t \, 
\Delta_{12q2'1'}/p_g^+ \ j_\mu d^{\mu\nu} \bar j_\nu \ \tilde \delta_{12.1'2'} 
\ b^\dagger_{1}  d^\dagger_{2} d_{2'} b_{1'} \ .
\eeq
Contribution of this term to the eigenvalue Eq.~(\ref{EV2}), {\it cf.} Eq.~\ref{testmatrixelement}), in the form
\beq
\langle 12| H_{\rm 2body \, exch}| P_\phi \rangle 
\es
\int[2'1'] \, K_{\rm exch}(12,2'1')  \ P_\phi^+ \tilde \delta_{P_\phi.1'2'} \ \phi_{1' 2'}(x_{1'},k_{1'2'}^\perp)
\eeq
with the kernel of dimension ${1 \over + \perp^2}$,
\beq
\label{exchkernel}
K_{\rm exch}(12,2'1') \es g^2 C_{|2\rangle} \, f_{\underline 1 g, \overline 1}^t f_{\overline 2,\underline 2 g}^t \, 
\Delta_{12q2'1'}/p_g^+ \, j_\mu d^{\mu\nu} \bar j_\nu \, \tilde \delta_{12.1'2'} \ , \\
\Delta_{12q2'1'}/p_g^+ \es - \, \2 \left( {1 \over \rho_1} + {1 \over \rho_2 } \right) \ , \\
d^{\mu\nu} \es - g^{\mu \nu} + \eta^\mu \eta^\nu (\rho_1 + \rho_2)/ (2p_g^{+2}) \ .
\eeq
The exchange interaction is finite for finite $s$ despite that it includes the factor $1/q^{+2}$. It is finite because of 
the presence of $m_g > 0$ in the factors $f_{\underline 1 g, \overline 1}^t f_{\overline 2,\underline 2 g}^t$. Discussion
of the 2-body eigenvalue equation in the limit $m_g \to 0$ requires inclusion of the self-interaction terms, which also 
diverge in the limit $m_g \to 0$.

\vskip.1in \centerline{\bf The self-interaction terms}  \vskip.1in 

The self-interactions are one-body operators which I reevaluate below.
\beq
H_{\rm 2body \, \Sigma} \es \left[ - f_{LI} f_{IR} \, \Delta_{LIR} \, \cH_0^{(1)}  \cH_0^{(1)} \right]_\Sigma \ .
\eeq 
The products contributing are
\beq
&&
g \int[pp'q] \ \tilde \delta_{c.a} \, f^t
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\ a^\dagger_{q} b^\dagger_{p} b_{p'}  
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\ a^\dagger_{q} d^\dagger_{p} d_{p'}  \right]
\nt
g \int[pp'q] \ \tilde \delta_{c.a} \, f^t
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\ a^\dagger_{q} b^\dagger_{p} b_{p'}  
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\ a^\dagger_{q} d^\dagger_{p} d_{p'}  \right] \ ,
\eeq
but for the self-interaction count only
\beq
&& g \int[pp'q] \ \tilde \delta_{c.a} \, f^t
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\   b^\dagger_{p} b_{p'}  a_{q}
 \right] 
\
g \int[pp'q] \ \tilde \delta_{c.a} \, f^t
\left[  \bar u_{p} \gamma_\mu \varepsilon^\mu_{q} u_{p'} \ \chi^\dagger_p T^q \chi_{p'} 
\ a^\dagger_{q} b^\dagger_{p} b_{p'}  
 \right]
\np
g \int[pp'q] \ \tilde \delta_{c.a} \, f^t
\left[  
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\  d^\dagger_{p} d_{p'} a_{q} \right] 
\ g \int[pp'q] \ \tilde \delta_{c.a} \, f^t
\left[  
-        \bar v_{p'} \gamma_\mu \varepsilon^\mu_{q} v_{p} \ \chi^\dagger_{p'} T^q \chi_{p} 
\ a^\dagger_{q} d^\dagger_{p} d_{p'}  \right] \ .
\eeq
Using the parallel with the exchange calculation in Eq.~(\ref{contractionexch}), the notation in terms of 123 yields
\beq
&& g \int[11'3] \ \tilde \delta_{1.1'3} \, f^t
\left[  \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} \ \chi^\dagger_1 T^3 \chi_{1'} 
\   b^\dagger_{1} b_{1'}  a_{3}
 \right] 
\
g \int[22'\tilde 3] \ \tilde \delta_{2\tilde 3.2'} \, f^t
\left[  \bar u_{2} \gamma_\mu \varepsilon^\mu_{\tilde 3} u_{2'} \ \chi^\dagger_2 T^{\tilde 3} \chi_{2'} 
\ a^\dagger_{\tilde 3} b^\dagger_{2} b_{2'}  
 \right] 
\np
g \int[22'3] \ \tilde \delta_{2.2'3} \, f^t
\left[  
-        \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2} \ \chi^\dagger_{2'} T^3 \chi_{2} 
\  d^\dagger_{2} d_{2'} a_{3} \right] 
\ g \int[11'\tilde 3] \ \tilde \delta_{1\tilde 3.1'} \, f^t
\left[  
-        \bar v_{1'} \gamma_\mu \varepsilon^\mu_{\tilde 3} v_{1} \ \chi^\dagger_{1'} T^{\tilde 3} \chi_{1} 
\ a^\dagger_{\tilde 3} d^\dagger_{1} d_{1'}  \right] \ .
\eeq
Contraction of $\tilde 3$ with 3 and ignoring $(-1)^2$ for antiquarks, 
\beq
&& g \int[11'3] \ \tilde \delta_{1.1'3} \, f^t
\left[  \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} \ \chi^\dagger_1 T^3 \chi_{1'} 
\   b^\dagger_{1} b_{1'} 
 \right] 
\
g \int[22'] \ \tilde \delta_{23.2'} \, f^t
\left[  \bar u_{2} \gamma_\mu \varepsilon^\mu_{3} u_{2'} \ \chi^\dagger_2 T^{3} \chi_{2'} 
\ b^\dagger_{2} b_{2'}  
 \right] 
\np
g \int[22'3] \ \tilde \delta_{2.2'3} \, f^t
\left[  \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2} \ \chi^\dagger_{2'} T^3 \chi_{2} 
\  d^\dagger_{2} d_{2'} \right] 
\ g \int[11'] \ \tilde \delta_{13.1'} \, f^t
\left[  \bar v_{1'} \gamma_\mu \varepsilon^\mu_{3} v_{1} \ \chi^\dagger_{1'} T^{3} \chi_{1} 
\ d^\dagger_{1} d_{1'}  \right] \ .
\eeq
Contractions of 2 with 1', and 1 with 2', produce
\beq
&& g \int[11'3] \ \tilde \delta_{1.1'3} \, f^t
\left[  \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} \ \chi^\dagger_1 T^3 \chi_{1'} 
\   b^\dagger_{1} 
 \right] 
\
g \int[2'] \ \tilde \delta_{1'3.2'} \, f^t
\left[  \bar u_{1'} \gamma_\mu \varepsilon^\mu_{3} u_{2'} \ \chi^\dagger_{1'} T^{3} \chi_{2'} 
\ b_{2'}  
 \right]
\np
g \int[22'3] \ \tilde \delta_{2.2'3} \, f^t
\left[  \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2} \ \chi^\dagger_{2'} T^3 \chi_{2} 
\  d^\dagger_{2} \right] 
\ g \int[1'] \ \tilde \delta_{2'3.1'} \, f^t
\left[  \bar v_{1'} \gamma_\nu \varepsilon^\nu_{3} v_{2'} \ \chi^\dagger_{1'} T^{3} \chi_{2'} 
\ d_{1'}  \right] \ .
\eeq
Integrations over 1'3 and 2'3 yield, replacing the second $\mu$ by $\nu$
\beq
&& g^2 \sum_{31'} \int[12'] \int [x_3 k_{31'}] \ \tilde \delta_{1.2'}/p_1^+ \, f^t
\left[  \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} \ \chi^\dagger_1 T^3 \chi_{1'} 
\   b^\dagger_{1} \right]  \ f^t
\left[  \bar u_{1'} \gamma_\mu \varepsilon^\mu_{3} u_{2'} \ \chi^\dagger_{1'} T^{3} \chi_{2'} 
\ b_{2'}  
 \right] 
\np
g^2 \sum_{32'} \int[21'] \int [ x_3 k_{32'}] \ \tilde \delta_{2.1'} /p_2^+ \, f^t
\left[  \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2} \ \chi^\dagger_{2'} T^3 \chi_{2} 
\  d^\dagger_{2} \right] \  f^t
\left[  \bar v_{1'} \gamma_\nu \varepsilon^\nu_{3} v_{2'} \ \chi^\dagger_{1'} T^{3} \chi_{2'} 
\ d_{1'}  \right] \ .
\eeq
Sum over colors, integrations over 2' and 1', and rearrangement, produce
\beq
&& g^2 C_F \sum_{31'12'} \int[1] \int [x_3 k_{31'}] \ {1 \over p_1^{+2}} \, f^{2t}
\  \bar u_{1} \gamma_\mu \varepsilon^\mu_{3} u_{1'} 
\  \bar u_{1'} \gamma_\nu \varepsilon^\nu_{3} u_{2'} 
\ b^\dagger_{1} b_{1} 
\np
g^2 C_F \sum_{32'21'} \int[2] \int [ x_3 k_{32'}] \ {1 \over p_2^{+2} } \, f^{2t}
\  \bar v_{2'} \gamma_\mu \varepsilon^\mu_{3} v_{2}  
\  \bar v_{1'} \gamma_\nu \varepsilon^\nu_{3} v_{2'}  
\ d^\dagger_{2} d_{2} \ .
\eeq
Sum over intermediate spins yields
\beq
&& g^2 C_F \sum_{11'2'} \int[1] \int [x_3 k_{31'}] \ {1 \over p_1^{+2}} \, f^{2t}
\  j_{11'\mu} d^{\mu \nu} j_{1'2'\nu}  \ b^\dagger_{1} b_{1}
\np
g^2 C_F \sum_{22'1'} \int[2] \int [ x_3 k_{32'}] \ {1 \over p_2^{+2} } \, f^{2t}
\  \bar j_{1'2' \nu} d^{\mu \nu} \bar j_{2'2} \ d^\dagger_{2} d_{2} \ .
\eeq
With this result for the product $\cH_0^{(1)}  \cH_0^{(1)}$, the self-interaction operator is 
\beq
H_{\rm 2body \, \Sigma} \es \left[ - f_{LI} f_{IR} \, \Delta_{LIR} \, \cH_0^{(1)}  \cH_0^{(1)} \right]_\Sigma \\
\es
- g^2 C_F \sum_{11'2'} \int[1] \int [x_3 k_{31'}] \ {1 \over p_1^{+2}} \, f_1^{2t}
\  {j_{11'\mu} d^{\mu \nu} j_{1'2'\nu} \over p_1^- - p_{1'}^- - p_3^-} \ b^\dagger_{1} b_{1} 
\nm
g^2 C_F \sum_{22'1'} \int[2] \int [ x_3 k_{32'}] \ {1 \over p_2^{+2} } \, f_2^{2t}
\  {\bar j_{1'2' \nu} d^{\mu \nu} \bar j_{2'2} \over p_2^- - p_{2'}^- - p_3^-} \ d^\dagger_{2} d_{2} \ .
\eeq 
The matrix element of quark self-interactions in the eigenvalue equation is
\beq
\langle 12| H_{\rm 2body \, \Sigma}| P_\phi \rangle 
\es
\langle 12|
\left[
- g^2 C_F \sum_{11'2'} \int[1] \int [x_3 k_{31'}] \ {1 \over p_1^{+2}} \, f_1^{2t}
\  {j_{11'\mu} d^{\mu \nu} j_{1'2'\nu} \over p_1^- - p_{1'}^- - p_3^-} \ b^\dagger_{1} b_{1} 
\right.
\nm
\left.
g^2 C_F \sum_{22'1'} \int[2] \int [ x_3 k_{32'}] \ {1 \over p_2^{+2} } \, f_2^{2t}
\  {\bar j_{1'2' \nu} d^{\mu \nu} \bar j_{2'2} \over p_2^- - p_{2'}^- - p_3^-} \ d^\dagger_{2} d_{2} 
\right]
\nt
\int[\tilde 2' \tilde 1']
\ P_\phi^+ \tilde \delta_{P_\phi. \tilde 1' \tilde 2'} \ \phi_{\tilde 1' \tilde 2'}(x_{\tilde 1'},k_{\tilde 1'\tilde 2'}^\perp)
|\tilde 1' \tilde 2' \rangle \ .
\eeq
Evaluation of the matrix element proceeds by
\beq
\langle 12| H_{\rm 2body \, \Sigma}| P_\phi \rangle 
\es
\left[
-g^2 C_F \sum_{1'2'} \int [x_3 k_{31'}] \ {1 \over p_1^{+2}} \, f_1^{2t}
\  {j_{11'\mu} d^{\mu \nu} j_{1'2'\nu} \over p_1^- - p_{1'}^- - p_3^-} 
-
g^2 C_F \sum_{2'1'} \int [ x_3 k_{32'}] \ {1 \over p_2^{+2} } \, f_2^{2t}
\  {\bar j_{1'2' \nu} d^{\mu \nu} \bar j_{2'2} \over p_2^- - p_{2'}^- - p_3^-} 
\right]
\nt
\langle 12| \int[\tilde 2' \tilde 1']
\ P_\phi^+ \tilde \delta_{P_\phi. \tilde 1' \tilde 2'} \ \phi_{\tilde 1' \tilde 2'}(x_{\tilde 1'},k_{\tilde 1'\tilde 2'}^\perp)
|\tilde 1' \tilde 2' \rangle \\
\es
\left[
-g^2 C_F \sum_{1'2'} \int [x_3 k_{31'}] \ {1 \over p_1^{+2}} \, f_1^{2t}
\  {j_{11'\mu} d^{\mu \nu} j_{1'2'\nu} \over p_1^- - p_{1'}^- - p_3^-} 
-
g^2 C_F \sum_{2'1'} \int [ x_3 k_{32'}] \ {1 \over p_2^{+2} } \, f_2^{2t}
\  {\bar j_{1'2' \nu} d^{\mu \nu} \bar j_{2'2} \over p_2^- - p_{2'}^- - p_3^-} 
\right]
\nt
P_\phi^+ \tilde \delta_{P_\phi. 12} \ \phi_{1'2'}(x_1,k_{12}^\perp) \ .
\eeq
The self-interaction factors preserve spin. The quark contribution of  $m_1$ is multiplied by $\bar u_1 u_1 = 1$
and the antiquark contribution of $-m_2$ is multiplied by $\bar v_2 v_2 = -1$. Therefore, the self-interactions 
of the quark and antiquark differ only by the mass.
\beq
\langle 12| H_{\rm 2body \, \Sigma}| P_\phi \rangle 
\es
\left( {\Sigma_1 \over p_1^+} + {\Sigma_2 \over p_2^+} \right) \
P_\phi^+ \tilde \delta_{P_\phi. 12} \ \phi_{12}(x_1,k_{12}^\perp) \ ,
\eeq
where for the quark or antiquark of mass $m$ and momentum $p$ the self-iteraction contribution reads
\beq
\Sigma
\es
- g^2 C_F \int [x k] \ e^{-2t^2(p^- - p_q^- - p_g^-)^2}
\  {j_\mu d^{\mu \nu} j_\nu \over p^+(p^- - p_q^- - p_g^-)}  \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip.1in 
\centerline{\bf Summary of the 2-body exchange and self-interaction terms} 
\vskip.1in
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beq
\langle 12| H_{\rm 2body \, exch}| P_\phi \rangle        
\es
\int[2'1'] \, K_{\rm exch}(12,2'1')  \ P_\phi^+ \tilde \delta_{P_\phi.1'2'} \ \phi_{1' 2'}(x_{1'},k_{1'2'}^\perp)
\eeq
with the kernel of dimension ${1 \over + \perp^2}$,
\beq
\label{exchkernel}
K_{\rm exch}(12,2'1') \es g^2 C_{|2\rangle} \, f_{\underline 1 g, \overline 1}^t f_{\overline 2,\underline 2 g}^t \, 
\Delta_{12q2'1'}/p_g^+ \, j_\mu d^{\mu\nu} \bar j_\nu \, \tilde \delta_{12.1'2'} \ , \\
\Delta_{12q2'1'}/p_g^+ \es - \, \2 \left( {1 \over \rho_1} + {1 \over \rho_2 } \right) \ , \\
d^{\mu\nu} \es - g^{\mu \nu} + \eta^\mu \eta^\nu (\rho_1 + \rho_2)/ (2p_g^{+2}) \ .
\eeq
and
\beq
\langle 12| H_{\rm 2body \, \Sigma}| P_\phi \rangle 
\es
\left( {\Sigma_1 \over p_1^+} + {\Sigma_2 \over p_2^+} \right) \
P_\phi^+ \tilde \delta_{P_\phi. 12} \ \phi_{1'2'}(x_1,k_{12}^\perp) \ ,
\eeq
where for the quark of mass $m$ and momentum $p$ the self-iteraction contribution reads
\beq
\Sigma
\es
-g^2 C_F \int [x k] \ e^{-2t^2(p^- - p_q^- - p_g^-)^2}
\  {j_\mu d^{\mu \nu} j_\nu \over p^+(p^- - p_q^- - p_g^-)}  \ .
\eeq
The self-interaction $\Sigma$ is positive and kernel $K_{\rm exch}$ is negative. Both diverge in the limit $m_g \to 0$, 
see below. The question is if the divergences can cancel out in the eigenvalue equation.

\vskip.1in 
\centerline{\bf Using total-momentum conservation in the eigenvalue equation} 
\vskip.1in 

First I eliminate the total eigenstate momentum from everywhere except factors $f^t$ where it is stuck in ratio $t/P_\phi^+$.
I divide the matrix elements by the factor $P_\phi^+ \tilde \delta_{P_\phi. 12}$ so that the eigenvalue-side of the 2-body 
eigenvalue equation is $(P_\phi^{\perp 2} + M^2)/P^+_\phi$, and  
\beq
{ \langle 12| H_{\rm 2body \, exch}| P_\phi \rangle  \over  P_\phi^+ \tilde \delta_{P_\phi. 12}}   
\es
{1 \over P^+_\phi}  
\int[x_{1'}k_{1'2'}] \, K^{xk}_{\rm exch}(12,2'1')  \ \phi_{1' 2'}(x_{1'},k_{1'2'}^\perp) \ , \\
{ \langle 12| H_{\rm 2body \, \Sigma}| P_\phi \rangle \over  P_\phi^+ \tilde \delta_{P_\phi. 12}} 
\es
\left( {\Sigma_{m_1} \over p_1^+} + {\Sigma_{m_2} \over p_2^+} \right) \ \phi_{12}(x_1,k_{12}^\perp) \ ,
\eeq
where 
\beq
\label{exchkernelfinal1}
K^{xk}_{\rm exch}(12,2'1') \es - g^2 C_{|2\rangle} \
e^{-t^2(p_{\overline 1}^- - p_{\underline 1}^- - p_g^-)^2} 
e^{-t^2(p_{\overline 2}^- - p_{\underline 2}^- - p_g^-)^2} \
\left( {1 \over 2\rho_1} + {1 \over 2\rho_2 } \right) 
\nt
j_\mu  \left[ - g^{\mu \nu} + \eta^\mu \eta^\nu (\rho_1 + \rho_2)/ (2p_g^{+2}) \right] \bar j_\nu   \ , \\
\Sigma_m
\es
g^2 C_F \int [x k] \ e^{-2t^2(p^- - p_q^- - p_g^-)^2}
\  {x \over \rho} \ j_\mu \left[ - g^{\mu \nu} + \eta^\mu \eta^\nu \rho / p_g^{+2} \right] j_\nu  \ .
\eeq
I use identities
\beq
\label{rhoident}
p^- - p_q^- - p_g^- \es {- \rho \over p_g^+} \ , \ 
p_{\overline 1}^- - p_{\underline 1}^- - p_g^- \rs  {- \rho_1 \over p_g^+} \ , \
p_{\overline 2}^- - p_{\underline 2}^- - p_g^- \rs  {- \rho_2 \over p_g^+} \ ,
\eeq
to obtain
\beq
\label{exchkernelfinal2}
K^{xk}_{\rm exch}(12,2'1') \es - g^2 C_{|2\rangle} \
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{\rho_1 + \rho_2  \over 2\rho_1 \rho_2} \
j_\mu  \left[ - g^{\mu \nu} + \eta^\mu \eta^\nu (\rho_1 + \rho_2)/ (2p_g^{+2}) \right] \bar j_\nu   \ , \\
\Sigma_m
\es
g^2 C_F \int [x k] \ e^{-2t^2 \rho^2/p_g^{+2}}
\  {x \over \rho} \ j_\mu \left[ - g^{\mu \nu} + \eta^\mu \eta^\nu \rho / p_g^{+2} \right] j_\nu  \ .
\eeq
I introduce 
\beq
\Sigma_{m_i} \es \Sigma_{m_i}^{-g} + \Sigma_{m_i}^\eta \ , \\
K^{xk}_{\rm exch} \es K^{xk \, -g}_{\rm exch} + K^{xk \, \eta}_{\rm exch} \ ,
\eeq
where terms with superscript $-g$ are the ones containing $- g^{\mu \nu}$
and terms with the superscript $\eta$ are the ones containing $\eta^\mu \eta^\nu$.
\vskip.1in
\centerline{ \bf Analysis of the terms with  $-g^{\mu \nu}$ }
\vskip.1in
The terms with $- g^{\mu \nu}$ are 
\beq
\label{eqdiv-g}
\left( {\Sigma_{m_1}^{-g} \over p_1^+} + {\Sigma_{m_2}^{-g} \over p_2^+} \right) \ \phi_{12}(x_1,k_{12}^\perp)
+
{1 \over P^+_\phi}  
\int[x_{1'}k_{1'2'}] \, K^{xk \, -g}_{\rm exch}(12,2'1')  \ \phi_{1' 2'}(x_{1'},k_{1'2'}^\perp) 
 \ ,
\eeq
where
\beq
\label{sigma-g}
\Sigma^{-g}_m
\es
- g^2 C_F \int [x k] \ e^{-2t^2 \rho^2/p_g^{+2}}
\  {x \over \rho} \ j^\mu  j_\mu  \ , \\
\label{exchkernelmg}
K^{xk \, - g}_{\rm exch}(12,2'1') \es g^2 C_{|2\rangle} \
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ \rho_1 + \rho_2 \over 2 \rho_1 \rho_2 } \
j^\mu  \bar j_\mu \ .
\eeq
The products of quark currents are finite for quark states with finite invariant masses. The $\rho$s depend only on the 
quark momenta. Large values of $\rho$s are exponentially suppressed. The issue is what happens when $\rho$ get small,
or when $q^\perp$ or $z$ get small. I need to express $\rho$ and $p_g^+$ in terms of useful integration variables. 

\vskip.1in 
\centerline{\bf The case of $\Sigma^{-g}_m$} 
\vskip.1in

\beq
{m^2 - \cM_{qg}^2 \over p^+} \es p_g^+(p^- - p_q^- - p_g^-)/p_g^+ \rs {q^2 - m_g^2 \over p_g^+} \rs -\rho/p_g^+ \ , \\
\cM_{qg}^2 - m^2 \es \rho/x \ .
\eeq
\beq
x \es p_g^+ /p^+ \ , \ k^\perp \rs p_g^\perp - x p^\perp \ .
\eeq
In the quark-gluon rest frame,
\beq
p_g^+ \es E_g + k^z \ , \ p_q^+ \rs E_q - k^z \ , \ E = E_g + E_q  \ , \\
E_g \es \sqrt{ m_g^2 + k^{\perp 2} + k^{z2}} \ , \ E_q \rs \sqrt{ m^2 + k^{\perp 2} + k^{z2}} \ , \\
\int [ x k ] \es \int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x) }
\rs
\int  {d^3 k \over  2(2\pi)^3 } \, \left( {1\over E_g} + {1 \over E_q} \right) \ .
\eeq
The $g^{\mu \nu}$ terms contribute to the quark self-interaction
\beq
\label{sigma-g1}
\Sigma^{-g}_m
\es
- g^2 C_F 
\int  {d^3 k \over  2(2\pi)^3 } \, \left( {1\over E_g} + {1 \over E_q} \right) 
\ e^{-2(t/p^+)^2(\cM_{qg}^2 - m^2)^2 }
\ { j^\mu  j_\mu  \over \cM_{qg}^2 - m^2} \ .
\eeq
The fermion and antifermion factors are the same, 
\beq
j^\mu j_\mu \es
\bar u_p \gamma^\mu( / \hspace{-6pt} p_q + m ) \gamma_\mu u_p
\rs
\bar u_p (-2 / \hspace{-6pt} p_q + 4m ) u_p
\rs
- 4 p p_q + 8m^2 \\
\es
-2(2m^2-q^2)+ 8m^2 \rs 2 q^2 + 4m^2 \rs -2(m_g^2 - q^2)  + 2m_g^2 + 4m^2 \\
\es -2\rho  + 2m_g^2 + 4m^2  \rs  - 2x(E^2-m^2) + 2m_g^2 + 4m^2 \ .
\eeq
Therefore,
\beq
\label{sigma-g2}
\Sigma^{-g}_m
\es
- g^2 C_F 
\int  {d^3 k \over  2(2\pi)^3 } \, \left( {1\over E_g} + {1 \over E_q} \right) 
\ e^{-2(t/p^+)^2(E^2 - m^2)^2 }
\ { - 2x(E^2-m^2) + 2m_g^2 + 4m^2 \over E^2 - m^2} \ .
\eeq
The term with $- 2x(E^2-m^2)$ provides in the limit $m_g \to 0$ the angle dependent contribution,
\beq
-2x \es -2(E_g+ k^z)/E ~\to~ -2(k+k^z)/E  \rs -2k(1+ \cos \theta)/E \ ,
\eeq
implying
\beq
\label{sigma-g3}
\Sigma^{-g}_m
\es
- g^2 C_F 
\int  {d^3 k \over  2(2\pi)^3 } \, \left( {1\over E_g} + {1 \over E_q} \right) 
\ e^{-2(t/p^+)^2(E^2 - m^2)^2 }
\ \left[ -2k(1+ \cos \theta)/E + {2m_g^2 + 4m^2 \over E^2 - m^2} \right] \ .
\eeq
Integration over angle $\theta$ eliminates the term with $\cos \theta$. Therefore,
\beq
\label{sigma-g4}
\Sigma^{-g}_m
\es
{- g^2 C_F 4\pi \over   2(2\pi)^3} 
\int  {k^2 dk } \, \left( {1\over E_g} + {1 \over E_q} \right) 
\ e^{-2(t/p^+)^2(E^2 - m^2)^2 }
\ \left[ -2k/E + {2m_g^2 + 4m^2 \over E^2 - m^2} \right] \\
\es
{- g^2 C_F 4\pi \over   2(2\pi)^3} \2
\int  2k dE  \ e^{-2(t/p^+)^2(E^2 - m^2)^2 } \ \left[ -2k/E + {2m_g^2 + 4m^2 \over E^2 - m^2} \right] \ ,
\eeq
and, using the trick in Eq.~(\ref{trickE}),
\beq 
2k \es (E^2 - m^2)/E \ ,
\eeq
I get
\beq
\label{sigma-g5}
\Sigma^{-g}_m
\es
{- g^2 C_F 4\pi \over   2(2\pi)^3} \2
\int {E dE \over E^2} (E^2 - m^2) \ e^{-2(t/p^+)^2(E^2 - m^2)^2 } \ 
\left[ - {E^2 - m^2 \over E^2} + {2m_g^2 + 4m^2 \over E^2 - m^2} \right] \ ,
\eeq
or, using $u = E^2 -m^2$,
\beq
\label{sigma-g6}
\Sigma^{-g}_m
\es
{- g^2 C_F 4\pi \over   2(2\pi)^3} \4
\int_0^\infty {du \over u+m^2}  \ e^{-2(t/p^+)^2 u^2 } \ 
\left[ - {u^2 \over u+m^2} + 2m_g^2 + 4m^2  \right] \ ,
\eeq
which is finite when $m_g \to 0$.

\vskip.1in
\centerline{\bf The case of  ${1 \over P^+_\phi}  \int[x_{1'}k_{1'2'}] \, K^{xk \, -g}_{\rm exch}(12,2'1')  \ \phi_{1' 2'}(x_{1'},k_{1'2'}^\perp)$}
\vskip.1in

The kernel of Eq.~(\ref{exchkernelmg}) involves spin dependence, which contributes to the result of integration
with an unknown eigenfunction. But we can assume that the wave function falls off exponentially for extreme 
quark momenta, because the RGPEP factors force it to have such behavior. The integral to inspect is
\beq
\label{exchkernelmg1}
{1 \over P^+_\phi}  I_{\rm exch}
\es
 {1 \over P^+_\phi}  \int[x_{1'}k_{1'2'}] K^{xk \, - g}_{\rm exch}(12,2'1') \ \phi_{1' 2'}(x_{1'},k_{1'2'}^\perp) \\
\es {1 \over P^+_\phi}  \int[x_{1'}k_{1'2'}] \
g^2 C_{|2\rangle} \ e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
\2 \left( { 1 \over \rho_1 } + { 1 \over \rho_2 } \right) \
j^\mu_{11'}   \phi_{1' 2'}(x_{1'},k_{1'2'}^\perp)  \bar j_{2'2 \mu} \ .
\eeq
The gluon mass $m_g$ appears explicitly in $\rho_1$ and $\rho_2$ only,
\beq
\label{rho1general}
\rho_1 \es m_g^2 - q_1^2 \rs m_g^2 + 
{1 \over x_{1'} x_1 }\left[  \left( x_1 {k'}^\perp - x_{1'} k^\perp\right)^2 + m_1^2 \left( x_{1'}-x_{1} \right)^2 \right] \ , \\
\label{rho2general}
\rho_2 \es m_g^2 - q_2^2 \rs m_g^2 + 
{1 \over x_{2'} x_2 }\left[  \left( x_{2'} k^\perp - x_2 {k'}^\perp \right)^2 + m_2^2 \left( x_2 - x_{2'} \right)^2 \right]
\eeq
The question is if the integral is finite for $m_g=0$. There is no issue for large $\rho$s, given the exponential damping.
The issue is what happens for small $\rho$s appearing in the denominator. For $m_g \to 0$,
using 
\beq
x_{1'} \es x_1+ z \ , \ k_{1'2'}^\perp = k_{12}^\perp+ q^\perp \ ,
\eeq
I get 
\beq
\label{rho1ktext}
\rho_1 \es {m_1^2 (z/x_1)^2 + ( q^\perp - k_{12}^{\perp} z/x_1 )^2 \over 1 + z / x_1 } \ , \\
\label{rho2ktext}
\rho_2 \es {m_2^2 (z/x_2)^2 + ( q^\perp + k_{12}^{\perp} z/x_2 )^2 \over 1 - z / x_2 } \ .
\eeq
In the integration region where $\rho_i$ is small, $z/x_i$ is small and $q^\perp$ is close to the small vectors 
$\pm z k_{12}^\perp/x_i$. I neglect $z/x_1$ in denominator. For small $z$, $q^\perp$ is short, since it is close 
to a short vector. The integral is
\beq
\label{exchkernelmg2}
I_{\rm exch}
\es
{ g^2 C_{|2\rangle}  \over 4(2\pi)^3} 
\int_{-x_1}^{x_2} { dz  \over  (x_1+z)(x_2-z)} \int d^2q^\perp \
e^{-(t/P^+_\phi)^2(\rho_1^2 + \rho_2^2)/z^2} \
\left( { 1 \over \rho_1 } + { 1 \over \rho_2 } \right) 
\nt
j^\mu_{11'}   \phi_{1' 2'}(x_{1}+z,k_{12}^\perp + q^\perp)  \bar j_{2'2 \mu} \ .
\eeq
In the small-$\rho$ regions, the factor $j^\mu_{11'}   \phi_{1' 2'}(x_{1}+z,k_{12}^\perp + q^\perp)  \bar j_{2'2 \mu}$ 
is a finite $4 \times 4$ table of spin amplitudes. Consider small $\rho_1$. Substitute 
\beq
u^3 \es z \, m_1/x_1 \ , \ u^1 \rs q^1 - z k^1_{12} /x_1 \ , \ u^2 \rs q^2 - z k^2_{12} /x_1 \ , \\
z \es u^3 \, x_1/m_1 \ , \ q^1 \rs u^1 + z  k^1_{12} /x_1 \ , \ q^2 \rs u^2 + z k^2_{12} /x_1 \ , \\
dz \, d^2q \es {x_1 \over m1} \, d^3 u \ , \ \rho_1 \sim \vec u\,^2 \ .
\eeq
Thus, for small $\rho_1$, the essential form of the integral is  
\beq
\label{d^3u}
\int_{-x_1}^{x_2} dz  \int d^2q^\perp \ { 1 \over \rho_1 }
& \sim & \int {d^3 u \over \vec u\,^2} \ .
\eeq
Such integral is finite in the limit $m_g \to 0$. Its limiting value does not depend on $m_g$. The only terms
that are sensitive to $m_g \to 0$ are the terms with $\eta^\mu \eta^\nu$.

{\blue \bf There is a strong argument for $q_i^2$ to never approach 0 if the momenta of fermions are limited
and $q_i$ itself is not 0. $q = p' - p$ is a space-like four-vector that can be looked at in any frame. Let
\beq
p \es \left[ {p^{\perp 2} +m^2 \over p^+}, p^+, p^\perp \right] \ , \ p' \rs \left[ m, m,0^\perp\right] \ .
\eeq
Then,
\beq
q^2 \es (p-p')^2 \rs  ( p^+-m) \left(  {p^{\perp 2} +m^2 \over p^+} - m \right) - p^{\perp 2} 
\rs    
p^+{p^{\perp 2} +m^2 \over p^+}  - m {p^{\perp 2} +m^2 \over p^+} - p^+m +m^2 - p^{\perp 2} \\
\es 
m^2  - m {p^{\perp 2} +m^2 \over p^+} - p^+m +m^2 
\rs
m \left[  2m  - {p^{\perp 2} +m^2 \over p^+} - p^+ \right]
\rs 
m(2m - 2E) \rs - 2m(E-m) \\
\es
-2m(E^2 - m^2)/E+m) \rs - 2m \, \vec q\,^2 / (E+m) < - {m \vec q\,^2 \over  \sqrt{ m^2 + \vec q\,^2} } 
\ .
\eeq
Hence $q^2$ approaches 0 from below only when $\vec q$ approaches 0. For $|\vec q \, | \ll m$,
\beq
q^2 < - \vec q^2
\eeq
for a particle kicked from rest. For $|\vec q \, | \gg m$,
\beq
q^2 < - m |\vec q \, | \ .
\eeq

}


\vskip.1in
\centerline{ \bf Analysis of the terms with  $\eta^\mu \eta^\nu$ }
\vskip.1in
The terms with $\eta^\mu \eta^\nu$ are
\beq
\label{eqdiv1}
\left( {\Sigma_{m_1}^\eta \over p_1^+} + {\Sigma_{m_2}^\eta  \over p_2^+} \right) \ \phi_{12}(x_1,k_{12}^\perp)
+
{1 \over P^+_\phi}  
\int[x_{1'}k_{1'2'}] \, K^{xk \, \eta}_{\rm exch}(12,2'1')  \ \phi_{1' 2'}(x_{1'},k_{1'2'}^\perp) 
 \ ,
\eeq
where
\beq
\label{sigmamg}
\Sigma^\eta_m
\es
g^2 C_F \int [x k] \ e^{-2t^2 \rho^2/p_g^{+2}}
\  {x \rho \over \rho p_g^{+2}} \ j^+  j^+  \ , \\
\label{exchkernelmgeta}
K^{xk \, \eta}_{\rm exch}(12,2'1') \es - g^2 C_{|2\rangle} \
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2 p_g^{+2}} \
j^+  \bar j^+ \ .
\eeq
The currents $j^+$ and $\bar j^+$ conserve spin and provide $p^+$-dependent factors,
$j_1^+ j_2^+ = 2\sqrt{ p_{1}^+ p_{1'}^+}  2 \sqrt{p_{2}^+ p_{2'}^+} $.
Therefore,
\beq
\label{sigmamg1}
\Sigma^\eta_m
\es
g^2 C_F \int [x k] \ e^{-2t^2 \rho^2/p_g^{+2}}
\  {x \rho \over \rho p_g^{+2}} \ 
4 p_{\overline m}^+ p_{\underline m}^+ 
\ , \\
\label{exchkernelmg1}
K^{xk \, \eta}_{\rm exch}(12,2'1') \es - g^2 C_{|2\rangle} \
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2 p_g^{+2}} \
4 \sqrt{ p_{1}^+ p_{1'}^+ p_{2}^+ p_{2'}^+} \ .
\eeq
Simplifying,
\beq
\label{sigmamg2}
\Sigma^\eta_m
\es
g^2 C_F \int [x k] \ e^{-2t^2 \rho^2/p_g^{+2}}
\ {4 (1-x) \over x}
\ , \\
\label{exchkernelmg2}
K_{122'1'} \es K^{xk \, \eta}_{\rm exch}(12,2'1') \rs
- g^2 C_{|2\rangle} \
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2 x_g^{+2}} \
4 \sqrt{ x_{1} x_{1'} x_{2} x_{2'}} \ , 
\eeq
with the potentially diverging terms, in the limit $m_g \to 0$,
\beq
\label{eqdiv2}
&&
\left( {\Sigma_{m_1}^\eta \over p_1^+} + {\Sigma_{m_2}^\eta \over p_2^+} \right) \ \phi(x_1,k_{12}^\perp)
+
{1 \over P^+_\phi}  
\int[x_{1'}k_{1'2'}] \, K_{122'1'} \ \phi(x_{1'},k_{1'2'}^\perp) \ ,
\eeq
where spin labels are dropped because they are conserved by the singular terms. The eigenvalue-equation terms 
are of the form
\beq
\label{cancellation}
\Sigma^\eta \phi + {1 \over P^+_\phi} \int K^\eta \phi' 
\es
\left( \Sigma^\eta + {1 \over P^+_\phi} \int K^\eta \right) \phi +  {1 \over P^+_\phi} \int K^\eta \left( \phi' - \phi \right) \ . 
\eeq

\vskip.1in
\centerline{\bf The case of $\int K^\eta \left( \phi' - \phi \right)$}
\vskip.1in

The first question is if the integral $I = \int K^\eta \left( \phi' - \phi \right)$ depends on $m_g$ when $m_g \to 0$.
For fixed $x_1$ and $k_{12}^\perp$, I change variables according to
\beq
x_{1'} \es x_1+ z \ , \ k_{1'2'}^\perp = k_{12}^\perp+ q^\perp \ ,
\eeq
and obtain
\beq
\label{ExchPsi2POM}
I \es  
{- g^2 C_{|2\rangle}  \over 2 (2\pi)^3}  \int_{-x_1}^{x_2} dz \int d^2q^\perp 
 \
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2 \, z^2 } \
{ 4 \sqrt{ x_{1} x_{1'} x_{2} x_{2'}} \over x_{1'} x_{2'} } \ (\phi'-\phi) \ .
\eeq
Using $p_1 = x_1 P +k$, $p_{1'} = x_{1'} P + k'$ etc., and introducing 
\beq
\label{qzktext}
q^\perp \es \sqrt{|z|} \, k^\perp \ ,
\eeq
according to Eq.~(\ref{qzk}), I get, following Eqs.~(\ref{rho1k}) and (\ref{rho2k}),
\beq
\label{rho1ktext}
\rho_1 \es m_g^2 + {m_1^2 (z/x_1)^2 + 
( \sqrt{|z|} k^\perp - k_{12}^{\perp} z/x_1 )^2 \over 1 + z / x_1 } \ , \\
\label{rho2ktext}
\rho_2 \es m_g^2 + {m_2^2 (z/x_2)^2 + 
( \sqrt{|z|} k^\perp + k_{12}^{\perp} z/x_2 )^2 \over 1 - z / x_2 } \ .
\eeq
Then,
\beq
\rho_1  \es m_g^2 + |z| { (m_1/x_1)^2 |z| + ( k^\perp  - s_z \sqrt{|z|} \, k_{12}^{\perp}/x_1 )^2 \over 1 + z/x_1 } \ , \\
\rho_2  \es m_g^2 + |z| { (m_2/x_2)^2 |z| + ( k^\perp + s_z \sqrt{|z|} \, k_{12}^{\perp}/x_2 )^2 \over 1  - z/x_2) } \ .
\eeq
It is visible that $\rho_1$ and $\rho_2$ are non-negative,
they can be 0 only when $m_g=0$ and $z=0$, and they 
both approach $m_g^2$ plus terms that tend to 0 
as $|z| k^{\perp 2}$ when $z \to 0$ and $|k^\perp| > 0$
is kept fixed.  Consequently,
\beq
\rho_1/|z| \es {m_g^2 \over |z|} + {m_1^2 |z|/x_1^2 + (k^\perp  - k_{12}^{\perp} s_z \sqrt{|z|} /x_1 )^2 \over 1 + z / x_1 } \ , \\
\rho_2/|z| \es {m_g^2 \over |z|} + {m_2^2 |z|/x_2^2 + (k^\perp + k_{12}^{\perp} s_z \sqrt{|z|} /x_2 )^2 \over 1  - z / x_2 } \ .
\eeq
It is also visible that $\rho_1/|z|$ and $\rho_2/|z|$ behave for fixed $z$ and large $k^\perp$ as $k^{\perp 2}/(1+z/x_1)$ 
and $k^{\perp 2}/(1-z/x_2)$, respectively. The range of integration over $k^{\perp 2}$ is limited from above by a finite 
value when $s$ is finite and $x_1$ and $x_{1'}$ are limited, which is the case in the Hamiltonian matrix elements between 
effective 2-body states of finite invariant mass. {\red {\bf Is the inspection and conclusions below correct?} I inspect the integral in Eq.~(\ref{ExchPsi2POM}),
\beq
\label{ExchPsi2POM1}
I \es  
{- g^2 C_{|2\rangle}  \over 2 (2\pi)^3}  \int_{-x_1}^{x_2} dz \int |z| \, d^2k^\perp 
 \
e^{-(t/P^+_\phi)^2(\rho_1^2 + \rho_2^2)/z^2} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2 \, z^2 } \
{ 4 \sqrt{ x_{1} x_{1'} x_{2} x_{2'}} \over x_{1'} x_{2'} } \ (\phi'-\phi) \ .
\eeq
The singularity for small $z$ is changed from $1/z^2$ to $1/|z|$. But the difference $\phi'-\phi$ is expandable in the
Taylor series and the leading terms are
\beq
\phi(x_{1'},k_{1'2'}^\perp) - \phi(x_{1},k_{12}^\perp) 
\es
\phi(x_{1}+ z, k_{12}^\perp + q^\perp) - \phi(x_{1},k_{12}^\perp)  
\rs
z \phi_z + q^\perp \phi_\perp + O(z^2, z q^\perp, q^{\perp 2})  \\
\es
z \phi_z + \sqrt{|z|} k^\perp \phi_\perp + O(z^2, |z|^{3/2}k^\perp, |z| k^{\perp 2})  
\rs O(\sqrt{|z|}) \ ,
\eeq
which means that the integrand behaves as $1/\sqrt{|z|}$. The integral is finite for $m_g \to 0$.
Therefore, in Eq.~\ref{cancellation}), repeated here, 
\beq
\label{cancellation1}
\Sigma^\eta \phi + {1 \over P^+_\phi} \int K^\eta \phi' 
\es
\left( \Sigma^\eta + {1 \over P^+_\phi} \int K^\eta \right) \phi +  {1 \over P^+_\phi} \int K^\eta \left( \phi' - \phi \right) \ , 
\eeq
the second term on the right-hand side is finite and independent of $m_g$ in the limit $m_g \to 0$.
\bf The question of what to do with $1/(\rho_1 \rho_2)$ that leads to the 4th power of momentum transfer
in denominator requires inclusion of the RGPEP form factors. 20250906 20:24 sob San Dimas: the reasoning 
and conclusion are correct. }

\vskip.1in
\centerline{ \bf Subtraction of wave function with  the full kernel}
\vskip.1in

The purpose of the subtraction is to reason in terms of the potential, which applies only in the non-relativistic systems.
In Eq.~(\ref{exchkernelfinal2}), repeated below,
\beq
\label{exchkernelfinal3}
K^{xk}_{\rm exch}(12,2'1') \es - g^2 C_{|2\rangle} \
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{\rho_1 + \rho_2  \over 2\rho_1 \rho_2} \
j_\mu  \left[ - g^{\mu \nu} + \eta^\mu \eta^\nu (\rho_1 + \rho_2)/ (2p_g^{+2}) \right] \bar j_\nu  \ ,
\eeq
the kernel can be written as 
\beq
\label{exchkernelfinal4}
K^{xk}_{\rm exch}(12,2'1') \es - g^2 C_{|2\rangle} \
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{(\rho_1 + \rho_2)^2  \over 4\rho_1 \rho_2} \
j_\mu  \left[ {- 2 g^{\mu \nu} \over \rho_1 + \rho_2} + {\eta^\mu \eta^\nu \over p_g^{+2}} \right] \bar j_\nu  \ .
\eeq
In the non-relativistic limit, valid because the RGPEP form factors limit the quark relative momenta in the
bound-state rest frame, the current factors $j$ reduce to $2m_1$ and $2m_2$, preserving spin projections
on the $z$-axis,
\beq
\rho_1 \es m_g^2 + z^2 m_1^2/x_1^2 + q^{\perp 2} \ , \
\rho_2 \rs m_g^2 + z^2 m_2^2/x_2^2 + q^{\perp 2} \ , \
p_g^+ \rs z ( m_1+m_2 ) \ , \\
x_i \es {m_i \over m_1 + m_2} \ , \
\rho_1 \rs \rho_2 \rs m_g^2 + z^2 (m_1 + m_2)^2 + q^{\perp 2} \rs \rho \rs m_g^2 + \vec q\,^2 \ , \\
q_z \es z(m_1 + m_2) \ ,
\eeq
and the kernel becomes
\beq
\label{exchkernelfinal5}
K^{xk}_{\rm exch}(12,2'1') \es - g^2 C_{|2\rangle} \
e^{-2(t \vec q\,^2 /q_z)^2} \ 4 m_1 m_2  \ 
\left[ {- 1 \over \vec q\,^2} + {1 \over q_z^2} \right]  \ .
\eeq
The right-hand side of the eigenvalue equation is $M^2/P^+_\phi$. The exchange interaction term can thus be written 
in the non-relativistic limit as
\beq
{1 \over P^+_\phi} \int K \left( \phi' - \phi \right) 
\es
{1 \over P^+_\phi}  
\int[x_{1'}k_{1'2'}] \, K^{xk \, m_g}_{\rm exch}(12,2'1')  \ \left[ \phi_{1' 2'}(\vec k + \vec q\,) - \phi_{1' 2'}(\vec k\,)  \right] \\
\es
- {g^2 C_{|2\rangle} \over P^+_\phi}  
\int { d^3 q \over 2(2\pi)^3}\left( {1 \over m_1} + {1 \over m_2}\right)
e^{-2(t \vec q\,^2 /q_z)^2} \ 4 m_1 m_2  
\nt 
\left[  {1 \over q_z^2} - {1 \over \vec q\,^2} \right]
\
\left[ \2 q^i q^j \partial_i \partial_j \phi_{12}(\vec k\,) + O(q^4 \partial_k^4 \phi)\right] \ ,
\eeq
where $\partial$ denotes the gradient in $\vec k$. This result matches Eqs.~(67) in \cite{Serafin:2023pkf} and
(95) in \cite{Glazek:2003ky}.

\vskip.1in
\centerline{\bf The case with $\left( \Sigma^\eta + {1 \over P^+_\phi} \int K^\eta \right) \phi$ }
\vskip.1in

The sum of integrals that multiply the wave function $\phi_{12}(x_1, k_{12}^\perp)$ is
\beq
\label{Seta1}
S^\eta \es
{g^2 C_F \over p_2^+} \int [x k] \ e^{-2t^2 \rho_2^2/p_g^{+2}}
\ {4 (1-x) \over x}
+
{g^2 C_F \over p_1^+} \int [x k] \ e^{-2t^2 \rho_1^2/p_g^{+2}}
\ {4 (1-x) \over x}
\nm
{g^2 C_{|2\rangle} \over P_\phi^+} 
\int [x_{1'}k_{1'2'}]
e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2 x_g^{+2}} \
4 \sqrt{ x_{1} x_{1'} x_{2} x_{2'}} \ .
\eeq
The range of integration variables is illustrated in Fig.~\ref{fig:selfexch}. (20250824 09:10 nie Bukowina) 
I take advantage of Sec.~\ref{comparisonsigmaexch}, beginning with Eq.~(\ref{SigmaPsi1pom1}). The 
antiquark, 2, self-interaction term and the corresponding exchange term in the considered sum, using 
$z = x_{1'}- x_1$, are $g^2 (C_F {\rm ~or~} C_{|2\rangle})/[2(2\pi)^3]$ times
\beq
\label{Sigma2K2 1a}
\Sigma_{m_2}^\eta 
\es
 {1 \over p_2^+} \int_0^1 {dx \, d^2k^\perp \over x(1-x)} \ 
\ e^{-2t^2 \rho_2^2/p_g^{+2}} \ {4 (1-x) \over x} \ , \\
\label{Sigma2K2 1b}
K_2^\eta \es  { -1 \over P^+ }
\int_{x_1}^1{ dx_{1'} d^2k_{1'2'}^\perp \over x_{1'} x_{2'} } \ e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2 z^{+2}} \ 4\sqrt{ x_{1}x_{1'} x_2 x_{2'} }  \ .
\eeq
In the integral $K_2^\eta$, one can equally well integrate over $x_{2'}= 1- x_{1'}$, with $dx_{1'} = - dx_{2'}$, and 
\beq
\int_{x_1}^1{ dx_{1'} d^2k_{1'2'}^\perp \over x_{1'} x_{2'} }
\es
\int_0^{x_2 }{ dx_{2'} d^2k_{1'2'}^\perp \over x_{1'} x_{2'} } \ .
\eeq
The momenta in $K_2^\eta$ are
\beq
x_g \es x_2-x_{2'} \rs x x_2 \ , \ p_g^\perp \rs x(-k_{12}^\perp) + k^\perp \ , \\
x_q \es x_{2'} \rs (1-x) x_2 \ , \ p_q^\perp \rs p_{2'}^\perp \rs - k_{1'2'}^\perp \rs (1-x)(-k_{12}^\perp) - k^\perp
\eeq
The Jacobian is
\beq
{\partial (x_{2'}, k^1_{1'2'}, k^2_{1'2'} \over \partial (x, k^1, k^2)} = -x_2 \ .
\eeq
Therefore,
\beq
\int_{x_1}^1{ dx_{1'} d^2k_{1'2'}^\perp \over x_{1'} x_{2'} }
\es
\int_0^{x_2 }{ dx_{2'} d^2k_{1'2'}^\perp \over x_{1'} x_{2'} } 
\rs x_2 \int_0^1{ dx \, d^2k^\perp \over (x_{1} +x x_2) (1-x)x_{2} }
\rs  \int_0^1{ dx \, d^2k^\perp \over (x_{1} +x x_2) (1-x) } \ .
\eeq
The integrals to compare are
\beq
\label{Sigma2K2 2a}
\Sigma_{m_2}^\eta 
\es
 {1 \over p_2^+} \int_0^1 {dx \, d^2k^\perp \over x(1-x)} \ 
\ e^{-2t^2 \rho_2^2/p_g^{+2}} \ {4 (1-x) \over x} \ , \\
\label{Sigma2K2 2b}
K_2^\eta \es  { -1 \over P^+ } 
\int_0^1{ dx \, d^2k^\perp \over (x_{1} +x x_2) (1-x) }
 \ e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2 x^2 x_2^2 } \ 4\sqrt{ x_{1}(x_{1} +x x_2) x_2 (1-x)x_{2} }  \ .
\eeq
Simplifying,
\beq
\label{Sigma2K2 3a}
\Sigma_{m_2}^\eta 
\es
 {1 \over p_2^+} \int_0^1 dx \, d^2k^\perp \ 
\ e^{-2t^2 \rho_2^2/p_g^{+2}} \ {4 \over x^2} \ , \\
\label{Sigma2K2 3b}
K_2^\eta \es  { -1 \over P^+ x_2} 
\int_0^1{ dx \, d^2k^\perp \, \sqrt{ x_{1} }\over \sqrt{ (x_{1} +x x_2) (1-x)} }
 \ e^{-t^2(\rho_1^2 + \rho_2^2)/p_g^{+2}} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2  }  \ { 4  \over x^2}\ .
\eeq
In both integrals,
\beq
p_g^+ \es x p_2^+ \ .
\eeq
So,
\beq
\label{Sigma2K2 4a}
\Sigma_{m_2}^\eta 
\es
 {1 \over p_2^+} \int_0^1 dx \, d^2k^\perp \ 
\ e^{-2(t/p_2^+)^2 \rho_2^2/x^2} \ {4 \over x^2} \ , \\
\label{Sigma2K2 4b}
K_2^\eta \es  { -1 \over p_2^+} 
\int_0^1{ dx \, d^2k^\perp \, \sqrt{ x_{1} }\over \sqrt{ (x_{1} +x x_2) (1-x)} }
 \ e^{-(t/p_2^+)^2(\rho_1^2 + \rho_2^2)/x^2} \
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2  }  \ { 4  \over x^2}\ .
\eeq
I have to work out $\rho$s in both integrals. In $\Sigma$, I have
\beq
\rho_2 \es m_g^2 - (p_2 - p_{2'})^2 \rs m_g^2 - p_g^+(p_2^- - p_{2'}^-) + p_g^{\perp 2} 
\rs m_g^2 + {m_2^2 x^2 + k^{\perp 2} \over 1-x} \ .
\eeq
In $K$, I have 
\beq
\rho_2 \es m_g^2 - (p_2 - p_{2'})^2 \rs m_g^2 + {m_2^2 x^2 + k^{\perp 2} \over 1-x} \ .
\eeq
\beq
\rho_1 \es m_g^2 - (p_{1'} - p_{1})^2 \rs m_g^2 - p_g^+(p_{1'}^- - p_{1}^-) + p_g^{\perp 2} \\
\es m_g^2 - x [p_2^+ = (x_2/x_1) p_1^+] 
\left[ { p_{1'}^{\perp 2} + m_1^2 \over p_{1'}^+}  - { p_1^{\perp 2} +m_1^2 \over p_1^+}   \right]   + p_g^{\perp 2} \\
\es
m_g^2 - (x x_2/x_1) \left[ { k_{1'2'}^{\perp 2} + m_1^2 \over (x_{1'}/x_1)}  - k_{12}^{\perp 2} - m_1^2   \right]   
+ (x k_{12}^\perp - k^\perp )^2 \\
\es
m_g^2 - (x x_2/x_1) \left[ { ( (1-x) k_{12}^\perp + k^\perp)^2 + m_1^2 \over (x_{1'}/x_1)}  - k_{12}^{\perp 2} - m_1^2   \right]   
+ (x k_{12}^\perp - k^\perp )^2 \ .
\eeq
$m_1^2$:
\beq
- (x x_2/x_1) \left[ { 1 \over (x_{1'}/x_1)}  - 1   \right]   \es
- (x x_2/x_1) \left( { x_1 \over x_{1'} }  - 1 \right)  \rs (x x_2/x_1)z/x_{1'} \rs (x x_2/x_1) x x_2 /(x_1+xx_2) \\
\es { x^2 x_2^2 \over x_1 (x_1+xx_2) } \ .
\eeq
$k^{\perp 2}$:
\beq
- (x x_2/x_1) \left[ { x_1 \over x_{1'}}    \right]   + 1 \rs  1 - x x_2 { 1 \over x_1+xx_2} \rs { x_1 \over x_1+xx_2} \ .
\eeq
$k_{12}^{\perp 2}$:
\beq
- (x x_2/x_1) \left[ { (1-x)^2 \over (x_{1'}/x_1)} - 1 \right] + x^2 
\es  xx_2 \left[ 1/x_1 - (1-x)^2/x_{1'} \right] + x^2 
\rs  xx_2 \left[ 1/x_1 - (1-x)^2 / (x_1 + x x_2) \right] + x^2 \\
\es  xx_2 { x_1 + x x_2 - x_1(1-x)^2 \over x_1 (x_1 + x x_2) } + x^2
\rs  x^2 \left[ x_2 { x_2 + 2 x_1 - x_1 x \over x_1 (x_1 + x x_2) } + 1 \right] \\
\es x^2 { x_2^2 + 2 x_1x_2 - x_1 x_2 x + x_1^2 + x x_1 x_2 \over x_1 (x_1 + x x_2) } 
\rs { x^2 \over x_1 (x_1 + x x_2) } \ .
\eeq
$k^\perp k_{12}^\perp$:
\beq
- (x x_2/x_1) \left[ { 2 (1-x) \over (x_{1'}/x_1)} \right] -2 x 
\es 
-2 x \left[ x_2 (1-x) / x_{1'} + 1 \right]
\rs
-{ 2 x \over x_1 + x x_2}  \left[ x_2 (1-x) + x_1 + x x_2 \right] \\
\es -{ 2 x \over x_1 + x x_2}  \ .
\eeq
The result for $\rho_1$ is
\beq
\rho_1 \es m_g^2 +
m_1^2 { x^2 x_2^2 \over x_1 (x_1+xx_2) }
+
k^{\perp 2} { x_1 \over x_1+xx_2} 
+
k_{12}^{\perp 2} { x^2 \over x_1 (x_1 + x x_2) } 
-
2 k^\perp k_{12}^\perp { x \over x_1 + x x_2} \ .
\eeq
or
\beq 
x_1 (x_1+xx_2) (\rho_1 - m_g^2)
\es
m_1^2  x^2 x_2^2 
+
k^{\perp 2} x_1^2 
+
k_{12}^{\perp 2}  x^2 
-
2 k^\perp k_{12}^\perp  x x_1 \ .
\eeq
It looks right because of the simplicity.
\beq 
 \rho_1 
\es m_g^2 +
{  m_1^2  x^2 x_2^2 
+
(x_1 k^\perp  - x k_{12}^\perp )^2 
\over 
x_1 (x_1+xx_2) } \ .
\eeq
{\red I am sure this is a correct expression. 20250825 00:23 pon Bukowina $\to$ 20:36 pon Ekologiczna}
Now the integrals, $g^2 (C_F {\rm ~or~} C_{|2\rangle})/[2(2\pi)^3] \times $
\beq
\label{Sigma2K2 5a}
\Sigma_{m_2}^\eta 
\es
 {1 \over p_2^+} \int_0^1 dx \, d^2k^\perp \ 
\ e^{-2(t/p_2^+)^2 \rho_2^2/x^2} \ {4 \over x^2} \ , \\
\label{Sigma2K2 5b}
K_2^\eta \es  { -1 \over p_2^+} 
\int_0^1{ dx \, d^2k^\perp \, e^{-(t/p_2^+)^2(\rho_1^2 + \rho_2^2)/x^2}
 \ {  \sqrt{ x_{1} }  \over \sqrt{ (x_{1} +x x_2) (1-x)} } }\
{ ( \rho_1 + \rho_2)^2  \over 4 \rho_1 \rho_2  }  \ { 4  \over x^2}\ .
\eeq
\beq
\rho_2 \es m_g^2 + {m_2^2 x^2 + k^{\perp 2} \over 1-x} \ , \\
\rho_1 \es m_g^2 + {  m_1^2  x^2 x_2^2 + (x_1 k^\perp  - x k_{12}^\perp )^2 \over x_1 (x_1+xx_2) } \ .
\eeq
The question is how the sum, 
\beq
\label{Seta2}
S_2^\eta \es \Sigma_{m_2}^\eta  + K_2^\eta  \ ,
\eeq
behaves in the limit $m_g \to 0$. 

{\red \bf 

**********

I may have an error in my logic in Eq.~(\ref{seagull factor 1}) etc.
\beq
\label{seagull factor 1 error}
\Gamma_{\rm sg} \es { - \delta_{\sigma_1 , \sigma_{1'}} \delta_{\sigma_{2'} , \sigma_2} \over q^{+2} }  
4 \sqrt{p_1^+ p_{1'}^+ p_{2'}^+ p_2^+}  \ , \\
\label{exch factor 1 error}
\Gamma_{\rm exch} \es { + \delta_{\sigma_1 , \sigma_{1'}} \delta_{\sigma_{2'} , \sigma_2}  \over q^{+2} } 
4 \sqrt{p_1^+ p_{1'}^+ p_{2'}^+ p_2^+} \, {(\rho_1 + \rho_2)^2 \over 4 \rho_1 \rho_2} .
\eeq
Both terms involve the gluon spin singular factor $1/x^2$. One can subtract and add 1,
\beq
{(\rho_1 + \rho_2)^2 \over 4 \rho_1 \rho_2} - 1 + 1 \es {(\rho_1-\rho_2)^2 \over 4 \rho_1 \rho_2} + 1.
\eeq
The term $+1$ on the right-hand side leads to the same singularity that appears in the seagull. The difference between 
$\rho_1$ and $\rho_2$ is
\beq
\label{rho1-rho2 error}
\rho_1 - \rho_2 \es 2y (\cM_{12}^2 - \cM_{1'2'}^2 ) \sim y\ .
\eeq
In the limit $y \to 0$ I have
\beq
\rho_1 \es m_g^2 + {4y^2 m_1^2 + 4 (x l^{\perp} - y k^\perp )^2 \over x^2-y^2 }
\to
m_g^2 + 4 l^{\perp 2} - 8 y l^\perp k^\perp/x  
 \ , \\
\rho_2 \es m_g^2 + {4y^2 m_2^2 + 4 [(1-x) l^{\perp} + y k^\perp ]^2 \over (1-x)^2-y^2 } 
\to 
m_g^2 + 4 l^{\perp 2} + 8 y l^\perp k^\perp/(1-x)   \ , 
\eeq
In the quark-gluon rest frame,
\beq
p_g^+ \es E_g + k^z \ , \ p_q^+ \rs E_q - k^z \ , \ E = E_g + E_q  \ , \\
E_g \es \sqrt{ m_g^2 + k^{\perp 2} + k^{z2}} \ , \ E_q \rs \sqrt{ m^2 + k^{\perp 2} + k^{z2}} \ , \\
\int [ x k ] \es \int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x) }
\rs
\int  {d^3 k \over  2(2\pi)^3 } \, \left( {1\over E_g} + {1 \over E_q} \right) \ .
\eeq

There is another relation worth knowing and remembering. Namely,
\beq
q_1^2 - q_2^2 
\es
(p_{1'} - p_1)^2 - (p_2 - p_{2'})^2 \\
\es
(p_{1'} - p_1 -  p_2 + p_{2'}) (p_{1'} - p_1 +  p_2 - p_{2'}) \\
\es
(P_{1'2'} - P_{12}) (q_1 +  q_2) \ .
\eeq
But $P_{1'2'}$ and $P_{12}$ differ only in their minus components,
while $q_1^+ = q_2^+ = z P_{12}^+$. Therefore,
\beq
q_1^2 - q_2^2  \es \2 { \cM_{1'2'}^2 - \cM_{12}^2 \over P_{12}^+} 2 z P_{12}^+ \ ,
\eeq
which amounts to the universal relation
\beq
q_1^2 - q_2^2  \es z \, (\cM_{1'2'}^2 - \cM_{12}^2) \ .
\eeq
Generally, I change the definitions of momentum transfers to 
\beq
q_1 \es p_{\overline 1} - p_{\underline 1} \ , \  q_2 \rs p_{\overline 2} - p_{\underline 2} \ . 
\eeq
These momenta match the exchanged gluon momentum except for the minus component.
\beq
q_1^2 - q_2^2 \es (q_1 - q_2)(q_1 + q_2) \rs 
( p_{\overline 1} - p_{\underline 1} - p_{\overline 2} + p_{\underline 2}) (q_1 + q_2)\\
\es
( p_{\overline 1} + p_{\underline 2} - p_{\underline 1} - p_{\overline 2} ) (q_1 + q_2) \ .
\eeq
***********}

Substitution $k^\perp = \sqrt{x} \,  q^\perp$ yields, with $t_p = t/p_2^+$,
the sum in the form \\
$g^2 (C_F {\rm ~or~} C_{|2\rangle})/[2(2\pi)^3] \times $
\beq
\label{Seta3}
S_2^\eta \es
{1 \over p_2^+} 
\int_0^1 dx \, d^2q^\perp \ e^{-t_p^2 \sigma_2^2} \ {4 \over x} \
\left[  
e^{- t_p^2 \sigma_2^2} 
-  
e^{- t_p^2 \sigma_1^2} 
\ 
{  \sqrt{ x_1 }  \over \sqrt{ (x_1 +x x_2) (1-x)}   }
\ 
{ ( \sigma_1 + \sigma_2)^2  \over 4 \sigma_1 \sigma_2  } 
\right]  \ , \\
\sigma_2 \es \rho_2/x \rs {m_g^2\over x} + {m_2^2 x + q^{\perp 2} \over 1-x} \ , \\
\sigma_1 \es \rho_1/x \rs {m_g^2\over x} + { m_1^2  x x_2^2 + (x_1  q^\perp  - \sqrt{x} \, k_{12}^\perp )^2 \over x_1 (x_1+xx_2) } \ .
\eeq
Because of the exponential factors, $|q^\perp|$ is limited. The dominant behaviors of $\sigma$s near $x=0$ are the same,
\beq
\sigma_2 \es {m_g^2\over x} +  q^{\perp 2} \ , \\
\sigma_1 \es {m_g^2\over x} +  q^{\perp 2}  \ .
\eeq
Therefore, for small $x$, the large square bracket vanishes and the small-$x$ logarithmic divergence is actually absent.
{\red But there is also a divergence due to a small $q^\perp$ in the denominator. Ide spac 20250826 wto 01:10 Ekologiczna.}

{\blue 20250901 16:42 pon San Dimas: I need to solve the small denominator problems but now I continue with evaluation of
$S_2^\eta$ for a moment.} 
I recall the option to write 
\beq
{ ( \sigma_1 + \sigma_2)^2  \over 4 \sigma_1 \sigma_2  } \es { ( \sigma_1 + \sigma_2)^2  \over 4 \sigma_1 \sigma_2  }  - 1 + 1
\rs  { ( \sigma_1 - \sigma_2)^2  \over 4 \sigma_1 \sigma_2  } + 1 \ .
\eeq
The difference $\sigma_1 - \sigma_2$ vanishes for $x \to 0$. Therefore, the integral with that difference is not diverging
when $x \to 0$, which means I could set $m_g^2$ to zero in this term as far as small $x$ is concerned. The remaining term 
of 1 contributes to the difference involving the form factors and a factor with square roots that also tends to 1 for $x \to 0$,
which implies the logarithmic small-$x$ divergence is canceled. 

