os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\,.$ (160)

The two terms in the numerator on the right-hand side (in square brackets)
should be treated in practice as two separate integrals. The
$\mathcal{T}$-dependent integral can be calculated numerically given knowledge
of $\overline{\mathcal{K}}^{\sf os}(P,p,p^{\prime})$ and $g$. We can then for
$\mathcal{M}^{\sf df}(P,p,p^{\prime})$ using standard integral-equation
techniques, with the $\mathcal{T}$-dependent integral and
$\overline{\mathcal{K}}^{\sf os}$ forming the driving term.

For completeness, we also provide the angular-momentum-projected version of
eq. (160):

$\displaystyle\begin{split}\widetilde{\mathcal{M}}^{\sf df}_{\ell
m,\ell^{\prime}m^{\prime}}(P,|\bm{p}^{\star}|,|\bm{p}^{\prime\star}|)&=\overline{\mathcal{K}}^{\sf
os}_{\ell m,\ell^{\prime}m^{\prime}}(P)\\\\[5.0pt]
&+\frac{1}{2}\int_{0}^{\infty}\\!\frac{d|\bm{k}^{\star}|}{(2\pi)^{3}}\frac{4\pi|\bm{k}^{\star}|^{2\ell^{\prime\prime}+2}H(\bm{k}^{\star})\,g^{2}\mathcal{T}_{\ell
m,\ell^{\prime\prime}m^{\prime\prime}}(P,|\bm{p}^{\star}|,|\bm{k}^{\star}|)\,{\mathcal{K}}^{\mathcal{T}}_{\ell^{\prime\prime}m^{\prime\prime},\ell^{\prime}m^{\prime}}(P,|\bm{k}^{\star}|,|\bm{p}^{\prime\star}|)}{4\omega_{N}(\bm{k}^{\star})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\,\\\\[5.0pt]
&-\frac{1}{2}\int_{0}^{\infty}\\!\frac{d|\bm{k}^{\star}|}{(2\pi)^{3}}\frac{4\pi|\bm{k}^{\star}|^{2\ell^{\prime\prime}+2}H(\bm{k}^{\star})\widetilde{\mathcal{M}}^{\sf
df}_{\ell
m,\ell^{\prime\prime}m^{\prime\prime}}(P,|\bm{p}^{\star}|,|\bm{k}^{\star}|)\,{\mathcal{K}}^{\mathcal{T}}_{\ell^{\prime\prime}m^{\prime\prime},\ell^{\prime}m^{\prime}}(P,|\bm{k}^{\star}|,|\bm{p}^{\prime\star}|)}{4\omega_{N}(\bm{k}^{\star})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\,.\end{split}$ (161)

In this version we have separated out the driving term (first two lines) more
explicitly.

Having obtained $\mathcal{M}^{\sf df}(P,p,p^{\prime})$, one can recover the
amplitude $\mathcal{M}(P,p,p^{\prime})$ by combining eqs. (145) and (159)

$\mathcal{M}(P,p,p^{\prime})=\frac{1}{2}\left[\mathcal{M}^{\sf
df}(P,p,p^{\prime})+\mathcal{M}^{\sf
df}(P,p,P-p^{\prime})\right]+g^{2}\mathcal{E}(P,p,p^{\prime})\,.$ (162)

We can see that, when putting external arguments on shell, the
$\mathcal{E}(P,p,p^{\prime})$ term (as defined in eq. (106)) is responsible
for generating the $t$\- and $u$-channel $\pi$ exchange poles present in the
amplitude, and the divergence-free auxiliary amplitudes does not carry any
singular behaviour until one reaches the two-pion-exchange cut.

