aainstitutetext: Higgs Centre for Theoretical Physics, School of Physics and
Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK

# Finite-volume scattering on the left-hand cut

A. Baião Raposo a and M. T. Hansen<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The two-particle finite-volume scattering formalism derived by Lüscher and
generalized in many subsequent works does not hold for energies far enough
below the two-particle threshold to reach the nearest left-hand cut. The
breakdown of the formalism is signaled by the fact that a real scattering
amplitude is predicted in a regime where it should be complex. In this work,
we address this limitation by deriving an extended formalism that includes the
nearest branch cut, arising from single particle exchange. We focus on two-
nucleon ($NN\to NN$) scattering, for which the cut arises from pion exchange,
but give expressions for any system with a single channel of identical
particles. The new result takes the form of a modified quantization condition
that can be used to constrain an intermediate K-matrix in which the cut is
removed. In a second step, integral equations, also derived in this work, must
be used to convert the K-matrix to the physical scattering amplitude. We also
show how the new formalism reduces to the standard approach when the $N\to
N\pi$ coupling is set to zero.

##  1 Introduction

A powerful method for reliably predicting properties of quantum chromodynamics
(QCD), is the application of Monte Carlo importance sampling to numerically
evaluate the imaginary-time, discretized, finite-volume QCD path integral.
This approach, called lattice QCD, delivers estimates of imaginary-time,
discretized, finite-volume correlation functions, and various theoretical
frameworks are then applied to relate this data to physical observables.

One example application of this general approach is the extraction of finite-
volume energies in a given spatial volume, with periodicity $L$, defined with
a particular set of internal quantum numbers and a specified total momentum
$\bm{P}$ in the finite-volume frame. Generally speaking, the numerical values
of such finite-volume energies depend on the interaction strength of the
hadrons in the channel. Following the seminal work of Lüscher Luscher:1986pf
and subsequent generalizations Rummukainen:1995vs ; He:2005ey ; Christ:2005gi
; Kim:2005gf ; Lage:2009zv ; Bernard:2010fp ; Fu:2011xz ; Briceno:2012yi ;
Hansen:2012tf ; Guo:2012hv ; Briceno:2014oea , this can be used to constrain
the infinite-volume hadronic scattering amplitudes.

Such relations between energies and amplitudes necessarily come with kinematic
restrictions. Denoting the $n^{\text{th}}$ finite-volume energy for a given
periodicity and momentum by $E_{n}(L,\bm{P})$, the original work of Lüscher
and the extension to non-zero $\bm{P}$ of refs. Rummukainen:1995vs ;
Kim:2005gf ; Christ:2005gi  apply only for two-pion elastic scattering, i.e.
for energies satisfying

$0<E_{n}(L,\bm{P})^{2}-\bm{P}^{2}<(4M_{\pi})^{2}\,,$ (1)

where $M_{\pi}$ is the physical, infinite-volume pion mass. Because $G$-parity
prevents the coupling between even- and odd-number multi-pion states, the
lowest inelastic threshold is $4M_{\pi}$, as indicated. The lower bound of eq.
(1) is completely irrelevant in practical calculations, because the lowest
finite-volume energy is generically either near or above $2M_{\pi}$ or else
near some bound state mass that, while perhaps well below $2M_{\pi}$, is still
well above zero.

For this work it is nevertheless instructive to recall the origin of the lower
bound. Two reasons can be given (see also ref. Kim:2005gf ): One is that the
boost matrices to the center-of-mass (CM) frame become ill-defined, or at the
very least require a subtle analytic continuation, if the four-vector
$P^{\mu}=(E,\bm{P})$ becomes either light-like or space-like. The other reason
is that the two-to-two amplitude has a left-hand branch cut, with a branch
point at Madelstam $s=P^{2}=E^{2}-\bm{P}^{2}=0$ and this is not taken into
account in the derivation.

In the extension to $NN$ systems Briceno:2014oea , the analogous restriction
is

$(2M_{N})^{2}-M_{\pi}^{2}<E_{n}(L,\bm{P})^{2}-\bm{P}^{2}<(2M_{N}+M_{\pi})^{2}\,,$
(2)

where $M_{N}$ is the nucleon mass. The upper bound here is simply the lowest-
lying inelastic threshold, that of $NN\pi$ production. The lower bound is due
to a left-hand cut from single-pion exchange and is the focus of this work.

Over the last decade, extending the range of validity for finite-volume
scattering formulae has received much attention. One aspect of this is the
generalization to three-particle amplitudes Polejaeva:2012ut ; Hansen:2014eka
; Hansen:2015zga ; Briceno:2017tce ; Guo:2017ism ; Hammer:2017uqm ;
Hammer:2017kms ; Mai:2017bge ; Doring:2018xxx ; Briceno:2018mlh ; Klos:2018sen
; Briceno:2018aml ; Guo:2018ibd ; Blanton:2019igq ; Pang:2019dfe ; Romero-
Lopez:2019qrt ; Hansen:2020zhy ; Blanton:2020gha ; Blanton:2020jnm ;
Guo:2020spn ; Pang:2020pkl ; Romero-Lopez:2020rdq ; Blanton:2020gmf ;
Muller:2020vtt ; Muller:2020wjo ; Hansen:2021ofl ; Blanton:2021mih ;
Muller:2021uur ; Blanton:2021eyf ; Muller:2022oyw ; Jackura:2022xml ;
Severt:2022jtg ; Baeza-Ballesteros:2023ljl ; Draper:2023xvu ; Bubna:2023oxo .
This has progressed rapidly in recent years and a theoretical framework is now
in place, along with first numerical lattice QCD determinations of three-to-
three scattering amplitudes Beane:2007es ; Detmold:2008fn ; Woss:2018irj ;
Romero-Lopez:2018rcb ; Mai:2018djl ; Horz:2019rrn ; Blanton:2019vdk ;
Mai:2019fba ; Culver:2019vvu ; Fischer:2020jzp ; Alexandru:2020xqf ;
Hansen:2020otl ; Brett:2021wyd ; Blanton:2021llb ; Mai:2021nul ;
Garofalo:2022pux ; Baeza-Ballesteros:2022bsn ; Draper:2023boj . With respect
to (2), this can be understood as extending beyond the upper cutoff of
$(2M_{N}+M_{\pi})^{2}$.111For this particular system, the generalization is
still outstanding. Given the recent work treating non-identical and non-
degenerate particles Blanton:2020gmf  and particles with intrinsic spin
Draper:2023xvu , no fundamental issues are expected in deriving the relevant
formalism. In this article, we are concerned with extending the regime of
validity below the lower cutoff of $s=(2M_{N})^{2}-M_{\pi}^{2}$. The latter
arises due to single pion exchange between the nucleons as shown in figure
1(a). This generates a pole in amplitudes with fixed external momenta. When
the amplitudes are projected to definite angular momentum the pole becomes a
branch cut with a branch point at $s=(2M_{N})^{2}-M_{\pi}^{2}$.

Figure 1: Examples of pion exchanges leading to left-hand cuts for various
processes: (a) single-pion exchange in $NN\to NN$, leading to the branch cut
in angular-momentum projected $NN\to NN$ amplitudes treated in this work, (b)
two-pion exchange, leading to a branch cut starting at
$s=4M_{N}^{2}-4M_{\pi}^{2}$, _not_ treated in this work, (c) single-pion
exchange in $DD^{*}\to DD^{*}$.

Remarkably, recent lattice calculations have extracted finite-volume energies
violating the lower bound of eq. (2), and it is currently unclear how to use
these energies to constrain the scattering amplitude. As already discussed in
ref. Green:2021qol , the simplest way to see that the standard formalism
breaks down is to note that the latter always predicts a real-valued
scattering amplitude for $s<(2M_{N})^{2}$ whereas, for
$s<(2M_{N})^{2}-M_{\pi}^{2}$, it is known that the angular-momentum-projected
amplitude must be complex.

The realness of the scattering amplitude is a symptom of the problem, but does
not directly reveal the underlying reason. As we discuss in detail in section
3, the fundamental issue is subtle. In short, the relation between finite-
volume energies and infinite-volume scattering amplitudes is derived by
studying an all orders diagrammatic expansion of a finite-volume correlation
function. At various steps, subdiagrams with four-external legs are identified
and, because these are inside a larger diagram, the momenta they carry take on
all values and do not satisfy the on-shell condition $k^{2}=M_{N}^{2}$.
However, at a particular step in the derivation, one can show that the
replacement $k^{2}\to M_{N}^{2}$ is valid and only leads to neglected
$L$-dependence of the form $e^{-\mu(s)L}$ for some (possibly energy-dependent)
characteristic scale $\mu(s)$.

For $s<(2M_{N})^{2}-M_{\pi}^{2}$, however, this on-shell replacement is
invalid. In this work we resolve the issue by separating out the pion
exchanges in all diagrams and avoiding the problematic step for these
contributions. The result is a generalized formalism that relates finite-
volume energies to scattering amplitudes in an extended region:

$(2M_{N})^{2}-(2M_{\pi})^{2}<E_{n}(L,\bm{P})^{2}-\bm{P}^{2}<(2M_{N}+M_{\pi})^{2}\,,$
(3)

where the new lower limit arises from two-pion exchanges, such as that shown
in figure 1(b). These are not treated in our generalization.

It should be emphasized that neither Lüscher nor subsequent authors claimed
that the scattering relations would hold on the cut. However, it was only
recently stressed in ref. Green:2021qol  that the subthreshold analytic
continuation is both invalid as one approaches the cut and also needed due to
lattice-determined energies in this regime.

We further comment that the inequality (2) does not define strict boundaries
of validity. For example, as one approaches the inelastic threshold at
$s=(2M_{N}+M_{\pi})^{2}$ from below, $\mu(s)$ tends to zero so that the
neglected $e^{-\mu(s)L}$ terms become enhanced. In this work, we find that the
same breakdown occurs as one approaches the lower limit of
$s=(2M_{N})^{2}-M_{\pi}^{2}$ from above. Thus, it is prudent to apply the
formulas derived here, even for finite-volume energies above the branch point,
to study the effect of potential enhancement of neglected exponentials. An
analogous issue is that, for infinite-volume scattering amplitudes, the
convergence of the partial-wave expansion becomes arbitrarily bad as one
approaches $s=(2M_{N})^{2}-M_{\pi}^{2}$. This is also addressed with the
formalism presented in this work, since partial-wave projection is only
required for an intermediate infinite-volume quantity in which the offending
cut is absent.

The basic workflow resulting from our generalized scattering formalism closely
mimics that used in three-particle finite-volume scattering calculations. In a
first step, the finite-volume energies are used to constrain the $NN\pi$
coupling together with an intermediate K-matrix, denoted
$\overline{\mathcal{K}}^{\sf os}$, where the bar indicates a non-standard
quantity and the label ${\sf os}$ stands for on shell, emphasizing the
physical kinematic dependence. Then, in a second step, integral equations are
solved to relate the $NN\pi$ coupling and $\overline{\mathcal{K}}^{\sf os}$ to
the physical two-to-two scattering amplitude. Still, the results presented
here are not just some version of the three-particle formalism in disguise.
For one thing, the present method is manifestly invalid for
$s>(2M_{N}+M_{\pi})^{2}$.

At the same time, it has been recently understood that the three-particle
formalism of refs. Hansen:2014eka ; Hansen:2015zga  can also provide a
solution to the left-hand cut problem. If one of the scatterers in the target
two-particle channel can be represented as a pole in a two-to-two subprocess
amplitude, then it is possible to analytically continue the three-to-three
relations below threshold and infer a two-to-two formalism in which the cut is
properly treated. This was first identified in ref. Dawid:2023jrj , in the
context of a toy system of three identical spin-zero particles. In this
important work, the authors predict both the two-to-two scattering amplitude
and the finite-volume spectrum from the underlying three-particle formalism.
They then show numerically that the two are clearly related by standard two-
particle formalism for energies away from the left-hand cut, but that the
relation breaks as expected on the cut.

While the present article focuses on $NN\to NN$, it is applicable for any
single-channel two-to-two system with identical scattering particles. In
particular, the relations are provided for any value of intrinsic spin. The
extension to non-identical and non-degenerate particles will be considered in
future work, and is needed to include systems such as $DD^{*}\to DD^{*}$, i.e.
the vector-pseudoscalar channel with the internal quantum numbers of two charm
quarks. This sector is of great interest due to the experimental discovery of
a doubly charmed tetraquark $T_{cc}(3875)^{+}$ by LHCb, as presented in refs.
LHCb:2021vvq ; LHCb:2021auc .

In fact, a lattice QCD calculation of the relevant finite-volume spectrum for
$DD^{*}\to DD^{*}$ was already performed and presented in ref.
Padmanath:2022cvl , including the prediction of virtual bound state at the
heavier than physical pion mass of $M_{\pi}\approx 280\,\text{MeV}$, for which
the $D^{*}$ is a stable particle. The authors argue that the virtual bound
state is likely associated with the $T_{cc}^{+}$. Subsequently, ref.
Du:2023hlu  argued that the left-hand cut, resulting from the pion exchange
shown in figure 1(c), could qualitatively change the result. In light of this,
it would clearly be valuable to make use of the finite-volume energies that
overlap the branch cut. One formal method to address this is to extend the
three-particle formalism to $DD\pi$; this is underway. The alternative method
of extending the work here to non-identical and non-degenerate particles, and
the relation of the latter to the three-particle approach, will also be
considered in a future publication.

Two other publications, of relevance for the left-hand cut in the finite
volume context, have come to our attention during the completion of this work.
These are ref. Sato:2007ms , concerning exponentially suppressed corrections
to finite-volume scattering formula for $NN\to NN$, as well as ref.
Meng:2021uhz , concerning a plane-wave basis for the quantization condition.
The detailed relation of the formalism presented here to the earlier work is
also a topic for future investigation.

The remainder of this manuscript is organized as follows: In the next section,
we review properties and relations for the infinite-volume $NN\to NN$
scattering amplitude in the region given by the inequality (3). Then, in
section 3, we turn to the finite-volume system. Over numerous subsections, we
detail the breakdown in the usual derivation and provide the resolution,
summarized in section 3.8. As we explain there, this depends on an
intermediate quantity that is related to the scattering amplitude via integral
equations in section 4. Finally, in section 5, we explore the result in
various ways, e.g. by recovering the standard formalism in the limiting case
when the $NN\pi$ coupling, leading to the left-hand branch cut, vanishes. In
section 6 we briefly conclude. We also include two appendices collecting
technical details of the derivation.

##  2 Infinite-volume scattering

Here we review aspects of infinite-volume scattering theory that are relevant
to this work. In the following subsection, we recall a derivation of the
relation between the scattering amplitude and the K-matrix. We give the
relation using all-orders diagrammatic arguments, and this serves as a warm-up
for the finite-volume derivation presented in section 3. In section 2.2, we
discuss the analytic continuation of the amplitude below threshold and the
origin of the left-hand cut. To avoid clutter of notation, we first give the
discussion for spin-zero nucleons. Section 2.3 then provides the details to
extend all results to generic spin, including the physical spin-half case.

###  2.1 Scattering amplitude, K-matrix, and phase space

Consider a generic relativistic quantum field theory that has one heavy spin-
zero particle, with mass $M_{N}$, and one light spin-zero particle, with mass
$M_{\pi}$, such that $M_{\pi}\ll M_{N}$. Refer to the light particle as a pion
(with associated quantities labelled $\pi$) and the heavy particle as a
nucleon (with associated quantities labelled $N$). Further require that the
nucleon is charged under a $U(1)$ symmetry such that the single-particle
states can be written $|N,\bm{p},\pm\rangle$ where the second label is spatial
momentum and the third is charge. In QCD, baryon number plays the role of this
conserved charge. In this set-up we consider a single flavor of pion,
corresponding to the $\pi^{0}$ in nature. The field is neutral under the
$U(1)$ charge, i.e. has baryon number zero.

The infinite-volume single-particle states are normalized as

$\displaystyle\langle N,\bm{p}^{\prime},q^{\prime}|N,\bm{p},q\rangle$
$\displaystyle=\delta_{q^{\prime}q}2\omega_{N}(\bm{p})(2\pi)^{3}\delta^{3}(\bm{p}^{\prime}-\bm{p})\,,$
(4) $\displaystyle\langle\pi,\bm{p}^{\prime}|\pi,\bm{p}\rangle$
$\displaystyle=2\omega_{\pi}(\bm{p})(2\pi)^{3}\delta^{3}(\bm{p}^{\prime}-\bm{p})\,,$
(5)

where

$\displaystyle\omega_{N}(\bm{p})=\sqrt{\bm{p}^{2}+M_{N}^{2}}\,,\qquad\omega_{\pi}(\bm{p})=\sqrt{\bm{p}^{2}+M_{\pi}^{2}}\,.$
(6)

We allow the theory to have generic interactions respecting the conservation
of baryon number. In particular, we include $N\to N\pi$ couplings of the form

$\mathcal{L}_{N^{\dagger}\\!N\pi}(x)=N^{\dagger}(x)N(x)\mathcal{F}\big{(}\pi(x)\big{)}\,,$
(7)

where $\mathcal{F}$ is a generic polynomial in the pion field, starting with a
linear dependence on $\pi(x)$.

For both the pion and the scalar nucleon, the momentum-space dressed scalar
propagator takes the form

$\Delta_{x,i\epsilon}(k)=\frac{i}{k^{2}-M_{x}^{2}-\Pi_{x}(k^{2})+i\epsilon}\,,$
(8)

for $x\in\\{\pi,N\\}$. We require the self-energy $\Pi_{x}(k^{2})$ to satisfy
$\Pi_{x}(M_{x}^{2})=\Pi^{\prime}_{x}(M_{x}^{2})=0$ where the prime indicates a
derivative. This implies that the residue of the single-particle pole has
magnitude one and that $M_{x}$ is the physical (pole) mass. Throughout this
work, we use Minkowski signature four-vectors satisfying
$k^{2}=(k^{0})^{2}-\bm{k}^{2}$ where $\bm{k}$ is a spatial three-vector. The
use of Minkowski signature may seem surprising, in particular in the finite-
volume analysis of section 3, since the purpose of this derivation is to
interpret Euclidean-signature lattice calculations. However, as the finite-
volume energies are metric-independent, this detail of the numerical
calculation is irrelevant here.

Figure 2:  (a) Diagrammatic representation of the $NN$ Bethe-Salpeter kernel
$B(P,p,p^{\prime})$, built from all $NN\to NN$ diagrams that are two-nucleon
irreducible in the $s$-channel. (b) Diagrammatic representation of the nucleon
self-energy $\Pi_{N}(p^{2})$ and of the dressed nucleon propagator. (c)
Representation of the infinite-volume integral equations: eqs. (10) and (11)
of the main text.

