# Partial-wave projection of the one-particle exchange in three-body
scattering amplitudes

Andrew W. Jackura<EMAIL_ADDRESS>Department of Physics, William & Mary,
Williamsburg, VA 23187, USA  Raúl A. Briceño<EMAIL_ADDRESS>Department
of Physics, University of California, Berkeley, CA 94720, USA  Nuclear Science
Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

###### Abstract

As the study of three-hadron physics from lattice QCD matures, it is necessary
to develop proper analysis tools in order to reliably study a variety of
phenomena, including resonance spectroscopy and nuclear structure.
Reconstructing the three-particle scattering amplitude requires solving
integral equations, which can be written in terms of data-constrained
dynamical functions and physical on-shell quantities. The driving term in
these equations is the so-called one-particle exchange, which leads to a
kinematic divergence for particles on-mass-shell. A vital component in
defining three-particle amplitudes with definite parity and total angular
momentum, which are used in spectroscopic studies, is to project the one-
particle exchange into definite partial waves. We present a general procedure
to construct exact analytic partial wave projections of the one-particle
exchange contribution for any system composed of three spinless hadrons. Our
result allows one full control over the analytic structure of the projection,
which we explore for some low-lying partial waves with applications to three
pions.

##  I Introduction

Applications of reaction theory to three-body systems have seen a resurgence
due to modern theoretical hadronic spectroscopy. The success of two-hadron
resonance studies from Quantum Chromodynamics (QCD) using numerical lattice
QCD Dudek  _et al._ (2011); Beane  _et al._ (2012); Pelissier and Alexandru
(2013); Dudek  _et al._ (2013); Liu  _et al._ (2013); Beane  _et al._ (2013);
Wilson  _et al._ (2015a); Dudek  _et al._ (2014); Orginos  _et al._ (2015);
Berkowitz  _et al._ (2017); Lang  _et al._ (2015); Wilson  _et al._ (2015b);
Dudek  _et al._ (2016); Briceño  _et al._ (2017); Moir  _et al._ (2016);
Bulava  _et al._ (2016); Hu  _et al._ (2016); Alexandrou  _et al._ (2017);
Bali  _et al._ (2017); Wagman  _et al._ (2017); Andersen  _et al._ (2018);
Briceno  _et al._ (2018); Woss  _et al._ (2018); Brett  _et al._ (2018);
Werner  _et al._ (2020); Mai  _et al._ (2019); Woss  _et al._ (2019); Wilson
_et al._ (2019); Cheung  _et al._ (2021); Rendon  _et al._ (2020); Woss  _et
al._ (2021); Hörz  _et al._ (2021) in conjunction with non-perturbative
mappings between finite-volume spectra and reaction amplitudes has allowed the
community to pursue implementing such an analysis strategy for excited hadrons
which have coupling to three-hadron decay modes.

The framework to compute non-perturbative reaction amplitudes from QCD relies
on a methodology first presented by Lüscher Luscher (1986a, b, 1991) for two-
particle systems Rummukainen and Gottlieb (1995); Kim  _et al._ (2005); He
_et al._ (2005); Davoudi and Savage (2011); Hansen and Sharpe (2012); Briceño
and Davoudi (2013); Briceño  _et al._ (2013); Briceño (2014); Romero-Lopez
_et al._ (2018), with extensions to three-body systems developed in the last
decade Hansen and Sharpe (2014, 2015, 2017); Briceno  _et al._ (2017); Briceño
_et al._ (2018); Briceno  _et al._ (2019); Briceño  _et al._ (2019); Blanton
_et al._ (2019); Hansen  _et al._ (2020); Blanton and Sharpe (2020); Polejaeva
and Rusetsky (2012); Guo and Gasparian (2018, 2017); Hammer  _et al._ (2017a,
b); Meng  _et al._ (2018); Pang  _et al._ (2019); Müller  _et al._ (2022);
Briceno  _et al._ (2017); Blanton and Sharpe (2021a, b); Hansen  _et al._
(2020); Blanton and Sharpe (2020, 2021a, 2021b); Jackura (2023); Mai  _et al._
(2017); Jackura  _et al._ (2019); Mikhasenko  _et al._ (2019); Dawid and
Szczepaniak (2021). The procedure for three-hadrons is given as follows.
Finite-volume correlation functions of operators with non-zero overlap to the
desired quantum numbers are computed via numerical Monte Carlo methods and the
subsequent spectrum is determined by novel techniques within lattice QCD. The
finite-volume energy spectrum is then used in conjunction with formalisms
known as _quantization conditions_ which relate short-distance dynamical
objects known as $K$ matrices to the spectrum through geometric functions
characterizing the distortions due to the periodic, finite-volume.
Practically, one uses this avenue to constrain the $K$ matrices which seed
into a set of integral equations which describe the on-shell scattering of the
three hadrons. Examples of this computational procedure are given in Briceño
_et al._ (2018); Blanton  _et al._ (2020); Romero-López  _et al._ (2019);
Fischer  _et al._ (2021); Jackura  _et al._ (2021); Hansen  _et al._ (2021).

A major challenge in the study of three-particle reactions via lattice QCD is
the last stage of the analysis, where physical amplitudes are reconstructed
from the data-constrained $K$ matrices. For spectroscopy, one usually desires
the resulting scattering amplitudes to be of definite spin-parity $J^{P}$ so
that one may search for the spectral content by means of analytic
continuation. Although there has been substantial progress on this end Briceño
_et al._ (2018); Blanton  _et al._ (2020); Romero-López  _et al._ (2019);
Fischer  _et al._ (2021); Jackura  _et al._ (2021); Hansen  _et al._ (2021);
Dawid  _et al._ (2023a, b), most studies have focused on the restricted
scenario where all the particles are identical spinless bosons in which all
angular momenta are projected to $S$ wave. 111An exception is the exploratory
study of an isovector $a_{1}$ meson Mai  _et al._ (2021); Sadasivan  _et al._
(2022), which numerically projected the scattering equations into
$J^{P}=1^{+}$ and neglected all other partial wave channels except
$a_{1}\to\rho\pi(^{3}S_{1})$.

In this work, we focus on lifting this technical restriction by presenting the
operations needed to project the $\mathbf{3}\to\mathbf{3}$ 222We use the
notation $\mathbf{n}\to\mathbf{m}$ to indicate a reaction involving
$\mathbf{n}$ incoming and $\mathbf{m}$ outgoing stable hadrons. scattering
amplitude into any definite $J^{P}$ partial wave. We consider the partial wave
expansion of the scattering amplitude of three arbitrary spinless particles,
that is the particles can be identical or distinguishable. The exact details
of the relativistic $\mathbf{3}\to\mathbf{3}$ scattering amplitude can be
found in Sec. II, as well as the introduction of relevant kinematic variables.
In Sec. III, we review key concepts used for partial-wave projecting the
scattering amplitude. As is emphasized there, the procedure followed is to
define amplitudes within the helicity basis that are then projected to
definite $J^{P}$.

At the center of our analysis is the one-particle exchange (OPE) process, a
known kinematic function central to the integral equations Fleming (1964);
Holman (1965); Kamal (1965); Jackura (2023). The OPE has a complicated angular
dependence which arises when two of the particles couple to some definite spin
before recoiling against the third, making it the most challenging amplitude
to project to definite partial waves. Schematically, the exchange propagator
of the OPE, denoted by $\mathcal{G}$, takes the form

$\displaystyle\mathcal{G}=\frac{\mathcal{H}}{u-m_{e}^{2}}\,,$

where $\mathcal{H}$ is a dense matrix in the angular momentum of the incoming
and outgoing pairs which we call the _spin-helicity matrix_ , and $u$ is the
momentum-squared of the exchange particle which has mass $m_{e}$. The
functions $\mathcal{H}$ and $u$ depend on the kinematics of the exchanged
spectator particles, including the scattering angle. The main goal of this
work is to provide a generic procedure to obtain an analytic representation of
the partial wave projection of $\mathcal{G}$. Since our focus is primarily for
lattice QCD applications, although this procedure can also be used in
phenomenological studies, we use the definition of $\mathcal{H}$ a presented
in Ref. Hansen and Sharpe (2014, 2015); Jackura (2023).

Details of the analytic partial wave projection of the OPE are given in Sec.
IV, which makes use the procedure outlined in Sec. III to derive a generic
result for the partial wave OPE for any target $J^{P}$ quantum number. Our
result can be expressed in terms of entirely known functions, taking the form

$\displaystyle\mathcal{G}^{J^{P}}=\mathcal{K}_{\mathcal{G}}^{J^{P}}+\mathcal{T}^{J^{P}}\,Q_{0}(\zeta_{pk})\,,$
(1)

where $\mathcal{G}^{J^{P}}$ is the exchange propagator projected to definite
spin-parity $J^{P}$, $\mathcal{K}_{\mathcal{G}}^{J^{P}}$ and
$\mathcal{T}^{J^{P}}$ are functions of external kinematics and include
Clebsch-Gordan coefficients which couple the system to $J^{P}$. The functions
$\mathcal{K}_{\mathcal{G}}^{J^{P}}$ and $\mathcal{T}^{J^{P}}$ are matrices in
space of partial waves which contribute to a particular $J^{P}$, and are
completely determined by the spin-helicity matrix $\mathcal{H}$, as shown in
Sec. IV. The $\mathcal{G}^{J^{P}}$ amplitude contains a branch cut in the
complex energy plane which is due to on-shell particle exchange. This non-
analytic behavior of the OPE is encoded entirely in $Q_{0}$, the zero-degree
Legendre functions of the 2nd kind, which depends on external kinematic
variables through the function $\zeta_{pk}$, which is defined in the main body
of the text. Our result allows one to control the entire analytic behavior of
the amplitude which is vital in the analytic continuation of three-body
amplitudes to complex energy planes Dawid  _et al._ (2023a, b).

In Sec. V, we use our main equation (1) to provide explicit expressions for
the OPE amplitude for key low-lying partial waves. Applications of these
results are given in Sec. VI, where we show numerical results for relevant
channels in $3\pi$ systems to illustrate some of the analytic properties of
these functions as discussed in the main text. Our procedure is summarized in
Sec. VII. To aid the reader, we provide three technical Appendices, App. A, B,
and C, that include details of various special functions that are used
throughout this work, a derivation of a key integral used in the analytic
partial wave projection, and an alternative version of our approach using
arbitrary reference frames. For the reader who wishes to use our explicit
partial wave projected OPE amplitudes directly in their analyses, a fourth
Appendix, App. D, collects the cases presented in Sec. V along with brief
explanations of the required kinematic variables.

##  II Amplitudes & Kinematics

In the following, we consider the scattering of three spinless particles. In
this work, we do not restrict the particles to be degenerate or identical,
however, we do not consider any additional internal symmetries. e.g. hadronic
flavor quantum numbers. 333It is straightforward to include restrictions due
to additional symmetries, e.g. by including the appropriate SU(2) Clebsch-
Gordan coefficients for three hadrons with isospin symmetry, cf. Refs. Blanton
and Sharpe (2021a, b); Hansen  _et al._ (2020). Since our focus is ultimately
on the on-shell exchange mechanism, we find this generalization benefits
future applications as we provide a generic result to accommodate not only
cases such as elastic $\pi\pi\pi$ scattering, but also those such as
$K\bar{K}\pi\to K\bar{K}\pi$ where $K\bar{K}\to\eta\pi$ allows for $\eta$
meson exchanges between $K\bar{K}$ pairs.

Therefore, we consider a three-body reaction of the form

$\displaystyle\varphi_{k}(k)+\varphi_{a}(a)+\varphi_{a^{\prime}}(a^{\prime})\to\varphi_{p}(p)+\varphi_{b}(b)+\varphi_{b^{\prime}}(b^{\prime})\,,$

where $\varphi_{k}(k)$ represents a single spinless particle carrying a four-
momentum $k=(\omega_{k},\mathbf{k})$ with its energy $\omega_{k}$ fixed by its
mass $m_{k}$ and momentum $\mathbf{k}$ through the usual relativistic on-shell
dispersion relation $\omega_{k}^{2}=m_{k}^{2}+\mathbf{k}^{2}$. Similar
definitions hold for the other particles. Here we adopt the notation that the
mass of the particle will be labeled by its momentum. We normalize the single
particle state by the usual Lorentz invariant measure

$\displaystyle\braket{\mathbf{k}^{\prime}}{\mathbf{k}}=(2\pi)^{3}\,2\omega_{k}\,\delta^{(3)}(\mathbf{k}^{\prime}-\mathbf{k})\,,$

where $\delta^{(3)}$ is the three-dimensional Dirac delta distribution. The
initial system carries a total four-momentum $P=(E,\mathbf{P})$, where $E$ is
the total energy and $\mathbf{P}$ is the total momentum, which in terms of the
constituent momenta is $P=k+a+a^{\prime}$. Similarly,
$P^{\prime}=p+b+b^{\prime}=(E^{\prime},\mathbf{P}^{\prime})$ for the final
state four-momentum. A three-particle state is constructed by the usual tensor
product of single-particle states, which we denote as
$\ket{P,\mathbf{k},\mathbf{a}}$. Here we trade the momentum
$\mathbf{a}^{\prime}$ for the total momentum as it is conserved in reactions
and $\mathbf{a}^{\prime}=\mathbf{P}-\mathbf{k}-\mathbf{a}$.

Figure 1: The fully connected $\mathbf{3}\to\mathbf{3}$ amplitude
$i\mathcal{M}$ with momentum assignments. All external legs represent incoming
and outgoing on-shell particles constrained by total momentum conservation.

The $\mathbf{3}\to\mathbf{3}$ scattering amplitude $\mathcal{M}$, depicted in
Fig. 1, is defined as the fully connected $S$ matrix element

$\displaystyle\braket{P^{\prime},\mathbf{p},\mathbf{b};\mathrm{out}}{P,\mathbf{k},\mathbf{a};\mathrm{in}}_{\mathrm{conn.}}=(2\pi)^{4}\delta^{(4)}(P^{\prime}-P)\,i\mathcal{M}(\mathbf{p},\mathbf{b};\mathbf{k},\mathbf{a})\,,$
(2)

where “conn.” indicates only the fully connected contribution is to be taken,
and “in/out” refer to the asymptotically far past/future. We have also
factored out a Dirac delta from the amplitude which ensures total momentum is
conserved, $P=P^{\prime}$. The amplitude depends on the total three-body
center-of-momentum (CM) frame energy $E=\sqrt{s}$, where $s\equiv
P^{2}=P^{\mu}P_{\mu}=E^{2}-\mathbf{P}^{2}$ is the Mandelstam invariant. The
physical scattering threshold is given by
$E^{(\mathrm{thr.})}=\max(m_{k}+m_{a}+m_{a^{\prime}},m_{p}+m_{b}+m_{b^{\prime}})$.
In this work, we suppress the dependence of $s$ for all amplitudes to simplify
the notation. The amplitude depends on seven more kinematic variables which
are formed from the set of initial and final state momenta.

In order to construct useful kinematic variables, it is convenient to consider
the kinematic configuration of the three-body system as one consisting of two
particles in a _pair_ with an associated _spectator_ being the third particle.
In most of this work, we choose to label the initial state spectator with
momentum $k$, while the associated pair is composed of the particles with
momenta $a$ and $a^{\prime}$. Likewise, for the final state, the spectator has
momentum $p$ and the pair consists of the particles with momenta $b$ and
$b^{\prime}$.

Each pair has a four-momentum given by $P_{k}=(E_{k},\mathbf{P}_{k})\equiv
P-k$ and $P_{p}=(E_{p},\mathbf{P}_{p})\equiv P-p$ for the initial and final
state, respectively, where the subscripts $k$ and $p$ indicate which spectator
is associated with the pair. The invariant mass-squared of the pairs is given
by

$\displaystyle\sigma_{k}\equiv P_{k}^{2}=(P-k)^{2}\,,\qquad\sigma_{p}\equiv
P_{p}^{2}=(P-p)^{2}\,.$ (3)

Focusing first on the initial state, for a fixed $s$ the physical region of
the pair invariant mass is limited to
$\sigma_{k}^{(\mathrm{thr.})}\leq\sigma_{k}\leq(\sqrt{s}-m_{k})^{2}$, where
$\sigma_{k}^{(\mathrm{thr.})}$ is the physical scattering threshold for that
pair, $\sigma_{k}^{(\mathrm{thr.})}=(m_{a}+m_{a^{\prime}})^{2}$. Momentum
conservation constrains the pair invariant masses through the usual Mandelstam
condition,

$\displaystyle\sigma_{k}+\sigma_{a}+\sigma_{a^{\prime}}=s+m_{k}^{2}+m_{a}^{2}+m_{a^{\prime}}^{2}\,,$
(4)

where $\sigma_{a}$ and $\sigma_{a^{\prime}}$ are the pair invariant masses
considering $a$ and $a^{\prime}$ as spectators, respectively. The physical
scattering region of the three particles is therefore bounded by the condition
$\Phi(k,a)\geq 0$, where $\Phi(k,a)$ is the Kibble boundary function defined
as Kibble (1960); Byckling and Kajantie (1973); Collins (1971)

$\displaystyle\Phi(k,a)=\sigma_{k}\sigma_{a}\sigma_{a^{\prime}}$
$\displaystyle-\sigma_{k}(s-m_{a}^{2})(m_{k}^{2}-m_{a^{\prime}}^{2})-\sigma_{a}(s-m_{k}^{2})(m_{a}^{2}-m_{a^{\prime}}^{2})\,$
$\displaystyle-(sm_{a^{\prime}}^{2}-m_{a}^{2}m_{k}^{2})(s+m_{a^{\prime}}^{2}-m_{a}^{2}-m_{k}^{2})\,.$
(5)

Similar restrictions hold for the final state particles, with expressions
given by the substitution $\\{k,a,a^{\prime}\\}\to\\{p,b,b^{\prime}\\}$ in the
above conditions.

In Sec. II.1, we specify three reference frames which we use to define
additional kinematic variables used in the partial wave projection.

###  II.1 Reference Frames

Figure 2: Reference frames for the $\mathbf{3}\to\mathbf{3}$ amplitude as
described in the text. Shown in blue is the initial three-body state plane, in
red the final three-body state, and in gray the reaction plane. Kinematics in
the reaction plane are shown in the total CM frame ($\mathbf{P}=\mathbf{0})$,
while the initial state planes are shown for both the initial pair CM frame
($\mathbf{P}_{k,k}^{\star}=\mathbf{0}$) and the total CM frame. The final
state planes are shown in the total CM frame and the final pair CM frame
($\mathbf{P}_{p,p}^{\star}=\mathbf{0}$).

Three reference frames are required in our analysis of the partial wave
projection of the $\mathbf{3}\to\mathbf{3}$ amplitude. Here we define the
essential characteristics of these frames, and will refer to these in our
constructions of partial waves in Sec. III and give additional kinematic
relations when we discuss the application to the exchange propagator in Sec.
IV. These reference frames are illustrated in Fig. 2, and are designated the
“initial pair CM frame”, the “final pair CM frame”, and the “total CM frame”.
We define these frames as follows:

  1. 1.

