# Three-body scattering and quantization conditions
from $S$ matrix unitarity

Andrew W. Jackura<EMAIL_ADDRESS>Thomas Jefferson National Accelerator
Facility, 12000 Jefferson Avenue, Newport News, VA 23606, USA  Department of
Physics, Old Dominion University, Norfolk, Virginia 23529, USA

(August 27, 2024)

###### Abstract

Two methodologies have been presented in the literature which connect
relativistic three-particle scattering amplitudes with lattice QCD spectra –
the “relativistic effective field theory” approach and the “finite-volume
unitarity” method. While both methods have been shown to be equivalent in
various works, it has not been shown how to arrive at the relativistic
effective field theory results directly from $S$ matrix unitarity. In this
work, we provide a simple proof of the relativistic effective field theory
form of the scattering equations directly from unitarity. Motivated by the
finite-volume unitarity approach, we then postulate a set of quantization
conditions which relate the finite-volume energy spectra to the $K$ matrices
which drive the short-distance physics in the scattering equations, obtaining
all previously known results for three identical particles. This work also
presents new relations which provide a new pathway to generalize the results
to arbitrary systems.

††preprint: JLAB-THY-22-3664

##  I Introduction

Calculating few-body dynamics from first principles Quantum Chromodynamics
(QCD) has become increasingly necessary to resolve many outstanding problems
in nuclear and particle physics. Algorithmic and theoretical advancements in
lattice QCD have made it possible to compute a number of non-perturbative
observables relevant for the study of low-energy phenomena. While lattice QCD
is the central tool for accessing strongly interacting few-body systems,
additional concepts are needed to link the resulting finite-volume observables
to infinite-volume scattering and transition amplitudes. Fortunately, a
synergistic approach between lattice QCD and scattering theory has resulted in
a rigorous pathway to connect the two classes of observables.

The key insight, first proposed by Lüscher Luscher (1986a, b, 1991), involves
constructing non-perturbative mappings between finite-volume observables and
scattering amplitudes. These mappings, often called _quantization conditions_
, have been developed for any $\mathbf{2}\to\mathbf{2}$ 111We adopt the
notation $\mathbf{n}\to\mathbf{m}$ which indicates a scattering process with
$\mathbf{n}$ incoming and $\mathbf{m}$ outgoing stable hadrons. scattering
processes 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), and have been successfully applied to access various hadronic
reactions from 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._ (2018a); 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._ (2020); Rendon  _et
al._ (2020); Woss  _et al._ (2021); Hörz  _et al._ (2021). 222See Ref. Briceno
_et al._ (2018b) for a recent review on the subject. In addition to standard
two-particle systems, there has been headway in determining matrix elements
involving external currents Lellouch and Luscher (2001); Kim  _et al._ (2005);
Christ  _et al._ (2005); Hansen and Sharpe (2012); Briceño  _et al._ (2015a);
Briceño and Hansen (2015); Agadjanov  _et al._ (2016); Bernard  _et al._
(2012); Briceño and Davoudi (2013); Briceño and Hansen (2016); Baroni  _et
al._ (2019); Briceño  _et al._ (2019a, b, 2021a, 2020, 2021b); Sherman  _et
al._ (2022); Lozano  _et al._ (2022), with applications for decays and
resonance transition amplitudes literature Blum  _et al._ (2012); Boyle  _et
al._ (2013); Feng  _et al._ (2015); Blum  _et al._ (2015); Bai  _et al._
(2015); Briceño  _et al._ (2015b, 2016); Feng  _et al._ (2019); Alexandrou
_et al._ (2018).

At the forefront of this venture is analyzing three-particle systems, which
has seen rapid theoretical developments within the past decade. 333Recent
reviews on this emerging topic include Refs. Hansen and Sharpe (2019); Mai
_et al._ (2021a). Much of the research thus far has been on constructing the
quantization conditions to link the three particle spectrum to
$\mathbf{3}\to\mathbf{3}$ scattering amplitudes. A complication that arises is
the number of degrees-of-freedom compared to the simpler two-particle case.
Because of this, the framework for computing $\mathbf{3}\to\mathbf{3}$
amplitudes is considerably more challenging, involving multiple intermediary
steps such the need to solve integral equations for the on-shell amplitude.

Two methodologies have emerged in the literature to derive the quantization
conditions and the relativistic integral equations. 444In addition to these
apporoaches, Refs. 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) have looked at the problem using generic non-
relativistic effective field theories. Since our focus is on the relativistic
framework, we will not discuss them further. The first is the relativistic
effective field theory (RFT) method, which generates the condition by summing
on-shell projected generalized Feynman diagrams to all-orders Hansen and
Sharpe (2014, 2015, 2017); Briceno  _et al._ (2017); Briceño  _et al._ (2018);
Briceno  _et al._ (2019); Briceño  _et al._ (2019c); Blanton  _et al._ (2019);
Hansen  _et al._ (2020); Blanton and Sharpe (2020a). On-shell physics is
driven by singularities in the physical energy region of interest, which in
turn cause the leading finite-volume effects. By isolating the volume
dependence, a set of quantization conditions are derived as well as a set of
integral equations in the infinite-volume limit. The original derivation
focused on three-identical particles, but has since been extended to include
coupled $\mathbf{2}\to\mathbf{3}$ processes Briceno  _et al._ (2017), as well
as for distinguishable scalar particles Blanton and Sharpe (2021a, b), three
pions with any isospin Hansen  _et al._ (2020), and transition amplitudes
Hansen  _et al._ (2021a). The latest results for non-degenerate scalars use a
modified form of the RFT method which uses time ordered perturbation theory
(TOPT) instead of covariant perturbation theory Blanton and Sharpe (2020a,
2021a, 2021b). Applications of this framework have appeared in the literature
Briceño  _et al._ (2018); Blanton  _et al._ (2020); Romero-López  _et al._
(2019); Fischer  _et al._ (2020); Jackura  _et al._ (2020), including the
first determination of the energy-dependent $3\pi^{+}\to 3\pi^{+}$ amplitude
Hansen  _et al._ (2021b).

The second approach taken to derive the scattering and finite-volume
frameworks is the finite-volume unitarity (FVU) method. Motivated in the fact
that on-shell physics is responsible for the dominant finite-volume effects,
the FVU approach derives an on-shell representation for the scattering
amplitude directly from $S$ matrix unitarity, which constrains all possible
on-shell intermediate states Mai  _et al._ (2017); Jackura  _et al._ (2019a);
Mikhasenko  _et al._ (2019); Dawid and Szczepaniak (2021). Unitarity has long
been used to reconstruct the non-analytic structure of scattering amplitudes
Eden  _et al._ (1966); Gribov (2008), with applications for three-body systems
primarily in the phenomenological study of hadronic decays Fleming (1964);
Holman (1965); Aaron and Amado (1966); Grisaru (1966); Aitchison and Pasquier
(1966); Ascoli and Wyld (1975); Mikhasenko  _et al._ (2018). The resulting
scattering integral equation is driven by the $K$ matrix, which is an unknown
object that contains all short-distance physics which cannot go on-mass-shell.
The on-shell representation is then used to suggest a finite-volume
counterpart, where all continuous integrations are replaced by discrete sums
over quantized momenta, giving the quantization condition as the point where
the finite-volume amplitude has a pole singularity Mai and Doring (2017);
Döring  _et al._ (2018); Mai and Doring (2019). As with the RFT formalism, the
FVU equations have been used in numerical studies of various systems Mai and
Doring (2019); Culver  _et al._ (2020); Hörz and Hanlon (2019); Sadasivan  _et
al._ (2020); Alexandru  _et al._ (2020), which includes the first calculation
of the $a_{1}(1260)$ resonance pole position from lattice QCD Mai  _et al._
(2021b).

Although both approaches use on-shell physics as a basis for their frameworks,
the resulting scattering equations and quantization conditions appear
structurally different. Since their original inceptions, it has been shown
that both the integral equations Jackura  _et al._ (2019b) and the finite-
volume conditions Blanton and Sharpe (2020b) are equivalent by fixing the
$\mathbf{3}\to\mathbf{3}$ amplitude as the observable and deducing relations
between the different short-distance objects that appear in each framework.
While equivalent, it has yet to be shown how to derive the RFT form of the
equations by using $S$ matrix unitarity alone as in the FVU approach. The
absence of such a direct proof has prevented a conceptual unification of both
approaches, even though both rely on on-shell physics.

Our objective here is to provide that conceptual unification, where we show
how to directly access the RFT form of the integral equations and quantization
conditions by considering the consequences of $S$ matrix unitarity alone.
While we follow the FVU methodology, we present a modification which allows us
to immediately access the scattering equations using the known results from
$\mathbf{2}\to\mathbf{2}$ scattering as the structural template. Our viewpoint
is that the RFT and FVU methods should not be thought of two separate
approaches, but rather adjacent procedures which could be used
interchangeably. Following this procedure, we generate new relations for the
scattering integral equations and quantization conditions, which may prove
useful for numerical applications

In this work, we strengthen the relationship between the FVU and RFT methods
by using the the FVU approach to derive the RFT result. In doing so, we
present a streamlined procedure to obtain the infinite-volume scattering
amplitudes and finite-volume quantization conditions, which provides an
outline on how it can be generalized to accommodate systems with any number of
incoming and outgoing particles. We present an overview of the main results in
Sec. II, highlighting new forms of the integral equations and quantization
conditions. In Sec. III, we define the $\mathbf{3}\to\mathbf{3}$ $S$ matrix
elements and kinematic constraints, and give the unitarity constraint in Sec.
III.1. From the unitarity constraint, in Sec. III.2 we propose an on-shell
representation for the $\mathbf{3}\to\mathbf{3}$ scattering amplitude in terms
of an integral equation. A new form for the integral equation presented here,
which may prove useful for numerical applications. The infinite-volume
scattering equations provide the necessary structure to derive the
quantization conditions in terms of the unknown $K$ matrices.

We then embed the system in a finite cubic volume in Sec. IV, and write the
corresponding finite-volume amplitudes which replace all on-shell
singularities by their finite-volume counterparts which give the dominant
volume dependence. Quantizations conditions follow by identifying the pole
structure of the finite-volume amplitudes, and agree with the ones derived
using the TOPT version of the RFT method. Section V provides new
representation of the scattering equations and quantization conditions by
performing a few simple operations on the result given in Sec. III.2.
Additionally, we show how this new form arises naturally from the unitarity
relations of the connection $\mathbf{3}\to\mathbf{3}$ amplitude in Sec. V.1,
and provides the key physical insight to the difference between the original
FVU and RFT formalisms. This new set of equations are closely related to the
original framework proposed by Ref. Hansen and Sharpe (2014, 2015), and we
show how to recover them by using simple symmetrization identities in Sec.
V.2. Finally we summarize our approach in Sec. VI.

##  II Overview

Before detailing the derivation of the results, we present a summary of the
main points and introduce new forms for the $\mathbf{3}\to\mathbf{3}$ integral
equations and quantization conditions. We seek elements of the $S$ matrix
subject to the unitarity constraint $S^{{\dagger}}S=1$. The $S$ matrix can be
decomposed as $S=1+iT$, where the interacting $T$ matrix has fully
disconnected contributions removed. 555Fully disconnected contributions are
those pieces of $S$ matrix where all particles are causally separated such
that they do not interact. For $\mathbf{3}\to\mathbf{3}$ processes, the $T$
matrix contains additional disconnected terms where two of the particles
interact and the third remains causally separated Weinberg (2005), which is
discussed at the end of this section. The unitarity condition for the $T$
matrix is given by

$\displaystyle T-T^{{\dagger}}=iT^{{\dagger}}T\,,$ (1)

which constrains the analytic structure of scattering amplitudes Eden  _et
al._ (1966). Matrix elements of Eq. (1) lead to a non-linear relation between
the imaginary part of the scattering amplitude and a product of amplitudes
with an infinite number of intermediate processes. If we limit the allowed
total energy to the elastic scattering region, in our case energies above the
three-particle threshold and below the first inelastic threshold, then higher
multi-particle intermediate states are kinematically inaccessible, and the
unitarity condition contains only a finite number of terms.

We exploit Eq. (1) to construct on-shell representations of the
$\mathbf{3}\to\mathbf{3}$ scattering amplitude. As detailed in Sec. III, the
resulting constraints for the $T$ matrix elements are satisfied by linear
integral equations driven by an unknown dynamical function called the $K$
matrix,

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(2)

where we have written the $T$ matrix element in the $k\ell m_{\ell}$ basis,
meaning two of the particles are coupled into a pair with definite angular
momentum $\ell$ and projection $m_{\ell}$, while the third remains a spectator
with momentum $\mathbf{k}$. 666The amplitude is also a function of the total
momentum of the system $P$, which we leave implicit. Both the initial and
final states for all objects in Eq. (II) are expressed in this basis, and thus
the relation is a matrix equation in angular momentum space (with indices
implicit), while it is an integral equation in the spectator momenta where we
have introduced the shorthand notation for integrations

$\displaystyle\int_{k}\equiv\int\\!\frac{\textrm{d}^{3}\mathbf{k}}{(2\pi)^{3}\,2\omega_{k}}\,.$

Explicitly, the $T$ matrix elements are defined as

$\displaystyle\bra{P^{\prime},\mathbf{p},\ell^{\prime}m_{\ell^{\prime}}}T\ket{P,\mathbf{k},\ell
m_{\ell}}\equiv(2\pi)^{4}\delta^{(4)}(P^{\prime}-P)\,\mathcal{T}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$

where $P=P^{\prime}$ is the total energy-momentum of the three-body system and
is conserved. We use the $(u,u)$ notation of Refs. Hansen and Sharpe (2014,
2015), which indicate that the amplitude is an asymmetric object, meaning we
have singled out the momenta $\mathbf{k}$ and $\mathbf{p}$ as the only
possible spectators. For our purposes, the asymmetric amplitude in the pair-
spectator basis is the central object, however the fully symmetric
$\mathbf{3}\to\mathbf{3}$ amplitude can be recovered by contracting the
angular momentum indices over spherical harmonics of the pair decay angles,
and summing over all possible spectators as described in Sec. III.

The $T$ matrix element contains a known singularity associated with the on-
shell spectator particle which does not interact with the pair, forcing the
separation

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\delta(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})+\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$
(3)

where $\mathcal{M}_{2}$ is the connected $\mathbf{2}\to\mathbf{2}$ amplitude
of the pair, $\sigma_{k}\equiv(P-k)^{2}$ is the invariant mass squared of the
pair, and $\mathcal{M}_{3}^{(u,u)}$ is the connected $\mathbf{3}\to\mathbf{3}$
scattering amplitude. We have also introduced the convenient shorthand
$\delta(\mathbf{p},\mathbf{k})\equiv(2\pi)^{3}\,2\omega_{k}\,\delta^{(3)}(\mathbf{p}-\mathbf{k})$
for the spectator with $\omega_{k}=\sqrt{m^{2}+\mathbf{k}^{2}}$ being its on-
shell energy. The $\mathbf{2}\to\mathbf{2}$ scattering amplitude
$\mathcal{M}_{2}$ can be written in terms of the two-particle $K$ matrix
$\mathcal{K}_{2}$ and the kinematic function $\mathcal{I}$ by the on-shell
representation $\mathcal{M}_{2}^{-1}=\mathcal{K}_{2}^{-1}+\mathcal{I}$. We
define $\mathcal{I}$ explicitly in Sec. III.2, but note here that it is a
known function associated with the two-particle kinematic phase space
describing on-shell pair-production.