###  4.4 Analytic continuation

Evaluating the on-shell scattering amplitude below threshold requires
analytically continuing the external momenta to be complex, such that
$|\bm{p}^{\star}|=|\bm{p}^{\prime\star}|=i\sqrt{M_{N}^{2}-s/4\,}$. One method
to achieve this is to first solve the integral equation (160) for real
external momenta, hence obtaining the amplitude, and then compute the sub-
threshold on-shell values using the relation

$\displaystyle\begin{split}\mathcal{M}(P,p,p^{\prime})&=\overline{\mathcal{K}}^{\sf
os}(P,p,p^{\prime})+ig^{2}\mathcal{E}(P,p,p^{\prime})\\\
&+\frac{1}{2}\int\\!\frac{d^{3}\bm{k}}{(2\pi)^{3}}\frac{H(\bm{k}^{\star})\,{\mathcal{K}}^{\mathcal{T}}(P,p,k)\big{[}\overline{\mathcal{K}}^{\sf
os}(P,k,p^{\prime})+ig^{2}\mathcal{E}(P,k,p^{\prime})\big{]}}{4\omega_{N}(\bm{k})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\\\
&-\frac{1}{4}\int\\!\frac{d^{3}\bm{k}}{(2\pi)^{3}}\int\\!\frac{d^{3}\bm{k}^{\prime}}{(2\pi)^{3}}\frac{H(\bm{k}^{\star})\,{\mathcal{K}}^{\mathcal{T}}(P,p,k)\,\mathcal{M}(P,k,k^{\prime})}{4\omega_{N}(\bm{k})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\frac{H(\bm{k}^{\prime\star}){\mathcal{K}}^{\mathcal{T}}(P,k^{\prime},p^{\prime})}{4\omega_{N}(\bm{k}^{\prime})\big{[}({k^{\prime}}_{\sf
os}^{\star})^{2}-({\bm{k}^{\prime}}^{\star})^{2}+i\epsilon\big{]}}\,.\end{split}$
(163)

The momenta $p$, $p^{\prime}$, $P-p$ and $P-p^{\prime}$ can be put on shell
and taken below threshold here. The key point is that the amplitude on the
right-hand side is nested between two loops and thus its arguments are always
real. Knowledge of $\mathcal{M}$ for real momenta can thus be exploited to
obtain its subthreshold on-shell value. These methods can similarly be applied
to continue the amplitude into the complex plane.

##  5 Exploring the new formalism

In this section we study the new quantization condition and integral equations
that we have derived. In section 5.1, we present an alternative form of the
quantization condition that is superficially more similar to the standard
formalism. Then, in section 5.2, we demonstrate that the standard Lüscher
formalism is exactly recovered, also for $s<4M_{N}^{2}-M_{\pi}^{2}$, in the
limit where the $NN\pi$ coupling is set to zero. Finally, in section 5.3, we
give expressions in the case where all partial-wave components of
$\overline{\mathcal{K}}^{\sf os}$, besides the $S$-wave, are negligible.

###  5.1 Alternative form

An instructive rewriting of the quantization condition is reached by first
noting that eq. (122) is satisfied whenever the following matrix has a
divergent eigenvalue:

$\Xi_{L}(P,L)=\frac{1}{1+\xi^{\dagger}\,\overline{\mathcal{K}}^{\sf
os}(P)\,\xi\big{[}S(P,L)^{-1}+2g^{2}\mathcal{T}(P)\big{]}^{-1}}\xi^{\dagger}\,\overline{\mathcal{K}}^{\sf
os}(P)\,\xi\,.$ (164)

Expanding order by order and rearranging the factors of $\xi$, one finds that
$\Xi_{L}(P,L)$ can be exactly rewritten as

$\Xi_{L}(P,L)=\xi^{\dagger}\frac{1}{\overline{\mathcal{K}}^{\sf
os}(P)^{-1}+\xi\big{[}S(P,L)^{-1}+2g^{2}\mathcal{T}(P)\big{]}^{-1}\xi^{\dagger}}\xi\,.$
(165)

We thus reach the following alternative to eq. (122):

$\det_{\ell m}\left[\overline{\mathcal{K}}^{\sf
os}(P_{j})^{-1}+F^{\mathcal{T}}(P_{j},L)\right]=0\,,$ (166)

where we have introduced

$\displaystyle F^{\mathcal{T}}(P,L)=\xi
S(P,L)\frac{1}{1+2g^{2}\mathcal{T}(P)S(P,L)}\xi^{\dagger}\,.$ (167)

Crucially, this quantization condition is now defined as a determinant only on
the orbital angular momentum, exactly as in refs. Luscher:1986pf ; Kim:2005gf
. We have included the $\ell m$ subscript on the determinant to emphasize this
point. One nice feature of this representation is that it allows one to use
the standard technology for projection to irreps. See, for example, refs.
Luscher:1986pf ; Dudek:2010ew .