Next, introduce $B(P,p,p^{\prime})$ as the Bethe-Salpeter kernel for $NN\to
NN$ scattering, formally defined as the sum of all amputated two-to-two
diagrams that are two-nucleon irreducible with respect to cuts intersecting
the flow of total energy and momentum, see figure 2(a). As shown in the
figure, the incoming nucleons carry four-momenta $p$ and $P-p$ and the
outgoing nucleons carry $p^{\prime}$ and $P-p^{\prime}$. So $P$ is the total
four-momentum in a given frame, related to the total energy and momentum via
$P^{\mu}=(E,\bm{P})$. We also define $s$ and $E^{\star}$ via

$s=E^{\star 2}=P^{2}=E^{2}-\bm{P}^{2}\,.$ (9)

In terms of the fully dressed nucleon propagator and the Bethe-Salpeter
kernel, the scattering amplitude $\mathcal{M}$ and the K-matrix $\mathcal{K}$
for $NN\to NN$ scattering can be expressed, respectively, via the integral
equations

$\displaystyle i\mathcal{M}(P,p,p^{\prime})$
$\displaystyle=iB(P,p,p^{\prime})+\frac{1}{2}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\,iB(P,p,k)\Delta_{N,i\epsilon}(P-k)\Delta_{N,i\epsilon}(k)i\mathcal{M}(P,k,p^{\prime})\,,$
(10)
$\displaystyle\begin{split}i\mathcal{K}(P,p,p^{\prime})&=iB(P,p,p^{\prime})\\\\[3.0pt]
&+\frac{1}{2}\,{\sf
pv}\\!\int\\!\\!\frac{d^{3}\bm{k}}{(2\pi)^{3}}\bigg{[}\int\\!\frac{dk^{0}}{2\pi}iB(P,p,k)\Delta_{N,i\epsilon}(P-k)\Delta_{N,i\epsilon}(k)i\mathcal{K}(P,k,p^{\prime})\bigg{]}_{\epsilon\to
0^{+}}\end{split}\,,$ (11)

where the label ${\sf pv}$ indicates that the principal-value pole
prescription is used to define the K-matrix. In eq. (11) the $i\epsilon$
prescription is only required to evaluate the $k^{0}$ integral, as indicated.
See figure 2(b) for a representation of these equations.

Iteratively substituting $\mathcal{M}$ and $\mathcal{K}$ on the right-hand
sides of these equations allows us to write a geometric series. Introducing a
compact notation, this can be written as

$\displaystyle i\mathcal{M}(s)$
$\displaystyle=iB(s)+iB(s)\circ_{i\epsilon}i\mathcal{M}(s)=\sum_{n=0}^{\infty}iB(s)[\circ_{i\epsilon}\,iB(s)]^{n}\,,$
(12) $\displaystyle i\mathcal{K}(s)$ $\displaystyle=iB(s)+iB(s)\circ_{\sf
pv}i\mathcal{K}(s)=\sum_{n=0}^{\infty}iB(s)[\circ_{\sf pv}\,iB(s)]^{n}\,,$
(13)

where the $\circ$ indicates that the quantities are combined via an integral
as

$\displaystyle iB(s)\circ_{i\epsilon}iB(s)$
$\displaystyle=\frac{1}{2}\int\\!\frac{d^{4}k}{(2\pi)^{4}}iB(P,p,k)\Delta_{N,i\epsilon}(P-k)\Delta_{N,i\epsilon}(k)iB(P,k,p^{\prime})\,,$
(14) $\displaystyle iB(s)\circ_{\sf pv}iB(s)$ $\displaystyle=\frac{1}{2}\,{\sf
pv}\\!\int\\!\\!\frac{d^{3}\bm{k}}{(2\pi)^{3}}\bigg{[}\int\\!\frac{dk^{0}}{2\pi}iB(P,p,k)\Delta_{N,i\epsilon}(P-k)\Delta_{N,i\epsilon}(k)iB(P,k,p^{\prime})\bigg{]}_{\epsilon\to
0^{+}}\,.$ (15)

To make use of these expressions, we now set all external four-momenta on
shell so that they satisfy $p^{2}=M_{N}^{2}$, etc. We also restrict attention
to the regime of elastic scattering, defined via
$(2M_{N})^{2}<s<(2M_{N}+M_{\pi})^{2}$. Then one can show

$iB(s)\circ_{i\epsilon}iB(s)-iB(s)\circ_{{\sf
pv}}iB(s)=\frac{1}{2}\int\\!\frac{d^{3}\bm{k}}{(2\pi)^{3}}iB(P,p,k)\frac{\pi\delta(E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k}))}{2\omega_{N}(\bm{k})2\omega_{N}(\bm{P}-\bm{k})}iB(P,k,p^{\prime})\,.$
(16)

This holds because, when the $k^{0}$ integral is evaluated via contour
integration, various contributions cancel between the principal-value and
$i\epsilon$ integrals. In particular, contributions that are analytic in
$\bm{k}$ and do not require a pole prescription will be identical between the
two terms on the left-hand side. The exception is the contribution arising
from the pole at $k^{0}=\omega_{N}(\bm{k})$. For this term, only the real part
of the $i\epsilon$ contribution cancels, leaving the imaginary part to
generate the term shown.

Next we project both sides of eq. (16) to definite orbital angular momentum.
To do so define $\bm{p}^{\star}$ as the spatial part of the four-momentum
resulting from boosting $p^{\mu}=(\omega_{N}(\bm{p}),\bm{p})$ with boost
velocity $\bm{\beta}=-\bm{P}/E$. [See also eq. (77) below.] Then the
expressions above can be expressed in terms of $\bm{p}^{\prime\star}$ and
$\bm{p}^{\star}$ and one can define

$\big{[}i\widetilde{B}(s)\circ_{i\epsilon}i\widetilde{B}(s)\big{]}_{\ell
m,\ell^{\prime}m^{\prime}}=\frac{1}{|\bm{p}^{\star}|^{\ell}\,|\bm{p}^{\prime\star}|^{\ell^{\prime}}}\int\\!\frac{d\Omega_{\hat{\bm{p}}^{\star}}}{4\pi}\\!\int\\!\frac{d\Omega_{\hat{\bm{p}}^{\prime\star}}}{4\pi}\,Y_{\ell
m}(\hat{\bm{p}}^{\star})\,[iB\circ_{i\epsilon}iB](\bm{p}^{\star},\bm{p}^{\prime\star})\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{p}}^{\prime\star})\,,$
(17)

where the $Y_{\ell m}$ are standard spherical harmonics. Here we have also
divided out a factor of
$|\bm{p}^{\prime\star}|^{\ell^{\prime}}\,|\bm{p}^{\star}|^{\ell}$. This is the
leading-order behavior of this combination as
$|\bm{p}^{\prime\star}|,|\bm{p}^{\star}|\to 0$ such that, by removing it, we
have defined a quantity that approaches a constant in this limit. We use the
tilde to indicate this rescaling. (See also, e.g. ref. Morningstar:2017spu .)

From here it follows that

$\displaystyle\big{[}i\widetilde{B}(s)\circ_{i\epsilon}i\widetilde{B}(s)-i\widetilde{B}(s)\circ_{{\sf
pv}}i\widetilde{B}(s)\big{]}_{\ell m,\ell^{\prime}m^{\prime}}$
$\displaystyle=\big{[}i\widetilde{B}(s)\,\widetilde{\rho}(s)\,i\widetilde{B}(s)\big{]}_{\ell
m,\ell^{\prime}m^{\prime}}\,,$ (18)

where, on the right-hand side, each of the three-factors is a diagonal matrix,
with each entry depending only on $s$. In particular

$\displaystyle\widetilde{\rho}_{\ell m,\ell^{\prime}m^{\prime}}(s)$
$\displaystyle=2\pi(k_{\sf
os}^{\star})^{\ell+\ell^{\prime}}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}Y_{\ell
m}(\hat{\bm{k}}^{\star})\frac{\pi\delta(E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k}))}{2\omega_{N}(\bm{k})2\omega_{N}(\bm{P}-\bm{k})}Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\star})\,,$
(19) $\displaystyle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,(k_{\sf
os}^{\star})^{2\ell}\,\rho(s)\,,$ (20)

where $\rho(s)$ is the volume of phase space

$\rho(s)=\frac{k^{\star}_{\sf os}}{16\pi\sqrt{s}}\,,\qquad\text{for}\qquad
k^{\star}_{\sf os}=\sqrt{s/4-M_{N}^{2}}\,,$ (21)

where ${\sf os}$ stands for on shell.

A non-trivial feature of eq. (18) is that the $B$ factors on the right-hand
side no longer depend on the magnitude of integrated momentum. This follows
from re-expressing quantities in terms of
$\bm{k}^{\star}=|\bm{k}^{\star}|\hat{\bm{k}}^{\star}$. Then the Dirac delta
function sets $|\bm{k}^{\star}|=k^{\star}_{\sf os}$ and the directional
degrees of freedom are decomposed into the spherical harmonics.

Equation (18) holds for any factors on either side of $\circ_{i\epsilon}$ and
$\circ_{{\sf pv}}$ and can be interpreted as an identity relating the pole
prescriptions: $\circ_{i\epsilon}=\circ_{{\sf pv}}+\rho$. Combining this with
eqs. (12) and (13) then allows one to relate the scattering amplitude and the
K-matrix via

$\displaystyle\widetilde{\mathcal{M}}(s)$
$\displaystyle=\sum_{n=0}^{\infty}\widetilde{B}(s)\big{[}\big{(}\circ_{{\sf
pv}}+\widetilde{\rho}(s)\big{)}\,i\widetilde{B}(s)\big{]}^{n}$ (22)
$\displaystyle=\sum_{n=0}^{\infty}\widetilde{\mathcal{K}}(s)\big{[}i\widetilde{\rho}(s)\,\widetilde{\mathcal{K}}(s)\big{]}^{n}\,,\qquad\qquad\qquad\text{for}\quad\widetilde{\mathcal{K}}(s)=\sum_{n^{\prime}=0}^{\infty}\widetilde{B}(s)\big{[}\circ_{{\sf
pv}}\,i\widetilde{B}(s)\big{]}^{n^{\prime}}\,,$ (23)
$\displaystyle=\frac{1}{\widetilde{\mathcal{K}}(s)^{-1}-i\widetilde{\rho}(s)}\,.$
(24)

Taking the standard parametrization

$\widetilde{\mathcal{K}}_{\ell
m,\ell^{\prime}m^{\prime}}(s)=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}(k^{\star}_{\sf
os})^{-2\ell}\,\mathcal{K}^{(\ell)}(s)=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}(k^{\star}_{\sf
os})^{-2\ell}\,16\pi\sqrt{s}\,\frac{\tan\delta_{\ell}(k^{\star}_{\sf
os})}{k^{\star}_{\sf os}}\,,$ (25)

then gives

$\mathcal{M}^{(\ell)}(s)=\frac{16\pi\sqrt{s}}{k^{\star}_{\sf
os}\cot\delta_{\ell}(k^{\star}_{\sf os})-ik^{\star}_{\sf os}}\,,$ (26)

where $\widetilde{\mathcal{M}}_{\ell
m,\ell^{\prime}m^{\prime}}(s)=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}(k^{\star}_{\sf
os})^{-2\ell}\mathcal{M}^{(\ell)}(s)$ defines the $\ell^{\text{th}}$ partial-
wave component of the scattering amplitude.

So far we have assumed $(2M_{N})^{2}<s<(2M_{N}+M_{\pi})^{2}$. The case of
$s>(2M_{N}+M_{\pi})^{2}$ requires explicit treatment of the $NN\pi$ states
within the Bethe-Salpeter kernel. This leads to significantly more complicated
unitarity constraints that have, however, received significant attention
recently and are by now well understood Dawid:2020uhn ; Jackura:2020bsk ;
Jackura:2022gib ; Dawid:2023jrj ; Dawid:2023kxu .

In this work, we are instead interested in the other direction:
$s<(2M_{N})^{2}$. In the next subsection, we describe the properties of the
infinite-volume scattering amplitude and K-matrix continued into this region,
with a particular focus on the left-hand cut. This will set the background and
the notation for our finite-volume discussion of this region in section 3.

###  2.2 Analytic continuation and the left-hand cut

The K-matrix is generally expected to be a meromorphic function in a region of
the complex plane containing $(k^{\star}_{\sf os})^{2}=0$, corresponding to
$s=(2M_{N})^{2}$. As a result, one can expand $k\cot\delta_{\ell}(k)$ about
this point as a series of polynomials and poles

$k^{\star}_{\sf os}\cot\delta_{\ell}(k^{\star}_{\sf
os})=-\frac{1}{a_{\ell}(k^{\star}_{\sf
os})^{2\ell}}\Big{[}1+\sum_{j=1}^{\infty}c_{j}(k^{\star}_{\sf
os})^{2j}\Big{]}+\sum_{n=1}^{P}\sum_{j=0}^{\infty}\frac{A_{n,j}(k^{\star}_{\sf
os})^{j}}{(k^{\star}_{\sf os})^{2}-C_{n}}\,.$ (27)

This then defines a sub-threshold analytic continuation for the K-matrix.

In the case where interactions of the form $\pi(x)N^{\dagger}(x)N(x)$,
$\pi(x)^{2}N^{\dagger}(x)N(x)$ are included, as in eq. (7), this expansion
breaks down due to branch cuts, generically referred to as left-hand cuts,
that run along the negative real axis of the complex $s$ plane. The nearest
cut starts at $s=4M_{N}^{2}-M_{\pi}^{2}$ and arises due to single pion
exchanges.

Figure 3: Single-meson exchange contribution to the $NN\to NN$ scattering
amplitude and K-matrix.

To see this in more detail, we consider the tree-level $t$\- and $u$-channel
exchange diagrams shown in figure 3. First assuming a fully dressed pion
propagator and general vertex functions, these diagrams have the form

$\displaystyle D_{t}(P,p,p^{\prime})$ $\displaystyle\equiv G(p,p^{\prime})\
\Delta_{\pi,i\epsilon}(p^{\prime}-p)\ G(P-p,P-p^{\prime})\,,$ (28)
$\displaystyle D_{u}(P,p,p^{\prime})$ $\displaystyle\equiv G(p,P-p^{\prime})\
\Delta_{\pi,i\epsilon}(P-p^{\prime}-p)\ G(p,P-p^{\prime})\,,$ (29)

where the middle propagator factor in each expression was introduced in eq.
(8). Here we have also introduced the form factor $G(p,p^{\prime})$ defined as
the amputated sum of all $N\to N\pi$ diagrams or, equivalently, via the matrix
element

$G(p,p^{\prime})\equiv\Delta_{\pi,i\epsilon}(p^{\prime}-p)^{-1}\langle
N,\bm{p}^{\prime},q^{\prime}|\pi(0)|N,\bm{p},q\rangle\,.$ (30)

Because our concern is the singularities in the diagrams
$D_{t}(P,p,p^{\prime})$ and $D_{u}(P,p,p^{\prime})$, it will be convenient to
identify simpler expressions that contain these relevant features but differ
from the full diagram by analytic terms. To define these first note that, for
the case where all fields have spin zero, $G(p,p^{\prime})$ is a Lorentz
scalar and only depends on Lorentz scalar momentum combinations. Further
noting that $p^{2}=p^{\prime 2}=M_{N}^{2}$ in eq. (30), we see that the
function in fact only depends on the momentum transfer. This allows us to
define $\mathcal{G}$ as the same function of a single Lorentz invariant

$\mathcal{G}\Big{(}(p^{\prime}-p)^{2}\Big{)}=G(p,p^{\prime})\,.$ (31)

We then define the coupling as

$g\equiv\lim_{q^{2}\to M_{\pi}^{2}}\mathcal{G}(q^{2})\,.$ (32)

With this coupling defined we now observe that the differences

$\displaystyle\delta D_{t}(P,p,p^{\prime})$ $\displaystyle\equiv
D_{t}(P,p,p^{\prime})-ig^{2}\mathcal{T}(P,p,p^{\prime})\,,$ (33)
$\displaystyle\delta D_{u}(P,p,p^{\prime})$ $\displaystyle\equiv
D_{u}(P,p,p^{\prime})-ig^{2}\mathcal{U}(P,p,p^{\prime})\,,$ (34)

are analytic in the vicinity of the pole, where

$\displaystyle ig^{2}\mathcal{T}(P,p,p^{\prime})$
$\displaystyle\equiv-\frac{ig^{2}}{(p^{\prime}-p)^{2}-M_{\pi}^{2}+i\epsilon}=-\frac{ig^{2}}{t-M_{\pi}^{2}+i\epsilon}\
,$ (35) $\displaystyle ig^{2}\mathcal{U}(P,p,p^{\prime})$
$\displaystyle\equiv-\frac{ig^{2}}{(P-p^{\prime}-p)^{2}-M_{\pi}^{2}+i\epsilon}=-\frac{ig^{2}}{u-M_{\pi}^{2}+i\epsilon}\
,$ (36)

correspond precisely to the $t$\- and $u$-channel exchange poles. Here we have
also made use of the Mandelstam invariants $t$ and $u$. Recall that, for
incoming particles with momenta $p$ and $P-p$ scattering to outgoing particles
with $p^{\prime}$ and $P-p^{\prime}$ these are defined as

$\displaystyle s$ $\displaystyle=(p+P-p)^{2}=P^{2}\,,$ (37) $\displaystyle t$
$\displaystyle=(p^{\prime}-p)^{2}\,,$ (38) $\displaystyle u$
$\displaystyle=(P-p^{\prime}-p)^{2}\,,$ (39)

and satisfy

$s+t+u=p^{2}+p^{\prime 2}+(P-p)^{2}+(P-p^{\prime})^{2}\underset{{\sf on\
shell}}{=}4M_{N}^{2}\,,$ (40)

where the second equality only holds when all momenta are on the mass shell.

For the following section, it will be convenient to define versions of the
exchanges of eqs. (35) and (36) in which the four-vectors $p$ and $p^{\prime}$
are on shell and the corresponding three-momenta, as well as $P$, are
expressed in the two-particle centre-of-mass (CM) frame, e.g. for the
$t$-channel exchange

$\mathcal{T}(s,\bm{p}^{\star},\bm{p}^{\prime\star})\equiv\mathcal{T}(P,p,p^{\prime})\Big{|}_{p^{0}=\omega_{N}(\bm{p}),\
p^{\prime 0}=\omega_{N}(\bm{p}^{\prime})}\,.$ (41)

Here we abuse notation by letting the nature of the arguments distinguish the
two functions. This can then be decomposed in partial waves as

$\mathcal{T}(s,\bm{p}^{\star},\bm{p}^{\prime\star})=\sum_{\ell=0}^{\infty}\mathcal{T}_{\ell}(s,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\,(2\ell+1)\,P_{\ell}(\cos\theta^{\star})\,,$
(42)

where
$\cos\theta^{\star}\equiv(\bm{p}^{\star}\cdot\bm{p}^{\prime\star})/(|\bm{p}^{\star}||\bm{p}^{\prime\star}|)$
and $P_{\ell}(x)$ is a standard Legendre polynomial.

