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It is natural to consider $S$-matrix theory techniques for identifying universal properties of seemingly distinct physical systems. A central lesson of the $S$-matrix program is that low-energy phenomena are governed by near-threshold singularities (such as branch points and poles) of scattering amplitudes considered as analytic functions of complexified kinematical variables. The existence and structure of these singularities can be analyzed largely independently of the details of an underlying microscopic model because they are constrained by general principles, like probability conservation, causality, or crossing symmetry~\citep{Chew:1962mpd, Eden:1966dnq}. While this approach has limited predictive power and is not, by itself, a substitute for field-theoretic descriptions, it can provide a useful set of analytic tools for isolating universal phenomena; see the contribution by M. Mai for further perspectives.

In this short highlight, we summarize results in this spirit for relativistic $3 \to 3$ scattering. We explain how the analytic structure of the amplitude, implied by $S$-matrix unitarity, is implemented through a $K$-matrix-like parameterization and used to study the Efimov phenomenon~\citep{Efimov:1970zz, Efimov:1973awb}. In particular, we describe the emergence of infinitely many three-body states and the evolution of Efimov resonances as the two-body scattering length $a$ is varied. A key ingredient in our analysis is the three-body threshold singularity, which is logarithmic (rather than algebraic) and therefore generates an infinitely sheeted Riemann surface in the complex energy plane. This topology has concrete consequences for the organization and trajectories of three-body bound-state and resonance poles.

This contribution is based on Refs.~\citep{Dawid:2023jrj, Dawid:2023kxu} and new results from an unpublished, upcoming follow-up article~\citep{Dawid:2026}.

\bigskip

\begin{center}
\textbf{Three-body scattering from $S$-matrix unitarity}
\end{center}

\medskip

We study the Efimov phenomenon by exploring simple elastic $s$-wave scattering of three identical spinless particles of mass $m$. Let $\mathcal M_3$ be the connected $3 \to 3$ reaction amplitude that describes the probability for the reaction. For convenience, we describe a three-particle state as a quasi-two-body pair-spectator system and assume that only $s$-waves contribute in the two-body pair subchannels as well. In the center-of-momentum (c.m.) frame of three particles, their total energy is $E$, spectator's momentum magnitude is $k = |\k|$, its energy is $\omega_k = \sqrt{k^2 + m^2}$, and the corresponding pair's invariant mass squared is $\sigma_k = (E - \omega_k)^2 - k^2 $.

The $s$-wave pair-spectator amplitude is constrained by the on-shell unitarity of the $S$ matrix to satisfy an integral equation~\cite{ Blankenbecler:1965gx, Aaron:1968aoz, Hansen:2015zga, Mai:2017vot, Jackura:2018xnx} which is a version of the three-body $K$-matrix parametrization~\citep{Jackura:2022gib},
    %%%%%
    \begin{align}
    \label{eq:M3_int_eq}
    \Mc_3(p,k)
    =
    - \Mc_2(p) \, B(p,k) \, \Mc_2(k)
    - \Mc_2(p)
    \int\limits_0^{k_{\rm max}} \! \frac{{\rm d} k' \, k'^2}{(2 \pi)^2 \omega_{k'}}
    \, B(p,k')
    \, \Mc_3(k',k)  \, ,
    \end{align}
    %%%%%
where an implicit $E$ dependence is assumed and the upper limit of the integration, $k_{\rm max}$, is set such that the integral avoids UV singularities of the kernel~\citep{Hansen:2014eka}. Here $B(p,k) = G(p,k) + C(p,k)$, where $G$ is an amplitude for an on-shell one-particle exchange (OPE) and $C(p,k)$ encodes three-body short-range interactions. The term $\Mc_2$ is an on-shell pair scattering amplitude governing interactions in the two-body subchannel. For concreteness, we use a low-energy approximation,
    %%%%%
    \beq
    \label{eq:M2_general}
    \Mc_2(k) = \frac{16 \pi \sqrt{\sigma_{k}}}{-1/a - i \sqrt{\sigma_{k} / 4 - m^2} } \, ,
    \eeq
    %%%%%
where $a$ is the two-body scattering length and the only parameter of the model. From the $S$-matrix theory point of view, this is the simplest form of the general amplitude satisfying two-body unitarity, in which the corresponding two-body $K$ matrix has no singularities and is approximated by a constant. The amplitude $\Mc_2(k)$ itself has a threshold branch point at,
    %%%%%
    \begin{align}
    \label{eq:branch_point}
    k = k_{\rm th}(E) &= \lambda^{1/2}(E^2, m^2, (2m)^2)/2 E \, ,
    \end{align}
    %%%%%
where $\lambda(x,y,z) = x^2 + y^2 + z^2 - 2 xy - 2 y z - 2 z x$. We orient the associated cut along the real $\sigma_{k}$ axis spanning the interval $\sigma_{k} \in [4m^2, \infty)$. Note that $k_{\textrm{th}}$ will move in the $k$ plane as a function of $E$. When the total energy is equal to the three-body threshold energy, $E = E_{\rm th}$, we have $k_{\rm th} = 0$, an important behavior discussed in more detail below. Due to the square-root-type singularity, $\mathcal{M}_2$ is properly defined on a Riemann surface in $\sigma_{k}$ that consists of two sheets. For $a>0$, $\Mc_2$ has a bound-state pole on the first (or ``physical") Riemann sheet, while for $a<0$ it has a virtual bound-state pole on the second (or ``unphysical") sheet.