{\red 20250901 17:30 pon San Dimas:  To complete the thesis of the paper I need to do 2 things; explain what happens for 
small denominators, which is the genuine infrared problem, and calculate how the gluon interacts with quarks in the 3-body 
component.}





\begin{figure}[ht!]
          \caption{Effective gluon exchange and self-interaction terms in the two-body component of a heavy quarkonium
with the RGPEP vertex factors $f_{LI}f_{IR}$ represented by the black dots.}
          \label{fig:selfexch}
\begin{center}
         \includegraphics[width=0.6\textwidth]{ExchangeSigmaHs}
\end{center} 
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Inclusion of quarkonium components with effective gluons} 
\label{includinggluons}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Two-body component is coupled to the three-body states containing an effective gluon in addition to the quark pair, 
\beq
\label{QbarQGstates}
|Q_{s1} \bar Q_{s2} G_{s3}\rangle \es b_{s1}^\dagger d_{s2}^\dagger a_{s3}^\dagger |0\rangle \ ,
\eeq
and to the states with more effective gluons. States including additional quark-antiquark pairs are assumed so heavy 
that they do not significantly contribute to the ground and low excited eigenstates. All these components  correspond 
to the basis states $| i \rangle_s$ in Eq.~(\ref{Hteppsi}), while their coefficients $\psi_i(s)$ are the eigenstate wave 
functions. Thus, the effective eigenvalue problem differs qualitatively from the equations derived using the Tamm-Dancoff approximation~\cite{Tamm:1945qv,Dancoff:1950ud,Perry:1990mz,Perry:1991ny}. Namely, the Hamiltonian $H_s$ is 
independent of the regularization parameter $r \to 0$. Instead, the effective interaction terms in $H_s$ include the vertex 
form factors $f^s_{\bar p,\bar q}$ computed according to Eqs.~(\ref{cH1t}) to (\ref{Tft}). As a result, the eigenvalue problem 
for $H_s$ does not require the so-called sector-dependent counterterms~\cite{Perry:1990mz}. Nevertheless, the eigenstates 
have components with different numbers of effective constituents.
$|\psi\rangle$ can be written as
\beq
|\psi \rangle \es |2\rangle + |3 \rangle + . . . \ ,
\eeq
where dots indicate the components $|n\rangle$ with $n> 3$. The interaction that couples the component $|3\rangle$ to 
the component $|2\rangle$, is proportional to the coupling constant. Therefore, the contribution of the component 
$|3\rangle$ to the dynamics of component $|2\rangle$ is at least of order $g^2$. The components $|n\rangle$ with 
$n > 3$ contribute terms of at least 4th order. When the coupling constant $g$ is very small, the effective coupling
constant $g_s$ is also very small, since the difference between them is of the third order. Up to the 2nd order, the
coupling constants $g$ and $g_s$ are the same.




Self-interactions: the same Eq.~(\ref{selfinteraction1quark333full}) for quarks, and Eq.~(\ref{SigmaTgluon}) for gluons.

Seagulls: similar to Eq.~(\ref{seagull}) for quarks but with different color factor, and a new seagull for gluon-quark interaction.

Terms $f-ff$: for qoark-antiquark interaction with different color factor, and new for the gluon-quark interaction

Seagull of quarks: These are new due to the instantaneous quarks

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{2nd-order seagull interactions in the 3-body space}
\label{H_3body seagull etc}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\red 2025006 21:11 sob San Dimas - Start here tomorrow : In the 3-body space I have the 2nd-order interaction terms . . .