_Initial pair CM frame_ – The initial pair CM frame is defined by
$\mathbf{P}_{k}=\mathbf{P}-\mathbf{k}=\mathbf{0}$. It is common to introduce a
notation to indicate a given kinematic variable is evaluated in some specific
reference frame. Commonly in the literature one uses $\star$ superscript to
indicate such a situation for the CM frame. In our case, however, we need to
be careful as there are three CM frames of interest. Therefore, we adopt a
notation for the initial pair rest frame that a $\star$ superscript _along
with_ a $k$ subscript indicating that the kinematic variable is evaluated in
this frame. While this results in slightly cumbersome notation, we feel this
will alleviate future confusion for implementing the results of this work. As
an example, the defining relation for this frame can be written as
$\mathbf{P}_{k,k}^{\star}=\mathbf{P}_{k}^{\star}-\mathbf{k}_{k}^{\star}=\mathbf{0}$,
where $\mathbf{P}_{k,k}^{\star}$ indicates the initial state pair momentum is
evaluated in its rest frame. 444An example where this notation is vital is for
$\mathbf{P}_{p,k}^{\star}$, which is the final state pair momentum evaluated
in the initial pair rest frame. Such evaluations become necessary as detailed
in Sec. IV.

In this frame, the pair has back-to-back momentum
$\mathbf{a}_{k}^{\star}=-\mathbf{a}_{k}^{\prime\star}$, with its magnitude
fixed by the pair invariant mass 555The difference between the four-momentum
$a=(\omega_{a},\mathbf{a})$ and the magnitude of its three-momentum
$a=\lvert\mathbf{a}\rvert$ is clear from context.

$\displaystyle
a_{k}^{\star}\equiv\lvert\mathbf{a}_{k}^{\star}\rvert=\frac{1}{2\sqrt{\sigma_{k}}}\,\lambda^{1/2}(\sigma_{k},m_{a}^{2},m_{a^{\prime}}^{2})\,,$
(6)

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+yz+zx)$ is the Källén triangle
function. Note that $\lambda(x,y,z)$ is symmetric under interchange of the
variables $x,y,z$. Note also that in the case where $m_{a}=m_{a^{\prime}}=m$,
then $a_{k}^{\star}$ reduces to $a_{k}^{\star}=\sqrt{\sigma_{k}/4-m^{2}}$ .

As illustrated in Fig. 2, we define a coordinate system with a $z_{k}$-axis
for the initial state defined to be anti-parallel to the spectator momentum,
i.e. $\mathbf{\hat{z}}_{k}\equiv-\mathbf{\hat{k}}$. 666We use the notation
$\mathbf{\hat{r}}$, the unit vector of $\mathbf{r}$, to indicate the polar and
azimuthal angles, $(\theta_{r},\varphi_{r})$. Note that we use the standard
convention for the domain of the polar and azimuthal angles,
$\theta_{r}\in[0,\pi]$, and $\varphi_{r}\in[0,2\pi)$.  Thus, the first
particle in the pair (taken to by $\varphi_{a}$) has its momentum oriented at
a polar angle $\vartheta_{k}^{\star}$ with respect to this $z_{k}$-axis.
Furthermore, the three momenta form a plane (the initial state plane) defined
by the normal vector $\mathbf{k}\times\mathbf{a}_{k}$, oriented with respect
to the _reaction plane_ (with a coordinate system $XYZ$ which is defined
later) by an azimuthal angle $\psi_{k}^{\star}$. This angle is preserved upon
Lorentz boosts along $z_{k}$, i.e. $\psi_{k}=\psi_{k}^{\star}$ from the total
CM frame to the initial pair CM frame. The boost velocity from the initial
pair CM frame to the total CM frame is given by
$\boldsymbol{\beta}_{k}=\mathbf{P}_{k}/E_{k}$.

  2. 2.

_Final pair CM frame_ – The final pair CM frame, defined by
$\mathbf{P}_{p,p}^{\star}=\mathbf{0}$, is constructed analogously to the
initial pair CM frame. The notation of a $\star$ superscript _with_ a $p$
subscript indicates kinematic variables are in this frame. Another body-fixed
coordinate system is assigned to this frame, with its $z_{p}$-axis is defined
by $\mathbf{\hat{z}}_{p}\equiv-\mathbf{\hat{p}}$ and the “final state plane”
defined with a normal vector $\mathbf{p}\times\mathbf{b}_{p}$, which is
depicted in Fig. 2. The pairs polar and azimuthal angles are
$\vartheta_{p}^{\star}$ and $\psi_{p}^{\star}$, respectively. The azimuthal
angle is again invariant under boost along $\mathbf{\hat{z}}_{p}$,
$\psi_{p}^{\star}=\psi_{p}$. The final pair momenta are defined back-to-back,
$\mathbf{b}_{p}^{\star}=-\mathbf{b}_{p}^{\prime\star}$, with a magnitude fixed
by $\sigma_{p}$

$\displaystyle
b_{p}^{\star}\equiv\lvert\mathbf{b}_{p}^{\star}\rvert=\frac{1}{2\sqrt{\sigma_{p}}}\,\lambda^{1/2}(\sigma_{p},m_{b}^{2},m_{b^{\prime}}^{2})\,.$
(7)

  3. 3.

_Total CM frame_ – The final reference frame in our analysis is the total CM
frame, defined by $\mathbf{P}=\mathbf{0}$. Unlike the initial and pair CM
frames, we do not include a special notation to indicate a kinematic variable
is evaluated in the total CM frame. This frame proves convenient to define the
_reaction plane_ , which connects the initial three-particle state to the
final state. Both the initial and final state momenta are equally evaluated in
this frame. Specifically, the magnitudes of the initial and final spectator
momenta are fixed by their pair invariant masses,

$\displaystyle
p\equiv\lvert\mathbf{p}\rvert=\frac{1}{2\sqrt{s}}\,\lambda^{1/2}(s,\sigma_{p},m_{p}^{2}),\qquad
k\equiv\lvert\mathbf{k}\rvert=\frac{1}{2\sqrt{s}}\,\lambda^{1/2}(s,\sigma_{k},m_{k}^{2})\,.$
(8)

These relations follow from Eq. (3), where the inverse relations are readily
given

$\displaystyle\sigma_{k}=s+m_{k}^{2}-2\sqrt{s}\,\omega_{k}\,,\quad\sigma_{p}=s+m_{p}^{2}-2\sqrt{s}\,\omega_{p}\,,$
(9)

where recall that $\omega_{k}=\sqrt{m^{2}+k^{2}}$ and
$\omega_{p}=\sqrt{m_{p}^{2}+p^{2}}$. The angular degrees of freedom are not
fixed. Instead of specifying the angles of the spectators, it proves more
convenient to consider the angles of the pair momenta $P_{k}$ and $P_{p}$. To
define the momentum orientations, we introduce a _space-fixed_ coordinate
system denoted by $XYZ$. This coordinate system allows us to define our
reaction plane, and allows us to think about the pair-spectator scattering
system as a _quasi-two-body_ reaction. This quasi-two-body reaction is
depicted in Fig. 3, which for some fixed invariant masses $\sqrt{\sigma_{k}}$
and $\sqrt{\sigma_{p}}$ is specified by the total CM frame energy $\sqrt{s}$
and scattering angle between the spectators.

Figure 3: Orientations of the initial and final state pair momenta defined
with respect to the external space-fixed coordinate system (denoted by $XYZ$).
The angle between the initial and final momentum $\mathbf{P}_{k}$ and
$\mathbf{P}_{p}$, respectively, is the effective CM frame scattering angle
$\theta_{pk}$.

Without loss of generality, 777In App. C we lift this choice of coordinates
and illustrate the partial wave expansion with respect to a generic externally
fixed coordinate system. Although important in future analyses, as discussed
in App. C, we find that working in a generic coordinate system is not vital to
reach our results in this work. Therefore, we invite the interested reader to
view App. C. we define the initial pair momentum $P_{k}$ to be aligned with
the $+Z$-axis of some space-fixed coordinate system (with axes $XYZ$), that is
$\mathbf{\hat{Z}}\equiv\mathbf{\hat{P}}_{k}$. Since $\mathbf{P}=\mathbf{0}$,
the initial spectator momentum is then aligned with the $-Z$-axis. We then
choose the final pair momentum $\mathbf{P}_{p}$ to lie in the $XZ$-plane, i.e.
the quasi-two-body reaction lies in the reaction plane. This plane is defined
with the $+Y$-axis proportional to
$\mathbf{k}\times\mathbf{p}=\mathbf{P}_{k}\times\mathbf{P}_{p}$. We denote the
total CM frame _scattering angle_ by $\theta_{pk}$, which is defined in the
usual way

$\displaystyle\cos\theta_{pk}$
$\displaystyle\equiv\mathbf{\hat{P}}_{p}\cdot\mathbf{\hat{P}}_{k}=\mathbf{\hat{p}}\cdot\mathbf{\hat{k}}\,.$
(10)

Notice that $\theta_{pk}$ is simply the angle of $\mathbf{P}_{p}$ with respect
to the $Z$-axis, $\cos\theta_{pk}=\mathbf{P}_{p}\cdot\mathbf{\hat{Z}}$, with
respect to our space-fixed coordinate system. Since we use the standard
convention that $\theta_{pk}\in[0,\pi]$, this means that we can specify
$\theta_{pk}$ completely with just $\cos\theta_{pk}$.

Additionally, as mentioned in the previous reference frame definitions, the
reaction plane serves as a convenient reference for the azimuthal angles of
the initial and final three-body planes, $\psi_{k}$ and $\psi_{p}$,
respectively.

To conclude this section, we summarize the eight necessary kinematic variables
relevant to project the $\mathbf{3}\to\mathbf{3}$ system to definite partial
waves. For energy variables, we choose the total CM energy $s$, as well as the
initial and final pair invariant mass-squares $\sigma_{k}$ and $\sigma_{p}$,
respectively. An alternative to $\sigma_{k}$ and $\sigma_{p}$ is the
magnitudes of the associated spectator momenta $k$ and $p$. Through Eq. (8) at
a fixed $s$, these are completely interchangeable. We freely use either the
set $\sigma_{k},\sigma_{p}$ or $k,p$ where convenient, either for ease of
notation or exploiting some physical relation. The final five variables orient
our system, four of which are the initial and final pair polar and azimuthal
angles defined in their respective rest frames, $\mathbf{\hat{a}}_{k}^{\star}$
and $\mathbf{\hat{b}}_{p}^{\star}$, respectively. The last variable is the
total CM frame scattering angle $\theta_{pk}$. In the following section, we
construct partial wave $\mathbf{3}\to\mathbf{3}$ amplitudes by integrating
over the angular degrees of freedom with appropriate angular momentum weight
functions.

##  III Partial Wave Projection

Our first task is to define the generic partial wave projection for
$\mathbf{3}\to\mathbf{3}$ scattering amplitudes. The scheme we follow is
similar to that of Ref. Mikhasenko  _et al._ (2020), where we first couple the
three-particle system to a definite total angular momentum $J$ through the
helicity framework. Then, we re-couple the helicity partial wave to ones of
definite parity using spin-orbit or $LS$ coefficients. The reason for going
through this two-step process is that helicity transforms simply under Lorentz
transformations compared to spin-projections against some space-fixed
$z$-axis. Doing so makes the projection of the exchange amplitude simpler, as
the angles in the total CM frame are simply related to those in either pair
rest frame.

###  III.1 Helicity Projection

The starting point is to project the amplitude in _helicity partial waves_ ,
that is partial waves of definite total angular momentum where the pairs have
their spin projected quantized along their momentum direction. Following the
decomposition used in Refs. Hansen and Sharpe (2014, 2015); Jackura (2023), we
proceed by first partial wave projecting the pair into a definite angular
momentum state

$\displaystyle\ket{P,\mathbf{k},\mathbf{a}}\propto\sqrt{4\pi}\,\sum_{\ell,\lambda}\ket{P,\mathbf{k},\ell\lambda}\,Y_{\ell\lambda}^{*}(\mathbf{\hat{a}}_{k}^{\star})\,,$

where $\ell$ is the angular momentum of the pair, $\lambda$ is its projection
along the $z_{k}$-axis defined by the opposite sense of the spectator momentum
$k$ (see Fig. 4.), and
$Y_{\ell\lambda}(\mathbf{\hat{a}}_{k}^{\star})=Y_{\ell\lambda}(\chi_{k}^{\star},\psi_{k}^{\star})\equiv\braket{\mathbf{\hat{a}}_{k}^{\star}}{\ell\lambda}$
are the usual spherical harmonics. Since the spin-quantization axis is along
the direction of the pair, we interpret $\lambda$ as the pair helicity. 888In
the recent three-particle finite-volume frameworks for lattice QCD analyses,
the pair angular momentum projection $m$ usually has a quantization axis taken
to be some fixed $z$-axis of a volume. If one starts with this definition,
then converting to a helicity quantization with $\lambda$ amounts to a unitary
rotation of the pair state $\ket{\mathbf{P}_{k},\ell\lambda}$,
$\displaystyle\ket{\mathbf{P}_{k},\ell\lambda}=\sum_{m}D_{m\lambda}^{(\ell)}(\mathbf{\hat{\mathbf{P}}}_{k})\ket{\mathbf{P}_{k},\ell
m}\,,$ where $D_{m\lambda}^{(\ell)}$ are the Wigner $D$ matrix elements which
are discussed in App. A.

The normalization of the state is not relevant for our discussion, as we
freely absorb this factor into the definition of the amplitude. As we work
with scattering states of three scalars, only $\ell\in\mathbbm{N}_{0}$ is
allowed, with $\lambda\in\mathbbm{Z}$ which spans $-\ell\leq\lambda\leq\ell$
for a given $\ell$. For the scattering amplitude, we arrive at the expansion

$\displaystyle\mathcal{M}(\mathbf{p},\mathbf{b};\mathbf{k},\mathbf{a})=4\pi\sum_{\ell^{\prime},\lambda^{\prime}}\sum_{\ell,\lambda}Y_{\ell^{\prime}\lambda^{\prime}}(\mathbf{\hat{b}}_{p}^{\star})\,\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})\,Y_{\ell\lambda}^{*}(\mathbf{\hat{a}}_{k}^{\star})\,,$
(11)

where the factor of $4\pi$ is convention. Given the full amplitude, the
projection is found by using the orthonormality of the spherical harmonics
999In this work we make frequent use of identities of mathematical special
functions. For convenience, we collected a set of useful properties and
appropriate references in App. A.

$\displaystyle\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})=\frac{1}{4\pi}\int\\!\textrm{d}\mathbf{\hat{b}}_{p}^{\star}\int\\!\textrm{d}\mathbf{\hat{a}}_{k}^{\star}\,Y_{\ell^{\prime}\lambda^{\prime}}^{*}(\mathbf{\hat{b}}_{p}^{\star})\,\mathcal{M}(\mathbf{p},\mathbf{b};\mathbf{k},\mathbf{a})\,Y_{\ell\lambda}(\mathbf{\hat{a}}_{k}^{\star})\,,$
(12)

where the integration measure is
$\textrm{d}\mathbf{\hat{a}}_{k}^{\star}\equiv\textrm{d}\psi_{k}^{\star}\,\textrm{d}\cos\vartheta_{k}^{\star}$
with the integration domain being over $\vartheta_{k}^{\star}\in[0,\pi]$ and
$\psi_{k}^{\star}\in[0,2\pi]$.

Figure 4: Kinematic configuration and spin-projection definitions for the
initial (blue) and final (red) state pairs in their associated rest frame.
Each pair has its spin-projection along the opposite direction of their
associated spectators, giving $\lambda$ and $\lambda^{\prime}$ a helicity
interpretation.

It is useful to consider the amplitude
$\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}$ as one describing
the reaction of a spinless particle of mass $m_{k}$ and a _quasi-particle_ of
mass $\sqrt{\sigma_{k}}$, spin $\ell$, and helicity $\lambda$, which
transitions to a spinless particle of mass $m_{p}$ recoiling against another
_quasi-particle_ of mass $\sqrt{\sigma_{p}}$, spin $\ell^{\prime}$, and
helicity $\lambda^{\prime}$. We represent the quasi-two-body reaction as

$\displaystyle\xi_{k}^{(\ell)}(P_{k},\lambda)+\varphi_{k}(k)\to\xi_{p}^{(\ell^{\prime})}(P_{p},\lambda^{\prime})+\varphi_{p}(p)\,,$

where $\xi_{k}^{(\ell)}$ represents the quasi-particle of spin $\ell$. Note
that this effective $\mathbf{2}\to\mathbf{2}$ processes only knows about
particles $\varphi_{a}$, $\varphi_{a^{\prime}}$ and $\varphi_{b}$,
$\varphi_{b^{\prime}}$ through formation and decay, thereby only restricting
the threshold of the invariant mass. Thus the details of the kinematic
configurations for these particles are not relevant in the rest of this
construction. However, since the amplitude depends on the pair invariant
masses, it contains an angular momentum barrier suppression as the energies of
the pairs approach their threshold. For example, as the initial state pair
invariant mass-squared $\sigma_{k}\to\sigma_{k}^{\mathrm{(thr.)}}$, then

$\displaystyle\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})\sim\left(a_{k}^{\star}\right)^{\ell}\,,$

with a similar behavior for the final state,
$\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}\sim\left(b_{p}^{\star}\right)^{\ell^{\prime}}$
as $\sigma_{p}\to\sigma_{p}^{\mathrm{(thr.)}}$.

Once we have the effective helicity amplitude
$\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}$, we can now couple
the initial and final state to those of definite total angular momentum $J$
and projection $m_{J}$ defined with respect to the space-fixed $Z$-axis. The
quasi-two-body state has the helicity partial wave expansion

$\displaystyle\ket{P,\mathbf{k},\ell\lambda}\propto\sum_{J,m_{J}}\sqrt{2J+1}\,\ket{P,k,Jm_{J},\ell\lambda}\,D_{m_{J}\lambda}^{(J)}(\mathbf{\hat{P}}_{k})\,,$

where $D_{m_{J}\lambda}^{(J)}$ are Wigner $D$ matrix elements (cf. App. A.).
Again, we do not specify a normalization as we absorb this kinematic factor
into the definition of the amplitude. Since we chose the pair momentum $P_{k}$
to have its momentum along the $+z_{k}$-axis, the angles we consider are those
of this momentum, and not the spectator. We choose the phase convention of the
Wigner D matrix elements such that

$\displaystyle D_{m_{J}\lambda}^{(J)}(\mathbf{\hat{P}}_{k})$
$\displaystyle=D_{m_{J}\lambda}^{(J)}(\phi_{k},\theta_{k},0)\,,$
$\displaystyle=e^{-im_{J}\phi_{k}}\,d_{m_{J}\lambda}^{(J)}(\theta_{k})\,,$
(13)

where $\theta_{k}$, and $\phi_{k}$ are the polar and azimuthal angles of the
momentum $\mathbf{P}_{k}$, respectively, and
$d_{m_{J}\lambda}^{(J)}(\theta_{k})$ are the $d$ matrix elements which are
real for physical $\theta_{k}$, i.e. $-1\leq\cos\theta_{k}\leq+1$, see Eq.
113. Applying this basis expansion on both the initial and final states of Eq.
(12) yields

$\displaystyle\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})=\sum_{J^{\prime},m_{J^{\prime}}}\sum_{J,m_{J}}\sqrt{2J^{\prime}+1}\,D_{m_{J^{\prime}}\lambda^{\prime}}^{(J^{\prime})\,*}(\mathbf{\hat{P}}_{p})\,\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J^{\prime}m_{J^{\prime}},Jm_{J}}(p,k)\,\sqrt{2J+1}\,D_{m_{J}\lambda}^{(J)}(\mathbf{\hat{P}}_{k})\,.$
(14)

The normalization of Eq. (14) is chosen such that for spinless pairs,
$\ell^{\prime}=\ell=0$, then (14) simplifies and the resulting expression has
same normalization as Eq. (11). 101010See Eq. (122) in App. A for the
relations between the spherical harmonics and Wigner $D$ matrix elements.