This rewriting also holds for the case of particles with intrinsic spin. The
only adjustment is that the spin indices remain in the final determinant
condition, directly inherited from the quantities introduced in section 3.9.
We summarize this as

$\det_{\ell mrs}\left[\overline{\mathcal{K}}^{\sf
os}(P_{j})^{-1}+F^{\mathcal{T}}(P_{j},L)\right]=\det_{Jm_{J}\ell
S}\left[\overline{\mathcal{K}}^{\sf
os}(P_{j})^{-1}+F^{\mathcal{T}}(P_{j},L)\right]=0\,.$ (168)

Here we have indicated two of the possible bases for particles with spin. In
the first expression, the labels $r$ and $s$ refer to the components of
individual particle spin, as in eq. (136). For example, for two spin-half
particles, the allowed values are given by $r,s=\pm 1/2$. The middle
expression is labelled in terms of total angular momentum $J$ as well as
orbital angular momentum $\ell$ and total spin $S$. To reach this form one
uses Clebsch-Gordon coefficients to first combine the individual spin states
to definite total spin, labelled as $S,m_{S}$. Then, in a second step,
$\ell,m$ and $S,m_{S}$ are combined to $J,m_{J},\ell,S$. This just amounts to
changes of basis on the matrices, as described, for example, in refs.
Briceno:2014oea ; Briceno:2015csa . As in the spin-zero cases, the projections
to definite irreps is inherited from the standard formalism.

Also common to the standard approach is the mixing of angular momenta due to
the reduced symmetry of the finite-volume system. This means that, even after
projection to a given irrep, the matrices entering the determinant are
formally infinite dimensional. As a result, numerical evaluation is only
possible with truncation: one sets $\overline{\mathcal{K}}^{\sf os}_{\ell
m\ell^{\prime}m^{\prime}}=0$ for $\ell,\ell^{\prime}>\ell^{\sf max}$. Having
truncated $\overline{\mathcal{K}}^{\sf os}$ in this way, one can do the same
for $F^{\mathcal{T}}(P,L)$ without any additional approximations. It is,
however, an additional approximation to truncate $S(P,L)$ and $\mathcal{T}(P)$
within $F^{\mathcal{T}}(P,L)$.

Further work is needed to investigate the numerical convergence in the
evaluation of components of $F^{\mathcal{T}}(P,L)$ as a function of the
truncations of its building blocks. At the same time, we view the separation
of the quantization condition into $\overline{\mathcal{K}}^{\sf os}(P)$,
$S(P,L)$, and $\mathcal{T}(P)$ as optimal, given the expectations of
underlying physical system. $\overline{\mathcal{K}}^{\sf os}(P)$ is a generic
analytic function and requires parametrization to extract scattering
predictions from finite-volume energies. In that sense, this is the most
challenging input and the one for which partial-wave convergence is most
important. For this reason it is important that the single-exchange left-hand
cut is removed from $\overline{\mathcal{K}}^{\sf os}(P)$, such that this
object does not suffer the partial wave convergence problems of the amplitude
itself.

By contrast, while the convergence for $S(P,L)$ and $\mathcal{T}(P)$ is less
obvious, no new free parameters are introduced if one varies the size of these
quantities. One can thus envision cases where only the $S$-wave component of
$\overline{\mathcal{K}}^{\sf os}(P)$ is kept but many more components of the
other factors. This will still lead to an $(n+1)$-parameter description of the
finite-volume energies, with $n$ being the number of free parameters used in
the $S$ component of $\overline{\mathcal{K}}^{\sf os}(P)$.