2025007 12:18 nie San Dimas}

The initial expression for the quark-gluon seagull term is given in Eq.~\ref{HJJ}, repeated here
\beq
\label{HJJ POM 1}
H_{JJ} \es \int dx^- d^2x^\perp \ 
\2 (J_{\psi f}^{a+} + J_{A f}^{a+}) 
{-1 \over \partial^{+2}} 
(J_{Af}^{a+} + J_{\psi f}^{a+}) \ ,
\eeq
The quark-gluon seagull is only
\beq
\label{HJJ POM 2}
H_{sqg} \es \int dx^- d^2x^\perp \ 
J_{A f}^{a+} {-1 \over \partial^{+2}} J_{\psi f}^{a+} \ .
\eeq
The seagull term structure in our notation is illustrated in Fig.~\ref{fig:qgs}. Using Eqs.~(\ref{jquark1}) and (\ref{jquark2}),
\beq
\label{jquark1 POM 3}
:J_{\psi f}^{a +}: \es -g
\sum_{c_1, c_2} \sum_{\sigma_1, \sigma_2} \ 
\chi_{c_1}^\dagger T^a \chi_{c_2} 
\int[p_1p_2]  \ 2 \sqrt{p_1^+ p_2^+} \ \left[ :\{ ~~ \}: \right]^r \ , \\ 
\left[ :\{ ~~ \}: \right]^r
\es
            b_1^\dagger 
            b_2 e^{i(p_1-p_2)x}
            \left[  \theta_{1-2} f^r_{1,2(1-2)} + \theta_{2-1}f^r_{2,1(2-1)} \right]
            \delta_{\sigma_1 , \sigma_2}
\nm                    
            d^\dagger_2 
            d_1 e^{i(p_2-p_1)x}
            \left[ \theta_{2-1} f^r_{2,1(2-1)} + \theta_{1-2}f^r_{1,2(1-2)} \right]
            \delta_{\sigma_1 , \sigma_2} \ .
\eeq
I dropped the pair creation and annihilation terms from $\left[ :\{ ~~ \}: \right]^r$. The gluon current is given in
Eq.~(\ref{gluoncurrent}), or 
\beq
\label{gluoncurrent POM}
J_{A f}^{a\mu} \es ig [  \partial^\mu A_{f \nu}, A_f^\nu ]^a \ , \ 
J_{A f}^{a+} \rs - ig [  \partial^+ A_f^\perp, A_f^\perp]^a \rs -i g f^{abc} (i \partial^+ A_f^{b\perp}) A_f^{c\perp} \ . 
\eeq
Using Eq.~(\ref{bareAf}), knowing that $x^+$ is set to 0,
\beq
\label{bareAf POM}
(i \partial^+ A_f^{b \perp}) A_f^{c\perp}
\es
\sum_{12}  \int [12] \, p_1^+ \,
\left[  \varepsilon^\perp_1 a_{1 b} e^{-ip_1x} - \varepsilon^{\perp *}_1 a^\dagger_{1b} e^{ip_1x} \right]
\left[  \varepsilon^\perp_2 a_{2 c} e^{-ip_2x} +  \varepsilon^{\perp *}_2 a^\dagger_{2c} e^{ip_2x}\right] \ .
\eeq
For the seagulls in the 3-body space, I only keep terms of the type $a^\dagger a$. The gluon spin is conserved.
Its spin conservation is assured by the products 
\beq
\varepsilon^{\perp *}_1 \varepsilon^\perp_2  \rs - \varepsilon^{*}_1 \varepsilon_2
~~~~ {\rm or } ~~~~ \varepsilon^{\perp}_1 \varepsilon^{\perp *}_2 \rs - \varepsilon_1 \varepsilon^{*}_2\ . 
\eeq
Since components $+$ of $\varepsilon$ are 0. Further,
\beq
\label{bareAf POM 1}
(i \partial^+ A_f^{b \perp}) A_f^{c\perp}
& \to &
\sum_\sigma  \int [12] \, p_1^+ \,
\left[   a^\dagger_{2 \sigma c} a_{1 \sigma b} e^{-ip_1x} e^{  ip_2x}
       -  a^\dagger_{1 \sigma b} a_{2 \sigma c} e^{  ip_1x} e^{-ip_2x} \right] \ .
\eeq
The current is normal-ordered and regulated, a la Eqs.~(\ref{jquark1}) and (\ref{jquark2}). I can simplify notation, keeping 1 
for quarks, 2 for antiquarks and 3 for gluons. 
\beq
J_{A f}^{a+} \es -i g f^{abc} (i \partial^+ A_f^{b\perp}) A_f^{c\perp} \\
& \to &
-i g f^{abc}
\sum_\sigma  \int [33'] \, p_3^+ \,
\left[   a^\dagger_{3' \sigma c} a_{3 \sigma b} e^{-ip_3x} e^{  ip_{3'}x}
       -  a^\dagger_{3 \sigma b} a_{3' \sigma c} e^{  ip_3x} e^{-ip_{3'}x} \right] 
      f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} \ . 
\eeq
I change $3 \leftrightarrow 3'$ in the first term.
\beq
J_{A f}^{a+} 
& \to &
-i g f^{abc}
\sum_\sigma  \int [33'] \, 
\left[   p_{3'}^+ a^\dagger_{3 \sigma c} a_{3' \sigma b} e^{-ip_{3'}x} e^{  ip_{3}x}
       -  p_3^+ a^\dagger_{3 \sigma b} a_{3' \sigma c} e^{  ip_3x} e^{-ip_{3'}x} \right] 
      f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} \ . 
\eeq
I change $b \leftrightarrow c$ in the first term.
\beq
J_{A f}^{a+} 
& \to &
-i g f^{abc}
\sum_\sigma  \int [33'] \, 
\left[  - p_{3'}^+ a^\dagger_{3 \sigma b} a_{3' \sigma c} e^{-ip_{3'}x} e^{  ip_{3}x}
       -  p_3^+ a^\dagger_{3 \sigma b} a_{3' \sigma c} e^{  ip_3x} e^{-ip_{3'}x} \right] 
      f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} \ . 
\eeq
The result is 
\beq
J_{A f}^{a+} 
& \to &
i g f^{abc}
\sum_\sigma  \int [33'] \, 
\left[  p_{3'}^+ a^\dagger_{3 \sigma b} a_{3' \sigma c} e^{-ip_{3'}x} e^{  ip_{3}x}
       + p_3^+ a^\dagger_{3 \sigma b} a_{3' \sigma c} e^{  ip_3x} e^{-ip_{3'}x} \right] 
      f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} \\
\es
i g f^{abc}
\sum_\sigma  \int [33'] \, (p_3^+ + p_{3'}^+) \, a^\dagger_{3 \sigma b} a_{3' \sigma c} e^{-i(p_{3'} - p_{3} ) x}
             f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} \ . 
\eeq
\begin{figure}[ht!]
          \caption{Hamiltonian seagull interaction term of a gluon (3) and a quark (1), with a momentum transfer $p_4$.}
          \label{fig:qgs}
\begin{center}
         \includegraphics[width=0.4\textwidth]{quark-gluon-seagull}
\end{center} 
\end{figure}
The seagull term, including both quarks and antiquarks, is
\beq
\label{Hsqg1}
H_{sqg} \es \int dx^- d^2x^\perp \ 
i g f^{abc}
\int [33'] \, (p_3^+ + p_{3'}^+) \ a^\dagger_{3b} a_{3'c} \ e^{-i(p_{3'} - p_{3} ) x}
\ \delta_{\sigma_3 \sigma_{3'}}
\ f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} 
\nt 
{1 \over (p_{3'}^+ - p_3^+)^2} 
\nt 
(-g) \  \chi_{1}^\dagger T^a \chi_{1'} 
\int[11']  \ 2 \sqrt{p_1^+ p_{1'}^+} \
\left[ b_1^\dagger  b_{1'} e^{i(p_1-p_{1'} ) x} - d^\dagger_{1'} d_1 e^{i ( p_{1'}-p_1 ) x} \right]
\nt
\delta_{\sigma_1  \sigma_{1'}} \ f^r_{\overline 1,\underline 1 (\overline 1 - \underline 1)} \ .
\eeq
Compacted, the gluon-quark seagull is
\beq
\label{Hsqg2}
H_{sqg} \es -g^2 \int [33'11'] \ \tilde \delta_{c.a} \ \delta_{\rm spins} 
\ f^r_{\overline 1,\underline 1 (\overline 1 - \underline 1)} 
\ f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} 
\
{ ( p_3^+ + p_{3'}^+ )  \ 2 \sqrt{p_1^+ p_{1'}^+}  \over (p_{3'}^+ - p_3^+)^2} 
\ \chi_{1}^\dagger T^a \chi_{1'} \, i f^{abc} \  a^\dagger_{3b} a_{3'c}  b_1^\dagger  b_{1'} \ ,
\eeq
and the gluon-antiquark seagull is
\beq
\label{Hsantiqg2}
H_{s\bar qg} \es+ g^2 \int [33'22'] \ \tilde \delta_{c.a} \ \delta_{\rm spins}
\ f^r_{\overline 2,\underline 2 (\overline 2 - \underline 2)} 
\ f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} 
\
{ ( p_3^+ + p_{3'}^+ ) \ 2 \sqrt{p_{2}^+ p_{2'}^+} \over (p_{3'}^+ - p_3^+)^2} 
\ \chi_{2'}^\dagger T^a \chi_{2} \, i f^{abc} \ a^\dagger_{3b} a_{3'c}  d^\dagger_{2} d_{2'}  \ .
\eeq
The wave-function color factor for the globally colorless 3-body states is  
\beq
\label{colorfactors3body1}
C_{\rm 3body}^{123} \es \chi^\dagger_1 T^3 \chi_2 \ .
\eeq
The color action of the quark seagull $H_{sqg}$ on the globally colorless 3-body states is described by 
\beq
\chi_1^\dagger T^s \chi_{1'}  i f^{s33'} \  \chi_{1'}^\dagger T^{3'} \chi_2
\es i f^{s33'} \chi_1^\dagger T^s  T^{3'} \chi_2
\rs {i \over 2}  f^{s33'} \chi_1^\dagger [ T^s , T^{3'} ] \chi_2
\rs {i \over 2}  f^{s33'} if^{as3'} \ \chi_1^\dagger T^a \chi_2 \\
\es - \2  f^{s33'} f^{as3'} \ \chi_1^\dagger T^a \chi_2
\rs \2  f^{3s3'} f^{as3'} \ \chi_1^\dagger T^a \chi_2 
\rs \2 N_c \ \chi_1^\dagger T^3 \chi_2 \ ,
\eeq
which means multiplication by $N/2$. The color action of the antiquark seagull $H_{s \bar qg}$ on the same globally colorless 
3-body states is described by 
\beq
\chi_{2'}^\dagger T^s \chi_2  i f^{s33'} \  \chi_{1}^\dagger T^{3'} \chi_{2'}
\es
 i f^{s33'} \ \chi_{2'}^\dagger T^s \chi_2 \  \chi_{1}^\dagger T^{3'} \chi_{2'}
\rs
 i f^{s33'} \ \chi_{1}^\dagger T^{3'}  T^s \chi_2
\rs
{i \over 2} f^{s33'} i f^{3'sa} \ \chi_{1}^\dagger T^a \chi_2 \\
\es
\label{colorfactors3body2}
-\2 f^{s33'}  f^{3'sa} \ \chi_{1}^\dagger T^a \chi_2 
\rs
-\2 f^{3's3}  f^{3'sa} \ \chi_{1}^\dagger T^a \chi_2
\rs
-\2 N_c \ \chi_{1}^\dagger T^3 \chi_2
\eeq
which means multiplication by $-N/2$. Including the sign factors in front of the operators, the net factor is the same 
in both cases, $-1/x^2$. The sign of the seagull term in the colorless 3-body states is the same as in the colorless 
2-body states. The strength is slightly increased from 4/3 to 3/2, by the factor 9/8.

{\red  20250908 21:27 pon San Dimas: Start tomorrow from checking what the gluon exchange does.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{2nd-order quark-gluon interactions through gluon exchange in the 3-body space}
\label{H_3body current-current etc}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The interaction of a gluon with a quark or an antiquark in the 3-body space, proceeding through the exchange of a gluon, 
results from the double action of the term $ - (J_{\psi f}^{a \mu}  + J_{A f}^{a \mu}) A_{f\mu}^a$ in $\cH_{\rm QCD}$ in 
Eq.~(\ref{cHcanQCD}) and the term $m_g \phi^a {1 \over \partial^+} (J_{Af}^{a+} + J_{\psi f}^{a+})$ in Eq.~(\ref{cHcanQCDmg}), 
using the currents 
\beq
\label{quarkcurrent exch}
J_{\psi f}^{a \mu} \es -g \bar \psi_f T^a \gamma^\mu  \psi_f \ , \\
\label{gluoncurrent exch}
J_{A f}^{a\mu} \es ig [  \partial^\mu A_{f \nu}, A_f^\nu ]^a \ .
\eeq
The exchange term formula in our notation is illustrated in Fig.~\ref{fig:3bg3q1}.
\begin{figure}[ht!]
          \caption{Hamiltonian interaction term of a gluon (3) with a quark (1) through the exchange of a gluon 
                         carrying the momentum transfer $p_4$.}
          \label{fig:3bg3q1}
\begin{center}
         \includegraphics[width=0.4\textwidth]{3bodyexch}
\end{center} 
\end{figure}
The formula follows from $\cH_f$ and the renormalized interaction Hamiltonian terms given in Eq.~(\ref{cH2t}). The gluon exchange in 
the 3-body space is contained in
\beq
\label{cH2t3body}
\cH^{(2)}
\es
f_{LR} \, \cH^{(2)}_0
+ 
(f_{LR} - f_{LI} f_{IR}) \
\Delta_{LIR} \
\cH^{(1)}_0 \, \cH^{(1)}_0 \ ,
\eeq
where we recognize analogs of the 2-body terms present in Eqs.~(\ref{HLR1pom1text}) and (\ref{cH2t scheme01}). We identify the 
the 3-body-space gluon exchange terms between a gluon and a quark in 
\beq
\cH_s \es
\label{cH3scheme01}
\cH_f +  f_{LR} \, \left[ \Sigma_{g \, \rm sing} + \delta m^2_{g \, \rm ln} + \Sigma_{q \, \rm sing} + \delta m^2_{q \, \rm ln} 
+  f_r f_r H_{\rm sggq} + F_r S_{\rm sggq} + F_r S_{\rm exch}   \right]
\np
 (f_{LR} - f_{LI} f_{IR}) \ \Delta_{LIR} \ f_r \cH^{(1)}_0 \, f_r \cH^{(1)}_0  \ .
\eeq
Question: Do the divergences of gluon-quark seagull $f_{LR} f_r f_r H_{\rm sggq}$ and gluon exchange in 
$f_{LR} \ \Delta_{LIR} \ f_r \cH^{(1)}_0 \, f_r \cH^{(1)}_0$ cancel each other? Gluon-quark seagull operator is 
given in Eq.~(\ref{Hsqg2}). The gluon exchange interaction between a gluon and a quark follows from the
product $\cH^{(1)}_0 \, \cH^{(1)}_0$, in which $\cH_0^{(1)}$ includes the following terms from Eq.~(\ref{cHcanQCDmg}),
\beq
\cH^{(1)}_0 \es \int_F \left[ - (J_{\psi f}^{a \mu}  + J_{A f}^{a \mu}) A_{f\mu}^a
+m_g \phi^a {1 \over \partial^+} (J_{Af}^{a+} + J_{\psi f}^{a+})  \right] \ .
\eeq
Integration by parts yields
\beq
\cH^{(1)}_0 
\eqv
- \int_F \left( J_{\psi f}^{a \mu} + J_{A f}^{a \mu} \right) \left( A_{f\mu}^a + {m_g \eta_\mu  \over \partial^+} \phi^a \right) \ .
\eeq
I called a file toolong on 20250912 16:00 in San Dimas and start from scratch evaluation of the interaction terms.
There are 4 terms to evaluate
\beq
\cH_{\psi A} \es - \int_F J_{\psi f}^{a \mu}   A_{f\mu}^a  \ , \\
\cH_{\psi \phi} \es - \int_F J_{\psi f}^{a \mu}  {m_g \eta_\mu  \over \partial^+} \phi^a \ , \\
\cH_{A A} \es - \int_F  J_{A f}^{a \mu}  A_{f\mu}^a \ , \\
\cH_{A \phi} \es - \int_F J_{A f}^{a \mu}  {m_g \eta_\mu  \over \partial^+} \phi^a \ .
\eeq
Reminder
\beq
{m_g \eta_\mu  \over \partial^+} \phi^a \rs {i m_g \eta_\mu  \over i \partial^+} \phi^a
\es
\int [4]
 {m_g \eta_\mu  \over p_4^+} \left( a_{4a} e^{-ip_4x} + a^\dagger_{4a} e^{ip_4x} \right) \ .
\eeq

\vskip.1in
\centerline{\bf Evaluation of $\cH_{\psi A}$ }
\vskip.1in

\beq
\cH_{\psi A} \es - \int_F J_{\psi f}^{a \mu}   A_{f\mu}^a  \ , \\
\es
- g_{\mu \nu} (-g)
\int[124]  \, \tilde \delta_{c.a} \, f
\left[  u_1 \chi_1 b_1 + v_1 \chi_1 d^\dagger_1 \right]^\dagger \gamma^0 \gamma^\mu
\left[  \varepsilon^\nu_4 T^4 a_4 + \varepsilon^{\nu *}_4 T^4 a^\dagger_4 \right]
\left[  u_2 \chi_2 b_2 + v_2 \chi_2 d^\dagger_2 \right]
\eeq
In the 3-body space, two terms count: quark or antiquark emitting and absorbing a gluon, while the pair creation or annihilation are 
ignored as too far off shell.
\beq
\cH_{\psi A} \rs - \int_F J_{\psi f}^{a \mu}   A_{f\mu}^a  
\es
g
\int[11'4]  \, \tilde \delta_{c.a} \, f 
\
\left[  u_1 \chi_1 b_1  \right]^\dagger \gamma^0 \gamma_\nu
\left[  \varepsilon^\nu_4 T^4 a_4 + \varepsilon^{\nu *}_4 T^4 a^\dagger_4 \right]
\left[  u_{1'} \chi_{1'} b_{1'} \right]
\np
g
\int[22'4]  \tilde \delta_{c.a} f 
\
\left[   v_{2'} \chi_{2'} d^\dagger_{2'} \right]^\dagger \gamma^0 \gamma_\nu
\left[  \varepsilon^\nu_4 T^4 a_4 + \varepsilon^{\nu *}_4 T^4 a^\dagger_4 \right]
\left[   v_2 \chi_2 d^\dagger_2 \right] \ .
\eeq
\beq
\cH_{\psi A} 
\es
g
\int[11'4] \, \tilde \delta_{c.a} \, f^r_{\overline 1, \underline 1 4} 
\ \chi_1^\dagger  T^4 \chi_{1'}  \ \bar u_1 \gamma_\nu u_{1'}       
\left(  \varepsilon^\nu_4 \ b_1^\dagger  b_{1'}  a_4 + \varepsilon^{\nu *}_4  a^\dagger_4 b_1^\dagger  b_{1'}  \right)
\nm
g
\int[22'4] \, \tilde \delta_{c.a} \, f^r_{\overline 2, \underline 2 4} 
\ \chi_{2'}^\dagger T^4 \chi_2 \ \bar v_{2'} \gamma_\nu v_2 \  
\left(  \varepsilon^\nu_4 \ d^\dagger_2 d_{2'} a_4 + \varepsilon^{\nu *}_4  \ a^\dagger_4 d^\dagger_2 d_{2'} \right) \ .
\eeq

\vskip.1in
\centerline{\bf Evaluation of $\cH_{\psi \phi} $ }
\vskip.1in

\beq
\cH_{\psi \phi} \es - \int_F J_{\psi f}^{a \mu}  {m_g \eta_\mu  \over \partial^+} \phi^a \\
\es
- g_{\mu \nu} (-g)
\int[12\tilde 4] \, \tilde \delta_{c.a} \, f
\left[  u_1 \chi_1 b_1 + v_1 \chi_1 d^\dagger_1 \right]^\dagger \gamma^0 \gamma^\mu
{m_g \eta^\nu \over p_{\tilde 4}^+}T^{\tilde 4}
\left[  a_{\tilde 4} + a^\dagger_{\tilde 4} \right]
\left[  u_2 \chi_2 b_2 + v_2 \chi_2 d^\dagger_2 \right]
\eeq
Again, in the 3-body space, two terms count: quark or antiquark emitting and absorbing a gluon, while the pair creation or annihilation are 
ignored as too far off shell.
\beq
\cH_{\psi \phi} \rs - \int_F J_{\psi f}^{a \mu}  {m_g \eta_\mu  \over \partial^+} \phi^a 
\es
g
\int[11'\tilde 4] \, \tilde \delta_{c.a} \, f^r_{\overline 1, \underline 1 \tilde 4} 
\ \chi_1^\dagger T^{\tilde 4} \chi_{1'} 
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g  \over p_{\tilde 4}^+}
\left( b_1^\dagger b_{1'}  a_{\tilde 4} + a^\dagger_{\tilde 4} b_1^\dagger b_{1'}  \right) 
\nm
g
\int[22'\tilde 4] \, \tilde \delta_{c.a} \, f^r_{\overline 2, \underline 2 \tilde 4}  
\ \chi_{2'}^\dagger T^{\tilde 4} \chi_2
\ \bar v_{2'}    \gamma^+ v_2
\ {m_g  \over p_{\tilde 4}^+}
\left( d^\dagger_2d_{2'}  a_{\tilde 4} + a^\dagger_{\tilde 4}d^\dagger_2d_{2'}  \right) \ .
\eeq
{\red \bf Below, I tend to omit regularization factors $f^r_{\overline x, \underline x  y}$ till I come to the final expression 
for the gluon-exchange interaction between the gluon and quark in the 3- body space.}

\vskip.1in
\centerline{\bf Evaluation of $\cH_{A A}$ }
\vskip.1in

\beq
\cH_{A A} \es - \int_F  J_{A f}^{a \mu}  A_{f\mu}^a \ , \\
A_{f\mu}^a \es 
\sum_{3'} \int [3'] \left[  \varepsilon_{3'\mu}  a_{3'a} e^{-ip_{3'}x} + \varepsilon_{3' \mu}^*  a^\dagger_{3'a} e^{ip_{3'}x} \right] \ , \\
J_{A f}^{a \mu} \es g [ i \partial^\mu A_{f \nu}, A_f^\nu ]^a 
                        \rs i g  f^{abc} (i \partial^\mu A_{f \nu}^b ) A_f^{c \nu}  \\ 
\es
i g  f^{abc} \sum_{34}  \int [34] \ 
p_3^\mu \left[  \varepsilon_3 a_{3b} e^{-ip_3x} - \varepsilon^*_3 a^\dagger_{3 b} e^{ip_3x} \right]
                \left[  \varepsilon_4 a_{4c} e^{-ip_4x} +\varepsilon^*_4 a^\dagger_{4c} e^{ip_4x} \right]
\eeq
\beq
\cH_{A A} \es 
- \int_F i g  f^{abc} \sum_{33'4}  \int [33'4] \ 
\left[  \varepsilon_3 a_{3b} e^{-ip_3x} - \varepsilon^*_3 a^\dagger_{3 b} e^{ip_3x} \right]
\left[  \varepsilon_4 a_{4c} e^{-ip_4x} +\varepsilon^*_4 a^\dagger_{4c} e^{ip_4x} \right]
\nt           
\left[  p_3 \varepsilon_{3'}  a_{3'a} e^{-ip_{3'}x} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a} e^{ip_{3'}x} \right] \\
\es
- i g  f^{abc} \sum_{33'4}  \int [33'4] \, \tilde \delta_{c.a}
\left[  \varepsilon_3 a_{3b}  - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  +\varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right]
\ .
\eeq
There are 8-2=6 terms to evaluate. 
\beq
&& 
112
\left[  \varepsilon_3 a_{3b}   \right]
\left[  \varepsilon_4 a_{4c}  \right]         
\left[   p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right]
+
121
\left[  \varepsilon_3 a_{3b}   \right]
\left[ \varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a}  \right]
+
122
\left[  \varepsilon_3 a_{3b}   \right]
\left[ \varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right]
\np
211
\left[   - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}   \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} \right]
+
212
\left[    - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  \right]         
\left[  p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right]
+
221
\left[    - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a}  \right] \ .
\eeq
Clean.
\beq
\cH_{A A} \es - i g  f^{abc} \sum_{33'4}  \int [33'4] \, \tilde \delta_{c.a} \ X \ , \\
X \es
    \varepsilon_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \ a_{3b} a_{4c} a^\dagger_{3'a}    
+ \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a_{3b} a^\dagger_{4c} a_{3'a}    
+ \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}^*  \ \ a_{3b} a^\dagger_{4c} a^\dagger_{3'a}    
\nm
    \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a_{4c} a_{3'a}   
-  \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \  a^\dagger_{3 b} a_{4c} a^\dagger_{3'a}    
-  \varepsilon^*_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a^\dagger_{4c} a_{3'a}  \ .
\eeq
Separate terms with 2 daggers and 1 dagger. 
\beq
X \es
    \varepsilon_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \ a_{3b} a_{4c} a^\dagger_{3'a}    
+ \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a_{3b} a^\dagger_{4c} a_{3'a}    
-  \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a_{4c} a_{3'a}   
\np
   \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}^*  \ \ a_{3b} a^\dagger_{4c} a^\dagger_{3'a}    
-  \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \  a^\dagger_{3 b} a_{4c} a^\dagger_{3'a}    
-  \varepsilon^*_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a^\dagger_{4c} a_{3'a}  \ .
\eeq

check
\beq
X \es
    \varepsilon_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \ a_{3b} a_{4c} a^\dagger_{3'a}    
-  \varepsilon^*_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a^\dagger_{4c} a_{3'a}
\np
   \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a_{3b} a^\dagger_{4c} a_{3'a}    
-  \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \  a^\dagger_{3 b} a_{4c} a^\dagger_{3'a}  
 \np
   \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}^*  \ \ a_{3b} a^\dagger_{4c} a^\dagger_{3'a}    
-  \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a_{4c} a_{3'a}  \ .
\eeq
\beq
-X \es
   \varepsilon^*_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a^\dagger_{4c} a_{3'a}
 - \varepsilon_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \ a_{3b} a_{4c} a^\dagger_{3'a}    
 \np
   \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \  a^\dagger_{3 b} a_{4c} a^\dagger_{3'a}  
 - \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a_{3b} a^\dagger_{4c} a_{3'a}    
 \np
   \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a_{4c} a_{3'a} 
 - \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}^*  \ \ a_{3b} a^\dagger_{4c} a^\dagger_{3'a}    \ .
\eeq

Arrange h.c.
\beq
- X \es
    \varepsilon^*_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a^\dagger_{4c} a_{3'a} 
-  \varepsilon_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \ a_{3b} a_{4c} a^\dagger_{3'a}     
\np
   \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \  a^\dagger_{3 b} a_{4c} a^\dagger_{3'a}
-  \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a_{3b} a^\dagger_{4c} a_{3'a}     
\np
   \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}^*  \ \ a_{3b} a^\dagger_{4c} a^\dagger_{3'a} 
- \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a_{4c} a_{3'a} \ .
\eeq
Rewrite.
\beq
- X \es
    \varepsilon^*_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a^\dagger_{4c} a_{3'a} 
+ \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \  a^\dagger_{3 b} a_{4c} a^\dagger_{3'a}
+ \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}^*  \ \ a_{3b} a^\dagger_{4c} a^\dagger_{3'a} 
\nm
   \varepsilon_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \ a_{3b} a_{4c} a^\dagger_{3'a}    
-  \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a_{3b} a^\dagger_{4c} a_{3'a}
- \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3 b} a_{4c} a_{3'a} \ .
\eeq
Notation.
\beq
\cH_{A A} \es g \sum_{33'4}  \int [33'4] \, \tilde \delta_{c.a} \ i f^{c_{3'}c_3 c_4} \ Z_{33'4}  + h.c. \ , \\
Z_{33'4} \es
    \varepsilon^*_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}  \ \ a^\dagger_{3} a^\dagger_{4} a_{3'} 
+ \varepsilon^*_3 \varepsilon_4 \ p_3 \varepsilon_{3'}^* \ \  a^\dagger_{3} a_{4} a^\dagger_{3'}
+ \varepsilon_3 \varepsilon^*_4 \ p_3 \varepsilon_{3'}^*  \ \ a_{3} a^\dagger_{4} a^\dagger_{3'} \ . 
\eeq
I change notation in the terms second and third to make them look like the first term. I rename $Z_{33'4}$ 
to $Y_{343'}$ including $i f^{c_3 c_4 c_{3'}}$. In the second term, I change $4 \leftrightarrow 3'$, and in the 
third term I change $3 \leftrightarrow 3'$. The result is
\beq
\cH_{A A} \es g \sum_{33'4}  \int [33'4] \, \tilde \delta_{c.a} \  Y_{343'}  + h.c. \ , \\
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \
\varepsilon^*_3 \varepsilon^*_4    \ p_3 \varepsilon_{3'}     \ a^\dagger_{3} a^\dagger_{4}                a_{3'} 
- i f^{c_3 c_4 c_{3'} } \
\varepsilon^*_3      \varepsilon_{3'} \ p_3 \varepsilon_{4}^* \ a^\dagger_{3}                 a_{3'} a^\dagger_{4}
{\red +}  i f^{c_{3} c_4 c_{3'}} \
\varepsilon_{3'}      \varepsilon^*_4 \ p_{3'} \varepsilon_{3}^* \ a_{3'}                   a^\dagger_{4}  a^\dagger_{3} \ . 
\eeq
Normal ordering.
\beq
\cH_{A A} \es g \sum_{343'}  \int [343'] \, \tilde \delta_{c.a} \  Y_{343'}  + h.c. \ , \\
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \left(
   \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}  p_3                 
- \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* p_3
{\red +}  \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^* p_{3'} 
\right) \ a^\dagger_{3} a^\dagger_{4}  a_{3'} \ . 
\eeq
{\bf crosscheck of $\cH_{AA}$}
\beq
\cH_{A A} \es 
- \int_F i g  f^{abc} \sum_{343'}  \int [343'] \ 
\left[  \varepsilon_3 a_{3b} e^{-ip_3x} - \varepsilon^*_3 a^\dagger_{3 b} e^{ip_3x} \right]
\left[  \varepsilon_4 a_{4c} e^{-ip_4x} +\varepsilon^*_4 a^\dagger_{4c} e^{ip_4x} \right]
\nt           
\left[  p_3 \varepsilon_{3'}  a_{3'a} e^{-ip_{3'}x} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a} e^{ip_{3'}x} \right] \\
\es
- i g  f^{abc} \sum_{343'}  \int [343'] \, \tilde \delta_{c.a}
\left[  \varepsilon_3 a_{3b}  - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  +\varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right]
\ .
\eeq
\beq
X \es \left[  \varepsilon_3 a_{3b}  - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  +\varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right] \\
\es
112 \left[  \varepsilon_3 a_{3b}  \right]
\left[  \varepsilon_4 a_{4c}   \right]         
\left[ p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right] 
+
121\left[  \varepsilon_3 a_{3b}   \right]
\left[ \varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a}   \right] 
+
122\left[  \varepsilon_3 a_{3b}    \right]
\left[  \varepsilon^*_4 a^\dagger_{4c} \right]         
\left[   p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right] 
\nm
211\left[  \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a}   \right] 
-
212\left[  \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  \right]         
\left[   p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right] 
-
221\left[   \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[ \varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} \right] \ .
\eeq
Clean check:
\beq
X \es \left[  \varepsilon_3 a_{3b}  - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  +\varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right] \\
\es
112 \ \    \varepsilon_3 a_{3b}    
    \varepsilon_4 a_{4c}              
   p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}     
+
121 \ \   \varepsilon_3 a_{3b}     
   \varepsilon^*_4 a^\dagger_{4c}            
    p_3 \varepsilon_{3'}  a_{3'a}      
+
122 \ \   \varepsilon_3 a_{3b}      
    \varepsilon^*_4 a^\dagger_{4c}            
     p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}     
\nm
211 \ \    \varepsilon^*_3 a^\dagger_{3 b}    
    \varepsilon_4 a_{4c}             
    p_3 \varepsilon_{3'}  a_{3'a}      
-
212 \ \    \varepsilon^*_3 a^\dagger_{3 b}    
    \varepsilon_4 a_{4c}             
   p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}     
-
221 \ \ \varepsilon^*_3 a^\dagger_{3 b}    
    \varepsilon^*_4 a^\dagger_{4c}      
    p_3 \varepsilon_{3'}  a_{3'a}  \ .
\eeq
Move of operators and change of sign:
\beq
-X \es -\left[  \varepsilon_3 a_{3b}  - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  +\varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right] \\
\es
- 112 \ \    \varepsilon_3 \varepsilon_4                
   p_3 \varepsilon_{3'}^* \ a_{3b} a_{4c} a^\dagger_{3'a}     
-
121 \ \   \varepsilon_3 \varepsilon^*_4             
    p_3 \varepsilon_{3'} \ a_{3b} a^\dagger_{4c} a_{3'a}      
-
122 \ \   \varepsilon_3 \varepsilon^*_4            
     p_3 \varepsilon_{3'}^*  a_{3b} a^\dagger_{4c} a^\dagger_{3'a}     
\np
211 \ \    \varepsilon^*_3 \varepsilon_4             
    p_3 \varepsilon_{3'} \ a^\dagger_{3 b} a_{4c} a_{3'a}      
+
212 \ \    \varepsilon^*_3  \varepsilon_4             
   p_3 \varepsilon_{3'}^* \ a^\dagger_{3 b} a_{4c} a^\dagger_{3'a}     
+
221 \ \ \varepsilon^*_3 \varepsilon^*_4       
    p_3 \varepsilon_{3'} \ a^\dagger_{3 b} a^\dagger_{4c} a_{3'a}  \ .
\eeq
Renaming color scripts  $a \to c_{3'}$, $b \to c_3$, $c \to c_4$:
\beq
\cH_{A A} \es g \sum_{343'}  \int [343'] \, \tilde \delta_{c.a} \ f^{ c_3 c_4 c_{3'}} (-iX_{343'})
\eeq
\beq
-iX_{343'} \es - i \left[  \varepsilon_3 a_{3b}  - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  +\varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right] \\
\es
- 112 \ i \    \varepsilon_3 \varepsilon_4 \                
   p_3 \varepsilon_{3'}^* \ a_{3} a_{4} a^\dagger_{3'}     
-
121 \ i \   \varepsilon_3 \varepsilon^*_4 \             
    p_3 \varepsilon_{3'} \ a_{3} a^\dagger_{4} a_{3'}      
-
122 \ i \   \varepsilon_3 \varepsilon^*_4 \            
     p_3 \varepsilon_{3'}^*  a_{3} a^\dagger_{4} a^\dagger_{3'}     
\np
211 \ i \    \varepsilon^*_3 \varepsilon_4 \             
    p_3 \varepsilon_{3'} \ a^\dagger_{3} a_{4} a_{3'}      
+
212 \ i \    \varepsilon^*_3  \varepsilon_4 \             
   p_3 \varepsilon_{3'}^* \ a^\dagger_{3} a_{4} a^\dagger_{3'}     
+
221 \ i \ \varepsilon^*_3 \varepsilon^*_4 \       
    p_3 \varepsilon_{3'} \ a^\dagger_{3} a^\dagger_{4} a_{3'}  \ .
\eeq
All terms are always normal-ordered and the changes of order are meant to be under the sign of normal ordering. 
Collecting h.c.
\beq
-iX_{343'} \es - i \left[  \varepsilon_3 a_{3b}  - \varepsilon^*_3 a^\dagger_{3 b}  \right]
\left[  \varepsilon_4 a_{4c}  +\varepsilon^*_4 a^\dagger_{4c} \right]         
\left[  p_3 \varepsilon_{3'}  a_{3'a} + p_3 \varepsilon_{3'}^*  a^\dagger_{3'a}  \right] \\
\es
- 112 \ i \    \varepsilon_3 \varepsilon_4 \                
    \varepsilon_{3'}^*p_3 \ a_{3} a_{4} a^\dagger_{3'}     
-
121 \ i \   \varepsilon_3 \varepsilon^*_4 \             
     \varepsilon_{3'} p_3 \ a_{3} a^\dagger_{4} a_{3'}      
-
122 \ i \   \varepsilon_3 \varepsilon^*_4 \            
      \varepsilon_{3'}^* p_3 \ a_{3} a^\dagger_{4} a^\dagger_{3'}     
\np
221 \ i \ \varepsilon^*_3 \varepsilon^*_4 \       
     \varepsilon_{3'}p_3 \ a^\dagger_{3} a^\dagger_{4} a_{3'}
+
212 \ i \    \varepsilon^*_3  \varepsilon_4 \             
    \varepsilon_{3'}^*p_3 \ a^\dagger_{3} a_{4} a^\dagger_{3'}
+
211 \ i \    \varepsilon^*_3 \varepsilon_4 \             
     \varepsilon_{3'} p_3 \ a^\dagger_{3} a_{4} a_{3'}  \ .
\eeq
\beq
-iX_{343'} \es
221 \ i \ \varepsilon^*_3 \varepsilon^*_4 \       
     \varepsilon_{3'} p_3 \ a^\dagger_{3} a^\dagger_{4} a_{3'}
+
212 \ i \    \varepsilon^*_3  \varepsilon_4 \             
   \varepsilon_{3'}^* p_3  \ a^\dagger_{3} a_{4} a^\dagger_{3'}
-
122 \ i \   \varepsilon_3 \varepsilon^*_4 \            
     \varepsilon_{3'}^* p_3 \ a_{3} a^\dagger_{4} a^\dagger_{3'}     
+ h.c.  \ .
\eeq
\beq
\cH_{A A} \es g \sum_{343'}  \int [343'] \,  \tilde \delta_{c.a} \ Z_{343'} \ , \  Z_{343'} \rs f^{ c_3 c_4 c_{3'}} (-iX_{343'}) \ , \\
Z_{343'} \es 
i f^{ c_3 c_4 c_{3'}}\ \varepsilon^*_3 \varepsilon^*_4 \       
     \varepsilon_{3'} p_3 \ a^\dagger_{3} a^\dagger_{4} a_{3'}
+i f^{ c_3 c_4 c_{3'}}\    \varepsilon^*_3  \varepsilon_4 \             
   \varepsilon_{3'}^* p_3  \ a^\dagger_{3} a_{4} a^\dagger_{3'}
- i f^{ c_3 c_4 c_{3'}}\   \varepsilon_3 \varepsilon^*_4 \            
    \varepsilon_{3'}^* p_3 \ a_{3} a^\dagger_{4} a^\dagger_{3'}     
+ h.c.  \ .
\eeq
I change notation $4 \leftrightarrow 3'$ in the second term and $3 \leftrightarrow 3'$ in the third one.
\beq
Z_{343'} \es 
i f^{ c_3 c_4 c_{3'}}\ \varepsilon^*_3 \varepsilon^*_4 \       
     \varepsilon_{3'} p_3 \ a^\dagger_{3} a^\dagger_{4} a_{3'}
+i f^{ c_3 c_{3'} c_4}\    \varepsilon^*_3  \varepsilon_{3'} \             
   \varepsilon_{4}^* p_3  \ a^\dagger_{3} a_{3'} a^\dagger_{4}
- i f^{ c_{3'} c_4 c_{3}}\   \varepsilon_{3'} \varepsilon^*_4 \            
    \varepsilon_{3}^* p_{3'} \ a_{3'} a^\dagger_{4} a^\dagger_{3}     
+ h.c.  \\
\es
  i f^{ c_3 c_4 c_{3'}}\ \varepsilon^*_3 \varepsilon^*_4 \       
     \varepsilon_{3'} p_3 \ a^\dagger_{3} a^\dagger_{4} a_{3'}
- i f^{ c_3 c_4 c_{3'} }\    \varepsilon^*_3  \varepsilon_{3'} \             
   \varepsilon_{4}^* p_3  \ a^\dagger_{3} a_{3'} a^\dagger_{4}
+ i f^{  c_{3}  c_4 c_{3'}}\   \varepsilon_{3'} \varepsilon^*_4 \            
    \varepsilon_{3}^* p_{3'} \ a_{3'} a^\dagger_{4} a^\dagger_{3}     
+ h.c. \ .
\eeq
{\red In the third term, I also change $4 \leftrightarrow 3$ to have $\varepsilon^*_4$ contracted with $p_{3'}$.
As a result,
\beq
Z_{343'} \es 
  i f^{ c_3 c_4 c_{3'}}\ \varepsilon^*_3 \varepsilon^*_4 \       
     \varepsilon_{3'} p_3 \ a^\dagger_{3} a^\dagger_{4} a_{3'}
- i f^{ c_3 c_4 c_{3'} }\    \varepsilon^*_3  \varepsilon_{3'} \             
   \varepsilon_{4}^* p_3  \ a^\dagger_{3} a_{3'} a^\dagger_{4}
- i f^{  c_{3}  c_4 c_{3'}}\   \varepsilon_{3'} \varepsilon^*_3 \            
    \varepsilon_{4}^* p_{3'} \ a_{3'} a^\dagger_{3} a^\dagger_{4}     
+ h.c. \ .
\eeq}
Normal ordering, harmless because $\int_F$ reduces terms with normal-order commutators to zero modes, which are 
eliminated in the regularization procedure, yields:
\beq
Z_{343'} \es i f^{ c_3 c_4 c_{3'}} \ Y_{343'} \ a^\dagger_{3} a^\dagger_{4} a_{3'} + h.c. \ , \\
Y_{343'} \es
   \varepsilon^*_3 \varepsilon^*_4 \ \varepsilon_{3'} p_3        
-  \varepsilon^*_3  \varepsilon_{3'} \ \varepsilon_{4}^* p_3    
{\red -  \varepsilon_{3'} \varepsilon^*_3 \ \varepsilon_{4}^* p_{3'}  }    
+ h.c. \ .
\eeq
Result prior to the check:
\beq
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \left(
   \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}  p_3                 
- \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* p_3
{\red -  \varepsilon^*_3 \varepsilon_{3'}  \ \varepsilon_{4}^* p_{3'}  }  
 \right) \ ,
\eeq
{\red which is also 
\beq
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \left[
   \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}  p_3                 
- \varepsilon^*_3 \varepsilon_{3'}   \   \varepsilon_{4}^* (p_3 +  p_{3'} )
\right] \\
\es  i f^{c_3 c_4 c_{3'}} \left[
   \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}  p_3 + \delta_{\sigma_3 \sigma_{3'}}   \   \varepsilon_{4}^* (p_3 +  p_{3'} )
\right]\ ,
\eeq
{\red Above {\black black text} is verified as correct}

** new version check for $p_3 + p_{3'}$
\beq
\varepsilon_{4}^* p_3 \es \varepsilon_{4}^*( p_{3'} - p_4 ) \rs \varepsilon_{4}^* p_{3'} \ . 
\eeq
\beq
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \left(
   \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}  p_3   
 - \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* p_{\red 3}
{\red -  \varepsilon^*_3 \varepsilon_{3'}  \ \varepsilon_{4}^* p_{3'}  } \right) \ . 
\eeq
Now I use the usual relative momentum variables for components $+$ and $\perp$,
\beq
y_4 \es x_4/x_{3'} \ , \ y_3 \rs x_3/x_{3'} \ , \  k_{43} \rs y_3 p_4 - y_4 p_3 \ , \\
p_4 \es y_4 p_{3'} + k_{43} \ , \ p_3 \rs y_3 p_{3'} - k_{43} \ .
\eeq
In these variables,
\beq
\varepsilon_{3'}    p_3    \es \varepsilon_{3'}(y_3 p_{3'} - k_{43}) \rs - \varepsilon_{3'}k_{43} \ , \\
\varepsilon_{4}^* p_{\red 3} \es \varepsilon_{4}^* p_{\red 3'} \rs \2 \varepsilon_{4}^{*-} p_{3'}^+ -  \varepsilon_{4}^{*\perp} p_{3'}^\perp
\rs \2 2 \varepsilon_{4}^{*\perp }p_4^\perp/ y_4 -  \varepsilon_{4}^{*\perp} p_{3'}^\perp \\
\es  \varepsilon_{4}^{*\perp} ( y_4 p_{3'}^\perp + k_{43}^\perp ) / y_4 -  \varepsilon_{4}^{*\perp} p_{3'}^\perp
\rs   \varepsilon_{4}^{*\perp} k_{43}^\perp / y_4 \rs - \varepsilon_{4}^* k_{43}/ y_4 \ , \\
\varepsilon_{\red 4}^* p_{3'} \es \2 2 \varepsilon_{\red 4}^{*\perp }p_{\red 4}^\perp/ y_{\red 4}  -  \varepsilon_{\red 4}^{*\perp} p_{3'}^\perp 
\rs \varepsilon_{\red 4}^{*\perp }(y_4 p_{3'}^\perp {\red +} k_{43}^\perp)/ y_{\red 4}  -  \varepsilon_{\red 4}^{*\perp} p_{3'}^\perp \\
\es {\red +} \varepsilon_{\red 4}^{*\perp } k_{43}^\perp)/ y_{\red 4} \rs {\red -}\varepsilon_{\red  4}^* k_{43}/ y_{\red 4} \ .
\eeq
\beq
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \left(
 - \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}k_{43}
 + \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* k_{43}/ y_4 
	{\red  +  \varepsilon^*_3 \varepsilon_{3'}   \   \varepsilon_{\red 4}^* k_{43}/ y_4}
\right) \ a^\dagger_{3} a^\dagger_{4}  a_{3'} \ . 
\eeq
Changing notation $3 \leftrightarrow 4$, one has to remember that $k_{34} = - k_{43}$. 



*** old version verified as correct
\beq
\varepsilon_{4}^* p_3 \es \varepsilon_{4}^*( p_{3'} - p_4 ) \rs \varepsilon_{4}^* p_{3'} \ . 
\eeq
\beq
\label{exampleY343'1}
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \left(
   \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}  p_3   
 - \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* p_{3'}
{\red +}  \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^* p_{3'} 
\right) \ a^\dagger_{3} a^\dagger_{4}  a_{3'} \ . 
\eeq
Now I use the usual relative momentum variables for components $+$ and $\perp$,
\beq
y_4 \es x_4/x_{3'} \ , \ y_3 \rs x_3/x_{3'} \ , \  k_{43} \rs y_3 p_4 - y_4 p_3 \ , \\
p_4 \es y_4 p_{3'} + k_{43} \ , \ p_3 \rs y_3 p_{3'} - k_{43} \ .
\eeq
In these variables,
\beq
\varepsilon_{3'}    p_3    \es \varepsilon_{3'}(y_3 p_{3'} - k_{43}) \rs - \varepsilon_{3'}k_{43} \ , \\
\varepsilon_{4}^* p_{3'} \es \2 \varepsilon_{4}^{*-} p_{3'}^+ -  \varepsilon_{4}^{*\perp} p_{3'}^\perp
\rs \2 2 \varepsilon_{4}^{*\perp }p_4^\perp/ y_4 -  \varepsilon_{4}^{*\perp} p_{3'}^\perp \\
\es  \varepsilon_{4}^{*\perp} ( y_4 p_{3'}^\perp + k_{43}^\perp ) / y_4 -  \varepsilon_{4}^{*\perp} p_{3'}^\perp
\rs   \varepsilon_{4}^{*\perp} k_{43}^\perp / y_4 \rs - \varepsilon_{4}^* k_{43}/ y_4 \ , \\
\varepsilon_{3}^* p_{3'} \es \2 2 \varepsilon_{3}^{*\perp }p_3^\perp/ y_3  -  \varepsilon_{3}^{*\perp} p_{3'}^\perp 
\rs \varepsilon_{3}^{*\perp }(y_3 p_{3'}^\perp - k_{43}^\perp)/ y_3  -  \varepsilon_{3}^{*\perp} p_{3'}^\perp \\
\es - \varepsilon_{3}^{*\perp } k_{43}^\perp)/ y_3 \rs \varepsilon_{3}^* k_{43}/ y_3 \ .
\eeq
\beq
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \left(
 - \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}k_{43}
 + \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* k_{43}/ y_4 
{\red +}  \varepsilon^*_4 \varepsilon_{3'}   \   \varepsilon_{3}^* k_{43}/ y_3
\right) \ a^\dagger_{3} a^\dagger_{4}  a_{3'} \ . 
\eeq
Changing notation $3 \leftrightarrow 4$, one has to remember that $k_{34} = - k_{43}$. }

****


\noindent {\blue In verified conclusion,
\beq
\cH_{A A} \es g \sum_{433'}  \int [433'] \ a^\dagger_{4} a^\dagger_{3}  a_{3'} \  Y_{433'} \, \tilde \delta_{43.3'} + h.c. \ , \\
Y_{433'} \es i f^{c_4 c_3 c_{3'}} \left(
   \varepsilon^*_4 \varepsilon^*_3   \  \varepsilon_{3'} k_{43}  
- \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^*k_{43}/y_3
 - \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^*k_{43}/y_4
  \right) \ ,
\eeq 
which matches Eqs.~(\ref{HA3k12}) and (\ref{Y123}). Also, changing notation $3 \leftrightarrow 4$, I obtain 
\beq
\label{exampleY343'}
Y_{343'} \es
    i f^{c_3 c_4 c_{3'}} \left(
   \varepsilon^*_3 \varepsilon^*_4  \  \varepsilon_{3'}  p_3                 
- \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* p_{3'}
{\red +}  \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^* p_{3'} 
\right) \ a^\dagger_{3} a^\dagger_{4}  a_{3'} \ ,
\eeq
whose equivalent is
\beq
Y_{433'} \es
    i f^{c_4 c_3 c_{3'}} \left(
   \varepsilon^*_4 \varepsilon^*_3  \  \varepsilon_{3'}  p_4                 
-  \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^* p_{3'}
{\red +} \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* p_{3'} 
\right) \ a^\dagger_{4} a^\dagger_{3}  a_{3'} \ . 
\eeq
Using
\beq
\varepsilon_{4}^* p_{3'} \es \2 \varepsilon_{4}^{*-} p_{3'}^+ -  \varepsilon_{4}^{*\perp} p_{3'}^\perp
\rs \2 2 \varepsilon_{4}^{*\perp }p_4^\perp/ y_4 -  \varepsilon_{4}^{*\perp} p_{3'}^\perp \\
\es  \varepsilon_{4}^{*\perp} ( y_4 p_{3'}^\perp + k_{43}^\perp ) / y_4 -  \varepsilon_{4}^{*\perp} p_{3'}^\perp
\rs   \varepsilon_{4}^{*\perp} k_{43}^\perp / y_4 \rs - \varepsilon_{4}^* k_{43}/ y_4 \ , \\
\varepsilon_{3}^* p_{3'} \es \2 2 \varepsilon_{3}^{*\perp }p_3^\perp/ y_3  -  \varepsilon_{3}^{*\perp} p_{3'}^\perp 
\rs \varepsilon_{3}^{*\perp }(y_3 p_{3'}^\perp - k_{43}^\perp)/ y_3  -  \varepsilon_{3}^{*\perp} p_{3'}^\perp \\
\es - \varepsilon_{3}^{*\perp } k_{43}^\perp)/ y_3 \rs \varepsilon_{3}^* k_{43}/ y_3 \ ,
\eeq
one obtains
\beq
Y_{433'} \es
    i f^{c_4 c_3 c_{3'}} \left[
   \varepsilon^*_4 \varepsilon^*_3  \  \varepsilon_{3'}  k_{43}             
-  \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^*  k_{43}/ y_3
{\red +} \varepsilon^*_3 \varepsilon_{3'}   \   (- \varepsilon_{4}^* k_{43}/ y_4)
\right] \ a^\dagger_{4} a^\dagger_{3}  a_{3'} \ ,
\eeq
or
\beq
Y_{433'} \es
    i f^{c_4 c_3 c_{3'}} \left(
   \varepsilon^*_4 \varepsilon^*_3  \  \varepsilon_{3'}  k_{43}             
-  \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^*  k_{43}/ y_3
{\red -} \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* k_{43}/ y_4
\right) \ a^\dagger_{4} a^\dagger_{3}  a_{3'} \ .
\eeq
Blue text above is verified. }

{\green \bf I check if a change in Eq.~(\ref{HA3k12}) that leads to $p_3^+ + p_{3'}^+$
agrees with the above blue and red results.} {\green

\beq
\label{HA3k12 orig}
H_{A^3} \es g 
                     \sum_{123}^2 \int [123] 
\, a_1^\dagger a_2^\dagger a_3 \ Y_{123} \, \tilde \delta_{12.3} + h.c. \ , \\
\label{Y123 orig}
Y_{123} \es i f^{c_1 c_2 c_3}
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right] 
\eeq
\beq
\label{HA3k12 check 1}
H_{A^3} \es g \sum_{343'} \int [343'] 
\, a_3^\dagger a_4^\dagger a_{3'} \ Y_{343'} \, \tilde \delta_{34.3'} + h.c. \ , \\
\label{Y123 check 1}
Y_{343'} \es i f^{c_3 c_4 c_{3'}}
\left[- \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} k_{43} 
+
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* k_{43}/y_4 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* k_{43}/y_3 \right]  \ . ~~~~~~~{\rm verified}
\eeq
Switch to momenta $p$, 
\beq
k_{43} \es (y_3 p_{4} - y_4 p_3) \ , \\
Y_{343'} \es i f^{c_3 c_4 c_{3'}}
\left[- \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'}  (y_3 p_{4} - y_4 p_3)
+
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* (y_3 p_{4} - y_4 p_3)/y_4 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* (y_3 p_{4} - y_4 p_3)/y_3 \right]  \\
\es
i f^{c_3 c_4 c_{3'}}
\left[- \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'}  (- p_3)
+
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* (- y_4 p_3)/y_4 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* (y_3 p_{4} )/y_3 \right] \\
\es
i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* p_3 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}  \right]  \\
\es
i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* p_{3'} 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}  \right]\ ,
\eeq
matching Eqs.~(\ref{exampleY343'}) and (\ref{exampleY343'1}) (remember $\varepsilon_4^* p_{3'}  = \varepsilon_4^* p_3 $). 
In the sums and integrals of all terms,  one can rename summed and integrated variables in each term of $Y$ independently. 
So, one can relabel $3 \leftrightarrow 4$ in the 3rd term. Creation operators retain their form, they only change order which is arbitrary anyway. Factor $i f^{c_3 c_4 c_{3'}}$ changes sign. The result is the same formula for $H_{A^3}$ with 
\beq
\label{HA343'} 
H_{A^3} \es 
g \sum_{343'} \int [343'] 
\, a_3^\dagger a_4^\dagger a_{3'} \ Y_{343'} \, \tilde \delta_{34.3'} + h.c. \ , \\
Y_{343'} \es 
i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* ( p_{3} + p_{3'} )  \right] \ .
\eeq
When 4 corresponds to the exchanges gluon, the only divergent part of  $H_{A^3}$ when $y_4 \to 0$ is 
\beq
\label{HA343'div4} 
H_{A^3}^{\rm div} \es 
g \sum_{343'} \int [343'] 
\, a_3^\dagger a_4^\dagger a_{3'} \ Y_{343'}^{\rm div} \, \tilde \delta_{34.3'} + h.c. \ , \\
Y_{343'}^{\rm div} \es 
- i f^{c_3 c_4 c_{3'}} \, \varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* ( p_{3} + p_{3'} ) \ .
\eeq
The divergence comes from $\epsilon_4^{*-}$. Now I can check the whole calculation of the gluon-quark interaction. 
The full expression for $H_{AA} = H_{A^3}$ can be written in two equivalent forms.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\bf Summary of the 3-transverse-gluons term}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
Equalities mean equivalence when all indices and momenta are fully summed and integrated, respectively. 
\beq
\cH_{A^3} \rs \cH_{A A} \es - \int_F  J_{A f}^{a \mu}  A_{f\mu}^a  \ ,  \\
{\green
\label{HA343'check3} 
H_{AA } }\rs {\green H_{A^3} } \es  
{\green g \sum_{343'} \int [343'] 
\, a_3^\dagger a_4^\dagger a_{3'} \ Y_{343'} \, \tilde \delta_{34.3'} 
+ 
{\red 
g \sum_{3'43} \int [3'43] 
\,a_{3'}^\dagger  a_4 a_3 \ Y_{343'}^* \, \tilde \delta_{3'.43} } } \\
\es
{\green g \sum_{343'} \int [343'] 
\, a_3^\dagger a_4^\dagger a_{3'} \ Y_{343'} \, \tilde \delta_{34.3'} 
+ 
{\red 
g \sum_{343'} \int [343'] 
\,a_{3}^\dagger  a_4 a_{3'} \ Y_{3'43}^* \, \tilde \delta_{3.43'} } } \ , \\
Y_{343'} \es i f^{c_3 c_4 c_{3'}}
\left[- \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} k_{43} 
+
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* k_{43}/y_4 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* k_{43}/y_3 \right]  \ , ~~~~ k_{43} \rs (y_3 p_{4} - y_4 p_3) \ , \\
Y_{343'} 
\es
i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* p_{3'} 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}  \right] \ , \\
\label{HA343'check4}
{\green Y_{343'} } \eqv
{\green i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3 
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* ( p_{3'} + p_{3} )  \right] } \ , \\
{\red Y_{3'43}^*} 
\es
 i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_4 p_{3} 
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_{4} p_{3'}  \right] \\
\eqv
i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3}^* \varepsilon_{3'}  \, \varepsilon_4 (p_{3} + p_{3'})  \right] \ .
\eeq
The identities above include the change $3 \leftrightarrow 4$ in the last terms of $Y$,
and from $\epsilon_4 p_4=0$ after adding $p_4$ to $p_3$ in the product with $\varepsilon_4$.
One can change notation under the sign of full sum and integral in each term separately.
\beq
{\red Y_{3'43}^*} \es 
\left\{ i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3
     -  \varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* p_{3'} 
     + \varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}  \right] ~~ 3 \leftrightarrow 3'  \right\}^* \\
% \es
% - i f^{c_{3'} c_4 c_3}
% \left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
%      -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_4 p_{3} 
%      + \varepsilon_4    \varepsilon_{3}^* \, \varepsilon_{3'} p_{4}  \right] \\
\es
 i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_4 p_{3} 
     + \varepsilon_4    \varepsilon_{3}^* \, \varepsilon_{3'} p_{4}  \right]
\eeq
Changing notation $4 \leftrightarrow 3'$ under signs of full sums and integrals in the last term alone, one gets
\beq
{\red Y_{3'43}^*} 
\es
 i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_4 p_{3} 
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_{4} p_{3'}  \right] \\
\es
i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3}^* \varepsilon_{3'}  \, \varepsilon_4 (p_{3} + p_{3'})  \right] \ .
\eeq

\beq
&& {\red \bf There \, was \, the \, error: \, confusion \, of \, p \, with \, k} \\
Y_{433'} \es i f^{c_4 c_3 c_{3'}} \left(
   \varepsilon^*_4 \varepsilon^*_3   \  \varepsilon_{3'} k_{43}  
- \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^*k_{43}{\red /y_3} 
 - \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^*k_{43}{\red /y_4} 
  \right) \\
\es  i f^{c_4 c_3 c_{3'}} \left(
   \varepsilon^*_4 \varepsilon^*_3  \  \varepsilon_{3'}  p_4                 
-  \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^* p_{3'}
{\red +} \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* p_{3'} 
\right) \ a^\dagger_{4} a^\dagger_{3}  a_{3'}\ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{ \bf End of the summary of the 3-transverse-gluons term}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
}



\vskip.1in
\centerline{\bf Evaluation of $\cH_{A \phi}$ }
\vskip.1in

\beq
\cH_{A \phi} \es - \int_F J_{A f}^{a \mu}  {m_g \eta_\mu  \over \partial^+} \phi^a \ , \\
J_{A f}^{a \mu} \es 
 \sum_{343'}  \int [343'] \ i g  f^{a c_3 c_{3'}}
p_3^\mu \left[  \varepsilon_3 a_{3} e^{-ip_3x} - \varepsilon^*_3 a^\dagger_{3} e^{ip_3x} \right]
                \left[  \varepsilon_{3'} a_{3'} e^{-ip_{3'}x} +\varepsilon^*_{3'} a^\dagger_{3'} e^{ip_{3'}x} \right] \ , \\
{m_g \eta_\mu  \over \partial^+} \phi^a 
\es \int [4]
 {m_g \eta_\mu  \over p_4^+} \left( \tilde a_{4a} e^{-ip_4x} + \tilde a^\dagger_{4a} e^{ip_4x} \right) \ , \\
\cH_{A \phi} \es 
- \int_F 
\sum_{343'}  \int [343'] \ i g  f^{c_4 c_3 c_{3'}}
p_3^\mu \left[  \varepsilon_3 a_{3} e^{-ip_3x} - \varepsilon^*_3 a^\dagger_{3} e^{ip_3x} \right]
                \left[  \varepsilon_{3'} a_{3'} e^{-ip_{3'}x} +\varepsilon^*_{3'} a^\dagger_{3'} e^{ip_{3'}x} \right]
\nt
 {m_g \eta_\mu  \over p_4^+} \left( \tilde a_{4} e^{-ip_4x} + \tilde a^\dagger_{4} e^{ip_4x} \right) 
\eeq
Calculation.
\beq
\cH_{A \phi} \es 
- \sum_{343'}  \int [343'] \, \tilde \delta_{c.a} \ i g  f^{c_4 c_3 c_{3'}} {m_g p_3^+ \over p_4^+} \,
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right) \ .
\eeq
Again, there are 8 - 2 = 6 terms.
\beq
&&
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right) \\
\es
112
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right)
\np
121
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right)
\np
122
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right)
\np
211
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right)
\np
212
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right)
\np
221
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right] \ .
\eeq
\beq
&&
112 \
  \varepsilon_3  
 \varepsilon_{3'} \ a_{3} a_{3'}  \tilde a^\dagger_{4}  
+
121 \
  \varepsilon_3 
 \varepsilon^*_{3'} \ a_{3}  a^\dagger_{3'} \tilde a_{4}
-
211 \
 \varepsilon^*_3  
  \varepsilon_{3'} \ a^\dagger_{3} a_{3'}   \tilde a_{4}  
\nm
221 \
\varepsilon^*_3 
 \varepsilon^*_{3'} \ a^\dagger_{3} a^\dagger_{3'}   \tilde a_{4}   
-
212 \
  \varepsilon^*_3  
  \varepsilon_{3'} \ a^\dagger_{3} a_{3'}   \tilde a^\dagger_{4} 
+
122 \
 \varepsilon_3
 \varepsilon^*_{3'} \ a_{3} a^\dagger_{3'}   \tilde a^\dagger_{4}  
\eeq
\beq
X \es
\left[  \varepsilon_3 a_{3} - \varepsilon^*_3 a^\dagger_{3} \right]
\left[  \varepsilon_{3'} a_{3'} +\varepsilon^*_{3'} a^\dagger_{3'}  \right] \left( \tilde a_{4}  + \tilde a^\dagger_{4}  \right) \\
\es
-
221 \
\varepsilon^*_3 
 \varepsilon^*_{3'} \ a^\dagger_{3} a^\dagger_{3'}   \tilde a_{4}   
-
212 \
  \varepsilon^*_3  
  \varepsilon_{3'} \ a^\dagger_{3} a_{3'}   \tilde a^\dagger_{4} 
+
122 \
 \varepsilon_3
 \varepsilon^*_{3'} \ a_{3} a^\dagger_{3'}   \tilde a^\dagger_{4}  - h.c. \ .
\eeq
Normal ordering to $\tilde a^\dagger_{4} a^\dagger_{3}  a_{3'}$.
\beq
\cH_{A \phi} \es g m_g
\sum_{433'}  \int [433'] \, \tilde \delta_{c.a} \ Z \ , \ Z \rs  f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+} \ (-i X) \ + \ h.c. \ , \\
Z \es   f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+} \ (-i X) \\
   \es
   221 f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+} \ i \, \varepsilon^*_3 \varepsilon^*_{3'} \ a^\dagger_{3} a^\dagger_{3'}  \tilde  a_{4}   
+212 f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+}  \ i \, \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'}    
- 122 f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+} \ i \, \varepsilon_3     \varepsilon^*_{3'} \ \tilde a^\dagger_{4}  a^\dagger_{3'}  a_{3}   \ .
\eeq
Here, I am interested in terms that do not change the number of transverse gluons,
\beq
Z \tl   f^{c_4 c_3 c_{3'}} { p_3^+ \over p_4^+} \ (-i X) \\
   \es
212 f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+}  \ i \, \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'}    
- 122 f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+} \ i \, \varepsilon_3     \varepsilon^*_{3'} \ \tilde a^\dagger_{4}  a^\dagger_{3'}  a_{3}   \ .
\eeq
I change notation in the second term, $3 \leftrightarrow 3'$.
\beq
Z \tl   f^{c_4 c_3 c_{3'}} { p_3^+ \over p_4^+} \ (-i X) \\
   \es
f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+}  \ i \, \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'}    
- f^{c_4 c_{3'} c_3} {p_{3'}^+ \over p_4^+} \ i \, \varepsilon_{3'}     \varepsilon^*_{3} \ \tilde a^\dagger_{4}  a^\dagger_{3}  a_{3'} \\
\es 
f^{c_4 c_3 c_{3'}} {p_3^+ \over p_4^+}  \ i \, \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'}    
+ f^{c_4  c_3 c_{3'}} {p_{3'}^+ \over p_4^+} \ i \, \varepsilon^*_{3} \varepsilon_{3'}     \varepsilon^*_{3} \ \tilde a^\dagger_{4}  a^\dagger_{3}  a_{3'}  \\
\es 
f^{c_4 c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_4^+}  \ i \, \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'}    
\ .
\eeq
{\blue The final result reads
\beq
\cH_{A \phi} \es g m_g
\sum_{433'}  \int [433'] \, \tilde \delta_{c.a} \ i \, f^{c_4 c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'}    \  + \ h.c. \ ,
\eeq}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\bf Summary of terms for gluon-quark interaction in the 3-body space}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Integration symbols include sums.
\beq
\cH^{(1)}_0 \eqv
- \int_F \left( J_{\psi f}^{a \mu} + J_{A f}^{a \mu} \right) \left( A_{f\mu}^a + {m_g \eta_\mu  \over \partial^+} \phi^a \right) 
\rs
\cH_{\psi A} + \cH_{\psi \phi} + \cH_{A A} + \cH_{A \phi} \ , \\
\cH_{\psi A} \rs - \int_F J_{\psi f}^{a \mu}   A_{f\mu}^a  
\es
g
\int[11'4] \, \tilde \delta_{c.a} \, f^r_{\overline 1, \underline 1 4} 
\ \chi_1^\dagger  T^4 \chi_{1'}  \ \bar u_1 \gamma_\nu u_{1'}       
\left(  \varepsilon^\nu_4 \ b_1^\dagger  b_{1'}  a_4 + \varepsilon^{\nu *}_4  a^\dagger_4 b_1^\dagger  b_{1'}  \right)
\nm
g
\int[22'4] \, \tilde \delta_{c.a} \, f^r_{\overline 2, \underline 2 4} 
\ \chi_{2'}^\dagger T^4 \chi_2 \ \bar v_{2'} \gamma_\nu v_2 \  
\left(  \varepsilon^\nu_4 \ d^\dagger_2 d_{2'} a_4 + \varepsilon^{\nu *}_4  \ a^\dagger_4 d^\dagger_2 d_{2'} \right) \ , \\
\cH_{\psi \phi} \rs - \int_F J_{\psi f}^{a \mu}  {m_g \eta_\mu  \over \partial^+} \phi^a 
\es
g
\int[11'\tilde 4] \, \tilde \delta_{c.a} \, f^r_{\overline 1, \underline 1 \tilde 4} 
\ \chi_1^\dagger T^{\tilde 4} \chi_{1'} 
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g  \over p_{\tilde 4}^+}
\left( b_1^\dagger b_{1'}  a_{\tilde 4} + a^\dagger_{\tilde 4} b_1^\dagger b_{1'}  \right) 
\nm
g
\int[22'\tilde 4] \, \tilde \delta_{c.a} \, f^r_{\overline 2, \underline 2 \tilde 4}  
\ \chi_{2'}^\dagger T^{\tilde 4} \chi_2