Rotational invariance of the entire three-body system imposes that total
angular momentum $J$ is conserved, and the amplitude is independent of the
projection $m_{J}$,

$\displaystyle\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J^{\prime}m_{J^{\prime}},Jm_{J}}=\delta_{J^{\prime}J}\delta_{m_{J^{\prime}}m_{J}}\,\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}\,.$
(15)

Since the helicity partial wave amplitude is block diagonal in each $J$
sector, we can reduce the sums in the expansion to

$\displaystyle\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})$
$\displaystyle=\sum_{J=J_{\min}}^{\infty}(2J+1)\,\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}(p,k)\,\sum_{m_{J}=-J}^{J}D_{m_{J}\lambda^{\prime}}^{(J)\,*}(\mathbf{\hat{P}}_{p})\,D_{m_{J}\lambda}^{(J)}(\mathbf{\hat{P}}_{k})\,,$
(16)

where $J_{\min}=\max(\lvert\lambda^{\prime}\rvert,\lvert\lambda\rvert)$ is the
minimum $J$ value for the sum. The helicity partial wave amplitudes
$\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}$ depend on the
three kinematic variables, total CM energy $\sqrt{s}$, and the initial and
final spectator momenta $k$ and $p$, respectively (or alternatively the pair
invariant masses $\sqrt{\sigma_{k}}$ and $\sqrt{\sigma_{p}}$).

Recall from Sec. II.1 that with respect to our chosen coordinate system,
$\mathbf{\hat{P}}_{k}=\mathbf{\hat{Z}}$. Thus, for all $J$ the initial Wigner
$D$ matrix element simplifies to

$\displaystyle
D_{m_{J}\lambda}^{(J)}(\mathbf{\hat{Z}})=\delta_{m_{J}\lambda}\,,$

allowing us to trivially perform the sum to find

$\displaystyle\sum_{m_{J}=-J}^{J}D_{m_{J}\lambda^{\prime}}^{(J)\,*}(\mathbf{\hat{P}}_{p})\,D_{m_{J}\lambda}^{(J)}(\mathbf{\hat{P}}_{k})$
$\displaystyle=d_{\lambda\lambda^{\prime}}^{(J)}(\theta_{pk})\,,$ (17)

where we recall from Eq. (10) that the CM frame scattering angle is defined by
$\cos\theta_{pk}\equiv\mathbf{\hat{P}}_{p}\cdot\mathbf{\hat{P}}_{k}$, and
since the pair momenta lie in the $XZ$-plane, there is no azimuthal angular
dependence. Therefore, the helicity partial wave expansion is given by

$\displaystyle\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})=\sum_{J}(2J+1)\,\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}(p,k)\,d_{\lambda\lambda^{\prime}}^{(J)}(\theta_{pk})\,,$
(18)

with the projection given by

$\displaystyle\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}(p,k)=\frac{1}{2}\int_{-1}^{+1}\\!\textrm{d}\cos\theta_{pk}\,d_{\lambda\lambda^{\prime}}^{(J)}(\theta_{pk})\,\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})\,.$
(19)

The helicity partial wave amplitudes do not possess definite parity Jacob and
Wick (1959), and we must take appropriate linear combinations to recover
definite parity amplitudes. In the following section, we construct definite
parity amplitudes by connecting to the spin-orbit basis. 111111One could of
course define a partial wave projection directly into the spin-orbit basis
without going through the helicity basis first. However, since our goal is
partial wave projection the OPE contribution to the three-body amplitude, we
find it more convenient to first project it into the helicity basis, and then
form linear combinations of definite parity states. The reasoning is due to
the complicated angular dependence of the OPE function, and the helicity basis
allows us to easily define relations between the different reference frames
which impact the OPE definition, which will be detailed in Sec. IV.

###  III.2 Spin-Orbit Projection

Spin-orbit amplitudes are those of definite spatial parity. These amplitudes
are important to construct for the spectroscopy as hadrons appear as resonant
states of amplitudes, and hadrons have definite spin-parity $J^{P}$. Given the
helicity partial wave projections for the $\mathbf{3}\to\mathbf{3}$ amplitude
in Sec. III.1, we can easily construct amplitudes of definite parity by taking
appropriate linear combinations. We use the fact that the amplitudes
$\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}$ can be interpreted
as a quasi-two-body amplitude where one particle has a helicity. We can
therefore use standard techniques Jacob and Wick (1959) from the partial wave
projection of a two-body helicity state to spin-orbit state to obtain

$\displaystyle\ket{P,k,^{2S+1}\\!L_{J},m_{J},\ell}\propto\sum_{\lambda}\ket{P,k,Jm_{J},\ell\lambda}\,\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})\,,$

which when applied to our helicity amplitude yields

$\displaystyle\mathcal{M}_{L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)=\sum_{\lambda^{\prime},\lambda}\mathcal{P}_{\lambda^{\prime}}^{(\ell^{\prime})}(^{2S^{\prime}+1}L^{\prime}_{J})\,\mathcal{M}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}(p,k)\,\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})\,.$
(20)

Here $S$ is the total intrinsic spin of the pair-spectator system, $L$ is the
orbital angular momentum between an initial state pair and its spectator in
their total CM frame, and $S^{\prime}$ and $L^{\prime}$ are similarly defined
for the final state. The total angular momentum $J$ of the three-body system
therefore has values $\lvert L-S\rvert\leq J\leq L+S$ and $\lvert
L^{\prime}-S^{\prime}\rvert\leq J\leq L^{\prime}+S^{\prime}$. The parity of
the three-particle state with a total angular momentum $J$ is

$\displaystyle P=\eta(-1)^{S+L}=\eta(-1)^{S^{\prime}+L^{\prime}}\,,$

where $\eta$ is the product of the intrinsic parities of the three particles,
e.g. for three pseudoscalar pions the product of intrinsic parities is
$\eta=(-1)^{3}=-1$. Since the strong interaction conserves parity, only
transitions where $S+L$ and $S^{\prime}+L^{\prime}$ are both even or both odd
are allowed.

To couple the helicity basis to the spin-orbit basis, we have introduced the
_spin-orbit coupling coefficients_ $\mathcal{P}_{\lambda}^{(\ell)}$, which are
defined in terms of Clebsch-Gordan coefficients as

$\displaystyle\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})=\sqrt{\frac{2L+1}{2J+1}}\,\braket{J\lambda}{L0,S\lambda}\,\delta_{\ell
S}\,.$ (21)

The Kronecker delta enforces that the total spin is equal to that of the pair,
$S=\ell$, as expected for our three spinless particles. 121212In anticipation
of extensions for external particles with spin we define the spin-orbit
coupling coefficients with the redundant $\ell=S$, which in the case for
particles with spin the Kronecker delta will be replaced with an additional
Clebsch-Gordan coefficient which couples the pair and spectator spins to total
$S$. From the completeness relation of the Clebsch-Gordan coefficients, one
immediately sees that the spin-orbit couplings satisfy

$\displaystyle\sum_{\lambda}\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L^{\prime}_{J})\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})=\delta_{L^{\prime}L}\,.$
(22)

The spin-orbit amplitudes describe the transition
${}^{2S+1}L_{J}\,\to\,^{2S^{\prime}+1}L^{\prime}_{J}$ for the quasi-two-body
reaction
$\xi_{k}^{(\ell)}+\varphi_{k}\to\xi_{p}^{(\ell^{\prime})}+\varphi_{p}$.
Therefore, for fixed $\sigma_{k}$ and $\sigma_{p}$, the amplitudes have the
usual threshold behavior from orbital angular momentum barrier suppression,

$\displaystyle\mathcal{M}_{L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)\sim
p^{L^{\prime}}k^{L}\,,$

as $p,k\to 0$ for fixed $\sigma_{k},\sigma_{p}$.

As an example of the form of spin-orbit coupling coefficients, let us consider
a system of three pions. For a pair of pions in relative $S$ wave, then
$\ell=0$ and the only allowed $L$ is $L=J$. So, the spin-orbit coefficient is
simply

$\displaystyle\mathcal{P}_{\lambda}^{(0)}(^{1}J_{J})=\delta_{\lambda 0}\,.$
(23)

If the pair of pions is in an angular momentum state $\ell=1$, i.e. the
resonant $P$ wave channel, then the pair-spectator system is then a triple
state with $S=1$. For some target total angular momentum $J$, the allowed
orbital angular momenta are $L=J-1,J,J+1$. Therefore, the spin-orbit
coefficients can be simplified to the form

$\displaystyle\mathcal{P}_{\lambda}^{(1)}(^{3}L_{J})=\begin{cases}\,\sqrt{\dfrac{J+1}{2(2J+1)}}\,\lvert\lambda\rvert+\sqrt{\dfrac{J}{2J+1}}\,\delta_{\lambda,0}\,,&L=J-1\,,\\\\[20.0pt]
\,-\dfrac{\lambda}{\sqrt{2}}\,,&L=J\,,\\\\[20.0pt]
\,\sqrt{\dfrac{J}{2(2J+1)}}\,\lvert\lambda\rvert-\sqrt{\dfrac{J+1}{2J+1}}\,\delta_{\lambda,0}\,,&L=J+1\,.\end{cases}$
(24)

Since the pion is a pseudoscalar, the product of the intrinsic parities is
$\eta=-1$, and therefore the total parity of the system is $P=(-1)^{L}$. For a
target $J^{P}=1^{+}$ and the initial and final pairs both being vectors
$\ell=\ell^{\prime}=1$, then only $S$ and $D$ waves contribute giving an two-
dimensional amplitude with ${}^{3}S_{1}$ and ${}^{3}D_{1}$.

##  IV One Particle Exchange Amplitude

We construct analytic representations of the $\mathbf{3}\to\mathbf{3}$
scattering amplitude by enforcing $S$ matrix unitarity on Eq. (2). One can
show that a driving kinematic singularity of the amplitude is due to the
exchange of an on-shell particle with mass $m_{e}$ and momentum $P-k-p$
between two-body sub-processes Jackura (2023). The imaginary part of the
$\mathbf{3}\to\mathbf{3}$ amplitude at this kinematic point, specifically for
the $\mathbf{k}$ and $\mathbf{p}$ spectators, is

$\displaystyle\operatorname{Im}\,\mathcal{M}(\mathbf{p},\mathbf{b};\mathbf{k},\mathbf{a})\supset\mathcal{M}_{2}^{*}(\sigma_{p};\mathbf{\hat{b}}_{p}^{\star},\mathbf{\hat{k}}_{p}^{\star})\,\pi\,\delta(u_{pk}-m_{e}^{2})\,\mathcal{M}_{2}(\sigma_{k};\mathbf{\hat{p}}_{k}^{\star},\mathbf{\hat{a}}_{k}^{\star})\,,$

where the angles $\mathbf{\hat{k}}_{p}^{\star}$ and
$\mathbf{\hat{p}}_{k}^{\star}$ correspond to the orientations of the spectator
in the rest frame of the _opposite_ pair indicated, e.g.
$\mathbf{\hat{k}}_{p}^{\star}$ is the unit vector of the initial spectator
defined in the final pair rest frame defined by its spectator $\mathbf{p}$.
131313Since the OPE involves pair-spectator systems in both its external and
intermediate states, the thresholds for the pair invariant masses extend to
the cases
$\sigma_{k}^{(\mathrm{thr.})}=\max(m_{a}+m_{a^{\prime}},m_{p}+m_{e})$ and
$\sigma_{p}^{(\mathrm{thr.})}=\max(m_{b}+m_{b^{\prime}},m_{k}+m_{e})$ for the
initial and final pair, respectively.  We also defined

$\displaystyle u_{pk}\equiv(P-k-p)^{2}\,,$

as the momentum-squared of the exchanged particle. We focus only on the $k$
and $p$ spectators here, but note that other spectator combinations will
result in similar contributions to the imaginary part.

Figure 5: The OPE contribution to the on-shell $\mathbf{3}\to\mathbf{3}$
amplitude $i\mathcal{M}$ with momentum assignments. The dashed line indicates
that we have removed short-distance contributions of the exchange propagator.
The $\mathbf{2}\to\mathbf{2}$ subprocesses
$i\,\,\overline{\\!\\!{\mathcal{M}}}_{2}$ are denoted by the grey-filled
circles on either side of the exchange propagator.

The aforementioned pole singularity of the $\mathbf{3}\to\mathbf{3}$ amplitude
is encoded in the OPE. Defining the momenta of the initial and final
spectators respectively to be $\mathbf{k}$ and $\mathbf{p}$, the OPE, depicted
diagrammatically in Figure 5, can in general be written as 141414Equation (25)
can be argued by constructing on-shell representations through either $S$
matrix unitarity Mai  _et al._ (2017); Jackura  _et al._ (2019); Jackura
(2023) or summing Feynman graphs to all-orders within some generalized
effective field theory and projecting intermediate states on their mass-shell
Hansen and Sharpe (2014, 2015). As with all on-shell representations, the OPE
is defined up to some real part in the physical region which is absorbed into
the global $K$ matrix which desribes short-distance three-body dynamics. For
example, in the resulting integral equations of the aforementioned references,
one usually includes a cutoff function to render the momentum integrals UV
finite. Since our focus here is on the partial wave projection of the
function, we omit the cutoff function for convenience.

$\displaystyle i\mathcal{M}(\mathbf{p},\mathbf{b};\mathbf{k},\mathbf{a})$
$\displaystyle\supset
i\,\,\overline{\\!\\!{\mathcal{M}}}_{2}(\sigma_{p};\mathbf{\hat{b}}_{p}^{\star},\mathbf{\hat{k}}_{p}^{\star})\,\frac{i}{u_{pk}-m_{e}^{2}+i\epsilon}\,i\,\,\overline{\\!\\!{\mathcal{M}}}_{2}(\sigma_{k};\mathbf{\hat{p}}_{k}^{\star},\mathbf{\hat{a}}_{k}^{\star})\equiv
i\mathcal{M}_{\mathrm{OPE}}(\mathbf{p},\mathbf{b};\mathbf{k},\mathbf{a})\,.$
(25)

On either side of the exchange propagator is a modified
$\mathbf{2}\to\mathbf{2}$ amplitude, $\,\,\overline{\\!\\!{\mathcal{M}}}_{2}$.
The modification chosen, which is not unique, assures that
$\,\,\overline{\\!\\!{\mathcal{M}}}_{2}$ agrees with ${\mathcal{M}}_{2}$ in
the limit that the exchanged particle goes on shell, while assuring that no
unphysical kinematic singularities are introduced. Explicitly, these
amplitudes are defined through the following angular momentum expansion,

$\displaystyle\,\,\overline{\\!\\!{\mathcal{M}}}_{2}(\sigma_{k};\mathbf{\hat{p}}_{k}^{\star},\mathbf{\hat{a}}_{k}^{\star})=4\pi\sum_{\ell^{\prime},\lambda^{\prime}}\sum_{\ell,\lambda}\left(\frac{p_{k}^{\star}}{q_{k}^{\star}}\right)^{\ell^{\prime}}\,Y_{\ell^{\prime}\lambda^{\prime}}(\mathbf{\hat{p}}_{k}^{\star})\,\mathcal{M}_{2,\ell^{\prime}\lambda^{\prime};\ell\lambda}(\sigma_{k})\,Y_{\ell\lambda}^{*}(\mathbf{\hat{a}}_{k}^{\star})\,,$
(26a)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{M}}}_{2}(\sigma_{p};\mathbf{\hat{b}}_{k}^{\star},\mathbf{\hat{k}}_{p}^{\star})=4\pi\sum_{\ell^{\prime},\lambda^{\prime}}\sum_{\ell,\lambda}Y_{\ell^{\prime}\lambda^{\prime}}(\mathbf{\hat{b}}_{p}^{\star})\,\mathcal{M}_{2,\ell^{\prime}\lambda^{\prime};\ell\lambda}(\sigma_{p})\,\left(\frac{k_{p}^{\star}}{q_{p}^{\star}}\right)^{\ell}\,Y_{\ell\lambda}^{*}(\mathbf{\hat{k}}_{p}^{\star})\,.$
(26b)

Angular momentum barrier factors are included to suppress the kinematic
divergence induced by the spherical harmonics as $p_{k}^{\star}$ and
$k_{p}^{\star}$ go to zero in their respective amplitudes. These momenta are
defined in the CM frame of the pair of the _opposite_ spectator, specifically
one can show

$\displaystyle
p_{k}^{\star}=\frac{1}{2\sqrt{\sigma_{k}}}\,\lambda^{1/2}(\sigma_{k},m_{p}^{2},u_{pk}),\qquad
k_{p}^{\star}=\frac{1}{2\sqrt{\sigma_{p}}}\,\lambda^{1/2}(\sigma_{p},m_{k}^{2},u_{pk})\,.$
(27)

The barrier factors are chosen to be unity at the on-shell point
$u_{pk}=m_{e}^{2}$, where we define the momenta

$\displaystyle q_{k}^{\star}$ $\displaystyle\equiv
p_{k}^{\star}\,\Big{\rvert}_{u_{pk}=m_{e}^{2}}=\frac{1}{2\sqrt{\sigma_{k}}}\,\lambda^{1/2}(\sigma_{k},m_{p}^{2},m_{e}^{2})\,,$
$\displaystyle q_{p}^{\star}$ $\displaystyle\equiv
k_{p}^{\star}\,\Big{\rvert}_{u_{pk}=m_{e}^{2}}=\frac{1}{2\sqrt{\sigma_{p}}}\,\lambda^{1/2}(\sigma_{p},m_{k}^{2},m_{e}^{2})\,,$
(28)

Finally note that rotational invariance of the two-body subsystems diagonalize
their respective $\mathbf{2}\to\mathbf{2}$ partial wave amplitude
$\mathcal{M}_{2,\ell^{\prime}\lambda^{\prime};\ell\lambda}(\sigma)=\delta_{\ell^{\prime}\ell}\,\delta_{\lambda^{\prime}\lambda}\,\mathcal{M}_{2,\ell}(\sigma)$.