The analytic structure of the $T$ matrix in Eq. (2) is governed by the
kinematic function $\Gamma$, which is a known scheme-dependent kinematic
function which has a fixed imaginary part in the physical region as imposed by
unitarity. This function contains two contributions which describe the non-
analyticities arising from on-shell intermediate states,

$\displaystyle\Gamma(\mathbf{p},\mathbf{k})\equiv\delta(\mathbf{p},\mathbf{k})\,H(\sigma_{k})\,\mathcal{I}(\sigma_{k})+\mathcal{G}(\mathbf{p},\mathbf{k})\,,$
(4)

where $H$ is a cutoff function introduced to render the integrals finite,
$\mathcal{I}$ is the two-body kinematic function, and $\mathcal{G}$ is a known
function detailing on-shell particle exchange processes between pairs, that is
it switches which particle is the spectator in the intermediate state. We
define these functions in detail in Sec. III.2, and stress here that these
functions are not unique as only their imaginary part is fixed by the
necessary non-analytic structure enforced by unitarity.

The three-body $K$ matrix is an unknown real function which contains dynamical
information of the scattering process. Since $\Gamma$ is a scheme-dependent
function, we require that the $K$ matrices are also scheme dependent in order
to compensate the behavior to ensure that the physical scattering amplitude is
scheme-independent. The scheme dependence originates from the fact that
unitarity constraint only effects on-shell physics, and we can shift off-shell
behavior from the kinematic function to the unknown $K$ matrix by redefining
the object to absorb these effects. Compared to the two-particle case,
additional scheme-dependencies come from the integration limits, which must in
general be regulated to avoid ultra-violet (UV) divergences, however these
play the same roles and we adopt the notion that any definition of the $K$
matrix is a scheme-dependent choice which does not affect the physical
amplitude. Similar to the $T$ matrix element, the $\mathbf{3}\to\mathbf{3}$
$K$ matrix also has a disconnected structure

$\displaystyle\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\delta(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}(\sigma_{k})+\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$
(5)

where the first term contains the usual $\mathbf{2}\to\mathbf{2}$ connected
$K$ matrix, and the connected $\mathbf{3}\to\mathbf{3}$ $K$ matrix is denoted
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}$, where the df stands for _divergence
free_ which is a notation borrowed from the original RFT derivation in Ref.
Hansen and Sharpe (2014). In Sec. V we show that the connected three-body $K$
matrix is the same object first introduced in Ref. Hansen and Sharpe (2014).
Using the aforementioned relations, we find a new integral equation for the
connected $\mathbf{3}\to\mathbf{3}$ amplitude

$\displaystyle\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{K}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})$
$\displaystyle-\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\,\Gamma(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})-\mathcal{K}_{2}(\sigma_{p})\int_{k^{\prime}}\Gamma(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(6)

As we show in Sec. V, the new relation Eq. (II) can be simply related to all
other previously obtained forms, most directly that of Ref. Hansen and Sharpe
(2015). This new representation allows for one to reconstruct the amplitude
for given $K$ matrices with a single integral equation, instead of the set
presented in Ref. Hansen and Sharpe (2015), which in turn allows for a more
direct computational approach. In Sec. V it is shown how to recover the
original set of integral equations of Ref. Hansen and Sharpe (2015), where we
follow a procedure of isolating all rescattering effects which do not include
three-body interactions governed by $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$.
Then, we use symmetrization identities to recover the symmetric form
$\mathcal{K}_{3,\textrm{df}}$ of Ref. Hansen and Sharpe (2014). This
connection allows us to write a similar relation to Eq. (II) in terms of the
symmetric $K$ matrix, shown in Sec. V.2.

Once on-shell representations such as Eq. (II) are constructed, we can then
embed the system in a finite cubic volume of size $L$ subject to periodic
boundary conditions. All momenta are quantized accordingly as

$\displaystyle\mathbf{k}=\frac{2\pi}{L}\mathbf{n}\,,$

where $\mathbf{n}\in\mathbbm{Z}^{3}$. Therefore, all integrals in the on-shell
relations are replaced by discrete sums over quantized momenta. We find in
Sec. IV that using the FVU approach with $T$ matrix elements yields the TOPT
version of the RFT quantization condition, which is given by

$\displaystyle\det\Big{[}\,1+\left(\widetilde{\mathcal{K}}_{2,L}+\mathcal{K}_{3,\textrm{df}}^{(u,u)}(P)\right)\cdot\left(\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)\right)\,\Big{]}_{E=E_{\mathfrak{n}}}=0\,.$
(7)

which is a determinant in $k\ell m_{\ell}$ space where $\mathbf{k}$ is fixed
by the momentum quantization $\mathbf{k}=2\pi\mathbf{n}/L$. Here
$\widetilde{F}_{2,L}$ and $\widetilde{G}_{L}$ are known functions which are
finite-volume analogues of the kinematic functions $\mathcal{I}$ and
$\mathcal{G}$, respectively, while
$\widetilde{\mathcal{K}}_{2,L}=2\omega_{k}L^{3}\,\mathcal{K}_{2}(\sigma_{k})\rvert_{\mathbf{k}=2\pi\mathbf{n}/L}$.
For a given total momentum $\mathbf{P}$, the Eq. (8) holds at the finite-
volume spectrum $E=E_{\mathfrak{n}}$. Alternatively, we can look at the
spectrum of the finite-volume analogue of $\mathcal{M}_{3}^{(u,u)}$. The
reshuffling of rescattering terms cause by the disconnected two-particle
subprocesses gives a different structure to the quantization condition, which
we find

$\displaystyle\det\Big{[}\,1+\mathcal{K}_{3,\textrm{df}}^{(u,u)}(P)\cdot
F_{3,L}^{(u,u)}(P)\,\Big{]}_{E=E_{\mathfrak{n}}}=0\,,$ (8)

where the determinant is again over $k\ell m_{\ell}$ space and
$F_{3,L}^{(u,u)}$ is a new function which depends both on the geometric nature
of the finite cubic volume as well as the dynamics of the two-particle
subsystem,

$\displaystyle
F_{3,L}^{(u,u)}(P)\equiv\left[\,\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)\,\right]\left(\,1-\frac{1}{{\widetilde{\mathcal{K}}_{2,L}}^{\,-1}+\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)}\cdot\left[\,\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)\,\right]\right)\,.$
(9)

Equation (8) is a new form of the three-particle quantization condition in
terms of the asymmetric $K$ matrix, which bears a similar structure to the one
by Ref. Hansen and Sharpe (2014) in terms of the symmetric three-body $K$
matrix. Using the same symmetrization identities on the $K$ matrices as was
done for the infinite-volume scattering amplitude, we show in Sec. V.2 that we
can recover the original symmetric version of the three-body $K$ matrix and
quantization condition.

##  III Scattering amplitudes

We consider a single scattering channel consisting of three identical spinless
particles with mass $m$, e.g. $3\pi^{+}$ elastic scattering. While the
procedure works for arbitrary types of particles, cf. Refs. Jackura  _et al._
(2019a); Mikhasenko  _et al._ (2019), we focus on the identical particle case
in order to present various relations to all previously known results. In some
general Lorentz frame 777We work in a $3+1$ Minkowski spacetime with metric
signature $\mathrm{diag}(+1,-1,-1,-1)$., the three-particle system carries a
total four-momentum $P=(E,\mathbf{P})$ with energy $E$ and momentum
$\mathbf{P}$. To suppress contributions from other multiparticle states, we
assume that (i) even-to-odd number particle transitions are forbidden by the
symmetry of the underlying interaction, e.g.
$\mathbf{2}\leftrightarrow\mathbf{3}$ transitions are prohibited, and (ii) the
total energy is restricted to a kinematic domain such that only three-particle
states can go on-shell, i.e. $(3m)^{2}\leq s<(5m)^{2}$ where $s$ is the
invariant-mass-squared of the three-particle system,

$\displaystyle s\equiv P^{2}=E^{2}-\mathbf{P}^{2}\,.$

Each particle has a four-momentum $k_{j}=(\omega_{k_{j}},\mathbf{k}_{j})$ with
$j=1,2,3$ denoting the particle, and has an energy fixed by the usual
relativistic dispersion relation $\omega_{k}=\sqrt{m^{2}+\mathbf{k}^{2}}$. For
a fixed $P$, the momentum of the system is constrained as
$P=k_{1}+k_{2}+k_{3}$ from momentum conservation.

We define the $T$ matrix element for $\mathbf{3}\to\mathbf{3}$ scattering,
$\mathcal{T}$, 888Note that Ref. Hansen and Sharpe (2015) and subsequent
references use $\mathcal{T}$ to denote a different scattering quantity, which
is discussed in Sec. V, and should not be confused here with the $T$ matrix
elements. as

$\displaystyle\bra{\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3}}T\ket{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}=(2\pi)^{4}\delta^{(4)}(P^{\prime}-P)\,\mathcal{T}\,,$
(10)

where $\ket{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}$ is the initial
three particle state of total momentum $P$, which is defined as the
symmetrized direct product over the three individual single particle states
$\ket{\mathbf{k}_{j}}$ 999States of three identical particles are defined by
the usual tensor product
$\ket{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}\equiv\frac{1}{\sqrt{3!}}\sum_{\pi\in
S_{3}}\ket{\mathbf{k}_{\pi(1)}}\otimes\ket{\mathbf{k}_{\pi(2)}}\otimes\ket{\mathbf{k}_{\pi(3)}}\,,$
where $S_{3}$ is the permutation group of order three., and
$\ket{\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3}}$ is the final state with
total momentum $P^{\prime}$ defined analogously. The amplitude depends on
eight kinematic variables constructed from the set of initial and final state
momenta
$\mathcal{P}_{k}\equiv\\{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}\\}$ and
$\mathcal{P}_{p}\equiv\\{\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3}\\}$,
respectively, with total momentum conservation enforced by the Dirac delta
function. Instead of specifying a set of eight variables, we show the
amplitude as a function of the initial and final sets of momenta. We fix the
normalization of the amplitude from the single particle states, which are
chosen to be normalized in the standard relativistic way

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

where the latter definition will prove as a convenient shorthand. Since we
consider the scattering of three identical scalars, the amplitude is a Lorentz
scalar and is symmetric under interchange of the particles due to Bose
statistics. Additionally, the amplitude is symmetric under time reversal
symmetry since we assume we work with hadrons within QCD. Note that one can
further link the scattering amplitude defined in Eq. (10) to quantum
correlation functions via the Lehmann-Symanzik-Zimmermann (LSZ) reduction
formula. Since our goal is to work solely with on-shell amplitudes via the
unitarity constraint Eq. (1), it is not necessary to specify the any details
of quantum fields for a generalized set of Feynman rules as is done in Refs.
Hansen and Sharpe (2014, 2015).

Figure 1: Scattering diagrams for (a) the $\mathbf{3}\to\mathbf{3}$ scattering
amplitude defined as the $T$ matrix element in Eq. (10) and (b) the pair-
spectator amplitude $\mathcal{T}^{(u,u)}$ defined in (12). In these and all
subsequent diagrams, momentum flow is from right to left. The single line on
the amplitude in (b) indicates the spectator whereas the double line
represents that pair coupled to a definite angular momentum state.

To simplify the degrees of freedom, it is convenient to work in a pair-
spectator basis, defined such that two of the particles couple in a pair
101010The pair associated with the spectator is also called an isobar or dimer
in the literature. with definite angular momentum $\ell$ and projection
$m_{\ell}$ which scatters against the third particle which is called the
spectator. We define such a state by singling out one particle, e.g. particle
$1$, to be the spectator and carry a momentum $\mathbf{k}$. The other two
particle constitute the pair, i.e. particles $2$ and $3$, in which we choose a
cyclic convention to fix the first particle of the pair, i.e. $2$ is the first
particle in the pair. We label the momentum of the first particle in the pair
as $\mathbf{a}_{k}$, and the second particle has a momentum
$\mathbf{b}_{k}\equiv\mathbf{P}-\mathbf{k}-\mathbf{a}_{k}$ by momentum
conservation. The subscripts $k$ serve to inform that this pair belongs to the
spectator with momentum $\mathbf{k}$. We now Lorentz boost to the pair center-
of-momentum (CM) frame, defined when
$\mathbf{P}^{\star}-\mathbf{k}^{\star}=\mathbf{0}$, where the $\star$
indicates that the kinematic variables are taken in this frame. Figure 2 shows
a cartoon of this reference frame. Then,
$\mathbf{a}_{k}^{\star}=-\mathbf{b}_{k}^{\star}\equiv
q_{k}^{\star}\,\mathbf{\hat{\mathbf{a}}}_{k}^{\star}$, where
$q_{k}^{\star}\equiv\lvert\mathbf{a}_{k}^{\star}\rvert=\sqrt{\sigma_{k}/4-m^{2}}$
is the relative momentum of the pair which is fixed by its invariant mass and
$\mathbf{\hat{\mathbf{a}}}_{k}^{\star}$ is the orientation of the first
particle, which is not fixed. We can then introduce a partial wave expansion
on this state, and use the angle $\mathbf{\hat{\mathbf{a}}}_{k}^{\star}$ to
define the angular momentum of the pair. A three particle state
$\ket{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}$ has three spectators,
which we enforce by summing over all permutations of the previous steps,
leading the to relation

$\displaystyle\ket{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}=\sqrt{4\pi}\,\sum_{\mathbf{k}\in\mathcal{P}_{k}}\sum_{\ell,m_{\ell}}\ket{P,\mathbf{k},\ell
m_{\ell}}Y_{\ell m_{\ell}}^{*}(\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,,$ (11)

where the sum is over the set of momenta
$\mathcal{P}_{k}=\\{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}\\}$ used to
recover all spectator contributions. The state $\ket{P,\mathbf{k},\ell
m_{\ell}}$ describes a system of spectator with momentum
$k=(\omega_{k},\mathbf{k})$ recoiling against a pair with momentum

$\displaystyle P_{k}\equiv P-k=(E-\omega_{k},\mathbf{P}-\mathbf{k})\,,$

which has an invariant-mass-squared

$\displaystyle\sigma_{k}\equiv
P_{k}^{2}=(P-k)^{2}=(E-\omega_{k})^{2}-(\mathbf{P}-\mathbf{k})^{2}\,.$

In a physical scattering process, the pair invariant mass can take values
$(2m)^{2}\leq\sigma_{k}\leq(\sqrt{s}-m)^{2}$. Bose symmetry imposes that
states with only even $\ell$ are allowed, hence the two particle state in the
pair CM frame has positive parity.