This concludes our discussion of the alternative form for the quantization
condition. In the following subsections we make use of eq. (166) in recovering
various instructive limiting cases, starting with the case that the $NN\pi$
coupling vanishes.

###  5.2 Recovering the standard formalism

We now show that, in the case of vanishing trilinear coupling: $g=0$, our
modified quantization condition, eq. (122), is equivalent to the well-known
Lüscher scattering formalism Luscher:1986pf  and its extensions to nonzero
spatial momentum in the finite-volume frame Rummukainen:1995vs ; Kim:2005gf .
To this end it is convenient to use the alternative form given in eq. (165).
In the $g\to 0$ limit, the result becomes

$\det_{\ell m}\left[\overline{\mathcal{K}}^{\sf os}(P_{j})^{-1}+\xi
S(P_{j},L)\xi^{\dagger}\right]=0\,,$ (169)

where we have substituted $F^{\mathcal{T}}(P,L)\underset{g\to
0}{\longrightarrow}\xi S(P,L)\xi^{\dagger}$, as follows directly from eq.
(167).

We next relate $\xi S(P,L)\xi^{\dagger}$ to a version of the Lüscher finite-
volume function, denoted here by $\widetilde{F}(P,L)$.999As above, the tilde
here indicates rescaling by powers of $k^{\star}_{\sf os}$. We do not use the
tilde for $S(P,L)$ since this is a non-standard quantity anyway, as has been
defined immediately with the rescaling. The key distinction between $\xi
S(P,L)\xi^{\dagger}$ and $\widetilde{F}(P,L)$ is simply that the latter is
defined with a sum-integral difference while $\xi S(P,L)\xi^{\dagger}$ is
defined only with a sum in isolation [see also eq. (90)]. Studying the
definitions carefully, one finds that the relation takes the form

$\displaystyle\xi S(P,L)\xi^{\dagger}$
$\displaystyle=\widetilde{F}(P,L)+\widetilde{I}(P)\,,$ (170)

where each of the quantities carries two set of angular-momentum indices, and
we have introduced

$\widetilde{I}_{\ell m,\ell^{\prime}m^{\prime}}(P)={\sf
p.v.}\frac{1}{2}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\,\frac{4\pi\,Y_{\ell
m}(\hat{\bm{k}}^{\star})\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\star})\,|\bm{k}^{\star}|^{\ell+\ell^{\prime}}\,H(\bm{k}^{\star})}{4\omega_{N}(\bm{k})\,\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}\big{]}}\,.$ (171)

This implicit definition of $\widetilde{F}(P,L)$ exactly matches the quantity
denoted $\widetilde{F}(E,\bm{P},L)$ in ref. Peterken:2023zwu . See e.g. eq.
(A1) of that work. Equivalently, it is related to the finite-volume function
defined by Kim, Sachrajda, and Sharpe in ref. Kim:2005gf  according to
$\widetilde{F}_{\ell m,\ell^{\prime}m^{\prime}}=(k^{\star}_{\sf
os})^{\ell+\ell^{\prime}}\text{Re}(iF^{\sf KSS}_{\ell
m,\ell^{\prime}m^{\prime}}/2)$, where we have added the superscript ${\sf
KSS}$ to distinguish the earlier definition.

The final missing ingredient in recovering the standard scattering formalism
from eq. (169) is the relation between $\overline{\mathcal{K}}^{\sf os}(P)$
and the scattering amplitude. This can mostly easily be recovered from the
integral equation, eq. (154), which simplifies in three ways for $g=0$: First,
the breaking of exchange symmetry in $\mathcal{M}^{\sf aux}$ is removed so
that this just directly becomes the scattering amplitude. Second, the off-
shell dependence of $\mathcal{M}^{\sf aux}$ on $|\bm{k}^{\star}|$, which is
inherited from $\mathcal{T}$, is removed when the coupling vanishes. Finally,
we can replace $\mathcal{K}^{\mathcal{T}}\to\overline{\mathcal{K}}^{\sf os}$
everywhere, since the difference between the two is proportional to $g^{2}$.
We reach