###  IV.1 Exchange Propagator

Given the OPE amplitude defined in Eq. (25), we manipulate it to be amenable
for an analytic partial wave projection to total angular momentum $J$. This
means isolating the dependence on the total scattering angle $\theta_{pk}$.
Using the on-shell representation defined in Eq. (25) with Eqs. (26a) and
(26b), we write the OPE amplitude as

$\displaystyle
i\mathcal{M}_{\mathrm{OPE}}=4\pi\sum_{\ell^{\prime},\lambda^{\prime}}\sum_{\ell,\lambda}Y_{\ell^{\prime}\lambda^{\prime}}(\mathbf{\hat{b}}_{p}^{\star})\,i\mathcal{M}_{\ell^{\prime}}(\sigma_{p})\,i\mathcal{G}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})\,i\mathcal{M}_{\ell}(\sigma_{k})\,Y_{\ell\lambda}^{*}(\mathbf{\hat{a}}_{k}^{\star})\,,$
(29)

which in effect performs the first partial wave expansion on the initial and
final state pairs as given in Eq. (11). Here we define the kinematic exchange
propagator $\mathcal{G}$ as

$\displaystyle\mathcal{G}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})\equiv\frac{\mathcal{H}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(\mathbf{p},\mathbf{k})}{u_{pk}-m_{e}^{2}+i\epsilon}\,\,,$
(30)

where the $\epsilon\to 0^{+}$ limit is understood, and $\mathcal{H}$ is the
spin-dependent numerator 151515We emphasize that we use the regular spherical
harmonics as opposed to the _real_ harmonics originally used in the original
derivation using the finite-volume framework Hansen and Sharpe (2014, 2015),
which are simply unitary transformations of the regular spherical harmonics,
$\mathbf{Y}_{\ell}=\mathbf{U}_{\ell}\cdot\mathbf{S}_{\ell}$ with
$\mathbf{S}_{\ell}$ denoting the real spherical harmonics of degree $\ell$.
which we define as the _spin-helicity matrix_ ,

$\displaystyle\mathcal{H}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(\mathbf{p},\mathbf{k})\equiv\left(\frac{k_{p}^{\star}}{q_{p}^{\star}}\right)^{\ell^{\prime}}4\pi\,Y_{\ell^{\prime}\lambda^{\prime}}^{*}(\mathbf{\hat{\mathbf{k}}}_{p}^{\star})Y_{\ell\lambda}(\mathbf{\hat{\mathbf{p}}}_{k}^{\star})\left(\frac{p_{k}^{\star}}{q_{k}^{\star}}\right)^{\ell}\,.$
(31)

From the properties of the spherical harmonics, the spin-helicity matrix,
obeys the reflection property

$\displaystyle\mathcal{H}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(\mathbf{p},\mathbf{k})$
$\displaystyle=(-1)^{\lambda^{\prime}+\lambda}\,\mathcal{H}_{-\lambda^{\prime}-\lambda}^{(\ell^{\prime}\ell)\,*}(\mathbf{p},\mathbf{k})\,.$
(32)

In order to analytically perform the partial wave projection, we manipulate
the exchange propagator (30) into a form to make explicit the dependence of
the angular variable $\theta_{pk}$. We therefore need to express Eqs. (30) and
(31) with respect to our reaction plane defined in the space-fixed coordinate
system illustrated in Fig. 3. For convenience we define $z_{pk}$ as the cosine
of the scattering angle,

$\displaystyle z_{pk}\equiv\cos\theta_{pk}\,,$

and work with $z_{pk}$. Upon inspection of the propagator of Eq. (30), we find
that the $z_{pk}$-dependence will reside in the pole term through
$u_{pk}=(P-k-p)^{2}$, and through the arguments of the spherical harmonics
which are related to the scattering angle by Lorentz transformations. The
dependence on $z_{pk}$ leads to singular behavior in both when the propagator
goes on the mass shell and through kinematic factors associated with the spin
of the pairs. In the following, we derive a generic form for the OPE which
identifies the angular dependence including the isolation of the singular
behavior of the function on $z_{pk}$.

The OPE is a $u$-channel process in the effective
$\xi_{k}^{(\ell)}(P_{k},\lambda)+\varphi_{k}(k)\to\xi_{p}^{(\ell^{\prime})}(P_{p},\lambda^{\prime})+\varphi_{p}(p)$
reaction. The invariant momentum transfer is related to the cosine of the
scattering angle in the usual way,

$\displaystyle u_{pk}$ $\displaystyle=(P-k-p)^{2}\,,$
$\displaystyle=\sigma_{k}+m_{p}^{2}-2E_{k}\omega_{p}-2pk\,z_{pk}\,,$
$\displaystyle\equiv u_{pk}^{(0)}-2pk(1+z_{pk})\,,$ (33)

where $E_{k}=\sqrt{s}-\omega_{k}$, and
$u_{pk}^{(0)}\equiv\sigma_{k}+m_{p}^{2}-2E_{k}\omega_{p}+2pk$ is the backward
limit ($z_{pk}=-1$) of $u_{pk}$. The value of $z_{pk}$ at the on-shell point,
$u_{pk}=m_{e}^{2}$, is given a special notation,

$\displaystyle\zeta_{pk}$ $\displaystyle\equiv
z_{pk}\,\Big{\rvert}_{u_{pk}=m_{e}^{2}}\,,$
$\displaystyle=-1-\frac{m_{e}^{2}-u^{(0)}_{pk}}{2pk}\,$
$\displaystyle=\frac{2s(\sigma_{k}+m_{p}^{2}-m_{e}^{2})-(s+\sigma_{k}-m_{k}^{2})(s+m_{p}^{2}-\sigma_{p})}{\lambda^{1/2}(s,\sigma_{k},m_{k}^{2})\lambda^{1/2}(s,\sigma_{p},m_{p}^{2})}\,,$
(34)

where we have used, along with $k$ and $p$ defined in Eq. (8), the relations

$\displaystyle
E_{k}=\sqrt{s}-\omega_{k}=\frac{1}{2\sqrt{s}}(s+\sigma_{k}-m_{k}^{2}),\qquad\omega_{p}=\frac{1}{2\sqrt{s}}(s+m_{p}^{2}-\sigma_{p}),$
(35)

which follow from the definition of $s$ in the total CM frame. Note that we
have not explicitly written the $+i\epsilon$ shift which avoids the pole.
However, one can include this shift by either substituting $m_{e}^{2}\to
m_{e}^{2}-i\epsilon$ or $\zeta_{pk}\to\zeta_{pk}+i\epsilon$. The OPE pole of
Eq. (30) in terms of $z_{pk}$ is then
$u_{pk}-m_{e}^{2}=2pk(\zeta_{pk}-z_{pk})$.

For the $z_{pk}$-dependence in the spin-helicity matrix, Eq. (31), we make use
of the Lorentz transformations between the total CM frame
($\mathbf{P}=\mathbf{0}$) where $\theta_{pk}$ is defined, and the pair CM
frames where the orientations of $\mathbf{k}_{p}^{\star}$ and
$\mathbf{p}_{k}^{\star}$ are defined. These Lorentz transformations are
illustrated in Figs. 6 and 7 for the initial pair and final pair rest frames,
respectively. Recall that in the reaction plane, i.e. the $XZ$-plane, the pair
momenta have zero azimuthal angle in the CM frame.

Focusing first on the initial pair rest frame where
$\mathbf{P}_{k,k}^{\star}=(\mathbf{P}-\mathbf{k})_{k}^{\star}=\mathbf{0}$, as
illustrated in Fig. 6 (a), we define $\chi_{k}^{\star}$ as the polar angle of
$\mathbf{p}_{k}^{\star}$ in the initial pair rest frame,
$\cos\chi_{k}^{\star}=\mathbf{p}_{k}^{\star}\cdot\mathbf{\hat{z}}_{k}=-\mathbf{p}_{k}^{\star}\cdot\mathbf{k}_{k}^{\star}$.
With respect to the external coordinate system, the azimuthal angle of
$\mathbf{p}_{k}^{\star}$ is $\pi$ since the vector is oriented with respect to
the negative $x_{k}$-axis. Therefore, the orientation of
$\mathbf{p}_{k}^{\star}$ is given by the angles $(\chi_{k}^{\star},\pi)$. In
the final pair rest frame
$\mathbf{P}_{p,p}^{\star}=(\mathbf{P}-\mathbf{p})_{p}^{\star}=\mathbf{0}$ as
shown in Fig. 7 (a), the polar angle of $\mathbf{k}_{p}^{\star}$ is
$\chi_{p}^{\star}$ which is defined as
$\cos\chi_{p}^{\star}=\mathbf{k}_{p}^{\star}\cdot\mathbf{\hat{z}}_{p}=-\mathbf{k}_{p}^{\star}\cdot\mathbf{p}_{p}^{\star}$.
With respect to our coordinate system, the azimuthal angle is zero, thus the
polar and azimuthal angles of $\mathbf{k}_{p}^{\star}$ in this frame are
$(\chi_{p}^{\star},0)$. Therefore, the angular dependence of the spin-helicity
matrix is of the form

$\displaystyle\mathcal{H}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(\mathbf{p},\mathbf{k})$
$\displaystyle\propto
Y_{\ell^{\prime}\lambda^{\prime}}^{*}(\chi_{p}^{\star},0)Y_{\ell\lambda}(\chi_{k}^{\star},\pi)\,,$
$\displaystyle=(-1)^{\lambda}\,Y_{\ell^{\prime}\lambda^{\prime}}(\chi_{p}^{\star},0)Y_{\ell\lambda}(\chi_{k}^{\star},0)\,.$
(36)

Figure 6: Kinematics of the OPE in (a) the initial state pair rest frame
($\mathbf{P}_{k,k}^{\star}=\mathbf{0}$) and (b) the total CM frame
($\mathbf{P}=\mathbf{0}$). The boost velocity from the
$\mathbf{P}_{k,k}^{\star}=\mathbf{0}$ frame to the $\mathbf{P}=\mathbf{0}$
frame is
$\boldsymbol{\beta}_{k}=\mathbf{P}_{k}/E_{k}=(\mathbf{P}-\mathbf{k})/(\sqrt{s}-\omega_{k})$.

The relation between the $\chi_{k}^{\star}$ and $\chi_{p}^{\star}$ and
$z_{pk}$ is found by Lorentz boosting between the frames in which these angles
are defined and the total CM frame. The Lorentz boost along the $z_{k}$ axis
from the initial pair rest frame to the total CM frame, cf. Fig. 6 (b), yields
the relation

$\displaystyle p_{k}^{\star}\cos\chi_{k}^{\star}$
$\displaystyle=\gamma_{k}\,\left[\,p\cos(\pi-\theta_{pk})-\omega_{p}\beta_{k}\,\right]\,,$
(37a) $\displaystyle p_{k}^{\star}\sin\chi_{k}^{\star}$
$\displaystyle=p\sin(\pi-\theta_{pk})\,,$ (37b)

where $\beta_{k}=\lvert\mathbf{P}_{k}\rvert/E_{k}=k/(\sqrt{s}-\omega_{k})$ is
the magnitude of the boost velocity and
$\gamma_{k}=(1-\beta_{k}^{2})^{-1/2}=(\sqrt{s}-\omega_{k})/\sqrt{\sigma_{k}}$.
Note that $\cos(\pi-\theta_{pk})=-\cos\theta_{pk}=-z_{pk}$ and
$\sin(\pi-\theta_{pk})=\sin\theta_{pk}=\sqrt{1-z_{pk}^{2}}$. A similar
analysis for the initial state spectator momentum in the final state pair rest
frame, $\mathbf{k}_{p}^{\star}$, yields the transformation (cf. Fig. 7)

$\displaystyle k_{p}^{\star}\cos\chi_{p}^{\star}$
$\displaystyle=\gamma_{p}\,\left[\,k\cos(\pi-\theta_{pk})-\omega_{k}\beta_{p}\,\right]\,,$
(38a) $\displaystyle k_{p}^{\star}\sin\chi_{p}^{\star}$
$\displaystyle=k\sin(\pi-\theta_{pk})\,,$ (38b)

where
$\beta_{p}=\lvert\mathbf{P}_{\mathbf{p}}\rvert/E_{p}=p/(\sqrt{s}-\omega_{p})$,
$\gamma_{p}=(1-\beta_{p}^{2})^{-1/2}=(\sqrt{s}-\omega_{p})/\sqrt{\sigma_{p}}$.
From Eqs. (37) and (38), we see that the spherical harmonics contain
dependencies on $s$, $p$, $k$, as well as $z_{pk}$.

Figure 7: Similar to Fig. 6 but for the (a) final state pair frame
($\mathbf{P}_{p,p}^{\star}=\mathbf{0}$) where the boost to the (b) total CM
frame is given by
$\boldsymbol{\beta}_{p}=\mathbf{P}_{p}/E_{p}=(\mathbf{P}-\mathbf{p})/(\sqrt{s}-\omega_{p})$.

The analytic structure of the $z_{pk}$-dependence of $\mathcal{H}$ can be
understood by its partial wave expansion. Since the helicity dependence of the
OPE is entirely contained in $\mathcal{H}$, it admits a partial wave expansion
similar to the full helicity amplitude as given in Eq (18). We write its
expansion as 161616In a slight abuse of notation, we express
$d_{\lambda\lambda^{\prime}}^{(J)}(\theta_{pk})$ as a function of
$z_{pk}=\cos\theta_{pk}$, $d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})$, so that
we can write everything as a function of $z_{pk}$.

$\displaystyle\mathcal{H}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(\mathbf{p},\mathbf{k})=\sum_{J}(2J+1)\,\mathcal{H}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell),J}(p,k)\,d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})$
(39)

In the complex $z_{pk}$-plane, the spin-helicity function contains
singularities associated with the Wigner $d$ functions as seen by Eq. (39). By
definition, cf. App. A and references therein, the $z_{pk}$ dependence in the
Wigner $d$ matrix elements is of the form

$\displaystyle
d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})\propto\xi_{\lambda\lambda^{\prime}}(z_{pk})\,P_{\rho}^{(\mu,\nu)}(z_{pk})\,,$
(40)

where $\xi_{\lambda\lambda^{\prime}}$ is called the _half-angle factor_ and
$P_{\rho}^{(\mu,\nu)}$ are the Jacobi polynomials with
$\mu=\lvert\lambda-\lambda^{\prime}\rvert$,
$\nu=\lvert\lambda+\lambda^{\prime}\rvert$, and
$\rho=J-(\mu+\nu)/2=J-J_{\min}$. The Jacobi polynomials
$P_{\rho}^{(\mu,\nu)}(z_{pk})$ are regular functions of $z_{pk}$ for all
indices $\mu$, $\nu$, and $\rho$, while the half-angle factor, defined by

$\displaystyle\xi_{\lambda\lambda^{\prime}}(z_{pk})=\left(\frac{1-z_{pk}}{2}\right)^{\frac{\lvert\lambda-\lambda^{\prime}\rvert}{2}}\left(\frac{1+z_{pk}}{2}\right)^{\frac{\lvert\lambda+\lambda^{\prime}\rvert}{2}}\,,$
(41)

contains potentially square-root singularities depending on the system of
helicities. In order to obtain an analytic representation for the OPE, we
isolate the singular dependencies in $z_{pk}$ as they will impact the partial
wave projection. Therefore, we conclude that the spin-helicity function for
any $\ell^{\prime},\ell$ has the generic structure, 171717This behavior has
been known from the scattering of two spinning particles, see for example Ref.
Martin and Spearman (1970) and references therein.

$\displaystyle\mathcal{H}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(\mathbf{p},\mathbf{k})\equiv\xi_{\lambda\lambda^{\prime}}(z_{pk})\,\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})\,,$
(42)

where $\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}$ is defined
to be a regular function in $z_{pk}$ for the physical kinematics. For example,
consider the scattering with $\ell^{\prime}=\ell=1$, with the helicities
$\lambda^{\prime}=+1$, $\lambda=1$. Then, as detailed in Sec. V, the spin-
helicity function behaves like
$\mathcal{H}_{+10}^{(11)}=\sin\chi_{p}^{\star}\cos\chi_{k}^{\star}$. This
function is non-analytic in $z_{pk}$ since
$\sin\chi_{p}^{\star}\propto\sin\theta_{pk}=\sqrt{1-z_{pk}^{2}}$ from the
Lorentz transformations Eq. (38). However,
$\xi_{01}(z_{pk})\propto\sqrt{1-z_{pk}^{2}}$, thus the spin-helicity function
factorizes into the non-analytic half-angle factor and a regular function in
$z_{pk}$.

It is further useful to define the $\mathcal{A}$ coefficient at the on-shell
point $z_{pk}=\zeta_{pk}$,

$\displaystyle\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\equiv\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})\,\Big{\rvert}_{z_{pk}=\zeta_{pk}}\,,$
(43)

which allows us to express the function as a term at the propagator pole and a
term which is the difference of the pole and non-pole term.

$\displaystyle\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})$
$\displaystyle=\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)+\left[\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})-\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\right]\,,$
(44)

Near the pole, the difference vanishes as $(\zeta_{pk}-z_{pk})$, thus it is
convenient to define a new function which is regular near this pole,

$\displaystyle\mathcal{B}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})\equiv\frac{1}{\zeta_{pk}-z_{pk}}\,\left[\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})-\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\right]\,.$
(45)

The introduction of the $\mathcal{B}$ coefficient allows us to completely
isolate the analytic behavior of the OPE in $z_{pk}$ into the generic form,

$\displaystyle\mathcal{G}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})=\frac{1}{2pk}\xi_{\lambda^{\prime}\lambda}(z_{pk})\,\left[\,\frac{\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)}{\zeta_{pk}-z_{pk}}+\mathcal{B}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})\,\right]\,.$
(46)

As discussed in the beginning of this section, the singular behavior of
$\mathcal{G}$ in the variable $z_{pk}=\cos\theta_{pk}$ comes from two
locations. First there is the overall half-angle factor, hidden in the spin-
helicity function, which exhibits kinematic singularities due to the spin of
the initial and final state pairs. Second, the OPE is singular where the
exchange particle goes on its mass-shell, which is encoded in the pole. The
remaining $z_{pk}$ behavior is analytic in the physical region we consider.

The $\mathcal{B}$ coefficients are constructed to be regular functions of
$z_{pk}$ on the interval $-1\leq z_{pk}\leq+1$, thus for a fixed
$\lambda^{\prime}$ and $\lambda$ we can freely expand it into Legendre
polynomials as

$\displaystyle\mathcal{B}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})=\sum_{j=0}^{\infty}(2j+1)\,\mathcal{B}_{j,\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,P_{j}(z_{pk})\,,$
(47)

where by the orthogonality of the Legendre polynomials we can obtain the
projected coefficients

$\displaystyle\mathcal{B}_{j,\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)=\frac{1}{2}\int_{-1}^{+1}\\!\textrm{d}z_{pk}\,P_{j}(z_{pk})\,\mathcal{B}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})\,.$
(48)

We stress here that although Eq. (47) is an expansion involving an infinite
number of terms, for a fixed $\ell^{\prime}$ and $\ell$ only a finite number
of projected coefficients Eq. (48) will exist for some target total angular
momentum $J$. In Sec. V, we give explicit examples of these coefficients for
$\ell^{\prime},\ell=\\{0,1\\}$.

Since the OPE is a known function, we can easily tabulate the $\mathcal{A}$
and $\mathcal{B}$ coefficients for the particular scattering channels of
interest by the procedure outlined above. While at first, this may seem like
extra computational steps given that $\mathcal{G}$ has a known form, we find
that this decomposition allows us to write down a generic analytic
representation for the definite-parity partial wave amplitudes of the OPE. In
doing so, we arrive at an exact result that isolates all the known singular
structures of the OPE in the $(\sigma_{p},s,\sigma_{k})$ variables, and a set
of coefficients which are determined from the identified $\mathcal{A}$ and
$\mathcal{B}$ coefficients.