The final step in reducing
$\mathcal{T}_{\ell}(s,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)$ is to note
that, when the particles are scattering with physical back-to-back momenta,
the magnitudes are constrained to satisfy
$|\bm{p}^{\star}|=|\bm{p}^{\prime\star}|=k^{\star}_{\sf os}$. For this reason,
the partial-wave projected amplitude reduces to a single coordinate function
that we denote by

$\mathcal{T}^{\sf os}_{\ell}(s)\equiv\mathcal{T}_{\ell}\big{(}s,k^{\star}_{\sf
os},k^{\star}_{\sf os}\big{)}\,.$ (43)

Equivalent expressions hold for the $u$-channel exchange.

Figure 4: Summary of the analytic structure of the $NN\to NN$ amplitude in the
complex $s$ plane: (a) at fixed CM scattering angle $\theta^{\star}$ and (b)
angular-momentum projected to a given partial wave. In both cases, we have the
usual $s$-channel cuts starting at the $NN$ and $NN\pi$ thresholds (indicated
as vertical dashed lines). Both poles and cuts, associated with the $t$-and
$u$-channel exchanges, arise below threshold for the fixed $\theta^{\star}$
case of (a). These give rise to the left-hand cuts shown in (b). The formalism
derived in this work holds for the region shown in cyan, i.e. for
$(2M_{N})^{2}-(2M_{\pi})^{2}\lesssim s\lesssim(2M_{N}+M_{\pi})^{2}$ where the
limits are not sharp due to an enhancement of neglected exponentially
suppressed terms near the branch points on either side.

Combining eqs. (35) and (42) then gives

$\displaystyle\mathcal{T}^{\sf os}_{\ell}(s)$
$\displaystyle=\frac{1}{2}\int_{-1}^{1}d(\cos\theta^{\star})\,\frac{P_{\ell}(\cos\theta^{\star})}{2(s/4-M_{N}^{2})(1-\cos\theta^{\star})+M_{\pi}^{2}-i\epsilon}\,,$
(44) $\displaystyle\mathcal{U}^{\sf os}_{\ell}(s)$
$\displaystyle=\frac{1}{2}\int_{-1}^{1}d(\cos\theta^{\star})\,\frac{P_{\ell}(\cos\theta^{\star})}{2(s/4-M_{N}^{2})(1+\cos\theta^{\star})+M_{\pi}^{2}-i\epsilon}\,,$
(45)

where we have used that $t=-2(s/4-M_{N}^{2})(1-\cos\theta^{\star})$. This
integral is straightforward to evaluate for any fixed $\ell$. For $\ell=0$,
for example, one finds

$\mathcal{T}^{\sf
os}_{0}(s)=\frac{1}{s-4M_{N}^{2}}\log\bigg{[}1+\frac{s-4M_{N}^{2}}{M_{\pi}^{2}}-i\epsilon\bigg{]}\,.$
(46)

The expression makes the logarithmic $t$-channel cut in the amplitude
manifest. See figure 4 for a summary of this analytic structure. Note that the
expression is regular for $s=4M_{N}^{2}$ with an expansion about that point
giving

$\mathcal{T}^{\sf
os}_{0}(s)=\frac{1}{M_{\pi}^{2}}-\frac{s-4M_{N}^{2}}{2M_{\pi}^{4}}+{\mathcal{O}}\\!\left(\left(s-4M_{N}^{2}\right)^{2}\right)\,.$
(47)

$\mathcal{U}^{\sf os}_{0}(s)$ satisfies the same expansion.

We stress that the partial wave expansion is expected to exhibit arbitrarily
poor convergence near $t=M_{\pi}^{2}$ or $u=M_{\pi}^{2}$. This is intuitive,
because the expansion must reproduce an angular dependence that diverges at
the pion exchange poles, but is otherwise finite. In addition, the $S$-wave
partial-wave-projected amplitudes diverge at the branch point
$s=4M_{N}^{2}-M_{\pi}^{2}$, as can be seen from eq. (46). This generically
does not match a divergence of the unprojected amplitude. In the following
sections, we will circumvent these issues by introducing an intermediate
quantity in which both the pole and the branch cut are removed. This is then
related to the scattering amplitude via integral equations that recover the
physical singularities as required.

###  2.3 Incorporating spin

The arguments given above go through with few modifications when intrinsic
spin is included for the nucleon field.

For spin-half nucleons, the first adjustment is that eq. (4) is modified to

$\displaystyle\langle\overline{N},\bm{p}^{\prime},s^{\prime}|N,\bm{p},s\rangle$
$\displaystyle=0\,,$ (48) $\displaystyle\langle
N,\bm{p}^{\prime},s^{\prime}|N,\bm{p},s\rangle$
$\displaystyle=\delta_{s^{\prime}s}\,2\omega_{N}(\bm{p})\,(2\pi)^{3}\delta^{3}(\bm{p}^{\prime}-\bm{p})\,,$
(49)

where the distinction between $N$ and $\overline{N}$ has replaced the $q$
label, and we have introduced the two-component spin label
$s\in\\{-1/2,\,1/2\\}$.

The second modification is that the scalar nucleon propagator is replaced with
a fully dressed Dirac propagator

$\Delta^{\alpha\beta}_{N,i\epsilon}(k)=\bigg{[}\frac{i}{A_{N}(k^{2})\not{k}-B_{N}(k^{2})+i\epsilon}\bigg{]}^{\alpha\beta}\,,$
(50)

where $\alpha$ and $\beta$ are Dirac indices. Instead of the single self-
energy function of the scalar case, here we have two self-energy functions
$A_{N}(k^{2})$ and $B_{N}(k^{2})$. These are constrained by renormalization
conditions to ensure that $M_{N}$ is the pole mass and that in an expansion
about the pole position one finds

$\Delta^{\alpha\beta}_{N,i\epsilon}(k)=\bigg{[}\frac{i(\not{k}+M_{N})}{k^{2}-M_{N}^{2}+i\epsilon}\bigg{]}^{\alpha\beta}+{\mathcal{O}}\left(\left(k^{2}-M_{N}^{2}\right)^{0}\right)\,.$
(51)

Equation (7) is similarly modified to

${\mathcal{L}}_{\overline{N}\\!N\pi}(x)=\overline{N}(x)\mathcal{F}\big{(}\pi(x)\big{)}N(x)\,,$
(52)

where $\mathcal{F}$ may now carry Dirac structures as allowed by the quantum
numbers.

Otherwise, with the exception of eq. (16) for which an explanation is given
below, eqs. (10)–(24) hold as written with the caveat that $B$, $\mathcal{K}$
and $\mathcal{M}$ all carry implicit indices that are contracted between
adjacent factors. As an example, eq. (14) can be written more explicitly as

$[iB\circ_{i\epsilon}iB]^{\sigma\delta,\sigma^{\prime}\delta^{\prime}}=\frac{1}{2}\int\\!\frac{d^{4}k}{(2\pi)^{4}}iB^{\sigma\delta,\alpha\beta}(P,p,k)\Delta^{\alpha\alpha^{\prime}}_{N,i\epsilon}(k)\Delta^{\beta\beta^{\prime}}_{N,i\epsilon}(P-k)iB^{\alpha^{\prime}\beta^{\prime},\sigma^{\prime}\delta^{\prime}}(P,k,p^{\prime})\,.$
(53)

The key relation of eq. (16) does require some modification. The result with
intrinsic spin takes the form

$\big{[}iB\circ_{i\epsilon}iB-iB\circ_{{\sf
pv}}iB\big{]}^{tv,t^{\prime}v^{\prime}}=\\\
\frac{1}{2}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}iB^{tv,rs}(P,p,k)\frac{\pi\delta(E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k}))\,\delta_{rr^{\prime}}\delta_{ss^{\prime}}}{2\omega_{N}(\bm{k})2\omega_{N}(\bm{P}-\bm{k})}iB^{r^{\prime}s^{\prime},t^{\prime}v^{\prime}}(P,k,p^{\prime})\,,$
(54)

where

$\displaystyle{B}^{tv,rs}(P,p,k)$
$\displaystyle=\overline{u}_{t}^{\sigma}(\bm{p})\overline{u}_{v}^{\delta}(\bm{P}-\bm{p})B^{\sigma\delta,\alpha\beta}(P,p,k)u_{r}^{\alpha}(\bm{k})u_{s}^{\beta}(\bm{P}-\bm{k})$
(55)

are spin-projected versions of the Bethe-Salpeter kernels. To define these we
have introduced the four-component spinors $\bar{u}_{r}^{\alpha}(\bm{k})$ and
$u_{r}^{\alpha}(\bm{k})$, which satisfy the relation

$[\not{k}+M_{N}]^{\alpha\beta}=\sum_{r}u_{r}^{\alpha}(\bm{k})\bar{u}_{r}^{\beta}(\bm{k})\,,$
(56)

provided that on-shell momentum $k=(\omega_{N}(\bm{k}),\bm{k})$ is used on the
left-hand side.

The key point here is that the spin indices appear through the Kronecker
deltas in eq. (54). This translates to a corresponding modification to the
phase-space factor, which can be written as

$\rho_{ss^{\prime}}(s)=\rho_{s_{1}s_{2},s_{1}^{\prime}s_{2}^{\prime}}(s)=\frac{k^{\star}_{\sf
os}}{16\pi\sqrt{s}}\big{[}\delta_{s_{1}s_{1}^{\prime}}\delta_{s_{2}s_{2}^{\prime}}+\delta_{s_{1}s_{2}^{\prime}}\delta_{s_{2}s_{1}^{\prime}}\big{]}\,,$
(57)

where we use the indices $s$ and $s^{\prime}$ as a shorthand for the pairs of
components, $s_{1}s_{2}$ and $s_{1}^{\prime}s_{2}^{\prime}$, respectively.
Completing the derivation as above then yields

$\displaystyle[\mathcal{M}]_{\ell s,\ell^{\prime}s^{\prime}}$
$\displaystyle=\frac{1}{[{\mathcal{K}^{-1}}]_{\ell
s,\ell^{\prime}s^{\prime}}-i\rho_{ss^{\prime}}\,\delta_{\ell\ell^{\prime}}}\,,$
(58)

where

$\mathcal{K}_{\ell
s,\ell^{\prime}s^{\prime}}(s)=16\pi\sqrt{s}\,\frac{\tan\delta_{\ell
s,\ell^{\prime}s^{\prime}}(k)}{k}\bigg{|}_{k=k^{\star}_{\sf os}}\,,$ (59)

and $\delta_{\ell s,\ell^{\prime}s^{\prime}}(k)$ is the phase shift for two-
to-two scattering with the spin components and orbital angular momentum as
labelled. Crucially, a consequence of spin is that orbital angular momentum is
no longer conserved so that two $\ell$ labels are needed.

A subsequent change of basis to total incoming and outgoing spin gives $\ell
SM_{S},\ell^{\prime}S^{\prime}M_{S}^{\prime}$, and one can further combine
these indices to label the amplitude in terms of total angular momentum. The
full index set then takes the form: $JM_{J},\ell S,\ell^{\prime}S^{\prime}$.
This is discussed in detail, for example in refs. Briceno:2014oea ;
Briceno:2015csa  and we do not give any more details here.

Having completed our discussion of $NN\to NN$ in the infinite-volume context,
we now turn to the effects of a finite, periodic, spatial volume.

##  3 Finite-volume formalism

In this section we review the analysis of finite-volume correlation functions,
originally worked out in refs. Luscher:1986pf ; Kim:2005gf , which leads to
the quantization condition relating two-particle finite-volume energies to
elastic scattering amplitudes. We then illustrate the problem with the
derivation that arises in the case of a subthreshold left-hand cut, and derive
a new formalism that can be applied for finite-volume energies extracted
either near or on the cut. In sections 3.1 – 3.4, we focus on energies in the
above-threshold elastic regime, while commenting occasionally on what changes
below threshold. We detail the issues that arise on the left-hand cut in
sections 3.5 and 3.6. The derivation of the new quantization condition is
presented in section 3.7 and the result is given in 3.8. Finally, the
incorporation of intrinsic spin is explained in section 3.9.

###  3.1 Skeleton expansion for the finite-volume correlator

Consider the system introduced in the previous section, now constrained to a
finite, periodic volume with periodicity $L$ in each of the three spatial
directions. Define a time-ordered Minkowski correlator

$C_{L}(P)\equiv\int
dx^{0}\int_{L}d^{3}\bm{x}\,e^{iEx^{0}}e^{-i\bm{P}\cdot\bm{x}}\langle
0|\text{T}\mathcal{A}(x)\mathcal{A}^{\dagger}(0)|0\rangle_{L}\ ,$ (60)

where we use the notation $x\equiv(x^{0},\bm{x})$ for the spatial coordinate
and $P\equiv(E,\bm{P})$ for the total four-momentum, as in the previous
section. The operators $\mathcal{A}(x)$ and $\mathcal{A}^{\dagger}(x)$ are
annihilation and creation operators, respectively, carrying the quantum
numbers of the $NN$ states of interest. The T symbol denotes standard time
ordering and the subscript $L$ on the integral indicates that integration is
performed over a single cell of the periodic volume, while the same subscript
on the matrix element emphasizes that this is evaluated in the finite-volume
theory.

In a finite volume the theory has a discrete energy spectrum. Looking at the
spectral representation of the correlator $C_{L}(P)$, it can be shown that it
has poles at the finite-volume energy levels of the system (see e.g. refs.
Luscher:1986pf ; Kim:2005gf ). In a suitably defined infinite-volume
($L\to\infty$) limit, these poles will accumulate to form the $s$-channel
branch cuts, matching the analytic structure of the amplitude shown in figure
4(a). However, the structure of the correlator below the (infinite-volume)
elastic threshold, in both finite and infinite-volume versions, differs
significantly from that of the amplitude: $C_{L}(P)$ contains no $t$\- or
$u$-channel poles or cuts. Particularly relevant for us later in this section
is the fact that the single-meson-exchange cut described in section 2.2 is not
present in the correlator, and only becomes an issue because of its presence
in the two-to-two scattering amplitude.

As with the scattering amplitude and the K-matrix in the infinite-volume case,
$C_{L}(P)$ admits a diagrammatic representation that we organize into a
skeleton expansion, shown in figure 5. The expansion employs the building
blocks introduced in the infinite-volume context of section 2.1, namely the
fully dressed propagator and the Bethe-Salpeter kernel. We denote these by
$\Delta_{N,L}(p)$ and $B_{L}(P,p,p^{\prime})$, respectively, in the finite-
volume theory. As is well known (e.g. from refs. Luscher:1986pf ; Kim:2005gf
), and will be recalled in some detail below, this decomposition is useful
because it exposes the power-like $L$ dependence of $C_{L}(P)$ in the elastic
regime: $(2M_{N})^{2}<s<(2M_{N}+M_{\pi})^{2}$.

Figure 5: Skeleton expansion for the finite-volume correlator $C_{L}(P)$,
built from Bethe-Salpeter kernels $B_{L}(P,p,p^{\prime})$, endcap functions
$\mathcal{A}_{L}(P,p)$ and $\mathcal{A}_{L}^{*}(P,p)$ and fully-dressed
nucleon propagators. Here we include the $L$-subscript on all quantities to
indicate that these are volume-dependent, albeit with exponentially suppressed
scaling in the elastic regime $(2M_{N})^{2}<s<(2M_{N}+M_{\pi})^{2}$. The
finite-volume two-particle loops are denoted $\sf fv$. Depending on the
details of the operators used in $C_{L}(P)$, one may also have a contribution
from a term with no two-particle loops (i.e. no intermediate $NN$ states),
which we omit here.

A consequence of the $L$ periodicity is that all spatial momenta are
discretized to values of type $\bm{k}={2\pi\bm{n}}/{L}$, with
$\bm{n}\in\mathbb{Z}^{3}$ a three-vector of integers. Thus, a diagrammatic
definition of $\Delta_{N,L}(p)$, $B_{L}(P,p,p^{\prime})$, and $C_{L}(P)$
itself, is given by taking a generic low-energy effective theory, formally
writing all diagrams to all orders in perturbation theory, and replacing the
integrals of spatial loop momenta with sums over the corresponding discretized
set:

$\int\\!\frac{d^{4}k}{(2\pi)^{4}}~{}\longrightarrow~{}\int\\!\frac{dk^{0}}{2\pi}\frac{1}{L^{3}}\\!\sum_{\bm{k}\in
2\pi\mathbb{Z}^{3}/L}\,,$ (61)

where we use the notation $k=(k^{0},\bm{k})$.

Beyond the $L$-dependence, the only new ingredients relative to the previous
section are the endcap “blobs” shown in figure 5. These represent functions in
momentum-space resulting from the creation and annihilation operators. We
write these as $\widetilde{\mathcal{A}}_{L}^{*}(P,p)$ and
$\widetilde{\mathcal{A}}_{L}(P,p)$, respectively. As we review in section
(3.2) based on the work of refs. Luscher:1986pf ; Kim:2005gf , these
quantities have only exponentially suppressed volume dependence of all
energies (including those on the left-hand cut). We will therefore drop the
$L$ subscripts in section 3.7, but keep them here to emphasize that the
functions are defined in the finite-volume theory.

The $n$-loop contributions to the skeleton expansion of the correlator can
thus be written as

$\displaystyle C^{(1)}_{L}(P)$
$\displaystyle=\frac{1}{2}\int\frac{dk_{1}^{0}}{(2\pi)}\frac{1}{L^{3}}\sum_{\bm{k}_{1}}\widetilde{\mathcal{A}}_{L}(P,k_{1})\,\Delta_{N,L}(k_{1})\,\Delta_{N,L}(P-k_{1})\widetilde{\mathcal{A}}_{L}^{*}(P,k_{1})\qquad\text{for
$n=1$}\,,$ (62) $\displaystyle C^{(n)}_{L}(P)$
$\displaystyle=\frac{1}{2^{n}}\int\frac{dk_{1}^{0}\,dk_{2}^{0}\,...\,dk_{n}^{0}}{(2\pi)^{n}}\frac{1}{(L^{3})^{n}}\sum_{\bm{k}_{1},\bm{k}_{2},...,\bm{k}_{n}}\widetilde{\mathcal{A}}_{L}(P,k_{1})\,\Delta_{N,L}(k_{1})\,\Delta_{N,L}(P-k_{1})$
$\displaystyle\times\prod_{j=2}^{n}\left[iB_{L}(P,k_{j-1},k_{j})\,\Delta_{N,L}(k_{j})\,\Delta_{N,L}(P-k_{j})\right]\widetilde{\mathcal{A}}_{L}^{*}(P,k_{n})\qquad\qquad\,\text{for
$n\geq 2$}\,.$ (63)

This can be expressed compactly by introducing $\circ_{\sf fv}$ as a shorthand
for the propagators and associated loop integration and summation operations,
in analogy to $\circ_{i\epsilon}$ and $\circ_{\sf pv}$ as used in eqs. (14)
and (15) above. The explicit definition for a single loop between two Bethe-
Salpeter kernels is

$B_{L}\circ_{\sf
fv}B_{L}=\frac{1}{2}\int\\!\frac{dk^{0}}{2\pi}\frac{1}{L^{3}}\\!\sum_{\bm{k}}B_{L}(P,p,k)\,\Delta_{N,L}(P-k)\,\Delta_{N,L}(k)\,B_{L}(P,k,p^{\prime})\,.$
(64)

Equation (63) can then be written as

$C_{L}^{(n)}(P)=\widetilde{\mathcal{A}}_{L}\,\circ_{\sf
fv}\left[iB_{L}\,\circ_{\sf
fv}\right]^{n-1}\widetilde{\mathcal{A}}_{L}^{\dagger}\qquad\qquad\text{for
$n\geq 1$}\,,$ (65)

and the full correlation function can be expressed compactly by summing over
the loops

$C_{L}(P)=C^{(0)}_{L}(P)+\sum_{n=0}^{\infty}\widetilde{\mathcal{A}}_{L}\,\circ_{\sf
fv}\left[iB_{L}\,\circ_{\sf
fv}\right]^{n}\widetilde{\mathcal{A}}_{L}^{\dagger}\,,$ (66)

where $C^{(0)}_{L}(P)$ is a potential contribution with no $NN$ intermediate
states. In the following subsections we illustrate the utility of this
expression for identifying power-like volume dependence in the correlator. To
do so, we first review the basic strategy for distinguishing power-like and
exponentially suppressed volume effects.