Figure 2: Kinematic assignment of a three particle state in the pair-spectator
basis in (a) a general moving frame with total momentum $\mathbf{P}$ and (b)
the pair CM frame $\mathbf{P}^{\star}-\mathbf{k}^{\star}=\mathbf{0}$. The
angular momentum of the pair ($\ell,m_{\ell}$) is defined in its CM frame with
respect to the angles of $\mathbf{a}_{k}^{\star}$, which has a magnitude fixed
by the pair invariant mass $\lvert\mathbf{a}_{k}^{\star}\rvert=q_{k}^{\star}$
as described in the text.

We use Eq. (11) for both the initial and final states to rewrite the symmetric
$\mathbf{3}\to\mathbf{3}$ amplitude in the pair-spectator basis,

$\displaystyle\mathcal{T}=\sum_{\mathbf{p}\in\mathcal{P}_{p}}\sum_{\mathbf{k}\in\mathcal{P}_{k}}\bigg{\\{}\,4\pi\sum_{\ell^{\prime},m_{\ell^{\prime}}}\sum_{\ell,m_{\ell}}Y_{\ell^{\prime}m_{\ell^{\prime}}}(\mathbf{\hat{\mathbf{a}}}_{p}^{\star})\,\mathcal{T}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{(u,u)}(\mathbf{p},\mathbf{k})\,Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,\bigg{\\}}\,,$ (12)

which separates the full amplitude into sum over nine pair-spectator
amplitudes $\mathcal{T}^{(u,u)}$, as shown in Fig. 3, which are defined as the
$T$ matrix elements

$\displaystyle\bra{P^{\prime},\mathbf{p},\ell^{\prime}m_{\ell^{\prime}}}T\ket{P,\mathbf{k},\ell
m_{\ell}}=(2\pi)^{4}\delta^{(4)}(P^{\prime}-P)\,\mathcal{T}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{(u,u)}(\mathbf{p},\mathbf{k})\,.$

Here we adopted the notation introduced in Ref. Hansen and Sharpe (2014) that
the superscripts $(u,u)$ indicate that this is an asymmetric amplitude
specified by spectators with momentum $\mathbf{k}$ and $\mathbf{p}$ for the
initial and final states, respectively. 111111This has been called an isobar
expansion, for instance in Ref. Mai  _et al._ (2017); Jackura  _et al._
(2019a); Dawid and Szczepaniak (2021). Four of the eight kinematic variables
are given by $\mathbf{\hat{\mathbf{a}}}_{k}^{\star}$ and
$\mathbf{\hat{\mathbf{a}}}_{p}^{\star}$, leaving four remaining degrees of
freedom to describe $\mathcal{T}^{(u,u)}$. One of those variables is the total
invariant mass $s$, or equivalently the total momentum $P$, which we suppress
while the other three can be formed from the two spectator momenta in an
appropriate frame. The amplitude $\mathcal{T}^{(u,u)}$ is similar to an
effective coupled $\mathbf{2}\to\mathbf{2}$ process of a spinless particle
scattering against a “particle” with spin $\ell$ and mass $\sqrt{\sigma_{k}}$,
which transitions to another “particle” with spin $\ell^{\prime}$ and mass
$\sqrt{\sigma_{p}}$, the primary exception being that the masses are variable.

Figure 3: The $\mathbf{3}\to\mathbf{3}$ amplitude expressed as a sum over
pair-spectator amplitudes as defined in Eq. (12). The vertices connecting the
two external lines to a double line represent the angular couplings to angular
momenta via spherical harmonics.

The $T$ matrix elements for $\mathbf{3}\to\mathbf{3}$ processes contain two
causally disconnected structures according to the cluster decomposition
principle. 121212The $\mathbf{3}\to\mathbf{3}$ $S$ matrix contains $3!=6$
fully disconnected contributions and $3\times 3=9$ disconnected contributions
when one of the particles remains causally separated Weinberg (2005). Since we
choose to work with pair-spectator amplitudes, Eq. (12), the nine combinations
of spectator momenta agrees with that of the partially disconnected terms,
giving us one term containing $\mathbf{2}\to\mathbf{2}$ amplitude in Eq. (3).
The first is where two of the particles undergo interactions while the third
remains a non-interacting spectator. The second term is the fully connected
$\mathbf{3}\to\mathbf{3}$ amplitude where there are interactions amongst all
three particles. For the pair-spectator amplitude $\mathcal{T}^{(u,u)}$, the
cluster decomposition principle imposes the structure Eq. (3), which we repeat
here

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})=\delta(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})+\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$

where $\mathcal{M}_{2}$ is the $\mathbf{2}\to\mathbf{2}$ partial wave
amplitude, $\delta(\mathbf{p},\mathbf{k})$ conserves the spectator momentum,
and $\mathcal{M}_{3}^{(u,u)}$ is the connected $\mathbf{3}\to\mathbf{3}$ pair-
spectator amplitude, cf. Fig. 4 for a diagram of this decomposition. If one
desires, a complete $\mathbf{3}\to\mathbf{3}$ connected scattering amplitude
$\mathcal{M}_{3}$ can be constructed by summing over all pair-spectator
combinations, yielding a similar relation as Eq. (12). The
$\mathbf{2}\to\mathbf{2}$ partial wave amplitude is the same as the usually
defined $T$ matrix element of the two-hadron angular momentum state with a
total momentum $P_{k}$,

$\displaystyle\bra{P_{p},\ell^{\prime}m_{\ell^{\prime}}}T\ket{P_{k},\ell
m_{\ell}}=(2\pi)^{4}\delta^{(4)}(P_{p}-P_{k})\,\mathcal{M}_{2;\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}(\sigma_{k})\,.$

The $\mathbf{2}\to\mathbf{2}$ amplitude is a diagonal matrix in angular
momentum space and independent of the projection $m_{\ell}$ due to rotational
invariance, $\mathcal{M}_{2;\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}=\delta_{\ell^{\prime}\ell}\delta_{m_{\ell^{\prime}}m_{\ell}}\,\mathcal{M}_{2;\ell}$.
Furthermore, it depends only on a single degree of freedom, which we choose as
the invariant $\sigma_{k}$. For a fixed physical $(3m)^{2}\leq s<(5m)^{2}$,
the $\mathbf{2}\to\mathbf{2}$ amplitude lies in the kinematic region
$(2m)^{2}\leq\sigma_{k}<(4m)^{2}$, which is precisely the physical domain
which closes off higher multi-particle states.

Figure 4: (a) The cluster decomposition of the $T$ matrix element Eq. (3) into
(b) the disconnected contribution involving the connected
$\mathbf{2}\to\mathbf{2}$ scattering amplitude $\mathcal{M}_{2}$, and (c) the
connected $\mathbf{3}\to\mathbf{3}$ amplitude $\mathcal{M}_{3}^{(u,u)}$.

###  III.1 Unitarity relations

Following the procedure discussed in Refs. Mai  _et al._ (2017); Jackura  _et
al._ (2019a), we form matrix elements of Eq. (1) using the basis defined by
Eq. (11) to obtain the unitarity relation for the pair-spectator amplitude.
Since energies are such that only three particle states are on-shell, we need
only consider the unitarity condition for elastic $\mathbf{3}\to\mathbf{3}$
scattering. Therefore, the unitarity relation for the amplitude
$\mathcal{T}^{(u,u)}$ is given by

$\displaystyle\operatorname{Im}\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{k^{\prime}}\,\mathcal{T}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{k}^{\prime})\,\rho(\sigma_{k^{\prime}})\,\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{T}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Delta(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(13)

which is an infinite dimensional matrix equation in angular momentum space
where the $\ell$ and $m_{\ell}$ indices are suppressed. For convenience, we
present the derivation of Eq. (III.1) is given in Appendix A. Two kinematic
functions are introduced in Eq. (III.1), the first being the usual two-body
phase space $\rho$, given by

$\displaystyle\rho_{\ell^{\prime}m_{\ell^{\prime}},\ell m_{\ell}}(\sigma_{k})$
$\displaystyle=\delta_{\ell^{\prime}\ell}\delta_{m_{\ell^{\prime}}m_{\ell}}\,\frac{\xi
q_{k}^{\star}}{8\pi\sqrt{\sigma_{k}}}\,\Theta(q_{k}^{\star})\,,$ (14)

where $\xi=1/2!$ is the symmetry factor for two identical particles. The
Heaviside function $\Theta(q_{k}^{\star})$ enforces that the intermediate
state exists when the pair CM energy allows for on-shell pair production,
which induces a square root branch cut in the $\sigma_{k}$-plane when
$\sqrt{\sigma_{k}}\geq 2m$. We call the first term of Eq. (III.1) the _direct_
term since the spectator remains unchanged in the intermediate state. The
second kinematic function is the single particle exchange pole, $\Delta$,
defined as

$\displaystyle\Delta_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}(\mathbf{p},\mathbf{k})=\pi\,$
$\displaystyle\delta\left((P-k-p)^{2}-m^{2}\right)\,$ $\displaystyle\times
4\pi
Y_{\ell^{\prime}m_{\ell^{\prime}}}^{*}(\mathbf{\hat{\mathbf{k}}}_{p}^{\star})Y_{\ell
m_{\ell}}(\mathbf{\hat{\mathbf{p}}}_{k}^{\star})\Big{\rvert}_{(P-k-p)^{2}=m^{2}}\,\Theta\left(q_{p}^{\star}\right)\Theta\left(q_{k}^{\star}\right)\,,$
(15)

which characterizes the singularity associated with exchanging an on-shell
particle between pairs. 131313The phases of the spherical harmonics cancel
each other in the unitarity condition as it sums over all possible
intermediate states. One can avoid keeping tack of all phase cancellations in
intermediate steps by considering real spherical harmonics, which respect the
same completeness and orthogonality relations as the standard one. The second
term of Eq. (III.1) will be referred to as the _exchange_ or _recoupling_ term
due to the switching of spectators in the intermediate state. The angles
associated with a spectators in each spherical harmonic are defined with
respect to the pair associated with the opposite spectators CM frame. For
example, $\mathbf{\hat{\mathbf{k}}}^{\star}_{p}$ is the orientation of the
initial state pair in the CM frame of the final state pair associated with the
spectator $p$. The two Heaviside functions enforce that the pair is on-shell,
while the Dirac delta function fixes the exchange particle to its mass-shell.
Note that the Heaviside functions limit the kinematics in the argument of the
delta function to forward propagation only, however we keep the covariant
representation for convenience. Figure 5 shows a diagrammatic representation
of Eq. (III.1), illustrating the the direct and exchange processes which cause
the non-analyticities in Eqs. (14) and (III.1), respectively. Equation (III.1)
is a non-perturbative statement for the on-shell amplitude for any interaction
satisfying the assumptions on particle content and allowed kinematic domain
discussed at the beginning of Sec. III.

Figure 5: Unitarity relation given in Eq. (III.1). The vertical dashed lines
indicate the direct (first term) and exchange (second term) cuts corresponding
to the functions $\rho(\sigma_{k^{\prime}})$ and
$\Delta(\mathbf{p}^{\prime},\mathbf{k}^{\prime})$, respectively. Each closed
loop represents a three-dimensional integral over the on-shell spectator
momentum, and we use the convention that the amplitude on the left of the
dashed line is conjugated.

In addition to unitarity for $T$ matrix elements, we can introduce separate
unitarity conditions for the connected amplitudes. Inserting Eq. (3) into
(III.1) results in the unitarity relation separating into two constraints,

$\displaystyle\operatorname{Im}\mathcal{M}_{2}(\sigma_{k})=\mathcal{M}_{2}^{{\dagger}}(\sigma_{k})\,\rho(\sigma_{k})\,\mathcal{M}_{2}(\sigma_{k})\,,$
(16)

which is the usual condition for the $\mathbf{2}\to\mathbf{2}$ partial wave
amplitude, and

$\displaystyle\operatorname{Im}\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{M}_{3}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{k^{\prime}}\,\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\Phi(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\,\mathcal{M}_{3}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})$
$\displaystyle+\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\Delta(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})\,,$
(17)

which is the condition for the connected $\mathbf{3}\to\mathbf{3}$ amplitude.
Here we have introduced a convenient shorthand, the matrix $\Phi$, which
contains both the direct and exchange kinematic functions

$\displaystyle\Phi(\mathbf{p},\mathbf{k})\equiv\delta(\mathbf{p},\mathbf{k})\,\rho(\sigma_{k})+\Delta(\mathbf{p},\mathbf{k})\,.$
(18)

Figure 6 presents a diagrammatic picture of the connected unitarity relations.
The first term of Eq. (III.1) is analogous to the $\mathbf{2}\to\mathbf{2}$
case with the addition of intermediate states particle exchanges, while the
other three terms are associated with rescattering effects of the two particle
sub-channels. The last term of Eq. (III.1) represents the pole singularity
associated with on-shell particle exchange between two
$\mathbf{2}\to\mathbf{2}$ sub-processes. Ultimately we aim to describe the
connected $\mathcal{M}_{2}$ and $\mathcal{M}_{3}$ amplitudes by constructing
suitable on-shell representations satisfying Eqs. (16) and (III.1),
respectively, as in e.g. Refs. Mai  _et al._ (2017); Jackura  _et al._
(2019a); Mikhasenko  _et al._ (2019). Instead, in the following section we
work directly with Eq. (III.1) to find an on-shell representation, and then
use Eq. (3) to find an expression for the $\mathcal{M}_{3}$ amplitude. This
direction proves to be a simpler, more direct route toward obtaining on-shell
representations for $\mathbf{3}\to\mathbf{3}$ amplitude, and builds on the
intuition learned from $\mathbf{2}\to\mathbf{2}$ scattering. In Section V, we
present alternative approaches which work with the connected unitarity
condition, and show the relations between the results found in Section III.1.

Figure 6: Diagrammatic representation of the connected unitarity conditions
for the (a) $\mathbf{2}\to\mathbf{2}$ amplitude Eq. (16), and (b) the
$\mathbf{3}\to\mathbf{3}$ amplitude Eq. (III.1).