$\widetilde{\mathcal{M}}_{\ell
m,\ell^{\prime}m^{\prime}}(P)=\overline{\mathcal{K}}^{\sf os}_{\ell
m,\ell^{\prime}m^{\prime}}(P)\\\ -\widetilde{\mathcal{M}}_{\ell
m,\ell^{\prime\prime}m^{\prime\prime}}(P)\frac{1}{2}\int_{0}^{\infty}\\!\frac{d|\bm{k}^{\star}|}{(2\pi)^{3}}\frac{4\pi|\bm{k}^{\star}|^{2\ell^{\prime\prime}+2}H(\bm{k}^{\star})}{4\omega_{N}(\bm{k}^{\star})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\overline{\mathcal{K}}^{\sf
os}_{\ell^{\prime\prime}m^{\prime\prime},\ell^{\prime}m^{\prime}}(P)\,.$ (172)

Switching to a matrix notation to drop indices and multiplying both sides by
$\mathcal{M}(P)^{-1}$ on the right and $\overline{\mathcal{K}}^{\sf
os}(P)^{-1}$ on the left, we find

$\overline{\mathcal{K}}^{\sf
os}(P)^{-1}=\widetilde{\mathcal{M}}(P)^{-1}-\widetilde{I}^{i\epsilon}(P)\,,$
(173)

where we have introduced

$\widetilde{I}^{i\epsilon}_{\ell
m,\ell^{\prime}m^{\prime}}(P)=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\frac{1}{2}\int_{0}^{\infty}\\!\frac{d|\bm{k}^{\star}|}{(2\pi)^{3}}\frac{4\pi|\bm{k}^{\star}|^{2\ell+2}H(\bm{k}^{\star})}{4\omega_{N}(\bm{k}^{\star})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\,.$ (174)

The notation for this quantity is well-chosen. As can be seen by evaluating
the angular integrals in eq. (171), $\widetilde{I}(P)$ and
$\widetilde{I}^{i\epsilon}(P)$ differ only in the pole prescription.

In particular, since the principal value defining $\widetilde{I}(P)$ is
equivalent to the real part of the $i\epsilon$ prescription, the difference in
integrals is just the imaginary part of $\widetilde{I}^{i\epsilon}(P)$

$\displaystyle\widetilde{I}^{i\epsilon}_{\ell
m,\ell^{\prime}m^{\prime}}(P)-\widetilde{I}_{\ell
m,\ell^{\prime}m^{\prime}}(P)$
$\displaystyle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}i\,\text{Im}\frac{1}{2}\int_{0}^{\infty}\\!\frac{d|\bm{k}^{\star}|}{(2\pi)^{3}}\frac{4\pi|\bm{k}^{\star}|^{2\ell+2}H(\bm{k}^{\star})}{4\omega_{N}(\bm{k}^{\star})\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}+i\epsilon\big{]}}\,,$ (175)
$\displaystyle=-\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}i\rho(s)(k_{\sf
os}^{\star})^{2\ell}=-i\widetilde{\rho}_{\ell
m,\ell^{\prime}m^{\prime}}(s)\,,$ (176)

where $\rho(s)$ is the usual phase space, defined in eq. (21) above, as is
$\widetilde{\rho}_{\ell m,\ell^{\prime}m^{\prime}}(s)$ in eq. (20).