###  3.2 Classifying finite-volume effects

The prescription of summing rather than integrating the spatial loop momenta
is the only distinction between the finite- and infinite-volume Feynman rules.
Thus, the relation between Feynman diagrams in the two contexts is
conveniently understood by studying sum-integral differences.

A first key observation in this direction is as follows: If, for any direction
of $\bm{k}$, a generic function, $f(\bm{k})$, has a strip of analyticity in
the complex $|\bm{k}|$ plane then one can use the Poisson summation formula to
show that

$\left[\frac{1}{L^{3}}\sum_{\bm{k}}-\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\right]f(\bm{k})={\mathcal{O}}\left(e^{-\mu
L}\right)\,,$ (67)

for a characteristic scale $\mu$, governed by the width of the strip. In
words, the function has exponentially suppressed volume effects.222An
alternative condition is that if $f(\bm{k})$ is smooth, i.e. infinitely
differentiable along the real axis, then the sum-integral difference falls
faster than any power of $1/L$. As is common to a large body of work
quantifying finite-volume effects for scattering states, we will neglect such
exponentially suppressed scaling throughout this work whenever $\mu\sim
M_{\pi}$.

If, instead, one considers a function, $g(\bm{k})$, with a singularity on the
real axis, then the difference has power-like scaling in $L$:

$\left[\frac{1}{L^{3}}\sum_{\bm{k}}-\,{\sf
presc.}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\right]g(\bm{k})={\mathcal{O}}(L^{-n})\
,$ (68)

for some integer $n$. This expression assumes that a suitable prescription has
been used to make the integral of $g(\bm{k})$ well defined, represented by the
abbreviated ${\sf presc.}$ acting on the integral. In the case of a single
pole, for example, both a principal-value and an $i\epsilon$ pole prescription
may be used. In our analysis of Feynman diagrams, we will mainly use a
principal value prescription, denoted by $\sf pv$ below. We stress that $n$
can also be negative, in which case the sum-integral difference has a
divergent infinite-volume limit.

Returning to the skeleton expansion of eq. (66), our task is to identify when
summands are analytic and when these are singular with respect to the summed
momentum coordinates $\bm{k}_{i}$. As has been shown in refs. Luscher:1985dn ;
Luscher:1986pf ; Kim:2005gf , the singularity condition is related to the
question of intermediate states going on shell. Whenever a set of internal
propagators can go on shell (i.e. can carry the total external energy and
momentum while satisfying $p^{2}=M^{2}$ for each propagator, with $M$ denoting
the appropriate propagator mass), then singularities arise in the loop
summands. By contrast, if a given propagator never appears in an on-shell set,
then the summand has a strip of analyticity in the spatial loop momenta, and
the sum can be replaced with an integral up to neglected $e^{-\mu L}$ effects.

Figure 6: Example application of time-ordered perturbation theory (TOPT) to a
simple diagram. This can represent a contribution to $C_{L}(P)$ (in which the
vertices are generated by the operators $\mathcal{A}(x)$ and
$\mathcal{A}^{\dagger}(x)$) or a contribution to the finite-volume amplitude
introduced in section 4 below. The diagram on the left has two possible time
orderings of the vertices, shown on the right. The cuts between consecutive
vertices are shown in blue. Defining the total energy flowing into the left
diagram in a given frame as $E=P^{0}$, we note that the energy flowing across
the cuts is $E_{\sf cut}=+E$ and $E_{\sf cut}=-E$ for the first and second
diagram on the right, respectively. TOPT gives a framework to identify the
singularities after $k^{0}$ integration. In the indicated case these occur at
$E=\omega_{N}(\bm{k})+\omega_{N}(\bm{P}-\bm{k})$ and
$-E=\omega_{N}(\bm{k})+\omega_{N}(\bm{P}-\bm{k})$.

As is discussed in refs. Hansen:2014eka ; Blanton:2020gha , both based on the
more general discussion in ref. sterman_1993 , the statement of the on-shell
condition can be made more precise via time-ordered perturbation theory
(TOPT). TOPT is a method for generating expressions from Feynman diagrams with
all $k_{i}^{0}$ integrals already performed. As is shown in figure 6, each
time ordering of the set of vertices corresponds to a distinct term. For a
given time ordering, one identifies all cuts between consecutive vertices, and
then the TOPT rules generate a factor of

$\frac{1}{E_{\sf cut}-\sum_{i}\omega_{f(i)}(\bm{p}_{i})}\,,$ (69)

associated to each cut. Here $E_{\sf cut}$ is the total energy flow across the
cut. This is defined (see eq. (9.71) of ref. sterman_1993 ) as the sum of
inflowing $p^{0}$ components, from all external propagators, into all vertices
to the left of (or “earlier than”) the cut location. In the subtracted term,
$i$ labels the $i^{\rm th}$ propagator intersected by the cut; $\bm{p}_{i}$ is
its corresponding momentum, and $f(i)$ is a flavor label. Note that
$\sum_{i}\bm{p}_{i}=\bm{P}$ by total momentum conservation.

As is described in refs Hansen:2014eka ; Blanton:2020gha , provided
$\mathcal{A}(x)$ and $\mathcal{A}^{\dagger}(x)$ are local in time, then any
diagram contributing to $C_{L}(P)$ only has two vertices with external energy
and momentum flow: $(E,\bm{P})$ flows in via $\mathcal{A}^{\dagger}(0)$ and
then back out via the Fourier transformed $\mathcal{A}(x)$. Thus $E_{\sf
cut}=\pm E$ or $0$ for any cut in the diagram. In this way, the condition of
an intermediate propagator set going on shell is simply reformulated as the
condition that the TOPT pole is hit for some real-valued $\bm{p}_{i}$. If this
is the case, then the summand is of the type of $g(\bm{k})$ in eq. (68),
leading to power-like $L$ scaling. If not, then the summand is like
$f(\bm{k})$ in eq. (67) and the $L$ dependence is neglected.

Restricting attention to the elastic scattering regime,
$(2M_{N})^{2}<s<(2M_{N}+M_{\pi})^{2}$, guarantees that only intermediate $NN$
states can go on shell, implying that all sums arising in self-energy diagrams
and Bethe-Salpeter kernels can be replaced by integrals. Thus we can replace
$B_{L}\to B$ and $\Delta_{N,L}\to\Delta_{N,i\epsilon}$, where the latter are
understood as infinite-volume objects. Analogous replacements can also be made
for the endcap factors.

Conversely, power-like $L$ dependence arises only from the sums over momenta
in $NN$ states and this is the motivation for the decomposition of eq. (66).
We emphasize the parallel between $\circ_{\sf fv}$ insertions leading to
volume dependence in $C_{L}(P)$ and $\circ_{i\epsilon}$ insertions leading to
imaginary contributions in the scattering amplitude. Both effects originate
from poles in the spatial momentum dependence.

While these arguments guarantee that all contributions to $C_{L}(P)$ are safe,
subtleties can arise when subdiagrams are considered in isolation. We come to
this issue in section 3.6.

###  3.3 Reduction of the finite-volume two-particle loop

To continue the analysis of power-like $L$ dependence in $C_{L}(P)$ for
$(2M_{N})^{2}<s<(2M_{N}+M_{\pi})^{2}$, we now analyze a single $NN$ loop,
arising at any location of any skeleton expansion term, as shown in figure 5.
We introduce generic functions to the left and right of the loop, denoted
$\mathcal{L}(P,k)$ and $\mathcal{R}^{*}(P,k)$ respectively.333Here the ∗ on
$\mathcal{R}^{*}(P,k)$ indicates complex conjugation. This convention is
useful when the functions represent the endcaps of a correlator constructed
from an operator and its hermitian conjugate, as in eq. (60). In this case,
$\mathcal{L}(P,k)=\mathcal{R}(P,k)$. These correspond to either endcap
functions or Bethe-Salpeter kernels, and we omit other possible momentum
arguments not relevant for the loop considered. The loop contribution can then
be written as

$\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}=\frac{1}{2}\int\frac{dk^{0}}{2\pi}\frac{1}{L^{3}}\sum_{\bm{k}}\mathcal{L}(P,k)\,\Delta_{N,i\epsilon}(k)\,\Delta_{N,i\epsilon}(P-k)\,\mathcal{R}^{*}(P,k)\,.$
(70)

The first step in simplifying the expression is to separate the propagator
into the simple pole and a remainder

$\displaystyle\Delta_{N,i\epsilon}(p)$
$\displaystyle=\frac{i}{p^{2}-M_{N}^{2}+i\epsilon}+R_{N}(p)\,,$ (71)
$\displaystyle R_{N}(p)$
$\displaystyle\equiv\frac{i}{p^{2}-M_{N}^{2}-\Pi_{N}(p^{2})+i\epsilon}-\frac{i}{p^{2}-M_{N}^{2}+i\epsilon}\,.$
(72)

The poles cancel in the second equation so that the remainder $R_{N}(p)$ is
analytic in a neighbourhood around $p^{2}=M_{N}^{2}$.444This relies on the on-
shell renormalization scheme established after eq. (8) which sets $M_{N}$ to
be the pole mass and the residue to be unity.

Substituting this decomposition of each propagator, and denoting the sum of
all terms containing at least one factor of $R_{N}$ by $\mathcal{I}^{[1]}(P)$,
we write

$\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}=\mathcal{I}^{[1]}(P)+\frac{1}{2}\int\frac{dk^{0}}{2\pi}\frac{1}{L^{3}}\sum_{\bm{k}}\frac{\mathcal{L}(P,k)\,i^{2}\,\mathcal{R}^{*}(P,k)}{[k^{2}-M_{N}^{2}+i\epsilon][(P-k)^{2}-M_{N}^{2}+i\epsilon]}\,.$
(73)

The term $\mathcal{I}^{[1]}(P)$ is defined via summands that are analytic with
respect to $|\bm{k}|$ in a strip including the real axis. Similar replacements
will be made below and the superscript on $\mathcal{I}^{[i]}(P)$ will be
incremented to indicate the change. In all cases, exponentially suppressed
volume effects are being dropped.

Performing the integration over $k^{0}$ (closing the contour in the lower half
of the complex plane) gives

$\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}=\mathcal{I}^{[2]}(P)+\sum_{\bm{k}}\frac{\mathcal{L}(P,k)\,i\,\mathcal{R}^{*}(P,k)}{2L^{3}\cdot
2\omega_{N}(\bm{k})\big{[}(E-\omega_{N}(\bm{k}))^{2}-\omega_{N}(\bm{P}-\bm{k})^{2}\big{]}}\,\Bigg{|}_{k^{0}=\omega_{N}(\bm{k})}\,.$
(74)

Note that, at this stage, $k$ has been set on shell by
$k^{0}=\omega_{N}(\bm{k})$, but $P-k$ is itself not on shell, i.e. does not
satisfy $(P-k)^{2}=M_{N}^{2}$.

Additional contributions from the $k^{0}$ integral can be picked up from poles
or branch cuts within $\mathcal{L}(P,k)$ and $\mathcal{R}^{*}(P,k)$, but these
lead to analytic functions of $\bm{k}$ that can be absorbed into the remainder
term via $\mathcal{I}^{[1]}(P)\to\mathcal{I}^{[2]}(P)$. This holds for all
objects that can appear in place of $\mathcal{L}(P,k)$ and
$\mathcal{R}^{*}(P,k)$, up to one important caveat that is discussed in detail
in appendix A. The issue has to do with the distinction between $t$\- and
$u$-channel loops within kernels, defined according to the choice of momentum
routing in the adjacent two-particle loops. The upshot is that it is always
possible to find a routing for which $\mathcal{I}^{[2]}(P)$,
$\mathcal{L}(P,k)$, and $\mathcal{R}^{*}(P,k)$ have the desired analyticity
and exponentially suppressed volume dependence. This also holds for the $t$\-
and $u$-channel single pion exchanges and for objects containing lower left-
hand cuts and is demonstrated in section 3.6 below, together with appendix A.

Returning to eq. (74), we next decompose the functions $\mathcal{L}(P,k)$ and
$\mathcal{R}^{*}(P,k)$, with on-shell $k=(\omega_{N}(\bm{k}),\bm{k})$, into
spherical harmonics with respect to the direction of the three-momentum in the
CM frame:

$\displaystyle\mathcal{L}(P,k)\big{|}_{k^{0}=\omega_{N}(\bm{k})}$
$\displaystyle\equiv\sqrt{4\pi}\,Y_{\ell
m}(\hat{\bm{k}}^{\star})|\bm{k}^{\star}|^{\ell}\,\widetilde{\mathcal{L}}_{\ell
m}(P,|\bm{k}^{\star}|)\ ,$ (75)
$\displaystyle\mathcal{R}^{*}(P,k)\big{|}_{k^{0}=\omega_{N}(\bm{k})}$
$\displaystyle\equiv\sqrt{4\pi}\,Y^{*}_{\ell
m}(\hat{\bm{k}}^{\star})|\bm{k}^{\star}|^{\ell}\,\widetilde{\mathcal{R}}^{*}_{\ell
m}(P,|\bm{k}^{\star}|)\ ,$ (76)

with sums over the indices $\ell,m$ left implicit. As introduced in section
2.1, the superscript $\star$ indicates the boost to the CM frame. In
particular $k^{\star}=(\omega_{N}(\bm{k}^{\star}),\bm{k}^{\star})$ is the
four-momentum reached by boosting $k=(\omega_{N}(\bm{k}),\bm{k})$ to with
boost velocity $\bm{\beta}=-\bm{P}/E$:

$\begin{pmatrix}\omega_{N}(\bm{k}^{\star})\\\
\bm{k}^{\star}\end{pmatrix}={\Lambda}(\bm{\beta})\begin{pmatrix}\omega_{N}(\bm{k})\\\
\bm{k}\end{pmatrix}\,,$ (77)

where $\Lambda(\bm{\beta})$ is a standard Lorentz boost matrix.

We include the factors of $|\bm{k}^{\star}|^{\ell}$ because $Y_{\ell
m}(\hat{\bm{k}}^{\star})$ without this factor is a singular function of
$\bm{k}^{\star}$.555For example
$Y_{10}(\hat{\bm{k}}^{\star})=\sqrt{\frac{3}{4\pi}}\,\frac{k_{z}^{\star}}{\sqrt{(k_{x}^{\star})^{2}+(k_{y}^{\star})^{2}+(k_{z}^{\star})^{2}}}\,,$
(78) vanishes for $k_{z}^{\star}\to 0$ with fixed
$(k_{x}^{\star},k_{y}^{\star})$ but can take on non-zero values if the origin
is approached from a different direction. As the functions on the left-hand
sides of eqs. (75) and (76) are non-singular, the coefficient of the spherical
harmonic must vanish with the corresponding power of
$|\bm{k}^{\star}|^{\ell}$. For the subsequent steps taken below, it is more
convenient to separate out this scaling in our definition of
$\widetilde{\mathcal{L}}_{\ell m}(P,|\bm{k}^{\star}|)$ and
$\widetilde{\mathcal{R}}^{*}_{\ell m}(P,|\bm{k}^{\star}|)$ as we have done.
The tilde notation denotes this rescaling, matching also the notation used in
section 2.1. A consequence of this definition is that
$\widetilde{\mathcal{L}}_{\ell m}(P,|\bm{k}^{\star}|)$ and
$\widetilde{\mathcal{R}}^{*}_{\ell m}(P,|\bm{k}^{\star}|)$ approach a constant
as $|\bm{k}^{\star}|\to 0$, rather than vanishing as
$|\bm{k}^{\star}|^{\ell}$.

It is also useful to introduce an index notation for the momenta. We find it
more convenient to index with $\bm{k}^{\star}$ rather than $\bm{k}$. It is
then understood that $\bm{k}^{\star}$ takes on a discrete set of allowed
values inherited from $\bm{k}\in(2\pi/L)\mathbb{Z}^{3}$, via the boost of eq.
(77). We write

$\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}=\mathcal{I}^{[3]}(P)+\widetilde{\mathcal{L}}_{\bm{k}^{\star}\ell
m}(P)\,iS_{\bm{k}^{\star}\ell
m,\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)\,\widetilde{\mathcal{R}}^{*}_{\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P)\
,$ (79)

where sums over all repeated indices (including momentum indices) are implied,
and we have introduced the matrix

$S_{\bm{k}^{\star}\ell
m,\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)=\frac{1}{2L^{3}}\,\frac{4\pi\,Y_{\ell
m}(\hat{\bm{k}}^{\star})\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\star})\,\delta_{\bm{k}^{\star}\bm{k}^{\prime\star}}\,|\bm{k}^{\star}|^{\ell+\ell^{\prime}}\,H(\bm{k}^{\star})}{4\omega_{N}(\bm{k})\,\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}\big{]}}\,.$ (80)

A few manipulations are required to reach the form shown here from the
preceding relations. These are detailed in appendix B.

We have also introduced a regulator function, $H(\bm{k}^{\star})$, which must
equal $1$ whenever

$(k_{\sf os}^{\star})^{2}-(\bm{k}^{\star})^{2}=0\,,$ (81)

and should decay exponentially, with increasing $(\bm{k}^{\star})^{2}$, to
ensure that our final results are numerically tractable. The value of
$1-H(\bm{k}^{\star})$ away from the pole (as well as a term resulting from the
manipulations leading to eq. (80)) leads to a smooth summand that is absorbed
by replacing $\mathcal{I}^{[2]}(P)\to\mathcal{I}^{[3]}(P)$, as shown.