###  IV.2 Partial Wave Projected Exchange Propagator

The OPE as given in Eq. (46) allows for explicit analytic partial wave
projection. First we project Eq. (46) into the helicity basis using Eq. (19),

$\displaystyle\mathcal{G}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}(p,k)$
$\displaystyle\equiv\frac{1}{2}\int_{-1}^{+1}\\!\textrm{d}\cos\theta_{pk}\,d_{\lambda\lambda^{\prime}}^{(J)}(\theta_{pk})\,\mathcal{G}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}(\mathbf{p},\mathbf{k})\,,$
$\displaystyle=\frac{1}{4pk}\,\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,\int_{-1}^{+1}\\!\textrm{d}z_{pk}\,\frac{\xi_{\lambda\lambda^{\prime}}(z_{pk})}{\zeta_{pk}-z_{pk}}\,d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})\,$
$\displaystyle\qquad+\frac{1}{4pk}\,\int_{-1}^{+1}\\!\textrm{d}z_{pk}\,\xi_{\lambda\lambda^{\prime}}(z_{pk})\,d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})\,\mathcal{B}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k,z_{pk})\,.$
(49)

The integral in the first term is entirely in terms of known functions
independent of spectator momenta, whereas the second integral involves the
$\mathcal{B}$ coefficient which depends on the momenta chosen pairs. By using
the expansion (47), we can remove the momentum dependence leaving an integral
over functions which depend solely on $z_{pk}$ for the second term of Eq.
(IV.2),

$\displaystyle(\mathrm{2\textsuperscript{nd}\,\,term})=\frac{1}{4pk}\sum_{j}(2j+1)\mathcal{B}_{j,\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,\int_{-1}^{+1}\\!\textrm{d}z_{pk}\,\xi_{\lambda\lambda^{\prime}}(z_{pk})\,d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})\,P_{j}(z_{pk})\,.$
(50)

Both of the resulting integrals can be computed analytically by recognizing
that the product the product
$\xi_{\lambda\lambda^{\prime}}(z_{pk})\,d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})$
is a regular function of $z_{pk}$ for any $J$, $\lambda^{\prime}$, and
$\lambda$ since
$d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})\propto\xi_{\lambda\lambda^{\prime}}(z_{pk})\,P^{\lvert\lambda-\lambda^{\prime}\rvert,\lvert\lambda+\lambda^{\prime}\rvert}_{J-J_{\min}}(z_{pk})$
and the singular behavior of the half-angle factor is removed since it is
squared. 181818An alternative approach to evaluating the first integral is to
use the rotational $e_{\lambda^{\prime}\lambda}^{(J)}$ functions as described
in Ref. Andrews and Gunson (2004). However, we find the approach presented in
this manuscript is ‘easier’ on the reader as we use the more commonly known
Legendre function $Q_{J}$. This allows us to perform the following expansion,

$\displaystyle\xi_{\lambda\lambda^{\prime}}(z_{pk})\,d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})=\sum_{j=0}^{\infty}(2j+1)\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}\,P_{j}(z_{pk})\,,$
(51)

where the expansion coefficients
$\mathcal{C}_{j,\lambda^{\prime}\lambda}^{(J)}$ are determined by

$\displaystyle\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}=\frac{1}{2}\int_{-1}^{+1}\\!\textrm{d}z_{pk}\,\xi_{\lambda\lambda^{\prime}}(z_{pk})\,d_{\lambda\lambda^{\prime}}^{(J)}(z_{pk})\,P_{j}(z_{pk})\,.$
(52)

We note here that this integral is precisely what appears in Eq. (50). So,
both the first and second terms of Eq. (IV.2) are related to the $\mathcal{C}$
coefficient Eq. (52). Equation (52) can be evaluated in closed form, with the
result being

$\displaystyle\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}$
$\displaystyle=\frac{(-1)^{\eta}}{2J_{\min}+1}\,\sqrt{\frac{\lvert\lambda-\lambda^{\prime}\rvert!\,\lvert\lambda+\lambda^{\prime}\rvert!}{(2J_{\min})!}}\,\braket{J_{\min}\lambda^{\prime}}{J\lambda^{\prime},j0}\braket{J_{\min}\lambda}{J\lambda,j0}\,,$
(53)

where $\eta=0$ if $\lambda^{\prime}\geq\lambda$ and
$\eta=\lambda^{\prime}-\lambda$ if $\lambda^{\prime}<\lambda$, and we recall
that $J_{\min}=\max(\lvert\lambda\rvert,\lvert\lambda^{\prime}\rvert)$. For
convenience, we provide a derivation of this result in App. B.

Using Eq. (52), we write the helicity partial wave projection of the OPE as

$\displaystyle\mathcal{G}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}(p,k)$
$\displaystyle=\frac{1}{4pk}\,\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,\sum_{j}(2j+1)\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}\,\int_{-1}^{+1}\\!\textrm{d}z_{pk}\,\frac{P_{j}(z_{pk})}{\zeta_{pk}-z_{pk}}\,$
$\displaystyle\qquad+\frac{1}{2pk}\sum_{j}(2j+1)\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}\,\mathcal{B}_{j,\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,.$
(54)

The final integral can be expressed in terms of the well-known Legendre
functions of the 2nd kind

$\displaystyle Q_{j}(\zeta_{pk})$
$\displaystyle=\frac{1}{2}\int_{-1}^{+1}\\!\textrm{d}z_{pk}\,\frac{P_{j}(z_{pk})}{\zeta_{pk}-z_{pk}}\,,$
(55)

where the analytic structure is fixed by $\zeta_{pk}\to\zeta_{pk}+i\epsilon$.
Properties and examples of the Legendre functions are given in App. A. The
Legendre $Q$ functions contain branch cuts when $\zeta_{pk}=\pm 1$, which
originate from the on-shell exchange $u_{pk}=m_{e}^{2}$. Since the
$\mathcal{A}$ and $\mathcal{B}$ coefficients are regular in the energy
variables, the $Q$ functions contain the entire non-analytic structure of the
partial wave projected OPE, which results in a branch cut in the complex
$s$-plane for fixed $\sigma_{k}$ and $\sigma_{p}$. The on-shell constraint
$\zeta_{pk}=\pm 1$ leads to an expression for the physical boundary region of
real-particle exchanges Byckling and Kajantie (1973), given by $\Phi(p,k)\geq
1$ where

$\displaystyle\Phi(p,k)$
$\displaystyle=\sigma_{k}\sigma_{p}(s+m_{k}^{2}+m_{p}^{2}+m_{e}^{2}-\sigma_{k}-\sigma_{p})\,$
$\displaystyle\quad-\sigma_{k}(s-m_{p}^{2})(m_{k}^{2}-m_{e}^{2})-\sigma_{p}(s-m_{k}^{2})(m_{p}^{2}-m_{e}^{2})\,$
$\displaystyle\quad-(sm_{e}^{2}-m_{p}^{2}m_{k}^{2})(s+m_{e}^{2}-m_{p}^{2}-m_{k}^{2})\,.$
(56)

Combining Eqs. (IV.2), (52), and (55), we find a compact expression for the
helicity partial wave projection of the OPE as

$\displaystyle\mathcal{G}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}(p,k)$
$\displaystyle=\frac{1}{2pk}\,\sum_{j}(2j+1)\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}\,\left[\,\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,Q_{j}(\zeta_{pk})+\mathcal{B}_{j,\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\right]\,,$
(57)

where $\zeta_{pk}=\zeta_{pk}(p,k)$ is defined in Eq. (IV.1). Note that the sum
is finite since the $\mathcal{C}$ coefficients are zero for $j$ outside the
range $\lvert J-J_{\min}\rvert\leq j\leq J+J_{\min}$. The pole term
multiplying the $\mathcal{A}$ coefficients results in singular behavior in the
energy variables, while the $\mathcal{B}$ coefficients are regular functions
of energies, thereby giving additional short-distance physics to the ones
already contained in the three-body $K$ matrix of the on-shell representations
as discussed in Ref. Jackura (2023).

Having this analytic representation for the helicity partial wave projection
of $\mathcal{G}$, we form the appropriate linear combinations to arrive at an
expression in the spin-orbit basis using Eqs. (20) and (21),

$\displaystyle\mathcal{G}_{L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)=\sum_{\lambda^{\prime},\lambda}\mathcal{P}_{\lambda^{\prime}}^{(\ell^{\prime})}(^{2S^{\prime}+1}L^{\prime}_{J})\,\mathcal{G}_{\ell^{\prime}\lambda^{\prime},\ell\lambda}^{J}(p,k)\,\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})\,.$
(58)

Inserting Eq. (57), we find the definite parity partial wave projections of
the OPE for the $\mathbf{3}\to\mathbf{3}$ scattering of external spinless
particles takes the form, as a matrix in $LS$-space,

$\displaystyle\mathcal{G}^{(\ell^{\prime}\ell),J}_{L^{\prime}S^{\prime},LS}(p,k)=\widetilde{\mathcal{K}}_{\mathcal{G},L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)+\sum_{j}\widetilde{\mathcal{T}}_{j,L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)\,Q_{j}(\zeta_{pk})\,,$
(59)

where $\widetilde{\mathcal{K}}_{\mathcal{G}}$ is a known short-distance
contribution given in terms of the $\mathcal{B}$ coefficients, whereas
$\widetilde{\mathcal{T}}_{j}$ are computed from a given set of $\mathcal{A}$
coefficients. The matrix elements of $\widetilde{\mathcal{K}}_{\mathcal{G}}$
are

$\displaystyle\widetilde{\mathcal{K}}_{\mathcal{G},L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)$
$\displaystyle\equiv\frac{1}{2pk}\,\sum_{j}(2j+1)\,$
$\displaystyle\qquad\qquad\times\sum_{\lambda^{\prime},\lambda}\mathcal{P}_{\lambda^{\prime}}^{(\ell^{\prime})}(^{2S^{\prime}+1}L^{\prime}_{J})\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}\,\mathcal{B}_{j,\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})\,,$
(60)

while the matrix elements of $\widetilde{\mathcal{T}}_{j}$ are

$\displaystyle\widetilde{\mathcal{T}}_{j,L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)\equiv\frac{1}{2pk}\,(2j+1)\,\sum_{\lambda^{\prime},\lambda}\mathcal{P}_{\lambda^{\prime}}^{(\ell^{\prime})}(^{2S^{\prime}+1}L^{\prime}_{J})\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}\,\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})\,.$
(61)

We stress here that Eq. (59) is a generic result for an exchange of a spinless
particle between pairs with any angular momentum that couple to some total
$J^{P}$. The matrices $\widetilde{\mathcal{K}}_{\mathcal{G}}$ and
$\widetilde{\mathcal{T}}_{j}$ are regular functions of the energies in the
physical region.

It is important to note that Eq. (59) is not a unique decomposition, as the
Legendre functions of the $2\textsuperscript{nd}$-kind can be written, cf.
Eqs. (102) and (103) in App. A, as

$\displaystyle Q_{j}(z)=P_{j}(z)\,Q_{0}(z)-W_{j-1}(z)\,,$ (62)

where $P_{j}$ are the Legendre polynomials and $W_{j-1}(z)$ is polynomial in
$z$, defined for $j>0$ by

$\displaystyle
W_{j-1}(z)=\sum_{n=1}^{j}\frac{1}{n}\,P_{n-1}(z)\,P_{j-n}(z)\,,$ (63)

whereas for $j=0$, $W_{-1}=0$. This relation allows one to shift the
definitions of $\widetilde{\mathcal{K}}_{\mathcal{G}}$ and
$\widetilde{\mathcal{T}}_{j}$ to absorb/remove terms regular in the kinematic
variables, which can further simplify expressions for the partial wave
projected OPE. Exploiting this relation, we reduce our result for the partial
wave OPE to one involving only $Q_{0}(\zeta_{pk})$,

$\displaystyle\mathcal{G}_{L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)=\mathcal{K}_{\mathcal{G},L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)+\mathcal{T}_{L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)\,Q_{0}(\zeta_{pk})$
(64)

which is the expression claimed in Eq. (1) as our primary result. The matrix
elements of $\mathcal{K}_{\mathcal{G}}$ are given by

$\displaystyle\mathcal{K}_{\mathcal{G},L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)$
$\displaystyle=\widetilde{\mathcal{K}}_{\mathcal{G},L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)-\sum_{j}\widetilde{\mathcal{T}}_{j,L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)\,W_{j-1}(\zeta_{pk})\,,$
$\displaystyle=\frac{1}{2pk}\,\sum_{j}(2j+1)\,\sum_{\lambda^{\prime},\lambda}\mathcal{P}_{\lambda^{\prime}}^{(\ell^{\prime})}(^{2S^{\prime}+1}L^{\prime}_{J})\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}\,$
$\displaystyle\qquad\qquad\times\left[\,\mathcal{B}_{j,\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)-W_{j-1}\,\mathcal{A}_{j,\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,\right]\,\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})\,,$
(65)

and the $\mathcal{T}$ coefficient is the sum over the
$\widetilde{\mathcal{T}}_{j}$ coefficients weighted by Legendre polynomials,

$\displaystyle\mathcal{T}_{L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)$
$\displaystyle\equiv\sum_{j}\widetilde{\mathcal{T}}_{j,L^{\prime}S^{\prime},LS}^{(\ell^{\prime}\ell),J}(p,k)\,P_{j}(\zeta_{pk})\,,$
$\displaystyle=\frac{1}{2pk}\,\sum_{j}(2j+1)P_{j}(\zeta_{pk})\,$
$\displaystyle\qquad\qquad\times\sum_{\lambda^{\prime},\lambda}\mathcal{P}_{\lambda^{\prime}}^{(\ell^{\prime})}(^{2S^{\prime}+1}L^{\prime}_{J})\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}\,\mathcal{A}_{\lambda^{\prime}\lambda}^{(\ell^{\prime}\ell)}(p,k)\,\mathcal{P}_{\lambda}^{(\ell)}(^{2S+1}L_{J})$
(66)

Equation (64), along with (IV.2) and (IV.2), are the main result of this work.
The only remaining task for the user is to construct the
$\mathcal{K}_{\mathcal{G}}$ and $\mathcal{T}$ matrices by identifying the
appropriate $\mathcal{A}$ and $\mathcal{B}$ coefficients. In the next section,
we illustrate the procedure for some low-spin cases.

##  V Case Studies

In this section we examine the consequence of our main result, Eq. (59), for
three cases of low-spin systems: pairs with $\ell^{\prime}=\ell=0$,
$\ell^{\prime}=\ell=1$, and the transition process $\ell^{\prime}=1,\ell=0$.
Any higher spin system can be found by following the procedure outlined in
this section. This procedure is easily amenable to symbolic computation with
software such as Mathematica, and such a notebook with examples is supplied in
the Supplemental Material.

The main tasks are to identify the $\mathcal{A}$ and $\mathcal{B}$
coefficients as defined in Eqs. (43) and (45), respectively, given for each
case. Once these coefficients are determined, we compute the
$\widetilde{\mathcal{K}}_{\mathcal{G}}$ and $\widetilde{\mathcal{T}}_{j}$
functions, Eqs. (IV.2) and (61), respectively. These matrices are then fed
into Eqs. (IV.2) and (IV.2) for $\mathcal{K}_{\mathcal{G}}$ and $\mathcal{T}$,
respectively, giving the analytic projection shown in Eq. (64).

Throughout this section, we denote $\eta=\eta_{k}\eta_{p}\eta_{e}$ as the
product of intrinsic parities of the initial and final spectators, and the
exchange particle. As our focus is on hadronic processes, we also assume
parity-conserving reactions with
$P=\eta(-1)^{L+S}=\eta(-1)^{L^{\prime}+S^{\prime}}$. Additionally, since
$S^{\prime}=\ell^{\prime}$ and $S=\ell$ always in this work, we introduce a
convenient notation for the partial wave OPE,

$\displaystyle\mathcal{G}(^{2S^{\prime}+1}L^{\prime}_{J}|^{2S+1}L_{J})\equiv\mathcal{G}_{L^{\prime}S^{\prime},LS}^{(S^{\prime}S),J}(p,k)\,,$

where the dependencies on kinematic variables are left implicit.

###  V.1 Pairs with $\boldsymbol{\ell^{\prime}=\ell=0}$

The simplest case is when both the incoming and outgoing pairs are in relative
$S$ waves, so that $\ell^{\prime}=\ell=0$. The complications with the spin-
helicity function are eliminated since $\lambda^{\prime}=\lambda=0$, i.e.
$\mathcal{H}_{\lambda^{\prime}\lambda}^{(00)}=\delta_{\lambda^{\prime}0}\delta_{\lambda
0}$. This in turn indicates that the coefficients are
$\mathcal{A}_{\lambda^{\prime}\lambda}^{(00)}=\delta_{\lambda^{\prime}0}\delta_{\lambda
0}$ and $\mathcal{B}_{\lambda^{\prime}\lambda}^{(00)}=0$. Therefore, the
exchange propagator (46) reduces to the simple form

$\displaystyle\mathcal{G}_{0\lambda^{\prime},0\lambda}=\frac{1}{2pk}\,\frac{\delta_{\lambda^{\prime}0}\delta_{\lambda
0}}{\zeta_{pk}-z_{pk}}\,.$ (67)

Recoupling to spin-orbit amplitudes is also trivial as only $S=\ell=0$ and
$S^{\prime}=\ell^{\prime}=0$, restricting the allowed orbital angular momentum
to be $J=L=L^{\prime}$. The parity of the system is then $P=\eta(-1)^{J}$.

From Eq. (IV.2), we see that
$\widetilde{\mathcal{K}}_{\mathcal{G}}(^{1}J_{J}|^{1}J_{J})=0$ because
$\mathcal{B}_{\lambda^{\prime}\lambda}^{(00)}=0$ for this case. Using Eq. (61)
and the identities for $\mathcal{P}_{\lambda}^{(0)}(^{1}J_{J})=\delta_{\lambda
0}$ and
$\,\mathcal{C}_{j,\lambda^{\prime}\lambda}^{J}=\delta_{Jj}\delta_{\lambda^{\prime}0}\delta_{\lambda
0}/(2J+1)$ found from Eqs. (21) and (53), respectively, one finds that the
partial wave OPE for a target $J^{P}$ simplifies to

$\displaystyle\mathcal{G}(^{1}J_{J}|^{1}J_{J})=\frac{1}{2pk}\,Q_{J}(\zeta_{pk})\,.$
(68)

As detailed in Eqs. (64), (IV.2), and (IV.2), we can express this amplitude in
terms of only $Q_{0}$. Explicitly, the $J=0$, and $1$ partial wave OPE
amplitudes are given by

$\displaystyle\mathcal{G}(^{1}S_{0}|^{1}S_{0})$
$\displaystyle=\frac{1}{2pk}\,Q_{0}(\zeta_{pk})\,,$ (69)
$\displaystyle\mathcal{G}(^{1}P_{1}|^{1}P_{1})$
$\displaystyle=-\frac{1}{2pk}+\frac{\zeta_{pk}}{2pk}\,Q_{0}(\zeta_{pk})\,.$
(70)

The $J=0$ OPE agrees with the well-known result found in many works on three-
body scattering processes, e.g. Refs. Jackura  _et al._ (2019); Dawid and
Szczepaniak (2021); Jackura  _et al._ (2021); Dawid  _et al._ (2023a).

We perform a simple check of Eq. (68) by verifying it satisfies the expected
behavior near threshold. As discussed in Sec. III.1, we can recognize that
$\mathcal{G}$ can be thought of as an effective two-body amplitude by
interpreting $\sqrt{\sigma_{k}}$ and $\sqrt{\sigma_{p}}$ as effective masses
of the external states. As a result, one would expect that such amplitude
$\mathcal{G}(^{1}J_{J}|^{1}J_{J})$ scales as $(pk)^{J}$ in the vicinity of the
nearest pair-spectator threshold
$\sqrt{s}\sim\sqrt{\sigma_{p}}+m_{p}\sim\sqrt{\sigma_{k}}+m_{k}$, cf. Sec.
III.2.