###  III.2 On-shell scattering equations

Nonlinear constraints such as Eq. (III.1) are solved by linear equations that
are driven by a kernel, known as a $K$ matrix, which is a real function within
the restricted kinematic region which we consider and represents the short-
distance dynamics of the scattering process which are not fixed by the
unitarity condition. As an example, we first review the well-known on-shell
representation for the $\mathbf{2}\to\mathbf{2}$ amplitude that satisfies Eq.
(16). We make an ansatz for the on-shell amplitude in the linear form

$\displaystyle\mathcal{M}_{2}(\sigma_{k})=\mathcal{K}_{2}(\sigma_{k})-\mathcal{K}_{2}(\sigma_{k})\,\mathcal{I}(\sigma_{k})\,\mathcal{M}_{2}(\sigma_{k})\,,$
(19)

where $\mathcal{K}_{2}$ is the $\mathbf{2}\to\mathbf{2}$ partial wave $K$
matrix, which is an unknown real function in the elastic kinematic domain, and
$\mathcal{I}$ is a kinematic function which must fulfill the constraint

$\displaystyle\operatorname{Im}\mathcal{I}(\sigma_{k})=-\rho(\sigma_{k})$

in order to satisfy Eq. (16). It is straightforward to show that Eq. (19)
satisfies unitarity (16). The $K$ matrix inherits the angular momentum
structure of $\mathcal{M}_{2}$, reducing Eq. (19) to a single algebraic
relation. Subject to these restrictions, one can easily verify that Eq. (19)
satisfies Eq. (16), and therefore a solution for a given particular
$\mathcal{K}_{2}$ constitutes an on-shell representation for the
$\mathbf{2}\to\mathbf{2}$ scattering amplitude valid for the kinematic domain
$(2m)^{2}\leq\sigma_{k}\leq(\sqrt{s}-m)^{2}$.

On-shell representations such as Eq. (19) have an intrinsic _scheme
dependence_ since the choice of $\mathcal{I}$, and hence the $K$ matrix, is
not unique. For any $\mathcal{I}$ satisfying the requisite
$\operatorname{Im}\mathcal{I}=-\rho$, the function $\mathcal{K}_{2}$ must
compensate the difference in order to ensure $\mathcal{M}_{2}$ is the physical
scattering amplitude. Choosing an alternative $\mathcal{I}^{\,\prime}$ results
in an associated $\mathcal{K}_{2}^{\prime}$, which can be related to
$\mathcal{K}$ through the invariance of $\mathcal{M}_{2}$ as

$\displaystyle{\mathcal{K}_{2}^{\,\prime}}^{-1}={\mathcal{K}_{2}}^{-1}+\mathcal{I}-\mathcal{I}^{\,\prime}\,.$

Note that the relation between $K$ matrices preserves that condition that
$\mathcal{K}_{2}$ is real in the restricted kinematic since we require
$\operatorname{Im}\mathcal{I}=\operatorname{Im}\mathcal{I}^{\prime}$. We will
use two different choices of $\mathcal{I}$ in this work. The first one is
trivially related to the simple phase space factor Eq. (14)

$\displaystyle\mathcal{I}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}(\sigma_{k})=-i\delta_{\ell^{\prime}\ell}\delta_{m_{\ell^{\prime}}m_{\ell}}\,\frac{\xi\,q_{k}^{\star}}{8\pi\sqrt{\sigma_{k}}}\,,$
(20)

With an associated $K$ matrix $\mathcal{K}_{2}$. Note that this term does not
depend on any explicit parameter that indicates a scheme dependence, however
it is still a choice that we can make which fixes the on-shell form for
$\mathcal{K}_{2}$. The second choice is given by an integral form

$\displaystyle\mathcal{I}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{\,\prime}(\sigma_{k})$
$\displaystyle=\int^{\Lambda}\\!\frac{\textrm{d}\sigma^{\prime}}{\pi}\,\frac{\rho_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}(\sigma^{\prime})}{\sigma_{k}-\sigma^{\prime}+i\epsilon}\left(\,\sqrt{\frac{\sigma^{\prime}-4m^{2}}{\sigma_{k}-4m^{2}}}\,\right)^{\ell^{\prime}+\ell}\,,$
(21)

with the associated $K$ matrix $\mathcal{K}_{2}^{\prime}$. This is the usual
Chew-Mandelstam dispersive form Chew and Mandelstam (1960) including barrier
factors to ensure the kinematic singularity induced by the partial wave
projection is removed. It is understood that $\epsilon\to 0^{+}$, and the
dispersive integral runs from $4m^{2}\leq\sigma^{\prime}<\infty$, with
$\Lambda$ indicating the need for regularization of the divergent behavior.
141414In phenomenological analyses, it is usual to perform subtractions to
tame the UV divergence. We do not make any specific choice for UV regulation
here as any choice which respects the kinematic requirements is valid for our
arguments. This shows an explicit scheme dependence in
$\mathcal{K}_{2}^{\,\prime}$, as it must depend on the regularization
$\Lambda$ in order to ensure that $\mathcal{M}_{2}$ is regularization
independent. By inspection, both Eqs. (20) and (21) have
$\operatorname{Im}\mathcal{I}=\operatorname{Im}\mathcal{I}^{\prime}=-\rho$
above the two-body threshold, and the relation between the two $K$ matrices is

$\displaystyle{\mathcal{K}_{2}^{\,\prime}}^{-1}={\mathcal{K}_{2}}^{-1}-\operatorname{Re}\mathcal{I}^{\,\prime}\,.$

The advantage of Eq. (21) is it ensures that non-physical singularities of Eq.
(20), at $\sigma_{k}=0$, are removed. While the dispersive form has much
better analytic properties, it is common to use both forms of $\mathcal{I}$ in
phenomenological and lattice QCD studies Briceno  _et al._ (2018b). In the
next section, we will use the dispersive form to form the finite-volume
analogue.

We follow a similar path as the $\mathbf{2}\to\mathbf{2}$ case for
constructing an on-shell representation for $\mathcal{T}$. Given the unitarity
relation Eq. (III.1), we make an ansatz for the $\mathbf{3}\to\mathbf{3}$ $T$
matrix element in the form of a linear equation such that (III.1) is
satisfied. Since Eq. (III.1) is a matrix in angular momentum space and
contains integrals over the spectator momenta, the linear equation will take
the form of a matrix-integral equation shown in Eq. (2), which we repeat here

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$

Here, $\mathcal{K}_{3}$ is the $\mathbf{3}\to\mathbf{3}$ $K$ matrix, which
represents the unknown short-distance scattering physics, and $\Gamma$ is a
kinematic function containing the non-analytic structures that satisfy
$\operatorname{Im}\Gamma(\mathbf{p},\mathbf{k})=-\Phi(\mathbf{p},\mathbf{k})$
in order to respect Eq. (III.1). We choose to define $\Gamma$ as in Eq. (4),
which we repeat here for convenience

$\displaystyle\Gamma(\mathbf{p},\mathbf{k})=\delta(\mathbf{p},\mathbf{k})\,H(\sigma_{k})\,\mathcal{I}(\sigma_{k})+\mathcal{G}(\mathbf{p},\mathbf{k})\,,$

where $\mathcal{I}$ is as in Eq. (20) which describes the direct production of
three on-shell intermediate states, $H$ is a cutoff function to be described
shortly, and $\mathcal{G}$ is the exchange propagator

$\displaystyle\mathcal{G}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}(\mathbf{p},\mathbf{k})=\frac{H(\sigma_{p})H(\sigma_{k})}{(P-k-p)^{2}-m^{2}+i\epsilon}\,\left(\frac{k_{p}^{\star}}{q_{p}^{\star}}\right)^{\ell^{\prime}}4\pi\,Y_{\ell^{\prime}m_{\ell^{\prime}}}^{*}(\mathbf{\hat{\mathbf{k}}}_{p}^{\star})Y_{\ell
m_{\ell}}(\mathbf{\hat{\mathbf{p}}}_{k}^{\star})\left(\frac{p_{k}^{\star}}{q_{k}^{\star}}\right)^{\ell}\,,$
(22)

which satisfies the relation
$\operatorname{Im}\mathcal{G}(\mathbf{p},\mathbf{k})=-\Delta(\mathbf{p},\mathbf{k})$
to capture the exchange singularity. The exchange function contains barrier
suppression factors, which are included to ensure regularity near zero
spectator momenta, and evaluate to unity at the on-shell point
$(P-k-p)^{2}=m^{2}$. Like $\mathcal{I}$, $\mathcal{G}$ is not unique as any
real function in the kinematic domain can be added so long as
$\operatorname{Im}\mathcal{G}=-\Delta$ in the physical region, e.g. a non-
covariant form as is used in Ref. Hansen and Sharpe (2014). Proving that $T$
matrix unitarity (III.1) holds for (2) has been more challenging than its
$\mathbf{2}\to\mathbf{2}$ counterpart, cf. Refs. Jackura  _et al._ (2019a);
Briceño  _et al._ (2019c), however we provide a simple proof in Appendix B.

In Eqs. (4) and (22), we have introduced a general cutoff function $H$ to
regulate potential UV divergences in the spectator momentum integrals.
Generally, $H$ must satisfy the condition that $H=1$ for $q_{k}^{\star}\geq
0$, and we choose it to zero at some kinematic point below the two-particle
threshold. The minimal choice one can make is
$H(\sigma_{k})=\Theta(q_{k}^{\star})$, as was used in Ref. Jackura  _et al._
(2019a). Although this choice is valid with respect to the unitarity
condition, Ref. Hansen and Sharpe (2014) advocates for a smoother cutoff to
avoid spurious finite volume effects that may arise, e.g. Eq. (29) of Ref.
Hansen and Sharpe (2014). While this is not easily seen from the FVU
viewpoint, one can use a smooth cutoff function so long as it is unity above
the two-particle threshold and the integration does not overlap with any below
threshold cuts. Any different choice of the cutoff functions amounts to
choosing a different definition of the $K$ matrices, effectively being part of
the chosen scheme dependence. In this work, we do not choose any specific
cutoff, and leave any further discussion on the validity between smooth and
hard cutoffs to future work.

The $K$ matrix inherits the same connectedness structure as
$\mathcal{T}^{(u,u)}$, cf. Eq. (3). Splitting this function according to
cluster decomposition, we define $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$
151515In the TOPT derivations of Ref. Blanton and Sharpe (2020a), they find an
expression involving this same $K$ matrix. However, the authors take two
different approaches to constructing the $K$ matrix form, and stress that some
properties of the resulting $K$ matrices are different, even though the on-
shell form are the same. From the point of view of $S$ matrix unitarity, if
the on-shell forms are identical, that is some scheme is taken to establish
the kinematic function for intermediate states, then the $K$ matrices which
encapsulate _all_ remaining short-distance physics must be the same. as the
fully connected three particle $K$ matrix which does not contain the
disconnected contribution specified by $\mathcal{K}_{2}$, i.e.

$\displaystyle\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\delta(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}(\sigma_{k})+\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$

which was shown in Eq. (5). Both $K$ matrices are real symmetric matrices in
the kinematic domain of interest. We use the notation as in Refs. Hansen and
Sharpe (2014, 2015); Blanton and Sharpe (2020a) where the connected three-body
$K$ matrix is indicated by df, meaning divergent free, which indicates it is
free of any kinematic singularities. Since we build in cutoff functions in the
$\mathbf{3}\to\mathbf{3}$ $T$ matrix element, the two-body $K$ matrix
implicitly has a dependence given by
$\mathcal{M}_{2}^{-1}(\sigma_{k})=\mathcal{K}_{2}^{-1}(\sigma_{k})+H(\sigma_{k})\,\mathcal{I}(\sigma_{k})$.
Although this slightly modifies the original “unprimed” form in Eqs. (19) and
(20) by

$\displaystyle\mathcal{K}_{2}^{-1}(\sigma_{k})\to\mathcal{K}_{2}^{-1}(\sigma_{k})+[1-H(\sigma_{k})]\,\mathcal{I}(\sigma_{k})\,,$

we use the same notation $\mathcal{K}_{2}$ to avoid introducing yet another
definition.

We use Eqs. (3) and (5) to separate Eq. (2) into disconnected and connected
contributions, giving two relations: one equation for $\mathcal{M}_{2}$ given
by Eq. (19), and an another for $\mathcal{M}_{3}^{(u,u)}$ given by Eq. (II),
repeated here for convenience

$\displaystyle\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{K}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})$
$\displaystyle-\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\,\Gamma(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})-\mathcal{K}_{2}(\sigma_{p})\int_{k^{\prime}}\Gamma(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$

Given a $\mathcal{K}_{2}$ and $\mathcal{K}_{3,\textrm{df}}$, Eq. (II) can be
solved for the $\mathbf{3}\to\mathbf{3}$ scattering amplitude. In the next
section we construct the quantization conditions which link the $K$ matrices
to finite-volume spectra of particles in a box. Note that compared with Refs.
Hansen and Sharpe (2015); Blanton and Sharpe (2020a), we have a single
integral for the connected $\mathbf{3}\to\mathbf{3}$ amplitude, instead of
two. We show in Sec. V an alternative derivation of the connected
$\mathbf{3}\to\mathbf{3}$ amplitude which allows us to recover the original
set of equations shown in Refs. Hansen and Sharpe (2015); Blanton and Sharpe
(2020a) and connects to Eq. (II).

##  IV Quantization conditions

We now construct the quantization conditions associated with the on-shell
scattering equations. The scattering equations provide the basis for the
structure of the finite-volume counterparts to the scattering amplitudes. The
system is put into a cubic volume with a side length $L$ and subject to
periodic boundary conditions. Consequently, rotational invariance is broken
and all momentum are quantized according to

$\displaystyle\mathbf{k}=\frac{2\pi}{L}\,\mathbf{n}\,,$

where $\mathbf{n}\in\mathbbm{Z}^{3}$. Therefore, all momentum integrals are
replaced by sums by the rule,

$\displaystyle\int\\!\frac{\textrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\longrightarrow\frac{1}{L^{3}}\sum_{\mathbf{k}\in(2\pi/L)\mathbbm{Z}^{3}}\,.$

Likewise, the Dirac delta functions will be replaced by the Kronecker
equivalent,

$\displaystyle(2\pi)^{3}\,\delta^{(3)}(\mathbf{p}-\mathbf{k})\longrightarrow
L^{3}\delta_{pk}\,.$

Since the system is confined in a box, the only singularities of the finite-
volume amplitudes are poles on the real energy axis, known as Lüscher poles,
which correspond to the spectrum of the system. The quantization condition
allows us to connect the finite-volume spectrum of three particles in a finite
spatial volume to the infinite volume scattering quantities governed by the
on-shell relations.

Since Lorentz invariance is broken, the resulting finite-volume amplitude will
depend not only on the invariant mass of the two particles, but also the
direction of the total momentum of the system. Total angular momentum is no
longer conserved in a finite-volume, and therefore is it not a suitable
quantum number to label for the two particle state, requiring the need to
subduce the system to an irreducible representation of the octahedral group or
one of its little groups for moving frames. We will not discuss the subduction
to the cubic irreps, and instead refer to Refs. Thomas  _et al._ (2012); Moore
and Fleming (2006) for the discussion.