One possible choice is the regulator used in ref. Kim:2005gf

$H(\bm{k}^{\star})=\exp\\!\Big{(}\\!\\!-\\!\alpha\big{[}(\bm{k}^{\star})^{2}-(k_{\sf
os}^{\star})^{2}\big{]}\Big{)}\,,$ (82)

where $\alpha>0$ should be chosen sufficiently small to not enhance the
neglected volume dependence. Alternatively, one can make use of a piecewise
definition of $H(\bm{k}^{\star})$ that is identically zero above some cutoff,
similar to that used in the relativistic field-theoretic three-particle
quantization condition Hansen:2014eka . This type of function necessarily has
no strip of analyticity in the complex-$|\bm{k}^{\star}|$ plane and thus
generically leads to neglected volume effects that are not exponentially
suppressed, though still decaying faster than any power of $1/L$. In contrast
to the three-particle formalism, in this application there is no reason to
require the function to vanish identically, and we expect a function as in eq.
(82) will be more useful. It may seem surprising here that
$H(\bm{k}^{\star})=1$ is only required at $(\bm{k}^{\star})^{2}=(k_{\sf
os}^{\star})^{2}$ and not, for example, in the subthreshold region where one
encounters the $t$\- and $u$-channel pion exchange poles. In a nutshell, this
is valid because it leads to a correct representation of the singularities
within $C_{L}(P)$ and the scattering amplitude.

Suppressing the indices, eq. (79) can be written in a compact matrix form

$\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}=\mathcal{I}^{[3]}(P)+\widetilde{\mathcal{L}}(P)\,iS(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}\
,$ (83)

with $S(P,L)$ denoting the matrix with elements $S_{\bm{k}^{\star}\ell
m;\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)$, and
$\widetilde{\mathcal{L}}(P)$ and $\widetilde{\mathcal{R}}(P)^{\dagger}$
denoting a row and column vector with components
$\widetilde{\mathcal{L}}_{\bm{k}^{\star}\ell m}(P)$ and
$\widetilde{\mathcal{R}}^{*}_{\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P)$,
respectively. We also define the notation

$\mathcal{L}\,\circ_{\sf
rm}\,\mathcal{R}^{\dagger}\equiv\mathcal{I}^{[3]}(P)\,,$ (84)

for the remainder term, such that we can write the loop contribution as

$\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}=\mathcal{L}\left[\circ_{\sf
rm}+iS(P,L)\right]\mathcal{R}^{\dagger}\,,$ (85)

with $S(P,L)$ encoding the singular part of the summand and $\circ_{\sf rm}$
the smooth part of the summand in the loop momentum sum. In a slight abuse of
notation, we have dropped the arguments on $\widetilde{\mathcal{L}}(P)$ and
$\mathcal{R}^{\dagger}(P)$ multiplying $S(P,L)$ so that we can group the
remainder term, $\circ_{\sf rm}$, with the singular factor as shown.

Our result for the contribution of a generic two-particle finite-volume loop
can be applied to all loops in the skeleton expansion of eq. (85), provided
that the momentum routing issue of appendix A is handled correctly. Note,
however, that the factors multiplying $S(P,L)$ in eq. (85) still depend on
off-shell values of $P-k$. In the next section, we discuss how to eliminate
this issue. We highlight this intermediate result because partial off-
shellness will be of great importance for treating the left-hand cuts, as
detailed in sections 3.6 and 3.7.

###  3.4 On-shell intermediate states

In the elastic regime, $(2M_{N})^{2}<s<(2M_{N}+M_{\pi})^{2}$, the singularity
in the summand of the loop contribution, expressed compactly in eqs. (80) and
(83), corresponds to the on-shell condition for the intermediate $NN$ state.
In terms of quantities in the finite-volume frame, this condition is
$E=\omega_{N}(\bm{k})+\omega_{N}(\bm{P}-\bm{k})$, and in the CM frame, this
reduces to $|\bm{k}^{\star}|=k^{\star}_{\sf os}$. Due to the singularity, the
$L$-dependence of
$\widetilde{\mathcal{L}}(P)\,S(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}$ is
dominated by the left and right functions evaluated at these on-shell
kinematics.

To express the on-shell-value dominance more explicitly, define

$\displaystyle\widetilde{\mathcal{L}}^{\sf os}_{\ell m}(P)$
$\displaystyle\equiv\widetilde{\mathcal{L}}_{\ell m}(P,k^{\star}_{\sf os})\,,$
(86) $\displaystyle\widetilde{\mathcal{R}}^{\sf os*}_{\ell m}(P)$
$\displaystyle\equiv\widetilde{\mathcal{R}}^{*}_{\ell m}(P,k^{\star}_{\sf
os})\,,$ (87)

and make the straightforward separation

$\widetilde{\mathcal{L}}(P)\,S(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}=\widetilde{\mathcal{L}}^{\sf
os}(P)\,\xi\,S(P,L)\,\xi^{\dagger}\,\widetilde{\mathcal{R}}^{\sf
os}(P)^{\dagger}\\\\[5.0pt]
+\Big{[}\widetilde{\mathcal{L}}(P)\,S(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}-\widetilde{\mathcal{L}}^{\sf
os}(P)\,\xi\,S(P,L)\,\xi^{\dagger}\,\widetilde{\mathcal{R}}^{\sf
os}(P)^{\dagger}\Big{]}\,.$ (88)

On the right-hand side we have introduced the trivial row vector $\xi$ in the
momentum index space, with elements $\xi_{\bm{k}^{\star}}=1$, and the vectors
${\mathcal{L}}^{\sf os}(P)$ and ${\mathcal{R}}^{\sf os}(P)^{\dagger}$ in
angular momentum index space, with components $\widetilde{\mathcal{L}}^{\sf
os}_{\ell m}(P)$ and $\mathcal{R}^{\sf os*}_{\ell m}(P)$. The vectors $\xi$
and $\xi^{\dagger}$ contract only with the momentum indices of $S(P,L)$, while
$\widetilde{\mathcal{L}}^{\sf os}(P)$ and $\widetilde{\mathcal{R}}^{\sf
os}(P)^{\dagger}$ contract with the angular momentum indices

$\displaystyle\widetilde{\mathcal{L}}^{\sf
os}(P)\,\xi\,S(P,L)\,\xi^{\dagger}\widetilde{\mathcal{R}}^{\sf
os}(P)^{\dagger}$ $\displaystyle=\widetilde{\mathcal{L}}^{\sf os}_{\ell
m}(P)\,\xi_{\bm{k}^{\star}}\,S_{\bm{k}^{\star}\ell
m;\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)\,\xi_{\bm{k}^{\prime\star}}\widetilde{\mathcal{R}}^{\sf
os*}_{\ell^{\prime}m^{\prime}}(P)\,,$ (89)
$\displaystyle=\widetilde{\mathcal{L}}^{\sf os}_{\ell
m}(P)\bigg{(}\sum_{\bm{k}^{\star},\,\bm{k}^{\prime\star}}S_{\bm{k}^{\star}\ell
m;\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)\bigg{)}\widetilde{\mathcal{R}}^{\sf
os*}_{\ell^{\prime}m^{\prime}}(P)\,.$ (90)

We note that the square-bracketed term in eq. (88) represents the sum over an
analytic summand, with the singularities within $S(P,L)$ canceling in the
difference between terms. As a result, the sum can again be replaced with an
integral up to exponentially suppressed terms, and the latter can be absorbed
by replacing $\mathcal{I}^{[3]}(P)\to\mathcal{I}^{[4]}(P)$. The loop
contribution can thus be written as

$\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}=\mathcal{I}^{[4]}(P)+\widetilde{\mathcal{L}}^{\sf
os}(P)\,\xi\,iS(P,L)\,\xi^{\dagger}\,\widetilde{\mathcal{R}}^{\sf
os}(P)^{\dagger}\,.$ (91)

Provided we are in the elastic regime, this expression is always valid up to
exponentially suppressed $L$-dependence absorbed in $\mathcal{I}^{[4]}(P)$.
However, in the case of a nearby left-hand cut present in the on-shell
projected left and right functions, the scale $\mu$ in the $e^{-\mu L}$
scaling is set by the cut and can be significantly smaller than $M_{\pi}$ such
that the neglected volume effects can be numerically large. The situation is
more dramatic in the subthreshold regime where the on-shell projection can
lead to neglected power-like effects, thus leading to a quantization condition
that is no longer valid. In both cases the generalized quantization condition
presented in section 3.8 is required.

In summary, on-shell dominance allows one to derive a simpler form for the
loop with the $L$-dependence contained completely in
$\xi\,S(P,L)\,\xi^{\dagger}$, provided the left and right functions can be
safely put on-shell. We now consider the effect of analytically continuing
below threshold, first on the $S(P,L)$ factor (in the next subsection) and
then on the Bethe-Salpeter kernel (in section 3.6).

###  3.5 Subthreshold regime

The skeleton expansion for the correlator is applicable in the subthreshold
region $s<(2M_{N})^{2}$, and employing the cutting rules of TOPT allows us to
conclude that no intermediate states can go on-shell. Consequently, the
summands of spatial loop momenta have the required strips of analyticity for
all finite-volume effects to be exponentially suppressed:

$C_{L}(P)-C_{\infty}(P)={\mathcal{O}}(e^{-\mu L})\,,\qquad\text{for
$s<(2M_{N})^{2}$}\,,$ (92)

for some, potentially $s$-dependent scale, $\mu$. Here $C_{\infty}(P)$ is the
correlator evaluated in infinite volume, defined through the same diagrammatic
series as $C_{L}(P)$, but with integrated loops. Because the relation is given
below threshold, no pole prescription is required. Alternatively the
definition is equivalent to the analytic continuation of the $i\epsilon$
prescription below threshold, on the physical Riemann sheet.

To make eq. (92) more concrete, we return to the partially off-shell loop
decomposition of eq. (83). For $(2M_{N})^{2}<s<(2M_{N}+M_{\pi})^{2}$, we have
argued that the terms generating $\mathcal{I}^{[3]}(P)$ have negligible
exponentially suppressed scaling, while the second term,
$\widetilde{\mathcal{L}}(P)\,S(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}$,
has power-like $L$ dependence. Below threshold, the latter term is given by

$\widetilde{\mathcal{L}}(P)\,S(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}=\\\
-\frac{1}{2}\frac{1}{L^{3}}\sum_{\bm{k}}\widetilde{\mathcal{L}}_{\ell
m}(P,|\bm{k}^{\star}|)\frac{4\pi\,Y_{\ell
m}(\hat{\bm{k}}^{\star})Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\star})\,|\bm{k}^{\star}|^{\ell+\ell^{\prime}}H(\bm{k}^{\star})}{4\omega_{N}(\bm{k})\,\big{[}\kappa(s)^{2}+(\bm{k}^{\star})^{2}\big{]}}\widetilde{\mathcal{R}}^{*}_{\ell
m}(P,|\bm{k}^{\star}|)\,,$ (93)

where we have introduced the subthreshold binding momentum

$\kappa(s)=\sqrt{M_{N}^{2}-s/4}\,,$ (94)

and the functions $\widetilde{\mathcal{L}}_{\ell m}(P,|\bm{k}^{\star}|)$ and
$\widetilde{\mathcal{R}}^{*}_{\ell m}(P,|\bm{k}^{\star}|)$ are defined as
analytic continuations in the energy of the above-threshold objects. In words,
the above-threshold pole at $|\bm{k}^{\star}|=k^{\star}_{\sf os}$ moves to the
complex plane at $\pm i\kappa(s)$ and the function is analytic for
$-\kappa(s)<\text{Im}(|\bm{k}^{\star}|)<\kappa(s)$. This leads to volume
effects satisfying

$\left[1-\lim_{L\to\infty}\,\right]\widetilde{\mathcal{L}}(P)\,S(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}=\mathcal{\mathcal{O}}(e^{-\kappa(s)L})\,,\hskip
50.0pt\text{for $s<(2M_{N})^{2}$}\,,$ (95)

where we have used that the $L\to\infty$ limit is well-defined without a pole
prescription for $s<(2M_{N})^{2}$.

A key point for the derivation of the sub-threshold quantization condition
below is that it is only well motivated if a hierarchy exists between the size
of volume effects we include (within
$\widetilde{\mathcal{L}}(P)\,S(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}$)
and those we neglect (within $\mathcal{I}^{[3]}(P)$). Roughly, this holds
provided the binding momentum is below the pion mass: $\kappa(s)<M_{\pi}\
\Longrightarrow\ e^{-M_{\pi}L}<e^{-\kappa(s)L}$.

The partially off-shell form of eq. (93) is valid, even for energies
overlapping the left-hand cut. However, the subsequent step of
$\widetilde{\mathcal{L}}_{\ell m}(P,k^{\star}_{\sf
os})\to\widetilde{\mathcal{L}}^{\sf os}_{\ell m}(P)$ and
$\widetilde{\mathcal{R}}^{*}_{\ell m}(P,k^{\star}_{\sf
os})\to\widetilde{\mathcal{R}}^{\sf os*}_{\ell m}(P)$, described in section
3.4, fails in this region, as we now describe.

###  3.6 Analytic structure of the Bethe-Salpeter kernel

The infinite-volume Bethe-Salpeter kernel, $B(P,p,p^{\prime})$, was defined in
section 2.1 (see figure 2) as the sum of all amputated $NN\to NN$ diagrams
that are two-particle irreducible in the $s$-channel. In our analysis of the
correlator $C_{L}(P)$ in sections 3.3–3.5, this kernel can appear as the left
or right function of the finite-volume loops, in different versions with
various on- and off-shell kinematics. This is of great importance as the
nature and location of the $t$\- and $u$-channel singularities depends on
these.

We identify five different configurations of relevance:

  1. 1.

Most general case: all external four-momenta
$p,\,P-p,\,p^{\prime},\,P-p^{\prime}$ are off shell;

  2. 2.

Partially on-shell: two of the external four-momenta, $p\equiv(p^{0},\bm{p})$
and $p^{\prime}\equiv(p^{\prime 0},\bm{p}^{\prime})$, are placed on-shell,
with $p^{0}=\omega_{N}(\bm{p})$ and $p^{\prime 0}=\omega_{N}(\bm{p}^{\prime})$
respectively, and the two remaining four-momenta are off-shell, leaving $P$,
$\bm{p}$ and $\bm{p}^{\prime}$ as the only free arguments;

  3. 3.

Fully on-shell: as in Case 2, but with the remaining external four-momenta,
$(P-p)$ and $(P-p^{\prime})$, also on-shell, constraining the magnitude of CM
momenta $|\bm{p}^{\star}|=|\bm{p}^{\prime\star}|=k_{\sf os}^{\star}$ and
leaving $P$, $\hat{\bm{p}}^{\star}$ and $\hat{\bm{p}}^{\prime\star}$ as free
arguments;

  4. 4.

Partially on-shell, angular momentum projected: Case 2 above, with
$\hat{\bm{p}}^{\star}$ and $\hat{\bm{p}}^{\prime\star}$ integrated with
spherical harmonics to derive projected components (labelled by $\ell
m,\ell^{\prime}m^{\prime}$) which are functions of $P$, $|\bm{p}^{\star}|$ and
$|\bm{p}^{\prime\star}|$;

  5. 5.

Fully on-shell, angular momentum projected: Case 3, projected in the same way
as Case 4. In this case the components (again labelled by $\ell
m,\ell^{\prime}m^{\prime}$) are functions only of Mandelstam $s$.

The kernel of Case 1 appears in our expression for the correlator in eq. (66).
Case 2 then arises from performing the $k^{0}$ contour integration leading to
eq. (74) while Case 4, the angular momentum projection of Case 2, arises in
eqs. (75) and (76). Finally, Case 5 arises from the on-shell separation
discussed in subsection 3.4. Case 3, the on-shell but not-projected kernel,
has not appeared in our analysis, but is useful as an intermediate step in
understanding the analytic structure of its angular-momentum-projected
counterpart.

The analytic continuation of the on-shell kernel below threshold has common
features with that of the full scattering amplitude, discussed in section 2.2
and shown in figure 4(a). In particular, the single $t$\- and $u$-channel pion
exchanges are included in the kernel and give poles at $t=M_{\pi}^{2}$ and
$u=M_{\pi}^{2}$. Similarly, multiple pion exchanges, also included in the
kernel, lead to branch cuts.

It is instructive to study the on-shell kernel as a function of Mandelstam $s$
at fixed CM scattering angle $\theta^{\star}$. The single exchange poles lie
at

$\displaystyle s$
$\displaystyle=4M_{N}^{2}-\frac{M_{\pi}^{2}}{\sin^{2}(\theta^{\star}/2)}\qquad\qquad\qquad\qquad\qquad(t\text{-channel\
pole})\,,$ (96) $\displaystyle s$
$\displaystyle=4M_{N}^{2}-\frac{M_{\pi}^{2}}{\cos^{2}(\theta^{\star}/2)}\qquad\qquad\qquad\qquad\qquad(u\text{-channel\
pole})\,.$ (97)

For both channels, the pole position varies between $s=-\infty$ and
$s=4M_{N}^{2}-M_{\pi}^{2}$ as $\theta^{\star}$ is scanned from $0$ to $\pi$.
The integration required to project to definite angular momentum therefore
generates a branch cut along $s\in(-\infty,\ 4M_{N}^{2}-M_{\pi}^{2}]$. The
multiple meson exchanges lead to cuts in both the fixed $\theta^{\star}$ and
the angular-momentum-projected Bethe-Salpeter kernel. For the latter, the
corresponding branch points sit at $s=4M_{N}^{2}-(nM_{\pi})^{2}$ for each
$n\geq 2$.

The remainder of this subsection is dedicated to two important, but somewhat
technical points. First we argue that partially off-shell Bethe-Salpeter
kernels (i.e. those with the kinematics of Cases 2 and 4) have no $t$-channel
singularities. That is, provided that the spatial momenta $\bm{p}$ and
$\bm{p}^{\prime}$ are real, the partial offshellness removes the $t$-channel
branch cut. In this sense, there is a clear distinction between the
subthreshold behavior of the on-shell and partially off-shell Bethe-Salpeter
kernels. Second, we explain that the story is more subtle for the $u$-channel
singularities, but can be addressed with a trick based on the invariance of
Feynman diagrams under rerouting of internal loop momenta.