It is exactly this phase space factor that appears in the relation between the
scattering amplitude $\widetilde{\mathcal{M}}$ and the standard two-particle
K-matrix

$\widetilde{\mathcal{K}}(P)^{-1}=\widetilde{\mathcal{M}}(P)^{-1}+i\widetilde{\rho}(P)\,.$
(177)

For convenience we also repeat the relation to the scattering phase shift
given in eq. (25)

$\widetilde{\mathcal{K}}_{\ell m,\ell^{\prime}m^{\prime}}(P)=\frac{1}{(k_{\sf
os}^{\star})^{2\ell}}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\frac{16\pi\sqrt{s}}{k_{\sf
os}^{\star}}\tan\delta^{(\ell)}(k_{\sf os}^{\star})\,.$ (178)

Finally, combining eqs. (173), (176) and (177), we deduce

$\displaystyle\overline{\mathcal{K}}^{\sf os}(P)^{-1}$
$\displaystyle=\widetilde{\mathcal{K}}(P)^{-1}-\widetilde{I}(P)\,.$ (179)

Together with eq. (170), this is the second key identity of this subsection.
Substituting the identities into eq. (169), one finally obtains

$\det_{\ell
m}\left[\widetilde{\mathcal{K}}(P_{j})^{-1}-\widetilde{I}(P_{j})+\widetilde{F}(P_{j},L)+\widetilde{I}(P_{j})\right]=\det_{\ell
m}\left[\widetilde{\mathcal{K}}(P_{j})^{-1}+\widetilde{F}(P_{j},L)\right]=0\,.$
(180)

This is the standard quantization condition of Lüscher, generalized to nonzero
momentum $\bm{P}$ in the finite-volume frame Luscher:1986pf ;
Rummukainen:1995vs ; Kim:2005gf . We have thus achieved the aim of this
section.

Before concluding, we briefly return to the case of $g\neq 0$. If we restrict
our attention to energies above the left-channel cut, i.e. with
$s>(2M_{N})^{2}-M_{\pi}^{2}$, then we can ignore the presence of the latter in
the on-shell Bethe-Salpeter kernel, up to exponentially suppressed
$L$-dependence, albeit dependence that can be enhanced if we are too close to
saturating the inequality. In this case, one can formally group the $g$
dependent term into $\overline{B}^{\mathfrak{T}}$ and apply an effective $g=0$
analysis to again deduce the usual formulas. It would be instructive to
analytically recover the standard formalism from our results in the above
threshold regime, and to quantify the neglected exponentially suppressed
corrections. We leave this for future work.

As mentioned above, our perspective is that, in a numerical calculation, one
should ensure that there are no statistically significant differences between
the scattering predictions arising from the standard and the improved
formalisms and, if there are, to use the latter.

###  5.3 $S$-wave dominance

In this section we consider the form of the quantization condition and the
integral equations in the case that only the $S$-wave component of
$\overline{\mathcal{K}}^{\sf os}$ is non-zero. Beginning with the quantization
condition, eq. (166) becomes

$\overline{\mathcal{K}}^{{\sf
os}(0)}(P_{j})^{-1}+F_{0}^{\mathcal{T}}(P_{j},L)=0\,,$ (181)

where we have introduced the $S$-wave quantities

$\displaystyle\overline{\mathcal{K}}^{{\sf os}(0)}(P_{j})$
$\displaystyle=\overline{\mathcal{K}}_{00,00}^{\sf os}(P_{j})\,,$ (182)
$\displaystyle F^{\mathcal{T}}_{0}(P,L)$
$\displaystyle=Z(P,L)-V^{\dagger}(P,L)\frac{1}{1+2g^{2}\mathcal{T}(P)S(P,L)}2g^{2}\mathcal{T}(P)V(P,L)\,,$
(183)

with the latter expressed in terms of two new building blocks, a scalar
function $Z(P,L)$ and a vector function $V(P,L)$:

$\displaystyle Z(P,L)$ $\displaystyle\equiv[\xi
S(P,L)\xi^{\dagger}]_{00,00}=\frac{1}{2L^{3}}\sum_{\bm{k}}\frac{e^{-\alpha[(\bm{k}^{\star})^{2}-(k_{\sf
os}^{\star})^{2}]}}{4\omega_{N}(\bm{k})\,\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}\big{]}}\,,$ (184) $\displaystyle
V^{\dagger}_{\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)$
$\displaystyle\equiv\xi_{\bm{k}^{\star}}S_{\bm{k}^{\star}00,\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P,L)=\frac{1}{2L^{3}}\,\frac{\sqrt{4\pi}\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}^{\prime}}^{\star})\,|\bm{k}^{\prime\star}|^{\ell^{\prime}}\,e^{-\alpha[(\bm{k}^{\prime\star})^{2}-(k_{\sf
os}^{\star})^{2}]}}{4\omega_{N}(\bm{k}^{\prime})\,\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\prime\star})^{2}\big{]}}\,.$ (185)