\ \bar v_{2'}    \gamma^+ v_2
\ {m_g  \over p_{\tilde 4}^+}
\left( d^\dagger_2d_{2'}  a_{\tilde 4} + a^\dagger_{\tilde 4}d^\dagger_2d_{2'}  \right) \ , \\
% ***
\cH_{A^3} \rs \cH_{A A} \es - \int_F  J_{A f}^{a \mu}  A_{f\mu}^a  \ ,  \\
{\green
\label{HA343'check3summaryfinal} 
H_{AA } }\rs {\green H_{A^3} } \es  
{\green g \sum_{343'} \int [343'] 
\, a_3^\dagger a_4^\dagger a_{3'} \ Y_{343'} \, \tilde \delta_{34.3'} 
+ 
{\red 
g \sum_{3'43} \int [3'43] 
\,a_{3'}^\dagger  a_4 a_3 \ Y_{343'}^* \, \tilde \delta_{3'.43} } } \\
\es
{\green g \sum_{343'} \int [343'] 
\, a_3^\dagger a_4^\dagger a_{3'} \ Y_{343'} \, \tilde \delta_{34.3'} 
+ 
{\red 
g \sum_{343'} \int [343'] 
\,a_{3}^\dagger  a_4 a_{3'} \ Y_{3'43}^* \, \tilde \delta_{3.43'} } } \ , \\
Y_{343'} \es i f^{c_3 c_4 c_{3'}}
\left[- \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} k_{43} 
+
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* k_{43}/y_4 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* k_{43}/y_3 \right]  \ , \ k_{43} \rs (y_3 p_{4} - y_4 p_3) \ , \\
Y_{343'} 
\es
i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* p_{3'} 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}  \right] \ , \\
\label{HA343'check4}
{\green Y_{343'} } \eqv
{\green i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3 
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* ( p_{3'} + p_{3} )  \right] } \ , \\
{\red Y_{3'43}^*} 
\es
 i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_4 p_{3} 
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_{4} p_{3'}  \right] \\
\eqv
i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3}^* \varepsilon_{3'}  \, \varepsilon_4 (p_{3} + p_{3'})  \right] \ , \\
% ****
\cH_{A \phi}  \rs - \int_F J_{A f}^{a \mu}  {m_g \eta_\mu  \over \partial^+} \phi^a 
\es g m_g
\sum_{433'}  \int [433'] \, \tilde \delta_{c.a} \  i f^{c_4 c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'}    \  + \ h.c. \, , \
{\rm cons. \, \# \perp.} 
\eeq
For gluon-quark interaction in the 3-body space, without gluon-antiquark interaction,
\beq
\cH^{(1)}_0 \es \cH_{\psi A} + \cH_{\psi \phi} + \cH_{A A} + \cH_{A \phi} 
\ , \\
\cH_{\psi A} \es g \int[11'4] \, \tilde \delta_{c.a} \, f^r_{\overline 1, \underline 1 4} 
\ \chi_1^\dagger  T^4 \chi_{1'}  \ \bar u_1 \gamma_\nu u_{1'}       
\left(  \varepsilon^\nu_4 \ b_1^\dagger  b_{1'}  a_4 + \varepsilon^{\nu *}_4  a^\dagger_4 b_1^\dagger  b_{1'}  \right)
\ , \\
\cH_{\psi \phi} \es g \int[11'\tilde 4] \, \tilde \delta_{c.a} \, f^r_{\overline 1, \underline 1 \tilde 4} 
\ \chi_1^\dagger T^{\tilde 4} \chi_{1'} 
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g  \over p_{\tilde 4}^+}
\left( b_1^\dagger b_{1'}  a_{\tilde 4} + a^\dagger_{\tilde 4} b_1^\dagger b_{1'}  \right) 
 \ , \\
H_{AA } \es  
{\green g \sum_{343'} \int [343'] 
\, a_3^\dagger a_4^\dagger a_{3'} \ Y_{343'} \, \tilde \delta_{34.3'} 
+ 
{\red 
g \sum_{343'} \int [343'] 
\,a_{3}^\dagger  a_4 a_{3'} \ Y_{3'43}^* \, \tilde \delta_{3.43'} } } \ , \\
{\green Y_{343'}} 
\es
i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* p_{3'} 
+
\varepsilon_4^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}  \right] \ , \\
\label{HA343'check4}
\eqv
i f^{c_3 c_4 c_{3'}}
\left[ \varepsilon_3^* \varepsilon_4^* \,\varepsilon_{3'} p_3 
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_4^* ( p_{3'} + p_{3} )  \right]  \ , \\
{\red Y_{3'43}^*}
\es
 i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_4 p_{3} 
     -  \varepsilon_{3'} \varepsilon_{3}^* \, \varepsilon_{4} p_{3'}  \right] \\
\eqv
i f^{ c_3 c_4 c_{3'} }
\left[ \varepsilon_{3'} \varepsilon_4 \,\varepsilon_{3}^* p_{3'}
     -  \varepsilon_{3}^* \varepsilon_{3'}  \, \varepsilon_4 (p_{3} + p_{3'})  \right] \ , \\
%%
\cH_{A \phi}  \es g m_g \int [433'] \, \tilde \delta_{c.a} \  
i f^{c_4 c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'}    \  + \ h.c. \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\bf  Exchange with $\theta_{1-1'}$ using file ``FirstLine20250927''}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf In conclusion from ``FirstLine20250927'':}
\beq
\label{L1final}
L1 \es g^2 \sum_4 \int[11'33'] \,  \theta_{1-1'} \, { \tilde \delta_{13.1'3'} \over p_4} 
\, \chi_1^\dagger  T^4 \chi_{1'} \,  \bar u_1 \gamma_\nu u_{1'} \ i f^{c_{3} c_4 c_{3'}} 
\ Y_{131'3'} \ b_1^\dagger  b_{1'} \, a_{3}^\dagger a_{3'}  \ , \\
Y_{131'3'} \es 2 \varepsilon^\nu_4 
\left[ \varepsilon_3^* \varepsilon_{4}^* \,\varepsilon_{3'} p_3
+
\varepsilon_{4}^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_{4}^* p_{3'} \right] \ .
\eeq
$L1$ is the expression obtained for 
\beq
L1 \es g^2
\int[11'4] \, \tilde \delta_{1.1'4} 
\ \chi_1^\dagger  T^4 \chi_{1'}  \ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 \ b_1^\dagger  b_{1'}  a_4 
\ \
\int [34'3'] \, a_3^\dagger a_{4'}^\dagger a_{3'} \ {\green  Y_{34'3'}} \, \tilde \delta_{34'.3'} \\
\es g^2
\int[11'4] \, \tilde \delta_{1.1'4} \theta_{1-1'}
\ \chi_1^\dagger  T^4 \chi_{1'}  \ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 
\nt
\int [34'3'] \, \tilde \delta_{34'.3'}  \theta_{3'-3} 
\ i f^{c_3 c_{4'} c_{3'}}
\left[ \varepsilon_3^* \varepsilon_{4'}^* \,\varepsilon_{3'} p_3
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_{4'}^* p_{3'} 
+
\varepsilon_{4'}^*\varepsilon_{3'} \, \varepsilon_{3}^* p_{4'}  \right] 
\ b_1^\dagger  b_{1'}  a_4 \ a_3^\dagger a_{4'}^\dagger a_{3'} \ . 
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf  Exchange of $\phi$ with $\theta_{1-1'}$ from Eq.~(\ref{L3orig}) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beq
\label{L3origCopy}
L3 \es g^2 
\int[11'4] \, \tilde \delta_{1.1'4} 
\ \chi_1^\dagger T^{4} \chi_{1'} 
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g  \over p_{4}^+} \ b_1^\dagger b_{1'} \tilde a_{ 4} 
\ \
m_g  \int [4'33'] \, \tilde \delta_{4'3.3'} \  i f^{c_{4'} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4'}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4'} a^\dagger_{3}    a_{3'}  \ .
\eeq
Commuting and integration over 4.
\beq
\label{L3-1}
L3 \es g^2
\int[11'433'] \, \tilde \delta_{1.1'4} \theta_{1-1'}
\ \chi_1^\dagger T^{4} \chi_{1'} 
\ \bar u_1 \gamma_\nu  u_{1'} 
\ {m_g  \eta^\nu \over p_{4}^+} \ b_1^\dagger b_{1'} 
\ \
m_g \, \tilde \delta_{43.3'} \  i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'} \  a^\dagger_{3}    a_{3'}  \\ 
\label{L3final}
 \es g^2
\int[11'33'] \, \theta_{1-1'} \, { \tilde \delta_{13.1'3'} \over p_4^+} 
\ \chi_1^\dagger T^{4} \chi_{1'} 
\ \bar u_1 \gamma_\nu u_{1'} 
\ {m_g^2 \eta^\nu \over p_{4}^{+2}}  
\ \ 
i f^{c_{4} c_3 c_{3'}} (p_3^+ + p_{3'}^+)  \ \varepsilon^*_3 \varepsilon_{3'}   \ b_1^\dagger b_{1'}   \, a^\dagger_{3}    a_{3'}  \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf  Exchange of $A$ and $\phi$ with $\theta_{1-1'}$ together with the seagull}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The exchanges of $A$ and $\phi$ work with $\Delta_{LIR}$ of Eq.~(\ref{Deltaqg}),
\beq
\label{Deltaqgsummary}
{1 \over p_4^+}\Delta_{LIR} \es 
{1 \over 2p_4^+}  \left( {1 \over p_3^- + p_1^- - P_{\rm three}^- } +  {1 \over p_{3'}^- + p_{1'}^- - P_{\rm three}^- }\right) 
\rs - \, \2 \left( {1 \over \rho_1} + {1 \over \rho_3 } \right) \ .
\eeq
The exchanged gluon $p_4^-$ in $P_{\rm three}^-$, 
\beq 
p_4^- \es (p_4^{\perp 2} + m_g^2)/p_4^+ \ ,
\eeq
is the largest denominator term when $p_4^+$ tends to 0. This implies 
\beq
{1 \over p_4^+}\Delta_{LIR} \tl {-1 \over p_4^{\perp 2} + m_g^2} \ .
\eeq
From Eqs.~(\ref{L1final}), (\ref{L3final}), and (\ref{Hsqg2}), {\red Start here 20250928 20:29 nie San Dimas}
\beq
\label{L1final}
L1 \es g^2 \sum_4 \int[11'33'] \,  \theta_{1-1'} \, { \tilde \delta_{13.1'3'} \over p_4} 
\, \chi_1^\dagger  T^4 \chi_{1'} \,  \bar u_1 \gamma_\nu u_{1'} \ i f^{c_{3} c_4 c_{3'}} 
\ Y_{131'3'} \ b_1^\dagger  b_{1'} \, a_{3}^\dagger a_{3'}  \ , \\
Y_{131'3'} \es 2 \varepsilon^\nu_4 
\left[ \varepsilon_3^* \varepsilon_{4}^* \,\varepsilon_{3'} p_3
+
\varepsilon_{4}^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_{4}^* p_{3'} \right] \ .
\eeq
\beq
L3 \es g^2
\int[11'33'] \, \theta_{1-1'} \, { \tilde \delta_{13.1'3'} \over p_4^+} 
\ \chi_1^\dagger T^{4} \chi_{1'} 
\ \bar u_1 \gamma_\nu u_{1'} 
\ {m_g^2 \eta^\nu \over p_{4}^{+2}}  
\ \ 
i f^{c_{4} c_3 c_{3'}} (p_3^+ + p_{3'}^+)  \ \varepsilon^*_3 \varepsilon_{3'}   \ b_1^\dagger b_{1'}   \, a^\dagger_{3}    a_{3'}  \ ,
\eeq
\beq
\label{Hsqg2 POM1}
H_{sqg} \es -g^2 \int [33'11'] \ \tilde \delta_{c.a} \ \delta_{\rm spins} 
\
{ ( p_3^+ + p_{3'}^+ )  \ \bar u_1 \gamma_\nu u_{1'} \eta^\nu  \over p_4^{+ 2}} 
\ \chi_{1}^\dagger T^{c_4} \chi_{1'} \, i f^{c_3 c_{3'} c_4} \    a^\dagger_{3} b_1^\dagger  b_{1'}  a_{3'} \ .
\eeq
Regularization factors $\ f^r_{\overline 1,\underline 1 (\overline 1 - \underline 1)} 
\ f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} $ are omitted. Only the part with $\theta_{1-1'}$ is
taken from the seagull. Collection of common factors yields
\beq
CF \es
g^2 \sum_4 \int[11'33'] \,  \theta_{1-1'} \, { \tilde \delta_{13.1'3'} \over p_4} 
\, \chi_1^\dagger  T^4 \chi_{1'} \,  \bar u_1 \gamma_\nu u_{1'} \ i f^{c_{3} c_4 c_{3'}} \ a^\dagger_{3} b_1^\dagger  b_{1'}  a_{3'} \ .
\eeq
For the 3-body Hamiltonian, Eqs.~(\ref{cH2t} ) or (\ref{cH2t3body}) imply
\beq
\label{cH2t Lines} 
\cH^{(2)} \es f_{LR} \, \cH^{(2)}_0 + (f_{LR} - f_{LI} f_{IR}) \ \Delta_{LIR} \ \cH^{(1)}_0 \, \cH^{(1)}_0 \ .
\eeq
Only the terms with $f_{LR}$ involve the gluon severe small-$x$ divergences. These divergences appear in the same way
with $\theta_{1-1'}$ and with $\theta_{1'-1}$. For $\theta_{1-1'}$, I have the  formula
\beq
\cH^{(2)} \es f_{LR} CF \left[ {1\over p_4^+}\Delta_{LIR} ( L_1 + L_3 ) + L_s  \right] \ .
\eeq
where $L_1$, $L_3$, and $L_s$ denote the remaining factors of the lines 1, 2, and the seagull, respectively.
\beq
L_1 \es  2 \sum_4 \varepsilon^\nu_4 
\left[ \varepsilon_3^* \varepsilon_{4}^* \,\varepsilon_{3'} p_3 + \varepsilon_{4}^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}
- \varepsilon_3^*\varepsilon_{3'} \, \varepsilon_{4}^* p_{3'} \right]  \ , \\
L_3 \es 
{m_g^2 \eta^\nu \over p_{4}^{+2}}  
\ (-1)_{\rm color}  (p_3^+ + p_{3'}^+)  \ \varepsilon^*_3 \varepsilon_{3'}  \ , \\
L_s \es
- [ \delta_{\rm spins} = - \varepsilon^*_3 \varepsilon_{3'} ] \
{ p_3^+ + p_{3'}^+  \over p_4^{+ 2}} \ \eta^\nu \ (-1)_{\rm color} \ .
\eeq
I need to see if the gluon severe small-$x$ singularity $1/p_4^{+2}$ cancels out. The only part of $L_1$ that counts is the third one,
\beq
L_1^{\rm div} \es 
- 2 \varepsilon_3^*\varepsilon_{3'}  \sum_4 \varepsilon^\nu_4  \, \varepsilon_{4}^* p_{3'} \\
\eqv 
- \varepsilon_3^*\varepsilon_{3'}  \sum_4 \varepsilon^\nu_4  \, \varepsilon_{4}^{*\mu}  (p_{3'} + p_3)_\mu \ . 
\eeq
The equivalence is a consequence of $\varepsilon_{4}^* p_{3'} =  \varepsilon_{4}^*  p_3$. The divergence originates solely
from the minus component of $\varepsilon_4$. I write
\beq
\varepsilon_4 \es \2 \eta \varepsilon_4^- + \varepsilon_4^\perp \ .
\eeq
The transverse part does not involve small-$p_4^+$. The singularity is contained in $\varepsilon_4^- = 
2 \varepsilon_4^\perp p_4^\perp/p_4^+$. Therefore, the singular part of $L_1$ has the form
\beq
L_1^{\rm div} \es 
- \varepsilon_3^*\varepsilon_{3'}  \sum_4 
(p_4^\perp \varepsilon^\perp _4/p_4^+) (\varepsilon_{4}^{*\perp}p_4^\perp/p_4^+)  \eta^\nu \eta^ \mu  (p_{3'} + p_3)_\mu \\
\es
- \varepsilon_3^*\varepsilon_{3'} \ { p_4^{\perp 2} \over p_4^{+2} } \, \eta^\nu \,  (p_{3'}^+ + p_3^+) \ .
\eeq
All terms with singularity $1/p_4^{+2}$ thus are 
\beq
\cH^{(2)}_{\rm div} \es f_{LR} CF 
\left\{  {-1 \over p_4^{\perp 2} + m_g^2}  \left[ \left( p_4^{\perp 2} \right)_{L_1} + \left( m_g^2 \right)_{L_3} \right] 
+ \left( 1 \right)_{L_s}  \right\} (- \varepsilon_3^*\varepsilon_{3'}) 
\, \eta^\nu \, {p_{3'}^+ + p_3^+ \over p_4^{+2} } \ .
\eeq
The curly bracket amounts to 0, which ends my struggle with massive gluons. {\bf 20250929 00:12 pon San Dimas}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Gluon-quark interaction dependence on $m_g \to 0$ in 3-body space 20251002 10:12}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Hamiltonian terms to analyze are
\beq
\label{H3body 1}
_s\langle 123| H_s | 1'2'3' \rangle_s
\es \langle 123|  \left\{ {\cH_f} + f_{LR} \, \cH_r  - f_{LI} f_{IR} \,  \Delta_{LIR} \, \cH_0^{(1)} \cH_0^{(1)} \right\}  | 1'2'3' \rangle \ , \\
\label{cH3body 2}
\cH_r \es \cH^{(2)}_0 + \Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0
\ .
\eeq
The terms that matter come only from $ - f_{LI} f_{IR} \,  \Delta_{LIR} \, \cH_0^{(1)} \cH_0^{(1)}$. They are finite for finite $m_g$.
I consider gluon-quark interaction and quark and gluon self-interactions. The self interactions are the same as for free particles.
The interactions include the RGPEP factors $f^t$. 

I consider two pairs of interaction terms. One pair is the interaction of quark 1 with gluon 3 involving $\theta_{1-1'}$ and quark 1 self-interaction. The other pair is the interaction of quark 1 with gluon 3 involving $\theta_{1'-1}$ and the gluon self-interaction. 
Color factors for these interaction terms are found in Eqs.~(\ref{colorfactors3body1}) to (\ref{colorfactors3body2}).
\begin{figure}[ht!]
          \caption{Interactions that depend logarithmically on $m_g$ in the 3-body space. In this figure, all vertices
include factors $f^t$ making all expressions finite but dependent on $s$, quark masses, $m_1$ and $m_2$, and 
the gluon mass $m_g$. The question is what expressions one obtains in the limit $m_g \to 0$.}
          \label{fig:selfexch3body}
\begin{center}
         \includegraphics[width=0.3\textwidth]{3-bodyGQbarQinteractions}
\end{center} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\bf Gluon-quark interaction in the 3-body space, new = 20251011 12:06 sob Ekologiczna}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Color factors} follow from the wave function
\beq
\label{colorfactors3body1}
C_{\rm 3body}^{123} \es \chi^\dagger_1 T^3 \chi_2 \ .
\eeq
\begin{itemize}
\item
{\bf -3/2} The quark-gluon seagull $H_{sqg}$, Eq.~(\ref{Hsqg2}), 
\beq
\chi_1^\dagger T^4 \chi_{1'}  i f^{343'} \  \chi_{1'}^\dagger T^{3'} \chi_2
\es i f^{343'} \chi_1^\dagger T^4  T^{3'} \chi_2
\rs {i \over 2}  f^{343'} \chi_1^\dagger [ T^4 , T^{3'} ] \chi_2
\rs {i \over 2}  f^{343'} if^{43'a} \ \chi_1^\dagger T^a \chi_2 \\
\es - \2  f^{343'} f^{43'a} \ \chi_1^\dagger T^a \chi_2
\rs - \2 N_c \ \chi_1^\dagger T^3 \chi_2 \rs - {3 \over 2} \ \chi_1^\dagger T^3 \chi_2 
\rs - {3 \over 2} \  C_{\rm 3body}^{123} \ . \nn
\eeq
\item
{\bf -3/2} The antiquark-gluon seagull $H_{s \bar qg}$, Eq.~(\ref{Hsantiqg2})
\beq
\chi_{2'}^\dagger T^4 \chi_2 (- i f^{343'} ) \  \chi_{1}^\dagger T^{3'} \chi_{2'} 
\es
- i f^{343'} \ \chi_{2'}^\dagger T^4\chi_2 \  \chi_{1}^\dagger T^{3'} \chi_{2'}
\rs
- i f^{343'} \ \chi_{1}^\dagger T^{3'}  T^4 \chi_2
\rs
- {i \over 2} f^{343'} i f^{3'4a} \ \chi_{1}^\dagger T^a \chi_2 \\
\es
\label{colorfactors3body2}
\2 f^{343'}  f^{3'4a} \ \chi_{1}^\dagger T^4 \chi_2 
\rs
-\2 f^{33'4}  f^{3'4a} \ \chi_{1}^\dagger T^a \chi_2
\rs
- {N_c \over 2} \ \chi_{1}^\dagger T^3 \chi_2 \rs - {3 \over 2} \  C_{\rm 3body}^{123} \ . \nn
\eeq
Multiplication by $-N/2$. The strength is negative and slightly increased from 4/3 to 3/2, by the factor 9/8.
\item
{\bf -3/2} Quark-gluon exchange of a gluon, from Eq.~(\ref{L1final}),
\beq
\label{L1final color}
L1 \ C_{\rm 3body}^{123} \es g^2 \sum_4 \int[11'33'] \,  \theta_{1-1'} \, { \tilde \delta_{13.1'3'} \over p_4} 
\, \chi_1^\dagger  T^4 \chi_{1'} \,  \bar u_1 \gamma_\nu u_{1'} \ i f^{c_{3} c_4 c_{3'}} 
\ Y_{131'3'} \ b_1^\dagger  b_{1'} \, a_{3}^\dagger a_{3'} \ , \\
Y_{131'3'} \es 2 \varepsilon^\nu_4 
\left[ \varepsilon_3^* \varepsilon_{4}^* \,\varepsilon_{3'} p_3
+
\varepsilon_{4}^*\varepsilon_{3'} \, \varepsilon_3^* p_{4}
-
\varepsilon_3^*\varepsilon_{3'} \, \varepsilon_{4}^* p_{3'} \right] \ , \\
&& \chi_1^\dagger  T^4 \chi_{1'} \ i f^{343'}  \chi_{1'}^\dagger T^{3'} \chi_2 \rs - {3 \over 2} \  C_{\rm 3body}^{123}  \ .
 \eeq

\item
{\bf -3/2} Antiquark-gluon exchange of a gluon, {\it a la} Eq.~(\ref{L1final}),
The same is then true for the gluon exchange between an antiquark and a gluon. 

\item
{\bf 4/3}  Quark self-interaction is the same 4/3 for the quark and antiquark self-interactions. 

\item
{\bf 4/3}  Antiquark self-interaction.

\item
{\bf 3}  for the gluon self-interaction is $N=3$, see App.~\ref{GluonSelfComputation}, especially its part~\ref{gluonselfinteractionApp}, and Eq.~(\ref{GluonSelfA+phi}). 
I remind myself that in computing the term with $f_{LR}$, after inclusion of the 2nd-order simultaneous counterterm 
to severe small-$x$ and quadratic ultraviolet divergences, the remaining divergence is only logarithmic, while the
remaining gluon self-interaction result is  proportional to the gluon mass squared, $m_g^2$. In the limit $m_g \to 0$
the remaining, logarithmic gluon self-interaction multiplied by $m_g^2$ tends to 0.

In the effective gluon self-interaction with factors $f^t$ instead of $f^r$, there are no counterterms to include 
and the entire gluon self-interaction needs to be accounted for. I need to compare factors in the gluon-exchange 
between quark and gluon and the gluon self-interaction factors without any counterterms. The interaction is 
finite for $m_g > 0$. 

\item
{\bf -1/6} Gluon exchange between the quark and antiquark. 
\beq
\chi_1^\dagger T^4 \chi_{1'}  \  \chi_{1'}^\dagger T^{3'} \chi_{2'} \ \chi_{2'}^\dagger T^{4} \chi_{2}
\es 
\chi_1^\dagger T^4 \, T^{3'} \, T^{4} \chi_{2} 
\rs
\chi_1^\dagger \left(  [T ^4 , T^{3'}] \, T^{4} + T^{3'} T^{4}T^{4} \right) \chi_{2} \\
\es
\chi_1^\dagger \left(  i f^{43'a} T^a T^{4} + T^{3'} T^{4}T^{4} \right) \chi_{2} \\
\es \chi_1^\dagger \left( \2 i f^{43'a} [ T^a , T^{4}] + T^{3'} (N^2-1)/(2N) \right) \chi_{2} \\
\es \chi_1^\dagger \left( \2 i f^{43'a} i f^{a4b} T^b + T^{3'} (N^2-1)/(2N) \right) \chi_{2} \\
\es \chi_1^\dagger \left( - \2 N \delta^{3'b}T^b + T^{3'} (N^2-1)/(2N) \right) \chi_{2}  \\
\es  \left( - \2 N  +  (N^2-1)/(2N) \right) \ \chi_1^\dagger T^{3'}  \chi_{2} 
\rs  -{1 \over 2N} \ \chi_1^\dagger T^{3'}  \chi_{2} \\
\es -{1 \over 6} \ \chi_1^\dagger T^{3'}  \chi_{2}  \rs -{1 \over 6} \  C_{\rm 3body}^{123}   \ .
\eeq

\end{itemize}

{\bf Summary of color factors in the colorless 3-body space }
\beq
C_{Q\bar Q \rm exch} \es C_{Q\bar Q \rm seag} \rs -1/6 \ , \ C_{QG \rm exch} \rs  C_{QG \rm seag} \rs -3/2 \ , \\
C_{\Sigma_Q} \es C_{\Sigma_{\bar Q}} \rs 4/3 \ , \ C_{\Sigma_G} = 3
\eeq



{\bf Vertex change} from quark-gluon to quark or gluon self-interaction.






\beq
\chi_1^\dagger T^4 \chi_{1'}  i f^{343'} \  \chi_{1'}^\dagger T^{3'} \chi_2 
\tl 
\eeq



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\bf Gluon-quark interaction in the 3-body space, old}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The seagull of Eq.~(\ref{Hsqg2}) contributes 
\beq
\label{Hsqg2 POM1}
H_{sqg} \es -g^2 \int [33'11'] \ \tilde \delta_{c.a} \ \delta_{\rm spins} 
\ f^r_{\overline 1,\underline 1 (\overline 1 - \underline 1)} 
\ f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)} 
\
{ ( p_3^+ + p_{3'}^+ )  \ 2 \sqrt{p_1^+ p_{1'}^+}  \over (p_{3'}^+ - p_3^+)^2} 
\ \chi_{1}^\dagger T^{c_4} \chi_{1'} \, i f^{c_3 c_{3'} c_4} \    a^\dagger_{3} b_1^\dagger  b_{1'}  a_{3'} \ ,
\eeq
The 2nd-order gluon-quark interaction contained in the gluon-exchange product 
\beq
\cH^{(1)}_0 \cH^{(1)}_0 
\es  \left( \cH_{\psi A} + \cH_{\psi \phi} + \cH_{A A} + \cH_{A \phi} \right)
       \left( \cH_{\psi A} + \cH_{\psi \phi} + \cH_{A A} + \cH_{A \phi} \right) \ .
\eeq
The terms that count must be products that involve $\psi$ only in one $\cH^{(1)}_0$. This condition is satisfied 
only by the terms
\beq
\left( \cH_{\psi A} + \cH_{\psi \phi} \right) \left(\cH_{A A} + \cH_{A \phi} \right) 
+      
\left(\cH_{A A} + \cH_{A \phi} \right)  \left( \cH_{\psi A} + \cH_{\psi \phi} \right) \ .
\eeq
If there is $\phi$ in one factor, there must be $\phi$ in the other. This condition allows only for 
\beq
\cH_{\psi A} \cH_{A A} + \cH_{A A} \cH_{\psi A} + \cH_{\psi \phi}  \cH_{A \phi}  + \cH_{A \phi} \cH_{\psi \phi}  \ .
\eeq
Each of the Hamiltonian terms is a sum of two terms, in the forms
\beq
\cH_{\psi A} \es ( \cH_{\psi a}  + \cH_{\psi a^\dagger} ) \ , \\
\cH_{\psi \phi} \es ( \cH_{\psi \tilde a} + \cH_{\psi \tilde a^\dagger} ) \ , \\
\cH_{A A} \es ( \cH_{A a^\dagger a^\dagger a} +  \cH_{A a^\dagger a a} ) \ , \\
\cH_{A \phi}  \es ( \cH_{A \tilde a} + \cH_{A \tilde a^\dagger} ) \ ,
\eeq
where in each case the 2nd term is the h.c. of the first. The terms that count are
\beq
&&
    ( \cH_{\psi a}  + \cH_{\psi a^\dagger} )( \cH_{A a^\dagger a^\dagger a} +  \cH_{A a^\dagger a a} ) 
+ ( \cH_{A a^\dagger a^\dagger a} +  \cH_{A a^\dagger a a} ) ( \cH_{\psi a}  + \cH_{\psi a^\dagger} )
\np
    ( \cH_{\psi \tilde a} + \cH_{\psi \tilde a^\dagger} ) ( \cH_{A \tilde a} + \cH_{A \tilde a^\dagger} ) 
+  ( \cH_{A \tilde a} + \cH_{A \tilde a^\dagger} ) ( \cH_{\psi \tilde a} + \cH_{\psi \tilde a^\dagger} ) \ .
\eeq
The gluon exchange interaction results from contraction of $a$ and $a^\dagger$ that leaves behind
$a^\dagger a$, just 4 terms.
\beq
    \cH_{\psi a} \cH_{A a^\dagger a^\dagger a}  + \cH_{A a^\dagger a a} \cH_{\psi a^\dagger} 
+    \cH_{\psi \tilde a}  \cH_{A \tilde a^\dagger} +  \cH_{A \tilde a}   \cH_{\psi \tilde a^\dagger} \ .
\eeq
I omit the regularization factors $f^r$. I need to write all the contributing terms  explicitly, as $g^2$ times 
\beq
&&
\int[11'4] \, \tilde \delta_{1.1'4} 
\ \chi_1^\dagger  T^4 \chi_{1'}  \ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 \ b_1^\dagger  b_{1'}  a_4 
\ \
\int [34'3'] \, a_3^\dagger a_{4'}^\dagger a_{3'} \ Y_{34'3'} \, \tilde \delta_{34'.3'} 
\np
\int [343'] \,a_{3}^\dagger  a_4 a_{3'} \ Y_{3'43}^* \, \tilde \delta_{3.43'}
\ \
\int[11'4'] \, \tilde \delta_{4'1.1'} 
\ \chi_1^\dagger  T^{4'} \chi_{1'}  \ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{4'}  a^\dagger_{4'} b_1^\dagger  b_{1'}  
\np
\label{L3orig}
\int[11'4] \, \tilde \delta_{1.1'4} 
\ \chi_1^\dagger T^{4} \chi_{1'} 
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g  \over p_{4}^+} \ b_1^\dagger b_{1'} \tilde a_{ 4} 
\ \
m_g  \int [4'33'] \, \tilde \delta_{4'3.3'} \  i f^{c_{4'} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4'}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4'} a^\dagger_{3}    a_{3'}  
\np
m_g  \int [433'] \, \tilde \delta_{3'.43} \ \left[  i f^{c_4 c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_3 \varepsilon_{3'}     \ \tilde a^\dagger_{4} a^\dagger_{3}    a_{3'} \right]^\dagger  
\ \
\int[11'4'] \, \tilde \delta_{4'1.1'} 
\ \chi_1^\dagger T^{4'} \chi_{1'} 
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g  \over p_{4'}^+} \ \tilde a^\dagger_{4'} b_1^\dagger b_{1'}  \ .
\eeq
{\red Start here}
The interaction between the quark (1) and gluon (3) via gluon exchange  (4) is given by the formula 
$g^2 $ times
\beq
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger T^{4} \chi_{1'} 
\ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 \ b_1^\dagger  b_{1'}  
\ \
\int [4'33'] \, \tilde \delta_{4'3.3'} \  Y_{4'33'} \ a_4  a^\dagger_{4'} a^\dagger_{3}  a_{3'} 
{\green  \int [34'3'] 
\ Y_{34'3'} \, \tilde \delta_{34'.3'}  \, a_4 a_3^\dagger a_{4'}^\dagger a_{3'} } 
\np
\int [433']  \, \tilde \delta_{3'.43} \  Y_{433'}^* \ a_{3'}^\dagger 
\ {\green  \int [343'] 
\, a_{3'}^\dagger   \ Y_{343'}^* \, \tilde \delta_{3'.43}  } 
\ \
\int[11'4'] \, \tilde \delta_{4'1.1'}  \ \chi_1^\dagger T^{4'} \chi_{1'} 
\ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{4'} \  {\green a_{3} a_{4} }  a^\dagger_{4'} b_1^\dagger  b_{1'}  
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger T^{4} \chi_{1'} 
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g  \over p_{4}^+} \ b_1^\dagger b_{1'} 
\ \
m_g  \int [4'33'] \, \tilde \delta_{4'3.3'} \  i f^{c_{4'} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4'}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\  \tilde a_{4}  \tilde a^\dagger_{4'} a^\dagger_{3}    a_{3'}  
\np
m_g  \int [433'] \, \tilde \delta_{3'.43} \ i f^{ c_{3'} c_3 c_4 } {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_{3'} \varepsilon_{3}     \     
a_{3'}^\dagger 
\ \
\int[11' 4'] \, \tilde \delta_{4'1.1'} \ \chi_1^\dagger T^{4'} \chi_{1'} 
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g  \over p_{4'}^+} \ a_{3} \tilde a_{4} \tilde a^\dagger_{4'} b_1^\dagger b_{1'}  \ .
\eeq
I contract $a$s with $a^\dagger$s in $g^2  \ b_1^\dagger  b_{1'} $ times
\beq
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 
\ \
{\green  \int [34'3'] 
\ Y_{34'3'} \, \tilde \delta_{34'.3'}  \, \left( \delta_{44'} a^\dagger_{3} + a_{4'}^\dagger \delta_{43} \right) a_{3'}   }
\np
 {\green  \int [343']  \ Y_{343'}^* \, \tilde \delta_{3'.43}  } 
\ \
\int[11'4'] \, \tilde \delta_{4'1.1'}  \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{4'} \ {\green a_{3'}^\dagger \left(  a_3 \delta_{44'}  +a_4 \delta_{34'} \right) }   
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g^2  \over p_{4}^+} 
\ \
\int [4'33'] \, \tilde \delta_{4'3.3'} \  i f^{c_{4'} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4'}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\  \delta_{44'} a^\dagger_{3}    a_{3'}  
\np
\int [433'] \, \tilde \delta_{3'.43} \ i f^{ c_{3'} c_3 c_4 } {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_{3'} \varepsilon_{3}   
\ \
\int[11' 4'] \, \tilde \delta_{4'1.1'} \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g^2  \over p_{4'}^+} \ a_{3'}^\dagger  a_{3} \delta_{44'}   \ .
\eeq
After expansion of the brackets, I have $g^2  \ b_1^\dagger  b_{1'} $ times
\beq
old ~~
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 
\ \
\int [4'33'] \, \tilde \delta_{4'3.3'} \  Y_{4'33'} \ \delta_{44'} a^\dagger_{3} a_{3'} 
\np
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 
\ \
\int [4'33'] \, \tilde \delta_{4'3.3'} \  Y_{4'33'} \  a_{4'}^\dagger \delta_{43} a_{3'} 
\np
\int [433']  \, \tilde \delta_{3'.43} \  Y_{433'}^* 
\ \
\int[11'4'] \, \tilde \delta_{4'1.1'}  \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{4'} \ a_{3'}^\dagger a_3 \delta_{44'}  
\np
\int [433']  \, \tilde \delta_{3'.43} \  Y_{433'}^* 
\ \
\int[11'4'] \, \tilde \delta_{4'1.1'}  \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{4'} \ a_{3'}^\dagger a_4 \delta_{34'} 
\np
\int[11' 4]  \, \tilde \delta_{1.1'4'} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g^2  \over p_{4}^+} 
\ \
\int [4'33'] \, \tilde \delta_{4'3.3'} \  i f^{c_{4'} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4'}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\  \delta_{44'} a^\dagger_{3}    a_{3'}  
\np
\int [433'] \, \tilde \delta_{3'.43} \ i f^{ c_{3'} c_3 c_4 } {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_{3'} \varepsilon_{3}   
\ \
\int[11' 4'] \, \tilde \delta_{4'1.1'} \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g^2  \over p_{4'}^+} \ a_{3'}^\dagger  a_{3} \delta_{44'}   \ .
\eeq
%
\beq
new ~~
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 
\ \
{\green  \int [34'3'] 
\ Y_{34'3'} \, \tilde \delta_{34'.3'}  \, \delta_{44'} a^\dagger_{3} a_{3'}   }
\np
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 
\ \
{\green  \int [34'3'] 
\ Y_{34'3'} \, \tilde \delta_{34'.3'}  \,  a_{4'}^\dagger \delta_{43} a_{3'}   }
\np
 {\green  \int [343']  \ Y_{343'}^* \, \tilde \delta_{3'.43}  } 
\ \
\int[11'4'] \, \tilde \delta_{4'1.1'}  \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{4'} \ {\green a_{3'}^\dagger   a_3 \delta_{44'}   }   
\np
 {\green  \int [343']  \ Y_{343'}^* \, \tilde \delta_{3'.43}  } 
\ \
\int[11'4'] \, \tilde \delta_{4'1.1'}  \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{4'} \ {\green a_{3'}^\dagger a_4 \delta_{34'}  }  
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g^2  \over p_{4}^+} 
\ \
\int [4'33'] \, \tilde \delta_{4'3.3'} \  i f^{c_{4'} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4'}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\  \delta_{44'} a^\dagger_{3}    a_{3'}  
\np
\int [433'] \, \tilde \delta_{3'.43} \ i f^{ c_{3'} c_3 c_4 } {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_{3'} \varepsilon_{3}   
\ \
\int[11' 4'] \, \tilde \delta_{4'1.1'} \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g^2  \over p_{4'}^+} \ a_{3'}^\dagger  a_{3} \delta_{44'}   \ .
\eeq
%
Now I can carry out the integration resulting from the contractions, leading to 
$g^2  \ b_1^\dagger  b_{1'} \ \bar u_1 \gamma_\nu u_{1'} $ combined with
\beq
old ~~\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} \ \varepsilon^\nu_4 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  Y_{433'} \ a^\dagger_{3} a_{3'} 
\np
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} \ \varepsilon^\nu_4 
\ \
\int [4'3'] \, \tilde \delta_{4'4.3'} \  Y_{4'43'} \  a_{4'}^\dagger a_{3'} ~~~~~~ 4' \to 3
\np
\int [433']  \, \tilde \delta_{3'.43} \  Y_{433'}^* 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'}    
\ \varepsilon^{\nu *}_{4} \ a_{3'}^\dagger a_3  ~~~~~~ 3' \leftrightarrow 3
\np
\int [433']  \, \tilde \delta_{3'.43} \  Y_{433'}^* 
\ \
\int[11'] \, \tilde \delta_{31.1'}  \ \chi_1^\dagger  T^{3} \chi_{1'} \ \varepsilon^{\nu *}_{3} \ a_{3'}^\dagger a_4 
~~~~~~ 3 \leftrightarrow 4 {\rm ~~ and ~ then ~} ~ 3' \leftrightarrow 3
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} \ {m_g^2 \eta^\nu \over p_{4}^+} 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\ a^\dagger_{3}    a_{3'}  
\np
\int [433'] \, \tilde \delta_{3'.43} \ i f^{ c_{3'} c_3 c_4 } {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_{3'} \varepsilon_{3}   
\ \
\int[11'] \, \tilde \delta_{41.1'} \ \chi_1^\dagger  T^{4} \chi_{1'}
\ {m_g^2 \eta^\nu \over p_{4}^+} \ a_{3'}^\dagger  a_{3}  ~~~~~~ 3' \leftrightarrow 3  \ .
\eeq
\beq
new ~~
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 
\ \
{\green  \int [33'] 
\ Y_{343'} \, \tilde \delta_{34.3'}  \,  a^\dagger_{3} a_{3'}   }
\np
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'} \ \varepsilon^\nu_4 
\ \
{\green  \int [4'3'] 
\ Y_{44'3'} \, \tilde \delta_{44'.3'}  \,  a_{4'}^\dagger a_{3'}   }
\np
 {\green  \int [343']  \ Y_{343'}^* \, \tilde \delta_{3'.43}  } 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{4} \ {\green a_{3'}^\dagger   a_3    }   
\np
 {\green  \int [343']  \ Y_{343'}^* \, \tilde \delta_{3'.43}  } 
\ \
\int[11'] \, \tilde \delta_{31.1'}  \ \chi_1^\dagger  T^{3} \chi_{1'}
\ \bar u_1 \gamma_\nu u_{1'}       
\ \varepsilon^{\nu *}_{3} \ {\green a_{3'}^\dagger a_4  }  
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g^2  \over p_{4}^+} 
\ \
\int [4'33'] \, \tilde \delta_{4'3.3'} \  i f^{c_{4'} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4'}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\  \delta_{44'} a^\dagger_{3}    a_{3'}  
\np
\int [433'] \, \tilde \delta_{3'.43} \ i f^{ c_{3'} c_3 c_4 } {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_{3'} \varepsilon_{3}   
\ \
\int[11' 4'] \, \tilde \delta_{4'1.1'} \ \chi_1^\dagger  T^{4'} \chi_{1'}
\ \bar u_1 \gamma^+ u_{1'} 
\ {m_g^2  \over p_{4'}^+} \ a_{3'}^\dagger  a_{3} \delta_{44'}   \ .
\eeq
I am now (20250918 10:08 czw San Dimas) making these changes in 
$g^2  \ b_1^\dagger  b_{1'} \ \bar u_1 \gamma_\nu u_{1'} $ combined with
\beq
old ~~\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} \ \varepsilon^\nu_4 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  Y_{433'} \ a^\dagger_{3} a_{3'} 
\np
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} \ \varepsilon^\nu_4 
\ \
\int [33'] \, \tilde \delta_{34.3'} \  Y_{343'} \  a_{3}^\dagger a_{3'} ~~~~~~ 4' \to 3
\np
\int [43'3]  \, \tilde \delta_{3.43'} \  Y_{43'3}^* 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'}    
\ \varepsilon^{\nu *}_{4} \ a_{3}^\dagger a_{3'}  ~~~~~~ 3' \leftrightarrow 3
\np
\int [3'43]  \, \tilde \delta_{3.3'4} \  Y_{3'43}^* 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'} \ \varepsilon^{\nu *}_{4} \ a_{3}^\dagger a_{3'} 
~~~~~~ 3 \leftrightarrow 4 {\rm ~~ and ~ then ~} ~ 3' \leftrightarrow 3
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} \ {m_g^2 \eta^\nu \over p_{4}^+} 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\ a^\dagger_{3}    a_{3'}  
\np
\int [43'3] \, \tilde \delta_{3.43'} \ i f^{ c_{3} c_{3'} c_4 } {p_{3'}^+ + p_{3}^+ \over p_4^+}  \ \varepsilon^*_{3} \varepsilon_{3'}   
\ \
\int[11'] \, \tilde \delta_{41.1'} \ \chi_1^\dagger  T^{4} \chi_{1'}
\ {m_g^2 \eta^\nu \over p_{4}^+} \ a_{3}^\dagger  a_{3'}  ~~~~~~ 3' \leftrightarrow 3  \ .
\eeq
%
{\green And in the checking calculation in green on 20250924 21:58 sro San Dimas, I make analogous notation changes
in $g^2  \ b_1^\dagger  b_{1'} \ \bar u_1 \gamma_\nu u_{1'} $ combined with}
\beq
new ~~
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \varepsilon^\nu_4 
\ \
{\green  \int [33'] 
\ Y_{343'} \, \tilde \delta_{34.3'}  \,  a^\dagger_{3} a_{3'}   }
\np
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
 \ \varepsilon^\nu_4 
\ \
{\green  \int [33'] 
\ Y_{433'} \, \tilde \delta_{43.3'}  \,  a_{3}^\dagger a_{3'}   } ~~~~~~~ {\rm from}~~4' \to 3
\np
 {\green  \int [3'43]  \ Y_{3'43}^* \, \tilde \delta_{3.43'}  } 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'}     
\ \varepsilon^{\nu *}_{4} \ {\green a_{3}^\dagger   a_{3'}    }   ~~~~~~~~ {\rm from}~~ 3' \leftrightarrow 3
\np
 {\green  \int [43'3]  \ Y_{43'3}^* \, \tilde \delta_{3.3'4}  } 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'}       
\ \varepsilon^{\nu *}_{4} \ {\green a_{3}^\dagger a_{3'}  }  
~~~{\rm from}~~3 \leftrightarrow 4 {\rm ~~ and ~ then ~} ~ 3' \leftrightarrow 3
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} 
\ {m_g^2  \over p_{4}^+} 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\  a^\dagger_{3}    a_{3'}  
\np
\int [43'3] \, \tilde \delta_{3.43'} \ i f^{ c_{3} c_{3'} c_4 } {p_{3'}^+ + p_{3}^+ \over p_4^+}  \ \varepsilon^*_{3} \varepsilon_{3'}   
\ \
\int[11'] \, \tilde \delta_{41.1'} \ \chi_1^\dagger  T^{4} \chi_{1'}
\ {m_g^2  \over p_{4}^+} \ a_{3}^\dagger  a_{3'}  \ .  ~~~ {\rm from}~~ 3' \leftrightarrow 3
\eeq
%
The result is 
$g^2$ $\chi_1^\dagger  T^4 \chi_{1'}$  $\bar u_1 \gamma_\nu u_{1'} $ $b_1^\dagger  b_{1'} \, a^\dagger_{3} a_{3'} $ combined with
\beq
old ~~ \cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4}  \ \varepsilon^\nu_4 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  Y_{433'} 
\np
\int[11'4] \, \tilde \delta_{1.1'4}  \ \varepsilon^\nu_4 
\ \
\int [33'] \, \tilde \delta_{34.3'} \  Y_{343'} 
\np
\int [43'3]  \, \tilde \delta_{3.43'} \  Y_{43'3}^* 
\ \
\int[11'] \, \tilde \delta_{41.1'}     
\ \varepsilon^{\nu *}_{4} 
\np
\int [3'43]  \, \tilde \delta_{3.3'4} \  Y_{3'43}^* 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \varepsilon^{\nu *}_{4} 
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ {m_g^2 \eta^\nu \over p_{4}^+} 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\np
\int [43'3] \, \tilde \delta_{3.43'} \ i f^{ c_{3} c_{3'} c_4 } {p_{3'}^+ + p_{3}^+ \over p_4^+}  \ \varepsilon^*_{3} \varepsilon_{3'}   
\ \
\int[11'] \, \tilde \delta_{41.1'} \ {m_g^2 \eta^\nu \over p_{4}^+}   \ .
\eeq
{\green The new result will be $g^2$  $\bar u_1 \gamma_\nu u_{1'} $ $b_1^\dagger  b_{1'} $ combined with}
\beq
new ~~
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \varepsilon^\nu_4 
\ \
{\green  \int [33'] 
\ Y_{343'} \, \tilde \delta_{34.3'}  \,  a^\dagger_{3} a_{3'}   }
\np
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
 \ \varepsilon^\nu_4 
\ \
{\green  \int [4'3'] 
\ Y_{44'3'} \, \tilde \delta_{44'.3'}  \,  a_{4'}^\dagger a_{3'}   } ~~~~~~~4' \to 3
\np
 {\green  \int [343']  \ Y_{343'}^* \, \tilde \delta_{3'.43}  } 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'}     
\ \varepsilon^{\nu *}_{4} \ {\green a_{3'}^\dagger   a_3    }   ~~~~~~~~ 3' \leftrightarrow 3
\np
 {\green  \int [343']  \ Y_{343'}^* \, \tilde \delta_{3'.43}  } 
\ \
\int[11'] \, \tilde \delta_{31.1'}  \ \chi_1^\dagger  T^{3} \chi_{1'}       
\ \varepsilon^{\nu *}_{3} \ {\green a_{3'}^\dagger a_4  }  
~~~~~~ 3 \leftrightarrow 4 {\rm ~~ and ~ then ~} ~ 3' \leftrightarrow 3
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} 
\ {m_g^2  \over p_{4}^+} 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\  a^\dagger_{3}    a_{3'}  
\np
\int [433'] \, \tilde \delta_{3'.43} \ i f^{ c_{3'} c_3 c_4 } {p_3^+ + p_{3'}^+ \over p_4^+}  \ \varepsilon^*_{3'} \varepsilon_{3}   
\ \
\int[11'] \, \tilde \delta_{41.1'} \ \chi_1^\dagger  T^{4} \chi_{1'}
\ {m_g^2  \over p_{4}^+} \ a_{3'}^\dagger  a_{3}  \ .  ~~~ 3' \leftrightarrow 3
\eeq
\beq
new ~~
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
\ \varepsilon^\nu_4 
\ \
{\green  \int [33'] 
\ Y_{343'} \, \tilde \delta_{34.3'}  \,  a^\dagger_{3} a_{3'}   }
\np
\int[11'4] \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'}
 \ \varepsilon^\nu_4 
\ \
{\green  \int [33'] 
\ Y_{433'} \, \tilde \delta_{43.3'}  \,  a_{3}^\dagger a_{3'}   } ~~~~~~~ {\rm from}~~4' \to 3
\np
 {\green  \int [3'43]  \ Y_{3'43}^* \, \tilde \delta_{3.43'}  } 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'}     
\ \varepsilon^{\nu *}_{4} \ {\green a_{3}^\dagger   a_{3'}    }   ~~~~~~~~ {\rm from}~~ 3' \leftrightarrow 3
\np
 {\green  \int [43'3]  \ Y_{43'3}^* \, \tilde \delta_{3.3'4}  } 
\ \
\int[11'] \, \tilde \delta_{41.1'}  \ \chi_1^\dagger  T^{4} \chi_{1'}       
\ \varepsilon^{\nu *}_{4} \ {\green a_{3}^\dagger a_{3'}  }  
~~~{\rm from}~~3 \leftrightarrow 4 {\rm ~~ and ~ then ~} ~ 3' \leftrightarrow 3
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} \ \chi_1^\dagger  T^4 \chi_{1'} 
\ {m_g^2  \over p_{4}^+} 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\  a^\dagger_{3}    a_{3'}  
\np
\int [43'3] \, \tilde \delta_{3.43'} \ i f^{ c_{3} c_{3'} c_4 } {p_{3'}^+ + p_{3}^+ \over p_4^+}  \ \varepsilon^*_{3} \varepsilon_{3'}   
\ \
\int[11'] \, \tilde \delta_{41.1'} \ \chi_1^\dagger  T^{4} \chi_{1'}
\ {m_g^2  \over p_{4}^+} \ a_{3}^\dagger  a_{3'}  \ .  ~~~ {\rm from}~~ 3' \leftrightarrow 3
\eeq
{\green The new result is $g^2$ $\chi_1^\dagger  T^4 \chi_{1'}$  $\bar u_1 \gamma_\nu u_{1'} $ $b_1^\dagger  b_{1'} \, a^\dagger_{3} a_{3'} $ combined with}
\beq
new ~~
\cH_{13 \, \rm exch} \es
\int[11'4] \, \tilde \delta_{1.1'4} 
\ \varepsilon^\nu_4 
\ \
{\green  \int [33'] 
\ Y_{343'} \, \tilde \delta_{34.3'}   }
\np
\int[11'4] \, \tilde \delta_{1.1'4} 
 \ \varepsilon^\nu_4 
\ \
{\green  \int [33'] 
\ Y_{433'} \, \tilde \delta_{43.3'}    } ~~~~~~~ {\rm from}~~4' \to 3
\np
 {\green  \int [3'43]  \ Y_{3'43}^* \, \tilde \delta_{3.43'}  } 
\ \
\int[11'] \, \tilde \delta_{41.1'}  
\ \varepsilon^{\nu *}_{4}    ~~~~~~~~ {\rm from}~~ 3' \leftrightarrow 3
\np
 {\green  \int [43'3]  \ Y_{43'3}^* \, \tilde \delta_{3.3'4}  } 
\ \
\int[11'] \, \tilde \delta_{41.1'}         
\ \varepsilon^{\nu *}_{4} 
~~~{\rm from}~~3 \leftrightarrow 4 {\rm ~~ and ~ then ~} ~ 3' \leftrightarrow 3
\np
\int[11' 4]  \, \tilde \delta_{1.1'4} 
\ {m_g^2  \over p_{4}^+} 
\ \
\int [33'] \, \tilde \delta_{43.3'} \  i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\np
\int [43'3] \, \tilde \delta_{3.43'} \ i f^{ c_{3} c_{3'} c_4 } {p_{3'}^+ + p_{3}^+ \over p_4^+}  \ \varepsilon^*_{3} \varepsilon_{3'}   
\ \
\int[11'] \, \tilde \delta_{41.1'} 
\ {m_g^2  \over p_{4}^+} \ .  ~~~ {\rm from}~~ 3' \leftrightarrow 3
\eeq
I simplify to
$g^2$ $\chi_1^\dagger  T^4 \chi_{1'}$  $\bar u_1 \gamma_\nu u_{1'} $ $b_1^\dagger  b_{1'} \, a^\dagger_{3} a_{3'} $ combined with
\beq
old ~~
\cH_{13 \, \rm exch} \es
\int[11'433'] \, \tilde \delta_{1.1'4} \tilde \delta_{43.3'}  \ \varepsilon^\nu_4 
\ \
 \left( Y_{433'}  +   Y_{343'} \right)
\np
\int[11'433'] \, \tilde \delta_{3.43'} \tilde \delta_{41.1'} \ \left( Y_{43'3}^*  +  Y_{3'43}^*  \right)
\ \
\varepsilon^{\nu *}_{4} 
\np
\int[11'433'] \, \tilde \delta_{1.1'4} \tilde \delta_{43.3'} \ {m_g^2 \eta^\nu \over p_{4}^+} 
\ \
i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\np
\int[11'433'] \, \tilde \delta_{3.43'} \tilde \delta_{41.1'} \ i f^{ c_{3} c_{3'} c_4 } {p_{3'}^+ + p_{3}^+ \over p_4^+}  \ \varepsilon^*_{3} \varepsilon_{3'}   
\ \
{m_g^2 \eta^\nu \over p_{4}^+}   \ .
\eeq
{\green I simplify to
$g^2$ $\chi_1^\dagger  T^4 \chi_{1'}$  $\bar u_1 \gamma_\nu u_{1'} $ $b_1^\dagger  b_{1'} \, a^\dagger_{3} a_{3'} $ combined with}
\beq
new ~~
\cH_{13 \, \rm exch} \es
\int[11'433'] \, \tilde \delta_{1.1'4} \tilde \delta_{34.3'} \ \ {\green \ \varepsilon^\nu_4  Y_{343'} \,    }
\np 
\int[11'433'] \, \tilde \delta_{1.1'4} \tilde \delta_{43.3'} \ \ {\green  \varepsilon^\nu_4  Y_{433'} \,     } 
\np
\int[11'433'] \, \tilde \delta_{41.1'} \tilde \delta_{3.43'} \ \ {\green \ \varepsilon^{\nu *}_{4}  Y_{3'43}^* \,  } 
\np
\int[11'433'] \, \tilde \delta_{41.1'} \tilde \delta_{3.3'4} \ \ {\green  \ \varepsilon^{\nu *}_{4} Y_{43'3}^* \,   } 
\np
\int[11'433'] \, \tilde \delta_{1.1'4} \tilde \delta_{43.3'} \ \ {m_g^2  \over p_{4}^+} 
\ \ i f^{c_{4} c_3 c_{3'}} {p_3^+ + p_{3'}^+ \over p_{4}^+}  \ \varepsilon^*_3 \varepsilon_{3'}     
\np
\int[11'433'] \, \tilde \delta_{41.1'}  \tilde \delta_{3.43'} \ \ i f^{ c_{3} c_{3'} c_4 } {p_{3'}^+ + p_{3}^+ \over p_4^+}  \ \varepsilon^*_{3} \varepsilon_{3'}  \ \ {m_g^2  \over p_{4}^+} \ .  
\eeq


There are two integrals over 4.
\beq
\int[4] \, \tilde \delta_{1.1'4} \tilde \delta_{43.3'} \es 
\int[4] \, 2(2\pi)^3 \delta_{1-1'-4} 2(2\pi)^3 \delta_{4+3-3'} \rs
\tilde \delta_{13.1'3'} \theta_{1-1'}/p_4^+ ~~~~~{\rm and}~~~~~p_4 \rs p_1-p_{1'} \ , \\
\int[4] \, \tilde \delta_{3.43'} \tilde \delta_{41.1'} \es 
\int[4] \, 2(2\pi)^3 \delta_{3-4-3'} 2(2\pi)^3  \delta_{4+1-1'} \rs
\tilde \delta_{13.1'3'} \theta_{1'-1}/p_4^+ ~~~~~{\rm and}~~~~~p_4 \rs p_{1'}-p_{1} \ .
\eeq
The result for the exchange interaction is 
\beq
\cH_{13 \, \rm exch} 
\es
g^2 \sum_{\rm tr} 
\int[11'33'] \, \tilde \delta_{13.1'3'} \ \chi_1^\dagger  T^4 \chi_{1'} \ \bar u_1 \gamma_\nu u_{1'} 
\, \left( \cJ_{\rm tr}^\nu/p_4^+ \right)  \ \ 
a^\dagger_{3} b_1^\dagger  b_{1'} a_{3'} \ , \ \ \ p_4 \rs p_{\overline 1} - p_{\underline 1} 
\np
g^2 
\int[11'33'] \, \tilde \delta_{13.1'3'} \ \chi_1^\dagger  T^4 \chi_{1'} \ \bar u_1 \gamma_\nu u_{1'} 
\, \left( \cJ_\eta^\nu/p_4^+ \right)  \ \ 
a^\dagger_{3} b_1^\dagger  b_{1'} a_{3'} \ , \ \ \ p_4 \rs p_{\overline 1} - p_{\underline 1}
\ , \\
\cJ_{\rm tr}^\nu 
\es
\theta_{1-1'} \, \varepsilon^\nu_4 \left( Y_{433'}  +   Y_{343'} \right) 
+
\theta_{1'-1} \, \varepsilon^{\nu *}_{4}  \left( Y_{43'3}^*  +  Y_{3'43}^* \right) \ , \\
\cJ_{\eta}^\nu \es
{m_g^2 \eta^\nu \over p_{4}^{+2}} \ i f^{c_{4} c_3 c_{3'}} (p_3^+ + p_{3'}^+) \ \varepsilon^*_3 \varepsilon_{3'} \ .
\eeq
\beq
Y_{433'}  +   Y_{343'} \es 2 Y_{433'} \ , \ Y_{43'3}^*  +  Y_{3'43}^*  \rs 2Y_{43'3}^* \ .
\eeq
\beq
Y_{433'} \es
    i f^{c_4 c_3 c_{3'}} \left(
   \varepsilon^*_4 \varepsilon^*_3  \  \varepsilon_{3'}  p_4                 
-  \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^* p_{3'}
+ \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^* p_{3'} 
\right) , ~~~~ {\rm to \ multiply \ by }~~~~\ a^\dagger_{4} a^\dagger_{3}  a_{3'} \ , \\
\es
 i f^{c_4 c_3 c_{3'}} \left(
   \varepsilon^*_4 \varepsilon^*_3   \  \varepsilon_{3'} k_{43}  
- \varepsilon^*_4 \varepsilon_{3'}   \  \varepsilon_{3}^*k_{43}/y_3
 - \varepsilon^*_3 \varepsilon_{3'}   \  \varepsilon_{4}^*k_{43}/y_4
  \right) \ , \\
Y_{43'3} \es i f^{c_4 c_{3'} c_{3}} 
 \left(
   \varepsilon^*_4 \varepsilon^*_{3'}  \  \varepsilon_{3}  p_4                 
-  \varepsilon^*_4 \varepsilon_{3}   \  \varepsilon_{3'}^* p_{3}
+ \varepsilon^*_{3'} \varepsilon_{3}   \  \varepsilon_{4}^* p_{3} 
\right) \ , \\
Y_{43'3}^* \es  - i f^{c_4 c_{3'} c_{3}} 
 \left(
   \varepsilon_4 \varepsilon_{3'}  \  \varepsilon_{3}^*  p_4                 
-  \varepsilon_4 \varepsilon_{3}^*   \  \varepsilon_{3'} p_{3}
+ \varepsilon_{3'} \varepsilon_{3}^*   \  \varepsilon_{4} p_{3} 
\right) \ .
\eeq
I can now pool out $i f^{c_4 c_{3} c_{3'}}$ and write (remember the regularization factors are included)
\beq
\label{exchpom}
\cH_{13 \, \rm exch} 
\es
g^2 \sum_{\rm tr} 
\int[11'33'] \, \tilde \delta_{13.1'3'} \ \chi_1^\dagger  T^4 \chi_{1'} \ \bar u_1 \gamma_\nu u_{1'} 
\, \left( \cJ_{\rm tr}^\nu/p_4^+ \right)  \, i f^{c_4 c_{3} c_{3'}} \ \ 
a^\dagger_{3} b_1^\dagger  b_{1'} a_{3'} \ , \ \ \ p_4 \rs p_{\overline 1} - p_{\underline 1} 
\np
g^2 
\int[11'33'] \, \tilde \delta_{13.1'3'} \ \chi_1^\dagger  T^4 \chi_{1'} \ \bar u_1 \gamma_\nu u_{1'} 
\, \left( \cJ_\eta^\nu/p_4^+ \right)  \, i f^{c_4 c_{3} c_{3'}} \ \ 
a^\dagger_{3} b_1^\dagger  b_{1'} a_{3'} \ , \ \ \ p_4 \rs p_{\overline 1} - p_{\underline 1} \ .
\eeq
\beq
\cJ_{\rm tr}^\nu 
\es
2 \theta_{1-1'} \, \varepsilon^\nu_4 Z_{433'}  
+
2 \theta_{1'-1} \, \varepsilon^{\nu *}_{4} Z_{43'3}^*  \ , \\
\cJ_{\eta}^\nu 
\es \eta^\nu  \ S_\eta \ , \\
Z_{433'} \es \varepsilon^{*\mu}_4 S_{433' \mu} \ , \\
S_{0433' \mu} \es 
\left(
   \varepsilon^*_{3\mu}  \  \varepsilon_{3'}  p_4                 
-  \varepsilon_{3'\mu}   \  \varepsilon_{3}^* p_{3'}
+ \varepsilon^*_3 \varepsilon_{3'}   \  p_{3'\mu} 
\right) ~~~~~~~ p_4^- ~ {\rm is ~ absent}\ ,  \\
Z_{43'3}^* \es 
 \varepsilon_4^\mu S_{43'3 \mu}^* \ , \\
S_{043'3 \mu}^* \es
 \left(
    \varepsilon_{3'\mu}  \  \varepsilon_{3}^*  p_4                 
-   \varepsilon_{3\mu}^*   \  \varepsilon_{3'} p_{3}
+  \varepsilon_{3'} \varepsilon_{3}^*   \  p_{3\mu} 
\right) ~~~~~~~ p_4^- ~ {\rm is ~ absent} \ , \\
S_\eta \es {m_g^2 \over p_{4}^{+2}} \  (p_3^+ + p_{3'}^+) \ \varepsilon^*_3 \varepsilon_{3'}\ .
\eeq
0 refers to the gluon mass 0, although the transverse polarization vectors do npot depend on the gluon mass. 
Then
\beq
\cJ_{\rm tr}^\nu 
\es
2 \theta_{1-1'} \, \sum_{\sigma_4} \varepsilon^\nu_4  \varepsilon^{*\mu}_4 S_{0433' \mu}
+
2 \theta_{1'-1} \, \sum_{\sigma_4} \varepsilon^{\nu *}_{4}  \varepsilon_4^\mu S_{043'3 \mu}^*  \ .
\eeq
Sum over spins of the transverse gluons yields
\beq
\cJ_{\rm tr}^\nu 
\es
2 \theta_{1-1'} \, d^{\nu \mu}_{04} \, S_{0433' \mu}
+
2 \theta_{1'-1} \, d^{\nu \mu}_{04} \, S_{043'3 \mu}^* \ ,
\eeq
where 
\beq
 d^{\nu \mu}_{04} \es -g^{\nu \mu} + {p_{04}^\nu \eta^\mu + \eta^\nu p_{04}^\mu \over p_4^+} \ ,
\eeq
and 0 refers again to the gluon mass set to 0. I recall Eq.~(\ref{Hsqg2}) for the quark-gluon seagull, not showing 
$f^r_{\overline 1,\underline 1 (\overline 1 - \underline 1)}$ $f^r_{\overline 3,\underline 3 (\overline 3 - \underline 3)}$, 
\beq
\label{Hsqg2 recall 1}
H_{sqg} \es -g^2 \int [33'11'] \ \tilde \delta_{c.a} \ \delta_{\rm spins} 
\
{ ( p_3^+ + p_{3'}^+ )  \ 2 \sqrt{p_1^+ p_{1'}^+}  \over (p_{3'}^+ - p_3^+)^2} 
\ \chi_{1}^\dagger T^{c_4} \chi_{1'} \, i f^{c_4 c_3 c_{3'}} \  a^\dagger_{3}  b_1^\dagger  b_{1'} a_{3'}\ ,
\eeq
while the full expression for the gluon exchange, including regulating factors $f$ that are not indicated, 
\beq
\label{cH2t3body recall}
\cH^{(2)}
\es
f_{LR} \, \cH^{(2)}_0
+ 
(f_{LR} - f_{LI} f_{IR}) \
\Delta_{LIR} \
\cH^{(1)}_0 \, \cH^{(1)}_0 \ ,
\eeq
reads, see Eq.~(\ref{exchpom}), 
\beq
{\rm Exch}
\es
(f_{LR} - f_{LI} f_{IR}) \
\Delta_{LIR} \
g^2 
\int[11'33'] \, \tilde \delta_{c.a} \ \chi_1^\dagger  T^{c_4} \chi_{1'} \ \bar u_1 \gamma_\nu  u_{1'}
\left( d_{04}^{\nu \mu} \cE_{\mu \rm tr} + \eta^\nu S_\eta \right) 
{1 \over p_4} \, i f^{c_4 c_{3} c_{3'}}  \ \ 
a^\dagger_{3} b_1^\dagger  b_{1'} a_{3'} \ , 
\eeq
where
\beq
\cE_{\mu \rm tr}
\es
2 \theta_{1-1'} \, S_{0433' \mu}
+
2 \theta_{1'-1}  \, S_{043'3 \mu}^* \ , \\
S_\eta \es
{m_g^2 \over p_{4}^{+2}} \  (p_3^+ + p_{3'}^+) \ \varepsilon^*_3 \varepsilon_{3'} \ , \\
\label{Deltaqg}
{1 \over p_4^+}\Delta_{LIR} \es 
{1 \over 2p_4^+}  \left( {1 \over p_3^- + p_1^- - P_{\rm three}^- } +  {1 \over p_{3'}^- + p_{1'}^- - P_{\rm three}^- }\right) 
\rs - \, \2 \left( {1 \over \rho_1} + {1 \over \rho_3 } \right) \ .
\eeq
The gluon four-momenta in the denominators come from $\cH_f$ and have mass $m_g$. Therefore, they do not carry
subscript 0 indicating mass 0. 
\beq
\theta_{1-1'}~\rightarrow~~~~
&& p_4^+  \left( p_3^- + p_1^- - P_{\rm three}^- \right) \rs
(p_{1}^+ - p_{1'}^+)  \left( p_3^- + p_1^- - p_3^- - p_4^- - p_{1'}^- \right) \\
\es
(p_{1}^+ - p_{1'}^+)  \left(  p_1^-  - p_{1'}^- - p_4^-  \right)  
\rs
(p_{1}^+ - p_{1'}^+) (  p_1^-  - p_{1'}^-) -  (p_{1}^\perp - p_{1'}^\perp)^2 - m_g^2 \\
\es q_1^2 - m_g^2 \rs - \rho_1 \ , \\
&& p_4^+  \left( p_{3'}^- + p_{1'}^- - P_{\rm three}^- \right) \rs
(p_{1}^+ - p_{1'}^+)  \left( p_{3'}^- + p_{1'}^- - p_3^- - p_4^- - p_{1'}^-\right) \\
\es
(p_{3'}^+ - p_{3}^+)  \left(  p_{3'}^-  - p_3^- - p_4^-  \right)  
\rs
(p_{3'}^+ - p_{3}^+) (  p_{3'}^-  - p_{3}^-)  -  (p_{3'}^\perp - p_{3}^\perp)^2 - m_g^2 \\
\es q_3^2 - m_g^2 \rs - \rho_3 \ , \\
\theta_{1'-1}~\rightarrow~~~~
&&
p_4^+  \left( p_{3}^- + p_{1}^- - P_{\rm three}^- \right) \rs
(p_{1'}^+ - p_{1}^+)  \left( p_{3}^- + p_{1}^- - p_{3'}^- - p_4^- - p_{1}^- \right) \\
\es 
(p_{3}^+ - p_{3'}^+)  \left(  p_{3}^- - p_{3'}^- - p_4^-  \right)
\rs
(p_{3}^+ - p_{3'}^+) (  p_{3}^- - p_{3'}^-) - (p_{3}^\perp - p_{3'}^\perp)^2 - m_g^2 \\
\es q_3^2 - m_g^2 \rs - \rho_3 \ , \\
&&
p_4^+  \left( p_{3'}^- + p_{1'}^- - P_{\rm three}^- \right) \rs
(p_{1'}^+ - p_{1}^+)  \left( p_{3'}^- + p_{1'}^- - p_{3'}^- - p_4^- - p_{1}^- \right) \\
\es 
(p_{1'}^+ - p_{1}^+)  \left(  p_{1'}^- - p_{1}^- - p_4^-  \right)
\rs
(p_{1'}^+ - p_{1}^+) (p_{1'}^- - p_{1}^-) - (p_{1'}^\perp - p_{1}^\perp)^2 - m_g^2 \\
\es q_1^2 - m_g^2 \rs - \rho_1 \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ \bf Products in gluon exchange between a quark and a gluon in the 3-body space}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Added first subscript 0 indicates that the relevant mass-squared  in minus component is 0.
\beq
\theta_{1-1'}~\rightarrow~~~~&&
q_{03}^\mu S_{0433' \mu} \rs (p_{03'} - p_{03})^\mu S_{0433' \mu} \\ 
\es (p_{03'} - p_{03})^\mu
\left(
   \varepsilon^*_{3\mu}  \  \varepsilon_{3'}  p_{04}                 
-  \varepsilon_{3'\mu}   \  \varepsilon_{3}^* p_{03'}
+ \varepsilon^*_3 \varepsilon_{3'}   \  p_{03'\mu} 
\right) \\
\es
(p_{03'} - p_{03})^\mu\varepsilon^*_{3\mu}  \  \varepsilon_{3'}  p_{04}                 
- (p_{03'} - p_{03})^\mu \varepsilon_{3'\mu}   \  \varepsilon_{3}^* p_{03'}
+ \varepsilon^*_3 \varepsilon_{3'}   \ (p_{03'} - p_{03})^\mu p_{03'\mu} \\
\es
p_{03'} ^\mu\varepsilon^*_{3\mu}  \  \varepsilon_{3'}  p_{04}                 
+
p_{03}^\mu \varepsilon_{3'\mu}   \  \varepsilon_{3}^* p_{03'}
-
\varepsilon^*_3 \varepsilon_{3'}   \ p_{03}^\mu p_{03'\mu} \\
\es
p_{03'} \varepsilon^*_{3}  \  \varepsilon_{3'}  p_{04}                 
+
p_{03} \varepsilon_{3'}   \  \varepsilon_{3}^* p_{03'}
-
\varepsilon^*_3 \varepsilon_{3'}   \ p_{03} p_{03'}  \\
\es
- p_{03'} \varepsilon^*_{3}  \  \varepsilon_{3'}  p_{03}                 
+
p_{03} \varepsilon_{3'}   \  \varepsilon_{3}^* p_{03'}
-
\varepsilon^*_3 \varepsilon_{3'}   \ p_{03} p_{03'} \ .
\eeq
\beq
\theta_{1-1'}~\rightarrow~~~~&&
q_{03}^\mu S_{0433' \mu} \rs -
\varepsilon^*_3 \varepsilon_{3'}   \ p_{03} p_{03'} \rs \delta_{3 \rm spin}\ p_{03} p_{03'} \ .
\eeq
\beq
\theta_{1'-1}~\rightarrow~~~~&&
q_{03}^\mu S_{043'3 \mu}^* \rs (p_{03} - p_{03'})^\mu S_{043'3 \mu} \\ 
\es (p_{03} - p_{03'})^\mu
\left(
    \varepsilon_{3'\mu}  \  \varepsilon_{3}^*  p_{04}                 
-   \varepsilon_{3\mu}^*   \  \varepsilon_{3'} p_{03}
+  \varepsilon_{3'} \varepsilon_{3}^*   \  p_{03\mu} 
\right) \\
\es
(p_{03} - p_{03'})^\mu    \varepsilon_{3'\mu}  \  \varepsilon_{3}^*  p_{04}                 
-  (p_{03} - p_{03'})^\mu \varepsilon_{3\mu}^*   \  \varepsilon_{3'} p_{03}
+ \varepsilon_{3'} \varepsilon_{3}^*   \  (p_{03} - p_{03'})^\mu p_{03\mu} \\
\es
p_{03}    \varepsilon_{3'}  \  \varepsilon_{3}^*  p_{04}                 
{\red +} p_{03'} \varepsilon_{3}^*   \  \varepsilon_{3'} p_{03}
-  \varepsilon_{3'} \varepsilon_{3}^*   \  p_{03'}p_{03} \\
\es
p_{03}    \varepsilon_{3'}  \  \varepsilon_{3}^*  p_{04}           
+ (p_{03}-p_{04}) \varepsilon_{3}^*   \  \varepsilon_{3'} p_{03}
-  \varepsilon_{3'} \varepsilon_{3}^*   \  p_{03'}p_{03} {\red ~ error?} \\
\es
p_{03}    \varepsilon_{3'}  \  \varepsilon_{3}^*  p_{04}                 
- p_{04} \varepsilon_{3}^*   \  \varepsilon_{3'} p_{03}
-  \varepsilon_{3'} \varepsilon_{3}^*   \  p_{03'}p_{03} {\red ~ no ~ error} \\
\es -  \varepsilon_{3'} \varepsilon_{3}^*   \  p_{03'}p_{03} \ .
\eeq
\beq
\theta_{1'-1}~\rightarrow~~~~&&
q_{03}^\mu S_{043'3 \mu}^* \rs -  \varepsilon_{3}^* \varepsilon_{3'}    \  p_{03} p_{03'} \rs \delta_{3 \rm spin}\ p_{03} p_{03'} \ .
\eeq
Now, 20250921 09:51 nie San Dimas, I can work out the $d_{04}^{\nu \mu}$. The factor $-  \varepsilon_{3}^* \varepsilon_{3'}$
is the gluon spin conservation factor $\delta_{\sigma_3} \sigma_{3'}$. The indices $\nu$ and $\mu$ are contracted with the quark 