To explain the first point, we consider the single-pion $t$-channel diagram in
isolation. This is given in eq. (28) for the infinite-volume case,and the
finite-volume analog can be written as

$\displaystyle D_{t,L}(P,p,p^{\prime})\equiv G_{L}(p,p^{\prime})\
\Delta_{\pi,L}(p^{\prime}-p)\ G_{L}(P-p,P-p^{\prime})\,,$ (98)

where we have also introduced finite-volume versions of the $NN\pi$ form
factors and the pion propagator. As discussed above, the $L$-depencence within
$G_{L}(p,p^{\prime})$ and $\Delta_{\pi,L}(p^{\prime}-p)$ is exponentially
suppressed as long as the energy and momentum flowing through the diagrams is
below the lowest multi-particle threshold. We can then additionally separate
out the singularity from a remainder, as in eqs. (33) and (35), to write

$\displaystyle
ig^{2}\mathcal{T}(P,p,p^{\prime})=-\frac{ig^{2}}{\big{(}\omega_{N}(\bm{p})-\omega_{N}(\bm{p}^{\prime})\big{)}^{2}-(\bm{p}-\bm{p}^{\prime})^{2}-M_{\pi}^{2}+i\epsilon}=-\frac{ig^{2}}{t-M_{\pi}^{2}+i\epsilon}\,,$
(99)

where we have already applied the kinematics of Case 2 as described above. As
in the infinite-volume section, $M_{\pi}$ is the physical pion mass and $g$ is
the residue of poles in the full scattering amplitude. This is the same
residue as in the Bethe-Salpeter kernel as follows from the fact that the pole
arises from the single-Bethe-Salpeter term of the skeleton expansion.

The first key observation for eq. (99) is that it simply has no $s$
dependence, and therefore cannot generate any singularities in the complex $s$
plane. In addition to this, the momenta entering the $t$-channel exchange here
guarantee that $t<0$ and thus that the singularity at $t=M_{\pi}^{2}$ is not
encountered. This, in turn, implies that no power-like finite-volume effects
arise in $C_{L}(P)$ due to this contribution. We will argue that this
statement generalizes to any diagram that can be expressed as some number of
$t$-channel exchanges.

We can also analyze the single $t$-channel exchange using TOPT. Following the
rules described in ref. sterman_1993 , we note that the diagram has two
vertices and one internal propagator (see the left diagram of figure 3). For
Case 2 kinematics, momentum
$p-p^{\prime}=\big{(}\omega_{N}(\bm{p})-\omega_{N}(\bm{p}^{\prime}),\bm{p}-\bm{p}^{\prime}\big{)}$
flows into one vertex and out of the other. Thus the energy flowing across the
cut is given by $E_{\sf cut}=\omega_{N}(\bm{p})-\omega_{N}(\bm{p}^{\prime})$
for one of the two time orderings, and by $E_{\sf
cut}=-\omega_{N}(\bm{p})+\omega_{N}(\bm{p}^{\prime})$ for the other. Combining
with the single pion propagator intersected by each cut, we expect the
denominators
$\omega_{N}(\bm{p})-\omega_{N}(\bm{p}^{\prime})-\omega_{\pi}(\bm{p}-\bm{p}^{\prime})$
and
$-\omega_{N}(\bm{p})+\omega_{N}(\bm{p}^{\prime})-\omega_{\pi}(\bm{p}-\bm{p}^{\prime})$
to appear. These combine with other factors in the TOPT construction to
reproduce eq. (99), which can be rewritten as

$ig^{2}\mathcal{T}(P,p,p^{\prime})=-\frac{ig^{2}}{2\omega_{\pi}(\bm{p}-\bm{p}^{\prime})}\bigg{[}\frac{1}{\omega_{N}(\bm{p})-\omega_{N}(\bm{p}^{\prime})-\omega_{\pi}(\bm{p}-\bm{p}^{\prime})+i\epsilon}\\\
+\frac{1}{-\omega_{N}(\bm{p})+\omega_{N}(\bm{p}^{\prime})-\omega_{\pi}(\bm{p}-\bm{p}^{\prime})+i\epsilon}\bigg{]}\,.$
(100)

More generally, for any diagram that contributes to the Bethe-Salpeter kernel,
we can enumerate all time orderings and then identify the (possibly empty) set
of cuts for which $E_{\sf
cut}=\pm\big{(}\omega_{N}(\bm{p})-\omega_{N}(\bm{p}^{\prime})\big{)}$. Each
will lead to a denominator of the form

$E_{\sf
cut}-\sum_{i}\omega_{f(i)}(\bm{p}_{i})=\pm\big{(}\omega_{N}(\bm{p})-\omega_{N}(\bm{p}^{\prime})\big{)}-\sum_{i}\omega_{f(i)}(\bm{p}_{i})\,,$
(101)

where the sum runs over all internal cut propagators, $f(i)$ is either $N$ or
$\pi$, and $\sum_{i}\bm{p}_{i}=\pm(\bm{p}-\bm{p}^{\prime})$, with the sign
matching that of $E_{\sf cut}$. For these cases, no singularity is encountered
for real three-momenta, since it would correspond to the decay of a nucleon
into a multi-particle state, which is always forbidden either by kinematics or
by the conserved quantum numbers.

Note the contrast of the discussion here to that in section 3.2 and refs.
Hansen:2014eka ; Blanton:2020gha . If we are only interested in the
singularities of the full correlator, $C_{L}(P)$, then the cuts always satisfy
$E_{\sf cut}=\pm E,0$. However, if we are also concerned with singularities of
the Bethe-Salpeter kernel itself, for various choices of external four-
momenta, then it is neccesary to revisit TOPT at the level of diagrams with
four-external legs (as in figure 3). Then new energies can flow across the
cuts, such as the case with $E_{\sf
cut}=\pm\big{(}\omega_{N}(\bm{p})-\omega_{N}(\bm{p}^{\prime})\big{)}$
considered above. If this were the only possibility, then the kernel of Case 2
would have no singularities on the real $s$-axis.

A caveat to the above discussion comes from contributions to the kernel that
depend on Mandelstam $u$ (such as the right diagram in figure 3). Such
diagrams have singularities when evaluated with Case 2 kinematics, i.e. with
$k=(\omega_{N}(\bm{k}),\bm{k})$, but with $P-k$ off shell. These manifest in
TOPT from orderings with the vertices that have inflowing $p$ and outflowing
$P-p^{\prime}$ both appearing either to the left or to the right of a
particular cut. Then one has $E_{\sf
cut}=\pm\big{(}\omega_{N}(\bm{p})-E+\omega_{N}(\bm{p}^{\prime})\big{)}$ and,
as a result, the denominator has the form

$E_{\sf
cut}-\sum_{i}\omega_{f(i)}(\bm{p}_{i})=\pm\big{(}\omega_{N}(\bm{p})-E+\omega_{N}(\bm{p}^{\prime})\big{)}-\sum_{i}\omega_{f(i)}(\bm{p}_{i})\,.$
(102)

As we discuss in more detail in appendix A, this can lead to singularities in
the partly off-shell Bethe-Salpeter kernel that do not correspond to
singularities of $C_{L}(P)$. This is an artifact of the partial off-shellness
and can be treated most easily by defining an auxiliary Bethe-Salpeter kernel,
denoted by $B^{\mathfrak{T}}(P,p,p^{\prime})$, in which all $u$-channel
exchanges are replaced by $t$-channel exchanges.

The explicit construction of this kernel is given in appendix A so we do not
provide the details here, but only emphasize three key properties: First,
$B^{\mathfrak{T}}(P,p,p^{\prime})$, like $\mathcal{T}(P,p,p^{\prime})$ in eq.
(99), has no singularities on the real $s$-axis when kinematics are assigned
in the partially off-shell manner of Case 2 and Case 4 (the angular-momentum
projected counterpart). This follows from the discussion for a generic
$t$-channel diagram above and is detailed in appendix A. Second,
$B^{\mathfrak{T}}(P,p,p^{\prime})$ does have singularities for fully on-shell,
e.g. Case 5 kinematics, in the subthreshold regime. For
$(2M_{N})^{2}-(2M_{\pi})^{2}<s<(2M_{N}+M_{\pi})^{2}$, these singularities are
completely captured by $\mathcal{T}(P,p,p^{\prime})$. Third, the auxiliary
kernel is equal to the original Bethe-Salpeter kernel after symmetrization:

$\displaystyle
B(P,p,p^{\prime})=\frac{1}{2}\Big{[}B^{\mathfrak{T}}(P,p,p^{\prime})+B^{\mathfrak{T}}(P,p,P-p^{\prime})\Big{]}\,.$
(103)

Returning to our two versions of the loop-reduction equation, (83) and (91):
If the left or right function is taken to be a $B^{\mathfrak{T}}$ insertion,
one encounters the kinematics of Case 4 in
$\widetilde{\mathcal{L}}(P)S(P,L)\widetilde{\mathcal{R}}(P)^{\dagger}$ within
eq. (83) and that of Case 5 in $\widetilde{\mathcal{L}}^{\sf os}(P)\xi
S(P,L)\xi^{\dagger}\widetilde{\mathcal{R}}^{\sf os}(P)^{\dagger}$ within eq.
(91). While the subthreshold cut is absent from the first version it is
present in the second, and therefore also present in the remainder term
$\mathcal{I}^{[4]}(P)$. Neglecting this singularity in the remainder
invalidates the subsequent steps in the standard derivation of a quantization
condition.

To address the issue caused by $|\bm{k}^{\star}|\to k^{\star}_{\sf os}$ on the
cut, we introduce

$\overline{B}^{\mathfrak{T}}(P,p,p^{\prime})\equiv
B^{\mathfrak{T}}(P,p,p^{\prime})-2g^{2}\mathcal{T}(P,p,p^{\prime})\,.$ (104)

By construction, this does not contain single-exchange poles and it is safe to
perform the angular momentum projections and set the arguments on shell. Each
partial-wave amplitude of this kernel is then a function of $s$ with an
analytic strip along the real axis for $s>(2M_{N})^{2}-(2M_{\pi})^{2}$. We
cannot do the same for $\mathcal{T}(P,p,p^{\prime})$ and thus reach a new
quantization condition by keeping this party off-shell, as we explain in the
next subsection.

Before turning to this, it is useful to define a subtracted version of the
full Bethe-Salpeter kernel

$\overline{B}(P,p,p^{\prime})\equiv
B(P,p,p^{\prime})-g^{2}\mathcal{E}(P,p,p^{\prime})\,,$ (105)

where $\mathcal{E}(P,p,p^{\prime})$ is the sum of the $t$\- and $u$-channel
single-pion exchanges

$\mathcal{E}(P,p,p^{\prime})\equiv\mathcal{T}(P,p,p^{\prime})+\mathcal{U}(P,p,p^{\prime})\,,$
(106)

and $\mathcal{U}(P,p,p^{\prime})$ is defined in eq. (36). A symmetrization
relation holds between the subtracted kernels, directly inherited from the
unsubtracted cases

$\overline{B}(P,p,p^{\prime})=\frac{1}{2}\Big{[}\overline{B}^{\mathfrak{T}}(P,p,p^{\prime})+\overline{B}^{\mathfrak{T}}(P,p,P-p^{\prime})\Big{]}\,.$
(107)

###  3.7 Full decomposition of the finite-volume correlator

We are finally in position to put together the results of the previous
sections into the final derivation. Recalling the expression for the finite-
volume correlator, eq. (66), and substituting the separation of the kernel
introduced in eq. (105) above, we reach

$\displaystyle C_{L}(P)$
$\displaystyle=C^{(0)}(P)+\sum_{n=0}^{\infty}\widetilde{\mathcal{A}}\,\circ_{\sf
fv}\,\left[\big{(}i\overline{B}+ig^{2}\mathcal{E}\big{)}\,\circ_{\sf
fv}\,\right]^{n}\,\widetilde{\mathcal{A}}^{\dagger}\,,$ (108)
$\displaystyle=C^{(0)}(P)+\sum_{n=0}^{\infty}\widetilde{\mathcal{A}}\,\circ_{\sf
fv}\,\left[\big{(}i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\big{)}\,\circ_{\sf
fv}\,\right]^{n}\,\widetilde{\mathcal{A}}^{\dagger}\,.$ (109)

Here and in the following we drop the $L$ subscripts of objects (such as
$\widetilde{\mathcal{A}}_{L}$) that are known to have exponentially suppressed
volume dependence. Since the loops and the endcaps are exchange-symmetric, we
have noted that both separations can be substituted for the Bethe-Salpeter
kernel.

To further reduce this expression, consider a generic $\mathcal{T}$ factor
with a finite-volume loop on both sides of it. In each loop, the contour
integration leading to eq. (74) places one of the two momenta on shell. The
resulting object is (cf. Case 2):

$\mathcal{T}(P,p,p^{\prime})\Big{|}_{\begin{subarray}{l}p^{0}=\omega_{N}(\bm{p}),\\\
p^{\prime
0}=\omega_{N}(\bm{p}^{\prime})\end{subarray}}=\frac{1}{2\omega_{N}(\bm{p}^{\star})\omega_{N}(\bm{p}^{\prime\star})-2|\bm{p}^{\star}||\bm{p}^{\prime\star}|\cos\theta^{\star}-2M_{N}^{2}+M_{\pi}^{2}-i\epsilon}\,.$
(110)

This quantity has a strip of analyticity about the real-axis in the
$|\bm{p}^{\star}|$ and $|\bm{p}^{\prime\star}|$ complex planes.666We note that
the contour integration also encircles the poles in
$\mathcal{T}(P,p,p^{\prime})$ itself, but this is not an issue as it simply
leads to an $NN\pi$ intermediate state pole, which is above the elastic region
and thus gives a smooth summand contribution that can be absorbed in the
definition of $\mathcal{I}^{[2]}$ in eq. (74). As a result, the loop
separation identity of eq. (85), $\circ_{\sf fv}=\circ_{\sf rm}+iS(P,L)$, can
be applied to two-particle loops involving $\mathcal{T}$ exchanges, and thus
to all finite-volume loops appearing in (109). Making the substitution, we
reach

$C_{L}(P)=C^{(0)}(P)+\sum_{n=0}^{\infty}\widetilde{\mathcal{A}}\,\big{[}\\!\circ_{\sf
rm}+\,iS(P,L)\big{]}\,\Big{[}\big{(}i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\big{)}\,\big{[}\\!\circ_{\sf
rm}+\,iS(P,L)\big{]}\,\Big{]}^{n}\,\widetilde{\mathcal{A}}^{\dagger}\,.$ (111)

Next, we rearrange the series by $S(P,L)$ insertions to find

$C_{L}(P)=\mathcal{I}^{[1]}_{C}(P)+\sum_{n=0}^{\infty}A^{[1]}(P)\,iS(P,L)\,\Big{[}\big{(}i\overline{\mathcal{K}}^{[1]}(P)+2ig^{2}\mathcal{T}(P)\big{)}\,iS(P,L)\Big{]}^{n}\,{A}^{[1]\dagger}(P)\,,$
(112)

where all $\circ_{\sf rm}$ operations are absorbed into the four new
$L$-independent quantities:

$\displaystyle\mathcal{I}^{[1]}_{C}(P)$
$\displaystyle=C^{(0)}(P)+\sum_{n=0}^{\infty}\widetilde{\mathcal{A}}\,\circ_{\sf
rm}\,\Big{[}\big{(}i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\big{)}\,\circ_{\sf
rm}\Big{]}^{n}\,\widetilde{\mathcal{A}}^{\dagger}\,,$ (113) $\displaystyle
A^{[1]}(P)$
$\displaystyle=\sum_{n=0}^{\infty}\widetilde{\mathcal{A}}\,\Big{[}\circ_{\sf
rm}\big{(}i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\big{)}\Big{]}^{n}\,,$
(114) $\displaystyle A^{[1]\dagger}(P)$
$\displaystyle=\sum_{n=0}^{\infty}\,\Big{[}\big{(}i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\big{)}\circ_{\sf
rm}\Big{]}^{n}\,\widetilde{\mathcal{A}}^{\dagger}\,,$ (115) $\displaystyle
i\overline{\mathcal{K}}^{[1]}(P)$
$\displaystyle=\sum_{n=0}^{\infty}\Big{[}\big{(}i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\big{)}\,\circ_{\sf
rm}\Big{]}^{n}\,\big{(}i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\big{)}-2ig^{2}\mathcal{T}\,.$
(116)

Here the [1] superscripts are used, as in section 3.3, to indicate
intermediate quantities.

In contrast to eq. (111), in eq. (112) the integral operator does not appear
explicitly. All factors are vectors or matrices on the $\bm{k}^{\star}\ell m$
space, multiplied with indices contracted in the usual way. In particular,
$\mathcal{T}(P)$ is defined from the projection of
$\mathcal{T}(P,p,p^{\prime})$, as defined in eq. (110), to definite angular
momentum [see also eq. (42)]:

$\displaystyle\mathcal{T}(P,p,p^{\prime})\Big{|}_{\begin{subarray}{l}p^{0}=\omega_{N}(\bm{p}),\\\
p^{\prime 0}=\omega_{N}(\bm{p}^{\prime})\end{subarray}}$ $\displaystyle\equiv
4\pi\,|\bm{p}^{\star}|^{\ell}\,Y^{*}_{\ell
m}(\hat{\bm{p}}^{\star})\,\mathcal{T}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\,|\bm{p}^{\prime\star}|^{\ell^{\prime}}\,Y_{\ell^{\prime}m^{\prime}}(\hat{\bm{p}}^{\prime\star})\,.$
(117)

This leads us to define

$\displaystyle\mathcal{T}_{\bm{k}^{\star}\ell
m,\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P)$
$\displaystyle=\mathcal{T}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{k}^{\star}|,|\bm{k}^{\prime\star}|)\,.$
(118)

It is straightforward to work out expressions for particular angular-momentum
components. The $S$-wave result, for example, is given by

$\mathcal{T}_{\bm{k}^{\star}00,\bm{k}^{\prime\star}00}(P)=\frac{1}{4|\bm{k}^{\star}||\bm{k}^{\prime\star}|}\log\left(\frac{2\omega_{N}(\bm{k}^{\star})\omega_{N}(\bm{k}^{\prime\star})+2|\bm{k}^{\star}||\bm{k}^{\prime\star}|-2M_{N}^{2}+M_{\pi}^{2}-i\epsilon}{2\omega_{N}(\bm{k}^{\star})\omega_{N}(\bm{k}^{\prime\star})-2|\bm{k}^{\star}||\bm{k}^{\prime\star}|-2M_{N}^{2}+M_{\pi}^{2}-i\epsilon}\right)\,,$
(119)

which matches eq. (46) when we set $|\bm{k}^{\star}|$ and
$|\bm{k}^{\prime\star}|$ to their complex, subthreshold on-shell values.

The final step is to note that all insertions of $A^{[1]}(P)$,
$\overline{\mathcal{K}}^{[1]}(P)$, and $A^{[1]\dagger}(P)$ can be set on
shell, as described in sections 3.4 and 3.5. This requires a complicated set
of redefinitions for the $L$-independent quantities, as detailed in appendix
C. The resulting expression is

$\displaystyle C_{L}(P)$
$\displaystyle=\mathcal{I}_{C}(P)+\sum_{n=0}^{\infty}A^{\sf
os}(P)\,\xi\,iS(P,L)\left[\left(\xi^{\dagger}\,i\overline{\mathcal{K}}^{\sf
os}(P)\,\xi+2ig^{2}\mathcal{T}(P)\right)iS(P,L)\right]^{n}\xi^{\dagger}\,A^{\sf
os}(P)^{\dagger}\,,$ (120)

where $\mathcal{I}_{C}(P)$, $A^{\sf os}(P)$, $A^{\sf os}(P)$, and
$\overline{\mathcal{K}}^{\sf os}(P)$, yet another set of infinite-volume
quantities, are the final objects entering our decomposition of $C_{L}(P)$.
Their detailed definitions are given in appendix C where we also prove eq.
(120). Note that the definitions are not particularly relevant to the main
derivation, since all quantities besides $\overline{\mathcal{K}}^{\sf os}(P)$
do not appear in the final result. The latter is more usefully defined through
its relation to the scattering amplitude, derived in section 4. One crucial
point, also addressed in appendix C, is that the asymmetry from
$\overline{B}^{\mathfrak{T}}$ is removed at this stage, so that
$\overline{\mathcal{K}}^{\sf os}(P)$ is defined with the same exchange
symmetry as the scattering amplitude.