The finite-volume scaling for short-distance objects will be such that they
are equivalent to the infinite-volume quantity up to some correction which is
exponenitally suppressed in the volume. For example, the finite-volume hadron
mass $M_{L}$, $M_{L}=M+\mathcal{O}(e^{-mL})$ where $m$ is the lightest mass
scale of the system, e.g. the pion in QCD Luscher (1986a). Similar arguments
hold for all connected $K$ matrices. Therefore, we can ignore the volume
correction for large enough box sizes and simply replace all short-distance
finite-volume objects with their infinite-volume counterparts. This argument
holds because any such object can be represented by an analytic function in
the kinematic region of study since all the relevant on-shell physics has been
isolated Kim  _et al._ (2005). Since the functions are analytic, any integral
representation will not contain singular behavior in the integrand $f$, and we
can freely replace discrete sums with continuous integrals up to the
exponentially suppressed behavior according to the Poisson summation formula,

$\displaystyle\frac{1}{L^{3}}\sum_{\mathbf{k}}\,f(\mathbf{k})=\int\frac{\textrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\,f(\mathbf{k})+\mathcal{O}(e^{-mL})\,.$

###  IV.1 Two particle quantization condition

First we establish the quantization condition for the two-particle subsystem.
Let us fix the spectator momentum $\mathbf{k}$, and work with the
$\mathbf{2}\to\mathbf{2}$ amplitude $\mathcal{M}_{2}(\sigma_{k})$. Since
rotational invariance is broken, the finite-volume counterpart amplitude,
denoted $\mathcal{M}_{2,L}$, will in general be a function of the total
momentum flow $P_{k}$. Additionally the amplitude is a dense matrix in angular
momentum space since angular momentum is no longer conserved. We write the
finite-volume equivalent of Eq. (19) as

$\displaystyle\mathcal{M}_{2,L}(P_{k})=\mathcal{K}_{2}^{\,\prime}(\sigma_{k})-\mathcal{K}_{2}^{\,\prime}(\sigma_{k})\,\mathcal{I}_{L}^{\,\prime}(P_{k})\,\mathcal{M}_{2,L}(P_{k})\,,$
(23)

where we have used the relation
$\mathcal{K}_{2,L}^{\,\prime}(P_{k})=\mathcal{K}_{2}^{\,\prime}(\sigma_{k})$
which is valid in the elastic two-particle scattering region and dropping the
exponentially suppressed correction with a large enough box size. Since
$\mathcal{M}_{2,L}$ must only contain real-axis pole singularities in the
complex energy plane, the explicit cut structure of Eq. (20) makes it
difficult to transition to a finite $L$ system. We have chosen the scheme
where we use the finite-volume counterpart of Eq. (21) since the dispersive
version is more amenable to the appropriate analytic structure of the finite-
volume amplitude. Note that we can rewrite Eq. (21) as

$\displaystyle\mathcal{I}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{\,\prime}(\sigma_{k})$
$\displaystyle=\frac{\xi}{2}\int^{\Lambda}\\!\frac{\textrm{d}^{3}\mathbf{a}}{(2\pi)^{3}}\left(\frac{a^{\star}}{q_{k}^{\star}}\right)^{\ell^{\prime}}\frac{4\pi\,Y_{\ell^{\prime}m_{\ell^{\prime}}}^{*}(\mathbf{\hat{\mathbf{a}}}^{\star})Y_{\ell
m_{\ell}}(\mathbf{\hat{\mathbf{a}}}^{\star})}{2\omega_{a}(q_{k}^{\star\,2}-a^{\star\,2}+i\epsilon)}\left(\frac{a^{\star}}{q_{k}^{\star}}\right)^{\ell}\,,$
(24)

which can be seen by integrating the above expression. The re-introduction of
the spherical harmonic is necessary as we will now replace the integration
with a discrete sum, breaking rotational invariance. The finite-volume
analogue of Eq. (24) is then

$\displaystyle\mathcal{I}_{L\,;\,\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{\,\prime}(P_{k})$
$\displaystyle=\frac{\xi}{2}\frac{1}{L^{3}}\sum_{\mathbf{a}}^{\Lambda}\left(\frac{a^{\star}}{q_{k}^{\star}}\right)^{\ell^{\prime}}\frac{4\pi\,Y_{\ell^{\prime}m_{\ell^{\prime}}}^{*}(\mathbf{\hat{\mathbf{a}}}^{\star})Y_{\ell
m_{\ell}}(\mathbf{\hat{\mathbf{a}}}^{\star})}{2\omega_{a}(q_{k}^{\star\,2}-a^{\star\,2})}\left(\frac{a^{\star}}{q_{k}^{\star}}\right)^{\ell}\,,$
(25)

We can easily solve the matrix relation Eq. (23) for $\mathcal{M}_{2,L}$, and
identify that the pole singularities occur when

$\displaystyle\det\Big{[}\,1+\mathcal{K}_{2}^{\,\prime}(\sigma_{k})\,\mathcal{I}_{L}^{\,\prime}(P_{k})\,\Big{]}_{E_{k}=E_{k,\mathfrak{n}}}=0\,,$
(26)

where the determinant is over angular momentum space. Equation (26) is the
Lüscher quantization condition, which holds at the finite volume energies
$E_{k,\mathfrak{n}}$. It is a determinant is over all angular momentum space,
and provides a mapping between the infinite-volume $K$ matrices and the finite
volume spectra. Formally, Eq. (26) holds in the kinematic region that the
scattering equation was constructed, i.e. for
$2m\leq\sqrt{\sigma_{k}}<4m^{2}$. While Eq. (26) is an adequate expression,
often one wants to find a regulator independent relation that is independent
on the cutoff $\Lambda$. We can relate $\mathcal{K}_{2}^{\,\prime}$ to
$\mathcal{K}_{2}$ via
$\mathcal{K}_{2}^{\prime\,-1}=\mathcal{K}_{2}^{-1}-\operatorname{Re}\mathcal{I}^{\,\prime}$,
where $\mathcal{K}_{2}$ is defined with the usual two-body phase space
function $\mathcal{I}$, Eq. (20). Substituting this relation in Eq. (23), we
find an alternative relation

$\displaystyle\mathcal{M}_{2,L}(P_{k})=\frac{1}{1+\mathcal{K}_{2}(\sigma_{k})\,F_{2,L}(P_{k})}\,\mathcal{K}_{2}(\sigma_{k})\,,$
(27)

which has a quantization condition

$\displaystyle\det\Big{[}\,1+\mathcal{K}_{2}(\sigma_{k})\,F_{2,L}(P_{k})\,\Big{]}_{E_{k}=E_{k,\mathfrak{n}}}=0\,.$

Here we have defined the finite-volume function

$\displaystyle F_{2,L\,;\,\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}(P_{k})$
$\displaystyle\equiv\mathcal{I}_{L\,;\,\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{\,\prime}(P_{k})-\operatorname{Re}\mathcal{I}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{\,\prime}(\sigma_{k})\,$
$\displaystyle=\frac{\xi}{2}\left[\,\frac{1}{L^{3}}\sum_{\mathbf{a}}^{\Lambda}-\mathcal{P}\int^{\Lambda}\\!\frac{\textrm{d}^{3}\mathbf{a}}{(2\pi)^{3}}\,\right]\,\left(\frac{a^{\star}}{q_{k}^{\star}}\right)^{\ell^{\prime}}\frac{4\pi\,Y_{\ell^{\prime}m_{\ell^{\prime}}}^{*}(\mathbf{\hat{\mathbf{a}}}^{\star})Y_{\ell
m_{\ell}}(\mathbf{\hat{\mathbf{a}}}^{\star})}{2\omega_{a}^{\star}(q_{k}^{\star\,2}-a^{\star\,2})}\left(\frac{a^{\star}}{q_{k}^{\star}}\right)^{\ell}\,,$
(28)

where $\mathcal{P}$ indicates the principle value prescription. 161616Note
that formally $F_{2,L}$ is independent of the cutoff $\Lambda$ as
$\Lambda\to\infty$, however in practice one usually requires some choice to
numerically compute the object. The function $F_{2,L}$ is equivalent to the
$F_{\textrm{pv}}$ functions of Refs. Hansen and Sharpe (2014), which can be
proven using the Poisson summation formula to generate equalities up to the
ignored exponentially suppressed corrections Kim  _et al._ (2005). We choose
to use $F$ instead of $F_{\textrm{pv}}$ in order to simplify the notation for
this work. In effect, defining $F_{2,L}$ in this way creates a rule for
replacing the two-body phase space by a finite-volume counterpart

$\displaystyle\mathcal{I}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}(\sigma_{k})=-i\delta_{\ell^{\prime}\ell}\delta_{m_{\ell^{\prime}}m_{\ell}}\,\frac{\xi
q_{k}^{\star}}{8\pi\sqrt{\sigma_{k}}}\longrightarrow
F_{2,L\,;\,\ell^{\prime}m_{\ell^{\prime}},\ell m_{\ell}}(P_{k})\,.$

###  IV.2 Three-particle quantization condition

We now construct the three-particle quantization condition. The most direct
way is to use the on-shell equation for the $T$ matrix element, Eq. (2). In
transitioning to the finite-volume, the finite-volume amplitudes in general
will become dense matrices in both angular momentum ($\ell,m_{\ell}$) and the
spectator momentum spaces ($k$). Thus every object will be cast into the
$(k\ell m_{\ell})$ basis. The spectator delta function
$\delta(\mathbf{p},\mathbf{k})$ must be replaced as

$\displaystyle\delta(\mathbf{p},\mathbf{k})\to
2\omega_{k}L^{3}\,\delta_{pk}=[\,2\omega L^{3}\,]_{pk}\,.$

The $[2\omega L^{3}]$ matrix is also trivially proportional to the identity
matrix in $(\ell,m_{\ell})$ space. Integration measures over the spectator
momentum then transition as

$\displaystyle\int_{k}=\int\\!\frac{\textrm{d}^{3}\mathbf{k}}{(2\pi)^{3}\,2\omega_{k}}\longrightarrow\sum_{\mathbf{k}}\,\frac{1}{2\omega_{k}L^{3}}\,.$

We denote the finite-volume $T$ matrix element as
$\mathcal{T}_{L}^{(u,u)}(P)$, which is a matrix in $(k\ell m_{\ell})$ space,
and a function of the total energy-momentum $P=(E,\mathbf{P})$. From Eq. (2),
we find that $\mathcal{T}_{L}^{(u,u)}(P)$ satisfies the relation

$\displaystyle\mathcal{T}_{L}^{(u,u)}(P)=\mathcal{K}_{3,L}^{(u,u)}(P)-\mathcal{K}_{3,L}^{(u,u)}(P)\cdot\frac{1}{2\omega
L^{3}}\cdot\Gamma_{L}(P)\cdot\frac{1}{2\omega
L^{3}}\cdot\mathcal{T}_{L}^{(u,u)}(P)$ (29)

where we have introduced the $\cdot$ notation to indicate the matrix product
over the combined ($k\ell m_{\ell}$) space. We also use the notation that any
object with an $L$ subscript depends explicitly on the box size. Implicitly
all three-body objects contain an $L$ dependence through the quantized
momentum, $\mathbf{k}=2\pi\,\mathbf{n}/L$. The three-body $K$ matrix has an
explicit $L$ dependence since it contains a spectator delta function,

$\displaystyle\mathcal{K}_{3,L}^{(u,u)}(P)=2\omega
L^{3}\cdot\mathcal{K}_{2}+\mathcal{K}_{3,\textrm{df}}^{(u,u)}(P)\,,$

where $\mathcal{K}_{2}$ is the infinite-volume two-body $K$ matrix interpreted
now as a diagonal matrix in ($k\ell m_{\ell}$) space where the $k$ index is
encoded in the argument $\sigma_{k}=(P-k)^{2}$, that is

$\displaystyle[\,\mathcal{K}_{2}\,]_{p\ell^{\prime}m_{\ell^{\prime}},k\ell
m_{\ell}}=\delta_{pk}\delta_{\ell^{\prime}\ell}\delta_{m_{\ell^{\prime}}m_{\ell}}\,\mathcal{K}_{2,\ell}(\sigma_{k})\,,$

and the infinite-volume connected three-body $K$ matrix
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}(P)$ is a dense matrix in ($k\ell
m_{\ell}$) space where the elements in momentum space are determined by
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$ with
$\mathbf{p},\mathbf{k}=2\pi\,\mathbf{n}/L$. We keep $\mathcal{K}_{2}$ an
implicit function of $P$ for convenience. Following Ref. Blanton and Sharpe
(2020a) we define a convenient matrix

$\displaystyle\widetilde{\mathcal{K}}_{2,L}\equiv 2\omega
L^{3}\cdot\mathcal{K}_{2}$

because the combination appears frequently.

The remaining finite-volume function $\Gamma_{L}$ is defined as

$\displaystyle\Gamma_{L}(P)=2\omega L^{3}\cdot H\cdot F_{2,L}+G_{L}(P)\,,$
(30)

which is found by replacing $\mathcal{I}$ with $F_{2,L}$ and taking $H$ as a
diagonal matrix in $k$ space with elements $H(\sigma_{k})$. The second
function is the finite-volume counterpart to $\mathcal{G}$, which is identical
to the infinite-volume since $\mathcal{G}$ contains only simple poles which
carry over to the finite-volume, i.e. $G_{L}=\mathcal{G}$. The matrix elements
of $G_{L}$ in momentum space are given by $\mathcal{G}(\mathbf{p},\mathbf{k})$
where $\mathbf{p},\mathbf{k}$ are evaluated for quantized momenta. Like
$\widetilde{\mathcal{K}}_{2,L}$, these function frequently appear with
multiplicative factors of $[2\omega L^{3}]^{-1}$. We therefore conveniently
define the functions

$\displaystyle\widetilde{\Gamma}_{L}(P)$ $\displaystyle\equiv\frac{1}{2\omega
L^{3}}\cdot\Gamma_{L}(P)\cdot\frac{1}{2\omega L^{3}}\,,$
$\displaystyle=\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)\,,$

where we have defined $\widetilde{F}_{2,L}$ by

$\displaystyle\widetilde{F}_{2,L}\equiv\frac{1}{2\omega L^{3}}\cdot H\cdot
F_{2,L}\,,$

and $\widetilde{G}_{L}$ as

$\displaystyle\widetilde{G}_{L}(P)\equiv\frac{1}{2\omega L^{3}}\cdot
G_{L}(P)\cdot\frac{1}{2\omega L^{3}}\,.$

The Lüscher poles of $\mathcal{T}_{L}^{(u,u)}$ occur whenever

$\displaystyle\det\Big{[}\,1+\mathcal{K}_{3,L}^{(u,u)}(P)\cdot\widetilde{\Gamma}_{L}(P)\,\Big{]}_{E=E_{\mathfrak{n}}}=0\,,$
(31)

or equivalently the result shown in Eq. (7), which we repeat here

$\displaystyle\det\Big{[}\,1+\left(\widetilde{\mathcal{K}}_{2,L}+\mathcal{K}_{3,\textrm{df}}^{(u,u)}(P)\right)\cdot\left(\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)\right)\,\Big{]}_{E=E_{\mathfrak{n}}}=0\,.$

The quantization condition is a determinant over $(k\ell m_{\ell})$ space, and
agrees with the TOPT result of Ref. Blanton and Sharpe (2020a). If one can
determine the spectrum $E_{\mathfrak{n}}$ of three particles in a box, say
from lattice QCD simulations, then Eq. (31) can be used to constrain the
connected $\mathbf{3}\to\mathbf{3}$ $K$ matrix given the
$\mathbf{2}\to\mathbf{2}$ $K$ matrix (or constrain both objects
simultaneously). Note that if instead of Eq. (2), we use (II) to construct the
finite-volume amplitude, we would arrive at the same quantization condition as
(31) since any other pole in the expression corresponds to the free three-
particle spectrum.