To reproduce this behavior, we note that near this threshold both $p$ and $k$
are small, $p,k\to 0$. Fixing $\sigma_{k}>\sigma_{k}^{(\mathrm{thr}.)}$ and
$\sigma_{p}>\sigma_{p}^{(\mathrm{thr}.)}$, we see from the definition of
$\zeta_{pk}$, Eq. (IV.1), that near threshold $\zeta_{pk}$ diverges as $1/pk$,

$\displaystyle\zeta_{pk}=\frac{\mathcal{N}}{2pk}+O\\!\left(pk^{-1},p^{-1}k\right)\,,$

where $\mathcal{N}=(\sqrt{\sigma_{k}}-m_{p})^{2}-m_{e}^{2}$ is a positive
constant. From the behavior of Legendre function for large arguments, Eq. 105,
we see that near threshold $Q_{J}(\zeta_{pk})\to(pk)^{J+1}$. Therefore, near
threshold Eq. (68) satisfies the expected behavior of
$\mathcal{G}(^{1}J_{J}|^{1}J_{J})\sim(pk)^{J}$. Explicitly, we write the
threshold expansions of Eqs. (69) and (70) by using the asymptotic expansion
of $Q_{0}(\zeta_{pk})$ for $\zeta_{pk}\to\infty$ as given in Eq. (106),

$\displaystyle Q_{0}(\zeta_{pk})$
$\displaystyle=\frac{1}{\zeta_{pk}}+\frac{1}{3\zeta_{pk}^{3}}+O\\!\left(\zeta_{pk}^{\,-5}\right)\,.$
(71)

Since $\zeta_{pk}$ appears as a reciprocal in this expansion, we can write the
Taylor series for $1/\zeta_{pk}$ for small $p$ and $k$ as

$\displaystyle\frac{1}{\zeta_{pk}}=\frac{2pk}{\mathcal{N}}+O\\!\left(p^{2}k,pk^{2}\right)\,.$

Then, the explicit threshold behavior for the ${}^{1}S_{0}\to^{1}S_{0}$ and
${}^{1}P_{1}\to^{1}P_{1}$ amplitudes is

$\displaystyle\mathcal{G}(^{1}S_{0}|^{1}S_{0})$
$\displaystyle=\frac{1}{\mathcal{N}}+O\\!\left(p,k\right)\,,$
$\displaystyle\mathcal{G}(^{1}P_{1}|^{1}P_{1})$
$\displaystyle=\frac{2}{3\mathcal{N}^{2}}\,pk+O\\!\left(p^{2}k,pk^{2}\right)\,,$

which is consistent with the expected behavior.

###  V.2 Pairs with $\boldsymbol{\ell^{\prime}=\ell=1}$

We turn to the case where the pairs carry non-zero angular momenta
$\ell^{\prime}=\ell=1$. In this case, the spin-helicity matrix has a non-
trivial structure, and we must work out the $\mathcal{A}$ and $\mathcal{B}$
coefficients. Recall that given the elements of the spin-helicity matrix
$\mathcal{H}$, we can solve for the $\mathcal{A}$ and $\mathcal{B}$
coefficients using Eqs. (43) and (45), respectively. Most readily, the
$\mathcal{A}$ coefficient is given by first isolating $\mathcal{A}(z_{pk})$
from Eq. (42),

$\displaystyle\,\,\overline{\\!\\!{\mathcal{A}}}_{\lambda^{\prime}\lambda}^{(11)}(p,k,z_{pk})=\frac{1}{\xi_{\lambda\lambda^{\prime}}(z_{pk})}\,\,\,\overline{\\!\\!{\mathcal{H}}}_{\lambda^{\prime}\lambda}^{(11)}(\mathbf{p},\mathbf{k})\,,$
(72)

and then setting $z_{pk}=\zeta_{pk}$. For convenience, we introduced the
$\ell^{\prime}=\ell=1$ spin-helicity function
$\,\,\overline{\\!\\!{\mathcal{H}}}$ and $\,\,\overline{\\!\\!{\mathcal{A}}}$
coefficient which has a common factor removed,

$\displaystyle\,\,\overline{\\!\\!{\mathcal{H}}}_{\lambda^{\prime}\lambda}^{(11)}$
$\displaystyle\equiv\frac{q_{p}^{\star}q_{k}^{\star}}{3pk}\,\mathcal{H}_{\lambda^{\prime}\lambda}^{(11)}\,,$
$\displaystyle\,\,\overline{\\!\\!{\mathcal{A}}}_{\lambda^{\prime}\lambda}^{(11)}$
$\displaystyle\equiv\frac{q_{p}^{\star}q_{k}^{\star}}{3pk}\,\mathcal{A}_{\lambda^{\prime}\lambda}^{(11)}\,.$

When both initial and final state pairs are in relative $P$ wave, the spin
structure is such that there are only five independent functions which results
from the reflection property Eq. (32). There is an accidental symmetry
relating the $\lambda^{\prime},\lambda=+1,+1$ and $+1,-1$ components, yielding
only four independent functions. In terms of the polar angles previously
defined, we obtain for the spin-helicity function

$\displaystyle\,\,\overline{\\!\\!{\mathcal{H}}}_{\pm 1\pm
1}^{(11)}=-\,\,\overline{\\!\\!{\mathcal{H}}}_{\pm 1\mp 1}^{(11)}$
$\displaystyle=-\frac{1}{2}\frac{k_{p}^{\star}p_{k}^{\star}}{pk}\,\sin\chi_{p}^{\star}\sin\chi_{k}^{\star}\,,$
(73a) $\displaystyle\,\,\overline{\\!\\!{\mathcal{H}}}_{\pm 10}^{(11)}$
$\displaystyle=\mp\frac{1}{\sqrt{2}}\frac{k_{p}^{\star}p_{k}^{\star}}{pk}\,\sin\chi_{p}^{\star}\cos\chi_{k}^{\star}\,,$
(73b) $\displaystyle\,\,\overline{\\!\\!{\mathcal{H}}}_{0\pm 1}^{(11)}$
$\displaystyle=\pm\frac{1}{\sqrt{2}}\frac{k_{p}^{\star}p_{k}^{\star}}{pk}\,\cos\chi_{p}^{\star}\sin\chi_{k}^{\star}\,,$
(73c) $\displaystyle\,\,\overline{\\!\\!{\mathcal{H}}}_{00}^{(11)}$
$\displaystyle=\frac{k_{p}^{\star}p_{k}^{\star}}{pk}\,\cos\chi_{p}^{\star}\cos\chi_{k}^{\star}\,.$
(73d)

The Lorentz transformations Eqs. (37) and (38) relate the polar angles
$\chi_{k}^{\star}$ and $\chi_{p}^{\star}$ to the total CM frame polar angle
$\theta_{pk}$. Note that if we compare Eq. (73) to expressions in Ref. Kamal
(1965), we find disagreement with respect to overall phase factors. This is
due to the azimuthal angle of $\pi$ for one of the momenta as detailed in Eq.
(IV.1), which was neglected by the author of Ref. Kamal (1965). One can
convince themselves that this phase factor is necessary and consistent with an
alternative model for the OPE which replaces the spin-helicity matrix as given
in Eq. (31) with polarization Lorentz tensors contracted with momenta, e.g.
$\mathcal{H}_{\lambda^{\prime}\lambda}^{(11)}\propto(\varepsilon^{*}(P_{p},\lambda^{\prime})\cdot
k)(\varepsilon(P_{k},\lambda)\cdot p)$, which can be seen by considering an
effective Lagrangian of vector field $V_{\mu}$ coupling to two scalars, e.g.
$\mathcal{L}\supset-
ig_{p}V_{p}^{\mu}\,\varphi_{e}\partial_{\mu}\varphi_{k}-ig_{k}V_{k}^{\mu}\,\varphi_{e}\partial_{\mu}\varphi_{p}$
where $g_{p}$, $g_{k}$ are effective couplings to the scalar $\varphi$ fields.

Recalling that $z_{pk}\equiv\cos\theta_{pk}$ and
$\sin\theta_{pk}=\sqrt{1-z_{pk}^{2}}$, the half-angle factor for each helicity
combination is given by

$\displaystyle\xi_{\pm 1\pm 1}(z_{pk})$ $\displaystyle=\frac{1+z_{pk}}{2}\,,$
$\displaystyle\xi_{\pm 1\mp 1}(z_{pk})$ $\displaystyle=\frac{1-z_{pk}}{2}\,,$
$\displaystyle\xi_{\pm 10}(z_{pk})=\xi_{0\pm 1}(z_{pk})$
$\displaystyle=\frac{\sqrt{1-z_{pk}^{2}}}{2}\,,$
$\displaystyle\xi_{00}(z_{pk})$ $\displaystyle=1\,.$

Combining Eqs. (72) and (73) and using the boost relations Eqs. (38) and (37)
gives for the $\,\,\overline{\\!\\!{\mathcal{A}}}(p,k,z_{pk})$ coefficients

$\displaystyle\,\,\overline{\\!\\!{\mathcal{A}}}_{\pm 1\pm 1}^{(11)}$
$\displaystyle=-1+z_{pk}\,,$ (74a)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{A}}}_{\pm 1\mp 1}^{(11)}$
$\displaystyle=1+z_{pk}\,,$ (74b)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{A}}}_{\pm 10}^{(11)}$
$\displaystyle=\pm\sqrt{2}\,\gamma_{k}\,\left(\frac{\beta_{k}\omega_{p}}{p}+z_{pk}\right)\,,$
(74c) $\displaystyle\,\,\overline{\\!\\!{\mathcal{A}}}_{0\pm 1}^{(11)}$
$\displaystyle=\mp\sqrt{2}\,\gamma_{p}\,\left(\frac{\beta_{p}\omega_{k}}{k}+z_{pk}\right)\,,$
(74d) $\displaystyle\,\,\overline{\\!\\!{\mathcal{A}}}_{00}^{(11)}$
$\displaystyle=\gamma_{p}\gamma_{k}\left(\frac{\beta_{p}\omega_{k}}{k}+z_{pk}\right)\left(\frac{\beta_{k}\omega_{p}}{p}+z_{pk}\right)\,.$
(74e)

The on-shell coefficients
$\,\,\overline{\\!\\!{\mathcal{A}}}(p,k)\equiv\,\,\overline{\\!\\!{\mathcal{A}}}(p,k,\zeta_{pk})$
are given by Eq. (74) with the substitution $z_{pk}\to\zeta_{pk}$. We again
define the $\,\,\overline{\\!\\!{\mathcal{B}}}$ coefficients in terms of
$\mathcal{B}$ coefficients as

$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{\lambda^{\prime}\lambda}^{(11)}$
$\displaystyle\equiv\frac{q_{p}^{\star}q_{k}^{\star}}{3pk}\,\mathcal{B}_{\lambda^{\prime}\lambda}^{(11)}\,,$

which are related to $\,\,\overline{\\!\\!{\mathcal{A}}}$ through Eq. (45),

$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{\lambda^{\prime}\lambda}^{(11)}(p,k,z_{pk})=\frac{1}{\zeta_{pk}-z_{pk}}\,\left[\,\,\,\overline{\\!\\!{\mathcal{A}}}_{\lambda^{\prime}\lambda}^{(11)}(p,k,z_{pk})-\,\,\overline{\\!\\!{\mathcal{A}}}_{\lambda^{\prime}\lambda}^{(11)}(p,k)\,\right]\,,$

Evaluating the difference and dividing by the pole gives
$\,\,\overline{\\!\\!{\mathcal{B}}}(p,k,z_{pk})$

$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{\pm 1\pm 1}^{(11)}$
$\displaystyle=-1\,,$ (75a)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{\pm 1\mp 1}^{(11)}$
$\displaystyle=-1\,,$ (75b)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{\pm 10}^{(11)}$
$\displaystyle=\mp\sqrt{2}\,\gamma_{k}\,,$ (75c)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{0\pm 1}^{(11)}$
$\displaystyle=\pm\sqrt{2}\,\gamma_{p}\,,$ (75d)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{00}^{(11)}$
$\displaystyle=-\gamma_{p}\gamma_{k}\left(\frac{\beta_{p}\omega_{k}}{k}+\frac{\beta_{k}\omega_{p}}{p}+z_{pk}+\zeta_{pk}\right)\,.$
(75e)

Finally, we require the projected $\mathcal{B}$ coefficients defined by Eq.
(48), repeated here for convenience,

$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{j,\lambda^{\prime}\lambda}^{(11)}(p,k)=\frac{1}{2}\int_{-1}^{+1}\\!\textrm{d}z_{pk}\,P_{j}(z_{pk})\,\,\,\overline{\\!\\!{\mathcal{B}}}_{\lambda^{\prime}\lambda}^{(11)}(p,k,z_{pk})\,.$

Upon substituting Eq. (75) we find only two non-zero terms in the expansion

$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{0,\lambda^{\prime}\lambda}^{(11)}$
$\displaystyle=-|\lambda\,\lambda^{\prime}|\,-\sqrt{2}\gamma_{k}\,\lambda^{\prime}\,\delta_{\lambda,0}+\sqrt{2}\gamma_{p}\,\lambda\,\delta_{\lambda^{\prime},0}\,$
$\displaystyle\qquad-\gamma_{p}\gamma_{k}\left(\frac{\beta_{p}\omega_{k}}{k}+\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)\,\delta_{\lambda^{\prime},0}\delta_{\lambda,0}\,,$
(76a)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{1,\lambda^{\prime}\lambda}^{(11)}$
$\displaystyle=-\frac{1}{3}\,\gamma_{p}\gamma_{k}\,\delta_{\lambda^{\prime},0}\delta_{\lambda,0}\,,$
(76b)
$\displaystyle\,\,\overline{\\!\\!{\mathcal{B}}}_{j,\lambda^{\prime}\lambda}^{(11)}$
$\displaystyle=0\,,\quad\mathrm{for}\,\,j>1\,.$ (76c)

Having found the $\mathcal{A}$ and $\mathcal{B}$ coefficients for
$\ell^{\prime}=\ell=1$, we now construct $\mathcal{K}_{\mathcal{G}}$ and
$\mathcal{T}$ matrices for some target $J^{P}$. Trivially $S^{\prime}=S=1$,
therefore if $J=0$ only $L^{\prime}=L=1$ contributes, while for $J>0$ the
allowed orbital angular momenta are $L=J-1,J,J+1$ and $L^{\prime}=J-1,J,J+1$
for the initial and final states, respectively.

####  V.2.1 Total $\boldsymbol{J=0}$

Let us first consider a target $J=0$, where only $L^{\prime}=L=1$ is allowed
and the corresponding parity is $\eta$. Thus, we need only compute a single
${}^{3}P_{0}$ amplitude with the spin-orbit recoupling from Eq. (21) being
$\mathcal{P}_{\lambda}^{(1)}(^{3}P_{0})=-\delta_{\lambda,0}$. Evaluating the
expression for $\widetilde{\mathcal{K}}_{\mathcal{G}}$ and
$\widetilde{\mathcal{T}}_{j}$ using Eqs. (IV.2) and (61) respectively, we find

$\displaystyle\widetilde{\mathcal{K}}_{\mathcal{G}}(^{3}P_{0}|^{3}P_{0})$
$\displaystyle=-\frac{3\gamma_{k}\gamma_{p}}{2q_{k}^{\star}q_{p}^{\star}}\left(\frac{\beta_{p}\omega_{k}}{k}+\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)\,,$
$\displaystyle\widetilde{\mathcal{T}}_{j}(^{3}P_{0}|^{3}P_{0})$
$\displaystyle=\delta_{j0}\,\frac{3\gamma_{k}\gamma_{p}}{2q_{k}^{\star}q_{p}^{\star}}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)\left(\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)\,.$

Since only $j=0$ contributes for this case, feeding these matrices into Eqs.
(IV.2) and (IV.2) gives trivially
$\mathcal{K}_{\mathcal{G}}=\widetilde{\mathcal{K}}_{\mathcal{G}}$ and
$\mathcal{T}=\widetilde{\mathcal{T}}_{0}$. Adding these contributions together
as dictated by Eq. (64), one finds

$\displaystyle\mathcal{G}(^{3}P_{0}|^{3}P_{0})$
$\displaystyle=-\frac{3\gamma_{k}\gamma_{p}}{2q_{k}^{\star}q_{p}^{\star}}\left(\frac{\beta_{p}\omega_{k}}{k}+\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)$
$\displaystyle\qquad+\frac{3\gamma_{k}\gamma_{p}}{2q_{k}^{\star}q_{p}^{\star}}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)\left(\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)Q_{0}(\zeta_{pk}).$
(77)

As before, we check the threshold behavior of this amplitude by fixing
$\sigma_{k}$ and $\sigma_{p}$ and expanding for small $p$ and $k$. Just as in
Sec. V.1, $\zeta_{pk}$ diverges as $\zeta_{pk}=\mathcal{N}/2pk$ as $p,k\to 0$,
thus $Q_{0}(\zeta_{pk})$ admits an expansion as Eq. (71). Moreover,
$\gamma_{p}=1+O(p^{2})$, $\omega_{p}=m_{p}+O(p^{2})$, and
$E_{p}=\sqrt{\sigma_{p}}+O(p^{2})$ as $p\to 0$, with similar expansions for
variables of the $k$ spectator as $k\to 0$. Near the pair-spectator
thresholds, one finds that the expansion of Eq. (V.2.1) for $p,k\to 0$ is
given by

$\displaystyle\mathcal{G}(^{3}P_{0}|^{3}P_{0})=\frac{1}{\mathcal{N}q_{p}^{\star}q_{k}^{\star}}\left(1+\frac{3m_{p}m_{k}}{\sqrt{\sigma_{p}}\sqrt{\sigma_{k}}}\right)\,pk+O\\!\left(p^{2}k,pk^{2}\right)\,,$

where the relative momenta $q_{p}^{\star}$ and $q_{k}^{\star}$ are finite
positive constants since $\sigma_{p}$ and $\sigma_{k}$ are fixed above their
respective thresholds. Therefore, as expected
$\mathcal{G}(^{3}P_{0}|^{3}P_{0})\sim pk$ near threshold.

