\contentsline {title}{Gluon mass and small-$x$ dynamics in hadrons}{1}{section*.2}%
\contentsline {abstract}{Abstract}{1}{section*.1}%
\contentsline {section}{\numberline {I}Verbatim}{2}{section*.3}%
\tocdepth@munge 
\contentsline {section}{\numberline {}Contents}{3}{section*.4}%
\tocdepth@restore 
\contentsline {section}{\numberline {II}Introduction}{5}{section*.5}%
\contentsline {section}{\numberline {III}Bare Hamiltonian}{6}{section*.6}%
\contentsline {subsection}{\numberline {A}FF Hamiltonian of QCD}{6}{section*.7}%
\contentsline {subsection}{\numberline {B}Gluon mass and auxiliary field}{8}{section*.8}%
\contentsline {subsection}{\numberline {C}Quantum Hamiltonian}{8}{section*.9}%
\contentsline {section}{\numberline {IV}Regularization}{9}{section*.10}%
\contentsline {subsection}{\numberline {A}Particle momenta and cutoff $\epsilon ^+$}{9}{section*.11}%
\contentsline {subsection}{\numberline {B}The weak-coupling issue of vacuum}{10}{section*.12}%
\contentsline {subsection}{\numberline {C}Regulating interaction terms}{10}{section*.13}%
\contentsline {subsubsection}{\numberline {1} Terms linear in $g$ }{11}{section*.14}%
\contentsline {subsubsection}{\numberline {2} Terms quadratic in $g$ }{12}{section*.15}%
\contentsline {subsubsection}{\numberline {3}Fully regulated bare Hamiltonian}{13}{section*.16}%
\contentsline {section}{\numberline {V}Gluon mass and small $x$ in scattering amplitudes}{14}{section*.17}%
\contentsline {subsection}{\numberline {A}Severe small-$x$ singularity}{14}{section*.18}%
\contentsline {subsection}{\numberline {B}Cancellation of $1/x^2$ in scattering amplitudes}{15}{section*.19}%
\contentsline {subsection}{\numberline {C}Small $x$ in self-interactions}{16}{section*.20}%
\contentsline {section}{\numberline {VI}Hamiltonian eigenvalue problem }{17}{section*.22}%
\contentsline {subsection}{\numberline {A}Elements of the RGPEP}{18}{section*.23}%
\contentsline {subsection}{\numberline {B}Renormalized effective eigenvalue problem}{19}{section*.24}%
\contentsline {section}{\numberline {VII}Weak-coupling expansion for $H_s$}{19}{section*.25}%
\contentsline {subsection}{\numberline {A}Solutions of lowest orders}{19}{section*.26}%
\contentsline {subsection}{\numberline {B}Counterterms}{21}{section*.27}%
\contentsline {section}{\numberline {VIII}Gluon mass and small $x$ in heavy quarkonia}{21}{section*.28}%
\contentsline {subsection}{\numberline {A}Eigenvalue problem for the dominant component}{21}{section*.29}%
\contentsline {subsection}{\numberline {B}Hamiltonian matrix for dominant component}{22}{section*.30}%
\contentsline {subsection}{\numberline {C}Cancellation of $1/x^2$ in bound states}{22}{section*.31}%
\contentsline {subsection}{\numberline {D}Finite effective particle interactions}{25}{section*.33}%
\contentsline {subsubsection}{\numberline {1}\bf Contribution of $- f_{LI} f_{IR} \, \Delta _{LIR} \, {\cal H} _0^{(1)} {\cal H} _0^{(1)}$ to Eq.~(\ref {HLR1pom1text})}{25}{section*.34}%
\contentsline {section}{\numberline {IX}Inclusion of quarkonium components with effective gluons}{39}{section*.36}%
\contentsline {subsection}{\numberline {A}2nd-order seagull interactions in the 3-body space}{40}{section*.37}%
\contentsline {subsection}{\numberline {B}2nd-order quark-gluon interactions through gluon exchange in the 3-body space}{42}{section*.39}%
\contentsline {subsection}{\numberline {C}\bf Summary of the 3-transverse-gluons term}{49}{section*.41}%
\contentsline {subsection}{\numberline {D}\bf Summary of terms for gluon-quark interaction in the 3-body space}{51}{section*.42}%
\contentsline {subsection}{\numberline {E}\bf Exchange with $\theta _{1-1'}$ using file ``FirstLine20250927''}{52}{section*.43}%
\contentsline {section}{\numberline {X} Gluon-quark interaction dependence on $m_g \to 0$ in 3-body space 20251002 10:12}{54}{section*.44}%
\contentsline {subsection}{\numberline {A}\bf Gluon-quark interaction in the 3-body space, new = 20251011 12:06 sob Ekologiczna}{54}{section*.46}%
\contentsline {subsection}{\numberline {B}\bf Gluon-quark interaction in the 3-body space, old}{56}{section*.47}%
\contentsline {subsection}{\numberline {C} \bf Products in gluon exchange between a quark and a gluon in the 3-body space}{62}{section*.48}%
\contentsline {subsection}{\numberline {D} \bf Evaluation of quark-gluon interaction in Eqs.~(~\ref {qg3bodyInteraction1}) and (\ref {qg3bodyInteraction2}) }{65}{section*.49}%