The $K$ matrices can parameterized by some analytic function which are fitted
to the spectra, and then in turn can be fed into the scattering equations such
as Eq. (2) or Eq. (II) to determine the infinite-volume
$\mathbf{3}\to\mathbf{3}$ scattering amplitude. Both the two and three
particle quantization conditions involve infinite-dimensional matrices in
angular momentum space, while the cutoff functions truncate the spectator
momentum space for the three body conditions. the barrier suppression of
partial wave amplitudes allow us to truncate the determinant by focusing on a
low-energy region and neglecting waves which have negligible. In each case
this leads to systematic effects in determining $\mathcal{K}_{2}$ and
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}$, but in the $\mathbf{3}\to\mathbf{3}$
case there exists additional systematic effects since unitarity is explicitly
broken since Eq. (III.1) and Eq. (III.1) are not diagonal in $(\ell m_{\ell})$
space. This gives another set of systematic errors which are similar to the
ones usually found in isobar truncations used in phenomenological analyses of
three-body decays Mikhasenko  _et al._ (2020); Albaladejo  _et al._ (2020,
2021). A possible course of action is to include enough waves to saturate the
system in some low-energy domain with the relevant physical channels, and
treat the breaking of unitarity as a systematic error in the determination of
the $\mathbf{3}\to\mathbf{3}$ amplitudes.

This concludes the main results, where we have used $S$ matrix unitarity to
construct both the infinite-volume scattering equations in terms of on-shell
quantities and some unknown short-distance object, and the finite-volume
quantization conditions which relate these short-distance objects to the two
and three particle spectra. In the following section, we present an
alternative derivation which is closely related to the approach taken by Ref.
Mikhasenko  _et al._ (2019), and connect the symmetric
$\mathcal{K}_{3,\textrm{df}}$ form of Ref. Hansen and Sharpe (2015) to the one
presented here.

##  V Alternative representations

In this section, we derive an alternative representation of the
$\mathbf{3}\to\mathbf{3}$ scattering amplitude and the subsequent quantization
condition. The idea is to separate physics that is dominated by on-shell
particle exchanges between two-body sub-channels from interactions which
involve short-distance three-body physics. By doing so, we find the resulting
framework to be similar to the one originally presented in Ref. Hansen and
Sharpe (2014, 2015), with the exception that we work with the asymmetric
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}$ instead of the symmetric form called
$\mathcal{K}_{3,\textrm{df}}$. Our starting point is Eq. (II), the on-shell
representation for the connected $\mathbf{3}\to\mathbf{3}$ amplitude, which is
repeated here for convenience

$\displaystyle\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{K}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})$
$\displaystyle-\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\,\Gamma(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})-\mathcal{K}_{2}(\sigma_{p})\int_{k^{\prime}}\Gamma(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$

We then multiply relation on the left by
$1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})$ and use Eq. (19) to
convert the two occurrences of $\mathcal{K}_{2}$ to $\mathcal{M}_{2}$, finding

$\displaystyle\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})-\mathcal{M}_{2}(\sigma_{p})\int_{k^{\prime}}\mathcal{G}(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\left[\,1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})\,\right]\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\left[\,\delta(\mathbf{k}^{\prime},\mathbf{k})-\Gamma(\mathbf{k}^{\prime},\mathbf{k})\mathcal{M}_{2}(\sigma_{k})\,\right]$
$\displaystyle-\left[\,1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})\,\right]\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(32)

We now introduce an amplitude $\mathcal{D}^{(u,u)}$, which is the solution for
$\mathcal{M}_{3}^{(u,u)}$ when no short-distance physics is included, i.e.
when $\mathcal{K}_{3,\textrm{df}}^{(u,u)}=0$ or equivalently

$\displaystyle\lim_{\mathcal{K}_{3,\textrm{df}}\to
0}\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k})\,.$

This of course is scheme dependent statement, as any redefinition of the $K$
matrices will result in a different interpretation of the short-distance
physics. However, here we fix to a specific scheme, and define
$\mathcal{D}^{(u,u)}$ with respect to this scheme. The $\mathcal{D}^{(u,u)}$
amplitude is often called the _ladder_ amplitude, as it represents an infinite
series of exchanges between $\mathbf{2}\to\mathbf{2}$ subprocesses. Therefore,
when $\mathcal{K}_{3,\textrm{df}}^{(u,u)}=0$, we say there is no short-
distance three-body interactions, and the system is dominated by on-shell
exchanges between pair-wise interactions. Setting
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}=0$ in Eq. (V), we find that
$\mathcal{D}^{(u,u)}$ is the solution to the integral equation

$\displaystyle\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k})=-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})-\mathcal{M}_{2}(\sigma_{p})\int_{k^{\prime}}\mathcal{G}(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(33)

which is completely controlled by two-body physics and is represented in Fig.
7.

Figure 7: Exchange dominated, or ladder $\mathbf{3}\to\mathbf{3}$ amplitude
$\mathcal{D}^{(u,u)}$ as defined in Eq. (33). The solid vertical lines
indicate that the intermediate states are represented by on-shell kinematic
functions.

We can remove the ladder contribution from the integral equation for
$\mathcal{M}_{3}^{(u,u)}$ by defining the divergence free amplitude
$\mathcal{M}_{3,\textrm{df}}^{(u,u)}$

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k})\,.$
(34)

Inserting this into Eq. (V) and using the ladder equation (33), we find an
integral equation for $\mathcal{M}_{3,\textrm{df}}^{(u,u)}$,

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=-\mathcal{M}_{2}(\sigma_{p})\int_{k^{\prime}}\mathcal{G}(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\left[\,1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})\,\right]\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{R}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle-\left[\,1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})\,\right]\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$

where $\mathcal{R}^{(u,u)}$ defined as

$\displaystyle\mathcal{R}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})-\int_{k^{\prime}}\Gamma(\mathbf{p},\mathbf{k}^{\prime})\,\Big{[}\,\delta(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k^{\prime}})+\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,\Big{]}\,.$
(35)

We interpret $\mathcal{R}^{(u,u)}$ as an initial state rescattering function,
which contains three distinct topological structures of three interacting
particles in the initial state before they succumb to any short-distance
three-body forces. The first term of Eq. (35) represents the case where the
initial state does not interact, the second indicating two of the particles
scatter either directly or with an initial exchange, and the third with pair-
wise exchanges. Figure 8 shows the different on-shell topologies which dress
the initial state.

We can define a similar rescattering function for the final state, denoted by
$\mathcal{L}^{(u,u)}$, which we define as

$\displaystyle\mathcal{L}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})-\int_{k^{\prime}}\Big{[}\,\delta(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{2}(\sigma_{k^{\prime}})+\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\,\Big{]}\,\Gamma(\mathbf{k}^{\prime},\mathbf{k})\,,$
(36)

and has a similar interpretation as the initial state rescattering function.
The functions $\mathcal{R}^{(u,u)}$ and $\mathcal{L}^{(u,u)}$ are closely
related to those presented in Ref. Hansen and Sharpe (2015), which we will
show in the following section.

The introduction of the initial and final state rescattering functions
motivates us to define an amputated divergence-free amplitude
$\mathcal{T}_{\textrm{df}}^{(u,u)}$, which is free of rescatterings in both
the initial and final state,

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{L}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{R}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(37)

Substituting this relation into the above integral equation, we find that

$\displaystyle\int_{p^{\prime}}\int_{k^{\prime}}$
$\displaystyle\left[\,1+\mathcal{M}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{p}^{\prime})\,\right]\,\mathcal{L}^{(u,u)}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle=\left[\,1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})\,\right]\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle-\left[\,1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})\,\right]\int_{p^{\prime}}\int_{q^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\,\mathcal{L}^{(u,u)}(\mathbf{q}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$

Combining the equations of $\mathcal{L}^{(u,u)}$, $\mathcal{D}^{(u,u)}$, and
$\mathcal{M}_{2}$, we find the identity

$\displaystyle\int_{k^{\prime}}\left[\,1+\mathcal{M}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{k}^{\prime})\,\right]\,\mathcal{L}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle=\left[\,1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})\,\right]\delta(\mathbf{p},\mathbf{k})\,.$

Therefore, all terms of the above integral equation have a common
$[1-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{I}(\sigma_{p})]$ factor on the final
state. Removing this factor, we arrive at the integral equation for
$\mathcal{T}_{\textrm{df}}^{(u,u)}$

$\displaystyle\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{q^{\prime}}\int_{k^{\prime}}\,\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\,\mathcal{L}^{(u,u)}(\mathbf{q}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(38)

The structure of Eq. (38) similar to the original one for the $T$ matrix
element Eq. (2), except explicit kinematic divergences are removed. This
includes initial and final state rescatterings, which also removes the
disconnected structure of $\mathcal{M}_{2}$ implicitly since it is part of the
rescattering functions. Equation (38) also shows explicit intermediate state
interactions with $\Gamma\cdot\mathcal{L}^{(u,u)}$. We can interpret
$\mathcal{T}_{\textrm{df}}$ as the $T$ matrix element where the initial and
final state rescatterings associated with two-body systems have been removed.
We have decomposed the single integral equation Eq. (II) into two separate
equations (33) and (38). Thus we can first study a system in the case where
there are no short-distance three-body interactions with solutions of Eq.
(33), and then feed that into (38) by introducing some
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}$. It is not yet known if working with the
two separate equations or directly with (II) directly is simpler, however it
presents an opportunity to study the stability of such solutions or examine
different physical cases, i.e. a small $\mathcal{K}_{3,\textrm{df}}$ which is
independent of the spectator momenta leads to an analytic solution for
$\mathcal{T}_{\textrm{df}}$, cf. the isotropic approximation of Ref. Hansen
and Sharpe (2014).

Figure 8: Diagrammatic representation of the rescattering function
$\mathcal{R}^{(u,u)}$ acting on the amputated amplitude
$\mathcal{T}_{\textrm{df}}^{(u,u)}$ (gray filled circle) to form the
divergence free amplitude $\mathcal{M}_{3,\textrm{df}}^{(u,u)}$ (black filled
circle).

In a finite-volume, we can write the counterpart to
$\mathcal{T}_{\textrm{df}}^{(u,u)}$ as

$\displaystyle\mathcal{T}_{\textrm{df},L}^{(u,u)}(P)=\mathcal{K}_{3,\textrm{df}}^{(u,u)}(P)-\mathcal{K}_{3,\textrm{df}}^{(u,u)}(P)\cdot\widetilde{\Gamma}_{L}(P)\cdot\widetilde{\mathcal{L}}_{L}^{(u,u)}(P)\cdot\mathcal{T}_{\textrm{df},L}^{(u,u)}(P)\,,$
(39)

where we have defined a new finite-volume function

$\displaystyle\widetilde{\mathcal{L}}_{L}^{(u,u)}(P)$
$\displaystyle\equiv\mathcal{L}_{L}^{(u,u)}(P)\cdot\frac{1}{2\omega L^{3}}\,,$
$\displaystyle\equiv\left[2\omega L^{3}-\left(2\omega
L^{3}\cdot\mathcal{M}_{2,L}+\mathcal{D}_{L}^{(u,u)}(P)\right)\cdot\frac{1}{2\omega
L^{3}}\cdot\Gamma_{L}(P)\right]\cdot\frac{1}{2\omega L^{3}}\,,$
$\displaystyle=1-2\omega
L^{3}\cdot\mathcal{M}_{2,L}\cdot\widetilde{\Gamma}_{L}(P)-\mathcal{D}_{L}^{(u,u)}(P)\cdot\widetilde{\Gamma}_{L}(P)\,,$

where $\mathcal{M}_{2,L}$ is given by Eq. (27) and $\mathcal{D}_{L}^{(u,u)}$
is defined by

$\displaystyle\mathcal{D}_{L}^{(u,u)}(P)$
$\displaystyle=-\mathcal{M}_{2,L}\cdot
G_{L}(P)\cdot\mathcal{M}_{2,L}-\mathcal{M}_{2,L}\cdot
G_{L}(P)\cdot\frac{1}{2\omega L^{3}}\cdot\mathcal{D}_{L}^{(u,u)}(P)\,,$ (40)

which follows from Eq. (33) as its finite-volume counterpart. One can show by
direct manipulation that the combination $2\omega
L^{3}\cdot\mathcal{M}_{2,L}+\mathcal{D}_{L}^{(u,u)}(P)$ can be simplified to

$\displaystyle 2\omega
L^{3}\cdot\mathcal{M}_{2,L}+\mathcal{D}_{L}^{(u,u)}(P)=\frac{1}{{\widetilde{\mathcal{K}}_{2,L}}^{\,-1}+\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)}\,,$

thus the combination $\Gamma_{L}\cdot\widetilde{\mathcal{L}}^{(u,u)}_{L}$ can
be written entirely in terms of $\widetilde{\mathcal{K}}_{2,L}$,
$\widetilde{F}_{2,L}$, and $\widetilde{G}_{L}$. We define a convenient the
function $F_{3,L}^{(u,u)}$ as

$\displaystyle F_{3,L}^{(u,u)}(P)$
$\displaystyle\equiv\widetilde{\Gamma}_{L}(P)\cdot\widetilde{\mathcal{L}}_{L}^{(u,u)}(P)\,,$
$\displaystyle=\left[\,\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)\,\right]\left(\,1-\frac{1}{{\widetilde{\mathcal{K}}_{2,L}}^{\,-1}+\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)}\cdot\left[\,\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)\,\right]\right)\,,$
(41)

which was given in Eq. (9).