####  V.2.2 Total $\boldsymbol{J=1}$, Parity $\boldsymbol{\eta}$

Next let us consider a target $J=1$ with a parity $\eta$, which enforces
$L^{\prime}=L=1$, i.e. ${}^{3}P_{1}\to^{3}P_{1}$. From Eq. (21) the spin-orbit
coupling is $\mathcal{P}_{\lambda}^{(1)}(^{3}P_{1})=-\lambda/\sqrt{2}$. From
Eq. (IV.2) we find that
$\widetilde{\mathcal{K}}_{\mathcal{G}}(^{3}P_{1}|^{3}P_{1})=0$, while from Eq.
(61) the $\widetilde{\mathcal{T}}_{j}$ factors are

$\displaystyle\widetilde{\mathcal{T}}_{j}(^{3}P_{1}|^{3}P_{1})=-\frac{1}{2q_{p}^{\star}q_{k}^{\star}}\left(\,2\delta_{j0}-3\zeta_{pk}\,\delta_{j1}+\delta_{j2}\,\right)\,,$

with all coefficients $j>2$ being zero. Thus, the partial wave OPE Eq. (59) is

$\displaystyle\mathcal{G}(^{3}P_{1}|^{3}P_{1})=-\frac{1}{q_{p}^{\star}q_{k}^{\star}}Q_{0}(\zeta_{pk})+\frac{3\zeta_{pk}}{2q_{p}^{\star}q_{k}^{\star}}Q_{1}(\zeta_{pk})-\frac{1}{2q_{p}^{\star}q_{k}^{\star}}Q_{2}(\zeta_{pk})\,.$
(78)

Before applying the simplifications of Eqs. (IV.2) and (IV.2), we first
perform an intermediary manipulation of Eq. (78) by using the Bonnet recursion
relation for the Legendre function $Q_{1}$, see Eq. 101, to simplify the
expression to

$\displaystyle\mathcal{G}(^{3}P_{1}|^{3}P_{1})=-\frac{1}{2q_{p}^{\star}q_{k}^{\star}}\Big{[}\,Q_{0}(\zeta_{pk})-Q_{2}(\zeta_{pk})\,\Big{]}\,.$
(79)

We note that this expressions agrees up to an overall sign with the result of
Ref. Kamal (1965), in the context of studying the binding of the $\omega$
meson via $\pi$-exchange between the $\pi$ and the resonating $\pi\pi\to\rho$
subsystems. The difference in overall sign is due to the error in Ref. Kamal
(1965) from not considering the correct azimuthal angle of one of the momenta
as discussed in the beginning of Sec. V.2. Finally, we use
$Q_{2}(\zeta_{pk})=P_{2}(\zeta_{pk})Q_{0}(\zeta_{pk})-3\zeta_{pk}/2$ to
express Eq. (79) in the form of Eq. (64),

$\displaystyle\mathcal{G}(^{3}P_{1}|^{3}P_{1})=-\frac{3}{4q_{p}^{\star}q_{k}^{\star}}\zeta_{pk}+\frac{3}{4q_{p}^{\star}q_{k}^{\star}}\left(\,\zeta_{pk}^{2}-1\,\right)\,Q_{0}(\zeta_{pk})\,.$
(80)

Near threshold we expect $\mathcal{G}(^{3}P_{1}|^{3}P_{1})\sim pk$, which is
verified by following the same procedure as in the previous cases, and finding

$\displaystyle\mathcal{G}(^{3}P_{1}|^{3}P_{1})=-\frac{1}{\mathcal{N}q_{p}^{\star}q_{k}^{\star}}\,pk+O\left(p^{2}k,pk^{2}\right)\,,$

which agrees with the expected behavior.

####  V.2.3 Total $\boldsymbol{J=1}$, Parity $\boldsymbol{-\eta}$

Our final example for this case is $J=1$ with parity $-\eta$. Here we
encounter a coupled channel system in $S$ and $D$ waves since
$L^{\prime},L=J\pm 1$. The spin-orbit factors are given by

$\displaystyle\mathcal{P}_{\lambda}^{(1)}(^{3}S_{1})$
$\displaystyle=\sqrt{\dfrac{1}{3}}\,,$
$\displaystyle\mathcal{P}_{\lambda}^{(1)}(^{3}D_{1})$
$\displaystyle=\sqrt{\dfrac{1}{6}}\,\lvert\lambda\rvert-\sqrt{\dfrac{2}{3}}\,\delta_{\lambda,0}\,.$

Feeding this, the $A$ and $B$ coefficients, and other building blocks into
Eqs. (IV.2) and (61), then through Eqs. (IV.2) and (IV.2), gives the following
expressions for the ${}^{3}S_{1}\to{{}^{3}S_{1}}$,
${}^{3}S_{1}\to{{}^{3}D_{1}}$, ${}^{3}D_{1}\to{{}^{3}S_{1}}$, and
${}^{3}D_{1}\to{{}^{3}D_{1}}$ OPE amplitudes:

$\displaystyle\mathcal{G}(^{3}L^{\prime}_{1}|^{3}L_{1})=\mathcal{K}_{\mathcal{G}}(^{3}L^{\prime}_{1}|^{3}L_{1})+\mathcal{T}(^{3}L^{\prime}_{1}|^{3}L_{1})\,Q_{0}(\zeta_{pk})$
(81)

with the following $\mathcal{K}_{\mathcal{G}}$ and $\mathcal{T}$ matrices:

$\displaystyle\mathcal{K}_{\mathcal{G}}(^{3}S_{1}|^{3}S_{1})$
$\displaystyle=\frac{1}{2q_{p}^{\star}q_{k}^{\star}}\Bigg{\\{}\gamma_{p}\left[\zeta_{pk}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)-\frac{2}{3}\right]+\gamma_{k}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)-\frac{2}{3}\right]$
$\displaystyle\quad-\gamma_{p}\gamma_{k}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\frac{\beta_{p}\beta_{k}\omega_{p}\omega_{k}}{pk}+\frac{1}{3}\right]-\zeta_{pk}^{2}+\frac{2}{3}\Bigg{\\}}\,,$
(82a) $\displaystyle\mathcal{K}_{\mathcal{G}}(^{3}D_{1}|^{3}S_{1})$
$\displaystyle=\frac{1}{2\sqrt{2}q_{p}^{\star}q_{k}^{\star}}\Bigg{\\{}-2\gamma_{p}\left[\zeta_{pk}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)-\frac{2}{3}\right]+\gamma_{k}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)-\frac{2}{3}\right]$
$\displaystyle\quad+2\gamma_{p}\gamma_{k}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\frac{\beta_{p}\beta_{k}\omega_{p}\omega_{k}}{pk}+\frac{1}{3}\right]-\zeta_{pk}^{2}+\frac{2}{3}\Bigg{\\}}\,,$
(82b) $\displaystyle\mathcal{K}_{\mathcal{G}}(^{3}D_{1}|^{3}D_{1})$
$\displaystyle=\frac{1}{4q_{p}^{\star}q_{k}^{\star}}\Bigg{\\{}-2\gamma_{p}\left[\zeta_{pk}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)-\frac{2}{3}\right]-2\gamma_{k}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)-\frac{2}{3}\right]$
$\displaystyle\quad-4\gamma_{p}\gamma_{k}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\frac{\beta_{p}\beta_{k}\omega_{p}\omega_{k}}{pk}+\frac{1}{3}\right]-\zeta_{pk}^{2}+\frac{2}{3}\Bigg{\\}}\,,$
(82c)

for $\mathcal{K}_{\mathcal{G}}$, and

$\displaystyle\mathcal{T}(^{3}S_{1}|^{3}S_{1})$
$\displaystyle=\frac{1}{2q_{p}^{\star}q_{k}^{\star}}\Bigg{\\{}\gamma_{p}(1-\zeta_{pk}^{2})\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\gamma_{k}(1-\zeta_{pk}^{2})\left(\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)$
$\displaystyle\quad+\gamma_{p}\gamma_{k}\,\zeta_{pk}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\frac{\beta_{p}\beta_{k}\omega_{p}\omega_{k}}{pk}\right]+\zeta_{pk}^{3}-\zeta_{pk}\Bigg{\\}}\,,$
(83a) $\displaystyle\mathcal{T}(^{3}D_{1}|^{3}S_{1})$
$\displaystyle=\frac{1}{2\sqrt{2}q_{p}^{\star}q_{k}^{\star}}\Bigg{\\{}-2\gamma_{p}(1-\zeta_{pk}^{2})\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\gamma_{k}(1-\zeta_{pk}^{2})\left(\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)$
$\displaystyle\quad-2\gamma_{p}\gamma_{k}\,\zeta_{pk}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\frac{\beta_{p}\beta_{k}\omega_{p}\omega_{k}}{pk}\right]+\zeta_{pk}^{3}-\zeta_{pk}\Bigg{\\}}\,,$
(83b) $\displaystyle\mathcal{T}(^{3}D_{1}|^{3}D_{1})$
$\displaystyle=\frac{1}{4q_{p}^{\star}q_{k}^{\star}}\Bigg{\\{}-2\gamma_{p}(1-\zeta_{pk}^{2})\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)-2\gamma_{k}(1-\zeta_{pk}^{2})\left(\frac{\beta_{k}\omega_{p}}{p}+\zeta_{pk}\right)$
$\displaystyle\quad+4\gamma_{p}\gamma_{k}\,\zeta_{pk}\left[\zeta_{pk}\left(\frac{\beta_{k}\omega_{p}}{p}+\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\frac{\beta_{p}\beta_{k}\omega_{p}\omega_{k}}{pk}\right]+\zeta_{pk}^{3}-\zeta_{pk}\Bigg{\\}}\,.$
(83c)

for the $\mathcal{T}$ matrices. The $\mathcal{K}_{\mathcal{G}}$ and
$\mathcal{T}$ coefficients for the ${}^{3}D_{1}\to{{}^{3}S_{1}}$ process are
found by interchanging $k\leftrightarrow p$ in the
${}^{3}S_{1}\to{{}^{3}D_{1}}$ coefficients, noting the symmetry
$\zeta_{pk}=\zeta_{kp}$ which can be seen from Eq. (IV.1).

Examining the threshold behavior as in previous cases, we find that the
${}^{3}S_{1}\to{{}^{3}S_{1}}$ OPE has the following expansion near threshold

$\displaystyle\mathcal{G}(^{3}S_{1}|^{3}S_{1})=\frac{1}{\mathcal{N}q_{p}^{\star}q_{k}^{\star}}\left(\frac{m_{p}}{\sqrt{\sigma_{k}}}\,k^{2}+\frac{m_{k}}{\sqrt{\sigma_{p}}}\,p^{2}\right)+O\\!\left(p^{3},k^{3},p^{2}k,pk^{2}\right)\,.$

If we fix $\sigma_{p}<\sigma_{k}$, then the threshold we approach first is
$\sqrt{s}\sim\sqrt{\sigma_{k}}+m_{k}$. As we approach this threshold, then the
amplitude does approach a constant, as $p$ will be finite this threshold.
However, if both $k$ and $p$ approach threshold simultaneous, e.g. in the case
where $m_{p}=m_{k}$ and $\sigma_{p}=\sigma_{k}$, then the amplitude scales as
$k^{2}$, which is faster than the requisite constant scaling we expect for $S$
waves. Although this behavior may be surprising, it is not inconsistent with
the requirement that the amplitude is equal to a finite constant at threshold.

Repeating this exercise for ${}^{3}S_{1}\to{{}^{3}D_{1}}$ and
${}^{3}D_{1}\to{{}^{3}D_{1}}$ waves, we find the following expansions

$\displaystyle\mathcal{G}(^{3}D_{1}|^{3}S_{1})$
$\displaystyle=-\frac{\sqrt{2}m_{k}}{\mathcal{N}q_{p}^{\star}q_{k}^{\star}\sqrt{\sigma_{p}}}\,p^{2}+O\\!\left(p^{3},p^{2}k\right)\,,$
$\displaystyle\mathcal{G}(^{3}D_{1}|^{3}D_{1})$
$\displaystyle=\frac{2}{15\mathcal{N}^{2}q_{p}^{\star}q_{k}^{\star}}\left(1+\frac{10m_{p}m_{k}}{\sqrt{\sigma_{p}}\sqrt{\sigma_{k}}}\right)\,p^{2}k^{2}+O\\!\left(p^{3}k^{2},p^{2}k^{3}\right)\,,$

where the threshold expansion of the ${}^{3}D_{1}\to{{}^{3}S_{1}}$ amplitude
is found by interchanging $k\leftrightarrow p$ on
$\mathcal{G}(^{3}D_{1}|^{3}S_{1})$. Both of these follow the expected
threshold behavior. This completes our set of examples for
$\ell^{\prime}=\ell=1$. Next, we will look at examples for transitions between
$\ell=0$ and $\ell^{\prime}=1$.

###  V.3 Pairs with $\boldsymbol{\ell^{\prime}=1}$ and $\boldsymbol{\ell=0}$

Here we consider an initial pair with spin $\ell=0$, and a final pair with
spin $\ell^{\prime}=1$. Such transitions are allowed in general, and observed
in nature, e.g. in $\sigma\pi\to\rho\pi$ in the $I(J^{P})=1(1^{+})$ channel of
$3\pi$ scattering. Repeating the same strategy as in the previous cases, the
spin-helicity matrix is given by

$\displaystyle\mathcal{H}_{\lambda^{\prime}\lambda}^{(10)}=\delta_{\lambda
0}\,\sqrt{4\pi}\,\left(\frac{k_{p}^{\star}}{q_{p}^{\star}}\right)\,Y_{1\lambda^{\prime}}^{*}(\mathbf{\hat{k}}_{p}^{\star})$
(84)

which corresponds to an $\mathcal{A}$ coefficient

$\displaystyle\mathcal{A}^{(10)}_{\pm 10}$
$\displaystyle=\mp\frac{\sqrt{6}k}{q_{p}^{\star}}\,,$ (85a)
$\displaystyle\mathcal{A}^{(10)}_{00}$
$\displaystyle=-\frac{\sqrt{3}\gamma_{p}k}{q_{p}^{\star}}\,\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)\,,$
(85b)

and a $B$ coefficient

$\displaystyle\mathcal{B}_{\lambda^{\prime}\lambda}^{(10)}$
$\displaystyle=\delta_{\lambda^{\prime}0}\delta_{\lambda
0}\,\frac{\sqrt{3}\gamma_{p}k}{q_{p}^{\star}}\,.$ (86)

Therefore, there is only one contribution to the $\mathcal{B}_{j}$ amplitudes,
$\mathcal{B}_{j,\lambda^{\prime}\lambda}^{(10)}=\delta_{j0}\,\mathcal{B}_{\lambda^{\prime}\lambda}^{(10)}$.
The coefficients are then fed into the expressions for the
$\mathcal{K}_{\mathcal{G}}$ and $\mathcal{T}$ matrices. Since total angular
momentum and parity are conserved, an initial state with $S=0$, $L=J$, and
$P=\eta(-1)^{J}$ restricts the final state, with $S^{\prime}=1$, to have an
$L^{\prime}$ quantum number be $L^{\prime}=1$ for $J=0$, and $L^{\prime}=J\pm
1$ for $J>0$.

####  V.3.1 Total $\boldsymbol{J=0}$

Following the same procedure as in the previous cases, we have for $J=0$, in
which the system parity is $\eta$, the ${}^{1}S_{0}\to{{}^{3}}P_{0}$ OPE is
given by

$\displaystyle\mathcal{G}(^{3}P_{0}|^{1}S_{0})$
$\displaystyle=-\frac{\sqrt{3}\gamma_{p}}{2pq_{p}^{\star}}+\frac{\sqrt{3}\gamma_{p}}{2pq_{p}^{\star}}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)Q_{0}(\zeta_{pk})\,,$
(87)

which has a threshold expansion

$\displaystyle\mathcal{G}(^{3}P_{0}|^{1}S_{0})$
$\displaystyle=\frac{\sqrt{3}m_{k}}{\mathcal{N}q_{p}^{\star}\sqrt{\sigma_{p}}}\,p+O\\!\left(p^{2},pk\right)\,,$

which agrees with the expected behavior.

####  V.3.2 Total $\boldsymbol{J=1}$

Our final example is for $J=1$, which must be in a $-\eta$ parity states due
to the initial ${}^{1}P_{1}$ state. There are two options for the transition,
either ${}^{1}P_{1}\to{{}^{3}S_{1}}$ or ${}^{1}P_{1}\to{{}^{3}D_{1}}$. The
partial wave OPE amplitudes for these transitions are

$\displaystyle\mathcal{G}(^{3}S_{1}|^{1}P_{1})$
$\displaystyle=\frac{1}{2pq_{p}^{\star}}\left(\gamma_{p}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)-\zeta_{pk}\right)\,$
$\displaystyle\qquad-\frac{1}{2pq_{p}^{\star}}\left[\gamma_{p}\zeta_{pk}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)-\zeta_{pk}^{2}+1\right]Q_{0}(\zeta_{pk})\,,$
(88) $\displaystyle\mathcal{G}(^{3}D_{1}|^{1}P_{1})$
$\displaystyle=-\frac{1}{2\sqrt{2}pq_{p}^{\star}}\left(2\gamma_{p}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\zeta_{pk}\right)\,$
$\displaystyle\qquad+\frac{1}{2\sqrt{2}pq_{p}^{\star}}\left[2\gamma_{p}\zeta_{pk}\left(\frac{\beta_{p}\omega_{k}}{k}+\zeta_{pk}\right)+\zeta_{pk}^{2}-1\right]Q_{0}(\zeta_{pk})\,.$
(89)

We note that the threshold behavior for fixed $\sigma_{k}$, $\sigma_{p}$ is

$\displaystyle\mathcal{G}(^{3}S_{1}|^{1}P_{1})$
$\displaystyle=-\frac{1}{\mathcal{N}q_{p}^{\star}}\,k+O\\!\left(k^{2},pk\right)\,,$
$\displaystyle\mathcal{G}(^{3}D_{1}|^{1}P_{1})$
$\displaystyle=\frac{2\sqrt{2}m_{k}}{3\mathcal{N}^{2}q_{p}^{\star}\sqrt{\sigma_{p}}}\,p^{2}k+O\\!\left(p^{3}k,p^{2}k^{2}\right)\,,$

as expected.

Transitions from $\ell=1$ and $\ell^{\prime}=0$ states can be obtained by
interchanging the initial and final state, $k\leftrightarrow p$ in the above
expressions. Any higher angular momentum state can be found by the same
procedure outlined here. In the next section, we examine some applications of
the above results for the scattering of three pions.

##  VI Application – $\boldsymbol{3\pi\to 3\pi}$

As a final illustration, we apply the results in Sec. V for $3\pi\to 3\pi$
scattering, and plot the partial wave OPEs for some selected allowed quantum
numbers of three pions in various kinematic regions. We limit the total energy
of the three pion system such that inelastic processes are forbidden, i.e.
$3m_{\pi}\leq\sqrt{s}<5m_{\pi}$ where $m_{\pi}$ is the pion mass. Therefore,
the exchange amplitude consists only of pion interactions,
$m_{k}=m_{p}=m_{e}=m_{\pi}$. Furthermore, we will only consider physical
scattering kinematics, so that the physical boundary is set by Eq. (IV.2)
where all masses are set to the pion mass, i.e. $\Phi(p,k)\geq 0$ with
$\Phi(p,k)=\sigma_{k}\sigma_{p}(s+3m_{\pi}^{2}-\sigma_{k}-\sigma_{p})-m_{\pi}^{2}(s-m_{\pi}^{2})^{2}$.