In the same spirit, we also adopt the simplest possible model for $C(p,k)$ consistent with $S$-matrix unitarity. It has to be a real, meromorphic function for physical scattering kinematics, so we take it to be a constant. For this work, we assume $C(p,k) = 0$, which does not disrupt the universal behavior of the low-energy spectrum of our model~\citep{Dawid:2023kxu}.

The last building block of Eq.~\eqref{eq:M3_int_eq} is the $s$-wave-projected OPE amplitude,
    %%%%%
    \begin{align}
    \label{eq:Gs_proj}
    G(p, k)
    & = - \frac{1}{4pk} \, \log\left( \frac{z(p, k) - 2pk + i \epsilon }{z(p, k) + 2pk + i \epsilon } \right) \, ,
    \end{align}
    %%%%%
where $z(p, k) = (E-\omega_{k} - \omega_p)^2 - k^2 - p^2 - m^2$. We note that the imaginary part of this amplitude is fixed by the three-body unitarity of the $S$ matrix.


\bigskip

\begin{center}
\textbf{Efimov resonances}
\end{center}

\medskip

\begin{wrapfigure}[30]{r}{0.5\textwidth}
    \centering
    \includegraphics[width=0.45\textwidth]{dawid_figure.pdf}
    \caption{Riemann surface of the three-body scattering amplitude and the schematic trajectories of Efimov resonances corresponding to the variation of the scattering length parameter. Figure adapted from Ref.~\citep{Dawid:2023kxu}.}
    \label{fig:1}
\end{wrapfigure}
The three-body amplitude $\Mc_3$ has a rich singularity structure reflecting both kinematic thresholds and dynamical effects. In particular, it possesses a branch point at the three-body threshold, $E=E_{\rm th}\equiv 3m$. As shown in the next section, this threshold singularity is logarithmic, implying that even for a single elastic channel, the corresponding Riemann surface contains a countable infinity of sheets.

The origin of the threshold singularity can be understood directly from the integral-equation representation, Eq.~\eqref{eq:M3_int_eq}. The integrand develops an energy-dependent (movable) branch point in the integration variable, located at $k_{\rm th}$ (Eq.~\eqref{eq:branch_point}). As mentioned above, this singularity coincides with the origin of the integration interval at $E = E_{\rm th}$. This generates non-analyticity of the solution in the complex $E$ variable~\citep{Eden:1966dnq, Dawid:2023jrj, Dawid:2023kxu}.

For energies away from the collision point, $k_{\rm th}\neq 0$, the singularity can be avoided by deforming the integration contour into the complex $k$ plane. By Cauchy's theorem, this provides an analytic continuation of $\Mc_3$ and allows one to access different Riemann sheets in $E$; the choice of sheet is specified by how the contour circumvents the singularity~\citep{PhysRev.137.B935, PhysRev.167.1505, Brayshaw:1968yia, Dawid:2023jrj, Dawid:2023kxu}. Practical contour prescriptions and numerical implementations are described in Ref.~\citep{Dawid:2023lcq} and in Sec.~IV.E of Ref.~\citep{Dawid:2023jrj}.