The Lüscher poles occur when the inverse of $\mathcal{M}_{3}^{(u,u)}$
vanishes. Since the rescattering functions do not contain the three-body $K$
matrix, the spectrum for three interacting particles can equally be found when
the inverse of $\mathcal{T}_{\textrm{df},L}^{(u,u)}$ vanishes. Thus, an
alternative yet equivalent form for the quantization condition is given by Eq.
(8), for convenience repeated here

$\displaystyle\det\Big{[}\,1+\mathcal{K}_{3,\textrm{df}}^{(u,u)}(P)\cdot
F_{3,L}^{(u,u)}(P)\,\Big{]}_{E=E_{\mathfrak{n}}}=0\,.$ (42)

In these preceding manipulations, it is presented in such a way which
introduces some physical content into the various definitions amplitudes, but
ultimately is just a way for us to derive a set of relations which are similar
to those in other works for which we know the answer. In the next section, we
shall explore how these equations arise naturally from the unitarity
relations, and present and alternative derivation of Eq. (38). At the end of
the alternative derivation, we make a connection to the form of the original
FVU result Mai  _et al._ (2017); Mai and Doring (2017), and show how the
difference manifest by considering different rescattering functions. An
advantage of this new form is it allows us to convert to the original results
of Refs. Hansen and Sharpe (2014, 2015), which use a symmetric form of the $K$
matrix called $\mathcal{K}_{3,\textrm{df}}$ instead of working with the
asymmetric $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$. We show how to obtain the
symmetric form of Eq. (38) in Sec. V.2.

###  V.1 Alternative derivation of the
$\mathcal{T}_{\textnormal{{df}}}^{(u,u)}$ integral equation

We now show how Eq. (38) arises directly from the unitarity condition for
$\mathcal{M}_{3}^{(u,u)}$, i.e. Eq. (II). We schematically follow the
derivation shown in Ref. Mikhasenko  _et al._ (2019), which presented a
variant of the $R$ matrix form from Refs. Mai  _et al._ (2017); Jackura  _et
al._ (2019a). We will note the underlying difference in deriving the $R$
matrix form instead of the $K$ matrix form discussed here at the end of this
section. The unitarity condition is repeated here for convenience,

$\displaystyle\operatorname{Im}\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{M}_{3}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{k^{\prime}}\,\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\Phi(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\,\mathcal{M}_{3}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})$
$\displaystyle+\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\Delta(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})\,,$

contains a pole singularity in the last term. Therefore, the amplitude
$\mathcal{M}_{3}^{(u,u)}$ must contain this singularity explicitly Eden  _et
al._ (1966). This is exactly the exchange pole we introduced in Eq. (22). Let
us denote the amplitude for a single particle exchange by
$\mathcal{D}_{0}^{(u,u)}$, and define it as

$\displaystyle\mathcal{D}_{0}^{(u,u)}(\mathbf{p},\mathbf{k})=-\mathcal{M}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})\,,$
(43)

We can now remove the single exchange pole from $\mathcal{M}_{3}^{(u,u)}$,
where we define

$\displaystyle\mathcal{M}_{3,0}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{D}_{0}^{(u,u)}(\mathbf{p},\mathbf{k})\,.$
(44)

We substitute Eq. (44) everywhere in (III.1), and use the imaginary part of
(43) given by

$\displaystyle\operatorname{Im}\mathcal{D}_{0}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\rho(\sigma_{p})\,\mathcal{D}_{0}^{(u,u)}(\mathbf{p},\mathbf{k})+\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\Delta(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})$
$\displaystyle+\mathcal{D}_{0}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{k})\,\rho(\sigma_{k})\,\mathcal{M}_{2}(\sigma_{k})\,,$
(45)

where Eq. (16) was used on the first and last term. The three terms in Eq.
(V.1) are removed, yielding a new unitarity relation for
$\mathcal{M}_{3,0}^{(u,u)}$

$\displaystyle\operatorname{Im}\mathcal{M}_{3,0}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{M}_{3,0}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3,0}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{M}_{3,0}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\Big{[}\,\delta(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})+\mathcal{D}_{0}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,\Big{]}$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\,\Big{[}\,\delta(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})+\mathcal{D}_{0}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Big{]}\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3,0}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{D}_{0}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{D}_{0}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{k^{\prime}}\,\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\Delta(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{D}_{0}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\,\mathcal{D}_{0}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Delta(\mathbf{p}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})\,.$
(46)

The kinematic singularity associated with the single particle exchange is
absent, however we have introduced explicit singularities from two particle
exchanges in the last three terms. The procedure of removing isolated
kinematic divergences is repeated, this time for the two particle exchange
amplitude

$\displaystyle\mathcal{D}_{1}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{M}_{2}(\sigma_{p})\,\int_{k^{\prime}}\,\mathcal{G}(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{2}(k^{\prime})\,\mathcal{G}(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})\,.$
(47)

The imaginary part of Eq. (47) gives five terms, three of which are contained
in the last three terms of Eq. (V.1), and two which are contained in the
second and third term of Eq. (V.1). Defining a new amplitude
$\mathcal{M}_{3,1}^{(u,u)}$ which does not contain $\mathcal{D}_{1}^{(u,u)}$,

$\displaystyle\mathcal{M}_{3,1}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle\equiv\mathcal{M}_{3,0}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{D}_{1}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$
$\displaystyle=\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{D}_{0}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{D}_{1}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$

and inserting into Eq. (V.1) gives a new unitarity relation now with all
kinematic singularities associated with two particle exchanges removed. This
will introduce new singularities arising from three particle exchanges, and
the processes can be repeated ad infinitum to generate the $n$ particle
exchange amplitudes $\mathcal{D}_{n}^{(u,u)}$ which are then subsequently
removed from the remaining connected $\mathbf{3}\to\mathbf{3}$ amplitude.

Removing all kinematic exchanges allows us to then extend the notion in Eq.
(44) to define the divergence free $\mathbf{3}\to\mathbf{3}$ amplitude,

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle\equiv\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\sum_{n=0}^{\infty}\mathcal{D}_{n}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$
(48)

where the infinite sum of all $n$ particle exchange amplitudes, defined by the
iterate

$\displaystyle\mathcal{D}_{n}^{(u,u)}(\mathbf{p},\mathbf{k})=(-1)^{n}\,\mathcal{M}_{2}(\sigma_{p})\,\bigg{[}\,\prod_{j=1}^{n}\int_{k_{j}}\,\bigg{]}\,\prod_{j=0}^{n}\bigg{(}\,\mathcal{G}(\mathbf{k}_{j},\mathbf{k}_{j+1})\,\mathcal{M}_{2}(\sigma_{k_{j+1}})\,\bigg{)}\Big{\rvert}_{\mathbf{k}_{0}=\mathbf{p},\mathbf{k}_{n+1}=\mathbf{k}}\,,$

are removed. Equation (48) is identical to (34) since the infinite sum of
exchanges is exactly the ladder amplitude

$\displaystyle\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\sum_{n=0}^{\infty}\mathcal{D}_{n}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$

which is the solution to the integral equation Eq. (33), which can be verified
by successive substitutions of the equation into itself. By construction, the
ladder amplitude satisfies the relation

$\displaystyle\operatorname{Im}\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{D}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{k^{\prime}}\,\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\Phi(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\,\mathcal{D}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})$
$\displaystyle+\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})\,\Delta(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k}),$
(49)

which contains all terms removed from the $\mathcal{M}_{3}^{(u,u)}$ unitarity
equation via Eq. (48).

All short distance three body interactions are contained within
$\mathcal{M}_{3,\textrm{df}}^{(u,u)}$ since the ladder amplitude represents
the contributions consisting of exchanges between two particle subprocesses.
After substituting Eq. (48) into (III.1) and using (V.1), we find that
$\mathcal{M}_{3,\textrm{df}}^{(u,u)}$ has the unitarity relation

$\displaystyle\operatorname{Im}\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{M}_{3,\textrm{df}}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{M}_{3,\textrm{df}}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\Big{[}\,\delta(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})+\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,\Big{]}$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\,\Big{[}\,\delta(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})+\mathcal{D}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Big{]}\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(50)

The second and third terms represent singularities associated with the physics
of initial and final state interactions, respectively, and are composed of
entirely $\mathbf{2}\to\mathbf{2}$ sub-processes. The unitarity relation (V.1)
implies that $\mathcal{M}_{3,\textrm{df}}^{(u,u)}$ must have an amputated
structure, where the initial and final state of three particle can undergo
pair wise interactions. The possible rescatterings are encoded in functions
$\mathcal{R}^{(u,u)}$ and $\mathcal{L}^{(u,u)}$ which we defined in Eqs. (35)
and (36), respectively. We remove these rescattering topologies by introducing
the amputated divergence free amplitude $\mathcal{T}_{\textrm{df}}^{(u,u)}$ as
defined in Eq. (37), and use the imaginary parts of $\mathcal{R}^{(u,u)}$ and
$\mathcal{L}^{(u,u)}$ given by

$\displaystyle\operatorname{Im}\mathcal{L}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\Big{[}\,\delta(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{M}_{2}^{{\dagger}}(p^{\prime})+\mathcal{D}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Big{]}\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{L}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
$\displaystyle\operatorname{Im}\mathcal{R}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{R}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\Big{[}\,\delta(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(k)+\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,\Big{]}\,,$

to reduce the unitarity constraint to one for the
$\mathcal{T}_{\textrm{df}}^{(u,u)}$ amplitude,

$\displaystyle\operatorname{Im}\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{T}_{\textrm{df}}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Xi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(51)

Here we have introduced yet another new kinematic function $\Xi$, which
accounts for the intermediate state singularities from all two-body
rescatterings

$\displaystyle\Xi(\mathbf{p},\mathbf{k})$
$\displaystyle\equiv\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{L}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{L}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
$\displaystyle=-\operatorname{Im}\left[\,\int_{p^{\prime}}\,\Gamma(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{L}^{(u,u)}(\mathbf{p}^{\prime},\mathbf{k})\,\right]\,.$
(52)

The first line follows directly from the substitution of Eq. (37) into (V.1),
and is equal to the second line by direct computation of the imaginary part
and using $\operatorname{Im}\mathcal{L}^{(u,u)}$. Equation (51) is exactly the
condition found in Ref. Briceño  _et al._ (2019c) when the authors showed that
their original integral equations satisfied unitarity.

The unitarity equation for $\mathcal{T}_{\textrm{df}}^{(u,u)}$ is similar in
structure to the condition for the $\mathbf{2}\to\mathbf{2}$ amplitude, or the
original $\mathbf{3}\to\mathbf{3}$ $T$ matrix unitarity relation. Therefore,
an appropriate on-shell representation is given by the previously derived
linear integral equation (38), which we recall as

$\displaystyle\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{q^{\prime}}\int_{k^{\prime}}\,\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\,\mathcal{L}^{(u,u)}(\mathbf{q}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}_{\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$

where $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$ is the $\mathbf{3}\to\mathbf{3}$
connected $K$ matrix as before. If one desires, we can reintroduce the initial
and final state rescattering functions to Eq. (38) to yield and integral
equation for $\mathcal{M}_{3,\textrm{df}}^{(u,u)}$ directly,

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{L}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{R}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle-\int_{p^{\prime}}\int_{q^{\prime}}\int_{k^{\prime}}\,\mathcal{L}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\,\Gamma(\mathbf{q}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(53)

It can be shown that Eq. (38) satisfies Eq. (51) using the techniques in
Appendix B.

####  V.1.1 $B$ matrix form

An aside, we summarize the derivation the $B$ matrix form (also called the $R$
or $C$ matrix forms in the literature) which was derived in Refs. Mai  _et
al._ (2017); Jackura  _et al._ (2019a); Mikhasenko  _et al._ (2019). Doing so
will reveal the physical, yet inconsequential, difference between the original
FVU and RFT frameowrks. Notice that Eq. (V.1), the unitarity relation for
$\mathcal{D}^{(u,u)}$, is exactly the same as for the original connected
$\mathbf{3}\to\mathbf{3}$ amplitude $\mathcal{M}_{3}$. It was argued that any
chosen $\mathcal{G}$ is valid so long as
$\operatorname{Im}\mathcal{G}=-\Delta$, which means any additional arbitrary
real function can be added to $\mathcal{G}$. Since it is a real function being
added, the resulting alternate integral equation to Eq. (33) will satisfy Eq.
(III.1). This is the basis for the $R$ matrix form, where the additional
unknown real function represents short-range three body physics. This new
integral equation is of the same form as Eq. (33), and is given by

$\displaystyle\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{M}_{2}(\sigma_{p})\,\mathcal{B}^{(u,u)}(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})+\mathcal{M}_{2}(\sigma_{p})\,\int_{k^{\prime}}\,\mathcal{B}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(54)

where
$\mathcal{B}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\mathcal{C}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{G}(\mathbf{p},\mathbf{k})$
where $\mathcal{C}^{(u,u)}$ represents the short-distance physics, that is it
plays the same role as $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$. 171717In Ref.
Jackura  _et al._ (2019a); Mikhasenko  _et al._ (2019); Jackura  _et al._
(2019b) the short-distance object was called $R$, while in Ref. Mai  _et al._
(2017) it was called $C$. We choose to not use $R$ here to avoid confusion
with the rescattering function. The generalization
$-\mathcal{G}(\mathbf{p},\mathbf{k})\longrightarrow\mathcal{B}^{(u,u)}(\mathbf{p},\mathbf{k})$
is allowed since we define $\mathcal{G}$ such that
$\operatorname{Im}\mathcal{G}=-\Delta$, therefore any real function ensures
unitarity holds. Therefore, the relation between $\mathcal{C}$ and
$\mathcal{K}_{3,\textrm{df}}$ is simply a choice of zero
$\mathcal{K}_{3,\textrm{df}}$ and choosing a different $\mathcal{G}$ function
which has an unknown real part, in which one recovers the original FVU
equations from the RFT versions. It can be shown, cf. Refs. Jackura  _et al._
(2019b); Blanton and Sharpe (2020b), that the
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}$ and $\mathcal{C}^{(u,u)}$ matrices are
related by an integral equation,

$\displaystyle\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{K}_{2}(\sigma_{p})\,\mathcal{C}^{(u,u)}(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}(\sigma_{k})+\mathcal{K}_{2}(\sigma_{p})\int_{k^{\prime}}\,\mathcal{C}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(55)

where we must use the same implicit cutoff dependence in order for the
relation to hold. Consequently, this relation shows that relating these $K$
matrix objects is a non-trivial task. Initial studies using the FVU Mai and
Doring (2019) and the RFT Blanton  _et al._ (2020) forms have shown a large
order of magnitude discrepancy between $\mathcal{C}$ and a related symmetric
form of $\mathcal{K}_{3,\textrm{df}}$, discussed in the next section. The
origin of this disparity may arise from the dressings of these short-distance
objects given in Eq. (55). Moreover, both analyses assume that these short-
distance functions are constants in energy, which clearly violates Eq. (55).
The finite-volume amplitude corresponding to Eq. (54) is given by

$\displaystyle\mathcal{M}_{3,L}^{(u,u)}(P)$
$\displaystyle=\mathcal{M}_{2,L}\cdot\mathcal{B}_{L}^{(u,u)}(P)\cdot\mathcal{M}_{2,L}+\mathcal{M}_{2,L}\cdot\mathcal{B}_{L}^{(u,u)}(P)\cdot\frac{1}{2\omega
L^{3}}\cdot\mathcal{M}_{3,L}^{(u,u)}(P)\,,$

where $\mathcal{B}_{L}^{(u,u)}=\mathcal{C}^{(u,u)}-G_{L}^{(u,u)}$ where
$G_{L}^{(u,u)}$ is as in Sec. IV.2. The quantization condition can then be
shown to be

$\displaystyle\det\Big{[}\,\widetilde{\mathcal{K}}_{2,L}^{\,-1}+\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)-\frac{1}{2\omega
L^{3}}\cdot\mathcal{C}^{(u,u)}(P)\cdot\frac{1}{2\omega
L^{3}}\,\Big{]}_{E=E_{\mathfrak{n}}}=0\,,$ (56)

which is the simple generalization of Ref. Mai and Doring (2017) for arbitrary
angular momenta.