and gluon currents, respectively.
\beq
 d^{\nu \mu}_{04} \es -g^{\nu \mu} + {p_{04}^\nu \eta^\mu + \eta^\nu p_{04}^\mu \over p_4^+} \ ,\\
p_{04}^\nu \es q_1^\nu + \2 \eta^\nu (p_{04}^- - q_1^-) 
\ , \
p_{04}^\mu \rs q_{03}^\mu + \2 \eta^\mu (p_{04}^- - q_{03}^-) \ .
\eeq
I need to write down the sum of the exchange and seagull terms. 
\beq
\label{Hsqg2 recall 2}
{\rm Seagull} \es -g^2 f_{LR} \int [33'11'] \ \tilde \delta_{c.a} \ \delta_{\rm spins} 
\
{ ( p_3^+ + p_{3'}^+ )  \ 2 \sqrt{p_1^+ p_{1'}^+}  \over (p_{3'}^+ - p_3^+)^2} 
\ \chi_{1}^\dagger T^{c_4} \chi_{1'} \, i f^{c_4 c_3 c_{3'}} \  a^\dagger_{3}  b_1^\dagger  b_{1'} a_{3'}\ ,
\eeq
\beq
{\rm Exchange}
\es
(f_{LR} - f_{LI} f_{IR}) \
\Delta_{LIR} \
g^2 
\int[11'33'] \, \tilde \delta_{c.a} \ \chi_1^\dagger  T^{c_4} \chi_{1'} \ \bar u_1 \gamma_\nu  u_{1'}
\left( d_{04}^{\nu \mu} \cE_{\mu \rm tr} + \eta^\nu S_\eta \right) 
{1 \over p_4} \, i f^{c_4 c_{3} c_{3'}}  \ \ 
a^\dagger_{3} b_1^\dagger  b_{1'} a_{3'} \ . \nn
\eeq
Since $f_{LI}f_{IR}$ makes the terms finite, I focus on terms $f_{LR}$. I extract common factors 
\beq
CF \es g^2 \ f_{LR} \ \int[11'33'] \ \tilde \delta_{c.a} \ \chi_1^\dagger  T^{c_4} \chi_{1'} \ i f^{c_4 c_3 c_{3'}} \ a^\dagger_{3}  b_1^\dagger  b_{1'} a_{3'} \ .
\eeq
The sum exchange plus seagull is
\beq
CF \left( {\rm  Seagull + Exchange} \right) \ ,
\eeq
where the remaining factors are
\beq
\label{Hsqg2 recall 2}
{\rm Seagull} \es - \delta_{\rm spins} 
\ { ( p_3^+ + p_{3'}^+ )  \ 2 \sqrt{p_1^+ p_{1'}^+}  \over (p_{3'}^+ - p_3^+)^2} \ , \\
{\rm Exchange}
\es
\ \bar u_1 \gamma_\nu  u_{1'}
\left( d_{04}^{\nu \mu} \cE_{\mu \rm tr} + \eta^\nu S_\eta \right) 
\left[ {\Delta_{LIR}  \over p_4} = -  \2 \left( {1 \over \rho_1} + {1 \over \rho_3 } \right) \right]\ .
\eeq
\beq
\cE_{\mu \rm tr}
\es
2 \theta_{1-1'} \, S_{0433' \mu}
+
2 \theta_{1'-1}  \, S_{043'3 \mu}^* \ , \\
S_\eta \es
{m_g^2 \over p_{4}^{+2}} \  (p_3^+ + p_{3'}^+) \ \varepsilon^*_3 \varepsilon_{3'} \ , \\
 d^{\nu \mu}_{04} \es -g^{\nu \mu} + {p_{04}^\nu \eta^\mu + \eta^\nu p_{04}^\mu \over p_4^+} \ ,\\
p_{04}^\nu \es q_1^\nu + \2 \eta^\nu (p_{04}^- - q_1^-) 
\ , \
p_{04}^\mu \rs q_{03}^\mu + \2 \eta^\mu (p_{04}^- - q_{03}^-) \ .
\eeq
I can write 
\beq
d^{\nu \mu}_{04} \es -g^{\nu \mu} + {1\over p_4^+}  \left[ q_1^\nu + \2 \eta^\nu (p_{04}^- - q_1^-)  \right]\eta^\mu
                                                         + {1\over p_4^+} \eta^\nu \left[ q_{03}^\mu + \2 \eta^\mu (p_{04}^- - q_{03}^-) \right]
\eeq
Contraction of $q_1$ with $\bar u_1 \gamma_\nu  u_{1'}$ yields 0. Contraction of $q_{03}$ with $\cE_{\mu \rm tr}$ gives
\beq
q_{03}^\mu \cE_{\mu \rm tr} \es
2 \theta_{1-1'} \,q_{03}^\mu S_{0433' \mu} + 2 \theta_{1'-1}  \, q_{03}^\mu S_{043'3 \mu}^*
\rs - {\red 2} \varepsilon^*_3 \varepsilon_{3'}   \ p_{03} p_{03'} \rs {\red 2} \delta_{\rm spin3}  \, p_{03} p_{03'} \ , \\
- 2 p_{03} p_{03'} \es (p_{03} - p_{03'})^2 
\rs
[ (p_3 - p_{3'}) + \eta m_g^2/(2p_{3'}^+) - \eta  m_g^2/(2p_{3}^+) ]^2  \\
\es
(p_3 - p_{3'})^2 + m_g^2(p_3^+ - p_{3'}^+) (1/p_{3'}^+ - /p_{3}^+) \\
\es
(p_3 - p_{3'})^2 + {m_g^2 \over p_{3'}^+ p_{3}^+}(p_3^+ - p_{3'}^+)^2 
\rs q_3^2 + {m_g^2 \over p_{3'}^+ p_{3}^+}(p_3^+ - p_{3'}^+)^2 \ .
\eeq
Therefore, using $j_{1\nu} = \bar u_1 \gamma_\nu  u_{1'}$,
\beq
j_{1\nu} \left( d_{04}^{\nu \mu} \cE_{\mu \rm tr} + \eta^\nu S_\eta \right) 
\es
 -j_{1\nu}  g^{\nu \mu} \cE_{\mu \rm tr}  
+  j_{1\nu}{1\over p_4^+}  \left[ \2 \eta^\nu (p_{04}^- - q_1^-)  \right] \eta^\mu\cE_{\mu \rm tr}  
\np j_{1\nu}{1\over p_4^+} \eta^\nu \left[ q_{03}^\mu + \2 \eta^\mu (p_{04}^- - q_{03}^-) \right]  \cE_{\mu \rm tr}  
 +  j_{1\nu} \eta^\nu S_\eta \\
\es
-j_{1\nu}  g^{\nu \mu} \cE_{\mu \rm tr}  
+ { j_1^+ \over p_4^+} \, \2  (p_{04}^- - q_1^-) \cE_{\rm tr}^+  
+ {j_1^+ \over p_4^+} \,\left[ q_{03}^\mu \cE_{\mu \rm tr}   + \2 (p_{04}^- - q_{03}^-) \cE_{\rm tr}^+ \right]   
 +  j_1^+ S_\eta \\
\es
-j_{1\mu}  \cE_{ \rm tr}^\mu  + { j_1^+ \over p_4^+}  T \ .
\eeq
The term $-j_{1\mu}  \cE_{ \rm tr}^\mu$ is free from small-$x$ singularities.
\beq
{\rm Exchange}
\es
j_{1\nu}
\left( d_{04}^{\nu \mu} \cE_{\mu \rm tr} + \eta^\nu S_\eta \right) 
\left[ {\Delta_{LIR}  \over p_4} = -  \2 \left( {1 \over \rho_1} + {1 \over \rho_3 } \right) \right] \\
\es
\left( -j_{1\mu}  \cE_{ \rm tr}^\mu  + { j_1^+ \over p_4^+}  T  \right)
\left[ {\Delta_{LIR}  \over p_4} = -  \2 \left( {1 \over \rho_1} + {1 \over \rho_3 } \right) \right] 
\eeq
\beq
T \es \2  (p_{04}^- - q_1^-) \cE_{\rm tr}^+
+  q_{03}^\mu \cE_{\mu \rm tr}   + \2 (p_{04}^- - q_{03}^-) \cE_{\rm tr}^+
+ p_4^+ S_\eta \ .
\eeq
\beq
S_{0433'}^+     \es  \varepsilon^*_3 \varepsilon_{3'}   \  p_{3'}^+ \rs - \delta_{\rm spin3} \, p_{3'}^+  
\ , \
S_{043'3}^{+*} \rs  \varepsilon_{3'} \varepsilon_{3}^*   \  p_{3}^+ \rs - \delta_{\rm spin3} \, p_{3}^+ \ , \\
\cE_{\rm tr}^+
\es
2 \theta_{1-1'} \, S_{0433'}^+
+
2 \theta_{1'-1}  \, S_{043'3}^{+*} 
\rs 
- 2 \theta_{1-1'} \  \delta_{\rm spin3} \, p_{3'}^+ 
- 2 \theta_{1'-1} \  \delta_{\rm spin3} \, p_{3}^+ \\
\es -2 p_{\overline 3}^+ \  \delta_{\rm spin3} \ , \\
q_{03}^\mu \cE_{\mu \rm tr} \es {\red 2} \delta_{\rm spin3}  \, p_{03} p_{03'} \ , \\
p_4^+ S_\eta \es
- {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \ \delta_{\rm spin3}  \ .
\eeq
\beq
T \es \2  (p_{04}^- - q_1^-) ( -2 p_{\overline 3}^+ \  \delta_{\rm spin3})
+ {\red 2}  \delta_{\rm spin3}  \, p_{03} p_{03'}   + \2 (p_{04}^- - q_{03}^-)( -2 p_{\overline 3}^+ \  \delta_{\rm spin3}) 
 - {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \ \delta_{\rm spin3} \ .
\eeq
\beq
T \es  T_s \ \delta_{\rm spin3} \ , \\
T_s \es 
- (p_{04}^- - q_1^- + p_{04}^- - q_{03}^-) p_{\overline 3}^+
+  {\red 2} p_{03} p_{03'} - {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \ .
\eeq

***
Remaining
\beq
\label{Hsqg2 recall 3}
{\rm Seagull} \es - \delta_{\rm spins} 
\ { ( p_3^+ + p_{3'}^+ )  \ 2 \sqrt{p_1^+ p_{1'}^+}  \over (p_{3'}^+ - p_3^+)^2} 
\rs 
- \delta_{\rm spins} 
\ { ( p_3^+ + p_{3'}^+ )  \ j_1^+  \over (p_{3'}^+ - p_3^+)^2} \ .
\eeq
\beq
{\rm Exchange}
\es
j_{1\nu} 
\left( d_{04}^{\nu \mu} \cE_{\mu \rm tr} + \eta^\nu S_\eta \right) 
\left[ {\Delta_{LIR}  \over p_4} = -  \2 \left( {1 \over \rho_1} + {1 \over \rho_3 } \right) \right]\\
\es
\left( -j_{1\mu}  \cE_{ \rm tr}^\mu  + { j_1^+ \over p_4^+}  T  \right)
\left( {1 \over -2 \rho_1} + {1 \over -2 \rho_3 } \right) 
\eeq
The term 
\beq
-j_{1\mu}  \cE_{ \rm tr}^\mu  
\left( {1 \over -2 \rho_1} + {1 \over -2 \rho_3 } \right) 
\eeq
is free from small-$x$ singularities. It does involve transverse momenta, limited
by the RGPEP factor $f_{LR}$ and the test wave functions. I include $\delta_{\rm spins}$ and $j_1^+$ in the common factors
\beq
CF \tl g^2 \ f_{LR} \ \int[11'33'] \ \tilde \delta_{c.a} \ \chi_1^\dagger  T^{c_4} \chi_{1'} \ i f^{c_4 c_3 c_{3'}} \ 
\delta_{\rm spins} \ j_1^+  \ a^\dagger_{3}  b_1^\dagger  b_{1'} a_{3'} \ .
\eeq
The terms that diverge for $p_4 \to 0$ in the sum Seagull + Exchange have the form 
\beq
\label{qg3bodyInteraction1}
&&
CF \left( {\rm  Seagull + Exchange} \right) 
\rs CF\left[ 
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} + { 1 \over p_4^+}  T_s \left( {1 \over -2 \rho_1} + {1 \over -2 \rho_3 } \right) \right] \\
\es CF\left\{
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} + { 1 \over p_4^+} \left[ - (p_{04}^- - q_1^- + p_{04}^- - q_{03}^-) p_{\overline 3}^+
+   {\red 2}p_{03} p_{03'} - {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \right]
\left( {1 \over -2 \rho_1} + {1 \over -2 \rho_3 } \right) \right\} \ .
\label{qg3bodyInteraction2}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ \bf Evaluation of quark-gluon interaction in Eqs.~(~\ref{qg3bodyInteraction1}) and (\ref{qg3bodyInteraction2}) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I rewrite $\{ . . . \}$ in Eq.(\ref{qg3bodyInteraction2}),
\beq
\{ . . . \} \es
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} +  \left[ (p_{04}^- - q_1^- + p_{04}^- - q_{03}^-) p_{\overline 3}^+
-   {\red 2}p_{03} p_{03'} + {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \right]
 {\rho_1 + \rho_3  \over 2 \rho_1 \rho_3 p_4^+} \\
\es \theta_{1-1'} A +  \theta_{1'-1} B \ .
\eeq
\beq
A \es 
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} +  \left[ (p_{04}^- - q_1^- + p_{04}^- - q_{03}^-) p_{3'}^+
-   {\red 2}p_{03} p_{03'} + {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \right]_A
 {\rho_1 + \rho_3  \over 2 \rho_1 \rho_3 p_4^+} \ , \\
B \es
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} +  \left[ (p_{04}^- - q_1^- + p_{04}^- - q_{03}^-) p_{3}^+
-   {\red 2}p_{03} p_{03'} + {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \right]_B
 {\rho_1 + \rho_3  \over 2 \rho_1 \rho_3 p_4^+} \ .
\eeq
\beq
[ . . . ]_A \es
\left(2p_4^- - 2 m_g^2/p_4^+ - q_1^-  - p_{3'}^- + m_g^2/p_{3'}^+  + p_{3}^- - m_g^2/p_{3}^+ \right) p_{3'}^+ - {\red 2}p_{03} p_{03'} + {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \\
\es
(2p_4^- - q_1^- - q_3^-)p_{3'}^+  - 2 m_g^2p_{3'}^+/p_4^+    + (m_g^2/p_{3'}^+   - m_g^2/p_{3}^+) p_{3'}^+  - {\red 2}p_{03} p_{03'} + {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \\
\es
(2p_4^- - q_1^- - q_3^-)p_{3'}^+ + {m_g^2 \over p_{3}^+} (p_{3}^+ - p_{3'}^+)   
- {\red 2}p_{03} p_{03'} - {m_g^2 \over p_{4}^{+}} p_4^+\\
\es
(p_4^- - q_1^- + p_4^- - q_3^-)p_4^+ {p_{3'}^+ \over p_4^+} - m_g^2 {p_{3'}^+ \over p_{3}^+} 
- {\red 2}p_{03} p_{03'} \\
\es
\left(m_g^2 + q_1^{\perp 2} - q_1^-q_1^+  + m_g^2 + q_3^{\perp 2} - q_3^- q_3^+ \right) {p_{3'}^+ \over p_4^+} - m_g^2 {p_{3'}^+ \over p_{3}^+} 
- {\red 2}p_{03} p_{03'} \\
\es
\left(\rho_1 + \rho_3 \right) {p_{3'}^+ \over p_4^+} - m_g^2 {p_{3'}^+ \over p_{3}^+} 
- {\red 2}p_{03} p_{03'} 
\eeq
\beq
[ . . . ]_B \es
(p_{04}^- - q_1^- + p_{04}^- - q_{03}^-) p_{3}^+
-   {\red 2}p_{03} p_{03'} + {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \\
\es
\left( 2p_{4}^- - 2 m_g^2/p_4^+ - q_1^-  - p_{3}^- + m_g^2/p_3^+  + p_{3'}^- - m_g^2/p_{3'}^+ \right) p_{3}^+
-   {\red 2}p_{03} p_{03'} + {m_g^2 \over p_{4}^{+}} \  (p_3^+ + p_{3'}^+) \\
\es
\left( 2p_{4}^-  - q_1^-  - p_{3}^- + p_{3'}^- \right) p_4^+ p_{3}^+/p_4^+ + m_g^2(p_{3'}^+ - p_3^+) /p_{3'}^+ 
-   {\red 2}p_{03} p_{03'} + {m_g^2 \over p_{4}^{+}} \  (- p_3^+ + p_{3'}^+) \\
\es
\left( 2p_{4}^-  - q_1^-  - q_{3}^- \right) p_4^+ p_{3}^+/p_4^+ + m_g^2(p_{3'}^+ - p_3^+) /p_{3'}^+ 
-   {\red 2}p_{03} p_{03'} - {m_g^2 \over p_{4}^{+}} p_4^+ \\
\es
\left( \rho_1 + \rho_3 \right) p_{3}^+/p_4^+ + m_g^2(p_{3'}^+ - p_3^+) /p_{3'}^+ 
-   {\red 2}p_{03} p_{03'} - {m_g^2 \over p_{4}^{+}} p_4^+ \\
\es
\left( \rho_1 + \rho_3 \right) {p_{3}^+ \over p_4^+} -  m_g^2 { p_3^+ \over p_{3'}^+} -   {\red 2}p_{03} p_{03'} \ .
\eeq
With these results for the brackets,
\beq
A \es 
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} +  \left[ \left(\rho_1 + \rho_3 \right) {p_{3'}^+ \over p_4^+} 
- m_g^2 {p_{3'}^+ \over p_{3}^+} - {\red 2}p_{03} p_{03'}  \right]_A
 {\rho_1 + \rho_3  \over 2 \rho_1 \rho_3 p_4^+} \ , \\
B \es
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} +  \left[ \left( \rho_1 + \rho_3 \right) {p_{3}^+ \over p_4^+} 
-  m_g^2 { p_3^+ \over p_{3'}^+} -   {\red 2}p_{03} p_{03'}  \right]_B
 {\rho_1 + \rho_3  \over 2 \rho_1 \rho_3 p_4^+} \ .
\eeq
In $A$, with $\theta_{1-1'}$,
\beq
- {\red 2}p_{03} p_{03'} \es -2 [p_3 - \eta m_g^2/2p_{3}^+)] [(p_{3'} - \eta m_g^2/(2p_{3'}^+)] 
\rs
-2 p_3 p_{3'} + m_g^2 { p_3^+ \over  p_{3'}^+ }   +  m_g^2{ p_{3'} \over p_{3}^+} \\
\es   (p_{3'} - p_3)^2 - 2 m_g^2 + m_g^2 { p_3^+ \over  p_{3'}^+ }  + m_g^2{ p_{3'} \over p_{3}^+} 
\rs  (p_{3'} - p_3)^2 + m_g^2 { (p_{3'}^+ - p_3^+ )^2 \over p_3^+ p_{3'}^+} \\
\es  (p_{3'} - p_3)^2 + m_g^2 { p_4^{+2} \over p_3^+ p_{3'}^+} 
\rs q_3^2 + m_g^2 { p_4^{+2} \over p_3^+ p_{3'}^+} \ .
\eeq
In $B$, with $\theta_{1'-1}$,
\beq
- {\red 2}p_{03} p_{03'} \es  (p_{3} - p_{3'})^2 + m_g^2 { p_4^{+2} \over p_3^+ p_{3'}^+} 
\rs q_3^2 + m_g^2 { p_4^{+2} \over p_3^+ p_{3'}^+} \ .
\eeq
Thus,
\beq
A \es 
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} +  \left[ \left(\rho_1 + \rho_3 \right) {p_{3'}^+ \over p_4^+} 
- m_g^2 {p_{3'}^+ \over p_{3}^+} + q_3^2 + m_g^2 { p_4^{+2} \over p_3^+ p_{3'}^+} \right]_A
 {\rho_1 + \rho_3  \over 2 \rho_1 \rho_3 p_4^+} \ , \\
B \es
- { p_3^+ + p_{3'}^+  \over p_4^{+2}} +  \left[ \left( \rho_1 + \rho_3 \right) {p_{3}^+ \over p_4^+} 
-  m_g^2 { p_3^+ \over p_{3'}^+} + q_3^2 + m_g^2 { p_4^{+2} \over p_3^+ p_{3'}^+} \right]_B
 {\rho_1 + \rho_3  \over 2 \rho_1 \rho_3 p_4^+} \ .
\eeq




****

********

\vskip1in
{\red *******}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Old text discussing  the 3-body space}
\label{H_3body}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
So far I developed an equation in the dominant 2-body component alone. But there also exist terms of 2nd order due to the components 
with one effective gluon. If I used perturbative expansion up to 2nd order only, I would return to the 2body equation like in QED
with the coupling constant $\alpha$ multiplied by the color factor. But gluons are different from photons. They are blocked already
in 2nd order by the gluon-quark interactions and more interactions . This 
is the same as if one omitted other components than the dominant one. But if we want terms of second order, we can do so with mass $m_g \to m_G$ in 3-body sector. I think I finally 
understand what I am doing.

The interaction terms in $H_s$ change the number of effective particles with much less strength than indicated solely by 
the smallness of the coupling constant. The reason is that the vertex form factors limit the range of interactions in the FF 
energy. Nevertheless, in addition to the dominant quark-antiquark component, a quarkonium state includes also the 
components containing a quark-antiquark pair and 1 effective gluon, 2 gluons, and so on. Components with more 
heavy quarks than 2 are insignificant when the quark mass is large. Together, all these components form a vector in 
an {\it a priori} infinite-dimensional space. Corrections to the dynamics of the dominant effective 2-body component 
due to these additional components can be computed using the general expression for an effective Hamiltonian acting 
only in a subspace of states~\cite{Wilson:1970tp}. We discuss the lowest order, using the 2nd-order of the weak-coupling 
expansion. {\bf \red Insert here a comment on $m_G$} 

\beq
_s\langle 12| H_s | 1'2' \rangle_s 
\es  
_s\langle 12| \left\{ H_f + H_s^{(2)} + {1 \over 2} H_s^{(1)} \left[ (P_{12}^- - H_f)^{-1} 
+ (P_{1'2'}^- - H_f)^{-1} \right] H_{t }^{(1)} \right\}|1'2'\rangle_s \ .
\eeq
The same matrix is obtained using the operator $\cH$ and bare states $|12\rangle = b_1^\dagger d_2^\dagger |0\rangle$,
\beq
\label{12cH12 POM}
\langle 12| \cH | 1'2' \rangle 
\es  
\langle 12| \left\{ \cH_f + \cH^{(2)} + {1 \over 2} \cH^{(1)} \left[ (P_{12}^- - \cH_f)^{-1} 
+ (P_{1'2'}^- - \cH_f)^{-1} \right] \cH^{(1)} \right\}|1'2'\rangle \ .
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Eigenvalue problems for $m_g \to 0$} 
\label{mgto0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The RGPEP form factor $f^t_{\bar p, \bar q}$ in Eq.~(\ref{frapprox1}),
associated with emission or absorption of a gluon with the FF energy 
$p_g^-$, behaves for extremely small $p_g^+$ as
\beq
\label{frapprox1gluon1}
f^t_{\bar p, \bar q} & \sim & e^{ - t^2 \, (p_g^- \, \pi_{ca}) ^2 } 
                              \rs e^{ - t^2 \, [ (p_g^{\perp 2} + m_g^2)\, \pi_{ca}/ p_g^+] ^2 } \ .
\eeq
For $|p_g^\perp| \gg m_g$, the limit $m_g \to 0$ presents no problem. 
But when for $|p_g^\perp| \lesssim m_g$, which is the infrared regime, the form 
factor suppresses contributions from gluons with small momentum $p_g^+$ by 
a factor of the type
\beq
\label{frapprox1gluon2}
f^t_{\bar p, \bar q} & \sim & 
e^{ - \epsilon^2/x^2 } \ , \
\epsilon \rs t \, m_g^2 \, \pi_{ca}/ P^+ \ ,
\eeq
where $P^+$ is the momentum of a system under 
consideration and $x$ is the fraction of that momentum 
carried by the gluon. The fraction $x$ is thus effectively 
limited from below by a number on the order of $\epsilon$. 
For a fixed gluon mass parameter $m_g$, the severe 
divergences due to the gluon factors $1/x^2$ are thus 
tamed in a way that is qualitatively illustrated by the formula 
\beq
\int_\epsilon^{x_0} dx/x^2 \es 1/\epsilon - 1/x_0 \ ,
\eeq
where $x_0$ is some positive number smaller than 1. This 
result is finite but diverges as $1/m_g^2$ when $m_g \to 0$. 
When the divergences due to the factors $1/x^2$ cancel out, 
one is left with a logarithmic dependence on $\epsilon$,
\beq
\int_\epsilon^{x_0} dx/x \es \log(x_0/\epsilon) \ .
\eeq
We inspect the logarithmic dependence of the quarkonium 
eigenvalue problem for the Hamiltonian $H_{LR}$ given in 
Eq.~(\ref{HLR1}). In particular, we examine the relationship 
between the quarkonium and single-quark dynamics. The 
relationship holds through the quark self-interactions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Single-quark eigenvalue problem for $m_g \to 0$}
\label{singleQ}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The single-quark eigenvalue problem involes self-interaction terms.
As observed in discussion of Eqs.~(\ref{sigmaqint}) and Eq.~(\ref{SigmaQBS}), 
the counterterm integrand $\sigma^{(2)} = 1/x^2$ is sufficient to remove the 
2nd-order severe small-$x$ divergence from the self-interactions in scattering 
amplitudes and bound-state eigenvalue problems when $m_g \neq 0$. Moreover, 
the modification $\sigma^{(2)} \to \sigma^{(2)} + 1/[2(1-x)]$ yields simultaneous 
cancellation of the ultraviolet transverse quadratic divergence. Thus the divergences 
$\sim 1/r$ in the quark self-interactions for $r \to 0$ are removed by the 
counterterm $\delta m_{i \, \rm sev}^2$ given in Eq.~(\ref{deltami2}). The 
remaining logarithmic divergence requires an additional counterterm 
$\delta m_{i \, \rm ln}^2$, see Sec.~\ref{deltamiln}. 

If free quarks with a finite, definite masses existed, one could demand that 
these masses solve the eigenvalue Eq.~(\ref{Htep}) for states with the 
single-quark quantum numbers. But the single quarks are not observed, 
which might be explained if the quark mass eigenvalues are sufficiently 
large. The single-quark eigenstates must have the form 
\beq
|\psi \rangle \es |1\rangle + |2\rangle + . . . ~\ ,
\eeq
where the component $|1\rangle$ is a state of one effective quark,
$|2\rangle$ is a state with addition of one effective gluon and
the dots indicate components with more effective particles. 
Using the single-quark counterpart of Eq.~(\ref{HLR1}), one obtains
the eigenvalue equation for the component $|1\rangle$,
\beq
\label{HLRQ}
H_{1} |1\rangle \es p_E^- |1\rangle \ ,
\eeq
where 
\beq
H_{1} \es H_f + H_{t 11}^{(2)} + H_{t12}^{(1)} \, (p^- - H_f)^{-1} H_{t21}^{(1)} \ .
\eeq
The effective quark in the state $|1\rangle$ carries the 
kinematic momentum with components $p^+$ and $p^\perp$.
So, the eigenvalue is written in the form
\beq
p_E^- \es (p^{\perp 2} + m_E^2)/p^+ \ .
\eeq
Given the RGPEP solution for the Hamiltonian, one obtains
the eigenvalue condition
\beq
\label{HLR1pompom}
&&
\cH_{f1} + f_{11} \, \left[ \cH^{(2)}_{0,11} + \Delta_{121}\right] \,  \cH^{(1)}_{0,12} \, \cH^{(1)}_{0,21}
\np
f_{12} \cH_{0,12}^{(1)} \left[ (p^- - \cH_{f2})^{-1}  - \Delta_{121}\right]  \, f_{21} \cH_{0,21}^{(1)} 
\rs p_E^- \ ,
\eeq
where 
\beq
\cH_{f1} \es p_1^- \rs  (m_i^2 + p^{\perp 2}) /p^+   ~~~~~ f_{11} \rs 1 \ , \\
\cH^{(2)}_{0,11} \es \delta m_{i \, \rm tot}^2 /p^+ ~~~~ {\rm see ~Eq.~(\ref{deltamitot}) } \ , \\
f_{12}f_{21} \es f_{12}^2 ~~~~ , ~~~~ \Delta_{121} \rs (p_1^- - p_q^- -p_g^-)^{-1} \ , \\
(p^- - \cH_{f2})^{-1} \es \left[ (p_E^- ~ {\rm or} ~p_1^-) - p_q^- - p_G^- \right]^{-1} \ .
\eeq
The eigenvalue $p_E^-$ in the denominator results from the TD procedure. 
If the eigenvalue were infinite, the corresponding term would vanish and the 
other terms would have to yield the infinite result for the eigenvalue. One can 
also consider the operation $R$, {\it e.g.} see [Wilson], which in second order
of expansion in powers of the coupling constant yields $p_1^-$ in place of 
$p_E^-$ in the denominator. We pursue the latter case since the operation 
$R$ is precisely defined order-by-order and does not require knowledge of
the eigenvalue. The notation $p_g$ and $p_G$ for the gluon momenta indicates
that $p_g^2 = m_g^2$ and $p_G^2 = m_{iG}^2$. The gluon-mass ansatz 
$m_{iG}$ carries the subscript $i$ to indicate its dependence on the quark 
mass $m_i$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gluon mass ansatz in single-quark eigenstates}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The single-quark eigenvalue condition of Eq.~(\ref{HLR1pompom}) implies
\beq
\label{sqev}
m_i^2  + \delta m_{i \, \rm tot} + \Sigma_q^{(2)} + p^+\Delta p_E^- \es m_E^2 \ ,
\eeq
where $\delta m_{i \, \rm tot} ^2$ is the counterterm given in Eq.~(\ref{deltamitot}),
$\Sigma_q^{(2)}$ is the self-interaction term of Eq.~(\ref{quarkCTSigma}), and
\beq
\label{HLR1if1}
\Delta p_E^- 
\es
f_{12} \cH_{0,12}^{(1)} 
\left( {1 \over p^- - p_q^- - p_G^- } - {1 \over p^- - p_q^- - p_g^- } \right)
f_{21} \, \cH_{0,21}^{(1)} \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Computation of $\Delta p_E^-$ dependence on $m_g$}
\label{DeltapE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In detail,
\beq
\label{SigmaSingleQuark2text}
\Delta p_E^- 
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x)p^+ } 
\left({N \ f^{t 2}_{p_q p_g,p} \over p^- - p_q^- - p_g^- - A/p^+} 
-      {N \ f^{t 2}_{p_q p_g,p} \over p^- - p_q^- - p_g^- }\right) \ ,
\eeq
\beq
A \es (m_{iG}^2 - m_g^2)/x \ , \  B \rs (1-x)(m_{iG}^2 - m_g^2) \rs x(1-x) A \ , \\
N \eqv N_g
\rs 4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x) \ ,
\eeq
\beq
N_G \eqv
     4 [ k^{\perp 2} + (1-x)m_{iG}^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x) \ ,\\
N_g \es N_G - 4 (1-x)(m_{iG}^2 - m_g^2)/x^2 \rs N_G - 4 B/x^2 \ , 
\eeq
\beq
D \eqv D_g
\rs
x(1-x) p^+( p^-  - p_q^- - p^-_g) \rs x(1-x) (m_i^2 - \cM_{qg}^2) \ , \\
D_g \es - [ k^{\perp 2} + x^2 m_i^2 + (1-x) m_g^2 ] \ , 
\eeq
\beq
D_G \eqv
x(1-x) p^+( p^-  - p_q^- - p^-_G) \rs x(1-x) (m_i^2 - \cM_{qG}^2) \\
\es D_g + x(1-x) p^+ (p_g-p_G) \rs D_g - B \ , \\
D_G \es - [ k^{\perp 2} + x^2 m_i^2 + (1-x) m_{iG}^2 ] \ .
\eeq 
Then
\beq
\label{SigmaSingleQuark3}
\Delta p_E^- 
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3  } 
\left({N_G - 4 B/x^2  \over D_G} 
-      {N_g \over D_g }\right) f^{t 2}_{p_q p_g,p} \\
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3  } 
\left({N_G \over D_G} 
-      {N_g \over D_g }  -  {4 \over x^2} \, {B \over D_G} \right) f^{t 2}_{p_q p_g,p} \ .
\eeq
Now I can use the formulas
\beq
\sigma_q^{(2)}  \es {1 \over x^2} + {1 \over 2(1-x)} \ , \\
{N_g \over D_g } + 4\sigma_q^{(2)} \es 
{ 4 m_i^2 + 2 m_g^2 \over - D_g } \ , \\
{N_G \over D_G } + 4\sigma_q^{(2)} \es 
{ 4 m_i^2 + 2 m_{iG}^2 \over  - D_G } \ .
\eeq
I write
\beq
\label{SigmaSingleQuarktrick1}
\Delta p_E^- 
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3  } 
\left( { 4 m_i^2 + 2 m_{iG}^2 \over  - D_G }
      - { 4 m_i^2 + 2 m_g^2 \over - D_g }  -  {4 \over x^2} \, {B \over D_G} \right) f^{t 2}_{p_q p_g,p} \ ,
\eeq
combined with the change of integration variables in Eq.~(\ref{CRFveckTEXT}), 
\beq
\label{CRFveckTEXTrepeat}
\int [ x k ] \es \int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x) }
\rs
\int  {d^3 k \over  2(2\pi)^3 } \, \left( {1\over E_g} + {1 \over E_q} \right) 
\rs 
\int  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ , \\
\eeq
\beq
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} 
\es
\int  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ x(1-x) \ .
\eeq
The change of variables is made only for gluons with $m_g$,
even though a more advanced calculations with running
gluon mass may lead to variables proper for the gluons
with effective mass. The factor $x(1-x)$ cancels out and the 
eigenvalue contribution is 
\beq
\label{SigmaSingleQuarktrick2}
\Delta p_E^- 
\es
{ g^2  C_F \over p^+}
\int_{m_i + m_g}^\infty  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2} 
\nt
\left[ { 4 m_i^2 + 2 m_{iG}^2 \over  \cM_{qG}^2 - m_i^2 }
      - { 4 m_i^2 + 2 m_g^2 \over   \cM_{qg}^2 - m_i^2 }  
     + {4 \over x^2} \, {B \over \cM_{qG}^2 - m_i^2 } \right] \ .
\eeq
In terms of $E$, with $M^2 = m_{iG}^2 - m_g^2$,
\beq
\label{SigmaSingleQuarktrick3}
\Delta p_E^- 
\es
{ g^2  C_F \over p^+}
\int_{m_i + m_g}^\infty  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2} 
\nt
\left[ { 4 m_i^2 + 2 m_{iG}^2 \over  E^2 - m_i^2 + M^2/x}
      - { 4 m_i^2 + 2 m_g^2 \over   E^2 - m_i^2 }  
     + {4 \over x^2} \, { (1-x) M^2 \over E^2 - m_i^2 + M^2/x} \right] \ .
\eeq
The question is how this correction depends on $m_g \to 0$. Does $\Delta p_E^-$ diverge?
\beq
\label{SigmaSingleQuarktrick4}
\Delta p_E^- 
\es
{ g^2  C_F \over p^+}
\int_{m_i + m_g}^\infty  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2} 
\nt
\left[ { x(4 m_i^2 + 2 m_{iG}^2) \over M^2 + x( E^2 - m_i^2) }
      - { 4 m_i^2 + 2 m_g^2 \over   E^2 - m_i^2 }  
      + { 4 M^2 (1-x)/x \over  M^2 + x(E^2 - m_i^2) } \right] \ .
\eeq
The key are the relations between $x$, $k$ and $E$.
\beq
x \es  (E_g + k \cos \theta) /E 
\rs
{ \sqrt{m_g^2 + k^2} + k \cos \theta \over E_g + E_q}
\rs  
{ \sqrt{m_g^2 + k^2} + k \cos \theta \over  \sqrt{m_g^2 + k^2} + \sqrt{m_i^2 + k^2} } \ .
\eeq
The momentum $k = |\vec k \, |$ is limited 
because $E$ is limited by the RGPEP form factor. 
There is also the bound  $E \ge m_i + m_g$.

The first term in the bracket has a finite limit when $m_g \to 0$
because $0 < x <1$. The last term can be simplified by dropping
$-4M^2$ in its numerator as not diverging. If there is a singularity 
it must come from the part of the square bracket,
\beq
\label{SigmaSingleQuarktrick5}
\Delta p_{E 2}^- 
\es
{ g^2  C_F \over p^+}
\int_{m_i + m_g}^\infty  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2} 
\nt
\left[ - { 4 m_i^2 + 2 m_g^2 \over   E^2 - m_i^2 }  + { 4 M^2/x \over  M^2 + x(E^2 - m_i^2) } \right] \ .
\eeq
The first term in the bracket can get large when $E \to m_i +m_g$.
This happens only when $k \to 0$, where
\beq
E^2 - m_i^2 \es (E + m_i)(E-m_i) \sim (2m_i + E_g)(E_g + E_q - m_i) \ .
\eeq
The leading denominator term for small $k$ is then $2 m_i E_g$, while in the numerator of the integral there is $k$.
The ratio $k/\sqrt{m_g^2 + k^2} = 1 /\sqrt{1 + m_g^2 /k^2} < 1$ for finite $k$. One is left only with the term
\beq
\label{SigmaSingleQuarktrick6}
\Delta p_{E1}^- 
\es
{ g^2  C_F \over p^+}
\int_{m_i + m_g}^\infty  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2} 
\nt
{ 4 M^2/x \over  M^2 + x(E^2 - m_i^2) } \ .
\eeq
The only source of divergence can be small $x$.
For sizable $M$, actually, only in the limit $M \to \infty$, the only source of divergence is
\beq
\label{SigmaSingleQuarktrick6}
\Delta p_{E0}^- 
\es
{ g^2  C_F \over p^+}
\int_{m_i + m_g}^\infty  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2} 
\ \left[ {4 \over x } = {4 E \over E_g + k \cos \theta} \right] \ .
\eeq
Carrying out the integrals over angles I get
\beq
\label{SigmaSingleQuarktrick7}
\Delta p_{E0}^- 
\es
{ g^2  C_F \over p^+}
\int_{m_i + m_g}^\infty  { 2 \pi \, 4 E dE \over  2(2\pi)^3 } \ e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2} 
\ \left[ \int_{-1}^1 {k dz  \over E_g + k z}  =  \ln {E_g + k \over E_g - k} \right] \ .
\eeq
\beq
\ln {E_g + k \over E_g - k} \es \ln { (E_g + k)^2 \over m_g^2}  \ , \
dE \rs (k/E_g + k/E_q) dk \to (1 + k/E_q) dk \ .
\eeq
In the limit $m_g \to 0$,
\beq
\label{SigmaSingleQuarktrick8}
\Delta p_{E0}^- 
\es
{ g^2  C_F \over p^+}
\int_{m_i}^\infty  { 2 \pi \, 4 E dE \over  2(2\pi)^3 } \ e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2} 
2 \ln { 2k \over m_g}  \ .
\eeq
In the limit I have 
\beq 
2k \es (E^2 - m_i^2)/E \ , \ u \rs E^2 - m_i^2 \ .
\eeq
\beq
\label{SigmaSingleQuarktrick9}
\Delta p_{E0}^- 
\es
{ g^2  C_F \over 2 \pi^2 p^+}
\int_0^\infty  du \, e^{ - 2 t^2 u^2 (\pi^+/p^+)^2} 
\ln { u \over m_g \sqrt{u+ m_i^2} } \ .
\eeq
Using $u = v m_i^2$ 
\beq
\label{SigmaSingleQuarktrick10}
\Delta p_{E0}^- 
\es
{ g^2  C_F \over 2 \pi^2 p^+} m_i^2  
\int_0^\infty  dv \, e^{ - 2 t^2 m_i^4 (\pi^+/p^+)^2 v^2} \ 
\ln { m_i v \over m_g \sqrt{1+ v^2} } \ .
\eeq
Using $a = \sqrt{2}\, t m_i^2 \pi^+/p^+$,
\beq
\label{SigmaSingleQuarktrick11}
\Delta p_{E0}^- 
\es
{ g^2  C_F \over 2 \pi^2 p^+} {m_i^2 \over a}
\int_0^\infty  a \, dv \, e^{ - (a v)^2} \ 
\ln { m_i v \over m_g \sqrt{1+ v^2} } \ .
\eeq
Finally, using $w = a v$,
\beq
\label{SigmaSingleQuarktrick12}
\Delta p_{E0}^- 
\es
{ g^2  C_F \over 2 \pi^2 p^+} {p^+/\pi^+  \over \sqrt{2}\, t } 
\int_0^\infty  dw \, e^{ - w^2} \ 
\left[  \ln { m_i \over m_g }  +  \ln { w \over \sqrt{a^2+ w^2} } \right]\ .
\eeq
There is a divergence when $m_g \to 0$ of the form
\beq
\label{SigmaSingleQuarktrick12}
\Delta p_{E0}^- 
\es
{ g^2  C_F \over 2 \pi^2 p^+} 
{p^+/\pi^+  \over \sqrt{2}\, t } \, 
{\sqrt{\pi} \over 2} \  \ln { m_i \over m_g } \ .
\eeq
Simplified,
\beq
\label{SigmaSingleQuarktrick13}
p^+ \Delta p_{E0}^- 
\es
{ g^2  C_F \over (2 \pi)^2 } 
{p^+ \over  \pi^+ \, t } \, 
\sqrt{\pi \over 2} \  \ln { m_i \over m_g } \ .
\eeq
This looks almost the same as Eq.~(51) for $\tilde \cM_i^2$ in 
[K. Serafin et al., Phys. Rev. D 109, 016017 (2024)], where the 
tilde stands for the slanted $\cM$ in Eq.~(51). But in a product
under the logarithm one can have various factors depending 
on the arrangement of other terms.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Other single-particle eigenvalue problems}
\label{sqep}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is shown in previous sections that in the case of a heavy single-quark 
eigenvalue problem, the eigenmass can tend to infinity when $m_g \to 0$. 
The divergence is logarithmic, $\ln (m_i/m_g)$. In the cases of a single-light 
quark and a single-gluon eigenvalue problems one faces the complication 
that the scale of $\Lambda_{\rm QCD}$ is much greater than the masses in 
$H_f$. In distinction from the heavy quark case, the assumption that a state 
of a single effective particle is the dominant component of a single-quark 
or single-gluon eigenstate is in doubt. The 2nd-order computation of the Hamiltonian that 
is reasonable for heavy quarkonia and the 3rd-order computation 
that demonstrates asymptotic freedom in the Hamiltonian, do 
not provide the required understanding of the binding dynamics. 
To address the issue of how the quark and gluon operators and
their interactions ``dress up'' in the RGPEP evolution, to find out
if for large $s$ a constituent picture emerges, one needs to solve 
Eq.~(\ref{RGPEPmaster}) including at least the 4th order terms. 
This is the lowest order in which the running coupling enters the 
Hamiltonian interaction terms that count in the eigenvalue problems. 
The need to carry out the 4th-order computations is precisely the 
reason for developing a method for handling the severe small-$x$ 
divergences in the presence of mass for gluons described in this 
paper.

The self-interaction terms in the single-quark eigenvalue problems
can be adjusted to remove the ultraviolet transverse divergences.
Then one can subtract the ultraviolet logarithmic divergences, to be
left with a finite eigenvalue condition that depends on $m_g$. The 
issue may appear similar to the Bloch-Nordsieck issue of photon 
divergence in the scattering amplitudes involving electrons, whose
cancellation requires the inclusion of the electron-photon contributions. 
{\red \bf Check Bloch-Nordsieck argument}
Correspondingly, in the electron eigenvalue problem the contribution 
with one photon is needed. But it does not include the photon mass 
and the term that diverges here is absent. There is no blocking of
the photon. In contrast with QED, the effective gluons are assumed
blocked by their non-Abelian interactions. The blocking is modeled  
here using the sizable gluon mass $m_G$ in the component with one 
gluon, possibly representing the effects of non-Abelian interactions 
in all components with more quanta. Therefore, the RGPEP and 
Wilsonian elimination no longer cancel each other in the quark 
self-interaction. The resulting term diverges in the limit $m_g \to 0$. 
Could this be a signal of how to handle confinement, visible already 
in 2nd order? One can assume this is the case and proceed to the 
limit $m_g \to 0$ in the case of quarkonium.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Elements of the quarkonium eigenvalue problem}
\label{DetailsEVforHLR}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The quark-antiquark Hamiltonian $H_{LR}$ in Eq.~(\ref{HLR1}) 
has 6 parts,
\beq
\label{HLR1pom1}
H_{LR} \es A + B + C \ .
\eeq
{\red 20250204 15:43 wto Ekologiczna} I indicate how these 3 terns come about.
\beq 
\label{HLRDelta}
H_{LR} \es [m^2]_A + \left[  f\delta S+ f S + f \delta \Sigma + f \delta G + (f-ff)\Delta \, (\Sigma + G) \right]_B 
+ \left[ ff R \, (\Sigma + G) \right]_C \ . \nn
\eeq
{\red \bf Here $\Delta$ does not include $\rho_1 + \rho_2$. That factor comes along with $\eta^\mu \eta^\nu$.} \\
$f \delta S$ cancels the $r$-divergence in $fS$, \\
$f \delta \Sigma$ cancels the $r$-divergence in $(f-ff)\Delta \, \Sigma$, \\ 
$f \delta G$ cancels the $r$-divergence in $(f-ff)\Delta \, G$, \\ 
I established that $\delta G = - \delta S$.
\beq
H_{LR} \es
m^2 + f S + f \delta \Sigma +  (f-ff)\Delta \, (\Sigma + G)  + ff R \, (\Sigma+ G)  \ .
\eeq
I change the order of terms. In case of $\Sigma$ there is no zero in denominator 
because the RGPEP $\Delta$ equals the inverse of the old-fashioned denominator 
that has one sign and is separated by at least $m_g^2$ from 0. The minimal
separation only happens when $k^\perp$ and  $x_g$ tend to 0. In the absence of 
0 in the denominator, I can separate $ff$ from $f$ in the self-interaction terms,
\beq
\Sigma_{\rm tot}  \es 
f \delta \Sigma + (f-ff)\Delta \, \Sigma  + ff R \, \Sigma  
\rs 
f (\delta \Sigma  + \Delta \, \Sigma ) + ff(-\Delta + R ) \, \Sigma \ .  
\eeq
So, 
\beq
H_{LR} \es
m^2 + f (\delta \Sigma  + \Delta \, \Sigma ) + ff(-\Delta + R ) \, \Sigma  + f S + (f-ff)\Delta \, G + ff R\, G  \ .
\eeq
In the exchange terms, the issue is that $(f-ff)\Delta$ is free from the denominator 0 while $\Delta$ itself is not.
By adding and subtracting $f R G$ to the exchange terms, 
\beq
G_{\rm tot} \es f S + (f-ff)\Delta \, G + ff R\, G \ ,
\eeq
one obtains
\beq
G_{\rm tot} \es f (S + RG ) + (ff - f)(-\Delta + R) \, G .
\eeq
Now the denominator 0 in $\Delta$ is always compensated by the numerator 0 in $ff-f$.
I can write $G = G_g + G_\eta$, where $G_g$ is the part with $-g^{\mu \nu}$
and $G_\eta$ is the part with $\eta^\mu \eta^\nu$.
\beq
G_{\rm tot} \es f (S + RG_g + RG_\eta) + (ff - f)(-\Delta + R) \, (G_g + G_\eta) \\
\es f R G_g + (ff - f)(-\Delta + R) \, G_g + f (S + R G_\eta)  + (ff - f)(-\Delta + R) \, G_\eta \ .
\eeq
So,
\beq
H_{LR} \es m^2 + \Sigma_{\rm tot} + G_{\rm tot}  \\
\es
m^2 + f (\delta \Sigma  + \Delta \, \Sigma ) + ff(-\Delta + R ) \, \Sigma
\np
f R G_g + (ff - f)(-\Delta + R) \, G_g + f (S + R G_\eta)  + (ff - f)(-\Delta + R) \, G_\eta \ .
\eeq
{\bf Hypotheses: \\
In $G_{\rm tot} $, the first term is the Coulomb, not sensitive to $m_g \to 0$ because $G_g$ is not; \\
The second term in $G_{\rm tot} $ is small because $G_g$ is not singular; \\
The third term shows how the singular seagull cancels out with $G_\eta$; \\ 
The fourth term is the exchange whose logarithmic dependence on $m_g$ 
is suspected to cancel with that of the second term in $\Sigma_{\rm tot}$,
which is $ff(-\Delta + R ) \, \Sigma$.}

{\red Stop 20250205 00:48 sro Ekologiczna, Start 20250205 22:53 sro Ekologiczna} 

The hoped-for cancellation of $\ln m_g$ between the self-energy and exchange 
concerns the sum of singular terms with $\eta$,
\beq
\label{Delta0}
(\Sigma G)_\eta \es ff(-\Delta + R ) \, \Sigma + (ff - f)(-\Delta + R) \, G_\eta \ .
\eeq
The factor $ff - f$ cancels the denominator 0 in $\Delta$. All I have done here
is merely to write the terms in such an order that the RGPEP small denominator 
cancellation is explicit. The difference between Eqs.~(\ref{Delta0}) and (\ref{C}),
the latter written in full detail in Eqs.~(\ref{SigmaPsi}) and (\ref{ExchPsi}), is that
now the denominator 0 in $\Delta$ is canceled by the corresponding numerator 0.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Analysis of the part $A$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The first part, $A$, describes the free $p^-$ of quarks. Their mass parameters 
need to be fitted to data [KSerafin, heavy quarkonia and heavy baryons]. 
Part $A$ does not depend on $m_g$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Analysis of the part $B$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The second part,
\beq
\label{B}
B \es  f_{LR}  H^{(2)}_0  + 
(f_{LR} - f_{L3}f_{3R}) \Delta_{L3R} \ 
\left\{ [ H^{(1)}_0 \, H^{(1)}_0 ]_\Sigma + [ H^{(1)}_0 \, H^{(1)}_0 ]_{\rm exch} \right\} \ ,
\eeq
results from solving the RGPEP equation. The term $H^{(2)}_0$ contains the RGPEP-tansformed 
seagull term and its counterterm, see Sec.~\ref{homogeneousCT}, whose contributions to the 
eigenvalue equation for the $Q\bar Q$ quarkonium-component wave function are denoted 
below in Eq.~(\ref{FullEvEqQbarQ}) by $B_{\rm s}$ and $\delta B_{\rm s}$, respectively. The terms 
$[ H^{(1)}_0 \, H^{(1)}_0 ]_\Sigma$ and $[ H^{(1)}_0 \, H^{(1)}_0 ]_{\rm exch}$ contribute the 
quark self-interactions and the gluon exchange, respectively. These contributions are denoted in 
Eq.~(\ref{FullEvEqQbarQ}) by $\Sigma^{(2)}_{q}$ and $B_{\rm exch}$. Both require counterterms, 
contained in $H^{(2)}_0$ and denoted by $\delta m_{i \rm tot}$ and $\delta B_{\rm exch}$ in Eq.~(\ref{FullEvEqQbarQ}), correspondingly.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Quark self-interactions in the part $B$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The bare 2nd-order quark self-interaction is given by Eq.~(\ref{quarkCTSigma}). 
It requires the counterterm $\delta m_{i \, \rm tot}$ of Eq.~(\ref{deltamitot}),
which is
\beq
\label{selfinteractionCTtot}
\delta m_{i \, \rm tot} \es \delta m^2_{i \, \rm sev} + \delta m_{i \, \ln }^2 \ ,
\eeq
where $\delta m^2_{i \, \rm sev}$ counters the severe small-$x$ 
and transverse quadratic divergences, and $\delta m_{i \, \ln }^2$ 
counters the remaining logarithmic dependence on $r$. The counterterm 
$\delta m_{i \, \rm tot}$ appears the same way in the single-quark 
eigenvalue problem {\red (in 2nd order there is no Lamb-shift effect,
besides a variation of the gluon mass ansatz to be discussed later on)} 
and in the quarkonium eigenvalue problem, as befits the RGPEP universal 
solution for a term in the Hamiltonian. There is no issue of zero denominator 
in $\Delta$ in the self-interactions and one can separate the diverging 
term with $f_{LR}$ from the $r$-independent term with $f_{L3} f_{3R}$ 
since $L=R$ in this case.

The full self-interaction term in part $B$ reads
\beq
B_\Sigma \es (f_{LR} - f_{L3}f_{3R}) \Delta_{L3R} \  [ H^{(1)}_0 \, H^{(1)}_0 ]_\Sigma \ ,
\eeq
where, according to Eq.~(\ref{quarkCTSigma}), the operator
$B_\Sigma$ leads to
\beq
\label{quarkCTSigmaPOM}
{\Sigma_q^{(2)} \over p^+}
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
  {N \over D}  \, f^{r 2}_{p_q p_g,p} (1 - f^{t 2}_{p_q p_g,p} ) \ , \\
\label{NumSigmaQPOM}
N
\es 4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x) \ ,\\
D
\es
x(1-x) p^+( p^-  - p_q^- - p^-_g) \rs x(1-x) (m_i^2 - \cM_{qg}^2) \ .
\eeq 
The calculation starts with Eq.~(\ref{cHSigma11check1}),
\beq
\label{cHSigma11check1text}
&& \cH^{\rm div}_{\Sigma \ p \sigma} \rs 
{ \left[ H_1^r H_1^r \right]_{\Sigma \ p \sigma} \over p_q^- - p_{qg}^-  } \\
\es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
  {\bar u_{p\sigma} \gamma_\mu ( p_q \hspace{-14.5pt} \not \hspace{12pt} + m) \gamma_\nu u_{p \sigma}
  \over  (1-x)p_g^+ (p^- - p_q^- - p_g^-)}  \, \left[ -g^{\mu \nu} + \eta^\mu \eta^\nu (p_g^-+p_q^- - p^-)/p_g^+ \right]  f^{r 2}_{p_q p_g,p} \nn
\es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \,  
\bar u_p \left\{
{ 2 p_q \hspace{-14.5pt} \not \hspace{12pt} - 4 m 
  \over  (1-x)p_g^+ (p^- - p_q^- - p_g^-)} 
-
  { 2 p_q^+ \gamma^+ 
  \over  (1-x)p_g^{+ 2} }
\right\} u_p  f^{r 2}_{p_q p_g,p} \\
 \es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \,  
 \left[
{ 4 p_q p - 8 m^2 
  \over  (1-x)p_g^+ (p^- - p_q^- - p_g^-)} 
-
  { 4 p^+ p_q^+ 
  \over  (1-x)p_g^{+ 2} }
\right] f^{r 2}_{p_q p_g,p} \ ,
\eeq
where the first term is from $-g^{\mu \nu}$ and the second from $\eta^\mu \eta^\nu$.
Together, they yield $N/D$, see Eq.~(\ref{cHSigma11check2}). After addition of the 
counterterm $\delta m^2_{i \, \rm sev}$, see  Eqs.~(\ref{deltami2}), (\ref{deltamitot}), 
(\ref{NDsigma}), and (\ref{deltamiln}), the factor $N/D$ is replaced by the remaining
\beq
{ 4 m_i^2 + 2 m_g^2 \over k^{\perp 2} + m_i^2 x^2 + m_g^2(1-x) } \ ,
\eeq
which leads to the logarithmic divergence in Eq.~(\ref{Deltai}),
\beq
\label{DeltaiPOMtext}
\Delta_i^2 \es
g^2  C_F 
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
{ 4 m_i^2 + 2 m_g^2 \over k^{\perp 2} + m_i^2 x^2 + m_g^2(1-x) }
\, f^{r 2}_{p_q p_g,p} \ ,
\eeq
and the logarithmic counterterm $\delta m^2_{i \, \rm ln}$ in 
Eq.~(\ref{deltamiln}), all independent of $m_g$.
Thus, according to Eqs.~(\ref{deltamitot}) to (\ref{deltamiln}), the counterterm
$\delta m^2_{i \, \rm tot}$ removes dependence on $r \to 0$ without producing
any dependence on $m_g$ from 1 in $(1 - f^{t 2}_{p_q p_g,p} )$. The reason is
that the integral is 3-dimensional and its integration measure makes it finite for 
$m_g=0$ after the severe divergences are countered by $\delta m^2_{i \, \rm sev}$.
I think about an analogy with the case of the Coulomb potential, where $d^3 q/q^2$ 
is finite for $q \to 0$. So, the self-interaction terms with $f=1$ in $f-ff$ are all 
eliminated and only $m_i^2$ is left in the eigenvalue equation for the $Q \bar Q$ 
component of the quarkonium.

The self-interaction due to the second term in the bracket, $- f^{t 2}_{p_q p_g,p}$,  
does not depend on $r \to 0$. Instead it depends on $s$ and does not induce 
any counterterms. The gluon mass in it is $m_g$. This contribution combines 
with the effective-particle  contribution to the self-interaction that includes the 
gluon mass ansatz $m_G$ in the part $C$, see below. Such ansatz mass, $m_G$, 
is not acceptable in QED because photons do not interact with electrons like gluons 
do with quarks. The combination of the self-interaction terms with $ff$, one in the 
part $B$ and the other one in part $C$, does not vanish because $m_G \neq m_g$. 
If $m_G$ were equal to $m_g$, which would be the case in QED, a theory without 
confinement, the combination would contribute 0 and the $i$th quark mass-squared 
eigenvalue would be just $m^2_i$. For $m_G \neq m_g$, the self-interaction cancellation 
does not occur. The non-cancellation leads to the divergent logarithmic dependence 
of the single-quark eigenvalue on $m_g$, shown in  Eq.~(\ref{SigmaSingleQuarktrick13}). 
The self-interactions in the quarkonium eigenvalue problem may differ from the 
self-interactions in the single-quark due to a different value of the gluon mass ansatz,
corresponding to different color dynamics in triplet and singlet states. {\bf Irrespective
of that ansatz difference, the question is if the self-interaction logarithm,  $\ln m_g$, 
can be canceled by the combination of the gluon exchange interactions in parts $B$ 
and $C$.} 
%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Gluon exchange in the part $B$}
%%%%%%%%%%%%%%%%%%%%%%%%
{\red Start 20250128 01:16 wto Ekologiczna}
The gluon-exchange term in the part $B$ contains the factor $\Delta_{L3R}$ 
generated by the RGPEP and requires a counterterm placed in $\cH_0^{(2)}$. 
To compute the divergence and the counterterm we proceed as in
Sec.~\ref{homogeneoussol}, see Sec.~\ref{homogeneousCT}. The exchange 
term matrix elements between quark-antiquark states of Eq.~(\ref{Pnt}),
\beq
\label{Pnt1}
|P n\rangle_s
\es
\sum_{12} \int[12] \,  \psi_P^n(1,2) \, b_{t1}^\dagger d_{t2}^\dagger |0\rangle \ ,
\eeq
read
\beq
_s \langle Pn| \, \cU_s^\dagger  [ (f_{LR} - f_{L3}f_{3R}) \Delta_{L3R} \cH^{(1)}_0 \cH^{(1)}_0 ]_{\rm exch} \, \cU_s |P'n' \rangle_s 
\es P^+ \tilde \delta_{P.P'} \sum_{122'1'} Y_{nn' 122'1'} \ , 
\eeq
where 
\beq
\label{standardvariablesY}
Y_{nn'122'1'}
\es 
\int [x_1 k_{12}^\perp] \int [x_{1'} k_{1'2'}^\perp]
\
(f-ff ) 
\nt  
\psi^{*n}_{12}(x_1,k_{12}^\perp) \, \left\{ ~~ \right\}_{\rm exch} \,
\psi^{n'}_{1' 2'}(x_{1'},k_{1'2'}^\perp)  \ ,      \\
(f-ff) \es (f^t_{12,2'1'} - f^t_{{\overline 1}, g{\underline 1}} f^t_{{\overline 2}, g{\underline 2}} ) \ ,
\eeq
and
\beq
\label{ ~~ exch}
\left\{ ~~ \right\}_{\rm exch} 
\es  
\Delta_{12g2'1'} \,
\sum_{c_g = 1}^8 j_{11' \mu}^{c_g} j_{22' \nu}^{c_g} \ d^{\mu \nu} {1 \over q_{m_g}^+} \ 
f^r_{{\overline 1}, g{\underline 1}} f^r_{{\overline 2}, g{\underline 2}} \\
\label{ ~~ exch POM}
\es  
\sum_{c_g = 1}^8 j_{11' \mu}^{c_g} j_{22' \nu}^{c_g} \ d^{\mu \nu} \ \Delta \ 
f^r_{{\overline 1}, g{\underline 1}} f^r_{{\overline 2}, g{\underline 2}} \ , \\
d^{\mu \nu} \es - g^{\mu \nu} 
+ {\eta^\mu q_{m_g}^\nu + q_{m_g}^\mu \eta^\nu \over q_{m_g}^+ } 
\rs
- g_{\mu \nu} 
+ \eta_\mu \eta_\nu 
{ \rho_1 + \rho_2 \over 2 q_{m_g}^{+2} } \ , \\
\Delta \es {\Delta_{12g2'1'} \over q_{m_g}^+} 
\rs -
                    { \rho_1 \pi_{\overline 1}^2 
                  +  \rho_2 \pi_{\overline 2}^2
\over 
                      \rho_1^2 \pi_{\overline 1}^2 
                  + \rho_2^2 \pi_{\overline 2}^2 
- (\rho_1 - \rho_2)^2 \pi_{12}^2 } \ ,
\eeq
while $j^{c_g}_{11' \mu} = - \bar u_1 \gamma_\mu u_{1'} T^{c_g}_{c_1 c_{1'}}$,
$j^{c_g}_{22' \nu} = \bar v_{2'} \gamma_\nu v_2 T^{c_g}_{c_{2'} c_2}$,
$\rho_i = m_g^2 - q_i^2$, and $q_i = p_{\overline i} - p_{\underline i}$.

The regulating factors have the form $f^r_1 f^r_2 ( f^t - f^t_1 f^t_2)$. 
One can keep a factor formally as different from 1 even if it is negligibly 
different from 1. In the limit $r \to 0$, the form factors $f^r_{{\overline 1}, g{\underline 1}} 
f^r_{{\overline 2}, g{\underline 2}}$ tend to 1 everywhere where the factors 
$f^t_{{\overline 1}, g{\underline 1}} f^t_{{\overline 2}, g{\underline 2}}$ are
not equivalent to 0. Dependence on $r$ due to $f^rf^r$ is removed from 
all the terms with $f^tf^t$ for fixed, positive $s$. 

I need to understand $\rho_i$ in terms of the variables defined in
Eqs.~(\ref{C1}) - (\ref{C14}). I need formulas for $q_i^2$. In the square, 
the sign of $q_i$ does no matter, which implies I can simply use the 
difference between momenta before and after exchange.
\beq
q_i^2 \es (p_{i'} - p_{i})^2
          \rs (p_{i'}^+ - p_{i}^+)
               (p_{i'}^- - p_{i}^- ) 
           -   (p_{i'} - p_{i})^{\perp 2} \\
           \es (x_{i'}-x_i) \left[ { (x_{i'} P^\perp + k_{i'}^\perp)^2 + m_i^2 \over x_{i'} } 
                          - { (x_{i } P^\perp + k_{i }^\perp)^2 + m_i^2 \over x_{i } } \right]
                          - (x_{i'} P^\perp + k_{i'}^\perp - x_{i } P^\perp - k_{i }^\perp)^2 \nn
\es  (x_{i'}-x_i)  \left[ { k_{i'}^{\perp 2} + m_i^2 \over x_{i'} } 
               - {  k_{i }^{\perp 2} + m_i^2 \over x_{i } } \right]
                          - (k_{i'}^\perp - k_{i }^\perp)^2
\rs - {z^2 m_i^2 + (x_i k_{i'}^{\perp} - x_{i'} k_{i}^{\perp})^2 \over x_{i'} x_i } \ . 
\eeq
{\red \bf Always true in $2 \leftarrow 2'$:
\beq
\label{qi2}
q_i^2 \es - {z^2 m_i^2 + (x_i k_{i'}^{\perp} - x_{i'} k_{i}^{\perp})^2 \over x_{i'} x_i } \ , \\
k_1 \es k_{12} \ , \ k_{1'} \rs k_{1'2'} \ , \ k_2 \rs - k_{12} \ , \ k_{2'} \rs - k_{1'2'}  \ .
\eeq}

\beq
\label{rho1pom}
q_1^2 \es - {4y^2 m_1^2 + 4 (x l^{\perp} - y k^\perp )^2 \over x^2-y^2 } \ , \\
\label{rho2pom}
q_2^2 \es - {4y^2 m_2^2 + 4 [(1-x) l^{\perp} + y k^\perp ]^2 \over (1-x)^2-y^2 } \ , \\
\label{rho11pom}
\rho_1 \es m_g^2 + {4y^2 m_1^2 + 4 (x l^{\perp} - y k^\perp )^2 \over x^2-y^2 } \ , \\
\label{rho22pom}
\rho_2 \es m_g^2 + {4y^2 m_2^2 + 4 [(1-x) l^{\perp} + y k^\perp ]^2 \over (1-x)^2-y^2 } \ .
\eeq
{\bf In the non-relativistic limit:
\beq
\rho_1 \es m_g^2 + z^2 m_1^2/x_1^2 + q^{\perp 2} \ , \
\rho_2 \rs m_g^2 + z^2 m_2^2/x_2^2 + q^{\perp 2} \ , \\
x_i \es {m_i \over m_1 + m_2} \ , \
\rho_1 \rs \rho_2 \rs m_g^2 + z^2 (m_1 + m_2)^2 + q^{\perp 2} \rs \rho \ .
\eeq
Note the sum $m_1 + m_2$! ~~ {\red I have just observed the mass sum on 20250128 at 04:47 wto Ekologiczna.}
In the non-relativistic limit, $\Delta = -1/\rho$, $\tilde \Delta = - 1$, $\tilde \Delta' = 0 $, and 
0 in the denominator of $\Delta$ does not appear in that limit, since $\rho_1 = \rho_2$. The factor 
$f^r_{{\overline 1}, g{\underline 1}} f^r_{{\overline 2}, g{\underline 2}}$ reduces to
\beq
f^t_{\overline 1, \underline 1 g} f^t_{\overline 2, \underline 2 g}
\es
e^{ - \left( t \, \rho_1 \, \pi_{\overline 1}/q_{m_g}^+ \right)^2 }   
e^{ - \left( t \, \rho_2 \, \pi_{\overline 2}/q_{m_g}^+ \right)^2 }
~ \sim ~ 
e^{ - t^2  \, \rho^2 (x_1^2 + x_2^2)/z^2}  \ .
\eeq 
For $t = 1/M^2$, where $M^2 = m_1^2 + m_2^2$, 
I obtain 
\beq
f^t_{\overline 1, \underline 1 g} f^t_{\overline 2, \underline 2 g}
~ \sim ~ 
\exp{ \left[ - \left( { \vec q\,^2 \over q_z M} \right)^2 \right] } 
\rs 
\exp{ \left[ - \left( { |\vec q\,| \over M \cos \theta} \right)^2 \right] } 
\ .
\eeq 
End of the non-relativistic limit.}

Start with the term that involves $f^r f^r f^t$. The first term in $d^{\mu \nu}$, 
equal $-g^{\mu \nu}$, produces regular matrix elements and does not require 
any counterterms. It yields the Coulomb potential with the Breit-Fermi terms. 
In QED, the factor $f^t$ in the potential could noticeably deviate from 1 only in 
the extreme atomic effects such as the ones associated with the measurement of 
the radius of proton in hydrogen. A few percent effect in the radius could occur 
when the value of the parameter $s$ in the non-relativistic Schroedinger equation for 
hydrogen is set to the value on the order of the proton mass [Glazek proton radius].

The second term in $d^{\mu \nu}$, the one proportional to $\eta^\mu \eta^\nu$, 
       produces a severe small-$x$ divergence in the matrix element,
\beq
\label{standardvariablesYsummary}
Y_{nn'122'1'}^{\rm div}
\es 
\int [x_1 k_{12}^\perp] \int [x_{1'} k_{1'2'}^\perp]
\ (f-ff) \  
\psi^{*n}_{12}(x_1,k_{12}^\perp) \, \left\{ ~~ \right\}_{\rm exch}^{\rm div} \,
\psi^{n'}_{1' 2'}(x_{1'},k_{1'2'}^\perp)  \ , \\
\label{ ~~ exch summary}
\left\{ ~~ \right\}_{\rm exch}^{\rm div} 
\es
- \, g^2 \, C_{11'2'2} \, \delta_{s f} \, 4 \sqrt{x_1 x_{1'} x_2 x_{2'}} \
{ \tilde \Delta \over z^2} \ f^r_{123} f^r_{31'2'}  \ , 
\eeq
where
\beq
\tilde \Delta \es \Delta \ {\rho_1 + \rho_2 \over 2} \rs -1 + \tilde \Delta' \ , \
\lim_{z \to 0} \tilde \Delta' \rs {z \over 2 c} \ (b_1-b_2) \ 
{ \pi_{\overline 1}^2 - \pi_{\overline 2}^2 \over  \pi_{\overline 1}^2 + \pi_{\overline 2}^2} \ , \\
c \es 4l^{\perp 2} + m_g^2 \ , \
b_1 \rs  - 4 k^\perp l^\perp/x \ , \
b_2 \rs 4 k^\perp l^\perp/(1-x) \ .
\eeq
{\bf Calculation of $\tilde \Delta'$:}
\beq
\tilde \Delta \es (\rho_1 + \rho_2)\Delta/2 
\rs - \2
                    { \rho_1^2 \pi_{\overline 1}^2 
                  +  \rho_2^2 \pi_{\overline 2}^2
+ \rho_1 \rho_2(\pi_{\overline 1}^2 + \pi_{\overline 2}^2) 
\over 
                      \rho_1^2 \pi_{\overline 1}^2 
                  + \rho_2^2 \pi_{\overline 2}^2 
- (\rho_1 - \rho_2)^2 \pi_{12}^2 } \ ,
\eeq
\beq
\tilde \Delta' \es \tilde \Delta + 1 
\rs
- \2
                    { \rho_1^2 \pi_{\overline 1}^2 
                  +  \rho_2^2 \pi_{\overline 2}^2
+ \rho_1 \rho_2(\pi_{\overline 1}^2 + \pi_{\overline 2}^2) 
\over 
                      \rho_1^2 \pi_{\overline 1}^2 
                  + \rho_2^2 \pi_{\overline 2}^2 
- (\rho_1 - \rho_2)^2 \pi_{12}^2 } + 1 \ .
\eeq
In the limit $y \to 0$ I have
\beq
\rho_1 \es m_g^2 + {4y^2 m_1^2 + 4 (x l^{\perp} - y k^\perp )^2 \over x^2-y^2 }
\to
m_g^2 + 4 l^{\perp 2} - 8 y l^\perp k^\perp/x  
 \ , \\
\rho_2 \es m_g^2 + {4y^2 m_2^2 + 4 [(1-x) l^{\perp} + y k^\perp ]^2 \over (1-x)^2-y^2 } 
\to 
m_g^2 + 4 l^{\perp 2} + 8 y l^\perp k^\perp/(1-x)   \ , 
\eeq
and I confirm 
\beq
\lim_{z \to 0} \tilde \Delta' \rs {z \over 2 c} \ (b_1-b_2) \ 
{ \pi_{\overline 1}^2 - \pi_{\overline 2}^2 \over  \pi_{\overline 1}^2 + \pi_{\overline 2}^2}  \ .
\eeq
The term $-1$ in $\tilde \Delta$ requires the counterterm opposite in sign 
to the one obtained in Sec.~\ref{homogeneousCT} for the homogeneous 
seagull term. The small-$z$ correction $\tilde \Delta'$ depends on the choice 
of $\pi_{ca}$ in the RGPEP generator in Eq.~(\ref{cHIpi}). For $\pi_{\overline i} 