%Analytic continuation of $\Mc_3$ to the nearest unphysical Riemann sheet enables one to identify Efimov three-body resonances as complex poles and to follow their evolution as the two-body scattering length $a$ is varied (here we focus on $a<0$, for which resonances appear in the spectrum).
By analytically continuing $\mathcal M_3$ to the nearest unphysical Riemann sheet, one can identify Efimov three-body resonances as complex poles and follow their evolution as $a$ is varied (here, we focus on $a < 0$, for which resonances appear in the spectrum). A schematic summary of the pole trajectories for the first few states is shown in Fig.~\ref{fig:1}, using $\Delta E\equiv E-E_{\rm th}$ as the energy coordinate. We show four Riemann sheets of the amplitude and highlight the trajectories of the first three trimers. Below, we briefly summarize our main observations, which generalize and support earlier studies~\citep{Bringas:2004zz, Hyodo:2013zxa}.

As expected, the complex energies of the resonant trimers are approximately related by the Efimov discrete scaling, $\Delta E(a)\approx \lambda^{2}\Delta E(\lambda a)$ with $\lambda\simeq 22.7$. In the relativistic setting, the particle mass $m$ provides an intrinsic scale that mildly breaks the discrete scaling symmetry, leading to visible deviations for states far from threshold (i.e., for states that have $\Delta E \approx m$). We also observe that the complex poles follow loop-like trajectories; as $|a|$ increases, the pole location evolves, starting from the three-body threshold, where the Efimov state resembles a virtual state. It then moves above the three-body threshold, becoming a narrow resonance, to eventually reach the threshold again.

A notable feature of the evolution is the synchronous behavior of neighboring levels near threshold. As the $(n+1)$th excited pole emerges at $E_{\rm th}$ on the nearest unphysical sheet for a particular value of $a$, the $n$th state simultaneously moves through threshold and becomes a bound-state pole on the physical sheet. In Ref.~\citep{Dawid:2023kxu}, this pattern was conjectured to be tied to the infinitely-sheeted topology induced by the logarithmic three-body threshold. The observed cyclic trajectories on the first few sheets likely represent only a small part of a global flow of poles across all sheets, wherein resonance poles ``descend'' from the higher to lower Riemann branches and become bound states in the unitarity limit. A related puzzle concerns the accompanying ``mirror'' poles implied by the Schwarz reflection property: as a resonance pole approaches threshold, its mirror partner approaches as well, yet only a single bound-state pole is seen to emerge on the physical sheet. In practice, the mirror poles seem to terminate at threshold. This is contrary to a behavior known from the two-body scattering amplitude analysis, where such a state would become an additional physical bound state or a virtual state.

A direct numerical test of this picture would require analytic continuation beyond the nearest unphysical sheet, which remains technically and conceptually challenging. In the next section, we instead provide analytic evidence for the underlying mechanism by deriving the universal form of the three-body threshold singularity and its consequences for pole conditions across the Riemann surface.

\bigskip

\begin{center}
\textbf{Three-body threshold and pole condition}
\end{center}

\medskip

In the following, we present the relevant result from Ref.~\citep{Dawid:2026}. Namely, we show how the logarithmic branch point emerges in the unitarity-preserving parametrization of the $3 \to 3$ scattering amplitude. The main point is to observe that both the integration kernel of Eq.~\eqref{eq:M3_int_eq} and the OPE amplitude are square integrable with respect to the measure $dk' \, k'^2/ (2\pi)^2 \omega_{k'}$. Namely, one can show that
    %%%%%
    \begin{align}
    \int\limits_0^{k_{\rm max}} \frac{dp \, p^2}{(2\pi)^2 \omega_p } \int\limits_0^{k_{\rm max}} \frac{dk \, k^2}{(2\pi)^2 \omega_k} \, \left| G(p,k)
    \, \Mc_2(k) \right|^2 < \infty \, ,
    \qquad {\rm and} \qquad
    \int\limits_0^{k_{\rm max}} \frac{dp \, p^2}{(2\pi)^2 \omega_p } \int\limits_0^{k_{\rm max}} \frac{dk \, k^2}{(2\pi)^2 \omega_k} \, \left| G(p,k) \right|^2 < \infty \, ,
    \end{align}
    %%%%%
inside a small complex disk centered at the three-body threshold energy $E_{\rm th} = 3m$, as long as $1/a \neq 0$~\citep{Amado:1971dlx}. This holds regardless of whether the $\Mc_2$ amplitude has a subthreshold bound-state pole---in such a case, one defines the kernel on the appropriate Riemann sheet by a contour deformation around the pole. Here, we simply assume no dimer poles, $a < 0$, since this is the case relevant for the Efimov resonances, and it allows us to ignore the additional two-body bound-state--particle threshold.