The key observation here is that $V(P,L)$ is still populated with all angular
momentum components, even though $F^{\mathcal{T}}(P,L)$ has been truncated to
the $S$-wave. This is because the exchanges, encoded in $\mathcal{T}(P)$,
depend on all partial waves, and the truncation of this object is logically
separate from the truncation of the quantization condition. In practice, the
approach is to evaluate $F^{\mathcal{T}}_{0}(P,L)$ for various truncations of
$\mathcal{T}(P)$ and look for saturation. The convergence will depend on the
values of $g,P,L$. Further investigation is needed to understand this in
detail.

To give one explicit example, we consider the case of zero total momentum and
the trivial irrep of the finite-volume symmetry group, called $A_{1}^{+}$.
Then the lowest-lying contaminating partial wave is $\ell=4$ and the $m$
component is removed by irrep projection such that one can define a two-
dimensional angular momentum space. The resulting building blocks for
$F^{\mathcal{T}}_{0}(P,L)$ are then given by

$\displaystyle V^{\dagger}_{\bm{k}}(P,L)$
$\displaystyle=\frac{1}{2L^{3}}\,\frac{\,e^{-\alpha[(\bm{k})^{2}-(k_{\sf
os})^{2}]}}{4\omega_{N}(\bm{k})\,\big{[}(k_{\sf
os})^{2}-(\bm{k})^{2}\big{]}}\Big{(}\begin{matrix}1\ \ &\ \
\mathcal{Y}_{4}^{A_{1}^{+}}(\bm{k})\end{matrix}\Big{)}\,,$ (186)
$\displaystyle\mathcal{T}_{\bm{k},\bm{k}^{\prime}}(P)$
$\displaystyle=\begin{pmatrix}\mathcal{T}^{(\ell=0),A_{1}^{+}}_{\bm{k},\bm{k}^{\prime}}(P)&0\\\
0&\mathcal{T}^{(\ell=4),A_{1}^{+}}_{\bm{k},\bm{k}^{\prime}}(P)\end{pmatrix}\,,$
(187) $\displaystyle S_{\bm{k},\bm{k}^{\prime}}(P,L)$
$\displaystyle=\frac{1}{2L^{3}}\,\frac{\delta_{\bm{k}\bm{k}^{\prime}}\,e^{-\alpha[(\bm{k})^{2}-(k_{\sf
os})^{2}]}}{4\omega_{N}(\bm{k})\,\big{[}(k_{\sf
os})^{2}-(\bm{k})^{2}\big{]}}\begin{pmatrix}1\\\\[5.0pt]
\mathcal{Y}_{4}^{A_{1}^{+}}(\bm{k})\end{pmatrix}\Big{(}\begin{matrix}1\ \ &\ \
\mathcal{Y}_{4}^{A_{1}^{+}}(\bm{k})\end{matrix}\Big{)}\,.$ (188)

where we have introduced the $S$-wave projected $G$-wave spherical harmonic:

$\mathcal{Y}_{4}^{A_{1}^{+}}(\bm{k})=\frac{\sqrt{21}}{4}\Big{(}5(k_{x}^{4}+k_{y}^{4}+k_{z}^{4})-3(\bm{k}^{2})^{2}\Big{)}\,.$
(189)

This is also used to define the $G$-wave component of $\mathcal{T}(P)$, which
is given explicitly by