= \pi^+$ for all $i$ equally, one obtains $\tilde \Delta' \to O(z^2)$ and no 
need for a logarithmic counterterm. For $\pi_{\overline i} = p_{\overline i}^+$,
\beq
\lim_{z \to 0} \tilde \Delta' \rs  - z \ { 8 k^\perp l^\perp \over c}
 \ { x - 1/2 \over x(1-x)[x^2 + (1-x)^2]} \ ,
\eeq
which might lead to a logarithmic small-$x$ divergence corresponding to 
the small-$z$ behavior of the integrand here. The divergence would appear 
if the other factors in the integrand of $Y_{nn'122'1'}^{\rm div}$ in 
Eq.~(\ref{standardvariablesYsummary}) were not smooth as functions of $z$ 
near $z=0$. However, the analysis of Sec.~\ref{homogeneousCT}, 
beginning with Eq.~(\ref{TreeGluonExchU1}), applies and they are smooth. 
The result is that the logarithmic dependence on $r$ is absent in the 
term $B$ of Eq.~(\ref{HLR1pom1}). Hence no logarithmic small-$x$ 
counterterm is required due to the term $B$ in the bound-state 
eigenvalue problem of the effective Hamiltonian in Eq.~(\ref{HLR1}). 
{\red I do not see yet that a stronger argument is excluded: writing $l^\perp 
= \sqrt{|z|}\, \tilde l^\perp$, I obtain $\rho/z$ in $f^r$ that involves $\tilde 
l^{\perp 2}$ in the exponential without $1/z$ and the difference $b_1-b_2 
\sim \sqrt{|z|}$, implying reduction of the 
singularity from $1/z$ to the integrable one $1/ \sqrt{|z|}$. }

{\bf General explanation of how to treat 0 in the denominator of $\Delta$ 
in calculating the counterterms to small-$x$ singularities is that, when the 
gluon $x \to 0$, the difference $\rho_1-\rho_2 \to 0$ and cannot nullify the 
denominator because the other terms in the denominator tend to 
$(m_g^2 + q^{\perp 2})^2 (\pi_1^2 + \pi_2^2) > 0$. In other words, one can
handle the limits $z \to 0$ for terms with $f$ and for terms with $ff$ separately.}
%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Analysis of the part $C$}
%%%%%%%%%%%%%%%%%%%%%%%%
The third part,
\beq
\label{C}    
C \es \2 f_{L3}  \, H^{(1)}_0 \left\{ -  2 \Delta_{L3R}  + \left[P_L^- - H_f' \right]^{-1} 
                        + \left[P_R^- - H_f'\right]^{-1} 
                         \right\} f_{3R} \, H^{(1)}_0 \ .
\eeq
is composed of two terms. The part with $-2\Delta_{L3R}$ is obtained by solving 
the RGPEP equation. The part with two denominators is obtained by reducing 
[Wilson R] the eigenvalue problem for $H_s$ to the eigenvalue problem for the 
quark-antiquark states using the expansion in powers of the interaction to the
2nd order. The reduction is done using the simplifying assumption that one 
can replace all components besides the quark-antiquark component by the 
component with a quark, an antiquark and a gluon whose mass is set to a 
hypothetical value $m_G$, called the gluon-mass ansatz [Kamil etc.].  The mass 
parameter $m_G$ is viewed as a multiple of $\Lambda_{\rm QCD}$ in the RGPEP 
scheme. As such, it would be 0 in the expansion in powers of the running coupling 
constant. The double action of the Hamiltonian term $H^{(1)}_0$ produces the 
quark self-interactions and the gluon exchange between the quarks. Both involve 
the singular expression $p_g^-/x_g$. The form factors $f^t$ prevent any dependence 
on $r$ from occurring. However, the resulting effective interaction depends on 
$m_g$. The question addressed below is how the self-interaction and gluon 
exchange terms in the part $C$ behave in the limit $m_g \to 0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Quark self-interactions in the part $C$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beq
fH \, \2 \left\{~~\right\} \, fH \tl  C_{\Sigma \, } \rs 
\int[1] {1 \over p_{1}^+ } \int [23] \ \tilde \delta_{1.23} \ d_{\mu \nu} \
        j_{12}^{c \mu}         j_{21}^{c \nu}  \ f^2 \ \Delta_{1 \Sigma} \ b_{1}^\dagger b_{1}  
        \np
\int[2] {1 \over p_{2}^+ } \int[13] \ \tilde \delta_{2.13} \ d_{\mu \nu} \
\bar j_{12}^{c \mu}  \bar j_{21}^{c \nu}  \ f^2 \ \Delta_{2 \Sigma} \ d_{2}^\dagger d_{2} \ .
\eeq
\beq
        j_{12}^{c \mu}  \es - g \bar u_1   \gamma^\mu  u_2 \ \chi_1^\dagger T^c  \chi_2 \ , \
\bar j_{12}^{c \mu}   \rs   g  \bar v_1   \gamma^\mu  v_2 \ \chi_1^\dagger T^c  \chi_2 \ ,
\eeq
\beq
q_1  \es p_{\overline 1} - p_{\underline 1} \rs q_2 \rs p_{\overline 2} - p_{\underline 2} \ , \\
d^{\mu \nu} 
\es  - g^{\mu \nu} + {1 \over 2  q_{m_g}^+} \eta^\mu \eta^\nu 
 \left( { \cM_{g \underline 1}^2 - m_1^2 \over p_{\overline 1}^+}
      + { \cM_{g \underline 2}^2 - m_2^2 \over p_{\overline 2}^+} \right) \\
\es  - g^{\mu \nu} + {1 \over 2  q_{m_g}^{+2}} \eta^\mu \eta^\nu 
 \left( m_g^2 - q_1^2 + m_g^2 - q_2^2 \right) \ .
\eeq
Using the pattern of Eqs.~(\ref{HLR1if1}) and (\ref{SigmaSingleQuark2text}), 
the factor $\Delta_{i \Sigma}$ is 
\beq
\Delta_{i \Sigma} \es
{1 \over 2 }   \, \left\{ -  2 \Delta_{iqgi}  + 
\left[ p^-_{i} - p_{qG}^- \right]^{-1}  
+ 
\left[ p^-_{i} - p_{qG}^- \right]^{-1}   \right\} \\
\es
- \Delta_{iqgi}  + \left[ p^-_{i} - p_{qG}^- \right]^{-1} 
\rs
{- 1\over p_i^- - p_q^- - p_g^- }  +  { 1 \over p_i^- - p_q^- - p_G^-} \ .
\eeq
The self-interaction terms $C_\Sigma$ in the part $C$ depend on the assumed 
quarkonium parameter $m_G$, which is likely to differ from the single-quark 
ansatz parameters $m_{iG}$. In contrast, the calculation in Sec.~\ref{DeltapE} shows in 
Eq.~(\ref{SigmaSingleQuarktrick13}) that the self-interaction logarithmic growth 
for $m_g \to 0$ does not depend on the single-quark gluon-mass ansatz $m_{iG}$. 
It thus occurs in the quarkonium eigenvalue problem in the same way as in a single 
quark eigenvalue problem even though $m_G$ may differ from $m_{iG}$. The 
unlimited growth of the quark self-interactions in the $Q \bar Q$ quarkonium 
component for $m_g \to 0$ appears in a sum 
\beq
\label{DeltaQbarQ}
C_\Sigma \tl  { C_{1 \Sigma } \over p_1^+} + { C_{2 \Sigma} \over p_2^+} \ ,
\eeq 
which multiplies the quarkonium wave function in the eigenvalue 
equation. The parts of the self-interaction terms $C_{1 \Sigma }$
and $C_{1 \Sigma }$ that do not diverge logarithmically with $m_g \to 0$ 
are sensitive to the ansatz masses $m_{iG}$ and $m_G$.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Gluon exchange in the part $C$}
%%%%%%%%%%%%%%%%%%%%%%%%%%
The gluon exchange interaction acts on the wave function 
$\psi_E(1,2)$ integrated in the eigenstate,
\beq
\label{Eigenstate}
|E\rangle
\es
\sum_{12} \int[12] \,  \psi_E(1,2) \, b_{t1}^\dagger d_{t2}^\dagger |0\rangle \ .
\eeq
For eigenvalues of the form $E = (P^{\perp 2} + M_E^2)/P^+$, the wave function 
may be set to have the form 
\beq
\psi_E(1,2) \es P^+ \tilde \delta_{P.12} \ \psi_{1 2}(x_1,k_{12}^\perp) \ ,
\eeq 
with the labels 1 and 2 including spins, flavors and colors as needed. 
The relative-motion variables $x_1$ and $k_{12}^\perp$ for the 
effective quarks 1 and 2 are defined in the same way as for the gluons 
in Eq.~(\ref{HA3k12}).  The eigenvalue problem for the Hamiltonian 
in Eq.~(\ref{HLR1pom1})  
\beq
\label{HLR1pom11}
H_{LR} \es A + B + C \ .
\eeq
in terms of matrix elements in plane-wave $Q \bar Q$ basis states reads
\beq
\label{projectedHepPOM}
\langle 1 2|H_{LR}|E\rangle 
\es 
\langle 1 2| A + B + C |E \rangle 
\rs 
P_E^- \langle 1 2|E \rangle \ .
\eeq
Using Eq.~(\ref{Hsquared}), I get
\beq
\langle 12|C_{\rm exch}|E \rangle \es
\sum_{c} \int[1'2'] \ \tilde \delta_{12.1'2'} 
\ d_{\mu \nu} 
\   j_{11'}^{\mu c}  \ \bar j_{2'2}^{\nu c} \  f^t_{\overline 1, \underline 1 q} f^t_{\overline 2, \underline 2 q}
\nt
{1 \over 2 q_{m_g}^+}   \, \left\{ -  2 \Delta_{12g2'1'}  + 
\left[ P^-_{12} - P_{3}^- \right]^{-1}  
+ 
\left[ P^-_{1'2'} - P_{3}^- \right]^{-1}   \right\} \psi_E(1',2') \ , \\
d_{\mu \nu} \es - g_{\mu \nu} 
+ \eta_\mu \eta_\nu 
{ \rho_1 + \rho_2 \over 2 q_{m_g}^{+2} } \ , \\
\Delta \es {\Delta_{12g2'1'} \over q_{m_g}^+} 
\rs -
                    { \rho_1 \pi_{\overline 1}^2 
                  +  \rho_2 \pi_{\overline 2}^2
\over 
                      \rho_1^2 \pi_{\overline 1}^2 
                  + \rho_2^2 \pi_{\overline 2}^2 
- (\rho_1 - \rho_2)^2 \pi_{12}^2 } \ , \\
P_{3}^- \es p_{\underline 1}^- +  p_{\underline 2}^- + q_{m_G}^- \ , \  \rho_i \rs m_g^2 - q_i^2 \ , \\
{1 \over q_{m_g}^+ } &&
\left[  \left(P^-_{12} - P_{3}^- \right)^{-1}  
+ 
\left( P^-_{1'2'} - P_{3}^- \right)^{-1} \right] \ ,  \\
\es
- \left[ {1 \over \rho_1 + (m_G^2 - m_g^2)}  + {1 \over \rho_2 + (m_G^2 - m_g^2) } \right] 
~,~~ {\rm see ~ Appendix}.
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$Q \bar Q$ wave-function eigenvalue equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\red 20250107 23:05 wto San Dimas Start tomorrow od analizy rownania}

The wave-function eigenvalue equation reads
\beq
\label{FullEvEqQbarQ}
&&
\left[ {m_1^2  + \Sigma^{(2)}_{q1} + \delta m_{1 \, \rm tot}^2 \over p_1^+ } 
+
         {m_2^2  + \Sigma^{(2)}_{q2} + \delta m_{2 \, \rm tot}^2 \over p_2^+ }\right]  
         \psi_{1 2}(x_1,k_{12}^\perp)
\np
\int[x_{1'} k_{1'2'}^\perp]
\left[ B_{\rm s} + \delta B_{\rm s} + B_{\rm exch} + \delta B_{\rm exch} \right] 
(x_1, k_{12}^{\perp}; x_{1'}, k_{1'2'}^{\perp}) \ \psi_{1'2'}(x_{1'},k_{1'2'})
\np
\left[ {C_{1 \Sigma} \over p_1^+ } + {C_{2 \Sigma} \over p_2^+ } \right] \psi_{12}(x_{1},k_{12})
+
\int[x_{1'} k_{1'2'}^\perp] C_{\rm exch} (x_1, k_{12}^{\perp}; x_{1'}, k_{1'2'}^{\perp})  \, \psi_{1'2'}(x_{1'},k_{1'2'}) \ , \nn
\eeq
where 
\beq 
C_{i \Sigma} 
\es  {1 \over p_i^+} \int [x k^\perp ] \ d_{\mu \nu} \
        j_{iq}^{c \mu}         j_{qi}^{c \nu}  \  (f^t_{i, qg})^2 
        \nt
{1 \over 2 }   \, \left\{ -  2 \Delta_{iqgi}  + 2 \left[ p^-_{i} - p_{qG}^- \right]^{-1}  \right\} \ , \\
C_{\rm exch} (x_1, k_{12}^{\perp}; x_{1'}, k_{1'2'}^{\perp}) 
\es
{1 \over q_{m_g}^+ P^+}
d_{\mu \nu} 
\ j_{11'}^{c \mu}  \ \bar j_{2'2}^{c \nu} \  f^t_{\overline 1, \underline 1 q} f^t_{\overline 2, \underline 2 q}
\nt
{1 \over 2 }   \, \left\{ -  2 \Delta_{12g2'1'}  + 
\left[ P^-_{12} - P_{3}^- \right]^{-1}  
+
\left[ P^-_{1'2'} - P_{3}^- \right]^{-1}   \right\} .
\eeq
The counterterms $\delta m_{i \, \rm tot}^2$ can be adjusted to cancel
$\Sigma_{qi}^{(2)}$ and make $m_i^2$ the quark mass parameters
that are fitted to the quarkonium spectra. The counterterms $\delta B_{\rm s}$ 
and $\delta B_{\rm exch}$ cancel each other, which is shown in Sec.~\ref{Qdynamics}. 
They will be omitted in further equations. When $m_g \to 0$, the sum 
of terms $B_{\rm s} + B_{\rm exch}$ provides the Coulomb potential 
with Breit-Fermi spin dependence, in the non-relativistic approximation for 
the relative motion of quarks. The self-interaction terms $C_{1 \Sigma }$ 
and $C_{2 \Sigma }$ grow proportionally to $\ln(m_i/m_g)$ when $m_g \to 0$, 
as they do in the single-quark eigenvalue equations rendering infinite masses 
to single quarks, see Eq.~(\ref{SigmaSingleQuarktrick13}), but perhaps using 
a potentially different gluon mass ansatz. Can the exchange cancel the infinite 
growth of the self-interactions in colorless quarkonia when $m_g \to 0$? 
{\red Start 20250105 09:21 -> 19:06 nie San Dimas} I focus on the cancellation
issue.

The self-interaction terms $C_{i\Sigma}$ multiply the wave function while the 
exchange term integrates it with the kernel $C_{\rm exch} (x_1, k_{12}^{\perp}; 
x_{1'}, k_{1'2'}^{\perp})$. The cancellation is possible if the kernel's logarithmic 
dependence on $m_g$ does not involve any change of the quark momenta. 
The logarithmic rise of $C_{i\Sigma}$ results from the term $\eta^\mu \eta^\nu 
\rho_i/p_g^{+2}$ in the sum over gluon polarizations, 
which blows up when the gluon $x$ tends to 0. The blowup of $C_{i\Sigma}$ 
is tamed by the RGPEP vertex factor squared, $(f^t_{i, qg})^2$. In the exchange, 
$(f^t_{i, qg})^2$ is replaced by the product $f^t_{\overline 1, \underline 1 g} 
f^t_{\overline 2, \underline 2 g}$, which works in a similar way. The RGPEP 
vertex factors depend on the gluon mass $m_g$ through $p_g^- = (p_g^{\perp 2} 
+ m_g^2)/p_g^+$. Therefore, the gluon mass $m_g$ contributes to their behavior only 
when $|k^\perp = p_g^\perp - p_g^+ p_{\overline i}^\perp/p_{\overline i} | 
\lesssim m_g$. Otherwise the gluon transverse momentum, relative to the quarks, 
regulates the gluon small-$x$ behavior and prevents dependence on $m_g \to 0$. 
Dependence on $m_g$ can only originate from the region of small transverse 
momentum of the gluon.

The self-interaction logarithmic rise for $m_g \to 0$ in Eq.~(\ref{FullEvEqQbarQ}) 
comes from
\beq
\left[ { C_{1 \Sigma } \over p_1^+} + { C_{2 \Sigma} \over p_2^+} \right]_{\ln m_g} 
\es
\sum_i  
 {1 \over p_i^+} \int [x k^\perp ] \ {\rho_i \over q_{m_g}^{+2}} 
        j_{iq}^{c +}         j_{qi}^{c +}  \  (f^t_{i, qg})^2 
\nt
{1 \over 2 p_i^+}   \, \left\{ -  2 \Delta_{iqgi}  + 2 \left[ p^-_{i} - p_{qG}^- \right]^{-1}  \right\} 
\eeq
multiplying $\psi_{12}(x_{1},k_{12})$. The exchange-part sensitive to $m_g \to 0$ comes  from
integrating the wave function,
\beq
&&
\int[x_{1'} k_{1'2'}^\perp] C_{\rm exch} (x_1, k_{12}^{\perp}; x_{1'}, k_{1'2'}^{\perp})  \, \psi_{1'2'}(x_{1'},k_{1'2'}) \\
\es
{ 1 \over P^+ }
\int[x_{1'} k_{1'2'}^\perp] \ { \rho_1 + \rho_2 \over 2 q_{m_g}^{+2} } \
\ j_{11'}^{c +}  \ \bar j_{2'2}^{c +} \  f^t_{\overline 1, \underline 1 q} f^t_{\overline 2, \underline 2 q}
\nt
{1 \over 2 q_{m_g}^+ }   \, \left\{ -  2 \Delta_{12g2'1'}  + 
\left[ P^-_{12} - P_{3}^- \right]^{-1}  
+
\left[ P^-_{1'2'} - P_{3}^- \right]^{-1}   \right\} \, \psi_{1'2'}(x_{1'},k_{1'2'}) \ .
\eeq
Inserting some formulas for some factors, such as for the masses squared in
emission of absorption of a gluon by a quark $i$,
\beq
\cM_{qg}^2 - m_i^2  \es {p_{\overline i}^+  \over q_{m_g}^+ } \ \rho_i  \rs {p_{\overline i}^+  \over q_{m_g}^+ } (m_g^2 + q_i^2) \ ,
\eeq
I arrive at the comparison
\beq
\tilde \Sigma \psi \rs \left[ { C_{1 \Sigma } \over p_1^+} + { C_{2 \Sigma} \over p_2^+} \right]_{\ln m_g} 
\psi_{12}(x_{1},k_{12})
\es
\sum_i  
 {1 \over p_i^+} \int [x k^\perp ] \ {\rho_i \over q_{m_g}^{+2}} 
        j_{iq}^{c +}         j_{qi}^{c +}  \  (f^t_{i, qg})^2 
\nt 
\left[ {1\over \cM_{qg}^2 - m_i^2}  -  { 1 \over \cM_{qG}^2 - m_i^2} \right] \psi_{12}(x_{1},k_{12}) \ , \\
{1\over \cM_{qg}^2 - m_i^2}  -  { 1 \over \cM_{qG}^2 - m_i^2} \es
{q_{m_g}^+ \over p_i^+} \left[ { 1 \over \rho_i}  -  { 1 \over \rho_i + m_G^2 - m_g^2} \right]
\eeq
\beq
\tilde C\psi \es
\int[x_{1'} k_{1'2'}^\perp] C_{\rm exch} (x_1, k_{12}^{\perp}; x_{1'}, k_{1'2'}^{\perp})  \, \psi_{1'2'}(x_{1'},k_{1'2'}) \\
\es  { 1 \over P^+ }
\int[x_{1'} k_{1'2'}^\perp] \ { \rho_1 + \rho_2 \over 2 q_{m_g}^{+2} } \
\ j_{11'}^{c +}  \ \bar j_{2'2}^{c +} \  f^t_{\overline 1, \underline 1 q} f^t_{\overline 2, \underline 2 q}
\nt
{1 \over 2 q_{m_g}^+ }   \, \left\{ -  2 \Delta_{12g2'1'}  + 
\left[ P^-_{12} - P_{3}^- \right]^{-1}  
+
\left[ P^-_{1'2'} - P_{3}^- \right]^{-1}   \right\} \, \psi_{1'2'}(x_{1'},k_{1'2'}) \ .
\eeq
\beq
&&
{-2 \Delta_{12g2'1'} \over 2 q_{m_g}^+} 
\rs 
                    { \rho_1 \pi_{\overline 1}^2 
                  +  \rho_2 \pi_{\overline 2}^2
\over 
                      \rho_1^2 \pi_{\overline 1}^2 
                  + \rho_2^2 \pi_{\overline 2}^2 
- (\rho_1 - \rho_2)^2 \pi_{12}^2 } \ , \\
&& {1 \over 2 q_{m_g}^+ }
\left[  \left(P^-_{12} - P_{3}^- \right)^{-1}  
+ 
\left( P^-_{1'2'} - P_{3}^- \right)^{-1} \right] \ ,  \\
\es
- \2 \left[ {1 \over \rho_1 + m_G^2 - m_g^2}  + {1 \over \rho_2 + m_G^2 - m_g^2 } \right] \ .
\eeq
{\red I need to integrate over [23]. Stop 20250105 21:40 nie San Dimas, 
Start 20250106 17:11 pon San Dimas, Start 20250107 13:30 wto San Dimas}
Now {\red 20250107 15:01 wto San Dimas} I write these two terms in abbreviated 
forms, using also $M^2 = m_G^2 - m_g^2$. 
\beq
\label{SigmaPsi}
\tilde \Sigma \psi 
\es
\sum_i  
 {1 \over p_i^+} \int [x k^\perp ] \ {\rho_i \over q_{m_g}^{+} p_i^+} 
        j_{iq}^{c +}         j_{qi}^{c +}  \  (f^t_{i, qg})^2 
\
\left[ {1\over \rho_i}  -  { 1 \over \rho_i + M^2} \right]
\psi_{12}(x_{1},k_{12}) \ , \\
\label{ExchPsi}
\tilde C\psi \es  { 1 \over P^+ }
\int[x_{1'} k_{1'2'}^\perp] \ { \rho_1 + \rho_2 \over 2 q_{m_g}^{+2} } \
\ j_{11'}^{c +}  \ \bar j_{2'2}^{c +} \  f^t_{\overline 1, \underline 1 q} f^t_{\overline 2, \underline 2 q}
\nt
 \left[
{ \rho_1     \pi_{\overline 1}^2 + \rho_2     \pi_{\overline 2}^2
\over 
  \rho_1^2 \pi_{\overline 1}^2 + \rho_2^2 \pi_{\overline 2}^2  - (\rho_1 - \rho_2)^2 \pi_{12}^2 } 
 - {1/2 \over \rho_1 + M^2}  - {1/2 \over \rho_2 + M^2 } 
\right] \, \psi_{1'2'}(x_{1'},k_{1'2'}) \ .
\eeq
{\bf \red 20250109 18:02 I am certain Eqs.~(\ref{SigmaPsi}) and (\ref{ExchPsi}) are correct.}

Now {\red (20250109 20:43)}, first I pay attention to the RGPEP form factors. They exclude
gluons with $x \to 0$ by the term $(p_g^{\perp 2} + m_g^2)/p_g^+$ in the 
exponent. So, the greater $|p_g^\perp|$ the stronger suppression of small  
$x_g$ and vice versa, the smaller $x_g$ the smaller $|p_g^\perp|$ must be for 
the factors not to vanish. But when $|p_g^\perp| \to 0$, gluon mass parameter 
$m_g$ takes over and regulates the gluon small-$x$ diverging factor. 

Also, if the gluon transverse momentum is small, for example, proportional to 
the square root of the gluon $x$, the integration measure provides the factor 
$x$. Can the exchange cancel the self-interaction divergence for $m_g \to 0$ 
in the colorless states? 

Since the exchange changes the relative momentum of quarks and the 
self-interaction does not, the question is if replacing $\psi_{1'2'}(x_{1'},k_{1'2'})$ 
by $\psi_{1'2'}(x_{1'},k_{1'2'}) -\psi_{12}(x_{1},k_{12})$ takes the dependence
on $m_g$ away from the integral with the wave function. 
{\red Stop 20250109 21:50 czw San Dimas, Start 20250110 19:47 pia San Dimas}
The currents $j^+$ conserve spin projections. The color factor is: $C_F$
for the self-interaction, $C_F$ for the gluon exchange in  quarkonium, 
and $C_A$ for the gluon exchange in the octet quarkonium. The only 
arguments of the wave functions that need to be considered are $x$ 
and $k^\perp$. Using  
\beq
\cM_{qg}^2 - m_i^2  \es {p_{\overline i}^+  \over q_{m_g}^+ } \ \rho_i  
\rs {p_{\overline i}^+  \over q_{m_g}^+ } (m_g^2 - q_i^2) \ ,
\eeq
the dependence of the RGPEP form factors on $q^+_{m_g}$ is fully displayed in
\beq
\label{ft2}
(f^t_{i, qg})^2 
\es 
e^{ - 2 \left\{ t \, [ (p_{\underline i} + p_g)^2 - m_1^2 ] \, \pi_{\overline i}/p_{\overline i}^+ \right\}^2 } \\
\es
e^{ - 2 \left[  t \, (m_g^2 - q_i^2) \, \pi_{\overline i}/q_{m_g}^+ \right]^2 }  
\rs 
e^{ - 2 \left( t \, \rho_i \, \pi_{\overline i}/q_{m_g}^+ \right)^2 } 
 \ , \\
\label{f1f2t}
f^t_{\overline 1, \underline 1 g} f^t_{\overline 2, \underline 2 g}
\es  
e^{ - \left\{ t \, [ (p_{\underline 1} + p_g)^2 - m_1^2 ] \, \pi_{\overline 1}/p_{\overline 1}^+ \right\}^2 }   
e^{ - \left\{ t \, [ (p_{\underline 2} + p_g)^2 - m_2^2 ] \, \pi_{\overline 2}/p_{\overline 2}^+ \right\}^2 } \\
\es
e^{ - \left[ t \, (m_g^2 - q_1^2) \, \pi_{\overline 1}/q_{m_g}^+ \right]^2 }   
e^{ - \left[ t \, (m_g^2 - q_2^2) \, \pi_{\overline 2}/q_{m_g}^+ \right]^2 } \\
\es
e^{ - \left( t \, \rho_1 \, \pi_{\overline 1}/q_{m_g}^+ \right)^2 }   
e^{ - \left( t \, \rho_2 \, \pi_{\overline 2}/q_{m_g}^+ \right)^2 } 
\ .
\eeq 
Using $z = x_{1'}-x_1$, the key comparison concerns $-g^2 \, \psi (C_F {\rm ~or~} C_A)/[2(2\pi)^3]$ times
\beq
\label{SigmaPsi1}
\tilde \Sigma 
\es
\sum_i  
 {1 \over p_i^+} \int {dx \, d^2k^\perp \over x(1-x)} \ {\rho_i \over x } 
   4 (1-x) \  (f^t_{i, qg})^2 
\
\left[ {1\over \rho_i}  -  { 1 \over \rho_i + M^2} \right] \ , \\
\label{ExchPsi1}
\tilde C \es  { 1 \over P^+ }
\int { dx_{1'} d^2k_{1'2'}^\perp \over x_{1'} x_{2'} } \ 
{ \rho_1 + \rho_2 \over 2 z^2 } \ (-4\sqrt{ x_{1}x_{1'} x_2 x_{2'} } ) 
\  f^t_{\overline 1, \underline 1 q} f^t_{\overline 2, \underline 2 q}
\nt
 \left[
{ \rho_1     \pi_{\overline 1}^2 + \rho_2     \pi_{\overline 2}^2
\over 
  \rho_1^2 \pi_{\overline 1}^2 + \rho_2^2 \pi_{\overline 2}^2  - (\rho_1 - \rho_2)^2 \pi_{12}^2 } 
 - {1/2 \over \rho_1 + M^2}  - {1/2 \over \rho_2 + M^2 } 
\right] \ ,
\eeq
provided that the exchange integral, 
\beq
\label{ExchPsi2}
\tilde C (\psi' - \psi) \es  { 1 \over P^+ }
\int_{-x_1}^{x_2} dz \int d^2q^\perp  \ 
{ \rho_1 + \rho_2 \over 2 z^2 } \ 4\sqrt{ x_{1} x_2 \over x_{1'} x_{2'}}  
\nt
 \left[
{ \rho_1     \pi_{\overline 1}^2 + \rho_2     \pi_{\overline 2}^2
\over 
  \rho_1^2 \pi_{\overline 1}^2 + \rho_2^2 \pi_{\overline 2}^2  - (\rho_1 - \rho_2)^2 \pi_{12}^2 } 
 - {1/2 \over \rho_1 + M^2}  - {1/2 \over \rho_2 + M^2 } 
\right] 
\nt \left[ \psi(x_1 + z, k_{12}^\perp + q^\perp) -\psi(x_{1},k_{12}^\perp) \right] 
\ 
e^{ - \left[ t \, (\rho_1/z) \, (\pi_{\overline 1}/P^+) \right]^2 }   
e^{ - \left[ t \, (\rho_2/z) \, (\pi_{\overline 2}/P^+) \right]^2 } 
\ , \nn
\eeq
has a finite limit for $m_g \to 0$. I need to check 
the above condition before I tackle the comparison.
{\red Start 20250121 13:39 wto Ekologiczna}
Convergence of $\tilde C (\psi' - \psi)$ is shown using 
a change of variables, rather than an explicit integration.
I use
\beq
\label{qzk}
q^\perp \es \sqrt{|z|} \, k^\perp \ ,
\eeq
and I obtain
\beq
\label{ExchPsi21}
\tilde C (\psi' - \psi) \es  { 1 \over P^+ }
\int_{-x_1}^{x_2} dz \int d^2k^\perp  \ 
{ \rho_1 + \rho_2 \over 2 |z| } \ 4\sqrt{ x_{1} x_2 \over x_{1'} x_{2'}}  
\nt
 \left[
{ \rho_1     \pi_{\overline 1}^2 + \rho_2     \pi_{\overline 2}^2
\over 
  \rho_1^2 \pi_{\overline 1}^2 + \rho_2^2 \pi_{\overline 2}^2  - (\rho_1 - \rho_2)^2 \pi_{12}^2 } 
 - {1/2 \over \rho_1 + M^2}  - {1/2 \over \rho_2 + M^2 } 
\right] 
\nt \left( \psi_z z + \psi_\perp \, k^\perp \, \sqrt{|z|} \right) 
\ 
e^{ - \left[ t \, (\rho_1/z) \, (\pi_{\overline 1}/P^+) \right]^2 }   
e^{ - \left[ t \, (\rho_2/z) \, (\pi_{\overline 2}/P^+) \right]^2 } 
\ ,
\eeq
where, using $p_1 = x_1 P +k$, $p_{1'} = x_{1'} P + k'$ etc., I have
\beq
\label{rho1k}
\rho_1 \es m_g^2 + {m_1^2 (z/x_1)^2 + 
( \sqrt{|z|} k^\perp - k_{12}^{\perp} z/x_1 )^2 \over 1 + z / x_1 } \ , \\
\label{rho2k}
\rho_2 \es m_g^2 + {m_2^2 (z/x_2)^2 + 
( \sqrt{|z|} k^\perp + k_{12}^{\perp} z/x_2 )^2 \over 1 - z / x_2 } \ .
\eeq
Then,
\beq
\rho_1  \es m_g^2 + |z| { (m_1/x_1)^2 |z| + ( k^\perp  - s_z \sqrt{|z|} \, k_{12}^{\perp}/x_1 )^2 \over 1 + z/x_1 } \ , \\
\rho_2  \es m_g^2 + |z| { (m_2/x_2)^2 |z| + ( k^\perp + s_z \sqrt{|z|} \, k_{12}^{\perp}/x_2 )^2 \over 1  - z/x_2) } \ .
\eeq
It is visible that $\rho_1$ and $\rho_2$ are non-negative,
they can be 0 only when $m_g=0$ and $z=0$, and they 
both approach $m_g^2$ plus terms that tend to 0 
as $|z| k^{\perp 2}$ when $z \to 0$ and $|k^\perp| > 0$
is kept fixed. For all values of momenta $k^\perp$ and fraction $z$,
\beq
\rho_1 - \rho_2 & \sim & a \, |z| \ ,
\eeq
where 

\beq
\rho_1/|z| \es {m_g^2 \over |z|} + {m_1^2 |z|/x_1^2 + (k^\perp  - k_{12}^{\perp} s_z \sqrt{|z|} /x_1 )^2 \over 1 + z / x_1 } \ , \\
\rho_2/|z| \es {m_g^2 \over |z|} + {m_2^2 |z|/x_2^2 + (k^\perp + k_{12}^{\perp} s_z \sqrt{|z|} /x_2 )^2 \over 1  - z / x_2 } \ .
\eeq
It is now visible that $\rho_1/|z|$ and $\rho_2/|z|$ behave for 
fixed $z$ and large $k^\perp$ as $k^{\perp 2}/(1+z/x_1)$ 
and $k^{\perp 2}/(1-z/x_2)$, respectively. In the integral
\beq
\label{ExchPsi22}
\tilde C (\psi' - \psi) \es  { 1 \over P^+ }
\int_{-x_1}^{x_2} dz \int d^2k^\perp  \ 
{ \rho_1 + \rho_2 \over 2 |z| } \ 4\sqrt{ x_{1} x_2 \over x_{1'} x_{2'}}  
\nt
 \left[
{ \rho_1     \pi_{\overline 1}^2 + \rho_2     \pi_{\overline 2}^2
\over 
  \rho_1^2 \pi_{\overline 1}^2 + \rho_2^2 \pi_{\overline 2}^2  - (\rho_1 - \rho_2)^2 \pi_{12}^2 } 
 - {1/2 \over \rho_1 + M^2}  - {1/2 \over \rho_2 + M^2 } 
\right] 
\nt \left( \psi_z z + \psi_\perp \, k^\perp \, \sqrt{|z|} \right) 
\ 
e^{ - \left[ t \, (\rho_1/z) \, (\pi_{\overline 1}/P^+) \right]^2 }   
e^{ - \left[ t \, (\rho_2/z) \, (\pi_{\overline 2}/P^+) \right]^2 } \ ,
\eeq
the range of integration over $k^{\perp 2}$ when $m_g^2=0$ 
is limited from above by about 
\beq
\Lambda_{k}^2 
\es 
\left\{ [t (\pi_{\overline 1}/P^+) (x_1/x_{1'})]^2 + [t (\pi_{\overline 2}/P^+) (x_2/x_{2'})]^2 \right\}^{-1/2} \\
\es
{1 \over t}
\left[ \left( {\pi_{\overline 1} x_1 \over P^+ x_{1'} } \right)^2 
     + \left( {\pi_{\overline 2} x_2 \over P^+ x_{2'} } \right)^2 \right]^{-1/2}  
     \rs 
     {1 \over t}
\left[ \left( x_{\overline 1} \, {x_1 \over x_{1'} } \right)^2 
     + \left( x_{\overline 2} \, {x_2 \over x_{2'} } \right)^2 \right]^{-1/2}  .
\eeq
This is a finite upper bound. 
For $m_g=0$ and $z \to 0$ one has
\beq
\Lambda_{k}^2 
\es 
 {1 \over t  \sqrt {x_{\overline 1}^2 + x_{\overline 2}^2 } } \ .
\eeq
The wave-function difference softens $1/|z|$ to $1/\sqrt{|z|}$,
which is integrable over $z$.
{\red Start 20250122 03:30 sro Ekologiczna, Continued 20250123 
from early morning czw Ekologiczna} 
The remaining question concerning large transverse momentum 
$k^\perp$ is if the limit $m_g \to 0$ exists and if it is the same when 
one starts with $m_g>0$. Regarding the exponential factors, their 
arguments are
\beq
\rho_i/|z| \es m_g^2/|z| + \rho_{i0}/|z| \ , \\
( \rho_i/|z| )^2 \es (m_g^2/z)^2 + 2 m_g^2 \rho_{i0}/z^2 + \rho_{i0}^2/z^2 \ ,
\eeq
and 
\beq
e^{-a( \rho_i/|z| )^2} \es e^{-a(m_g^2/z)^2} \, e^{-a 2 m_g^2\rho_{i0}/|z|} \, e^{-a (\rho_{i0}/z)^2} \ .
\eeq
The third exponential function is the same as the one for $m_g=0$.
The first and second exponential functions provide additional 
suppression of the integrand for small $z$ or large $k^\perp$.
The third exponential limits $z$ and $k^\perp$ as in the case 
of $m_g=0$. The additional suppression by the first and second 
exponentials goes away when $m_g \to 0$. The limit of the 
exponential factors is the same as the result for $m_g=0$.

The next question is what happens when $k^\perp$ is limited 
or tends to 0. The terms with $M^2$ in denominator are safe, 
because $M^2$ is positive and excludes zero in the denominator 
while $\rho_i$s in the numerator have definite limits when $m_g \to 0$. 
The issue is how $\Delta_{232}$ behaves for small $\rho_i$s. Given 
expressions for $\rho_i$s, I can replace every $\rho$ by $\tilde \rho 
= \rho/|z|$ in both the RGPEP $\Delta_{232}$ and the gluon spin 
factor $\rho_1+\rho_2$ by $\tilde \rho = \rho/|z|$,
\beq
\tilde \rho_1 \es
\rho_1/|z| \rs {m_g^2 \over |z|} + {m_1^2 |z|/x_1^2 + (k^\perp  - k_{12}^{\perp} s_z \sqrt{|z|} /x_1 )^2 \over 1 + z / x_1 } \ , \\
\tilde \rho_2 \es
\rho_2/|z| \rs {m_g^2 \over |z|} + {m_2^2 |z|/x_2^2 + (k^\perp + k_{12}^{\perp} s_z \sqrt{|z|} /x_2 )^2 \over 1  - z / x_2 } \ .
\eeq
I look at 
\beq
\Delta \es { (\tilde \rho_1+\tilde \rho_2)
(\tilde \rho_1     \pi_{\overline 1}^2 + \tilde \rho_2     \pi_{\overline 2}^2)
\over 
 \tilde \rho_1^2 \pi_{\overline 1}^2 + \tilde \rho_2^2 \pi_{\overline 2}^2  
 - (\tilde \rho_1 - \tilde \rho_2)^2 \pi_{12}^2 } 
 \rs
{  \tilde \rho_1^2 \pi_{\overline 1}^2 + \tilde \rho_2^2 \pi_{\overline 2}^2   
+ \tilde \rho_1 \tilde \rho_2 (\pi_{\overline 1}^2 + \pi_{\overline 2}^2)
\over 
 \tilde \rho_1^2 \pi_{\overline 1}^2 + \tilde \rho_2^2 \pi_{\overline 2}^2  
 - (\tilde \rho_1 - \tilde \rho_2)^2 \pi_{12}^2 } \ .
\eeq
Cancellation in the denominator {\red was thought to be} eliminated by the corresponding term in part $B$
of the eigenvalue equation. {\red Stop 20250123 18:23 czw Ekologiczna padam}
{\red Start 20250124 04:32 pia Ekologiczna} In the case of $m_g=0$ and all $\pi_i$ being the same, the 
factor to check is
\beq
\Delta \es { (\tilde \rho_1+\tilde \rho_2)^2
\over 
2 \tilde \rho_1 \tilde \rho_2 } \rs 
\2 \left( {\tilde \rho_1 \over \tilde \rho_2} + {\tilde \rho_2 \over \tilde \rho_1} + 2 \right)\ .
\eeq
Can only one $\tilde \rho$ vanish? This would require that $z$ vanishes
and consequently also that $k^\perp$ vanishes. Hence, both $\tilde \rho$
would vanish. If both vanish, one vanishes, too. So one vanishing is equivalent 
to both vanishing. The question is then if the ratio of two $\tilde \rho$s can
diverge as $1/k^{\perp 2}$ or $1/\sqrt{|z|}$. Consider the example of ratio
\beq
{\tilde \rho_1 \over \tilde \rho_2}
\es
{ {\red {m_g^2 \over |z|}} + m_1^2 |z|/x_1^2 + (k^\perp  - k_{12}^{\perp} s_z \sqrt{|z|} /x_1 )^2 
  \over
  {\red {m_g^2 \over |z|}} + m_2^2 |z|/x_2^2 + (k^\perp + k_{12}^{\perp} s_z \sqrt{|z|} /x_2 )^2 } \
{1  - z / x_2 \over 1 + z / x_1} \ .
\eeq
Since $z$ cannot approach $- x_1$ or $x_2$ too closely in the 
eigenvalue problem (the basis states would have too large 
invariant masses, which are suppressed by the RGPEP width 
limit on changes of invariant masses), I can drop the second 
factor. It is close to 1 and it cannot diverge. Dividing the 
numerator and denominator by $|z|$ I get
\beq
{\tilde \rho_1 \over \tilde \rho_2}
& \sim &
{ {\red {m_g^2 \over z^2}} + m_1^2 /x_1^2 + (k^\perp/\sqrt{|z|} - k_{12}^{\perp} s_z  /x_1 )^2 
  \over
  {\red {m_g^2 \over z^2}} + m_2^2 /x_2^2 + (k^\perp/\sqrt{|z|} + k_{12}^{\perp} s_z  /x_2 )^2 } \ .
\eeq
Now the behavior of the ratio $R(\xi^\perp) = \tilde \rho_1 / \tilde \rho_2$ is
reduced to a function of one variable $\xi^\perp = k^\perp/\sqrt{|z|}$ that 
can range from $-\infty$ to $+\infty$,
\beq
R(\xi^\perp) 
& \sim &
{ {\red {m_g^2 \over z^2}} + a_1 + (\xi^\perp - b_1^\perp )^2  
\over  
  {\red {m_g^2 \over z^2}} +a_2 + (\xi^\perp - b_2^\perp )^2  } \ .
\eeq
with finite positive $a_1$, $a_2$, $b_1^\perp$ and $b_2^\perp$. This function
can be written in terms of $\zeta^\perp = \xi^\perp - b_2^\perp$ in the 
form 
\beq
\tilde R(\zeta^\perp) 
& \sim &
{ {\red {m_g^2 \over z^2}} + a_1 + (\zeta^\perp + b^\perp )^2  \over  {\red {m_g^2 \over z^2}} + a_2 + \zeta^{\perp 2}  } 
\rs 
{ {\red {m_g^2 \over z^2}} + a_1 + \zeta_\perp^2 + (\zeta_{||} + b)^2  \over  {\red {m_g^2 \over z^2}} + a_2 + \zeta_\perp^2 + \zeta_{||}^2 } 
\ .
\eeq
It is visible that for large $\zeta^\perp$ one obtains $\tilde r=1$.
One can find the minimum and maximum but it is not necessary.
It suffices that the ratio $\tilde R(\zeta^\perp) $ has limited positive 
size for finite positive $a_1$, $a_2$ and limited $b^\perp$. This 
means that $1/\tilde R(\zeta^\perp) $ is also positive and limited. 
Therefore both $R(\xi^\perp)$ and $1/R(\xi^\perp)$ are limited.
Hence, $\Delta$ is limited and thus the integral $\tilde C (\psi' - \psi)$ 
in Eq.~(\ref{ExchPsi22}) is finite for $m_g = 0$. {\red When $m_g>0$,
I get $R=1$ for $z \to 0$ and there is no divergence. But there is 
a range of variation around 1.}  

Now the question is what happens when $\pi_i$ are not all equal.
I can still write $\Delta$ in terms of the ratio $R(\xi^\perp) = \tilde \rho_1 
/ \tilde \rho_2$,
\beq
\Delta \es { ( R + 1)
(R \pi_{\overline 1}^2 + \pi_{\overline 2}^2)
\over 
 R^2 \pi_{\overline 1}^2 + \pi_{\overline 2}^2  
 - (R- 1)^2 \pi_{12}^2 }  \ .
\eeq
Since $R$ is limited and positive, separated from 0, 
one could conclude that $\Delta$ is limited if the
denominator could not be zero. But it can be 0. 
However, the complete $Q \bar Q$  eigenvalue 
problem includes the part $B$ that effectively 
provides a factor that vanishes precisely when 
the denominator vanishes {\red (Is it true? YES, IT IS TRUE, 
see Eqs.~(\ref{HLRDelta}) to (\ref{Delta0})}. I inspect below
how the interplay between $\Delta$, the RGPEP
factors $f$ and the parts $B$ and $C$ eliminates
the denominator 0 and leads to the conclusion
that the subtraction $\psi' - \psi$ is a legitimate step.
{\red However, for $m_g>0$, I get for $z \to 0$ 
that $k^\perp$ is limited, $R=1$ and $\Delta =2$,
independently of 0 that may occur away from $z=0$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{The denominator 0 in $\Delta_{LIR}$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The $Q \bar Q$ eigenvalue Eq.~(\ref{HLR1pom1})
\beq
\label{HLR1pom11}
\cH_{LR} \es \cH_f + f_{LR} \, \left[ \cH^{(2)}_0 + \Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0 \right]
\np
{1 \over 2} f_{LI} f_{IR} \ \cH_0^{(1)}\left[ (P_L^- - \cH_f)^{-1} 
+ (P_R^- - \cH_f)^{-1} - 2 \Delta_{LIR} \right] \cH_0^{(1)} \ .
\eeq
The product $\cH^{(1)}_0 \, \cH^{(1)}_0$ provides the factor 
$(\rho_1 + \rho_2)/z^2$. The divergent dependence on $r$ 
occurs in the exchange with one $f$ due to $1/z^2$. In the 
case of exchange with $ff$, the same factor $(\rho_1 + \rho_2)/z^2$ 
appears but the factors $ff$ suppress small $z$. The wave function 
difference turns $1/z^2$ to $1/|z|$ or to $1/|z|^{3/2}$, which is seen using 
$q^\perp = \sqrt{|z|} k^\perp$. The issue is that in 
\beq
(\rho_1 + \rho_2) \, (f_{LR} - f_{LI} f_{IR}) \, \Delta_{LIR} 
\eeq
the denominator zero in $\Delta = (\rho_1 + \rho_2)\Delta_{LIR}$,
which comes from the denominator in $\Delta_{LIR}$, is compensated 
by the difference of $f$s in the numerator. Therefore, I need to verify 
if dealing with divergences involving $\Delta_{LIR}$ with $f_{LR}$ or 
$\Delta_{LIR}$ with $f_{LI} f_{IR}$ separately, as I did so far (20250125 05:46 
sob Ekologiczna), is legitimate. In other words, are zeros in the denominator 
of $\Delta_{LIR}$ invalidating my analysis and computation of counterterms.

First, I use $\rho_i$s to compare the functions 
$f_{LR}$ and  $f_{LI} f_{IR}$, the latter already displayed 
in Eq.~(\ref{f1f2t}). The former is $\exp\{ - [ t ( \cM_{12}^2 - \cM_{1'2'}^2 ) \pi_{12}/P^+]^2 \}$,
where
\beq
\cM_{12}^2 - \cM_{1'2'}^2
\es
(p_{1'}^+ + p_{2'}^+)(p_{1'}^- + p_{2'}^-) - (p_{1}^+ + p_2^+)(p_{1}^- + p_{2}^-) \\
\es
P^+ (p_{1'}^- + p_{2'}^- - p_{1}^+ - p_2^+) \rs s_z P^+(q_1^- - q_2^-) \\
\es (s_z P^+/q_1^+) (q_1^+ q_1^- - q_1^{\perp 2} - q_2^+ q_2^- + q_2^{\perp 2}) \\
\es (1/z)(q_1^2 - q_2^2) \rs (\rho_1 - \rho_2)/z \ .
\eeq
Therefore,
\beq
\label{ft2pom1}
f_{LR} \es e^{ - \left[ t (\rho_1/z - \rho_2/ z)\pi_{12} /P^+\right]^2 } 
 \ , \\
\label{f1f2tpom1}
f_{LI} f_{IR}
\es
e^{ - \left[ t \, (\rho_1 /z) \, \pi_{\overline 1}/P^+ \right]^2 }   
e^{ - \left[ t \, (\rho_2 /z) \, \pi_{\overline 2}/P^+ \right]^2 } 
\ .
\eeq 
In the case of my boost-ivariant version,
\beq
\label{ft2pom2}
f_{LR} \es e^{ - \left[ t (\rho_1 - \rho_2)/ z \right]^2 } 
 \ , \\
\label{f1f2tpom2}
f_{LI} f_{IR}
\es
e^{ - \left[ t \, \rho_1 \, x_{\overline 1}/z \right]^2 }   
e^{ - \left[ t \, \rho_2 \, x_{\overline 2}/z \right]^2 } 
\ .
\eeq 
Thus, I can change variables using $q^\perp = \sqrt{|z|} k^\perp$
that reduces the singularity for small $|z|$ to the integrable $1/\sqrt{|z|}$
while I use  
\beq
\tilde \rho_1 \es
\rho_1/|z| \rs {m_1^2 |z|/x_1^2 + (k^\perp  - k_{12}^{\perp} s_z \sqrt{|z|} /x_1 )^2 \over 1 + z / x_1 } \ , \\
\tilde \rho_2 \es
\rho_2/|z| \rs {m_2^2 |z|/x_2^2 + (k^\perp + k_{12}^{\perp} s_z \sqrt{|z|} /x_2 )^2 \over 1  - z / x_2 } \ .
\eeq
in
\beq
\Delta \es { (\tilde \rho_1+\tilde \rho_2)
(\tilde \rho_1     \pi_{\overline 1}^2 + \tilde \rho_2     \pi_{\overline 2}^2)
\over 
 \tilde \rho_1^2 \pi_{\overline 1}^2 + \tilde \rho_2^2 \pi_{\overline 2}^2  
 - (\tilde \rho_1 - \tilde \rho_2)^2 \pi_{12}^2 } 
\eeq
with
\beq
\label{ft2pom3}
f_{LR} \es e^{ - \left[ t (\tilde \rho_1 - \tilde \rho_2) \right]^2 } 
 \ , \\
\label{f1f2tpom3}
f_{LI} f_{IR}
\es
e^{ - \left[ t \, \tilde \rho_1 \, x_{\overline 1} \right]^2 }   
e^{ - \left[ t \, \tilde \rho_2 \, x_{\overline 2} \right]^2 } 
\ .
\eeq 
I can use the argument that $k^\perp$ is limited with $f-ff$
as well as I had done it for $f$ and for $ff$ separately.



Second, I note that in the self-interactions and mass counterterms, 
the factor $\Delta_{LIR}$ is not sensitive to $\pi_i$s. It only provides 
the denominator that is the same as in the old-fashioned Hamiltonian 
perturbation theory, no matter what $\pi_i$s I choose. In the 
self-interaction terms, there is no issue with denominator 0 in $\Delta$
because the invariant mass of quark and gluon is always greater than 
that of the quark, except the case when the gluon momentum is zero, 
which lies on the edge of the integration region. The singularity from
that region is accounted for and a counterterm is introduced.

In the exchange terms, one ought to consider
\beq
(f_{LR} - f_{LI} f_{IR}) \Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0 \ ,
\eeq
where the bracket factor cancels the denominator of $\Delta_{LIR}$
when the latter tends to 0. My analysis of the exchange was so
far (20250125 11:14 sob Ekologiczna) carried out for the two terms
separately, the term with the factor $f_{LR} \Delta_{LIR}$, which is 
singular for finite $s$ because of the gluon small-$x$ singularity, 
and the term with the factor $f_{LI} f_{IR} \Delta_{LIR}$ that for 
finite $s$ excludes small-$x$ singularities but depends on the gluon 
mass $m_g$. The question is how is each of these analyses altered
when the factors $f_{LR}$ or $f_{LI} f_{IR}$ are replaced by their
difference. In the case of part $B$, see Eq.~(\ref{B}), repeated here,
\beq
\label{Bpom1}
B \es  f_{LR}  \left[ H^{(2)}_0 + \Delta_{L3R} \ H^{(1)}_0 \, H^{(1)}_0 \right] \ ,
\eeq
I go back to Eq.~(\ref{B}) and check what changes in the subsequent
reasoning and equations for the exchange terms when I replace 
$f_{LR}$ by $f_{LR}-f_{LI} f_{IR}$. The seagull analysis is not changed at all
because it does not involve $\Delta$ . The self-interactions are free from 
the denominator 0 in $\Delta$ as already explained in a paragraph above;
the denominator 0 is absent. I need to identify the change in the computation 
of the ccounterterm to the diverging dependence on $r$ when instead 
of  $f_{LR} \Delta_{L3R} \ H^{(1)}_0 \, H^{(1)}_0$ I consider the dependence 
on $r$ of the exchange
\beq
\label{Bpom2}
(f_{LR} - f_{LI} f_{IR}) \Delta_{LIR} \  \cH^{(1)}_0 \, \cH^{(1)}_0 \ .
\eeq
The dependence on $r$ comes only from the first term.
One can set $r$ to 0 in the second term; the second term
has a finite limit when $r \to 0$. The counterterm to the first 
term contains $\delta(z)$. For $z=0$, the second term 
vanishes because of the factor $f_{LI} f_{IR}$ that vanishes 
for $m_g^2/z \to \infty$. 
{\red Start 20250126 06:50 nie Ekologiczna}
I go back to Eqs.~(\ref{rho1k}) and (\ref{rho2k}), repeated here
\beq
\label{rho1kpom1}
\rho_1 \es m_g^2 + {m_1^2 (z/x_1)^2 + 
( q^\perp + k_{12}^{\perp} z/x_1 )^2 \over 1 + z / x_1 } \ , \\
\label{rho2kpom1}
\rho_2 \es m_g^2 + {m_2^2 (z/x_2)^2 + 
( q^\perp + k_{12}^{\perp} z/x_2 )^2 \over 1 - z / x_2 } \ .
\eeq
It is clear that for $q^\perp$ limited by $f$s and $z \to 0$ 
I have $\rho_i = m_g^2 + q^{\perp 2}$ and there is no zero 
in the denominator of $\Delta = 2$.








{\red \bf 
************
 
************

************}

Higher orders

Extrapolation $g \to g_s$

Beyond perturbative expansion for $H_s$

Successive approximations

Increasing $m_g \to m_G$ 

No need for artificial potential and scaling of oscillator

Mass for confinement instead of potential

Gluons needed above the oscillator


\begin{appendix}


%%%%%%%%%%%%%%%%%%%%%%
\section{Gluon self-interaction computation}
\label{GluonSelfComputation}
%%%%%%%%%%%%%%%%%%%%%%
\beq
\label{bareAfpom}
\hat A_f^\mu
\es
\sum_{c=1}^8 
\sum_{\sigma =1}^2  \int [p]
\left[  \varepsilon^\mu_{p\sigma} T^c \hat a_{p\sigma c} e^{-ipx}
        +  \varepsilon^{\mu *}_{p\sigma} T^c \hat a^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0}
\ ,
\\
\label{hatphipom}
\hat \phi
\es
\sum_{c=1}^8 
\int [p]
\left[  -i T^c \hat a_{p3c} e^{-ipx}
        +  i T^c \hat a^\dagger_{p3c} e^{ipx}
\right]_{x^+=0} \ .
\eeq
%%%%%%%%%%%%%%%%%%%%
\subsection{ Computation of $H_{+1}$}
%%%%%%%%%%%%%%%%%%%%
\beq
J_A^{a+} \es - g [  i\partial^+ A^\perp , A^\perp ]^a \ , \\
\label{H+1pom}
\cH_{+1} 
\es
\left( \partial^\perp A_f^{a\perp} + m_g \phi^a \right) {1 \over \partial^+} J_A^{a+} 
\ \equiv \
\left[ {-1 \over i\partial^+} \left( i\partial^\perp A_f^{a\perp} + i m_g \phi^a \right) \right] J_A^{a+}  \ .
\eeq
Common factors.
\beq
\label{|N|^2}
{\Sigma_g^{(2)} \over p^+} \es {g^2 \over p^+} \int {d k^+ d^2k^\perp \over  2 (2\pi)^3 k^+ (p^+-k^+)} \
{ |N|^2 \over P^- - Q^- - p_1^- - p_2^- } \ .
\eeq
I need to properly write $N$.
{\red Start 20240926 07:33 -> 13:53 czw Cienega Ave.}
\beq
J_A^{a+} \tl - g i f^{12a} p_1^+ \left(\varepsilon_1^\perp a_1 - \varepsilon_1^{\perp *}a_1^\dagger \right)\left(\varepsilon_2^\perp a_2 + \varepsilon_2^{\perp *} a_2^\dagger \right) \\
\es
- g i f^{12a} \ p_1^+ \ \varepsilon_1^\perp a_1 \ \varepsilon_2^\perp a_2 
- g i f^{12a} \ p_1^+ \ \varepsilon_1^\perp a_1 \ \varepsilon_2^{\perp *} a_2^\dagger 
\np
  g i f^{12a} \ p_1^+ \ \varepsilon_1^{\perp *} a_1^\dagger \ \varepsilon_2^\perp a_2 
+ g i f^{12a} \ p_1^+ \ \varepsilon_1^{\perp *}a_1^\dagger \ \varepsilon_2^{\perp *} a_2^\dagger \ .
\eeq
Normal order and arrangement.
\beq
J_A^{3+} 
\tl 
- g i f^{123} \ p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
- g i f^{123} \ p_1^+ \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
\nm
  g i f^{123} \ p_2^+ \ \varepsilon_2^{\perp *}\varepsilon_1^\perp \ a_2^\dagger a_1 
+ g i f^{123} \ p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger \ .
\eeq
\beq
{-1 \over i\partial^+} \left( i\partial^\perp A_f^{3\perp} + i m_g \phi^3 \right) 
\tl
- \left[ \left( {p_3^\perp \varepsilon_3^\perp \over p_3^+} a_3 
              + {p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} a_3^\dagger \right)  
+
                {m_g \over p_3^+} \left( \alpha_3 + \alpha_3^\dagger \right)
\right] \ .
\eeq
Simplify.
\beq
J_A^{3+} 
\tl 
- g i f^{123} \ 
\left[ 
    p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
+ (p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
-  p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger 
\right] \ .
\eeq
\beq
{-1 \over i\partial^+} \left( i\partial^\perp A_f^{3\perp} + i m_g \phi^3 \right) 
\tl
- \left[ \left( {p_3^\perp \varepsilon_3^\perp \over p_3^+} a_3 
              + {p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} a_3^\dagger \right)  
+
                {m_g \over p_3^+} \left( \alpha_3 + \alpha_3^\dagger \right)
\right] \ .
\eeq
Multiplication.
\beq
H_{+1} \es \int_F
{-1 \over i\partial^+} \left( i\partial^\perp A_f^{3\perp} + i m_g \phi^3 \right) J_A^{3+} \\
\tl
- \sum_{123} \int[123] \ \tilde \delta_{c.a} \left( {p_3^\perp \varepsilon_3^\perp \over p_3^+} a_3 
              + {p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} a_3^\dagger 
+
                {m_g \over p_3^+} \alpha_3 + {m_g \over p_3^+} \alpha_3^\dagger \right)
\nt
- g i f^{123} \ 
\left[ 
    p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
+ (p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
-  p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger 
\right] 
\eeq
Simplify.
\beq
H_{+1} 
\tl
\sum_{123} \int[123] \ g i f^{123} \ \tilde \delta_{c.a} 
%      
\left( {p_3^\perp \varepsilon_3^\perp \over p_3^+} a_3 
     + {p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} a_3^\dagger 
+
                {m_g \over p_3^+} \alpha_3 + {m_g \over p_3^+} \alpha_3^\dagger \right)
\nt
 \ 
\left[ 
    p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
+ (p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
-  p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger 
\right] \\
\es
\sum_{123} \int[123] \ g i f^{123} \ \tilde \delta_{c.a} \  i g \ S \ .
\eeq
There are 12 terms to simplify.
\beq
S \es
f^{123}
{p_3^\perp \varepsilon_3^\perp \over p_3^+} a_3 
\left[ 
        (p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
      -  p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger 
\right] 
\np
f^{123}
{p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} a_3^\dagger 
\left[ 
        p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
    + (p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
\right] 
\np
f^{123}
{m_g \over p_3^+} \alpha_3
\left[ (p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
-  p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger 
\right] 
\np
f^{123}
{m_g \over p_3^+} \alpha_3^\dagger
\left[ 
    p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
+ (p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
\right] \ .
\eeq
Move.
\beq
S \es
f^{123}
{p_3^\perp \varepsilon_3^\perp \over p_3^+} 
(p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \    a_2^\dagger a_3 a_1
~~~~
2 \leftrightarrow 3
\nm 
f^{123}
{p_3^\perp \varepsilon_3^\perp \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \   a_1^\dagger a_2^\dagger a_3
%
\np
f^{123}
{p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} 
p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_3^\dagger  a_2 a_1 
\np
f^{123}
{p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} 
(p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_3^\dagger  a_2^\dagger a_1
~~~~
1 \leftrightarrow 3
%
\np
f^{123}
{m_g \over p_3^+} 
(p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \   a_2^\dagger \alpha_3 a_1
~~~~
2 \leftrightarrow 3
\nm  
f^{123}
{m_g \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \  a_1^\dagger a_2^\dagger \alpha_3
%
\np
f^{123}
{m_g \over p_3^+} 
    p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ \alpha_3^\dagger  a_2  a_1
\np
f^{123}
{m_g \over p_3^+} 
(p_1^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  \alpha_3^\dagger a_2^\dagger a_1 
~~~~
1 \leftrightarrow 3 \ .
\eeq
Changes.
\beq
S \es f^{123} S' \ , \\
S' \es
- 
{p_2^\perp \varepsilon_2^\perp \over p_2^+} 
(p_1^+ + p_3^+) \ \varepsilon_3^{\perp *} \varepsilon_1^\perp  \    a_3^\dagger a_2 a_1
\nm 
{p_3^\perp \varepsilon_3^\perp \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \   a_1^\dagger a_2^\dagger a_3
%
\np
{p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} 
p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_3^\dagger  a_2 a_1 
\nm
{p_1^\perp \varepsilon_1^{\perp *} \over p_1^+} 
(p_3^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp  \  a_1^\dagger  a_2^\dagger a_3
%
\nm
{m_g \over p_2^+} 
(p_1^+ + p_3^+) \ \varepsilon_3^{\perp *} \varepsilon_1^\perp  \   a_3^\dagger \alpha_2 a_1
\nm  
{m_g \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \  a_1^\dagger a_2^\dagger \alpha_3
%
\np
{m_g \over p_3^+} 
    p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ \alpha_3^\dagger  a_2  a_1
\nm
{m_g \over p_1^+} 
(p_3^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp  \  \alpha_1^\dagger a_2^\dagger a_3 
 \ .
\eeq
Collection.
\beq
S \es f^{123} S' \ , \\
S' \es
-
\left[
{p_3^\perp \varepsilon_3^\perp \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} 
+
{p_1^\perp \varepsilon_1^{\perp *} \over p_1^+} 
(p_3^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp 
\right]
\  a_1^\dagger  a_2^\dagger a_3
%
\np
\left[
{p_3^\perp \varepsilon_3^{\perp *} \over p_3^+} 
p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp 
+
{p_1^\perp \varepsilon_1^\perp \over p_1^+} 
(p_3^+ + p_2^+) \ \varepsilon_2^\perp \varepsilon_3^{\perp *}   
\right]
\ a_3^\dagger  a_2 a_1 
\nm
{m_g \over p_2^+} 
(p_3^+ + p_1^+) \ \varepsilon_1^\perp \varepsilon_3^{\perp *}   \   a_3^\dagger \alpha_2 a_1
+
{m_g \over p_2^+} 
(p_3^+ + p_1^+) \ \varepsilon_1^{\perp *} \varepsilon_3^\perp  \   a_1^\dagger \alpha_2^\dagger a_3 
\nm  
{m_g \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \   a_1^\dagger  a_2^\dagger \alpha_3
+
{m_g \over p_3^+} 
    p_1^+ \ \varepsilon_1^\perp \varepsilon_2^\perp \ \alpha_3^\dagger  a_2  a_1 \ .
\eeq
Result.
\beq
H_{+1} 
\es
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \ X_{+123}  + h.c. \ .
\eeq
\beq
X_{+123}
\es
i f^{c_1 c_2 c_3}
\left[
- {p_3^\perp \varepsilon_3^\perp \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} 
-
{p_1^\perp \varepsilon_1^{\perp *} \over p_1^+} 
(p_3^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp 
\right]
\  a_1^\dagger  a_2^\dagger a_3
\nm
i f^{c_1 c_2 c_3}
{m_g \over p_2^+} 
(p_3^+ + p_1^+) \ \varepsilon_1^\perp \varepsilon_3^{\perp *}   \   a_3^\dagger \alpha_2 a_1
\nm  
i f^{c_1 c_2 c_3}
{m_g \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \   a_1^\dagger  a_2^\dagger \alpha_3 \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%
\subsection{ Computation of $H_{\perp 1}$}
\label{Hperp1}
%%%%%%%%%%%%%%%%%%%%%%


{\red Stop 20240926 22:23 czw Cienega Ave.}

{\red Start 20240927 10:26 -> 10:42 pia Cienega Ave.}

The complementary transverse gluon interaction density term, the 3rd term on 
the right-hand side in Eq.~(\ref{HQCDmgphir}), is
\beq
(J_{\psi f}^{a j}  + J_{A f}^{a j}) A_f^{a j} \ ,
\eeq
of which the purely transverse gluon part reads
\beq
\cH_{\perp 1}
\es  A_f^{a j} J_{A f}^{a j} \ . 
\eeq
\beq
J_{A f}^{a j} \es - g [  i\partial^j A^\perp , A^\perp ]^a \ , \\
J_{A f}^{a j} 
\tl - g i f^{12a} p_1^j \left(\varepsilon_1^\perp a_1 - \varepsilon_1^{\perp *}a_1^\dagger \right)\left(\varepsilon_2^\perp a_2 + \varepsilon_2^{\perp *} a_2^\dagger \right) \\
\es
- g i f^{12a} \ p_1^j \ \varepsilon_1^\perp a_1 \ \varepsilon_2^\perp a_2 
- g i f^{12a} \ p_1^j \ \varepsilon_1^\perp a_1 \ \varepsilon_2^{\perp *} a_2^\dagger 
\np
  g i f^{12a} \ p_1^j \ \varepsilon_1^{\perp *} a_1^\dagger \ \varepsilon_2^\perp a_2 
+ g i f^{12a} \ p_1^j \ \varepsilon_1^{\perp *}a_1^\dagger \ \varepsilon_2^{\perp *} a_2^\dagger \ .
\eeq
Normal order and arrangement.
\beq
J_A^{3j} 
\tl 
- g i f^{123} \ p_1^j \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
- g i f^{123} \ p_1^j \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
\nm
  g i f^{123} \ p_2^j \ \varepsilon_2^{\perp *}\varepsilon_1^\perp \ a_2^\dagger a_1 
+ g i f^{123} \ p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger \ , \\
A_f^{3 j}  \tl \left( \varepsilon_3^j  a_3  + \varepsilon_3^{j *} a_3^\dagger \right)  \ .
\eeq
Simplify.
\beq
J_A^{3j} 
\tl 
- g i f^{123} \ 
\left[ 
    p_1^j \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
+ (p_1^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
-  p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger 
\right] \ .
\eeq
Multiplication.
\beq
H_{\perp 1} \es \int_F \ A_f^{3 j} J_A^{3+} 
~~ \to ~~ 
\sum_{123} \int[123] \ \tilde \delta_{c.a} \ (-ig) \ S \ , \\
S
\es f^{123}
( \varepsilon_3^j  a_3  + \varepsilon_3^{j *} a_3^\dagger )
[ 
    p_1^j \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
+ (p_1^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
-  p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger 
] .
\eeq
There are 4 terms to simplify.
\beq
S
\es f^{123}
\varepsilon_3^j  a_3 
[ (p_1^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1
-  p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger ] 
\np
f^{123}  
\varepsilon_3^{j *} a_3^\dagger 
[   p_1^j \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_1 a_2 
+ (p_1^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_1 ] \ .
\eeq
Move.
\beq
S
\es 
f^{123} \varepsilon_3^j  
(p_1^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_2^\dagger a_3  a_1
~~~~ 2 \leftrightarrow 3
\nm
f^{123} \varepsilon_3^j 
p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger  a_3  
\np
f^{123}  \varepsilon_3^{j *} 
p_1^j \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_3^\dagger  a_2 a_1
\np 
f^{123}  \varepsilon_3^{j *} 
(p_1^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_1^\perp  \  a_3^\dagger  a_2^\dagger a_1 
~~~~ 1 \leftrightarrow 3
\ .
\eeq
Changes.
\beq
S
\es 
- f^{123} \varepsilon_2^j  
(p_1^j + p_3^j) \ \varepsilon_3^{\perp *} \varepsilon_1^\perp  \  a_3^\dagger a_2  a_1
\np
f^{123}  \varepsilon_3^{j *} 
p_1^j \ \varepsilon_1^\perp \varepsilon_2^\perp \ a_3^\dagger  a_2 a_1
\nm
f^{123} \varepsilon_3^j 
p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ a_1^\dagger a_2^\dagger  a_3  
\nm
f^{123}  \varepsilon_1^{j *} 
(p_3^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp  \  a_1^\dagger  a_2^\dagger a_3 
\ .
\eeq
Collection.
\beq
S
\es f^{123}
\left[
     \varepsilon_1^j  (p_3^j + p_2^j ) \ \varepsilon_2^\perp  \varepsilon_3^{\perp *} 
+  \varepsilon_3^{j *} p_1^j \ \varepsilon_1^\perp \varepsilon_2^\perp 
\right] \ 
a_3^\dagger  a_2 a_1
\nm
f^{123} 
\left[
\varepsilon_1^{j *} (p_3^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp
+
\varepsilon_3^j p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} 
\right] \ 
a_1^\dagger  a_2^\dagger a_3 \ .
\eeq
Result.
\beq
H_{\perp 1} 
\es 
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \ X_{\perp 123}  + h.c. \ . 
\eeq
\beq
X_{\perp 123} 
\es
i f^{123} 
\left[
\varepsilon_1^{j *} (p_3^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp
+
\varepsilon_3^j p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} 
\right] \ 
a_1^\dagger  a_2^\dagger a_3 \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Summary of $H_{+1}$ and $H_{\perp 1}$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beq
\label{H+1summary}
H_{+1} 
\es
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \ X_{+123}  + h.c. \ , \\
X_{+123}
\es
i f^{c_1 c_2 c_3}
\left[
- {p_3^\perp \varepsilon_3^\perp \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} 
-
{p_1^\perp \varepsilon_1^{\perp *} \over p_1^+} 
(p_3^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp 
\right]
\  a_1^\dagger  a_2^\dagger a_3
\nm
i f^{c_1 c_2 c_3}
{m_g \over p_2^+} 
(p_3^+ + p_1^+) \ \varepsilon_1^\perp \varepsilon_3^{\perp *}   \   a_3^\dagger \alpha_2 a_1