Given that the kernel and the OPE are of the Hilbert-Schmidt type, one can represent $G(p,k)$ as an (infinite) sum of separable functions and write a formal algebraic solution to the integral equation,
    %%%%%
    \begin{align}
    \Mc_3(p,k) = \sum_{i,j = 1}^\infty \Mc_2(p) \, \phi_i(p)
    \left[ \Gc^{-1} + \Ic \right]_{ij}^{-1} \,
    \phi_j^*(k) \, \Mc_2(k) \, ,
    \end{align}
    %%%%%
where $\{ \phi_i \}_{i \in \mathbb{N}}$ are polynomials on the interval $k \in [0, k_{\rm max}]$, which are orthogonal with respect to the original integration weight. The energy-dependent operator is
    %%%%%
    \begin{align}
    \Gc_{ij} =
    \int\limits_0^{k_{\rm max}}  \frac{dp \, p^2}{(2\pi)^2 \omega_p }
    \int\limits_0^{k_{\rm max}}  \frac{dk \, k^2}{(2\pi)^2 \omega_k } \,  \phi_i^*(p) \, G(p,k) \, \phi_j(k) \, .
    \end{align}
    %%%%%
The operator $\Ic$ plays a role analogous to the phase space in the two-body $K$ matrix parametrization of the partial-wave amplitude, in the sense that it contains the threshold branch point singularity corresponding to the opening of the three-body scattering continuum:
    %%%%%
    \begin{align}
    \label{eq:Integral}
    \Ic_{ij} = \int\limits_0^{k_{\rm max}} \frac{dk' \, k'^2}{(2\pi)^2 \omega_{k'} } \, \phi_i^*(k') \, \Mc_2(k') \, \phi_j(k') \, .
    \end{align}
    %%%%%
The secular equation that determines the position of the poles of the three-body amplitude is
    %%%%%
    \begin{align}
    \label{eq:quantization}
    \det_{ij} \left[ \Gc^{-1} + \Ic  \right] = 0 \, .
    \end{align}
    %%%%%
Let us now determine the nature of the three-body threshold singularity at $E = E_{\rm th}$. We recall that it is generated by the collision of the movable branch point, Eq.~\eqref{eq:branch_point}, with the origin of the integration interval at $k' = 0$ as $E \to E_{\rm th}$. Thus, to determine this singularity, it is sufficient to study Eq.~\eqref{eq:Integral} with the integrand evaluated near the collision point, i.e., for $k' \approx 0$ and $\Delta E = E - E_{\rm th} \approx 0$, which corresponds to the nonrelativistic approximation of $\Ic$. Assuming the following ordering of scales, $\Delta E \sim k'^2/m$, one obtains,
    %%%%%
    \begin{align}
    \frac{{k'}^2}{(2\pi)^2 \, \omega_{k'} } \, \phi_i^*(k') \,
    \Mc_2(k') \,
    \phi_j(k') \approx
    %%%
    \frac{{k'}^2}{(2\pi)^2 \, m} \, \phi^*_i(0) \,
    \frac{16\pi (\Delta E + 2m - 3 {k'}^2/4m )}{- \frac{1}{a} + \sqrt{\frac{3}{4} {k'}^2 - \Delta E m } }
    \phi_j(0) \, ,
    \end{align}
    %%%%%
Given this approximation, one can show that,
    %%%%%
    \begin{align}
    \label{eq:log-expansion}
    \Ic_{ij} = - \big( \phi_i^*(0) \phi_j(0)  \big) \,
    \left( \frac{4 a^2}{3 \sqrt{3} \pi } \right) \,
    (m \Delta E)^2 \,
    \log \left(-\frac{m \Delta E}{\Lambda^2} \right)
    + \Rc_{ij} \, ,
    \end{align}
    %%%%%
where $\Rc_{ij}$ is a function that is analytic in a disc centered at $E_{\rm th}$ and $\Lambda = 4 m/3$ is a non-relativistic approximation of $k_{\rm max}$. We note that the threshold singularity of the integral is of the type $x^2 \log(x)$, with $x = m \Delta E$. The analytic continuation of the integral to the $n$-th Riemann sheet is obtained by a standard continuation of the logarithm, i.e., replacing
    %%%%%
    \begin{align}
    \log \left(-\frac{m \Delta E}{\Lambda^2} \right) \to \log \left(-\frac{m \Delta E}{\Lambda^2} \right) + 2 \pi i n \, .
    \end{align}
    %%%%%
Let us note that at the threshold, $\Delta E = 0$, the analytic continuation of the pole condition~\eqref{eq:quantization} becomes independent of $n$. Therefore, whenever a pole emerges at the threshold on sheet 0 for some value of $a$, it must be accompanied by threshold poles on every other sheet (at least one pole per sheet). Since the Riemann surface for the three-body amplitude has infinitely many sheets, that implies infinitely many three-body states which appear synchronously at the threshold.%%%
\footnote{We assume that at $\Delta E=0$ the secular equation has a nonvanishing analytic contribution, so when the sheet-dependent term $\propto (\Delta E)^2[\,\log(\Delta E)+2\pi i n\,]$ vanishes as $\Delta E\to 0$, there are non-trivial solutions of the pole condition for some value of $a$.}
%%%
This verifies the conjecture made in Ref.~\citep{Dawid:2023kxu}.