Summing the geometric series in eq. (120), we finally obtain

$\displaystyle C_{L}(P)$ $\displaystyle=\mathcal{I}_{C}(P)+A^{\sf
os}(P)\,\xi\frac{i}{S(P,L)^{-1}+\xi^{\dagger}\,\overline{\mathcal{K}}^{\sf
os}(P)\,\xi+2g^{2}\mathcal{T}(P)}\xi^{\dagger}\,A^{\sf os}(P)^{\dagger}\,.$
(121)

This is the main result of this subsection. Up to neglected exponentially
suppressed contributions, the volume dependence of $C_{L}(P)$ is contained
entirely in the second term, both through the explicit $L$ dependence of
$S(P,L)$ and the implicit dependence in the matrix multiplications, which
involve sums over discretized spatial momenta.

###  3.8 Result

For a given total spatial momentum $\bm{P}$ and volume $L$, let us denote the
finite volume energy levels by $E_{j}(\bm{P},L)$. Our main motivation for
examining $C_{L}(P)$ was that it has poles at these energy levels and from eq.
(121), we see that these poles arise at the values of the energy for which the
matrix in the denominator has a vanishing eigenvalue. This gives us the
following quantization condition:

$\det_{\bm{k}^{\star}\ell
m}\left[S(P_{j},L)^{-1}+\xi^{\dagger}\,\overline{\mathcal{K}}^{\sf
os}(P_{j})\,\xi+2g^{2}\mathcal{T}(P_{j})\right]=0\,,$ (122)

where $P_{j}=(E_{j}(\bm{P},L),\bm{P})$ and the determinant is taken over the
whole $\bm{k}^{\star}\ell m,\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}$ index
space, as indicated. This condition, together with the integral equations of
section 4, is the main result of this work. It can be used to constrain
$\overline{\mathcal{K}}^{\sf os}(P)$, as well as the trilinear coupling $g$,
via lattice-QCD-determined energies.777We note that, in certain cases, it may
be more feasible or practical to constrain $g$ from fits to form factors, e.g.
the nucleon axial form factor in the case of the physical $NN$ system. In all
cases, one must be certain that the extracted coupling satisfies the defining
relation of being the residue of the physical scattering amplitude at the
$t$\- and $u$-channel poles.

To make this a stand-alone section, we collect all definitions required to
numerically evaluate the quantization condition.

The matrix $\mathcal{T}(P)$, which encodes the angular-momentum projected
$t$-channel exchange, has matrix elements given by

$\mathcal{T}_{\bm{k}^{\star}\ell
m,\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P)=\frac{1}{4\pi|\bm{k}^{\star}|^{\ell}|\bm{k}^{\prime\star}|^{\ell^{\prime}}}\int
d\Omega_{\hat{\bm{k}}^{\star}}d\Omega_{\hat{\bm{k}}^{\prime\star}}Y_{\ell
m}(\hat{\bm{k}}^{\star})\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\prime\star})\\\
\times\bigg{[}\frac{1}{2\omega_{N}(\bm{k}^{\star})\omega_{N}(\bm{k}^{\prime\star})-2|\bm{k}^{\star}||\bm{k}^{\prime\star}|\cos\theta^{\star}-2M_{N}^{2}+M_{\pi}^{2}-i\epsilon}\bigg{]}\,,$
(123)

where
$\cos\theta^{\star}=\hat{\bm{k}}^{\star}\cdot\hat{\bm{k}}^{\prime\star}$. This
expression is reached by combining eqs. (110), (117) and (118) and using the
orthogonality of spherical harmonics.

The matrix $S(P,L)$, which encodes the intermediate two-nucleon on-shell pole,
was already defined in eq. (80). We repeat the expression here

$S_{\bm{k}^{\star}\ell
m,\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)=\frac{1}{2L^{3}}\,\frac{4\pi\,Y_{\ell
m}(\hat{\bm{k}}^{\star})\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\star})\,\delta_{\bm{k}^{\star}\bm{k}^{\prime\star}}\,|\bm{k}^{\star}|^{\ell+\ell^{\prime}}\,e^{-\alpha[(\bm{k}^{\star})^{2}-(k_{\sf
os}^{\star})^{2}]}}{4\omega_{N}(\bm{k})\,\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}\big{]}}\,,$ (124)

where we have substituted the possible form for $H(\bm{k}^{\star})$ from eq.
(82). Recall that $\bm{k}^{\star}$ takes on all values reached by boosting
$(\omega_{N}(\bm{k}),\bm{k})$ to the CM frame, where
$\bm{k}\in(2\pi/L)\mathbb{Z}^{3}$ is a finite-volume momentum. Recall also the
definition $k_{\sf os}^{\star}=\sqrt{s/4-M_{N}^{2}}$, given in eq. (21).

It remains to describe the middle term of eq. (122). First,
$\overline{\mathcal{K}}^{\sf os}(P)$ (the quantity one aims to determine from
the spectrum) is a diagonal matrix in angular-momentum space so that we can
write

$\overline{\mathcal{K}}_{\ell m,\ell^{\prime}m^{\prime}}^{\sf
os}(P)=\delta_{\ell\ell^{\prime}}\,\delta_{mm^{\prime}}\,\overline{\mathcal{K}}^{{\sf
os}(\ell)}(s)\,,$ (125)

where we have also used that each component is a Lorentz invariant, depending
only on the CM energy, encoded here via the Mandelstam variable $s=P^{2}$.
Unlike the standard K-matrix, this quantity is analytic in the extended range
$(2M_{N})^{2}-(2M_{\pi})^{2}<s<(2M_{N}+M_{\pi})^{2}$ and the usual
parametrizations involving polynomials and poles can be used in this region.
Finally, as introduced in section 3.4, $\xi$ is a trivial vector with
$\bm{k}^{\star}$ indices, defined as

$\xi_{\bm{k}^{\star}}=1\,,$ (126)

used to promote $\mathcal{K}^{\sf os}(P)$ from the $\ell m$ space to the
$\bm{k}^{\star}\ell m$ space.

This completes our discussion of the generalized quantization condition for
spin-zero nucleons. In section 4, we discuss how $g$ and
$\overline{\mathcal{K}}^{\sf os}(P)$ can then be used to then obtain the
$NN\to NN$ scattering amplitude. Before turning to this, we first discuss the
extension to spinning particles.

###  3.9 Incorporating spin

Thus far, our finite-volume analysis has been restricted to particles without
intrinsic spin. In this subsection, we describe how the derivation of (122)
can be readily modified to accommodate spinning particles. We first present
the extension to identical spin-half particles, relevant for physical two-
nucleon systems, and then briefly comment on the generalization to arbitrary
spin.

Consider a realistic, generic low-energy theory of QCD with Dirac spinors $N$
representing nucleons and anti-nucleons coupled to pseudoscalar fields $\pi$
representing the spin-zero pions. The interactions in the Lagrangian are
modified appropriately to contain all vacuum-quantum-number combinations of
$N$, $\overline{N}$, and $\pi$.

The dressed propagator of the nucleon in a finite volume can be written as

$\displaystyle\Delta_{N,L}(k)$
$\displaystyle=\frac{i(\not{k}+M_{N})}{k^{2}-M_{N}^{2}+i\epsilon}+R_{N,L}(k)\
,$ (127)

where $R_{N,L}(k)$ encodes the self-energy contributions. As above, we require
$M_{N}$ to be the physical pole mass and assume renormalization such that the
residue at the pole of $\Delta_{N,L}(k)$ is the same as that of the first term
on the right-hand side. This has the consequence that $R_{N,L}(k)$ is analytic
in a neighborhood of the pole, up to the exponentially suppressed volume
effects on the pole mass, which we neglect.

As in eq. (60), we define a correlator $C_{L}(P)$ of operators carrying the
quantum numbers of an $NN$ state with spin and consider the corresponding
skeleton expansion. Following the steps of section 3.3, the contribution from
a generic two-nucleon loop after $k^{0}$ integration is given by

$\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}=\mathcal{I}^{[1]}(P)+\sum_{\bm{k}}\frac{i}{2L^{3}}\frac{\mathcal{L}^{\alpha\beta}(P,k)(\not{k}+M_{N})^{\alpha\alpha^{\prime}}(\not{P}-\not{k}+M_{N})^{\beta\beta^{\prime}}\mathcal{R}^{\alpha^{\prime}\beta^{\prime}}(P,k)^{*}}{2\omega_{N}(\bm{k})\left[(E-\omega_{N}(\bm{k}))^{2}-\omega_{N}(\bm{P}-\bm{k})^{2}\right]}\bigg{|}_{k^{0}=\omega_{N}(\bm{k})}$
(128)

where we have absorbed the $R_{N,L}(k)$ dependent terms into
$\mathcal{I}^{[1]}(P)$, since these only generate exponentially suppressed
contributions. Here $\alpha,\beta,\alpha^{\prime},\beta^{\prime}$ are Dirac
spinor indices and sums over repeated indices are implied.

Noting that $k$ is on shell, we identify the spin sum relation

$(\not{k}+M_{N})^{\alpha\alpha^{\prime}}\Big{|}_{k^{0}=\omega_{N}(\bm{k})}=\sum_{r=\pm}u^{\alpha}_{r}(\bm{k})\bar{u}^{\alpha^{\prime}}_{r}(\bm{k})\
,$ (129)

where $u^{\alpha}_{r}(\bm{k})$ and $\bar{u}^{\alpha^{\prime}}_{r}(\bm{k})$ are
the Dirac spinors describing a free $N$ state with definite spin. The spin-up
and down states are labelled by $r=\pm$. By contrast, the four-vector
$(P-k)|_{k^{0}=\omega_{N}(\bm{k})}=(E-\omega_{N}(\bm{k}),\bm{P}-\bm{k})$ is
not on-shell. To address this we consider an on-shell vector $(P-k)_{\sf
os}\equiv(\omega_{N}(\bm{P}-\bm{k}),\bm{P}-\bm{k})$ such that

$\displaystyle(\not{P}-\not{k}+M_{N})^{\beta\beta^{\prime}}$
$\displaystyle=\left((\not{P}-\not{k})_{\sf
os}+M_{N}\right)^{\beta\beta^{\prime}}+\left[(\not{P}-\not{k})-(\not{P}-\not{k})_{\sf
os}\right]^{\beta\beta^{\prime}}\ ,$ (130)
$\displaystyle=\sum_{s=\pm}u^{\beta}_{s}(\bm{P}-\bm{k})\bar{u}^{\beta^{\prime}}_{s}(\bm{P}-\bm{k})+\left[E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k})\right](\gamma^{0})^{\beta\beta^{\prime}}\
,$ (131)

where we have used the spin sum relation on $(P-k)_{\sf os}$. This differs
from the on-shell point identified in refs. Briceno:2014oea ; Briceno:2015csa
.

Substituting these expressions for $(\not{k}+M_{N})^{\alpha\alpha^{\prime}}$
and $(\not{P}-\not{k}+M_{N})^{\beta\beta^{\prime}}$ into (128), we see that
the term
$\left[E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k})\right](\gamma^{0})^{\beta\beta^{\prime}}$
in (131) leads to a cancellation of the pole, so we can include this
contribution in the remainder term and obtain

$\displaystyle\begin{split}\mathcal{L}\,\circ_{\sf
fv}\,\mathcal{R}^{\dagger}&=\mathcal{I}^{[2]}(P)+\sum_{\bm{k}}\frac{i}{2L^{3}}\frac{{\mathcal{L}}_{rs}(P,\bm{k}^{\star})\,\delta_{rr^{\prime}}\,\delta_{ss^{\prime}}\,{\mathcal{R}}^{*}_{r^{\prime}s^{\prime}}(P,\bm{k}^{\star})}{2\omega_{N}(\bm{k})\left[(E-\omega_{N}(\bm{k}))^{2}-\omega_{N}(\bm{P}-\bm{k})^{2}\right]}\,,\end{split}$
(132)

where we have defined the objects

$\displaystyle\begin{split}&{\mathcal{L}}_{rs}(P,\bm{k}^{\star})\equiv\mathcal{L}^{\alpha\beta}(P,k)\big{|}_{k^{0}=\omega_{N}(\bm{k})}\,u^{\alpha}_{r}(\bm{k})u^{\beta}_{s}(\bm{P}-\bm{k})\
,\\\
&{\mathcal{R}}^{*}_{r^{\prime}s^{\prime}}(P,\bm{k}^{\star})\equiv\bar{u}^{\alpha^{\prime}}_{r^{\prime}}(\bm{k})\bar{u}^{\beta^{\prime}}_{s^{\prime}}(\bm{P}-\bm{k})\,\mathcal{R}^{*\alpha^{\prime}\beta^{\prime}}(P,k)\big{|}_{k^{0}=\omega_{N}(\bm{k})}\
,\end{split}$ (133)

which we choose to write as functions of the CM spatial loop momentum
$\bm{k}^{\star}$.

We next decompose these functions into spherical harmonics

$\displaystyle\begin{split}&{\mathcal{L}}_{rs}(P,\bm{k}^{\star})\equiv\sqrt{4\pi}\,Y_{\ell
m}(\hat{\bm{k}}^{\star})\,|\bm{k}^{\star}|^{\ell}\,\widetilde{\mathcal{L}}_{rs,\ell
m}(P,|\bm{k}^{\star}|)\ ,\\\
&{\mathcal{R}}^{*}_{r^{\prime}s^{\prime}}(P,\bm{k}^{\star})\equiv\sqrt{4\pi}\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\star})\,|\bm{k}^{\star}|^{\ell^{\prime}}\,\widetilde{\mathcal{R}}^{*}_{r^{\prime}s^{\prime},\ell^{\prime}m^{\prime}}(P,|\bm{k}^{\star}|)\,,\end{split}$
(134)

and rearrange the pole term in eq. (132) to reach

$\mathcal{L}\,\circ_{\sf fv}\,\mathcal{R}^{\dagger}=\mathcal{L}\circ_{\sf
rm}\mathcal{R}^{\dagger}+\widetilde{\mathcal{L}}(P)\,iS(P,L)\,\widetilde{\mathcal{R}}(P)^{\dagger}\,,$
(135)

where, as in the spin zero case, we have absorbed extra terms in the in a
redefinition of $\mathcal{I}^{[2]}(P)\to\mathcal{I}^{[3]}(P)$, and also
introduced $\mathcal{I}^{[3]}(P)\equiv\mathcal{L}\circ_{\sf
rm}\mathcal{R}^{\dagger}$. This result exactly matches eq. (85) above.

The objects $\widetilde{\mathcal{L}}(P)$ and
$\widetilde{\mathcal{R}}(P)^{\dagger}$ are vectors in the combined space of
spin, spatial loop momentum, and angular momentum. The vector elements can be
written explicitly as $\widetilde{\mathcal{L}}_{rs,\bm{k}^{\star}\ell m}(P)$
and
$\widetilde{\mathcal{R}}^{*}_{r^{\prime}s^{\prime},\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P)$,
respectively. The matrix $S(P,L)$, in turn, has elements

$S_{rs,\bm{k}^{\star}\ell
m;\,r^{\prime}s^{\prime},\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)=\frac{1}{2L^{3}}\,\frac{4\pi\,Y_{\ell
m}(\hat{\bm{k}}^{\star})\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\star})\,\delta_{rr^{\prime}}\,\delta_{ss^{\prime}}\,\delta_{\bm{k}^{\star}\bm{k}^{\prime\star}}\,|\bm{k}^{\star}|^{\ell+\ell^{\prime}}\,H(\bm{k}^{\star})}{4\omega_{N}(\bm{k})\,\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}\big{]}}\,.$ (136)

This matches eq. (80) up to the additional Kronecker deltas in the spin
components.

To give the effect of spin on $\mathcal{T}$, the off-shell Bethe-Salpeter
kernel is split into the contribution from the exchanges and a remainder, as
before. The off-shell spin projection described above appears in this
separation and leads to the following quantity:

$\begin{split}\mathcal{T}_{rs,r^{\prime}s^{\prime}}(P,\bm{p}^{\star},\bm{p}^{\prime\star})&\equiv-\bar{u}^{\alpha}_{r}(\bm{p})\bar{u}^{\beta}_{s}(\bm{P}-\bm{p})\,\gamma_{5}^{\alpha\alpha^{\prime}}\,\gamma_{5}^{\beta\beta^{\prime}}\,u^{\alpha^{\prime}}_{r^{\prime}}(\bm{p}^{\prime})u^{\beta^{\prime}}_{s^{\prime}}(\bm{P}-\bm{p}^{\prime})\\\
&\hskip 130.0pt\times\
\mathcal{T}(P,p,p^{\prime})\big{|}_{p^{0}=\omega_{N}(\bm{p}),\,p^{\prime
0}=\omega_{N}(\bm{p}^{\prime})}\ .\end{split}$ (137)

Here we have also included the $\gamma_{5}$ factors that appear in the $NN\pi$
vertex, since the exchanged pion is a pseudoscalar. Projecting this to
definite angular momentum, we reach a modified matrix of off-shell log
functions, denoted by $\mathcal{T}$ as in the spin-zero case. The elements
$\mathcal{T}_{rs,\bm{k}^{\star}\ell
m;\,r^{\prime}s^{\prime},\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P)$ of
the matrix correspond to the projected functions.

In addition to $S$ and $\mathcal{T}$, the matrix $\overline{\mathcal{K}}^{\sf
os}$ is also modified and the elements with spin included are denoted by
$\overline{\mathcal{K}}^{\sf os}_{rs,\ell
m;\,r^{\prime}s^{\prime},\ell^{\prime}m^{\prime}}$. The symmetry properties of
this quantity match those of a physical two-to-two amplitude with an incoming
spin state of orbital angular momentum $\ell^{\prime},m^{\prime}$ and spin
components $r^{\prime}s^{\prime}$ scattering to a state with $\ell,m$ and
$r,s$. As with the usual two-to-two scattering formalism with spin, once a
truncation in $\ell,\ell^{\prime}$ is set one can identify the non-zero
components and their relations. Here it may be useful to work in the basis of
total spin and total angular momentum $J$.

This completes our construction of eq. (122) for the case of spin-half
particles. A final missing point here is the projection to irreducible
representations (irreps) of the finite-volume symmetry group.