We argued that any scattering equation satisfying the unitarity constraint is
a valid on-shell representation, however there remains an apparent difference
in the forms for the short-distance object in Eqs. (2) and (54). While both
are valid representations, there is a physical difference in the initial and
final state rescatterings, as discussed in Ref. Jackura  _et al._ (2019b). To
examine the difference, we follow the approach taken in Ref. Mikhasenko  _et
al._ (2019) which is as we presented in the preceding section until we reach
Eq. (V.1), repeated here for convenience

$\displaystyle\operatorname{Im}\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{M}_{3,\textrm{df}}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{M}_{3,\textrm{df}}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\Big{[}\,\delta(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})+\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,\Big{]}$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\,\Big{[}\,\delta(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{M}_{2}^{{\dagger}}(\sigma_{p})+\mathcal{D}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Big{]}\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$

The difference in the $B$ matrix form comes in at this step, where Ref.
Mikhasenko  _et al._ (2019) defines a different rescattering function which
does not contains the possibility that the particles do not interact pairwise
before the short-distance three-body forces. Defining this rescattering
function as $\mathcal{X}^{(u,u)}$,

$\displaystyle\mathcal{X}^{(u,u)}(\mathbf{p},\mathbf{k})=\delta(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(k)+\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$

and the corresponding amputated amplitude
$\widetilde{\mathcal{T}}_{\textrm{df}}^{(u,u)}$,

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{X}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\widetilde{\mathcal{T}}_{\textrm{df}}^{(u,u)}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{X}^{(u,u)}\,(\mathbf{k}^{\prime},\mathbf{k})\,.$

Since an the central difference between the rescattering functions here and
the ones in Eqs. (35) and (36) is an identity element, 181818There also
remains convolutions with kinematic factors, however careful analysis shows
that this is also an inconsequential factor with respect to unitarity. which
does not contain any singularities, the difference in rescattering functions
does not affect unitarity. Therefore, any affects of the initial or final
state particles were absorbed by the remaining rescattering functions. We find
the unitarity condition

$\displaystyle\operatorname{Im}\widetilde{\mathcal{T}}_{\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})=\int_{p^{\prime}}\int_{k^{\prime}}\,\widetilde{\mathcal{T}}_{\textrm{df}}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\operatorname{Im}\mathcal{X}^{(u,u)}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\widetilde{\mathcal{T}}_{\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$

where $\operatorname{Im}\mathcal{X}^{(u,u)}$ is given by

$\displaystyle\operatorname{Im}\mathcal{X}^{(u,u)}(\mathbf{p},\mathbf{k})=\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{X}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Phi(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{X}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$

The solution of this unitarity condition is the integral equation

$\displaystyle\widetilde{\mathcal{T}}_{\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{C}^{(u,u)}(\mathbf{p},\mathbf{k})+\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{C}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{X}^{(u,u)}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\widetilde{\mathcal{T}}_{\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$

which was shown to be equivalent to Eq. (54) in Ref. Jackura  _et al._
(2019b). We conclude that the relation between the
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}$ and $\mathcal{C}^{(u,u)}$ is the
presence/absence of the possibility of no initial or final state rescattering,
which the integral equation relating the two quantities either removes or
inserts these effects depending on the desired short-distance function.

###  V.2 Symmetric $K$ matrix form

The final formulation we cast the scattering equations and quantization
conditions to is to one which uses a symmetric $\mathcal{K}_{3,\textrm{df}}$
form which was first introduced in Refs. Hansen and Sharpe (2015) by taking
the $L\to\infty$ limit of the finite volume of their results using the RFT
framework. We use similar identities that were used in Ref. Blanton and Sharpe
(2020a) for the finite-volume quantization conditions, but we adapt them here
for the infinite-volume scattering amplitudes. To construct the symmetric
$\mathcal{K}_{3,\textrm{df}}$, we first introduce an alternative version of
the asymmetric $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$ which is defined with a
different exchange propagator
$\mathcal{G}^{\,\prime}(\mathbf{p},\mathbf{k})\equiv\mathcal{G}(\mathbf{p},\mathbf{k})-\mathcal{G}_{\textrm{pv}}(\mathbf{p},\mathbf{k})$
where $\mathcal{G}_{\textrm{pv}}$ is the principal part of the exchange
propagator. Therefore, this alternative exchange is thus

$\displaystyle\mathcal{G}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle\equiv-i\pi\,\delta\left((P-k-p)^{2}-m^{2}\right)\,H(\sigma_{p})H(\sigma_{k})\,4\pi
Y_{\ell^{\prime}m_{\ell^{\prime}}}^{*}(\mathbf{\hat{\mathbf{k}}}_{p}^{\star})Y_{\ell
m_{\ell}}(\mathbf{\hat{\mathbf{p}}}_{k}^{\star})\Big{\rvert}_{(P-k-p)^{2}=m^{2}}\,,$

such that only the on-shell point contributes. The scattering equation for the
$T$ matrix element with this choice is given by

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma^{\,\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(57)

where
$\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})=\delta(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}(\sigma_{k})+\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})$
and

$\displaystyle\Gamma^{\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})\,H(\sigma_{k})\,\mathcal{I}(\sigma_{k})+\mathcal{G}^{\,\prime}(\mathbf{p},\mathbf{k})\,,$
$\displaystyle=\Gamma(\mathbf{p},\mathbf{k})-\mathcal{G}_{\textrm{pv}}(\mathbf{p},\mathbf{k})\,,$

with $\Gamma(\mathbf{p},\mathbf{k})$ being the original function we defined in
Eq. (4). Since $\mathcal{T}^{(u,u)}$ in both Eqs. (2) and (57) must be the
same given that they are physical amplitudes, we can relate the different $K$
matrices by equating these expressions in $\mathcal{T}^{(u,u)}$. In Appendix C
we demonstrate the procedure, giving here the final relation

$\displaystyle\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{C}_{L}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{C}_{R}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\int_{q^{\prime}}\int_{k^{\prime}}\mathcal{C}_{L}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\mathcal{G}_{\textrm{pv}}(\mathbf{q}^{\prime},\mathbf{k}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(58)

where

$\displaystyle\mathcal{C}_{L}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})+\mathcal{K}_{2}(\sigma_{p})\,\mathcal{G}_{\textrm{pv}}(\mathbf{p},\mathbf{k})\,,$
$\displaystyle\mathcal{C}_{R}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})+\mathcal{G}_{\textrm{pv}}(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}(\sigma_{k})\,.$

The integral equation (V.2) is structurally identical to an analogous
relationship derived in Ref. Blanton and Sharpe (2020a).

In order to derive the relations found in Ref. Hansen and Sharpe (2015), we
will not replace $\mathcal{G}$ by $\mathcal{G}\,^{\prime}$ everywhere, and
instead keep the original $\mathcal{G}$ in the definition of
$\mathcal{D}^{(u,u)}$ as given in Eq. (33). While this may seem awkward at
first, we can rationalize it by following the same steps in Sec. V.1 until we
reach the unitarity equation (V.1) for $\mathcal{M}_{3,\textrm{df}}^{(u,u)}$,
and as this point choose the new $\mathcal{G}\,^{\prime}$ to arrive at the
relations for $\mathcal{T}_{\textrm{df}}^{(u,u)\,\prime}$,

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{L}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{T}_{\textrm{df}}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{R}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(59a) $\displaystyle\mathcal{L}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})-\int_{k^{\prime}}\Big{[}\,\delta(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{2}(\sigma_{k^{\prime}})+\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k}^{\prime})\,\Big{]}\,\Gamma\,^{\prime}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(59b) $\displaystyle\mathcal{R}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})-\int_{k^{\prime}}\Gamma\,^{\prime}(\mathbf{p},\mathbf{k}^{\prime})\,\Big{[}\,\delta(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})+\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,\Big{]}\,,$
(59c)
$\displaystyle\mathcal{T}_{\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle-\int_{p^{\prime}}\int_{q^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\Gamma\,^{\prime}(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\mathcal{L}^{(u,u)\,\prime}(\mathbf{q}^{\prime},\mathbf{k}^{\prime})\mathcal{T}_{\textrm{df}}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(59d)

where again $\mathcal{D}^{(u,u,)}$ is given by Eq. (33). This procedure still
leads to a consistent on-shell representation with respect to unitarity, and
will be the basis for the following manipulations.

It is convenient to express $\mathcal{T}_{\textrm{df}}^{(u,u)\,\prime}$ as its
series in the number of $\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}$
insertions,

$\displaystyle\mathcal{T}_{\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})=\sum_{n=1}^{\infty}\mathcal{T}_{\textrm{df},n}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})\,,$

where the leading and next-to-leading order terms are

$\displaystyle\mathcal{T}_{\textrm{df},0}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})\,,$
$\displaystyle\mathcal{T}_{\textrm{df},1}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle=-\int_{p^{\prime}}\int_{q^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\Gamma\,^{\prime}(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\mathcal{L}^{(u,u)\,\prime}(\mathbf{q}^{\prime},\mathbf{k}^{\prime})\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,,$

where all other orders follow from successive substitutions of Eq. (59) into
itself. For our purposes, the lowest two orders provide the structure needed
to generalize to the infinite series generated by Eq. (59).

At leading order in $\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}$,
$\mathcal{M}_{3,\textrm{df}}^{(u,u)}$ has the form

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\Big{\rvert}_{\mathrm{LO}}=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{L}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{R}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(60)

Focusing on the operation of $\mathcal{R}^{(u,u)\,\prime}$ on the initial
state, i.e. the expression

$\displaystyle\int_{p^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\,$
$\displaystyle\mathcal{R}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{k})=\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})$
$\displaystyle-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma\,^{\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\left[\,\delta(\mathbf{k}^{\prime},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})+\mathcal{D}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,\right]\,,$
(61)

the goal is to use properties of the symmetrization and exchange cut to
simplify this expression and cast it into a form where the $K$ matrix is
symmetric under interchange of particles. Two identities are used, which are
derived in Refs. Hansen and Sharpe (2014, 2015); Blanton and Sharpe (2020a)
and summarized in Appendix D, to convert the $K$ matrix from the unsymmetrized
to a symmetrized form. For the first term of Eq. (V.2), we note that the
initial state ultimately would be summed over all partial waves and
symmetrized over all spectators,

$\displaystyle\mathcal{K}_{3,\textrm{df}}^{(u)}(\mathbf{p},\mathbf{k},\mathbf{a}_{k}^{\star})=\sqrt{4\pi}\sum_{\mathbf{k}\in\mathcal{P}_{k}}\sum_{\ell,m_{\ell}}Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\,,$
(62)

where the left-hand side is only asymmetric in the final state ($\mathbf{p}$).
Since all the spectators are involved on the right-hand side, we can use the
identity Eq. (92) to replace the $K$ matrix as

$\displaystyle\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})\longrightarrow\frac{1}{3}\,\left[\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})+2\mathcal{K}_{3,\textrm{df}}^{(u,s)\,\prime}(\mathbf{p},\mathbf{k})\,\right]\,,$

where $\mathcal{K}_{3,\textrm{df}}^{(u,s)\,\prime}$ are the projected $K$
matrices in the non-$\mathbf{k}$ spectators, and the factor of 2 is because
both spectators contribute identically since identical scalar particles only
have even partial waves contributions, cf. Eqs. (89) and (90). The $1/3$
factor arises from overcounting the initial state as we sum over all
spectators, see Eq. (D).

The second term of Eq. (V.2) can be symmetrized as well. We use the identity
(D) which relates the exchange $\mathcal{G}^{\prime}$ to $\mathcal{I}$ when
summing over all angular momenta and integrating over the entire spectator
momentum space against a short-distance function. We find that for the initial
state

$\displaystyle\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k}^{\prime})\Gamma\,^{\prime}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle=\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k}^{\prime})\left[\,\delta(\mathbf{k}^{\prime},\mathbf{k})\,\widetilde{\rho}(\sigma_{k})+\mathcal{G}\,^{\prime}(\mathbf{k}^{\prime},\mathbf{k})\,\right]\,,$
$\displaystyle=\left[\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})+2\mathcal{K}_{3,\textrm{df}}^{\prime\,(u,s)}(\mathbf{p},\mathbf{k})\,\right]\,\widetilde{\rho}(\sigma_{k})\,,$

where we have defined $\widetilde{\rho}(\sigma_{k})\equiv
H(\sigma_{k})\,\mathcal{I}(\sigma_{k})$.

Note that so far we have only symmetrized the initial state, we can apply the
same identities for the final state of Eq. (60). We then define a _symmetric_
$\mathcal{K}_{3,\textrm{df}}$ as

$\displaystyle\mathcal{K}_{3,\textrm{df}}(\mathbf{p},\mathbf{k})$
$\displaystyle\equiv\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})+2\mathcal{K}_{3,\textrm{df}}^{(u,s)\,\prime}(\mathbf{p},\mathbf{k})+2\mathcal{K}_{3,\textrm{df}}^{(s,u)\prime}(\mathbf{p},\mathbf{k})+4\mathcal{K}_{3,\textrm{df}}^{(s,s)\prime}(\mathbf{p},\mathbf{k})\,.$
(63)

Then the leading order $\mathcal{M}_{3,\textrm{df}}$ is can equivalently be
written as

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\Big{\rvert}_{\mathrm{LO}}=\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{L}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{R}(\mathbf{k}^{\prime},\mathbf{k})\,,$

where we have defined new symmetric initial and final state rescattering
functions

$\displaystyle\mathcal{L}(\mathbf{p},\mathbf{k})$
$\displaystyle=\left[\,\frac{1}{3}-\mathcal{M}_{2}(\sigma_{k})\,\widetilde{\rho}(\sigma_{k})\,\right]\delta(\mathbf{p},\mathbf{k})-\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k})\widetilde{\rho}(\sigma_{k})\,,$
$\displaystyle\mathcal{R}(\mathbf{p},\mathbf{k})$
$\displaystyle=\left[\,\frac{1}{3}-\widetilde{\rho}(\sigma_{p})\,\mathcal{M}_{2}(\sigma_{p})\,\right]\delta(\mathbf{p},\mathbf{k})-\widetilde{\rho}(\sigma_{p})\,\mathcal{D}^{(u,u)}(\mathbf{p},\mathbf{k})\,.$

The next-to-leading order term of $\mathcal{M}_{3,\textrm{df}}^{(u,u)}$