The behavior near the three-body threshold differs qualitatively from the familiar two-body square-root case. In the latter, the threshold can be uniformized by a finite local coordinate, relative momentum $q \propto \sqrt{\Delta E}$. Poles pass through the threshold in a one-to-one manner, and the Cauchy's argument principle is satisfied in the complex $q$ plane for contours encircling point $q=0$. In the three-body case, however, infinitely many sheets meet at the threshold. The uniformization can be obtained by $m\Delta E=e^z$, which allows one to represent these sheets on a single $z$-plane. However, the threshold $\Delta E=0$ is sent to $\Re z\to -\infty$, not to a regular finite point. Thus the threshold is not a local crossing point analogous to $q=0$ in the two-body problem. Poles may accumulate at, or disappear at, this logarithmic end, so there is no requirement that the number of poles emerging on a given finite set of sheets equal the number entering from another finite set.

That said, one may still define a partial uniformization by defining a piecewise representation of several sheets on a single complex plane through a map $m \Delta E = z^{n_s}$, where $n_s$ is the number of $\Delta E$-plane Riemann sheets contained in a $z$ plane. A study of three-body contact-interaction toy-models, such as that of Ref.~\citep{Dawid:2020uhn}, shows that as poles approach the threshold from different sheets, they merge into a common threshold configuration and re-emerge in a re-paired manner. Heuristically, one may think of the threshold as redistributing sections of poles living on neighboring sheets, so that a pole emerging on $0$-th sheet is assembled from the incoming poles on two adjacent sheets. In this sense, the logarithmic threshold acts as a junction for an infinite collection of pole trajectories, rather than as a simple two-sheet crossing point. We refer the reader to Ref.~\citep{Dawid:2026} for more details.


\bigskip

\begin{center}
\textbf{Conclusion}
\end{center}

\medskip

Unitarity of the $S$ matrix (a consequence of probability conservation) implies that the elastic three-body amplitude develops a logarithmic branch point at the three-body threshold. As a result, the amplitude is naturally defined on a Riemann surface with a countably infinite number of sheets. In this contribution, we emphasized that this topology has concrete consequences for the organization of three-body poles: in the Efimov system, pole trajectories extend across multiple sheets and, in the unitarity limit, generate the familiar tower of three-body bound states. Successive states approach the threshold in a correlated manner as the interaction parameter (the scattering length $a$) is varied.

Since the low-energy form of $\Mc_2$ used here describes a generic short-range two-body scattering, and since one-particle exchange is a kinematic mechanism always present in relativistic three-body dynamics, we expect that aspects of the near-threshold pole structure---in particular, the correlated appearance of shallow poles across adjacent sheets---persist beyond the specific Efimov setting and do not rely on discrete scale invariance. While realistic three-body reactions will not generically exhibit perfectly cyclic trajectories, the logarithmic threshold behavior and the resulting form of the secular equation can still be used to anticipate near-threshold virtual or resonant states accompanying shallow three-body bound states. These implications will be developed in more detail in Ref.~\citep{Dawid:2026}.

Finally, the example spectrum discussed here, together with the contour-deformation techniques for analytic continuation developed in Refs.~\citep{Dawid:2023jrj, Dawid:2023kxu}, provides practical guidance for amplitude analyses in ongoing three-body lattice-QCD calculations~\cite{Hansen:2020otl, Sadasivan:2021emk, Yan:2024gwp, Dawid:2025doq, Dawid:2025zxc, Briceno:2025yuq, Yan:2025mdm}.

\medskip

\begin{center}
\textbf{Acknowledgments}
\end{center}

\medskip

We thank the organizers for hosting the workshop ``Universality in strongly interacting systems: from QCD to atoms'' at ECT$^*$ in Trento. This work contributes to the goals of the USDOE ExoHad Topical Collaboration, contract DE-SC0023598.

\clearpage
\endgroup