We also assume the isospin limit for pions, that is the pions have a flavor
symmetry characterized by their isospin $I_{\pi}=1$ and $G$ parity
$G_{\pi}=-1$. Isospin symmetry restricts the allowed partial wave
contributions for the three-pion system. Two pions are in either $I=0$, $1$,
or $2$ states with positive $G$ parity. Bose symmetry restricts even partial
waves, e.g. $S$ and $D$ waves, to be in either an $I=0$ or $2$ state, whereas
odd waves, e.g. $P$ waves, can only be in the $I=1$ state. For three pions
systems, which have negative $G$ parity, the allowed total isospin
representations are $I_{3\pi}=0,1,2,3$. There are multiple contributing three-
pion partial waves per target $J^{P}$, we summarize the lowest allowed three-
pion waves in Tab. 1 for each total isospin $I_{3\pi}$, total angular momentum
$J\leq 1$, and up through two-pions in relative $P$ wave. We label a three-
pion partial wave with $([\pi\pi]_{\ell}^{I}\pi)_{L}$, where the two-pion
system is in a relative $\ell$ wave and isospin $I$, and the pair-spectator
pion is in an orbital angular momentum $L$, e.g. $([\pi\pi]_{P}^{1}\pi)_{S}$
describes a three pion system where two of the pions are in an isovector $P$
wave and the recoiling pion is in a relative $S$ wave with the pair.

Table 1: Contributions of $([\pi\pi]_{\ell}^{I}\pi)_{L}$ partial waves to $\pi\pi\pi$ in total isospin $I_{3\pi}$ and for total angular momentum $J\leq 1$. Lowest angular momenta are considered, where the two pion pairs are in $\ell\leq P$ wave, and the orbital angular momentum between the pair and the spectators is $L\leq D$ wave. $I_{3\pi}^{G}$ |  $J^{PC}$ |  $([\pi\pi]_{\ell}^{I}\pi)_{L}$
---|---|---
$3^{-}$ |  $0^{-+}$ |  $([\pi\pi]_{S}^{2}\pi)_{S}$
|  $1^{-+}$ |  none
|  $1^{++}$ |  $([\pi\pi]_{S}^{2}\pi)_{P}$
$2^{-}$ |  $0^{--}$ |  $([\pi\pi]_{S}^{2}\pi)_{S}$, $([\pi\pi]_{P}^{1}\pi)_{P}$
|  $1^{--}$ |  $([\pi\pi]_{P}^{1}\pi)_{P}$
|  $1^{+-}$ |  $([\pi\pi]_{S}^{2}\pi)_{P}$, $([\pi\pi]_{P}^{1}\pi)_{S}$, $([\pi\pi]_{P}^{1}\pi)_{D}$
$1^{-}$ |  $0^{-+}$ |  $([\pi\pi]_{S}^{0,2}\pi)_{S}$, $([\pi\pi]_{P}^{1}\pi)_{P}$
|  $1^{-+}$ |  $([\pi\pi]_{P}^{1}\pi)_{P}$
|  $1^{++}$ |  $([\pi\pi]_{S}^{0,2}\pi)_{P}$, $([\pi\pi]_{P}^{1}\pi)_{S}$, $([\pi\pi]_{P}^{1}\pi)_{D}$
$0^{-}$ |  $0^{--}$ |  $([\pi\pi]_{P}^{1}\pi)_{P}$
|  $1^{--}$ |  $([\pi\pi]_{P}^{1}\pi)_{P}$
|  $1^{+-}$ |  $([\pi\pi]_{P}^{1}\pi)_{S}$, $([\pi\pi]_{P}^{1}\pi)_{D}$
Figure 8: Real and imaginary parts of the OPE for $I_{3\pi}=3$ and total spin-
parity $J^{P}=0^{-}$ and $1^{+}$ for $\pi\pi$ sub-channels in an isotensor $S$
wave state. Each panel is plotted for a fixed initial
$\sqrt{\sigma_{k}}=2.1m_{\pi}$. Each row is plotted at a different
$\sigma_{p}$: $\sqrt{\sigma_{p}}=2.1m_{\pi}$ for the top row,
$\sqrt{\sigma_{p}}=2.2m_{\pi}$ for the middle row, and
$\sqrt{\sigma_{p}}=2.3m_{\pi}$ for the bottom row. The spectroscopic label
${}^{2S+1}L_{J}$ indicates the partial wave OPE contribution to the
$([\pi\pi]_{S}^{2}\pi)_{L}$ amplitude. Thresholds are indicated for the $3\pi$
production at $s=(3m_{\pi})^{2}$, the effective initial pair-spectator
$\xi_{k}\pi$ at $s=(\sqrt{\sigma_{k}}+m_{\pi})^{2}=(3.1m_{\pi})^{2}$, and the
effective initial pair-spectator $\xi_{p}\pi$ at
$s=(\sqrt{\sigma_{p}}+m_{\pi})^{2}$, which is located at $s=(3.1m_{\pi})^{2}$,
$(3.2m_{\pi})^{2}$, and $(3.3m_{\pi})^{2}$ for each $\sigma_{p}$ shown. Figure
9: Same as Fig. 8 for $I_{3\pi}=2$ and $J^{P}=0^{-}$. Here there are two kinds
of $\pi\pi$ pairs, one which is in an isotensor $S$ wave, with a pair-
spectator system in ${}^{1}S_{0}$, and another pair in an isovector $P$ wave
in which the pair-spectator system is ${}^{3}P_{0}$. These waves are allowed
to mix through total angular momentum and isospin conservation, leading to a
non-zero ${}^{1}S_{0}\to{{}^{3}P_{0}}$ amplitude. Figure 10: Same as Fig. 8
for $I_{3\pi}=1$ and $J^{P}=1^{+}$. Shown are the contributions from isovector
$P$ wave $\pi\pi$ pairs. Figure 11: Same as Fig. 8 for $I_{3\pi}=1$ and
$J^{P}=1^{+}$, where now the contributions from isoscalar $S$ wave $\pi\pi$
pairs as well as the mixing to isovector $P$ wave pairs are shown. Figure 12:
Same as Fig. 8 for $I_{3\pi}=0$ and $J^{P}=1^{-}$. Only isovector $P$ wave
$\pi\pi$ pairs are allowed to couple to ${}^{3}P_{1}$ amplitudes in this
channel.

Our results in Sec. V can be applied immediately to these partial waves, with
the exception of the inclusion of appropriate isospin recoupling coefficients.
These can be included in a straightforward manner as detailed in Refs. Ascoli
and Wyld (1975); Hansen  _et al._ (2020). The result is the OPE in Eq. (64)
contains three additional quantum numbers,

$\displaystyle\left[\mathcal{G}^{(\ell^{\prime}\ell),J}_{L^{\prime}S^{\prime},LS}\,\right]_{I^{\prime},I}^{I_{3\pi}}$
$\displaystyle=\mathcal{G}^{(\ell^{\prime}\ell),J}_{L^{\prime}S^{\prime},LS}\,\braket{I^{\prime},I_{3\pi}}{I,I_{3\pi}}$
(90)

where $I$ is the initial pair isospin, $I^{\prime}$ is the final pair isospin,
and $I_{3\pi}$ is the total isospin of the three pion system. The
mutiplicative factor $\braket{I^{\prime},I_{3\pi}}{I,I_{3\pi}}$ is the three
pion isospin recoupling coefficient, which is defined in terms of the Wigner
6-j symbol as Varshalovich  _et al._ (1988)

$\displaystyle\braket{I^{\prime},I_{3\pi}}{I,I_{3\pi}}$
$\displaystyle=\sqrt{(2I^{\prime}+1)(2I+1)}\,\begin{Bmatrix}1&1&I\\\
1&I_{3\pi}&I^{\prime}\end{Bmatrix}\,.$ (91)

Explicit values can be found in Refs. Ascoli and Wyld (1975); Hansen  _et al._
(2020), or by direct computation via Eq. (91). The isospin recoupling
coefficients introduce a weight factor for the particular isospin channel.

We plot a representative partial wave OPE of each isospin channel to show the
generic behavior. For each channel, we plot the OPE as a function of $s$ in
the range $9\leq s/m_{\pi}^{2}\leq 14$ at fixed
$\sqrt{\sigma_{k}}/m_{\pi}=2.1$ and for three values of $\sigma_{p}$,
$\sqrt{\sigma_{p}}/m_{\pi}=\\{2.1,2.2,2.3\\}$. In each plot, we highlight the
$3\pi$ threshold at $s=(3m_{\pi})^{2}$, the initial pair-spectator threshold
which we indicate by $\xi_{k}\pi$ at $s=(\sqrt{\sigma_{k}}+m_{\pi})^{2}$,
where we remind the reader that $\xi_{k}$ represents a quasi-particle of mass
$\sqrt{\sigma_{k}}$, and the final pair-spectator threshold $\xi_{p}\pi$ at
$s=(\sqrt{\sigma_{p}}+m_{\pi})^{2}$. Note that when both
$\sigma_{k}=\sigma_{p}$, then the initial and final pair-spectator thresholds
overlap.

In our numerical evaluation of the partial wave OPE, we ensure that we
approach the real energy axes by introducing an artificial imaginary shift. To
control the limit, we introduce for $\sigma_{k}$, and $\sigma_{p}$ a shift
$\sigma\to\sigma+i\epsilon_{\sigma}$, and for $s$ we introduce a shift $s\to
s+i\epsilon_{s}$, with the restriction that $\epsilon_{s}>\epsilon_{\sigma}$,
meaning we assume that we approach the real $\sigma$-axes first before
approach the real $s$-axis. This limiting procedure then gives a positive
imaginary part to the spectator momentum for unphysical energies. To ensure
the correct behavior required by $S$ matrix unitarity, we set
$\zeta_{pk}\to\zeta_{pk}+i\epsilon$ appearing in the argument of the $Q$
functions as discussed with Eq. (55). To ensure the proper behavior for the
imaginary part of $\mathcal{G}$ required by $S$ matrix unitarity, we restrict
$\epsilon>\epsilon_{s}>\epsilon_{\sigma}$.

First, we consider the $I_{3\pi}=3$ channel, which from Tab. 1 the lowest
waves include $J^{P}=0^{-}$ and $1^{+}$. The real and imaginary parts of both
the $\mathcal{G}(^{1}S_{0}|^{1}S_{0})$ and $\mathcal{G}(^{1}P_{1}|^{1}P_{1})$
amplitudes are shown in Fig. 8. There is a clear movable singularity in both
amplitudes which arise from when the exchanged pion goes on-mass-shell, that
is when $\zeta_{pk}=\pm 1$. The analytic structure of the OPE has been well
studied in the literature, see for example Ref. Jackura  _et al._ (2019), so
we only highlight a few important features. In the physical kinematic region,
the imaginary part of each partial wave OPE is constrained by $S$ matrix
unitarity, through the imaginary part of $Q_{0}$, giving

$\displaystyle\operatorname{Im}\mathcal{G}^{J^{P}}=-\frac{\pi}{2}\,\mathcal{T}^{J^{P}}\,\Theta(\Phi(p,k))\Theta(p)\Theta(k)\,,$
(92)

where $\Theta$ is the Heaviside step function. Including isospin, we multiply
Eq. (92) by the recoupling coefficient (91). For fixed $\sigma_{p}$ and
$\sigma_{k}$, we can solve $\zeta_{pk}=\pm 1$ for the branch points in $s$
which are given by

$\displaystyle s^{(\pm)}$
$\displaystyle=\frac{1}{2m_{\pi}^{2}}\Bigg{[}\,(\sigma_{k}-m_{\pi}^{2})(\sigma_{p}-m_{\pi}^{2})+m_{\pi}^{2}(\sigma_{p}+\sigma_{k}+m_{\pi}^{2})\,$
$\displaystyle\qquad\qquad\pm\lambda^{1/2}(m_{\pi}^{2},m_{\pi}^{2},\sigma_{k})\lambda^{1/2}(m_{\pi}^{2},m_{\pi}^{2},\sigma_{p})\,\Bigg{]}\,.$
(93)

These movable branch points correspond to the non-zero imaginary part of the
OPE above the highest pair-spectator threshold in Fig. 8. The existence of
these branch points is independent of the partial wave of the OPE, as seen in
the ${}^{1}S_{0}$ and ${}^{1}P_{1}$ amplitudes of Fig. 8.

Next we examine three pions in $I_{3\pi}=2$ in the $J^{P}=0^{-}$ channel,
which is shown in Fig. 9. According to Tab. 1, two types of pairs contributes
to this partial wave, $([\pi\pi]_{S}^{2}\pi)_{S}$ and
$([\pi\pi]_{P}^{1}\pi)_{P}$. Therefore, we have three contributing OPE
amplitudes, ${}^{1}S_{0}$ for isotensor pairs, ${}^{3}P_{0}$ for isovector
pairs, and a mixing amplitude between isotensor and isovector pairs in
${}^{1}S_{0}\to^{3}P_{0}$. As in the $I_{3\pi}=3$ case, each amplitude has a
non-analytic structure arising from on-shell pion exchange with branch points
given by Eq. (VI).

Figures 10 and 11 show partial wave OPE amplitudes which contribute to the
$J^{P}=1^{+}$ channel of $I_{3\pi}=1$. Figure 10 show the contributions coming
from isovector $P$ wave $\pi\pi$ pairs, while Fig. 11 shows contributions from
the isoscalar $S$ wave pairs and its mixing with isovector $P$ wave pion
pairs. Note that we do not plot contributions from isotensor pion pairs which
are also part of this wave as shown in Tab. 1. Physically, this case is most
relevant for $3\pi$ scattering in the isovector $a_{1}$ channel, which allows
for dynamical mixing between resonating $\rho\pi\leftrightarrow\sigma\pi$
systems, cf. the $a_{1}$ listing in Ref. Workman  _et al._ (2022) and
references therein. Notice that for the ${}^{3}S_{1}$ amplitude in Fig. 10
that in the top panel with $\sigma_{k}=\sigma_{p}$, the scaling behavior at
the pair-spectator threshold does grow from zero as $p^{2}$ as indicated in
our threshold expansion discussed in Sec. V.2.3. However, when
$\sigma_{p}\neq\sigma_{k}$ as in the middle and bottom panel, then the
threshold behavior does approach a constant at the $\xi_{p}\pi$ threshold.

Finally, we show in Fig. 12 the $I_{3\pi}=0$, $J^{P}=1^{-}$ channel. Table 1
list one entry for this channel, $([\pi\pi]_{P}^{1})_{P}$, therefore only the
${}^{3}P_{1}$ partial wave OPE contributes. A physical application of for this
amplitude is in the isoscalar $\omega$ meson, which couples strongly to the
$\rho\pi$ channel in $P$ wave.

##  VII Summary

We have shown a generic procedure to project relativistic scattering
amplitudes of three spinless particles to definite $J^{P}$ partial waves, with
focus on it application to the kinematic singularity arising from on-shell
particle exchanges between two-body sub-channel scattering processes. The
procedure as presented in Sec. IV, specifically in the final projection
results shown in Eqs. (64), (IV.2), and (IV.2) allow one to systematically
compute the contribution from the one-particle exchange, which was illustrated
in Sec. V for some low-lying spins of immediate interest, e.g. in the
scattering of three pions as discussed in Sec. VI. These results can then be
supplied into the corresponding integral equations Hansen and Sharpe (2015);
Jackura (2023), along with some parameterized and constrained three-body $K$
matrix, e.g. ones constrained from lattice QCD calculations with finite-volume
quantization conditions, to reconstruct the complete on-shell
$\mathbf{3}\to\mathbf{3}$ hadronic scattering amplitude.

Our resulting analytic representation for the one-particle exchange of
definite $J^{P}$ allows one to avoid performing a numerical integration over a
singular function, which is generally slowly convergent, and allows the
practitioner to have full control over the analytic behavior in the complex
$s$-, $\sigma_{p}$-, and $\sigma_{k}$-planes. Controlling the analytic
structure of aspects of three-body amplitudes has been shown to be important
for the analytic continuation of the $\mathbf{3}\to\mathbf{3}$ amplitude, e.g.
for searching for resonant structures in hadron spectroscopy Dawid  _et al._
(2023a, b). Looking forward, our results can be immediately used in the
community in further theoretical and phenomenological studies of three-hadron
resonance production. Furthermore, they can be extended to accomadate a more
general class of reactions, such as those with external particles with
arbitrary spin.

## Acknowledgements

The authors would like to thank S. Dawid, J. Dudek, M. Hansen, V. Mathieu, F.
Romero-López, S. Sharpe, and A. Szczepaniak for useful comments and
discussions. AWJ acknowledges the support of the USDOE ExoHad Topical
Collaboration, contract DE-SC0023598. RAB acknowledges the support of the
USDOE Early Career award, contract DE-SC0019229.

##  Appendix A Recapitulation of Angular Momentum Functions

This appendix is devoted to collecting useful identities and properties of the
Legendre functions and Wigner rotation matrix elements which we use throughout
this work. While these relations can be found in the literature (which we
refer to as appropriate), we feel that this summary serves to assist the
reader in understanding the technical aspects of our work.

###  A.1 Legendre Functions of the $\boldsymbol{1\textsuperscript{st}}$-kind

The Legendre functions of the 1st kind, $P_{\ell}(z)$, are the regular
solutions of Legendre’s differential equation Abramowitz and Stegun (1964),
which can be expressed explicitly for $\ell\in\mathbbm{N}_{0}$ and $-1\leq
z\leq+1$ by the Rodrigues’ formula,

$\displaystyle
P_{\ell}(z)=\frac{1}{2^{\ell}\,\ell!}\,\frac{\textrm{d}^{\ell}}{\textrm{d}z^{\ell}}(z^{2}-1)^{\ell}\,.$
(94)

We consider only $\ell\in\mathbbm{N}_{0}$, therefore the functions are
analytic in $z\in\mathbbm{C}$ for each $\ell$. The Legendre functions form an
orthogonal set of functions over the interval $-1\leq z\leq+1$,

$\displaystyle\int_{-1}^{+1}\textrm{d}z\,P_{\ell^{\prime}}(z)P_{\ell}(z)=\frac{2}{2\ell+1}\,\delta_{\ell^{\prime}\ell}\,.$
(95)

The first few Legendre functions are

$\displaystyle P_{0}(z)=1\,,\quad P_{1}(z)=z\,,\quad
P_{2}(z)=\frac{1}{2}(3z^{2}-1)\,.$

Given $P_{0}$ and $P_{1}$, all remaining $P_{\ell}$ can be generated through
the Bonnet recursion relation for $\ell>1$,

$\displaystyle\ell
P_{\ell}(z)=z(2\ell-1)\,P_{\ell-1}(z)-(\ell-1)\,P_{\ell-2}(z)\,.$ (96)

There are many additional properties and identities which can be found in Ref.
Abramowitz and Stegun (1964). Here we state one integral relation,

$\displaystyle\int_{-1}^{+1}\\!\textrm{d}z\,z^{\ell}P_{\ell}(z)=\frac{2^{\ell+1}(\ell!)^{2}}{(2\ell+1)!}\,,$
(97)

which is useful in the asymptotic expansion of the Legendre functions of the
2nd kind as discussed next in section.

###  A.2 Legendre Functions of the $\boldsymbol{2\textsuperscript{nd}}$-kind

A second class of solutions of Legendre’s differential equation are the
Legendre functions of the 2nd kind, $Q_{\ell}(z)$. For every
$\ell\in\mathbbm{N}_{0}$, the $Q_{\ell}$ functions are related to the
$P_{\ell}$ functions through the Neumann relation Abramowitz and Stegun
(1964),

$\displaystyle
Q_{\ell}(z)=\frac{1}{2}\int_{-1}^{+1}\\!\textrm{d}z^{\prime}\,\frac{P_{\ell}(z^{\prime})}{z-z^{\prime}}\,.$
(98)

The integral has endpoint singularities, leading to branch points in
$Q_{\ell}$ in the complex $z$-plane at $z=\pm 1$ for each $\ell$. We choose to