Having extended our results to spin-half particles, we now comment on the
generalization to identical particles of arbitrary spin. If we take $N$ to
have spin $S$, the dressed propagator can be expanded about the on-shell point
$k^{2}=M_{N}^{2}$ as

$\Delta_{N,L}(k)=\frac{i\mathcal{P}_{N}(\bm{k})}{k^{2}-M_{N}^{2}+i\epsilon}+R_{N,L}(k)\
,$ (138)

where $\mathcal{P}_{N}(\bm{k})$ is a volume-independent matrix carrying two
sets of field indices $A,A^{\prime}$. In the spin-half case,
$\mathcal{P}_{N}(\bm{k})$ corresponds to
$(\not{k}+M_{N})|_{k^{0}=\omega_{N}(\bm{k})}$. As in the Dirac spinor case,
this can be decomposed into projectors that relate the $A,A^{\prime}$ indices
to the spin state indices $r,r^{\prime}$,

$\mathcal{P}_{N}(\bm{k})^{AA^{\prime}}=U_{r}^{A}(\bm{k})\,\delta_{rr^{\prime}}\,\overline{U}_{r^{\prime}}^{A^{\prime}}(\bm{k})\,.$
(139)

where $r$ and $r^{\prime}$ are summed over the allowed values
$r,r^{\prime}=-S,-S+1,...,S-1,S$, and the bar on $\overline{U}$ denotes the
appropriate conjugation. Using eq. (139), we are left with the same form for
the loop contribution as the spin-half case in eq. (132) above.

The details of the $\mathcal{T}$ factor will depend on how the light exchanged
particle couples to the spin-$S$ nucleon. For example if envision the case of
a scalar (rather than a pseudoscalar) exchange particle coupled to two real
vectors, we find

$\begin{split}\mathcal{T}_{rs,r^{\prime}s^{\prime}}(P,\bm{p}^{\star},\bm{p}^{\prime\star})&\equiv\bar{\epsilon}^{\alpha}_{r}(\bm{p})\bar{\epsilon}^{\beta}_{s}(\bm{P}-\bm{p})\,\delta^{\alpha\alpha^{\prime}}\,\delta^{\beta\beta^{\prime}}\,\epsilon^{\alpha^{\prime}}_{r^{\prime}}(\bm{p}^{\prime})\epsilon^{\beta^{\prime}}_{s^{\prime}}(\bm{P}-\bm{p}^{\prime})\\\
&\hskip 140.0pt\times\
\mathcal{T}(P,p,p^{\prime})\big{|}_{p^{0}=\omega_{N}(\bm{p}),\,p^{\prime
0}=\omega_{N}(\bm{p}^{\prime})}\ .\end{split}$ (140)

##  4 Relating $\overline{\mathcal{K}}^{\sf os}$ to the scattering amplitude

In section 3.8, we presented a generalized quantization condition, eq. (122),
which allows one to constrain the volume-independent object
$\overline{\mathcal{K}}^{\sf os}(P)$ using the finite-volume spectrum. As
emphasized in previous sections, this quantity differs from the standard
K-matrix, for example due to its dependence on the cutoff function
$H(\bm{k}^{\star})$, appearing in the definition of $S(P,L)$. In this section,
we show how the scheme-independent $\overline{\mathcal{K}}^{\sf os}(P)$ is
related to the infinite-volume $NN\to NN$ scattering amplitude, denoted
$\mathcal{M}(P,p,p^{\prime})$ and thus also to more standard K-matrix
definitions. This method has been used extensively, for example in refs.
Hansen:2015zga ; Briceno:2017tce ; Briceno:2018mlh ; Briceno:2018aml .

###  4.1 Finite-volume auxiliary amplitude

We begin by defining a finite-volume quantity
$\mathcal{M}_{L}(P,p,p^{\prime})$, that we loosely refer to as a “finite-
volume amplitude”. Diagrammatically it is defined by the sum of all $NN\to NN$
four-leg amputated diagrams, but with all diagrams evaluated in the finite
volume, and with external arguments not necessarily set on shell. This
quantity is useful because we can formally recover the physical amplitude from
it by introducing an $i\epsilon$ prescription on the propagators and taking an
ordered double limit: first $L\to\infty$ at fixed $\epsilon$, causing the
spatial loop momentum sums to converge to the usual $i\epsilon$-prescription
Feynman integrals, followed by $\epsilon\to 0$ and the on-shell limit, to
reach the standard scattering amplitude.

Using the notation introduced in the previous section, we can write
$\mathcal{M}_{L}(P,p,p^{\prime})$ as

$i\mathcal{M}_{L}=\sum_{n=0}^{\infty}\left[iB\,\circ_{\sf fv}\right]^{n}iB\,,$
(141)

where, for simplicity, we omit the dependence on total four-momentum $P$ and
external momentum arguments $p$ and $p^{\prime}$.

As in the previous section, one can express this in terms of the $t$-exchange
modified kernel $B^{\mathfrak{T}}$ which can be separated into the subtracted
part $\overline{B}^{\mathfrak{T}}$ and the exchange term $2g^{2}\mathcal{T}$.
The only issue here is that, unlike with $C_{L}(P)$, the lack of exchange
symmetry within $B^{\mathfrak{T}}$ will have an effect on the amplitude
constructed with it. To handle this, we define a finite-volume auxiliary
amplitude $\mathcal{M}^{\sf aux}_{L}(P,p,p^{\prime})$, as follows:

$i\mathcal{M}^{\sf
aux}_{L}=\sum_{n=0}^{\infty}\left[iB^{\mathfrak{T}}\,\circ_{\sf
fv}\right]^{n}iB^{\mathfrak{T}}=\sum_{n=0}^{\infty}\left[\left(i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\right)\,\circ_{\sf
fv}\right]^{n}\left(i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\right)\,.$
(142)

We additionally define its infinite-volume counterpart via

$i\mathcal{M}^{\sf
aux}=\sum_{n=0}^{\infty}\left[iB^{\mathfrak{T}}\,\circ_{i\epsilon}\right]^{n}iB^{\mathfrak{T}}=\sum_{n=0}^{\infty}\left[\left(i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\right)\,\circ_{i\epsilon}\right]^{n}\left(i\overline{B}^{\mathfrak{T}}+2ig^{2}\mathcal{T}\right)\,,$
(143)

and observe this satisfies the ordered double limit described in the first
paragraph of this subsection

$i\mathcal{M}^{\sf aux}=\lim_{\epsilon\to 0}\lim_{L\to\infty}i\mathcal{M}^{\sf
aux}_{L}\Big{|}_{E\to E+i\epsilon}\,.$ (144)

Finally, to recover the standard amplitude from the auxiliary, one simply
symmetrizes under the exchange $p^{\prime}\leftrightarrow(P-p^{\prime})$:

$\mathcal{M}(P,p,p^{\prime})=\frac{1}{2}\left[\mathcal{M}^{\sf
aux}(P,p,p^{\prime})+\mathcal{M}^{\sf aux}(P,p,P-p^{\prime})\right]\
,\qquad\qquad\text{(bosons)}\,,$ (145)

with the corresponding result for fermions as required by the total spin
state.

In the remainder of this section, we make extensive use of the angular
momentum-projected versions of these amplitudes. The relations between
$\mathcal{M}$ and $\mathcal{M}^{\sf aux}$ and their spherical harmonic
projections are given in the usual way:

$\displaystyle\mathcal{M}(P,p,p^{\prime})$
$\displaystyle=4\pi\,|\bm{p}^{\star}|^{\ell}\,|\bm{p}^{\prime\star}|^{\ell^{\prime}}\,Y^{*}_{\ell
m}(\hat{\bm{p}}^{\star})\,Y_{\ell^{\prime}m^{\prime}}(\hat{\bm{p}}^{\prime\star})\,\widetilde{\mathcal{M}}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\,,$
(146) $\displaystyle\mathcal{M}^{\sf aux}(P,p,p^{\prime})$
$\displaystyle=4\pi\,|\bm{p}^{\star}|^{\ell}\,|\bm{p}^{\prime\star}|^{\ell^{\prime}}\,Y^{*}_{\ell
m}(\hat{\bm{p}}^{\star})\,Y_{\ell^{\prime}m^{\prime}}(\hat{\bm{p}}^{\prime\star})\,\widetilde{\mathcal{M}}^{\sf
aux}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\,.$
(147)

These projections have a straightforward relation to the partial wave
expansion coefficients denoted by a single $(\ell)$ superscript:

$\displaystyle\widetilde{\mathcal{M}}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)$
$\displaystyle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\frac{1}{|\bm{p}^{\star}|^{\ell}|\bm{p}^{\prime\star}|^{\ell}}\,{\mathcal{M}}^{(\ell)}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\,,$
(148) $\displaystyle\widetilde{\mathcal{M}}^{\sf aux}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)$
$\displaystyle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\frac{1}{|\bm{p}^{\star}|^{\ell}|\bm{p}^{\prime\star}|^{\ell}}\,{\mathcal{M}}^{{\sf
aux}(\ell)}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\,.$ (149)

The symmetry properties of the amplitude under momentum exchanges, mentioned
above, allows us to derive the following relations for identical bosons:

$\displaystyle{\mathcal{M}}^{(\ell)}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)$
$\displaystyle={\mathcal{M}}^{{\sf
aux}(\ell)}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\,,\qquad$
$\displaystyle\text{for even $\ell$}\,,$ (150)
$\displaystyle{\mathcal{M}}^{(\ell)}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)$
$\displaystyle=0\,,$ $\displaystyle\text{for odd $\ell$}\,.$ (151)

with the corresponding results for fermions as required by the total spin
state.

###  4.2 Integral equations

We now derive an integral equation relating $\overline{\mathcal{K}}^{\sf
os}(P)$ to the auxiliary amplitude $\mathcal{M}^{\sf aux}(P,p,p^{\prime})$,
making use of $\mathcal{M}^{\sf aux}_{L}$ to do so. The series given in eq.
(142) can be manipulated in much the same way as the correlator series of eq.
(66). All steps detailed in the previous section can be applied in a
straightforward manner and one obtains

$i\mathcal{M}^{\sf
aux}_{L}(P)\equiv\sum_{n=0}^{\infty}\Big{[}\left(\xi^{\dagger}\,i\overline{\mathcal{K}}^{\sf
os}(P)\,\xi+2ig^{2}\mathcal{T}(P)\right)iS(P,L)\,\Big{]}^{n}\left(\xi^{\dagger}\,i\overline{\mathcal{K}}^{\sf
os}(P)\,\xi+2ig^{2}\mathcal{T}(P)\right)\,.$ (152)

As with other, similarly notated quantities, $\mathcal{M}^{\sf aux}_{L}(P)$ is
the matrix with elements $\mathcal{M}^{\sf aux}_{L,\bm{p}^{\star}\ell
m;\bm{p}^{\prime\star}\ell^{\prime}m^{\prime}}(P)$. Here the $\bm{p}^{\star}$
and $\bm{p}^{\prime\star}$ labels are misleading here, since the quantity
actually only depends on the magnitudes of these vectors. The directional
dependence has gone into the angular-momentum projection leading to the
remaining indices.888A subtlety arises from the fact that the momenta in
$\mathcal{M}^{\sf aux}_{L}(P)$ are in the finite-volume set. Thus, one does
not have the continuous rotational symmetry for the angular-momentum
projection. This is resolved by noting that the dependence in fact only
appears in infinite-volume quantities after the reduction to eq. (152), so the
extension of $\mathcal{M}^{\sf aux}_{L}(P)$ to the continuous infinite-volume
set of momenta is straightforward. See also the discussion above eq. (35) of
ref. Hansen:2015zga .

We can remove the infinite sum by substituting eq. (152) into itself, yielding

$i\mathcal{M}^{\sf
aux}_{L}(P)=\left(\xi^{\dagger}\,i\overline{\mathcal{K}}^{\sf
os}(P)\,\xi+2ig^{2}\mathcal{T}(P)\right)+i\mathcal{M}^{\sf
aux}_{L}(P)\,iS(P,L)\left(\xi^{\dagger}\,i\overline{\mathcal{K}}^{\sf
os}(P)\,\xi+2ig^{2}\mathcal{T}(P)\right)\,.$ (153)

Starting from this equation, we replace the energy as $E\to E+i\epsilon$ in
the matrix $S(P,L)$, which introduces the $i\epsilon$ prescription to the
relevant two-particle intermediate state pole, and take the infinite-volume
limit $L\to\infty$. These steps lead to the following integral equation:

$\widetilde{\mathcal{M}}^{\sf aux}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)=\mathcal{K}^{\mathcal{T}}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\\\
-\frac{1}{2}\int_{0}^{\infty}\\!\frac{d|\bm{k}^{\star}|}{(2\pi)^{3}}\widetilde{\mathcal{M}}^{\sf
aux}_{\ell
m,\ell^{\prime\prime}m^{\prime\prime}}(P,|\bm{p}^{\star}|,|\bm{k}^{\star}|)\frac{4\pi|\bm{k}^{\star}|^{2\ell^{\prime\prime}+2}H(\bm{k}^{\star})}{4\omega_{N}(\bm{k}^{\star})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\mathcal{K}^{\mathcal{T}}_{\ell^{\prime\prime}m^{\prime\prime},\ell^{\prime}m^{\prime}}(P,|\bm{k}^{\star}|,|\bm{p}^{\prime\star}|)\,,$
(154)

where we have introduced

$\mathcal{K}^{\mathcal{T}}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)=\overline{\mathcal{K}}^{\sf
os}_{\ell m,\ell^{\prime}m^{\prime}}(P)+2g^{2}\mathcal{T}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)\,.$
(155)

Note that the $L\to\infty$ limit is performed with fixed external momentum,
requiring an extension of these to be outside the finite-volume set of a given
$L$. See footnote 8 and the indicated reference for further discussion.

Observe also that no $L$-depence appears in eq. (154) and that we have
replaced the indices indicating discrete CM spatial momenta with continuous
arguments. To reach this expression, we have also performed the angular
integral resulting from the infinite-volume limit applied to the sums over
$S(P,L)$.

As indicated at various points above, both $\overline{\mathcal{K}}^{\sf os}$
and $\mathcal{M}^{\sf aux}$ can be represented either as matrices with
angular-momentum indices or as single functions of the scattering momenta. The
two forms are connected in the usual way, already seen for $\mathcal{M}^{\sf
aux}$ in eq. (147). For $\overline{\mathcal{K}}^{\sf os}$, this is simply

$\overline{\mathcal{K}}^{\sf
os}(P,p,p^{\prime})=4\pi\,|\bm{p}^{\star}|^{\ell}\,|\bm{p}^{\prime\star}|^{\ell^{\prime}}\,Y^{*}_{\ell
m}(\hat{\bm{p}}^{\star})\,Y_{\ell^{\prime}m^{\prime}}(\hat{\bm{p}}^{\prime\star})\,\overline{\mathcal{K}}^{\sf
os}_{\ell m,\ell^{\prime}m^{\prime}}(P)\,,$ (156)

Combining these relations with eq. (154), we obtain a second type of integral
equation, directly for the unprojected auxiliary amplitude:

$\mathcal{M}^{\sf
aux}(P,p,p^{\prime})=\mathcal{K}^{\mathcal{T}}(P,p,p^{\prime})-\frac{1}{2}\int\\!\frac{d^{3}\bm{k}^{\star}}{(2\pi)^{3}}\frac{\mathcal{M}^{\sf
aux}(P,p,k)\,H(\bm{k}^{\star})\,{\mathcal{K}}^{\mathcal{T}}(P,k,p^{\prime})}{4\omega_{N}(\bm{k}^{\star})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\,,$ (157)

where

$\mathcal{K}^{\mathcal{T}}(P,p,p^{\prime})=\overline{\mathcal{K}}^{\sf
os}(P,p,p^{\prime})+2g^{2}\mathcal{T}(P,p,p^{\prime})\,,$ (158)

and where the four-momentum arguments $p,p^{\prime},k$ are all taken to be on
shell.

Though the relations given here completely solve the task of relating
$\overline{\mathcal{K}}^{\sf os}$ to $\mathcal{M}^{\sf aux}$ in principle, the
singularities within $\mathcal{M}^{\sf aux}$ and $\mathcal{T}$ could well make
numerical evaluation very difficult. For this reason, in the next section we
consider an alternative, based on a divergence-free intermediate quantity.

###  4.3 Divergence-free amplitude

To find the physical scattering amplitude, $\mathcal{M}(P,p,p^{\prime})$, from
the intermediate quantity, $\overline{\mathcal{K}}^{\sf os}(P,p,p^{\prime})$,
one needs to set all external four-momenta,
$(p,\,P-p,\,p^{\prime},\,P-p^{\prime})$, to their on-shell values. This, in
turn, requires setting the momenta within the auxiliary amplitude,
$\mathcal{M}^{\sf aux}(P,p,p^{\prime})$, on shell. However, this may lead to
an unstable numerical evaluation, associated with the fact that the angular-
momentum components of the auxiliary amplitude contain a branch point at
$s=4M_{N}^{2}-M_{\pi}^{2}$, and the unprojected amplitude contains a pole at
$t=M_{\pi}^{2}$. Since both of these singularities only arise for on-shell
momenta, extrapolating or interpolating to physical kinematics could be
challenging.

Moreover, the partial-wave expansion of $\mathcal{M}^{\sf
aux}(P,p,p^{\prime})$ is expected to be slowly-converging in the vicinity of
the $t$-channel pole and, for this reason, an order-by-order determination of
the partial-wave components through eq. (154) will not be useful near
$t=M_{\pi}^{2}$. This is partly addressed with the unprojected integral
equation, eq. (157). An alternative solution, again taking inspiration form
refs. Hansen:2014eka ; Briceno:2015tza , is to instead introduced a
divergence-free scattering amplitude, defined as follows:

$\mathcal{M}^{\sf df}(P,p,p^{\prime})\equiv\mathcal{M}^{\sf
aux}(P,p,p^{\prime})-2g^{2}\mathcal{T}(P,p,p^{\prime})\,.$ (159)

Projecting this quantity to spherical harmonics as in eq. (147) gives a faster
converging series, down to the neighbourhood of the second left-hand cut, i.e.
for $s\gtrsim(2M_{N})^{2}-(2M_{\pi})^{2}$. We can substitute the divergence-
free amplitude into eq. (157) to find

$\mathcal{M}^{\sf df}(P,p,p^{\prime})=\overline{\mathcal{K}}^{\sf
os}(P,p,p^{\prime})-\frac{1}{2}\int\\!\frac{d^{3}\bm{k}}{(2\pi)^{3}}\frac{H(\bm{k}^{\star})\big{[}\mathcal{M}^{\sf
df}(P,p,k)-2g^{2}\mathcal{T}(P,p,k)\big{]}{\mathcal{K}}^{\mathcal{T}}(P,k,p^{\prime})}{4\omega_{N}(\bm{k})\big{[}(k_{\sf