-
i f^{c_1 c_2 c_3}
{m_g \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \   a_1^\dagger  a_2^\dagger \alpha_3 \ , \\
H_{\perp 1} 
\es 
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \ X_{\perp 123}  + h.c. \ , \\
X_{\perp 123} 
\es
i f^{c_1 c_2 c_3}
\left[
\varepsilon_1^{j *} (p_3^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp
+
\varepsilon_3^j p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} 
\right] \ 
a_1^\dagger  a_2^\dagger a_3 \ .
\eeq
The combined result is 
\beq
\label{Htest}
H_{+1} + H_{\perp 1}
\es
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \  i f^{123}  \
\tilde Y_{123} \  a_1^\dagger  a_2^\dagger a_3 + h.c. 
\np
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \  i f^{123}
\left( \tilde Y_{A123} \ a_3^\dagger \alpha_2 a_1 + \tilde Y_{\phi 123} \   a_1^\dagger  a_2^\dagger \alpha_3 \right)
+ h.c. \ , \\
\tilde Y_{123}
\es
- {p_3^\perp \varepsilon_3^\perp \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} 
-
{p_1^\perp \varepsilon_1^{\perp *} \over p_1^+} 
(p_3^+ + p_2^+) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp
\np
\varepsilon_1^{j *} (p_3^j + p_2^j) \ \varepsilon_2^{\perp *} \varepsilon_3^\perp
+
\varepsilon_3^j p_1^j \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \ , \\
\label{YA123}
\tilde Y_{A123}
\es
- 
{m_g \over p_2^+} 
(p_3^+ + p_1^+) \ \varepsilon_1^\perp \varepsilon_3^{\perp *}    \ , \\
\label{Yphi123}
\tilde Y_{\phi 123}
\es
-
{m_g \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *}
\ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%
\subsection{Crosscheck of $\tilde Y_{123}$}
%%%%%%%%%%%%%%%%%%%%%%
\beq
\tilde Y_{123}
\es
\varepsilon_3^\perp p_1^\perp \cdot \varepsilon_1^{\perp *} \varepsilon_2^{\perp *}
- {p_3^\perp \varepsilon_3^\perp \over p_3^+} 
p_1^+ \cdot  \varepsilon_1^{\perp *} \varepsilon_2^{\perp *} 
-
{p_1^\perp \varepsilon_1^{\perp *} \over p_1^+} 
(p_3^+ + p_2^+) \cdot \varepsilon_2^{\perp *} \varepsilon_3^\perp
\np
\varepsilon_1^{\perp *} (p_3^\perp + p_2^\perp) \cdot \varepsilon_2^{\perp *} \varepsilon_3^\perp \ .
\eeq
The first two terms are symmetric in spins 1 and 2.
\beq
\tilde Y_{123}
\es
\varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \cdot
\varepsilon_3^\perp \left( p_1^\perp - x_1  p_3^\perp \right) 
-
{p_1^\perp \varepsilon_1^{\perp *} \over p_1^+} 
(p_3^+ + p_2^+) \cdot \varepsilon_2^{\perp *} \varepsilon_3^\perp
+
\varepsilon_1^{\perp *} (p_3^\perp + p_2^\perp) \cdot \varepsilon_2^{\perp *} \varepsilon_3^\perp \\
\es
\varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \cdot
\varepsilon_3^\perp k_{12}^\perp
+
\left[ - {p_1^\perp  \over p_1^+} (p_3^+ + p_2^+) + (p_3^\perp + p_2^\perp) \right] 
\varepsilon_1^{\perp *}
\cdot \varepsilon_2^{\perp *} \varepsilon_3^\perp \ .
\eeq
The square bracket can be written using $p_3 = p_1 + p_2$ for the $+$ and $\perp$ components.
\beq
- {p_1^\perp  \over p_1^+} (p_1^+ + 2p_2^+) + (p_1^\perp + 2 p_2^\perp) 
\es
- p_1^\perp - {2 p_2^+ \over p_1^+} p_1^\perp + p_1^\perp + 2 p_2^\perp \\
\es
- {2 p_2^+ \over p_1^+} p_1^\perp + 2 p_2^\perp 
\rs
{2 \over p_1^+} (- p_2^+ p_1^\perp + p_1^+ p_2^\perp) \\
\es
{2 \over x_1} (- x_2 p_1^\perp + x_1 p_2^\perp) \rs {- 2 k^\perp_{12} \over x_1} \ ,
\eeq
So,
\beq
i f^{123} \tilde Y_{123}
\es
i f^{123}\varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \cdot
\varepsilon_3^\perp k_{12}^\perp + S_{12} \ , 
\eeq
where 
\beq
S_{12} \es
-i f^{123}
{2 k^\perp_{12} \over x_1} \varepsilon_1^{\perp *} \cdot \varepsilon_2^{\perp *} \varepsilon_3^\perp \ .
\eeq
One can use 
\beq
S = \2 (S_{12} + S_{21})
\eeq
instead of $S_{12}$ in Eq.~(\ref{Htest}), because one sums over 1 and 2 there.
\beq
2 S \es
-i f^{123}
{2 k^\perp_{12} \over x_1} \varepsilon_1^{\perp *} \cdot \varepsilon_2^{\perp *} \varepsilon_3^\perp 
-i f^{213}
{2 k^\perp_{21} \over x_2} \varepsilon_2^{\perp *} \cdot \varepsilon_1^{\perp *} \varepsilon_3^\perp 
\ ,
\eeq
Using $f^{213} = - f^{123}$ and $k_{21} = - k_{12}$ I obtain 
\beq
i f^{123} \tilde Y_{123}
\es i f^{123} \left[
\varepsilon_1^{\perp *} \varepsilon_2^{\perp *} \cdot
\varepsilon_3^\perp k_{12}^\perp
- {k^\perp_{12} \varepsilon_1^{\perp *} \over x_1} \cdot \varepsilon_2^{\perp *} \varepsilon_3^\perp 
- { k^\perp_{21} \varepsilon_2^{\perp *} \over x_2}  \cdot \varepsilon_1^{\perp *} \varepsilon_3^\perp 
\right] \ ,
\eeq
which matches my old result
\beq
\label{Y123pom}
Y_{123} \es i f^{c_1 c_2 c_3}
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right] \ a_1^\dagger a_2^\dagger a_3 \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Evaluation of the gluon 2nd-order self-interaction}
\label{gluonselfinteractionApp}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I use $x_1= p_1^+/p_3^+$, $x_2 = p_2^+/p_3^+$ 
and $k_{12}^\perp = x_2 p_1^\perp - x_1 p_2^\perp$, so that
$p_1^\perp = x_1 p_3^\perp + k_{12}^\perp$, 
$p_2^\perp = x_2 p_3^\perp  - k_{12}^\perp$,
1 = transverse gluon and 2 = longitudinal gluon.
The Hamiltonian interaction term is given in Eq.~(\ref{Htest}),
\beq
\label{Htest pom}
H_{+1} + H_{\perp 1}
\es
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \  Y_{123} \  a_1^\dagger  a_2^\dagger a_3 + h.c. 
\np
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \ 
\left( Y_{A123} \ a_3^\dagger \alpha_2 a_1 + Y_{\phi 123} \   a_1^\dagger  a_2^\dagger \alpha_3 \right)
+ h.c. \ , \\
Y_{123}
\es
i f^{c_1 c_2 c_3}
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right] \ a_1^\dagger a_2^\dagger a_3 \ , \\
\label{YA123evaluation}
Y_{A123}
\es
-  i f^{123}
{m_g \over p_2^+} 
(p_3^+ + p_1^+) \ \varepsilon_1^\perp \varepsilon_3^{\perp *}    \ , \\
Y_{\phi 123}
\es
-  i f^{123}
{m_g \over p_3^+} 
p_1^+ \ \varepsilon_1^{\perp *} \varepsilon_2^{\perp *}
\ .
\eeq
{\red Start 20240929 11:35 -> 11:54 nie Cienega Ave.}
So, I have
\beq
\langle p'| H_1^2|p\rangle_{\perp g}
& \to &
\langle 0| 
Y^*_{{\tilde 1} {\tilde 2} } \
a_{\tilde 2}  a_{\tilde 1}  \
Y_{\tilde {\tilde 1} \tilde {\tilde 2}} \
a_{\tilde {\tilde 1}}^\dagger  a_{\tilde {\tilde 1}}^\dagger
|0\rangle \\
& \to &
Y^*_{\tilde 1 \tilde 2}  Y_{\tilde 1 \tilde 2} 
+
Y^*_{\tilde 1 \tilde 2 }  Y_{ \tilde 2 \tilde 1} \ .
\eeq
{\red Stop 20240929 20:44  nie Cienega Ave. Start 20240930 11:00 pon Cienega Ave.}
\beq
|N|^2 \es \sum_{\tilde 1 \tilde 2} 
\left( Y^*_{\tilde 1 \tilde 2}  Y_{\tilde 1 \tilde 2} 
+
Y^*_{\tilde 1 \tilde 2 }  Y_{ \tilde 2 \tilde 1} \right) 
\rs
\sum_{12} 
\left( Y^*_{12}  Y_{ 12} 
+
Y^*_{12}  Y_{21} \right) \\
\es
\sum_{12}
-i f^{123}
\left[ \varepsilon_1 \varepsilon_2 \,\varepsilon_3^* k_{12} 
-
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2 k_{12}/x_2 
-
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1 k_{12}/x_1 \right] 
\nt
i f^{123}
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right] 
\np
\sum_{12}
-i f^{123}
\left[ \varepsilon_1 \varepsilon_2 \,\varepsilon_3^* k_{12} 
-
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2 k_{12}/x_2 
-
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1 k_{12}/x_1 \right] 
\nt
i f^{123}
\left[ \varepsilon_2^*\varepsilon_1^* \,\varepsilon_3k_{12} 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 \right] \ ,
\eeq
where the last line results from $f^{213} = - f^{123}$ combined
with $k_{21}=-k_{12}$. Simplify.
\beq
\label{Ntocorrect}
|N|^2 \es 
2 N_c \sum_{12}
\left[ \varepsilon_1 \varepsilon_2 \,\varepsilon_3^* k_{12} 
-
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2 k_{12}/x_2 
-
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1 k_{12}/x_1 \right] 
\nt
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right] \ .
\eeq
The factor 2 comes from symmetry $Y_{12} = Y_{21}$.

{\red
\noindent 
*********************************************************************\\
Here starts my summing over polarizations of intermediate gluons
that is written in red because I now doubt that it is correct.
20240930 19:00 pon Cienega Ave.\\
*********************************************************************

Also, integration over $k_{12}^\perp$ is symmetric with respect
to change of transverse directions $(x,y) \leftrightarrow (y,x)$.
Therefore, $k_{12}^i k_{12}^j \equiv k_{12}^{\perp 2}/2$. So,
\beq
|N|^2 \es 
{2 N_c k_{12}^{\perp 2} \over 2} \sum_{12 \ k}
\left[ \varepsilon_1 \varepsilon_2 \,\varepsilon_3^{*k} 
-
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2^k /x_2 
-
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1^k /x_1 \right] 
\nt
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3^k 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^{*k} /x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^{*k} /x_1 \right]  \ .
\eeq
9 terms.
\beq
|N|^2 \es N_c k_{12}^{\perp 2} S \ , \\
S \es \sum_{12}
\varepsilon_1 \varepsilon_2 \,
\left[ \varepsilon_1^*\varepsilon_2^* \ \varepsilon_3^*\varepsilon_3 
-
\varepsilon_1^*\varepsilon_3 \ \varepsilon_3^* \varepsilon_2^* /x_2 
-
\varepsilon_2^*\varepsilon_3 \ \varepsilon_3^* \varepsilon_1^* /x_1 \right]  
\nm
\varepsilon_1 \varepsilon_3^*  
\left[ \varepsilon_1^*\varepsilon_2^* \ \varepsilon_2 \varepsilon_3 
-
\varepsilon_1^*\varepsilon_3 \ \varepsilon_2 \varepsilon_2^* /x_2 
-
\varepsilon_2^*\varepsilon_3 \ \varepsilon_2 \varepsilon_1^* /x_1 \right] /x_2 
\nm
\varepsilon_2 \varepsilon_3^* 
\left[ \varepsilon_1^*\varepsilon_2^* \  \varepsilon_1 \varepsilon_3
-
\varepsilon_1^*\varepsilon_3 \  \varepsilon_1 \varepsilon_2^* /x_2 
-
\varepsilon_2^*\varepsilon_3 \  \varepsilon_1 \varepsilon_1^* /x_1 \right] /x_1
\ .
\eeq
Factors in.
\beq
S \es \sum_{12}
\left[ \varepsilon_1 \varepsilon_2 \ \varepsilon_1^*\varepsilon_2^* \ \varepsilon_3^*\varepsilon_3 
-
\varepsilon_1 \varepsilon_2 \ \varepsilon_1^*\varepsilon_3 \ \varepsilon_3^* \varepsilon_2^* /x_2 
-
\varepsilon_1 \varepsilon_2 \ \varepsilon_2^*\varepsilon_3 \ \varepsilon_3^* \varepsilon_1^* /x_1 \right]  
\nm 
\left[ \varepsilon_1 \varepsilon_3^* \ \varepsilon_1^*\varepsilon_2^* \ \varepsilon_2 \varepsilon_3 
-
\varepsilon_1 \varepsilon_3^* \ \varepsilon_1^*\varepsilon_3 \ \varepsilon_2 \varepsilon_2^* /x_2 
-
\varepsilon_1 \varepsilon_3^* \ \varepsilon_2^*\varepsilon_3 \ \varepsilon_2 \varepsilon_1^* /x_1 \right] /x_2 
\nm
\left[ \varepsilon_2 \varepsilon_3^* \ \varepsilon_1^*\varepsilon_2^* \  \varepsilon_1 \varepsilon_3
-
\varepsilon_2 \varepsilon_3^* \ \varepsilon_1^*\varepsilon_3 \  \varepsilon_1 \varepsilon_2^* /x_2 
-
\varepsilon_2 \varepsilon_3^* \ \varepsilon_2^*\varepsilon_3 \  \varepsilon_1 \varepsilon_1^* /x_1 \right] /x_1
\ .
\eeq
Sum over 2.
\beq
S \es \sum_{1}
\left[ \varepsilon_1 \varepsilon_1^* \ \varepsilon_3^*\varepsilon_3 
-
\varepsilon_1^*\varepsilon_3 \ \varepsilon_3^* \varepsilon_1 /x_2 
-
\varepsilon_1 \varepsilon_3 \ \varepsilon_3^* \varepsilon_1^* /x_1 \right]  
\nm 
\left[ \varepsilon_1 \varepsilon_3^* \ \varepsilon_1^* \varepsilon_3 
-
\varepsilon_1 \varepsilon_3^* \ \varepsilon_1^*\varepsilon_3 /x_2 
-
\varepsilon_1 \varepsilon_3^* \ \varepsilon_1^* \varepsilon_3 /x_1 \right] /x_2 
\nm
\left[ \varepsilon_3^* \ \varepsilon_1^* \  \varepsilon_1 \varepsilon_3
-
\varepsilon_1^*\varepsilon_3 \  \varepsilon_1 \varepsilon_3^*  /x_2 
-
\varepsilon_3^* \varepsilon_3 \  \varepsilon_1 \varepsilon_1^* /x_1 \right] /x_1
\ .
\eeq
Sum over 1.
\beq
S \es \varepsilon_3^*\varepsilon_3 
\left[ 1 - 1/x_2 - 1/x_1 \right]  
\nm  \varepsilon_3^*\varepsilon_3 
\left[ 1 - 1/x_2 - 1 /x_1 \right] /x_2 
\nm  \varepsilon_3^*\varepsilon_3 
\left[ 1 - 1/x_2 - 1/x_1 \right] /x_1 \ .
\eeq
Result for $S$.
\beq
S \es \varepsilon_3^*\varepsilon_3 
\left[ 1 - 1/x_1 - 1/x_2 \right]^2 \ .  
\eeq
Result. WRONG
\beq
\label{wrong}
|N|^2 \es N_c k_{12}^{\perp 2} \ \left[ 1 - 1/x_1 - 1/x_2 \right]^2 \ \varepsilon_3^*\varepsilon_3 \ .
\eeq
}
{\green Here in green is my corrected sum over polarizations, 20240930 19:08 pon Cienega Ave..

From Eq.~(\ref{Ntocorrect}) I take the formula
\beq
|N|^2 \es 
2 N_c S \ , \\
S \es 
\sum_{12}
\left[ \varepsilon_1 \varepsilon_2 \,\varepsilon_3^* k_{12} 
-
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2 k_{12}/x_2 
-
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1 k_{12}/x_1 \right] 
\nt
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right] \ .
\eeq
9 terms.
\beq
S \es 
\sum_{12}
\varepsilon_1 \varepsilon_2 \,\varepsilon_3^* k_{12} 
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right] 
\nm
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2 k_{12}
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right]/x_2 
\nm
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1 k_{12}
\left[ \varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 \right]/x_1 
\ .
\eeq
Factors in.
\beq
S \es 
\sum_{12}
\left[ 
\varepsilon_1 \varepsilon_2 \,\varepsilon_3^* k_{12} \ 
\varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1 \varepsilon_2 \,\varepsilon_3^* k_{12} \ 
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_1 \varepsilon_2 \,\varepsilon_3^* k_{12} \ 
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 
\right] 
\nm
\left[ 
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2 k_{12} \ 
\varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2 k_{12} \ 
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_1 \varepsilon_3^* \, \varepsilon_2 k_{12} \ 
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 
\right]/x_2 
\nm
\left[ 
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1 k_{12} \ 
\varepsilon_1^*\varepsilon_2^* \,\varepsilon_3k_{12} 
-
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1 k_{12} \ 
\varepsilon_1^*\varepsilon_3 \, \varepsilon_2^* k_{12}/x_2 
-
\varepsilon_2 \varepsilon_3^* \, \varepsilon_1 k_{12} \ 
\varepsilon_2^*\varepsilon_3 \, \varepsilon_1^* k_{12}/x_1 
\right]/x_1.
\eeq
Sum over 2.
\beq
S \es 
\sum_{1}
\left[ 
\varepsilon_3^* k_{12} \ \varepsilon_1^* \varepsilon_1 \ \varepsilon_3k_{12} 
-
\varepsilon_3^* k_{12} \ \varepsilon_1^* \varepsilon_3 \ \varepsilon_1 k_{12}/x_2 
-
\varepsilon_3^* k_{12} \ \varepsilon_1 \varepsilon_3 \ \varepsilon_1^* k_{12}/x_1 
\right] 
\nm
\left[ 
\varepsilon_1 \varepsilon_3^* \ \varepsilon_1^* k_{12} \ \varepsilon_3k_{12} 
-
\varepsilon_1 \varepsilon_3^* \ \varepsilon_1^*\varepsilon_3 \ k_{12}^2/x_2 
-
\varepsilon_1 \varepsilon_3^* \ \varepsilon_3 k_{12} \ \varepsilon_1^* k_{12}/x_1 
\right]/x_2 
\nm
\left[ 
\varepsilon_1^* \varepsilon_3^* \ \varepsilon_1 k_{12} \ \varepsilon_3k_{12} 
-
\varepsilon_1 k_{12} \ \varepsilon_1^*\varepsilon_3 \ \varepsilon_3^* k_{12}/x_2 
-
\varepsilon_1 k_{12} \ \varepsilon_3^*\varepsilon_3 \ \varepsilon_1^* k_{12}/x_1 
\right]/x_1.
\eeq
Sum over 1.
\beq
S \es 
\varepsilon_3^* k_{12} \ 2 \ \varepsilon_3k_{12} 
-
\varepsilon_3^* k_{12} \ \varepsilon_3 k_{12}/x_2 
-
\varepsilon_3^* k_{12} \ \varepsilon_3 k_{12}/x_1 
\nm
\left[ 
\varepsilon_3^* k_{12} \ \varepsilon_3 k_{12} 
-
\varepsilon_3^* \varepsilon_3 \ k_{12}^2/x_2 
-
\varepsilon_3 k_{12} \ \varepsilon_3^* k_{12}/x_1 
\right]/x_2 
\nm
\left[ 
\varepsilon_3^* k_{12} \ \varepsilon_3k_{12} 
-
\varepsilon_3 k_{12} \ \varepsilon_3^* k_{12}/x_2 
-
 \varepsilon_3^*\varepsilon_3 \ k_{12}^2/x_1 
\right]/x_1.
\eeq
Remove brackets.
\beq
S \es 
\varepsilon_3^* k_{12} \ 2 \ \varepsilon_3k_{12} 
-
\varepsilon_3^* k_{12} \ \varepsilon_3 k_{12}/x_2 
-
\varepsilon_3^* k_{12} \ \varepsilon_3 k_{12}/x_1 
\nm
\varepsilon_3^* k_{12} \ \varepsilon_3 k_{12} /x_2 
+
\varepsilon_3^* \varepsilon_3 \ k_{12}^2/x_2^2  
+
\varepsilon_3 k_{12} \ \varepsilon_3^* k_{12}/(x_1 x_2)
\nm
\varepsilon_3^* k_{12} \ \varepsilon_3k_{12} /x_1
+
\varepsilon_3 k_{12} \ \varepsilon_3^* k_{12}/(x_2 x_1)
+
 \varepsilon_3^*\varepsilon_3 \ k_{12}^2/x_1^2 \ .
\eeq
Common factors.
\beq
S \es
\varepsilon_3^* k_{12} \ \varepsilon_3 k_{12}
\left[ 2 - 2/x_2 - 2/x_1 + 2/(x_1 x_2) \right]
\np
\varepsilon_3^* \varepsilon_3 \ k_{12}^2
\left( 1/x_2^2 + 1/x_1^2 \right) \ .
\eeq
Simplify.
\beq
S \es
2 \varepsilon_3^* k_{12} \ \varepsilon_3 k_{12}
+
\varepsilon_3^* \varepsilon_3 \ k_{12}^2
\left( 1/x_2^2 + 1/x_1^2 \right) \ .
\eeq
Azimuthal symmetry of integral over $k^\perp_{12}$.
\beq
S \es
\varepsilon_3^* \varepsilon_3 \ k_{12}^{\perp 2}
+
\varepsilon_3^* \varepsilon_3 \ k_{12}^{\perp 2}
\left( 1/x_1^2 + 1/x_2^2 \right) \ .
\eeq
So, instead of the wrong result in Eq.~(\ref{wrong}),
I now have the fixed result
\beq
|N|^2 \es 
2 N_c \varepsilon_3^* \varepsilon_3 \ k_{12}^{\perp 2}
\left[ 1 + 1/x_1^2 + 1/x_2^2 \right] \ .
\eeq
Below I use the fixed result.}
Since $\varepsilon_3^*\varepsilon_3$ belongs to the normalization, or equals 1,
coming back to Eq.~(\ref{|N|^2}), 
\beq
\label{|N|^2pom}
{\Sigma_{\perp g}^{(2)} \over p^+} 
\es 
{g^2 N \over p^+} \int {d k^+ d^2k^\perp \over  2 (2\pi)^3 k^+ (p^+-k^+)} \
{ 2 k^{\perp 2} \ \left[ 1 + 1/x^2 + 1/(1-x)^2 \right]  \over P^- - Q^- - p_1^- - p_2^- } \ .
\eeq
From Eq.~(\ref{H+1summary}), dissociation of $\perp$ gluon 
into one $\perp$ and one $+$ gluon proceeds via
\beq
g \sum_{123} \int[123] \ \tilde \delta_{c.a} \ f_{12.3}^r  \  i f^{123}
\ \tilde Y_{A123} \ a_3^\dagger \alpha_2 a_1  + h.c.
\eeq
where from Eq.~(\ref{YA123})
\beq
\tilde Y_{A123}
\es
- {m_g \over p_2^+} (p_3^+ + p_1^+) \ \varepsilon_1^\perp \varepsilon_3^{\perp *}    \ ,
\eeq
or from Eq.~(\ref{YA123evaluation})
\beq
Y_{A123}
\es
-  i f^{123}
{m_g \over p_2^+} 
(p_3^+ + p_1^+) \ \varepsilon_1^\perp \varepsilon_3^{\perp *}  \ .
\eeq
So,
\beq
\langle p'| H_{+1}^2|p\rangle_{+g}
& \to &
\langle 0| Y_{A123} \alpha_2 a_1 \ a_1^\dagger \alpha_2^\dagger Y_{A123}^*
|0\rangle \\
& \to &
Y_{A \tilde 1 \tilde 2}  Y_{A \tilde 1 \tilde 2}^* \ .
\eeq
Numerator factor.
\beq
|N_+|^2 \es Y_{A \tilde 1 \tilde 2}  Y_{A \tilde 1 \tilde 2}^* \\
\es
f^{123'} f^{123} \ {m_g^2 \over p_2^{+2}} \ (p_{3'}^+ + p_1^+) (p_3^+ + p_1^+)\
\varepsilon_{3'}^{\perp *} \varepsilon_1^\perp \
\varepsilon_1^\perp \varepsilon_3^{\perp *} \ .
\eeq
Color factors yield $N_c$. The sum over spin 1 of the transverse gluon 
yields $\varepsilon_{3'}^{\perp *} \varepsilon_3^{\perp *} $, which goes 
into normalization. I note there is no transverse momentum, but there 
is a denominator. In terms of $x = p_1^+/p_3^+$, with 1 corresponding 
to the transverse intermedaie gluon, I have 
\beq
|N_+|^2 \tl 
N \ {m_g^2 \over (1-x)^2 } \ (1 + x)^2 \ .
\eeq
Thus,
\beq
{\Sigma_{+g}^{(2)} \over p^+} 
\es {g^2 N \over p^+} \int {d k^+ d^2k^\perp \over  2 (2\pi)^3 k^+ (p^+-k^+)} \
{ m_g^2 (1 + x)^2 / (1-x)^2 
\over P^- - Q^- - p_1^- - p_2^- } \ ,
\eeq
and the combined gluon self-interaction due to the transverse 
and longitudinal intermediate gluons is

{\red Stop 20240930 20:53 pon Cienega Ave.}

{\red Start 20241001 12:17 wto Cienega Ave.}

Corrected gluon self-interaction is
\beq
{\Sigma_{\perp g}^{(2)} + \Sigma_{+g}^{(2)}\over p^+} \es {g^2 N \over p^+} \int {d k^+ d^2k^\perp \over  2 (2\pi)^3 k^+ (p^+-k^+)} \
{ 2 k^{\perp 2} \ \left[ 1 + {1 \over x^2} + {1 \over (1-x)^2} \right] 
+ m_g^2 \left({1 + x \over 1-x}\right)^2 
\over P^- - Q^- - p_1^- - p_2^- } \ .
\eeq
I can write it in the form 
\beq
{\Sigma_{A+\phi}^{(2)} \over p^+} \es {\Sigma_{\perp g}^{(2)} + \Sigma_{+g}^{(2)}\over p^+} \ , \\
{\Sigma_{A+\phi}^{(2)} \over p^+}
\es {g^2 N \over p^+} \int {dx d^2k^\perp \over  2 (2\pi)^3 } \
{ 2k^{\perp 2} \ \left[ 1 + {1 \over x^2} + {1 \over (1-x)^2 } \right] 
+ m_g^2 \left({1 + x \over 1-x}\right)^2 
\over x (1 - x)p^+(P^- - Q^-) - k^{\perp 2} - m_g^2 } \ .
\eeq
The mass term can be symmetrized using $f(x) \equiv [f(x) + f(1-x)]/2$
since the rest of the integrand is symmetric with respect to exchange
$x \leftrightarrow 1-x$. I have
\beq
{1 + x \over 1-x} \es {1 + 1 - (1-x)  \over 1-x} 
\rs {2 \over 1-x} - 1 \ , \\
\left( {1 + x \over 1-x} \right)^2
\eqv 
\2\left( {2 \over 1-x} -1 \right)^2 + \2\left( {2 \over x} -1 \right)^2 \\
\es
\2 \left[ 4 + {4 \over x^2} + {4 \over (1-x)^2} - {4 \over x(1-x)} - 2 \right] \\
\es
2 \left[ 1 + {1 \over x^2} + {1 \over (1-x)^2} \right] - \left[ {2 \over x(1-x)} + 1 \right] \ .
\eeq
Consequently,
\beq
\label{GluonSelfA+phi}
{\Sigma_{A+\phi}^{(2)} \over p^+}
\es {g^2 N \over p^+} \int {dx d^2k^\perp \over  2 (2\pi)^3 } \
{ 2(k^{\perp 2} + m_g^2) \ \left[ 1 + {1 \over x^2} + {1 \over (1-x)^2 } \right] 
-m_g^2 \left[ {2 \over x(1-x)} + 1 \right]
\over x (1 - x)p^+(P^- - Q^-) - k^{\perp 2} - m_g^2 } \ .
\eeq




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Change of variables for homogeneous counterterm computation}
\label{changevariables}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To compute the counterterm coefficients $\cU_{122'1'}$ that 
eliminate dependence on $r \to 0$ from matrix elements in 
Eq.~(\ref{standardvariablesX}), we change the integration variables 
$x_1, k_{12}^\perp, x_{1'}, k_{1'2'}^\perp$, to $x$, $y$, $k^\perp$ 
and $l^\perp$,
\beq
\label{C1}
x \es (x_1 + x_1')/2 \ , \ y \rs (x_{1'} - x_1)/2 \rs z/2 \ , \\
\label{x1y}
x_1 \es x - y \ , \ x_{1'} \rs x + y \ , \\ 
\label{x2y}
x_2 \es 1 - x + y \ , \ x_{2'} \rs 1 - x - y \ , \\
dx_1 \, dx_{1'} \es {\red 2} \, dx \, dy \ , \\
\label{intx1}
\int_0^1 dx_{1} \int_0^1 dx_{1'}
\es
2 \left[ \int_0^{1/2} dx \int_{-x}^x dy
+
  \int_{1/2}^1 dx \int_{x-1}^{1-x} dy \right] \ , \\
\label{C6}
k^\perp \es (k_{12}^\perp + k_{1'2'}^\perp)/2 \ , \
l^\perp  \rs (k_{1'2'}^\perp - k_{12}^\perp )/2 \ , \\
\label{C7}
k_{12}^\perp \es k^\perp - l^\perp \ , \
k_{1'2'}^\perp \rs k^\perp + l^\perp \ , \\
d^2k_{12}^\perp \, d^2 k_{1'2'}^\perp
\es
{\red 4} d^2 k^\perp \, d^2 l^\perp \ , \\
\sqrt{x_1 x_2 x_{1'} x_{2'}} \es
\sqrt{(x-y)(1-x+y)(x+y)(1-x-y)} \\
\es
\sqrt{(x^2-y^2)[(1-x)^2-y^2]} \ .
\eeq 
\beq
\int [12] \es \int[P] \int[xk] \ .
\eeq
For equal masses, $\cM = 2E$,
\beq
\int [ x k ] \es \int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x) }
\rs
\int  {d^3 k \over  (2\pi)^3 E} \ ,
\eeq
For different masses, like a quark and a gluon in the quark self-interaction, , $\cM = E_1 + E_2$ and
\beq
\label{CRFveck}
\int [ x k ] \es \int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x) }
\rs
\int  {d^3 k \over  2(2\pi)^3 } \, \left( {1\over E_1} + {1 \over E_2} \right) \ .
\eeq
\beq
\label{C14}
\int [ x_1 k_{12}^\perp ] \int [ x_{1'} k_{1'2'}^\perp ]
\es
{ 2^3  \over [2 (2\pi)^3 ]^2}  \int d^2k^\perp \int d^2l^\perp
\nt
\left[ \int_0^{1/2} dx \int_{-x}^x dy
+
  \int_{1/2}^1 dx \int_{x-1}^{1-x} dy \right] 
  \nt
  \left[ {1 \over x_1 x_2 x_{1'} x_{2'}} = 
  {1 \over (x^2-y^2)[(1-x)^2-y^2]} \right] \ .
\label{integral}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Formulas for tree-level counterterm computations} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The invariant masses are
\beq
\label{cM12}
\cM_{12}^2 \es {k_{12}^{\perp 2} + m_1^2 \over x_1}
+ {k_{12}^{\perp 2} + m_2^2 \over x_2} 
\rs
{ (k-l)^{\perp 2} + m_1^2 \over x-y}
+ 
{ (k-l)^{\perp 2} + m_2^2 \over 1-x+y}\ , \\
\cM_{1'2'}^2 \es {k_{1'2'}^{\perp 2} + m_1^2 \over x_{1'}}
+ {k_{1'2'}^{\perp 2} + m_2^2 \over x_{2'}} 
\rs
{ (k+l)^{\perp 2} + m_1^2 \over x+y}
+ 
{ (k+l)^{\perp 2} + m_2^2 \over 1-x-y}
\ .
\eeq
The exchanged-gluon fourmomentum has the kinematic three-momentum
components $q_3^{+,\perp} = p_{\overline 1}^{+,\perp} - p_{\underline 1}^{+,\perp}$
and $q_3^-=(q_3^{\perp 2} + m_g^2)/q_3^+$.
\beq
{ (p_{\underline 1} + q_3)^2 - m_1^2 \over x_{\overline 1} }
\es
{\rm sgn}(y) \left[ {(k-l)^{\perp 2} + m_1^2 \over x-y} 
                   + {4l^{\perp 2} + m_g^2 \over 2y} 
                    - {(k+l)^{\perp 2} + m_1^2 \over x+y} \right] \ , \\
{ (p_{\underline 2} + q_3)^2 - m_2^2 \over  x_{\overline 2} }
\es
{\rm sgn}(y) \left[ {(k+l)^{\perp 2} + m_2^2 \over 1-x-y} 
                   + {4l^{\perp 2} + m_g^2 \over 2y} 
                    - {(k-l)^{\perp 2} + m_2^2 \over 1-x+y} \right] \ ,
\eeq
where $s_y = {\rm sgn}(y)$. Also,
\beq
{ (p_{\underline 1} + q_3)^2 - m_1^2 \over x_{\overline 1} }
\es
   {4l^{\perp 2} + m_g^2 \over 2|y|} 
+ {(k  - s_y l)^{\perp 2} + m_1^2 \over x  - |y|} 
 - {(k + s_y l)^{\perp 2} + m_1^2 \over x + |y|} \ , \\
{ (p_{\underline 2} + q_3)^2 - m_2^2 \over  x_{\overline 2} }
\es
   {4l^{\perp 2} + m_g^2 \over 2|y|} 
+ {(k + s_y l)^{\perp 2} + m_2^2 \over 1-x  - |y|} 
 - {(k  - s_y l)^{\perp 2} + m_2^2 \over 1-x + |y|} \ .
\eeq
So, with $c = 4l^{\perp 2} + m_g^2$, 
\beq
\label{D7}
{ (p_{\underline 1} + q_3)^2 - m_1^2 \over x_{\overline 1} }
\es
{c \over 2 |y|} + a_1 \ , \
a_1 \rs {(k-s_yl)^{\perp 2} + m_1^2 \over x-|y|}  
        - {(k+s_yl)^{\perp 2} + m_1^2 \over x+|y|} \ , \\
\label{D8}
{ (p_{\underline 2} + q_3)^2 - m_2^2 \over  x_{\overline 2} }
\es
{c \over 2|y|} + a_2 \ , \
a_2 \rs {(k+s_y l)^{\perp 2} + m_2^2 \over 1-x-|y|} 
                    - {(k-s_y l)^{\perp 2} + m_2^2 \over 1-x+|y|} \ ,
\eeq
with the original exressions
\beq
\label{D9}
a_1 \es {\rm sgn}(y)  
\left[ 
   {(k -l)^{\perp 2} + m_1^2 \over x -y} 
 - {(k+l)^{\perp 2} + m_1^2 \over x+y} 
\right] \ , \\
\label{D10}
a_2 \es {\rm sgn}(y) 
\left[ 
   {(k+l)^{\perp 2} + m_2^2 \over 1-x-y} 
 - {(k-l)^{\perp 2} + m_2^2 \over 1-x+y} \right] \ ,
\eeq
and
\beq
\label{f1}
f^r_{\overline 1, \underline 1 3} 
\es  
\exp{ \left\{- r^2 \left[ {c \over 2 |y|} + a_1 \right]^2 (\pi_{\overline 1} /P_{12}^+)^2  \right\} }  \ , \\
\label{f2}
f^r_{\overline 2, \underline 2 3} 
\es  
\exp{ \left\{ - r^2 \left[ {c \over 2 |y|} + a_2 \right]^2 (\pi_{\overline 2} /P_{12}^+)^2 \right\} }   \ .
\eeq 
I have to eventually take into account that $c = 4 l^{\perp 2} + m_g^2 \to m_g^2 \to 0$.
I  note that the product
\beq
x_1 x_2 x_{1'} x_{2'} \es (x^2-y^2)[(1-x)^2 - y^2]
\eeq
is a function of $y^2$ and for small $y$ differs from $x^2(1-x)^2$ by terms on
the order of $y^2$. \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Source of homogeneous divergence in integral over $y \sim 0$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The test states used in evaluation of the 
Hamiltonian matrix elements involve significant contributions only 
from quarks with $x_1$ and $x_2$ in a limited range. I denote that 
range by $(x_0,1-x_0)$, it does not depend on $k^\perp$. The test 
wave functions are assumed to fall off quickly for $x_1$ or $x_2$ 
smaller than $x_0$. The same is assumed about the range of $x_{1'}$ 
and $x_{2'}$. Hence the range of significant contribution to integrals 
over $x$ and $y$ is limited, see Fig.~\ref{fig:intx1x1}. In that figure,
\begin{figure}[ht!]
          \caption{Schematic view of the ranges of integration over 
$x_{1'}$ and $x_1$ in the matrix elements in Eq.~(\ref{PP'}), 
{\it cf.} Eqs.~(\ref{intx1}) and (\ref{intx11}). The large square 
indicates the ranges $(0,1)$. The crossed diagonal lines correspond 
to $y = (x_{1'}-x_1)/2 = 0$ and $x=(x_{1'}+x_1)/2 = 1/2$. The 
two parallel lines near a diagonal correspond to $y=\pm y_0$ with 
$y_0 \ll x_0$. The test functions in the matrix elements of 
Eq.~(\ref{PP'}) are negligible near the edge of and outside the 
small square.}
          \label{fig:intx1x1}
\begin{center}
         \includegraphics[width=0.3\textwidth]{x1x1prime20240207.jpg}
\end{center}
          \end{figure}
\begin{figure}[ht!]
          \caption{Enlargements of the lower-left and upper-right corners of Fig.~\ref{fig:intx1x1},
                      which explain the meaning of parameter $y_0$. By construction, the results of 
                      integration do not depend on $y_0$.}
          \label{fig:intx1x1corners}
\begin{center}
         \includegraphics[width=0.6\textwidth]{x1x1primecorners20240208.jpg}
\end{center}
          \end{figure}
the two near-diagonal parallel lines are described by relations $x_{1'} = x_1 
\pm y_0$. Consequently, the integration over $x_{1'}$ and $x_1$ can 
be limited according to the replacement illustrated in Figs.~\ref{fig:intx1x1} 
and \ref{fig:intx1x1corners},
\beq
\label{intxyfirst}
\int_0^1 dx_{1} \int_0^1 dx_{1'}
\tl
\int_{xy} \ , 
\eeq
where
\beq
\int_{xy}
\es
2
\int_{x_0+y_0}^{1/2} dx 
\left[ \int_{x_0-x}^{-y_0} + \int_{-y_0 }^{y_0} + \int_{y_0}^{x-x_0} \right] dy 
\np  
2   
\int_{1/2}^{1-x_0-y_0} dx 
\left[ \int_{x_0-1+x}^{-y_0} + \int_{-y_0}^{y_0} + \int_{y_0}^{1-x_0-x} \right] dy
\ .
\label{intx11}
\eeq
The only source of dependence on $r \to 0$ is the region of $y$ close to 0.
Therefore, the integral $\int_{xy}$ is split up into two parts: a convergent 
one ($c$) and a divergent one ($d$);
\beq
\label{splitexpy0}
\int_{xy} \es \int_c + \int_d 
\ , \ \ \
\label{splitexpy0c}
\int_c
\rs
\int_{xy} \theta( |y|-y_0 )
\ , \ \ \
\int_d
\rs
\int_{xy} \theta( y_0 - |y| )
\label{splitexpy0d}
\ .
\eeq
The Jacobian factor 2 is included in $2^3=8$ in 
front of a final formula for the matrix elements. 
We also use the shorthand notation
\beq
\int_\perp \es { 8  \over [2 (2\pi)^3 ]^2}  \int d^2k^\perp \int d^2l^\perp \ .
\eeq

{\bf \large Review of the homogeneous CT}

After the change of variables, the matrix element of 
Eq.~(\ref{standardvariables}) with the complete integration 
factors from Eq.~~(\ref{integral}), reads
\beq
\label{simple01}
X_{nn'122'1'} 
\es
\int F(z)
\left[ {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} 
 \over z^2 } + \cU_{122'1'} \right] \ , \\
 \int \es 
{ 2^3  \over [2 (2\pi)^3 ]^2}  \int d^2k^\perp \int d^2l^\perp
\left( \int_0^{1/2} dx \int_{-x}^x dy
+
  \int_{1/2}^1 dx \int_{x-1}^{1-x} dy \right) \ , 
\eeq
where, after change of variables, $z = 2y$, including all factors,
\beq
\label{F(y)}
F(z) \es
f^t_{\bar c, \bar a} \
{ 4 \sqrt{x_1 x_{1'} x_2 x_{2'}} \over x_1 x_2 x_{1'} x_{2'}} \ \delta_{\sigma_1 \sigma_{1'}} \delta_{\sigma_2 \sigma_{2'}}  ( - T_{11'}^{\tilde c} T_{22'}^{\tilde c}) \ \delta_{f_1 f_{1'}} \delta_{f_2  f_{2'}}
\nt 
\psi^{*n}_{12}(x_1,k_{12}^\perp) \ \psi^{n'}_{1' 2'}(x_{1'},k_{1'2'}^\perp) \ ,
\eeq
is a smooth function of $z$ around $z=0$. Now I use the formula with prime 
meaning $d/dz$,
\beq
F(z) \es F(0) + F'(0) z + [F(z) - F(0) - F'(0) z] \ , 
\eeq
of which only the first two terms contribute to divergence when $r \to 0$.
The term in the square bracket yields an integral that has a finite limit 
when $r \to 0$. The part of the matrix element that determines the 
counterterm is
\beq
\label{Xdiv1}
X_{nn'122'1'}^{\rm div}
\es
\int \left\{
 \left[ F(0) + F'(0) z \right] \,  {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} 
 \over z^2 } + F(z) \, \cU_{122'1'} \right\} \ .
\eeq
The divergence is solely due to integration over $y$ near $z=2y=0$. I write
\beq
\label{Xdiv1}
X_{nn'122'1'}^{\rm div}
\es
\int \left[  F(0)  \,  {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} \over z^2 } 
            + F'(0)  \,  {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} \over z } 
  +   F(z) \, \cU_{122'1'} \right] \ ,
\eeq
to see that I need the counterterm kernel to be
\beq
\label{cU1r}
\lim_{r \to 0} \, \cU_{122'1'}
\es
- \delta(z) \, \cV_{122'1'} + \delta'(z) \, \cW_{122'1'} \ ,
\eeq
where
\beq
\cV_{122'1'}  \es  \lim_{r \to 0}
\left[  \theta_{1/2-x} \int_{-x}^x dy + \theta_{x-1/2} \int_{x-1}^{1-x} dy \right] 
dy \, {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} \over 4y^2 }   \ , \\
\cW_{122'1'} \es  \lim_{r \to 0}
\left[  \theta_{1/2-x} \int_{-x}^x dy + \theta_{x-1/2}\int_{x-1}^{1-x} dy \right] 
dy \, {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} \over 2y } \ .
\eeq
By comparison with the section below, I get in the review the following results.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Review of the canonical case of $\pi_{\overline i}/P_{12}^+=1$}
\label{canonicalpireview}
\beq
\label{cV1rpom}
4\cV_{122'1'}  \es   \theta_{1/2-x} \left[ {\sqrt{\pi/A} \over r }  - {2 \over x}       + O(r)  \right]
                         + \theta_{x-1/2} \left[ {\sqrt{\pi/A} \over r }  - {2 \over 1-x}   + O(r)  \right] \ .
\eeq
This result can be written as                          
\beq
\label{cV1rpom}
4\cV_{122'1'}  \es {\sqrt{\pi/A} \over r }  - {2 \over {\rm min}(x,1-x)}       + O(r)  \ , \\         
\label{cW1r}
\cW_{122'1'} \es  O(r) \ , \  A \rs \2 (4l^{\perp 2} + m_g^2)^2 \ .
\eeq
In summary,
\beq
\label{cV1r}
4\cV_{122'1'}  \es {\sqrt{2 \pi} \over r \, (4l^{\perp 2} + m_g^2)}  - {2 \over {\rm min}(x,1-x)}       + O(r)  \ .     
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Review of the boost-invariant case of $\pi_{\overline i}=p_{\overline i}^+$ } 
\label{boostinpireview}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The result can be written as
\beq
\label{cV1rrpom}
4\cV_{122'1'} \es  {\sqrt{\pi/A} \over r } - {2 \over {\rm min}(x,1-x)}   - {2 \over x^2 + (1-x)^2 } + O(r) \ , \\
\label{cW1rr}
\cW_{122'1'} \es  O(r) \ , \  A \rs \2 (4l^{\perp 2} + m_g^2)^2  [x^2 + (1-x)^2]/2 \ .
\eeq
In summary,
\beq
\label{cV1rr}
4\cV_{122'1'} \es  {\sqrt{2\pi a} \over r \,  (4l^{\perp 2} + m_g^2)} - {2 \over {\rm min}(x,1-x)} - a + O(r) \ , \\
a \es {2 \over x^2 + (1-x)^2 } \ .
\eeq



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Computation of the homogenous counterterms}
\label{homogeneousCTin App}
% I changed the reference to this section on 20241103
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The counterterm operator $\cU_{0 \, \rm seagull}^{q\bar q}$ in Eq.~(\ref{TreeGluonExch1}) 
is set to have the structure given in Eq.~(\ref{TreeGluonExchU1}) in order that the matrix elements of 
2nd-order $\cU_{t \, \rm seagull}^{q\bar q} = f^t_{\bar c, \bar a}  \cU_{0 \, \rm seagull}^{q\bar q} $ 
can counter the small-$r$ divergences identified in matrix elements $X_{nn'122'1'}$ of 
Eq.~(\ref{simple01}). The divergences originate from the integration over $y$ irrespective 
of the integration over variables $x$, $k^\perp$ and $l^\perp$. 
{\bf I see in the review that instead of integrating from $-y_0$ to $y_0$
I need to integrate from $-x$ to $x$ or from $x-1$ to $1-x$, which means 
the same formulas but with $y_0$ replaced with $x$ or $1-x$.
So, I check if the integration goes through and all I need to do 
is to make such replacements in the previously calculated integrals.
The previous calculation follows.}
\beq
\cV_{122'1'}  \es  \lim_{r \to 0}
\int_{-y_0}^{y_0}  {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} \over y^2 }   \ , \\
\cW_{122'1'} \es  \lim_{r \to 0}
\int_{-y_0}^{y_0}  {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} \over y } 
\rs
\lim_{r \to 0}
\int_{-y_0}^{y_0}  {  f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} \over y^2 } \ y \ .
\eeq
Computation of the tree-level counterterm amounts to evaluation of the functions
$\cV_{122'1'}$ and $\cW_{122'1'}$. Using Appendix Eqs.~(\ref{f1}) and (\ref{f2}), 
we have
\beq
\label{f1text}
f^r_{\overline 1, \underline 1 3} 
\es  
\exp{ \left\{- r^2 \left[ {c \over 2 |y|} + a_1 \right]^2 (\pi_{\overline 1} /P_{12}^+)^2  \right\} }  \ , \\
\label{f2text}
f^r_{\overline 2, \underline 2 3} 
\es  
\exp{ \left\{ - r^2 \left[ {c \over 2 |y|} + a_2 \right]^2 (\pi_{\overline 2} /P_{12}^+)^2 \right\} }   \ ,
\eeq 
where
\beq
c \es  4l^{\perp 2} + m_g^2 \ , \\
a_1 \es {-4 k^\perp l^\perp \over x} \, {\rm sgn}(y)  
+ {k^2 + l^2 + m_1^2 \over x^2} \, 2|y| + O(y^2) \ , \\
a_2 \es {4 k^\perp l^\perp \over 1-x} \, {\rm sgn}(y)  
+ {k^2 + l^2 + m_2^2 \over (1-x)^2} \, 2|y| + O(y^2)  \ . 
\eeq
{\red Start 20240824 17:50 sob Leszno}
Since all variables in the integration are limited, the sensitivity to $r \to 0$
comes solely from $y \lesssim r \ll y_0$, where $y_0$ is an arbitrarily small
but fixed number. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{The canonical case of $\pi_{\overline i}/P_{12}^+=1$}
\label{canonicalpi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The exponent in factors $f^r_{\overline 1, \underline 1 q} f^r_{\overline 2, \underline 2 q} $ reads
\beq 
E \es E_1^2 + E_2^2 \rs
- r^2[c/(2|y|) + a_1 ]^2 - r^2[c/(2|y|) +a_2]^2 \ .
\eeq
When $y$ increases from $y < 0$ to $y > 0$ 
I have $a_1$ jumping from 
$ a_{1L} = {(k+l)^{\perp 2} + m_1^2 \over x   } - {(k -l)^{\perp 2} + m_1^2 \over x}$ 
to $a_{1R} = -a_{1L}$ 
and $a_2$ jumping from
$ a_{2L} = {(k -l)^{\perp 2} + m_2^2 \over 1-x} - {(k+l)^{\perp 2} + m_2^2 \over 1-x}$ 
to  $a_{2R} = - a_{2L}$.
\beq
a_{1L} \es   4k^\perp l^\perp/x \ , \
a_{1R} \rs - 4k^\perp l^\perp/x \ , \\
a_{2L} \es - 4k^\perp l^\perp/(1-x) \ , \
a_{2R} \rs   4k^\perp l^\perp/(1-x) \ .
\eeq
{\red Stop 20240121 23:30  nie Ekologiczna}
{\red Start 20240122 09:59 pon Ekologiczna}
I have
\beq
a_1 \es {-4 k^\perp l^\perp \over x} \, {\rm sgn}(y)  
+ {k^2 + l^2 + m_1^2 \over x^2} \, 2|y| + O(y^2) \ , \\
a_2 \es {4 k^\perp l^\perp \over 1-x} \, {\rm sgn}(y)  
+ {k^2 + l^2 + m_2^2 \over (1-x)^2} \, 2|y| + O(y^2)  \ . 
\eeq
Using $s_y = {\rm sgn}(y)$, the exponent is obtained in the form
\beq
E/(-r^2) \es
\left[ {c \over 2|y|} - {4 k^\perp l^\perp \over x} \, s_y  
+ {k^2 + l^2 + m_1^2 \over x^2} \, 2|y| + O(y^2) \right]^2 
\np 
\left[ {c \over 2|y|} + {4 k^\perp l^\perp \over 1-x} \, s_y
+ {k^2 + l^2 + m_2^2 \over (1-x)^2} \, 2|y| + O(y^2)  \right]^2  \ .
\eeq
The squares yield
\beq
E/(-r^2) \es
{c^2 \over 2 y^2} + {8 c \, k^\perp l^\perp (x-1/2)\over x(1-x) \, y} + O(y^0) \\
\es
{(4l^{\perp 2} 
+ m_g^2)^2 \over 2 y^2} + {8 (4l^{\perp 2} 
+ m_g^2) \, k^\perp l^\perp (x-1/2)\over x(1-x) \, y} + O(y^0) \ .
\eeq
The sign of $y$ is canceled out in terms that grow when $y\to 0$.
The exponent thus has the form
\beq
E \es - r^2 \left( {A \over y^2} + {B \over y} + C \right) \ , \\
A \es \2 (4l^{\perp 2} + m_g^2)^2 \ , \ \
B \rs  {8 (4l^{\perp 2} + m_g^2) \, k^\perp l^\perp (x-1/2)\over x(1-x) } \ .
\eeq
$A$ is positive, sign of $B$ varies with $x$ and $C$ has a finite limit when $y \to 0$.
The diverging integrals read
\beq
\label{cV}
\cV_{122'1'}  \es  \lim_{r \to 0}
\int_{-y_0}^{y_0}  { dy \over y^2 } \ e^{ - r ^2 \left( A / y^2 + B / y + C \right)}   \ , \\
\label{cW}
\cW_{122'1'} \es  \lim_{r \to 0}
\int_{-y_0}^{y_0}  { dy \over y^2 } \ e^{ - r ^2 \left( A / y^2 + B / y + C \right)} \ y \ .
\eeq
The integrals extend over positive and negative $y$. The integrals to tackle are
\beq
\label{REVISED04}
[\cZ] \es \left[ \int_{-y_0}^0 {dy \over y^2} + \int_0^{y_0} {dy \over y^2} \right] 
\
e^{ - r ^2 \left( A / y^2 + B / y + C \right)}
\ [ 1, y ] \ .
\eeq
{\red Stop 20240824 20:22 sob Leszno I am not going to cheat myself that I can do much tonight.}
The exponent diverges as $A/y^2$ and $B/y$. $B$ may be negative. 
First I eliminate $C$ from consideration. 
\beq
e^{- r^2 (A/y^2 + B/y + C)} 
\es
e^{- r^2 (A/y^2 + B/y + C)} - e^{- r^2 (A/y^2 + B/y)} + e^{- r^2 (A/y^2 + B/y)} \\
\es
e^{- r^2 (A/y^2 + B/y)} \left( e^{- r^2 C} - 1 \right) + e^{- r^2 (A/y^2 + B/y)}   \ .
\eeq
The round bracket in the limit $r \to 0$ is on the order of $- r^2 C$ with a limited $C$, 
while the divergence is at most on the order of $1/r$. Therefore the first term, the one with the 
round bracket, tends to 0 when $r$ tends to 0. The diverging and finite dependence of the 
integrals $\cZ$ on $r$ originates entirely in $[\cZ]$, 
\beq
\label{REVISED05}
[\cZ] \es \left[ \int_{-y_0}^0 {dy \over y^2} + \int_0^{y_0} {dy \over y^2} \right] 
\
e^{ - r ^2 \left( A / y^2 + B / y \right)}
\ [ 1, y ] \ , \\
A \es \2 (4l^{\perp 2} + m_g^2)^2 \ , \ \
B \rs  {8 (4l^{\perp 2} + m_g^2) \, k^\perp l^\perp (x-1/2)\over x(1-x) }
\eeq
By changing integration variable $y$ to $-y$ in the integral over negative $y$, I get
\beq
\label{REVISED06}
[\cZ] \es [\cZ_{1} , \cZ_{y}] \ , \\
\cZ_{1} 
\es
\int_0^{y_0} {dy \over y^2}
\left[
   e^{ - r ^2 \left( A / y^2 + B / y \right)} 
+ e^{ - r ^2 \left( A / y^2 - B / y \right)}
\right] \ , \\
\cZ_{y} 
\es
\int_0^{y_0} {dy \over y^2}
\left[
   e^{ - r ^2 \left( A / y^2 + B / y \right)} 
-  e^{ - r ^2 \left( A / y^2  - B / y \right)}
\right] y
\ .
\eeq
{\red Stop 2024826 14:30 pon Pasteura}
{\red Start 2024827 18:10 wto Ekologiczna}
I change the integration variable to $u = 1/y$ and obtain
\beq
\cZ_{1} 
\es
\int_{1/y_0}^\infty du \ e^{ - r ^2 A u^2 }
\left(
   e^{ - r ^2 B u } 
+ e^{   r ^2 B u }
\right) \ , \\
\cZ_{y} 
\es
\int_{1/y_0}^\infty {du \over u} \ e^{ - r ^2 A u^2 }
\left(
   e^{ - r ^2 B u } 
-  e^{   r ^2 B u }
\right) 
\ .
\eeq
Next I change the variable $u$ to $v=ru$.
\beq
\cZ_{1} 
\es
{1 \over r} \int_{r/y_0}^\infty dv \ e^{ - A v^2 }
\left(
   e^{ - r B v } 
+ e^{   r B v }
\right) \ , \\
\cZ_{y} 
\es
\int_{r/y_0}^\infty {dv \over v} \ e^{ - A v^2 }
\left(
   e^{ - r B v } 
-  e^{   r B v }
\right) 
\ .
\eeq
The Gaussian integrand prevents $v$ from being much larger than $1/\sqrt{A}$,
which is always finite for finite $m_g$. Using the expansion
\beq
e^{\pm rBv} \es 1 \pm r B v + O(v^2 r^2) \ ,
\eeq
for $r \to 0$, I obtain
\beq
\cZ_{1} 
\es
{1 \over r} \int_{r/y_0}^\infty dv \ e^{ - A v^2 }
\left[ 2 + O(v^2 r^2) \right] \ , \\
\cZ_{y} 
\es
\int_{r/y_0}^\infty dv \ e^{ - A v^2 }
\left[ - 2 r B + O(v^2 r^3) \right] 
\ .
\eeq
In the limit $r \to 0$ I get the leading terms
\beq
\cZ_{1} 
\tl
{2 \over r \sqrt{A}} \int_{r \sqrt{A} /y_0}^\infty dz \ e^{ - z^2 } 
\rs
{2 \over r \sqrt{A}} \left[ {\sqrt{\pi} \over 2} - r \sqrt{A} /y_0 + O(r^2) \right] \ , \\
\cZ_{y} 
\tl O(r) \ .
\eeq
Hence, in Eqs.~(\ref{cV}) and (\ref{cW})
\beq
\label{cV1}
\cV_{122'1'}  \es  {\sqrt{\pi/A} \over r } - 2 /y_0  + O(r)  \ , \\
\label{cW1}
\cW_{122'1'} \es  O(r) \ .
\eeq
Inclusion of $2/y_0$ secures independence of the complete result 
from the chosen value of $y_0$ when one includes the complementary 
part $\int_c$ in Eq.~(\ref{simple01}) of the integral over $y$. However, 
since I used an arbitrary $y_0$ and all integrals with different values of 
$y_0$ exhibit the same dependence on $r$ in the limit $r \to 0$, the 
dependence on $r$ that needs to be removed is given by the term that 
does not depend on $y_0$. Consequently, after dropping $y_0$, the
counterterm in Eq.~(\ref{cU1r}) must be
\beq
\label{cU1result}
\lim_{r \to 0} 4 \cU_{122'1'}
\es
- \delta(y) \, {\sqrt{\pi/A} \over r }  
\rs    
{ - \delta(y) \,  \sqrt{2\pi} \over r(4l^{\perp 2} + m_g^2)} \ .
\eeq
{\red Stop 20240827 22:57 wto Ekologiczna}
{\red Start  20240828 21:51 sro Ekologiczna}
Using Eq.~(\ref{CT11}), I obtain the counterterm 
\beq
\label{CT11}
\cU_{0 \, \rm seagull}^{q\bar q} 
\es
\sum_{122'1'} \int[122'1']  \
 {4 \sqrt{p_1^+ p_{1'}^+ p_2^+ p_{2'}^+} \over P_{12}^{+2}} \ \tilde \delta_{c.a} \
\delta_{sf} \ (-T^a_{11'} T^a_{2'2} ) 
\nt
{ - \delta(y) \,  \sqrt{2\pi} \over 4 r \, (4l^{\perp 2} + m_g^2)} \ b_{1}^\dagger  d^\dagger_{2} d_{2'} b_{1'} \ ,
\eeq
where 
\beq
y \es (x_{1'} - x_1)/2 \ , \\
l^\perp  \es (k_{1'2'}^\perp - k_{12}^\perp )/2 \ ,
\eeq 
and
\beq
x_1 \es p_1^+/P_{12}^+ \ , \ x_{1'} \rs p_{1'}^+/P_{1'2'}^+ \ , \\
x_2 \es 1-x_1 \ , \ x_{2'} \rs 1- x_{1'} \ , \\
k_{12}^\perp \es x_{2} p_{1}^\perp - x_{1} p_{2}^\perp \ , \
k_{1'2'}^\perp \rs x_{2'} p_{1'}^\perp - x_{1'} p_{2'}^\perp \ .
\eeq
{\red Stop 20240828 24:00 sro Ekologiczna}
{\red Start 20240829 19:42 czw Ekologiczna}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{The boost-invariant case of $\pi_{\overline i}=p_{\overline i}^+$ } 
\label{boostinpi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Using Eqs.~(\ref{overunderpom11}) and (\ref{overunderpom12})
together with (\ref{x1y}) and (\ref{x2y}), I have 
\beq
x_{\overline 1} \es x + |y| \ , \ x_{\overline 2} \rs 1 - x +  |y| \ .
\eeq
and the exponent arguments are
\beq 
E \es
- r^2[c/(2|y|) + a_1 ]^2( x+|y|)^2 - r^2[c/(2|y|) +a_2]^2(1-x+|y|)^2 \ , \\
a_1 \es {\rm sgn}(y)  
\left[ 
   {(k -l)^{\perp 2} + m_1^2 \over x -y} 
 - {(k+l)^{\perp 2} + m_1^2 \over x+y} 
\right] \ , \\
a_2 \es {\rm sgn}(y) 
\left[ 
   {(k+l)^{\perp 2} + m_2^2 \over 1-x-y} 
 - {(k-l)^{\perp 2} + m_2^2 \over 1-x+y} \right] \ .
\eeq
Analysis of $a_1$ and $a_2$ proceeds in the same way as 
in the simplest case, so that
\beq
E/(-r^2) \es
\left\{
\left[ {c \over 2|y|} - {4 k^\perp l^\perp \over x} \, s_y  
+ {k^2 + l^2 + m_1^2 \over x^2} \, 2|y| + O(y^2) \right] (x+|y|) 
\right\}^2
\np 
\left\{
\left[ {c \over 2|y|} + {4 k^\perp l^\perp \over 1-x} \, s_y
+ {k^2 + l^2 + m_2^2 \over (1-x)^2} \, 2|y| + O(y^2)  \right] (1-x+|y|)
\right\}^2
\ .
\eeq
Multiplication before squaring produces
\beq
E/(-r^2) \es
\left[ {c x \over 2|y|} + {c \over 2} -  4 k^\perp l^\perp \, s_y  -  {4 k^\perp l^\perp \, y \over x}
+ {k^2 + l^2 + m_1^2 \over x} \, 2|y| + O(y^2) \right]^2
\np 
\left[ {c (1-x) \over 2|y|} + {c \over 2} + 4 k^\perp l^\perp s_y  +  {4 k^\perp l^\perp \over 1-x} \, y
+ {k^2 + l^2 + m_2^2 \over 1-x} \, 2|y| + O(y^2)  \right]^2
\ .
\eeq
Evaluating the squares I get
\beq
E/(-r^2) \es
{c^2 \over 4 y^2}[x^2 + (1-x)^2] 
\np
{c x \over |y|} \left( {c \over 2} -  4 k^\perp l^\perp \, s_y \right)
+
{c (1-x) \over |y|} \left( {c \over 2} + 4 k^\perp l^\perp s_y \right) + O(y^0) \ ,
\eeq
which simplifies to
\beq
E/(-r^2) \es
{c^2 \over 4 y^2}[x^2 + (1-x)^2] + {c^2 \over 2 |y|}
-
{8 c k^\perp l^\perp (x-1/2) \over y} + O(y^0) \ ,
\eeq
or
\beq
E/(-r^2) \es A / y^2 + B_1 / y + B_2/|y| + C \ ,
\eeq
where
\beq
A           \es {c^2 \over 4}[x^2 + (1-x)^2] \ , \\
B_1       \es - 8 c k^\perp l^\perp (x-1/2)  \ , \\
B_2       \es  {c^2 \over 2} \ .
\eeq
The new element is appearance of $O(1/y)$ as well as $O(1/|y|)$. 
As in Eq.~(\ref{REVISED04}), the divergence originates in 
the region of $|y| < y_0$.
\beq
\label{REVISED04b}
[\cZ ]\es \left[ \int_{-y_0}^0 {dy \over y^2} + \int_0^{y_0} {dy \over y^2} \right] 
\
e^{ - r ^2 \left( A / y^2 + B_1 / y + B_2/|y| + C \right)} \ [1, y ] \ .
\eeq
Eliminating $C$ as before, I proceed to changing integration 
variable $y$ to $-y$ in the integral over negative $y$, which 
leads to the source of divergent dependence on $r \to 0$ in
\beq
[\cZ ] \es [ \cZ_1, \cZ_y] \ , \\
\label{REVISED06b}
\cZ_1 \es 
\int_0^{y_0} {dy \over y^2} \
\left\{ e^{ - r ^2 \left[ A / y^2 + (B_2  - B_1)/y  \right]} 
    + e^{ - r ^2 \left[ A / y^2 + (B_2 + B_1)/y  \right]} \right\} \ , \\
\cZ_y \es
\int_0^{y_0} {dy \over y^2} \
\left\{ e^{ - r ^2 \left[ A / y^2 + (B_2 + B_1)/y  \right]}
     - e^{ - r ^2 \left[ A / y^2  + (B_2 - B_1)/y  \right]} \right\} 
y \  .
\eeq
Change of integration variable from $y$ to $u = 1/y$ yields
\beq
\label{REVISED07b}
\cZ_1 \es 
\int_{1/y_0}^\infty du \
\left\{ e^{ - r ^2 \left[ A u^2 + (B_2  - B_1) u  \right]} 
      + e^{ - r ^2 \left[ A u^2 + (B_2 + B_1) u  \right]} \right\} \ , \\
\cZ_y \es
\int_{1/y_0}^\infty {du \over u} \
\left\{ e^{ - r ^2 \left[ A u^2 + (B_2 + B_1) u  \right]}
      -  e^{ - r ^2 \left[ A u^2  + (B_2 - B_1) u  \right]} \right\} \ ,
\eeq
where
%\beq
%A \es \2 (4l^{\perp 2} + m_g^2)^2 \ , \ \
%B \rs {8 \, k^\perp l^\perp (x-1/2)\over x(1-x) } \ \sqrt{2A} \ .
%\eeq
\beq
A           \es \2 (4l^{\perp 2} + m_g^2)^2  [x^2 + (1-x)^2]/2 \ , \\
B_1       \es - 8 (4l^{\perp 2} + m_g^2) k^\perp l^\perp (x-1/2)  \ , \\
B_2       \es  \2 (4l^{\perp 2} + m_g^2)^2 \ .
\eeq
Subsequently, the change of integration variable to $v = r u$ yields
\beq
\label{REVISED08b}
\cZ_1 \es {1 \over r}
\int_{r/y_0}^\infty dv \ e^{ - A v^2} 
\left[ e^{ - r (B_2  - B_1) v} + e^{ - r (B_2 + B_1) v } \right]  \ , \\
\cZ_y \es
\int_{r/y_0}^\infty {dv \over v} \ e^{ - A v^2} 
\left[ e^{ - r (B_2 + B_1) v } - e^{ - r (B_2 - B_1) v  } \right] \ .
\eeq
As in the canonical case before, the coefficient $A$ is limited 
from below by $m_g^4/8$. Therefore, $v$ is limited from 
above by $\sim 1/\sqrt{A} \leq 2\sqrt{2}/ m_g^2$. In the
limit $r \to 0$, I can expand $e^{- r (B_2 \mp B_1) u}$ in a 
series of powers of $r(B_2 \mp B_1) u$. 
\beq
\label{REVISED09b}
\cZ_1 \es {1 \over r}
\int_{r/y_0}^\infty dv \ e^{ - A v^2} 
\left[ 2 - r (B_2  - B_1) v - r (B_2 + B_1) v + O(v^2 r^2) \right]  \ , \\
\cZ_y \es
\int_{r/y_0}^\infty dv \ e^{ - A v^2} 
\left[ - r (B_2 + B_1)  + r (B_2 - B_1) + O(v r^2) \right] \ .
\eeq
Neglecting terms $O(r)$ whose contribution vanishes
when $r \to 0$, I get
\beq
\label{REVISED10b}
\cZ_1 \es {2 \over r}
\int_{r/y_0}^\infty dv \ e^{ - A v^2} 
\left[ 1 - r B_2 v - O(v^2 r^2) \right]  \ , \\
\label{REVISED10b1}
\cZ_y \es
\int_{r/y_0}^\infty dv \ e^{ - A v^2} 
\left[ -2 r B_1 + O(v r^2) \right] \ .
\eeq
Evaluation yields
\beq
\label{REVISED11b}
\cZ_1 \tl {\sqrt{\pi / A} \over r} - 2/y_0 - B_2/A \ , \\
\cZ_y \tl 0 \ ,
\eeq
with
\beq
A           \es \2 (4l^{\perp 2} + m_g^2)^2  [x^2 + (1-x)^2]/2 \ , \\
B_1       \es - 8 (4l^{\perp 2} + m_g^2) k^\perp l^\perp (x-1/2)  \ , \\
B_2       \es  \2 (4l^{\perp 2} + m_g^2)^2 \ .
\eeq
Hence, as in the canonical case, in Eqs.~(\ref{cV}) and (\ref{cW})
\beq
\label{cV1}
\cV_{122'1'} \es  {\sqrt{\pi/A} \over r } - 2 /y_0  - {2 \over x^2 + (1-x)^2 } + O(r) \ , \\
\label{cW1}
\cW_{122'1'} \es  O(r) \ .
\eeq
Consequently, the counterterm in Eq.~(\ref{cU1}) must be
\beq
\label{cU1result}
\lim_{r \to 0} 4 \cU_{122'1'}
\es
- \delta(y) \, {\sqrt{\pi/A} \over r }  
\rs    
{ - \delta(y) \,  \sqrt{2\pi} \over r(4l^{\perp 2} + m_g^2)} \sqrt{2 \over x^2 + (1-x)^2 }\ ,
\eeq
which differs from the canonical case by the square-root factor.
According to Eq.~(\ref{CT1}), the counterterm is
\beq
\label{CT12}
\cU_{0 \, \rm seagull}^{q\bar q} 
\es
\sum_{122'1'} \int[122'1']  \
 {4 \sqrt{p_1^+ p_{1'}^+ p_2^+ p_{2'}^+} \over P_{12}^{+2}} \ \tilde \delta_{c.a} \
\delta_{sf} \ (-T^a_{11'} T^a_{2'2} ) 
\nt
{ - \delta(y) \,  \sqrt{2\pi} \over 4 r \, (4l^{\perp 2} + m_g^2)} 
\ \sqrt{2 \over x^2 + (1-x)^2 }
\ b_{1}^\dagger  d^\dagger_{2} d_{2'} b_{1'} \ ,
\eeq
where 
\beq
y \es (x_{1'} - x_1)/2 \ , \\
l^\perp  \es (k_{1'2'}^\perp - k_{12}^\perp )/2 \ , \\
x_1 \es p_1^+/P_{12}^+ \ , \ x_{1'} \rs p_{1'}^+/P_{1'2'}^+ \ , \\
x_2 \es 1-x_1 \ , \ x_{2'} \rs 1- x_{1'} \ , \\
k_{12}^\perp \es x_{2} p_{1}^\perp - x_{1} p_{2}^\perp \ , \
k_{1'2'}^\perp \rs x_{2'} p_{1'}^\perp - x_{1'} p_{2'}^\perp \ .
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Counterterm computation summary: extension to $H_0$}
\label{CTsummary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\red Start 20240907 13:41 sob Ekologiczna}

In Eq.~(\ref{CT11}) the canonical counterterm is
\beq
\label{canCT}
\cU_{0 \, \rm seagull}^{q\bar q}/F^r
\es
\sum_{122'1'} \int[122'1']  \
 {4 \sqrt{p_1^+ p_{1'}^+ p_2^+ p_{2'}^+} \over P_{12}^{+2}} \ \tilde \delta_{c.a} \
\delta_{sf} \ (-T^a_{11'} T^a_{2'2} ) 
\nt
{ - \delta(y) \,  \sqrt{2\pi} \over 4 r \, (4l^{\perp 2} + m_g^2)} \ b_{1}^\dagger  d^\dagger_{2} d_{2'} b_{1'} \ ,
\eeq
and in Eq.~(\ref{CT12}) the boost invariant counterterm is
\beq
\label{binvCT}
\cU_{0 \, \rm seagull}^{q\bar q} /F^r
\es
\sum_{122'1'} \int[122'1']  \
 {4 \sqrt{p_1^+ p_{1'}^+ p_2^+ p_{2'}^+} \over P_{12}^{+2}} \ \tilde \delta_{c.a} \
\delta_{sf} \ (-T^a_{11'} T^a_{2'2} ) 
\nt
{ - \delta(y) \,  \sqrt{2\pi} \over 4 r \, (4l^{\perp 2} + m_g^2)} 
\ \sqrt{2 \over x^2 + (1-x)^2 }
\ b_{1}^\dagger  d^\dagger_{2} d_{2'} b_{1'} \ ,
\eeq
where $F^r$ denotes a suitable regularization factor 
and the variables are 
\beq
y \es (x_{1'} - x_1)/2 \ , \
l^\perp  \rs (k_{1'2'}^\perp - k_{12}^\perp )/2 \ , \\
x_1 \es p_1^+/P_{12}^+ \ , \ x_2 \rs 1-x_1 \ , \ x_{1'} \rs p_{1'}^+/P_{1'2'}^+ \ , \ x_{2'} \rs 1- x_{1'} \ , \\
k_{12}^\perp \es x_{2} p_{1}^\perp - x_{1} p_{2}^\perp \ , \
k_{1'2'}^\perp \rs x_{2'} p_{1'}^\perp - x_{1'} p_{2'}^\perp \ .
\eeq
The regularization factor $F^r$ for $\cU_{0 \, \rm seagull}^{q\bar q}$
is set in the same way as in the third term in Eq.~(\ref{currentfr}) that
shows how the canonical interaction term in Eq.~(\ref{seagull}) is regulated.
Namely,
\beq
F^r \es f^r_{12,1+2} \ f^r_{1'+2',1'2'} \ ,
\eeq
where the mass parameter for the {\it gedanken} quanta of kinematic 
momenta $P_{12} = p_1+p_2$ and $P_{1'2'} = p_{1'}+p_{2'}$ is set 
to any finite, fixed value that is smaller than $m_1+m_2$. In the limit 
$r \to 0$, the choice of the mass parameter in $F^r$ does not matter 
because it is nullified by $r$ anyways.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Review of the RGPEP single-quark term}
%%%%%%%%%%%%%%%%%%%%%%%%%%

{\red Stop 20241115 23:42 pia Ekologiczna The whole evaluation of
single-quark eigenvalue condition needed precise verification!}

\noindent {\red Start 20241116 14:10 sob Ekologiczna}
{\bf I review the expressions, especially their signs, in the 
single-quark case that  suddenly appears so illuminating.}.

The RGPEP for 2nd-order terms, Eq.~(\ref{h32'text})
\beq
\label{h32'textpom}
\cH_I^{(2)'}
\es
%2%
   \left[ \left[ \cH_f ,\cH_I^{\pi(2)}  \right] , \cH_f \right]  
+ \left[ \left[ \cH_f , \cH_I^{\pi(1)} \right] , \cH_I^{(1)} \right]_I  \ .
\eeq
Written in terms of $p_q^-$, for which the derivative of $p^{\perp 2} + m^2$ 
vanishes, taking into account that there is no homogeneous term, I drop the 
sum over quark spins,
\beq
\label{h32'textpommt2}
\int [p_q] \ ( p_q^-)' \ b_{p_q}^\dagger b_{p_q}
\es
2 (p_q^- - p^-_{qg})\pi_q^2 \, e^{- 2 t (p_q^- - p_{qg}^-)^2 \pi_q^2} \left[ H_1^r H_1^r \right]_\Sigma \ .
\eeq
The momentum dependent factors are actually operators that 
insert functions in the integrand d
In $p_q^-$, there is a free term that does not depend on $s$,
\beq
p_q^- \es {p^{\perp 2} + m^2 \over p^+} + {\delta m^2_s \over p^+} \ .
\eeq
Therefore
\beq
( p_q^-)'  \es {  (\delta m^2_s)' \over p^+} \ .
\eeq
Eq.~(\ref{h32'textpommt2}) can be integrated explicitly.