\contentsline {subsection}{\numberline {E}Old text discussing the 3-body space}{66}{section*.50}%
\contentsline {section}{\numberline {XI}Eigenvalue problems for $m_g \to 0$}{67}{section*.51}%
\contentsline {section}{\numberline {XII}Single-quark eigenvalue problem for $m_g \to 0$}{67}{section*.52}%
\contentsline {subsection}{\numberline {A}Gluon mass ansatz in single-quark eigenstates}{68}{section*.53}%
\contentsline {subsection}{\numberline {B}Computation of $\Delta p_E^-$ dependence on $m_g$}{69}{section*.54}%
\contentsline {subsection}{\numberline {C}Other single-particle eigenvalue problems}{71}{section*.55}%
\contentsline {section}{\numberline {XIII}Elements of the quarkonium eigenvalue problem}{72}{section*.56}%
\contentsline {subsection}{\numberline {A}Analysis of the part $A$}{73}{section*.57}%
\contentsline {subsection}{\numberline {B}Analysis of the part $B$}{73}{section*.58}%
\contentsline {subsubsection}{\numberline {1}Quark self-interactions in the part $B$}{73}{section*.59}%
\contentsline {subsubsection}{\numberline {2}Gluon exchange in the part $B$}{75}{section*.60}%
\contentsline {subsection}{\numberline {C}Analysis of the part $C$}{77}{section*.61}%
\contentsline {subsubsection}{\numberline {1}Quark self-interactions in the part $C$}{77}{section*.62}%
\contentsline {subsubsection}{\numberline {2}Gluon exchange in the part $C$}{78}{section*.63}%
\contentsline {section}{\numberline {XIV}$Q \bar Q$ wave-function eigenvalue equation}{79}{section*.64}%
\contentsline {subsubsection}{\numberline {1}The denominator 0 in $\Delta _{LIR}$}{84}{section*.65}%
\appendix 
\contentsline {section}{\numberline {A}Gluon self-interaction computation}{86}{section*.66}%
\contentsline {subsection}{\numberline {1} Computation of $H_{+1}$}{86}{section*.67}%
\contentsline {subsection}{\numberline {2} Computation of $H_{\perp 1}$}{88}{section*.68}%
\contentsline {subsection}{\numberline {3}Summary of $H_{+1}$ and $H_{\perp 1}$}{90}{section*.69}%
\contentsline {subsection}{\numberline {4}Crosscheck of $\tilde Y_{123}$}{90}{section*.70}%
\contentsline {subsection}{\numberline {5}Evaluation of the gluon 2nd-order self-interaction}{91}{section*.71}%
\contentsline {section}{\numberline {B}Change of variables for homogeneous counterterm computation}{95}{section*.72}%
\contentsline {section}{\numberline {C}Formulas for tree-level counterterm computations}{95}{section*.73}%
\contentsline {section}{\numberline {D}Source of homogeneous divergence in integral over $y \sim 0$}{96}{section*.74}%
\contentsline {subsubsection}{\numberline {a}Review of the canonical case of $\pi _{\overline i}/P_{12}^+=1$}{98}{section*.77}%
\contentsline {subsubsection}{\numberline {b}Review of the boost-invariant case of $\pi _{\overline i}=p_{\overline i}^+$ }{99}{section*.78}%
\contentsline {subsection}{\numberline {1}Computation of the homogenous counterterms}{99}{section*.79}%
\contentsline {subsubsection}{\numberline {a}The canonical case of $\pi _{\overline i}/P_{12}^+=1$}{99}{section*.80}%
\contentsline {subsubsection}{\numberline {b}The boost-invariant case of $\pi _{\overline i}=p_{\overline i}^+$ }{102}{section*.81}%
\contentsline {subsubsection}{\numberline {c}Counterterm computation summary: extension to $H_0$}{104}{section*.82}%
\contentsline {section}{\numberline {E}Review of the RGPEP single-quark term}{105}{section*.83}%
\contentsline {subsubsection}{\numberline {a} \color [rgb]{1,0,0}\bf 20250627: Correction due to $p^- \to P^- - Q^-$ in self-interaction in scattering}{107}{section*.84}%
\contentsline {subsubsection}{\numberline {b}Evaluation of $m_E^2$}{108}{section*.85}%
\contentsline {section}{\numberline {F}Formulas for TDW with $m_G$ not cancelling RGPEP}{109}{section*.86}%
\contentsline {section}{\numberline {G}RGPEP form factors in gluon exchange terms}{111}{section*.87}%
\contentsline {subsubsection}{\numberline {a}Scheme for evaluating self-interactions in gauge theories}{111}{section*.88}%
\contentsline {subsubsection}{\numberline {b}Scheme for evaluating exchanges in gauge theories}{114}{section*.89}%
\contentsline {section}{\numberline {H} Evaluation of W denominators}{116}{section*.90}%
\contentsline {subsection}{\numberline {1}Gluon mass in bound states}{117}{section*.91}%
\contentsline {section}{\numberline {I}COMMENTS to include on technical issues}{117}{section*.92}%
\contentsline {section}{\numberline {}References}{119}{section*.93}%