\beq
\int_0^t d\tau \  
\int [p_q] \ {  (\delta m^2_\tau)' \over p^+} \ b_{p_q}^\dagger b_{p_q}
\es
2 (p_q^- - p^-_{qg})\pi_q^2 \, \int_0^t d\tau \ 
e^{- 2 \tau (p_q^- - p_{qg}^-)^2 \pi_q^2} \left[ H_1^r H_1^r \right]_\Sigma  \ , \\
\int [p_q]  { \delta m^2_s  -  \delta m^2_0 \over p^+}
\es
{ 2 (p_q^- - p^-_{qg}) \pi_q^2 
\over  - 2 (p_q^- - p_{qg}^-)^2 \pi_q^2 }
\  \left[ e^{- 2 t (p_q^- - p_{qg}^-)^2 \pi_q^2} - 1\right] \ \left[ H_1^r H_1^r \right]_\Sigma \ , \\
\int [p_q] { \delta m^2_s \over p^+} 
\es
\int [p_q] { \delta m^2_0 \over p^+}
+
\left[ 1 - e^{- 2 t (p_q^- - p_{qg}^-)^2 \pi_q^2} \right]
{ \left[ H_1^r H_1^r \right]_\Sigma \over p_q^- - p_{qg}^-  } \ .
\eeq
The divergences show up only in the term 1 in the square bracket; 
the exponential kills divergences. The counterterm in $\delta m_0^2$ 
is designed to remove the gluon severe small-$x$ divergences from 
\beq
{ \left[ H_1^r H_1^r \right]_{\Sigma \ p\sigma c f } \over p_q^- - p_{qg}^-  } \ .
\eeq
{\red The divergence is computed in $\Sigma_q$ in Eq.~(\ref{hatHfSigma}),
which is the same as $\delta m_s^2$ considered here. \bf This is false statement
because in the scattering $p^-$ is replaced by $P^- - Q^-$, where $P^-$ is the 
shell value and $Q^-$ are the spectators of the self-interaction.}  

{\red \bf 20250627: See correction in scattering in Sec.~\ref{offshellcorrectioninscattering}}

But besides the severely diverging terms there are also logarithmic terms,
not included in the previous discussion of the most severe terms 
in scattering and bound states. Nevertheless, the severe divergences 
suggest the definite counterterm to themselves. Using Eq.~(\ref{cHSigma}), 
\beq
\label{cHSigma11}
\cH^{\rm div}_{\Sigma} \es \left[ \Delta_{LIR} \, \cH^{(1)}_0 \, \cH^{(1)}_0 \right]_{\Sigma} 
\ \equiv \ 
{ \left[ H_1^r H_1^r \right]_\Sigma \over p_q^- - p_{qg}^-  } \ ,                    
\eeq
From $\cH^{\rm div}_{\Sigma}$, I had obtained the full result for $\Sigma_q^{(2)}$ 
in Eq.~(\ref{quarkCTSigma}), which reads 
\beq
\label{quarkCTSigmapom1}
{\Sigma_q^{(2)} \over p^+}
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
  {N \over D}  \, f^{r 2}_{p_q p_g,p} \ .
\eeq 
I also obtained the counterterm integrand to the severe small-$x$ 
divergence, see Eq.~(\ref{SigmaQBS}), and I discovered that the
ultraviolet quadratic transverse divergence can be also removed
if the counterterm integrand is changed to $1/x^2 + 1/[2(1-x)]$.
But then the discovery is that the numerator becomes very simple.
{\red Start 20241117 15:05 nie Ekologiczna} I check it now using $p^-$
and taking advantage of the quark current conservation.
\beq
\label{cHSigma11check1}
&& \cH^{\rm div}_{\Sigma \ p \sigma} \rs 
{ \left[ H_1^r H_1^r \right]_{\Sigma \ p \sigma} \over p_q^- - p_{qg}^-  } \\
\es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
  {\bar u_{p\sigma} \gamma_\mu ( p_q \hspace{-14.5pt} \not \hspace{12pt} + m) \gamma_\nu u_{p \sigma}
  \over  (1-x)p_g^+ (p^- - p_q^- - p_g^-)}  \, \left[ -g^{\mu \nu} + \eta^\mu \eta^\nu (p_g^-+p_q^- - p^-)/p_g^+ \right]  f^{r 2}_{p_q p_g,p} \nn
\es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \,  
\bar u_p \left\{
{ 2 p_q \hspace{-14.5pt} \not \hspace{12pt} - 4 m 
  \over  (1-x)p_g^+ (p^- - p_q^- - p_g^-)} 
-
  { 2 p_q^+ \gamma^+ 
  \over  (1-x)p_g^{+ 2} }
\right\} u_p  f^{r 2}_{p_q p_g,p} \\
 \es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \,  
 \left[
{ 4 p_q p - 8 m^2 
  \over  (1-x)p_g^+ (p^- - p_q^- - p_g^-)} 
-
  { 4 p^+ p_q^+ 
  \over  (1-x)p_g^{+ 2} }
\right] f^{r 2}_{p_q p_g,p}
\eeq
The small-$x$ severe divergence results from the last term, which 
means, from the current non-conservation in direction of $\eta$.
The intermediate fermion four-momentum is
\beq
p_q \es (1-x)p - k + \2 \eta[ p_q^- - (1-x)p^- ] \ ,
\eeq
where $k^+=k^-=0$. So, the integral of $pk$ gives 0 and 
\beq
\label{cHSigma11check2}
\cH^{\rm div}_{\Sigma\ p \sigma c f }   
\es
{4 g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} 
\
\left[ {\tilde N \over D} - { 1  \over x^2} \right] \ f^{r 2}_{p_q p_g,p}  \ , \\
\tilde N \es 
(1-x)m^2 + p^\perp k^\perp + \2 p^+[ p_q^- - (1-x)p^- ] - 2 m^2 \ , \\
D \es  x(1-x)p^+ (p^- - p_q^- - p_g^-) \ , \\
4\left( {\tilde N \over D} - { 1  \over x^2} \right) \es {N \over D} ~~ {\rm in ~Eq.}~(\ref{quarkCTSigma}) \ , \\
N
\es 4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x) \ ,\\
D
\es
x(1-x) p^+( p^-  - p_q^- - p^-_g) \rs x(1-x) (m_i^2 - \cM_{qg}^2) \ .
\eeq 
Now I verify what is obtained by adding the modified counterterm integrand 
$4 \sigma_q^{(2)} = 4/x^2 + 2/(1-x)$ to $N/D$. 
\beq
&& {N \over D} + 4/x^2 + 2/(1-x) \rs \left(  N + 4D \left\{ 1/x^2 + 1/[2(1-x)] \right\} \right)/D \ , \\
&& 
N + D \left[ 4/x^2 + 2/(1-x) \right] \\
\es
4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x)
\nm
x(1-x) (\cM_{qg}^2 - m_i^2) \left[ 4/ x^2 + 2/(1-x) \right] \\
\es
4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x)
\nm
x(1-x) \left[  (k^{\perp 2} + m_g^2)/ x  + (k^{\perp 2} + m_i^2)/(1-x) - m_i^2 \right] \, 
\left[ 4/ x^2 + 2/(1-x) \right] \\
\es
4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x)
\nm
\left[  k^{\perp 2} + (1-x) m_g^2 + x^2 m_i^2 \right] \, 
\left[ 4/ x^2 + 2/(1-x) \right] \\
\es
4 [ (1-x)m_g^2]/ x^2  + 2 x^2 m_i^2 / (1-x)
\nm
(1-x) m_g^2  4/ x^2  -  (1-x) m_g^2  2/(1-x)  -  x^2 m_i^2 4/ x^2   -  x^2 m_i^2 2/(1-x) \\
\es
4 (1-x)m_g^2/ x^2  + 2 x^2 m_i^2 / (1-x)
\nm
4 (1-x) m_g^2 / x^2  -  2 m_g^2  -  4 m_i^2  -  m_i^2 2x^2 /(1-x) \\
\es
-  ( 4 m_i^2 + 2 m_g^2 ) \ . 
\eeq
This confirms my original result. {\bf End of review {\red 20241117 22:32 nie Ekologiczna}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{ \red \bf 20250627: Correction due to $p^- \to P^- - Q^-$ in self-interaction in scattering}
\label{offshellcorrectioninscattering}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\beq
\label{cHSigma11check1 corrected for scattering}
&& \cH^{\rm div ~ scattering}_{\Sigma \ p \sigma} \rs 
{ \left[ H_1^r H_1^r \right]_{\Sigma \ p \sigma} \over P^- - Q^-  - p_{qg}^-  } \\
\es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
  {\bar u_{p\sigma} \gamma_\mu ( p_q \hspace{-14.5pt} \not \hspace{12pt} + m) \gamma_\nu u_{p \sigma}
  \over  (1-x)p_g^+ (\Delta^- + p^- - p_q^- - p_g^-)}  \, \left[ -g^{\mu \nu} + \eta^\mu \eta^\nu (p_g^-+p_q^- - p^-)/p_g^+ \right]  f^{r 2}_{p_q p_g,p} \nn
\es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \,  
\bar u_p \left\{
{ 2 p_q \hspace{-14.5pt} \not \hspace{12pt} - 4 m 
  \over  (1-x)p_g^+ (\Delta^- + d^-)} 
-
  { 2 p_q^+ \gamma^+ 
  \over  (1-x)p_g^{+ 2} } \ {d^- \over \Delta^- + d^-}
\right\} u_p  f^{r 2}_{p_q p_g,p},
\eeq
where
\beq
\Delta^- \es P^- - Q^- - p^- \ , \ d^- \rs p^- - p_q^- - p_g^- \ .
\eeq
Therefore,
\beq
\label{cHSigma11check1 corrected for scattering 1}
&& \cH^{\rm div ~scattering}_{\Sigma \ p \sigma} \rs 
{ \left[ H_1^r H_1^r \right]_{\Sigma \ p \sigma} \over \Delta^- - p_{qg}^-  } \\
 \es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \,  
 \left[
{ 4 p_q p - 8 m^2 
  \over  (1-x)p_g^+ (\Delta^- + d^-)} 
-
  { 4 p^+ p_q^+ 
  \over  (1-x)p_g^{+ 2} } \ {d^- \over  \Delta^- + d^-} 
\right] f^{r 2}_{p_q p_g,p}  \\
 \es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \,  
 \left[
{ 4 p_q p - 8 m^2 
  \over  (1-x)p_g^+ d^-} 
-
  { 4 p^+ p_q^+ 
  \over  (1-x)p_g^{+ 2} } 
\right] {d^- \over  \Delta^- + d^-}  f^{r 2}_{p_q p_g,p} \\
 \es
{g^2 C_F \over p^+}   \int {dx \, d^2k^\perp \over 2 (2\pi)^3} \,  
 \left[
{ 4 p_q p - 8 m^2 
  \over  (1-x)p_g^+ d^-} 
-
  { 4 p^+ p_q^+ 
  \over  (1-x)p_g^{+ 2} } 
\right] \left( 1 - {\Delta^- \over  \Delta^- + d^-} \right)  f^{r 2}_{p_q p_g,p} 
\eeq
The square bracket is as in the self-interaction in the RGPEP. The term with 1
in the round bracket can be treated by the same counterterm as before in my 
flawed formula for scattering off-shell. The term with 
\beq
{\Delta^- \over  \Delta^- + d^-}  \es {1 \over  1 + d^-/ \Delta^-}
\eeq 
tends to 0 like $\Delta^-/d^-$, which is inverse of the gluon $p_g^-$.
Such factor eliminates $1/x^2$ as well as quadratic transverse divergence.
I can return to the text on scattering and insert a comment about the 
off-shell result. {\bf \red 20250627: end of correction to quark $\Sigma$
due to off-shellness.}
%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Evaluation of $m_E^2$}
%%%%%%%%%%%%%%%%%%%%%%

The quark self-interaction and its counterterm in Eq.~(\ref{cH20+Delta}) 
enter the quarkonium eigenvalue problem for $H_{LR}$. The counterterm
cancels the severe small-$x$ divergence in Eq.~(\ref{quarkCTSigma})
\beq
\sigma_q^{(2)} \es 1/x^2 + 1/[2(1-x)] \ , \\
{\rm Num} 
\es 4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x) \ ,\\
{\rm Den}
\es
x(1-x) p^+( p^-  - p_q^- - p^-_g) \rs x(1-x) (m_i^2 - \cM_{qg}^2) 
\eeq 
\beq
\tilde \Sigma \es
{ {\rm Num} \over {\rm Den} } + 4/x^2 + 2/(1-x)
\rs
{ {\rm Num}  + [4/x^2 + 2/(1-x) ] {\rm Den}   \over {\rm Den} } \ .
\eeq
\beq
R \es {\rm Num}  + [4/x^2 + 2/(1-x) ] {\rm Den} \\
  \es
4 k^{\perp 2}/ x^2  + 2 k^{\perp 2}/ (1-x) 
+
4 (1-x)m_g^2/ x^2  + 2 x^2 m_i^2 / (1-x) 
\np
[4/x^2 + 2/(1-x) ] 
x(1-x) \left\{ m_i^2 - k^{\perp 2}/[x(1-x)]  - m_i^2/(1-x) - m_g^2/x  \right\} \ .
\eeq
\beq
R \es
4 k^{\perp 2}/ x^2  + 2 k^{\perp 2}/ (1-x) 
+
4 (1-x)m_g^2/ x^2  + 2 x^2 m_i^2 / (1-x) 
\np
[4/x^2 + 2/(1-x) ] \left[ - k^{\perp 2} - m_i^2 x^2 - m_g^2(1-x)  \right] \ .
\eeq
\beq
R \es
4 (1-x)m_g^2/ x^2  + 2 x^2 m_i^2 / (1-x) 
\nm
[4/x^2 + 2/(1-x) ] \left[  m_i^2 x^2 + m_g^2(1-x)  \right] \ .
\eeq
\beq
R \es
4 (1-x)m_g^2/ x^2  + 2 x^2 m_i^2 / (1-x) 
\nm
(4/x^2) \left[  m_i^2 x^2 + m_g^2(1-x)  \right] 
-
[2/(1-x) ] \left[  m_i^2 x^2 + m_g^2(1-x)  \right]  \\
\es
4 (1-x)m_g^2/ x^2  + 2 x^2 m_i^2 / (1-x) 
\nm
4 m_i^2 - 4 m_g^2(1-x)/x^2 
- 
2m_i^2 x^2 /(1-x)   - 2 m_g^2 \\
\es
m_i^2 \left[   2 x^2 / (1-x) - 4  - 2 x^2 /(1-x) \right]
\np
m_g^2 \left[ 4 (1-x) / x^2  - 4 (1-x)/x^2 - 2  \right] \ .
\eeq
\beq
R \es
-4 m_i^2 - 2 m_g^2 \ .
\eeq
\beq
\tilde \Sigma
\es
{ {\rm Num} \over {\rm Den} } + 4/x^2 + 2/(1-x) 
\rs
{ 4 m_i^2 + 2 m_g^2 \over x(1-x) ( \cM_{qg}^2 - m_i^2 ) } \\
\es
{ 4 m_i^2 + 2 m_g^2 \over x(1-x) \left\{  k^{\perp 2}/[x(1-x)] + m_i^2/(1-x) + m_g^2/x  - m_i^2 \right\}  } \\
\es
{ 4 m_i^2 + 2 m_g^2 \over x(1-x) \left\{  k^{\perp 2}/[x(1-x)] + m_i^2 x/(1-x) + m_g^2/x  \right\}  } \\
\es
\label{F44}
{ 4 m_i^2 + 2 m_g^2 \over k^{\perp 2} + m_i^2 x^2 + m_g^2(1-x) } \ .
\eeq

{\red Stop 20241113 01:53 sro Ekologiczna}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Formulas for TDW with $m_G$ not cancelling RGPEP}
\label{Self-interaction in EV problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
When the numerator factors are included, 
\beq
\label{SigmaSingleQuark}
&& f^t \cH \left( ~~ \right)f^t \cH \\
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
 {p_G^- - p_g^- \over p_G^- - p_g^- + p_g^- + p_q^- - p^- } \ {N \over x(1-x)p^+(p_g^- + p_q^- - p^-) } \
f^{t 2}_{p_q p_g,p} \ , \\
\label{NumSinglkeQuark}
N
\es 4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x) \ .
\eeq
I have $M^2 = m_G^2 - m_g^2$ and
\beq
\label{SigmaSingleQuark1}
&& f^t \cH \left( ~~ \right)f^t \cH \\
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3} \, 
{M^2 \over M^2 + p_g^+(p_g^- + p_q^- - p^-) } 
\ 
{N \ f^{t 2}_{p_q p_g,p} \over (1-x)p_g^+(p_g^- + p_q^- - p^-) } \  .
\eeq
This is the same equation as Eq.~(44) in [K. Serafin et al., Phys. Rev. D 109, 016017 (2024)].
{\red Stop 20241119 04:44 wto Ekologiczna} {\red Start 20241119 22:11 -> 23:13 wto Ekologiczna}
Before the subtraction is evaluated as above here, I have 
\beq
\label{SigmaSingleQuark2}
&& f^t \cH \left( ~~ \right)f^t \cH \\
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x)p^+ } 
\left({N \ f^{t 2}_{p_q p_g,p} \over p^- - p_q^- - p_g^- - A/p^+} 
-      {N \ f^{t 2}_{p_q p_g,p} \over p^- - p_q^- - p_g^- }\right) \ , \\
A \es (m_G^2 - m_g^2)/x \ , \\
N
\es 4 [ k^{\perp 2} + (1-x)m_g^2]/ x^2  + 2 (k^{\perp 2} + x^2 m_i^2) / (1-x) \ ,\\
D
\es
x(1-x) p^+( p^-  - p_q^- - p^-_g) \rs x(1-x) (m_i^2 - \cM_{qg}^2) \ ,
\eeq 
according to Eq.~(\ref{NumSigmaQ}). Then
\beq
\label{SigmaSingleQuark3}
f^t \cH \left( ~~ \right)f^t \cH 
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3  } 
\left[{N \over D - B} 
-      {N \over D }\right] f^{t 2}_{p_q p_g,p} \ , \\
B \es (1-x)(m_G^2 - m_g^2) \ .
\eeq
Now I can add and subtract the previously discovered 
integrand $\sigma_q^{(2)}$ that provided a counterterm 
to both $1/x^2$ and $k^{\perp 2}$ divergences, given 
in Eq.~(\ref{deltami2}) as $\sigma_q^{(2)} = 1/x^2 + 1/[2(1-x)]$.
\beq
\label{SigmaSingleQuark4}
f^t \cH \left( ~~ \right)f^t \cH 
\es
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3  } 
\left[{N \over D - B} + 4\sigma_q^{(2)}
-      \left( {N \over D } + 4\sigma_q^{(2)} \right)  \right] f^{t 2}_{p_q p_g,p} \ , \\
B \es (1-x)(m_G^2 - m_g^2) \ .
\eeq
Nothing has changed due to this adding and subtracting the same term.
I have the result from Eqs.~(\ref{NDsigma}), (\ref{Deltai}), and (\ref{F44}),
\beq
{N \over D } + 4\sigma_q^{(2)} \es {N + 4\sigma_q^{(2)} D  \over D } \rs
{ 4 m_i^2 + 2 m_g^2 \over - D } \ , \\
{N \over D - B } + 4\sigma_q^{(2)} \es {N + 4\sigma_q^{(2)} (D - B)\over D - B } \rs
{ 4 m_i^2 + 2 m_g^2  + 4\sigma_q^{(2)} B \over  - D + B} \ .
\eeq
Now I substitute $D$.
\beq
{N \over D } + 4\sigma_q^{(2)} \es 
{ 4 m_i^2 + 2 m_g^2 \over k^{\perp 2} + m_i^2 x^2 + m_g^2(1-x) } \ , \\
{N \over D - B } + 4\sigma_q^{(2)} \es 
{ 4 m_i^2 + 2 m_g^2  + 4\sigma_q^{(2)} B \over  k^{\perp 2} + m_i^2 x^2 + m_g^2(1-x) + B} \ ,
\eeq
and then I substitute $B$, using
\beq
4\sigma_q^{(2)} B \es
4\sigma_q^{(2)} (1-x)(m_G^2 - m_g^2)
\rs
{4(1-x) \over x^2}(m_G^2-m_q^2)  + 2m_G^2 - 2m_g^2 \ ,
\eeq
so that
\beq
{N \over D } + 4\sigma_q^{(2)} \es 
{ 4 m_i^2 + 2 m_g^2 \over k^{\perp 2} + m_i^2 x^2 + m_g^2(1-x) } \ , \\
{N \over D - B } + 4\sigma_q^{(2)} \es 
{ 4 m_i^2 + 2 m_G^2  + 4 (1-x)(m_G^2 - m_g^2)/x^2  \over  k^{\perp 2} + m_i^2 x^2 + m_G^2(1-x)  } \ ,
\eeq
The difference ${\rm Dif} = [ ~~]$ in Eq.~(\ref{SigmaSingleQuark4}) is
\beq
\label{SigmaSingleQuark5}
&& {\rm Dif} \rs
{N \over D - B} + 4\sigma_q^{(2)} - \left( {N \over D } + 4\sigma_q^{(2)} \right)  \\
\es
{ 4 m_i^2 + 2 m_g^2 \over k^{\perp 2} + m_i^2 x^2 + m_g^2(1-x) }
-
{ 4 m_i^2 + 2 m_G^2  + 4 (1-x)(m_G^2 - m_g^2)/x^2  \over  
k^{\perp 2} + m_i^2 x^2 + m_g^2(1-x) + (m_G^2- m_g^2) (1-x)  } \ ,
\eeq
or
\beq
\label{SigmaSingleQuark6}
&& {\rm Dif} \rs
{N \over D - B} + 4\sigma_q^{(2)} - \left( {N \over D } + 4\sigma_q^{(2)} \right)  \\
\es
{ 4 m_i^2 + 2 m_g^2 \over    x(1-x)(\cM^2 - m_i^2)  }
-
{ 4 m_i^2 + 2 m_G^2  + 4 (1-x)(m_G^2 - m_g^2)/x^2  \over  
x(1-x)(\cM^2 - m_i^2) +  (m_G^2- m_g^2) (1-x)x/x } \ .
\eeq
Calculational trick yields
\beq
\label{SigmaSingleQuark7}
x(1-x) {\rm Dif} \es
x(1-x) \left[ {N \over D - B} + 4\sigma_q^{(2)} - \left( {N \over D } + 4\sigma_q^{(2)} \right) \right]  \\
\es
{ 4 m_i^2 + 2 m_g^2 \over  \cM^2 - m_i^2 }
-
{ 4 m_i^2 + 2m_g^2 + (m_G^2- m_g^2) [2  + 4 (1-x)/x^2]  \over  
\cM^2 - m_i^2 +  (m_G^2- m_g^2)/x } \ .
\eeq
I tried some other ways of writing the same thing,
\beq
\label{SigmaSingleQuark8}
&&
x(1-x) {\rm Dif} \rs (m_G^2-m_g^2) \left[ ~~~~ \right] \ , \\
\left[ ~~~~ \right] 
\es
{ 4 m_i^2 + 2 m_g^2 \over x( \cM^2 - m_i^2) [ \cM^2 - m_i^2 +  (m_G^2- m_g^2)/x] }
-
{ 2  + 4 (1-x)/x^2  \over  \cM^2 - m_i^2 +  (m_G^2- m_g^2)/x } \ ,
\eeq
or
\beq
\label{SigmaSingleQuark9}
&&
x(1-x) {\rm Dif} \rs (m_G^2-m_g^2) \left[ ~~~~ \right] \ , \\
\left[ ~~~~ \right] 
\es
{ 4 m_i^2 + 2 m_g^2 \over x( \cM_g^2 - m_i^2) (\cM_G^2 - m_i^2) }
-
{ 2  + 4 (1-x)/x^2  \over  \cM_G^2 - m_i^2  } \ .
\eeq
The result to think about seems to be 
\beq
\label{SigmaSingleQuark10}
&& f^t \cH \left( ~~ \right)f^t \cH 
\rs
{ g^2  C_F \over p^+}
\int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x) }  \ x(1-x) {\rm Dif} \ f^{t 2}_{p_q p_g,p} \ , \\
&& x(1-x) {\rm Dif} 
\rs (m_G^2-m_g^2) \left[ ~~~~ \right] \ , \\
\left[ ~~~~ \right] 
\es
{ 4 m_i^2 + 2 m_g^2 \over x( \cM^2 - m_i^2) [ \cM^2 - m_i^2 +  (m_G^2- m_g^2)/x] }
-
{ 2  + 4 (1-x)/x^2  \over  \cM^2 - m_i^2 +  (m_G^2- m_g^2)/x } \ ,
\eeq
combined with Eq.~(\ref{CRFveckTEXT}), 
\beq
\label{CRFveckTEXTrepeat}
\int [ x k ] \es \int {dx \, d^2k^\perp \over 2 (2\pi)^3 x(1-x) }
\rs
\int  {d^3 k \over  2(2\pi)^3 } \, \left( {1\over E_g} + {1 \over E_q} \right) 
\rs 
\int  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ ,
\eeq
and
\beq
&& x(1-x) {\rm Dif}  \rs (m_G^2-m_g^2)  \left[ ~~~~ \right] \\ 
\es
{ (m_G^2-m_g^2)( 4 m_i^2 + 2 m_g^2) \over [(E_q+E_g)^2 - m_i^2] \{ x [(E_q+E_g)^2 - m_i^2] +  (m_G^2- m_g^2) \}  }
\nm
{ (m_G^2-m_g^2)[ 2x  + 4 (1-x)/x]  \over  x [(E_q+E_g)^2 - m_i^2] +  m_G^2- m_g^2 } \\
\es
{ m_G^2-m_g^2 \over  x [(E_q+E_g)^2 - m_i^2] +  m_G^2- m_g^2 }
\
\left[ { 4 m_i^2 + 2 m_g^2 \over (E_q+E_g)^2 - m_i^2 } + 4 - 2x  -  4 /x \right]   \ .
\eeq
In terms of $E$,
\beq
f^t \cH \left( ~~ \right)f^t \cH 
\es
{ g^2  C_F \over p^+} \int  {d\Omega \,  k \, dE \over  2(2\pi)^3 } \ A(E,x) \ 
e^{ - 2 t^2 (E^2 - m_i^2)^2 (\pi^+/p^+)^2}  \ , \\
A(E,x)
\es
{ m_G^2-m_g^2 \over  x (E^2 - m_i^2) +  m_G^2- m_g^2 }
\
\left[ { 4 m_i^2 + 2 m_g^2 \over E^2 - m_i^2 } + 4 - 2x  -  4 /x \right]  \ .
\eeq
\section{RGPEP form factors in gluon exchange terms}
From Eqs.~(\ref{fr1}) and (\ref{fr2}), in which $r$ is replaced 
by $s$, or from Eq.~(\ref{frapprox1gluon1}), I obtain
\beq
\label{frtot1}
f_1 \es f^t_{\overline 1, \underline 1 q} 
      \rs  
e^{ - t^2 \, [ (p_{\underline 1} + q)^2 - m_1^2 ]^2 \, (\pi_{\overline 1}/p_{\overline 1}^+)^2 }   \ , \\
\label{frtot2}
f_2 \es f^t_{\overline 2, \underline 2 q} 
\rs  
e^{ - t^2 \, [ (p_{\underline 2} + q)^2 - m_2^2 ]^2 \, (\pi_{\overline 2}/p_{\overline 2}^+)^2 }   \ .
\eeq 
From Eqs.~(\ref{D7}) and (\ref{D8}), with $z = 2y$ and
\beq
c \es 4l^{\perp 2} + m_g^2 \ , \ x_{\overline 1} \rs x + |y| \ , \ 
x_{\overline 2} \rs 1 - x +  |y| \ , \ s_y \rs {\rm sgn}(y) \ ,
\eeq
I have
\beq
\label{D7POMexch}
(p_{\underline 1} + q)^2  - m_1^2
\es
 x_{\overline 1} ( c/ |z| + a_1 ) \ , \
 a_1 \rs s_y
\left[ 
   {(k -l)^{\perp 2} + m_1^2 \over x -y} 
 - {(k+l)^{\perp 2} + m_1^2 \over x+y} 
\right] , \\
\label{D8POMexch}
(p_{\underline 2} + q)^2 - m_2^2
\es
 x_{\overline 2} ( c / |z| + a_2 ) \ , \
a_2 \rs s_y 
\left[ 
   {(k+l)^{\perp 2} + m_2^2 \over 1-x-y} 
 - {(k-l)^{\perp 2} + m_2^2 \over 1-x+y} \right] .
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
------------------------------------------------------------------------------------------------------------

************************************************************************
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Scheme for evaluating self-interactions in gauge theories}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First there is the Hamiltonian term
\beq
H \es \int_F - g_{\mu \nu} J^{a \mu} A^{a \nu} \ f  \ , \\
J^{a \mu} \es -g \bar \psi T^a \gamma^\mu  \psi \ ,
\eeq
and the field operators
\beq
\label{barepsi}
\psi \es
\sum_{c=1}^3 \sum_{\sigma = 1}^2  \int [p]
\left[  u_{p\sigma} \chi_c b_{p \sigma c} e^{-ipx}
      +    v_{p\sigma} \chi_c d^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0} \ , \\
\label{bareA}
A^\mu
\es
\sum_{c=1}^8 
\sum_{\sigma =1}^2  \int [p]
\left[  \varepsilon^\mu_{p\sigma} T^c a_{p\sigma c} e^{-ipx}
        +  \varepsilon^{\mu *}_{p\sigma} T^c a^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0} \ , \\
\label{barephi}
\phi
\es
\sum_{c=1}^8 
\int [p]
\left[  -i T^c a_{p3c} e^{-ipx} + i T^c a^\dagger_{p3c} e^{ipx}
\right]_{x^+=0} \ .
\eeq
Hence,
\beq
H \es  - g_{\mu \nu} (-g)
\int[123]  \tilde \delta_{c.a} f
\left[  u_1 \chi_1 b_1 + v_1 \chi_1 d^\dagger_1 \right]^\dagger \gamma^0 \gamma^\mu
\left[  \varepsilon^\nu_3 T^3 a_3 + \varepsilon^{\nu *}_3 T^3 a^\dagger_3 \right]
\left[  u_2 \chi_2 b_2 + v_2 \chi_2 d^\dagger_2 \right] 
 . \nn
\eeq
Self-interaction means no pairs and no exchange.
\beq
H \es  - g_{\mu \nu} 
\int[123] \ \tilde \delta_{c.a} \  \Sigma^{\mu \nu} \ f  \ , \\
 \Sigma^{\mu \nu}  \es (-g)
\left[ \bar u_1 \chi_1^\dagger  \gamma^\mu T^3  u_2 \chi_2 \  b_1^\dagger b_2 
-
        \bar v_1 \chi_1^\dagger  \gamma^\mu T^3  v_2 \chi_2 \ d^\dagger_2 d_1  \right]
\left[  \varepsilon^\nu_3 a_3 + \varepsilon^{\nu *}_3 a^\dagger_3 \right] \ .
\eeq
Rewritten,
\beq
\Sigma^{\mu \nu}  \es
\left[ - g \bar u_1   \gamma^\mu  u_2 \ \chi_1^\dagger T^3  \chi_2  \  b_1^\dagger b_2 
+
       g  \bar v_1   \gamma^\mu  v_2 \ \chi_1^\dagger T^3  \chi_2 \ d^\dagger_2 d_1  \right]
\left[  \varepsilon^\nu_3 a_3 + \varepsilon^{\nu *}_3 a^\dagger_3 \right] \\
\es
\left[ j_{12}^{3 \mu}   \  b_1^\dagger b_2 + \bar j_{12}^{3 \mu} \ d^\dagger_2 d_1  \right]
\left[  \varepsilon^\nu_3 a_3 + \varepsilon^{\nu *}_3 a^\dagger_3 \right] \ , 
\eeq
with
\beq
        j_{12}^{3 \mu}  \es - g \bar u_1   \gamma^\mu  u_2 \ \chi_1^\dagger T^3  \chi_2 \ , \
\bar j_{12}^{3 \mu} \rs  g  \bar v_1   \gamma^\mu  v_2 \ \chi_1^\dagger T^3  \chi_2 \ .
\eeq
{\red Stop 20241220 20:30 pia San Dimas} {\red Start 20241221 10:38 sob San Dimas}
I simplify $H$.
\beq
H \es  - 
\int[123] \ \tilde \delta_{c.a} f
\left[        j_{12}^{3} \varepsilon_3     \                       b_1^\dagger b_2   a_3  
    +         j_{12}^{3} \varepsilon_3^* \  a_3^\dagger b_1^\dagger b_2 
    + \bar j_{12}^{3} \varepsilon_3      \                       d_2^\dagger d_1   a_3    
    + \bar j_{12}^{3} \varepsilon_3^*  \ a_3^\dagger d_2^\dagger d_1  
\right]  \ .
\eeq
The self-interaction occurs in the square of $H$. Limiting $H^2$ to the self-interaction 
of fermions and anti-fermions, I have
\beq
H^2 \es
\int[1'2'3'] \ \tilde \delta_{c'.a'} f  \nt
\left[        j_{1'2'}^{3'} \varepsilon_{3'}     \                           b_{1'}^\dagger b_{2'}   a_{3'}  
    +         j_{1'2'}^{3'} \varepsilon_{3'}^* \  a_{3'}^\dagger b_{1'}^\dagger b_{2'} 
    + \bar j_{1'2'}^{3'} \varepsilon_{3'}      \                          d_{2'}^\dagger d_{1'}   a_{3'}    
    + \bar j_{1'2'}^{3'} \varepsilon_{3'}^*  \ a_{3'}^\dagger d_{2'}^\dagger d_{1'}  
\right]  
\nt
\int[1''2''3''] \ \tilde \delta_{c''.a''} f \nt
\left[        j_{1''2''}^{3''} \varepsilon_{3''}     \                           b_{1''}^\dagger b_{2''}   a_{3''}  
    +         j_{1''2''}^{3''} \varepsilon_{3''}^* \  a_{3''}^\dagger b_{1''}^\dagger b_{2''} 
    + \bar j_{1''2''}^{3''} \varepsilon_{3''}      \                          d_{2''}^\dagger d_{1''}   a_{3''}    
    + \bar j_{1''2''}^{3''} \varepsilon_{3''}^*  \ a_{3''}^\dagger d_{2''}^\dagger d_{1''}  
\right]   \nnn \\
\tl
\int[1'2'3'] \ \tilde \delta_{1'.2'3'} \
\left[   f     j_{1'2'}^{3'} \varepsilon_{3'}     \                           b_{1'}^\dagger b_{2'}   a_{3'}     \right]  
\
\int[1''2''3''] \ \tilde \delta_{3''1''.2''} \
\left[ f  j_{1''2''}^{3''} \varepsilon_{3''}^* \  a_{3''}^\dagger b_{1''}^\dagger b_{2''}   \right]  \nnn
\np
\int[1'2'3'] \ \tilde \delta_{2'.1'3'} \
\left[   f    \bar j_{1'2'}^{3'} \varepsilon_{3'}      \                          d_{2'}^\dagger d_{1'}   a_{3'}    \right]  
\
\int[1''2''3''] \ \tilde \delta_{3''2''.1''} \
\left[   f  \bar j_{1''2''}^{3''} \varepsilon_{3''}^*  \ a_{3''}^\dagger d_{2''}^\dagger d_{1''} \right]  
\ .
\eeq
Contraction of $3'$ with $3''$ yields
\beq
H^2 \tl
\int[1'2'3'] \ \tilde \delta_{1'.2'3'} \
\left[   f     j_{1'2'}^{3'} \varepsilon_{3'}     \    b_{1'}^\dagger b_{2'}    \right]  
\
\int[1''2'']  \ \tilde \delta_{3'1''.2''} \
\left[ f  j_{1''2''}^{3'} \varepsilon_{3'}^*       \  b_{1''}^\dagger b_{2''}   \right]  \nnn
\np
\int[1'2'3'] \ \tilde \delta_{2'.1'3'} \
\left[   f    \bar j_{1'2'}^{3'} \varepsilon_{3'}  \  d_{2'}^\dagger d_{1'}   \right]  
\
\int[1''2''] \ \tilde \delta_{3'2''.1''} \
\left[  f   \bar j_{1''2''}^{3'} \varepsilon_{3'}^* \ d_{2''}^\dagger d_{1''} \right]  
\ .
\eeq
Contraction of fermions yields 
{\red Continuing 20241227 13:16 pia San Dimas}
\beq
H^2 \tl
\int[1'2'3'] \ \tilde \delta_{1'.2'3'} \
\left[  f     j_{1'2'}^{3'} \varepsilon_{3'}     \    b_{1'}^\dagger   \right]  
\
\int[2'']  \ \tilde \delta_{3'2'.2''} \
\left[ f  j_{2'2''}^{3'} \varepsilon_{3'}^*       \  b_{2''}   \right]  \nnn
\np
\int[1'2'3'] \ \tilde \delta_{2'.1'3'} \
\left[   f    \bar j_{1'2'}^{3'} \varepsilon_{3'}  \  d_{2'}^\dagger   \right]  
\
\int[1''] \ \tilde \delta_{3'1'.1''} \
\left[   f  \bar j_{1''1'}^{3'} \varepsilon_{3'}^* \  d_{1''} \right]  
\ .
\eeq
Integration over $2''$ and $1''$ gives
\beq
H^2 \tl
\int[1'2'3'] \ \tilde \delta_{1'.2'3'} \
\left[    f    j_{1'2'}^{3'} \varepsilon_{3'}     \    b_{1'}^\dagger   \right]  
\
{1 \over p_{3'}^+ + p_{2'}^+}
\
\left[   f   j_{2'3'+2'}^{3'} \varepsilon_{3'}^*       \  b_{3'+2'}   \right]  \nnn
\np
\int[1'2'3'] \ \tilde \delta_{2'.1'3'} \
\left[   f    \bar j_{1'2'}^{3'} \varepsilon_{3'}  \  d_{2'}^\dagger   \right]  
\
{1 \over p_{3'}^+ + p_{1'}^+}
\
\left[   f   \bar j_{3'+1'1'}^{3'} \varepsilon_{3'}^* \  d_{3'+1'} \right]  
\ .
\eeq
Change of notation.
\beq
H^2 \tl
\int[1] {1 \over p_{1}^+ } \int [23] \ \tilde \delta_{1.23} \ f_1 f_2 \ \sum_{c \sigma} \
        j_{12}^{c \mu} \varepsilon_{\sigma \mu}  \         j_{21}^{c \nu} \varepsilon_{\sigma \nu }^*  
        \  b_{1}^\dagger b_{1}  \nnn
\np
\int[2] {1 \over p_{2}^+ } \int[13] \ \tilde \delta_{2.13} \ f_1 f_2  \ \sum_{c \sigma} \
\bar j_{12}^{c \mu} \varepsilon_{\sigma \mu}  \  \bar j_{21}^{c \nu} \varepsilon_{\sigma \nu}^*  
       \  d_{2}^\dagger d_{2} \ .
\eeq
Sum over the gluon polarizations and colors.
\beq
H^2 \tl
\int[1] {1 \over p_{1}^+ } \int [23] \ \tilde \delta_{1.23} \ d_{\mu \nu} \ f_1 f_2 \
        j_{12}^{c \mu}         j_{21}^{c \nu}  \  b_{1}^\dagger b_{1}  
        \np
\int[2] {1 \over p_{2}^+ } \int[13] \ \tilde \delta_{2.13} \ d_{\mu \nu} \ f_1 f_2 \
\bar j_{12}^{c \mu}  \bar j_{21}^{c \nu}  \  d_{2}^\dagger d_{2} \ .
\eeq
Knowing that the Dirac fermion currents, 
\beq
        j_{12}^{c \mu}  \es - g \bar u_1   \gamma^\mu  u_2 \ \chi_1^\dagger T^c  \chi_2 \ , \
\bar j_{12}^{c \mu}   \rs   g  \bar v_1   \gamma^\mu  v_2 \ \chi_1^\dagger T^c  \chi_2 \ ,
\eeq
are conserved and using FF rules of kinematic momentum conservation,
\beq
q_1  \es p_{\overline 1} - p_{\underline 1} \rs q_2 \rs p_{\overline 2} - p_{\underline 2} \ , \\
d^{\mu \nu} 
\es  - g^{\mu \nu} + {1 \over 2  q_{m_g}^+} \eta^\mu \eta^\nu 
 \left( { \cM_{g \underline 1}^2 - m_1^2 \over p_{\overline 1}^+}
      + { \cM_{g \underline 2}^2 - m_2^2 \over p_{\overline 2}^+} \right) \\
\es  - g^{\mu \nu} + {1 \over 2  q_{m_g}^{+2}} \eta^\mu \eta^\nu 
 \left( m_g^2 - q_1^2 + m_g^2 - q_2^2 \right) \ .
\eeq
Evaluation of matrix elements
\beq
&& \sigma_{23} \rs d_{\mu \nu} \ f_1 f_2 \  j_{12}^{c \mu}         j_{21}^{c \nu}  \ , \\
&&
\langle \tilde 1 \tilde 2 | \int [1] \int [23] \ \tilde \delta_{1.23} \ { \sigma_{23} \over p_1^+} \ b_{1}^\dagger b_{1} |\psi_E \rangle \\
\es
\langle \tilde 1 \tilde 2 | \int [1] \int [23] \ \tilde \delta_{1.23} \ \sigma_{23} \ b_{1}^\dagger b_{1} 
\int[56] \ P^+ \tilde \delta_{P.56} \ \psi(x_5,k_{56}^\perp) |56 \rangle \\
\es
\int [1] \int [23] \ \tilde \delta_{1.23} \ \sigma_{23} 
\int[56] \ P^+ \tilde \delta_{P.56} \ \psi(x_5,k_{56}^\perp) 
\nt
\left[ \langle \tilde 1 \tilde 2 | \ b_{1}^\dagger b_{1} \ |56 \rangle 
= 
\langle 0| d_{\tilde 2} b_{\tilde 1} \ b_{1}^\dagger b_{1} \ b_{5}^\dagger d_{6}^\dagger |0 \rangle 
\right] \\
\es
\int [1] \int [23] \ \tilde \delta_{1.23} \ \sigma_{23} 
\int[56] \ P^+ \tilde \delta_{P.56} \ \psi(x_5,k_{56}^\perp) \
\delta_{\tilde 2 6} \ \delta_{15} \ \delta_{\tilde 1 1} \\
\es
\int [23] \ \tilde \delta_{\tilde 1.23} \ \sigma_{23} 
\int[34] \ P^+ \tilde \delta_{P.56} \ \psi(x_5,k_{56}^\perp) \
\delta_{\tilde 2 6} \ \delta_{\tilde 15}  \\
\es
\int [23] \ \tilde \delta_{\tilde 1.23} \ \sigma_{23} 
\int[6] 
\ P^+ \tilde \delta_{P.\tilde 16} \ \psi(x_{\tilde 1},k_{\tilde 1 6}^\perp) \
\delta_{\tilde 2 6}  \\
 \es
\int [23] \ \tilde \delta_{\tilde 1.23} \ \sigma_{23} 
\ P^+ \tilde \delta_{P.\tilde 1 \tilde 2} \ \psi(x_{\tilde 1},k_{\tilde 1 \tilde 2}^\perp)  \\
 \es
 \int [x_2 k_{23}^\perp]  
 { 1  \over p_{\tilde 1}^+ } \sigma_{23} 
\ P^+ \tilde \delta_{P.\tilde 1 \tilde 2} \ \psi(x_{\tilde 1},k_{\tilde 1 \tilde 2}^\perp) \\
 \es
 { \Sigma_{\tilde 1}  \over p_{\tilde 1}^+ } \  \
\ P^+ \tilde \delta_{P.\tilde 1 \tilde 2} \ \psi(x_{\tilde 1},k_{\tilde 1 \tilde 2}^\perp) \ .
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
------------------------------------------------------------------------------------------------------------

************************************************************************
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Scheme for evaluating exchanges in gauge theories}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First there is the Hamiltonian term
\beq
H \es \int_F - g_{\mu \nu} J^{a \mu} A^{a \nu} \ f \ , \\
J^{a \mu} \es -g \bar \psi T^a \gamma^\mu  \psi \ ,
\eeq
and the field operators
\beq
\label{barepsi}
\psi \es
\sum_{c=1}^3 \sum_{\sigma = 1}^2  \int [p]
\left[  u_{p\sigma} \chi_c b_{p \sigma c} e^{-ipx}
      +    v_{p\sigma} \chi_c d^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0} \ , \\
\label{bareA}
A^\mu
\es
\sum_{c=1}^8 
\sum_{\sigma =1}^2  \int [p]
\left[  \varepsilon^\mu_{p\sigma} T^c a_{p\sigma c} e^{-ipx}
        +  \varepsilon^{\mu *}_{p\sigma} T^c a^\dagger_{p\sigma c} e^{ipx}
\right]_{x^+=0} \ , \\
\label{barephi}
\phi
\es
\sum_{c=1}^8 
\int [p]
\left[  -i T^c a_{p3c} e^{-ipx} + i T^c a^\dagger_{p3c} e^{ipx}
\right]_{x^+=0} \ .
\eeq
Hence,
\beq
H \es  - g_{\mu \nu} (-g)
\int[123]  \tilde \delta_{c.a} f
\left[  u_1 \chi_1 b_1 + v_1 \chi_1 d^\dagger_1 \right]^\dagger \gamma^0 \gamma^\mu
\left[  \varepsilon^\nu_3 T^3 a_3 + \varepsilon^{\nu *}_3 T^3 a^\dagger_3 \right]
\left[  u_2 \chi_2 b_2 + v_2 \chi_2 d^\dagger_2 \right] 
 . \nn
\eeq
Exchange means no pairs and no self-interaction.
\beq
H \es  - g_{\mu \nu} 
\int[123] \ \tilde \delta_{c.a} \  X^{\mu \nu} \ , \\
X^{\mu \nu} \es (-g) f
\left[ \bar u_1 \chi_1^\dagger  \gamma^\mu T^3  u_2 \chi_2 \  b_1^\dagger b_2 
-
        \bar v_1 \chi_1^\dagger  \gamma^\mu T^3  v_2 \chi_2 \ d^\dagger_2 d_1  \right]
\left[  \varepsilon^\nu_3 a_3 + \varepsilon^{\nu *}_3 a^\dagger_3 \right] \ .
\eeq
Rewritten,
\beq
X^{\mu \nu} \es f
\left[ - g \bar u_1   \gamma^\mu  u_2 \ \chi_1^\dagger T^3  \chi_2  \  b_1^\dagger b_2 
+
       g  \bar v_1   \gamma^\mu  v_2 \ \chi_1^\dagger T^3  \chi_2 \ d^\dagger_2 d_1  \right]
\left[  \varepsilon^\nu_3 a_3 + \varepsilon^{\nu *}_3 a^\dagger_3 \right] \\
\es f
\left[ j_{12}^{3 \mu}   \  b_1^\dagger b_2 + \bar j_{12}^{3 \mu} \ d^\dagger_2 d_1  \right]
\left[  \varepsilon^\nu_3 a_3 + \varepsilon^{\nu *}_3 a^\dagger_3 \right] \ , 
\eeq
with
\beq
        j_{12}^{3 \mu}  \es - g \bar u_1   \gamma^\mu  u_2 \ \chi_1^\dagger T^3  \chi_2 \ , \
\bar j_{12}^{3 \mu} \rs  g  \bar v_1   \gamma^\mu  v_2 \ \chi_1^\dagger T^3  \chi_2 \ .
\eeq
{\red Stop 20241220 20:30 pia San Dimas} {\red Start 20241221 10:38 sob San Dimas}
I simplify $H$.
\beq
H \es  - 
\int[123] \ \tilde \delta_{c.a} f
\left[        j_{12}^{3} \varepsilon_3     \                       b_1^\dagger b_2   a_3  
    +         j_{12}^{3} \varepsilon_3^* \  a_3^\dagger b_1^\dagger b_2 
    + \bar j_{12}^{3} \varepsilon_3      \                       d_2^\dagger d_1   a_3    
    + \bar j_{12}^{3} \varepsilon_3^*  \ a_3^\dagger d_2^\dagger d_1  
\right]  \ .
\eeq
The exchange occurs in the square of $H$. Limiting $H^2$ to the exchange between
a fermion and an anti-fermion, I have
\beq
H^2 \es
\int[1'2'3'] \ \tilde \delta_{c'.a'} \nt f
\left[        j_{1'2'}^{3'} \varepsilon_{3'}     \                           b_{1'}^\dagger b_{2'}   a_{3'}  
    +         j_{1'2'}^{3'} \varepsilon_{3'}^* \  a_{3'}^\dagger b_{1'}^\dagger b_{2'} 
    + \bar j_{1'2'}^{3'} \varepsilon_{3'}      \                          d_{2'}^\dagger d_{1'}   a_{3'}    
    + \bar j_{1'2'}^{3'} \varepsilon_{3'}^*  \ a_{3'}^\dagger d_{2'}^\dagger d_{1'}  
\right]  
\nt
\int[1''2''3''] \ \tilde \delta_{c''.a''} \nt f
\left[        j_{1''2''}^{3''} \varepsilon_{3''}     \                           b_{1''}^\dagger b_{2''}   a_{3''}  
    +         j_{1''2''}^{3''} \varepsilon_{3''}^* \  a_{3''}^\dagger b_{1''}^\dagger b_{2''} 
    + \bar j_{1''2''}^{3''} \varepsilon_{3''}      \                          d_{2''}^\dagger d_{1''}   a_{3''}    
    + \bar j_{1''2''}^{3''} \varepsilon_{3''}^*  \ a_{3''}^\dagger d_{2''}^\dagger d_{1''}  
\right]   \nnn \\
\tl
\int[1'2'3'] \ \tilde \delta_{1'.2'3'} \
\left[        j_{1'2'}^{3'} \varepsilon_{3'}     \                           b_{1'}^\dagger b_{2'}   a_{3'}     \right]  
\
\int[1''2''3''] \ \tilde \delta_{3''2''.1''} \
\left[\bar j_{1''2''}^{3''} \varepsilon_{3''}^*  \ a_{3''}^\dagger d_{2''}^\dagger d_{1''}  \right]  \nnn
\np
\int[1'2'3'] \ \tilde \delta_{2'.1'3'} \
\left[       \bar j_{1'2'}^{3'} \varepsilon_{3'}      \                          d_{2'}^\dagger d_{1'}   a_{3'}    \right]  
\
\int[1''2''3''] \ \tilde \delta_{3''1''.2''} \
\left[         j_{1''2''}^{3''} \varepsilon_{3''}^* \  a_{3''}^\dagger b_{1''}^\dagger b_{2''} \right]  
\ .
\eeq
Contraction of $3'$ with $3''$ yields
\beq
H^2 \tl
\int[1'2'3'] \ \tilde \delta_{1'.2'3'} \
\left[        j_{1'2'}^{3'} \varepsilon_{3'}     \    b_{1'}^\dagger b_{2'}    \right]  
\
\int[1''2''] \ \tilde \delta_{3'2''.1''} \
\left[\bar j_{1''2''}^{3'} \varepsilon_{3'}^*  \  d_{2''}^\dagger d_{1''}  \right] f f \nnn
\np
\int[1'2'3'] \ \tilde \delta_{2'.1'3'} \
\left[       \bar j_{1'2'}^{3'} \varepsilon_{3'}  \  d_{2'}^\dagger d_{1'}   \right]  
\
\int[1''2''] \ \tilde \delta_{3'1''.2''} \
\left[         j_{1''2''}^{3'} \varepsilon_{3'}^*  \  b_{1''}^\dagger b_{2''}  \right]  f f
\ .
\eeq
Integration over $3'$ yields 

{\red Stop 20241221 14:05 Miki przyjechal} {\red Start 20241223 13:26 pon San Dimas}
\beq
H^2 \tl
\sum_{3'} \int[1'2'] \  { \theta(1'-2') \over p_{1'}^+ - p_{2'}^+}
\left[        j_{1'2'}^{3'} \varepsilon_{3'}     \    b_{1'}^\dagger b_{2'}    \right]  
\
\int[1''2''] \ \tilde \delta_{c.a} \
\left[\bar j_{1''2''}^{3'} \varepsilon_{3'}^*  \  d_{2''}^\dagger d_{1''}  \right]  f f \nnn
\np
\sum_{3'} \int[1'2'] \ { \theta(2'-1') \over p_{2'}^+ - p_{1'}^+}
\left[       \bar j_{1'2'}^{3'} \varepsilon_{3'}  \  d_{2'}^\dagger d_{1'}   \right]  
\
\int[1''2''] \ \tilde \delta_{c.a} \
\left[         j_{1''2''}^{3'} \varepsilon_{3'}^*  \  b_{1''}^\dagger b_{2''}  \right]  f f
\ .
\eeq
Change of notation.
\beq
H^2 \tl
\sum_{c\sigma} \int[121'2'] \ {\theta_{1-1'} \tilde \delta_{12.1'2'} \over p_{\overline 1}^+ - p_{\underline 1}^+}
\         j_{11'}^{\mu c} \varepsilon_{\sigma \mu} 
\ \bar j_{2'2}^{\nu c} \varepsilon_{\sigma \nu}^*  \  b_{1}^\dagger  d_{2}^\dagger d_{2'}  b_{1'} f f \nnn
\np
\sum_{c\sigma} \int[121'2'] \ { \theta_{1'-1} \tilde \delta_{12.1'2'} \over p_{\overline 1}^+ - p_{\underline 1}^+}
\          j_{11'}^{\nu c} \varepsilon_{\sigma \nu} 
\  \bar j_{2'2}^{\mu c} \varepsilon_{\sigma \mu}^*  \   b_{1}^\dagger d_{2}^\dagger d_{2'}  b_{1'}   f f
\ .
\eeq
Sum over the gluon 3 polarizations.
\beq
H^2 \tl
\sum_{c} \int[121'2'] \ {\theta_{1-1'} \tilde \delta_{12.1'2'} \over p_{\overline 1}^+ - p_{\underline 1}^+}
\ d_{\mu \nu} \   j_{11'}^{\mu c} 
\ \bar j_{2'2}^{\nu c}   \  b_{1}^\dagger  d_{2}^\dagger d_{2'}  b_{1'} f f  \nnn
\np
\sum_{c} \int[121'2'] \ { \theta_{1'-1} \tilde \delta_{12.1'2'} \over p_{\overline 1}^+ - p_{\underline 1}^+}
\ d_{\mu \nu} \   j_{11'}^{\mu c} 
\ \bar j_{2'2}^{\nu c}   \   b_{1}^\dagger d_{2}^\dagger d_{2'}  b_{1'}  f f
\ .
\eeq
Knowing that the Dirac fermion currents, 
\beq
        j_{12}^{c \mu}  \es - g \bar u_1   \gamma^\mu  u_2 \ \chi_1^\dagger T^c  \chi_2 \ , \
\bar j_{12}^{c \mu}   \rs   g  \bar v_1   \gamma^\mu  v_2 \ \chi_1^\dagger T^c  \chi_2 \ ,
\eeq
are conserved and using FF rules of kinematic momentum conservation,
\beq
q_{m_g} \es p_{\overline 1} - p_{\underline 1} 
+ \2 \eta (q^-_{m_g} + p_{\underline 1} - p_{\overline 1}) 
\rs p_{\overline 2} - p_{\underline 2} 
+ \2 \eta (q^-_{m_g} + p_{\underline 2} - p_{\overline 2})  \ , \\ 
q_1  \es p_{\overline 1} - p_{\underline 1} \ , \ q_2 \rs p_{\overline 2} - p_{\underline 2} \ , \\
d^{\mu \nu} 
\es - g^{\mu \nu} + {\eta^\mu q_{m_g}^\nu + q_{m_g}^\mu \eta^\nu \over q_{m_g}^+ } \\
\eqv
 - g^{\mu \nu} + {1 \over 2  q_{m_g}^+} \eta^\mu \eta^\nu 
 (q^-_{m_g} + p_{\underline 1} - p_{\overline 1} + q^-_{m_g} + p_{\underline 2} - p_{\overline 2}) \\
\es  - g^{\mu \nu} + {1 \over 2  q_{m_g}^+} \eta^\mu \eta^\nu 
 \left( { \cM_{g \underline 1}^2 - m_1^2 \over p_{\overline 1}^+}
      + { \cM_{g \underline 2}^2 - m_2^2 \over p_{\overline 2}^+} \right) \\
\es  - g^{\mu \nu} + {1 \over 2  q_{m_g}^{+2}} \eta^\mu \eta^\nu 
 \left( m_g^2 - q_1^2 + m_g^2 - q_2^2 \right) \ .
\eeq
I universally have the $Q \bar Q$ gluon exchange term in the form
\beq
\label{Hsquared}
H^2 \tl
\sum_{c} \int[121'2'] \ \tilde \delta_{12.1'2'} 
\ {d_{\mu \nu} \over q_{m_g}^+} 
\   j_{11'}^{\mu c}  \ \bar j_{2'2}^{\nu c}  \ \  b_{1}^\dagger  d_{2}^\dagger d_{2'}  b_{1'} \ f_1 f_2 \ .
\eeq
This confirms my earlier results and conventions and ends the scheme of 
evaluating the Hamiltonian term of gluon exchange between Dirac fermions.

Evaluation of matrix elements. {\red Start 20250108 09:00 -> 09:20 czw San Dimas}
\beq
&& \sigma_{122'1'} \rs   {d_{\mu \nu} \over q_{m_g}^+} \ f_1 f_2 \
\   j_{11'}^{\mu c}  \ \bar j_{2'2}^{\nu c}  \ , \\
&&
\langle \tilde 1 \tilde 2 | \int [122'1'] \ \tilde \delta_{12.2'1'} \ \sigma_{122'1'}
\  b_{1}^\dagger  d_{2}^\dagger d_{2'}  b_{1'} \int [56] \ P^+ \tilde \delta_{P.56} \ \psi_{56}(x_5,k_{56}^\perp) |56 \rangle \\
\es
\int [122'1'] \ \tilde \delta_{12.2'1'} \ \sigma_{122'1'}
\   \int [56] \ P^+ \tilde\delta_{P.56} \ \psi_{56}(x_5,k_{56}^\perp) 
\nt
\langle 0  | d_{\tilde 2} b_{\tilde 1} \  b_{1}^\dagger  d_{2}^\dagger d_{2'}  b_{1'} \ b_5^\dagger d_6^\dagger |0 \rangle \\
\es
\int [122'1'] \ \tilde \delta_{12.2'1'} \ \sigma_{122'1'}
\   \int [56] \ P^+ \tilde\delta_{P.56} \ \psi_{56}(x_5,k_{56}^\perp) 
\
\delta_{1'5} \ \delta_{2'6} \ \delta_{\tilde 1 1} \ \delta_{\tilde 2 2} \\
\es
\int [2'1'] \ \tilde \delta_{\tilde 1\tilde 2.2'1'} \ \sigma_{\tilde 1\tilde 22'1'}
\  P^+ \tilde\delta_{P.1'2'} \ \psi_{1'2'}(x_{1'},k_{1'2'}^\perp) \\
\es
\left\{ \int [2'1'] = \int [P_{1'2'}] \int [x_{1'} k_{1'2'}^\perp] \right\} \ 
\tilde \delta_{\tilde 1\tilde 2.2'1'} \ \sigma_{\tilde 1\tilde 22'1'}
\  P^+ \tilde\delta_{P.1'2'} \ \psi_{1'2'}(x_{1'},k_{1'2'}^\perp) \\
\es
\int [P_{1'2'}] \int [x_{1'} k_{1'2'}^\perp] \ 
\tilde \delta_{\tilde 1\tilde 2.2'1'} \ \sigma_{\tilde 1\tilde 22'1'}
\  P^+ \tilde\delta_{P.1'2'} \ \psi_{1'2'}(x_{1'},k_{1'2'}^\perp) \\
\es
P^+ \tilde\delta_{P.\tilde 1 \tilde 2} 
\int [x_{1'} k_{1'2'}^\perp] \ {\sigma_{\tilde 1\tilde 22'1'}  \over P^+} \ \psi_{1'2'}(x_{1'},k_{1'2'}^\perp)
\eeq
%%%%%%%%%%%%%%%%%%%%%%
\section{ Evaluation of W denominators}
%%%%%%%%%%%%%%%%%%%%%%
Hypothesis:
\beq
\left[ P^-_{12} - P_{3}^- \right]^{-1}  
+ 
\left[ P^-_{1'2'} - P_{3}^- \right]^{-1}
\es
\left[ q_{1}^- - q_{m_G}^- \right]^{-1}  
+ 
\left[ q_{2}^- - q_{m_G}^- \right]^{-1} \ .
\eeq
Check:
\beq
&&  \left[ q_{1}^- - q_{m_G}^- \right]^{-1}  +  \left[ q_{2}^- - q_{m_G}^- \right]^{-1} \\
\es
\left[ p_{\overline 1}^- - p_{\underline 1}^- - q_{m_G}^- \right]^{-1}  
+  
\left[ p_{\overline 2}^- - p_{\underline 2}^- - q_{m_G}^- \right]^{-1} \\
\es
\theta_z \left[ {1\over p_{1'}^- - p_{1}^- - q_{m_G}^- }
                      +  
                      { 1 \over p_{2}^- - p_{2'}^- - q_{m_G}^-} \right]
                      \np
\theta_{-z} \left[ {1\over p_{1}^- - p_{1'}^- - q_{m_G}^- }
                      +  
                      { 1 \over p_{2'}^- - p_{2}^- - q_{m_G}^-} \right] \\
                      \es
\theta_z \left[ {1\over p_{1'}^- + p_{2'}^- - p_{1}^- - p_{2'}^- - q_{m_G}^- }
                      +  
                      { 1 \over p_{1}^- + p_{2}^- - p_{1}^- - p_{2'}^- - q_{m_G}^-} \right]
                      \np
\theta_{-z} \left[ {1\over p_{1}^- + p_{2}^- - p_{1'}^- - p_{2}^- - q_{m_G}^- }
                      +  
                      { 1 \over p_{1'}^- + p_{2'}^- - p_{1'}^- - p_{2}^- - q_{m_G}^-} \right] \\
                      \es
\theta_z \left[ {1\over P_{1'2'}^- - p_{\underline 1}^- - p_{\underline 2}^- - q_{m_G}^- }
                      +  
                      { 1 \over P_{12}^- - p_{\underline 1}^- - p_{\underline 2}^- - q_{m_G}^-} \right]
                      \np
\theta_{-z} \left[ {1\over P_{12}^- - p_{\underline 1}^- - p_{\underline 2}^- - q_{m_G}^- }
                      +  
                      { 1 \over P_{1'2'}^- - p_{\underline 1}^- - p_{\underline 2}^- - q_{m_G}^-} \right] \\
                      \es
                      \left[ {1\over P_{12}^- - p_{\underline 1}^- - p_{\underline 2}^- - q_{m_G}^- }
                      +  
                      { 1 \over P_{1'2'}^- - p_{\underline 1}^- - p_{\underline 2}^- - q_{m_G}^-} \right] \\
                      \es
\left[ P^-_{12} - P_{3}^- \right]^{-1}  
+ 
\left[ P^-_{1'2'} - P_{3}^- \right]^{-1}
\eeq
{\bf  Check completed above} Now {\red 20241226 15:15 czw San Dimas}
\beq
&& \left[ {1 \over q_{m_g}^+} = {2 \over 2 q_{m_g}^+} \right] \
\left[ {1 \over q_{1}^- - q_{m_G}^- }  + {1 \over q_{2}^- - q_{m_G}^- } \right] \\
\es 
\left[ {1 \over \2 (q_{1}^+ + q_{m_G}^+)( q_{1}^- - q_{m_G}^-) }  
      + {1 \over \2(q_{2}^+ + q_{m_G}^+ )(q_{2}^- - q_{m_G}^-) } \right] \\
\es
\left[ {1 \over ( q_{1} + q_{m_G} )( q_{1} - q_{m_G} ) }  
      + {1 \over ( q_{2} + q_{m_G} )(q_{2} - q_{m_G} ) } \right] \\
\es
\left[ {1 \over q_{1}^2  - q_{m_G}^2 }  + {1 \over q_{2}^2 - q_{m_G}^2 } \right] \rs
\left[ {1 \over q_{1}^2  - m_G^2 }  + {1 \over q_{2}^2 - m_G^2 } \right]
\\
\es
- \left[ {1 \over \rho_1 + (m_G^2 - m_g^2) }  + {1 \over \rho_2 + (m_G^2 - m_g^2) } \right] \ .
\eeq
{\red Stop 20241226 22:12 czw San Dimas po powrocie z Altadeny}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gluon mass in bound states}
\label{mginbs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Moreover, on 
the basis  of [K. Serafin, M. Gomez-Rocha, J. More and S. D. Glazek, 
{\it Dynamics of heavy quarks in the Fock space}, in preparation],
one can expect that the results for quarkonium states are not significantly
sensitive to the value of $m_G$ already when it exceeds $\sim 0.5$ GeV 
for $c \bar c$ or $\sim 0.8$ GeV for $b \bar b$ states. However, the 
Hamiltonian $H$ of Eq.~(\ref{HLR2}) depends on the gluon mass 
parameter $m_g$, which requires consideration of the limit $m_g \to 0$.

{\red Stop 20240907 23:41 sob Ekologiczna}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{COMMENTS to include on technical issues}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Estimates of error: symmetries of the spectrum, such as rotational symmetry.
Dependence on $s > 0$. Convergence of eigenvalues like in papers with 
Wilson and Mlynik.  Fate in QCD allows one to check ``the magnitude of
errors'' by comparison with data.


RGPEP  does more for us than the Melosh transformation did,
 {\it cf.}~\cite{Melosh:1974cu,Gell-Mann:1992zxl}, because it
provides the relation between the current and constituent quarks
no matter in what frame of reference one works.

We have opened access to small-$x$ logarithmic effects that point out how to look for 
confinement on the Hilbert space of QCD. How is this related to logarithmic running of 
the coupling constant?

Hamiltonian approach and relativity: The role of the fixed point.

Binding in quantum mechanics requires mass defect and I secure the mass defect by positive
mss squared counterterms that tend to infinity when $m_g \to 0$.


Relativity and counterterms, why would the results be relativistic? Only for relativistic 
value of the coupling constant?


The issue has a long history, starting with the 
pre-QCD treatment light quarks~\cite{Melosh:1974cu,Gell-Mann:1992zxl}. 

The transition amplitudes, obtained using 
perturbation theory, involve the quadratic, linear (for sharp cutoffs), and logarithmic dependence on 
$\Lambda$.

{\red \bf Jump to Sec.~\ref{seagullrecap}, Eq.~\ref{tostartwith}). 
Also: Explain the relationship to AdS/QCD inspired phenomenology
via $1/y$ in the integral over $y$ that results from writing $l^\perp = \sqrt{|y|} \, u^\perp$, or using the 
transverse momentum transfer variable $u^\perp = l^\perp \sqrt{|y|}$.}

The effective theory cannot be derived in perturbative expansion by any procedure that does not neutralizes 
small denominators because small-$x$ and large $k^\perp$ divergences are sensitive to the small denominators.
\beq
H^{(2)}_s \es f_s H^{(2)}_r + C^{(2)}_{r \, \rm m} + (f_s - f_s f_s) \Delta \, H^{(1)}_r H^{(1)}_r \\
\es             f_sH^{(2)}_r + C^{(2)}_{r \, \rm m} + (f_s - f_sf_s) \Delta \,  H^{(1)}_r H^{(1)}_r  - ff) H^{(2)}_r    \\
\es            ff H^{(2)}_r + C^{(2)}_{r \, \rm m}
                + (f - ff) \,\left[ \Delta \, H^{(1)} H^{(1)} + H^{(2)}_r  \right] \ , \\
H^{(2)}_r \es H_{r \, \rm se}^{(2)} + C^{(2)}_{r \, \rm ex} + C^{(2)}_{r \, \rm se} \ .
\eeq




\beq
H^{(2)}_s \es f_s H^{(2)}_r + C^{(2)}_{\rm m} + (f_s - f_s f_s) \Delta \, H^{(1)}_r H^{(1)}_r \\
\es             f_sH^{(2)}_r + C^{(2)}_{\rm m} + (f_s - f_sf_s) \left[ \Delta \,  H^{(1)}_r H^{(1)}_r + H^{(2)}_r \right] -  (f - ff) H^{(2)}_r    \\
\es            ff H^{(2)}_r + C^{(2)}_{\rm m}
                + (f - ff) \,\left[ \Delta \, H^{(1)} H^{(1)} + H^{(2)}_r  \right] \ , \\
H^{(2)}_r \es H_{\rm se}^{(2)} + C^{(2)}_{\rm ex} + C^{(2)}_{\rm se} \ .
\eeq
For $C^{(2)}_{\rm ex} = - C^{(2)}_{\rm se}$,
\beq
H^{(2)} \es f  H_{\rm se}^{(2)} + C^{(2)}_{\rm m}  + 
                  (f - ff) \Delta \, H^{(1)} H^{(1)} \\
\es             f H_{\rm se}^{(2)} + C^{(2)}_{\rm m} 
              + (f - ff) \left[ \Delta \,  H^{(1)} H^{(1)} + H_{\rm se}^{(2)} \right] 
              -  (f - ff) H_{\rm se}^{(2)}  \\
\es            ff H_{\rm se}^{(2)} + C^{(2)}_{\rm m} 
                + (f - ff) \,\left[ \Delta \, H^{(1)} H^{(1)} + H_{\rm se}^{(2)} \right] \ .
\eeq


Once the severe small-$x$ singularities $\sim 1/x^2$ are under control one can begin studies of the logarithmic
effects due to terms $\sim 1/x$. An example of such effects is provided in the case of heavy quarkonia
where the logarithmic divergences are capable to describe confinement in the following way: in the
limit $m_g \to 0$ the quark self-interaction terms diverge but the exchange cancels the divergence in 
colorless states.

 (More readable exposition, explanation of boost to IMF = larger $P^+$ means less evolution from bare 
theory, which explains Bjorken scaling and its violation.) 
%, using an example of the front-form Hamiltonian description of heavy quarkonia.
The infrared singularities in the limit of vanishing gluon mass are handled using the general features of 
the RGPEP. Namely, all small denominators are absent in perturbative computation of renormalized 
Hamiltonians.  The mechanism is somehwat similar to the one found in models in 1+1 dimensional QCD  
of a large number of colors, except that the transverse momenta are included and taken care of by the 
RGPEP factors. The distinct logarithmic small-x divergences, specific to the front form of QCD, can thus 
be identified and shown to cancel out. The cancellation occurs for globally colorless states and hints at 
a computationally verifiable scenario for describing confinement of color. Unlike in the lattice or perturbative 
forms of QCD, the scenario applies to the renormalized Hamiltonian operator acting in the space of states of 
virtual quarks and gluons. 

The separation of the small-x divergences from genuine infrared problems with emission or absorption of infrared gluons is
facilitated by the RGPEP design which eliminates small energy denominators from perturbative computation of renormalized Hamiltonian operators. 

The key to the cancellation of severe small-$x$ divergences, $\sim 1/x^2$, using gluon mass and 3 polarization states, is that 
the $\eta \eta$ part of the gluon exchange can be written as a sum of the term with $(\rho_1-\rho_2)^2$ and a term opposite 
to the seagull. The part with $(\rho1-\rho_2)^2$ provides the factor $x^2$ in the numerator. It also contains a small denominator.
But the small denominator is canceled by the RGPEP factor $f - ff$. The seagull and exchange divergences $\sim 1/x^2$ cancel
each other exactly. One could introduce counterterms for both. These would be opposite to each other. But they would also 
include small denominators with $k^{\perp 2} + m_g^2$. Such denominators would make the counterterms infrared divergent
when $m_g \to 0$. Using my trick with the 3rd gluon, I avoid the infrared transverse denominator that could cause logarithmic 
infrared divergence in the limit $m_g \to 0$, because of the restoration of the seagull-exchange cancellation of the $\eta\eta$ 
terms. 

Gluon mass $m_g$ is not used to regulate the infrared divergences, such $(\vec q\,^2 + m_g^2)^{-2}$ for small gluon momentum 
$\vec q$. These will be overcome using the RGPEP. But the gluon mass is helpful in handling the small-$x$ singularities that appear
at all scales simultaneously.

The degree of evolution depends on $P^+$. The larger $P^+$ the less evolved the effective particles.

`t Hooft model is no accident: it works because the leading singularities in QCD cancel out and produce the same output.

Separation of the most singular terms from the logarithmic dynamics is possible because of the identities implied by the 
current conservation, RGPEP analogs of the Slavnov-Taylor identities (Craig Roberts).

(More readable exposition, explanation of boost to IMF = larger $P^+$ means less evolution from bare theory, which explains Bjorken scaling and its violation.)
%, using an example of the front-form Hamiltonian description of heavy quarkonia.
The infrared singularities in the limit of vanishing gluon mass are handled using the general features of the RGPEP. Namely, all small denominators are absent in perturbative computation of renormalized Hamiltonians.  The mechanism is somehwat similar to the one 
found in models in 1+1 dimensional QCD  of a large number of colors, except that the transverse momenta are included and taken care 
of by the RGPEP factors. The distinct logarithmic 
small-x divergences, specific to the front form of QCD, can thus be identified and shown to cancel out. The cancellation occurs for globally colorless states and hints at a computationally verifiable scenario for describing confinement of color. Unlike in the lattice or perturbative 
forms of QCD, the scenario applies to the renormalized Hamiltonian operator acting in the space of states of virtual quarks and gluons. 

The separation of the small-x divergences from genuine infrared problems with emission or absorption of infrared gluons is
facilitated by the RGPEP design which eliminates small energy denominators from perturbative computation of renormalized Hamiltonian operators. 

The key to the cancellation of severe small-$x$ divergences, $\sim 1/x^2$, using gluon mass and 3 polarization states, is that 
the $\eta \eta$ part of the gluon exchange can be written as a sum of the term with $(\rho_1-\rho_2)^2$ and a term opposite 
to the seagull. The part with $(\rho1-\rho_2)^2$ provides the factor $x^2$ in the numerator. It also contains a small denominator.
But the small denominator is canceled by the RGPEP factor $f - ff$. The seagull and exchange divergences $\sim 1/x^2$ cancel
each other exactly. One could introduce counterterms for both. These would be opposite to each other. But they would also 
include small denominators with $k^{\perp 2} + m_g^2$. Such denominators would make the counterterms infrared divergent
when $m_g \to 0$. Using my trick with the 3rd gluon, I avoid the infrared transverse denominator that could cause logarithmic 
infrared divergence in the limit $m_g \to 0$, because of the restoration of the seagull-exchange cancellation of the $\eta\eta$ 
terms. 

Gluon mass $m_g$ is not used to regulate the infrared divergences, such $(\vec q\,^2 + m_g^2)^{-2}$ for small gluon momentum 
$\vec q$. These will be overcome using the RGPEP. But the gluon mass is helpful in handling the small-$x$ singularities that appear
at all scales simultaneously.

The degree of evolution depends on $P^+$. The larger $P^+$ the less evolved the effective particles.

`t Hooft model is no accident: it works because the leading singularities in QCD cancel out and produce the same output.

Separation of the most singular terms from the logarithmic dynamics is possible because of the identities implied by the 
current conservation, RGPEP analogs of the Slavnov-Taylor identities (Craig Roberts).



\end{appendix}
%%%%%%%%%%%%%%%%%
\bibliography{ReferencesMassiveGluons2026}
%%%%%%%%%%%%%%%%%
\end{document}