$\mathcal{T}^{(\ell=4),A^{+}_{1}}_{\bm{k},\bm{k}^{\prime}}(P)=\frac{21}{16(4\pi)^{2}|\bm{k}|^{4}|\bm{k}^{\prime}|^{4}}\int\\!d\Omega_{\hat{\bm{n}}}\int\\!d\Omega_{\hat{\bm{n}}}^{\prime}\,\\\
\times\frac{\big{(}5(\hat{n}_{x}^{4}+\hat{n}_{y}^{4}+\hat{n}_{z}^{4})-3\big{)}\big{(}5(\hat{n}^{\prime
4}_{x}+\hat{n}^{\prime 4}_{y}+\hat{n}^{\prime
4}_{z})-3\big{)}}{2\omega_{N}(\bm{k})\omega_{N}(\bm{k}^{\prime})-2|\bm{k}||\bm{k}^{\prime}|\hat{\bm{n}}\cdot\hat{\bm{n}}^{\prime}-2M_{N}^{2}+M_{\pi}^{2}-i\epsilon}\,,$
(190)

where $\hat{\bm{n}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ and
similar with primed coordinates. One can evaluate the integral to find

$\displaystyle\mathcal{T}^{(\ell=4),A^{+}_{1}}_{\bm{k},\bm{k}^{\prime}}(P)=\frac{\mathcal{G}\Big{(}|\bm{k}|/M_{N},|\bm{k}^{\prime}|/M_{N}\Big{)}}{M_{N}^{10}}\,,$
(191)

where we have introduced

$\displaystyle\begin{split}\mathcal{G}(x,y)&=\frac{5F(x,y)(-21F(x,y)^{2}+44x^{2}y^{2})}{384x^{8}y^{8}}\\\
&\hskip
20.0pt+\frac{35F(x,y)^{4}-120x^{2}y^{2}F(x,y)^{2}+48x^{4}y^{4}}{512x^{9}y^{9}}\log\left[\frac{F(x,y)-xy}{F(x,y)+xy}\right]\,,\end{split}$
(192) $\displaystyle F(x,y)$
$\displaystyle=-2\sqrt{1+x^{2}}\sqrt{1+y^{2}}+2-(M_{\pi}/M_{N})^{2}+i\epsilon\,.$
(193)

The second function also allows us to express the $S$-wave component in eq.
(119) in a more compact form:

$\mathcal{T}^{(\ell=0),A^{+}_{1}}_{\bm{k},\bm{k}^{\prime}}(P)=\frac{1}{M_{N}^{2}}\frac{1}{4xy}\log\left[\frac{F(x,y)-xy}{F(x,y)+xy}\right]\,.$
(194)

These expressions can be readily extended to higher dimensions, simply by
working out the trivial irrep harmonics and the integrals of the latter
entering $\mathcal{T}(P)$. In this way one can reliably estimate
$F^{\mathcal{T}}_{0}(P,L)$ and thereby determine the relation between
$\overline{\mathcal{K}}_{0}^{\sf os}(P)$, in a given parametrization, and the
finite-volume energies on the left-hand cut.

The second step of the procedure is then using the $S$-wave determined
$\overline{\mathcal{K}}_{0}^{\sf os}(P)$ together with $g$ to solve for
scattering amplitude. The strategies presented in refs. Dawid:2023kxu  will
clearly be useful here and we leave a detailed numerical exploration to future
work.

##  6 Conclusions

In this work, we have addressed the breakdown of the Lüscher scattering
formalism Luscher:1986pf  and extensions Rummukainen:1995vs ; He:2005ey ;
Christ:2005gi ; Kim:2005gf ; Lage:2009zv ; Bernard:2010fp ; Fu:2011xz ;
Briceno:2012yi ; Hansen:2012tf ; Guo:2012hv ; Briceno:2014oea  for finite-
volume energies that arise far enough below threshold to approach (or overlap)
the left-hand branch cut from single meson exchange. The initial motivation
for this investigation was a lattice calculation Green:2021qol  of baryon-
baryon scattering in the $H$-dibaryon sector with quark masses at the
$SU(3)$-flavor-symmetric point. The authors of that study found finite-volume
energies on the left-hand cut and highlighted the need for an extension of
finite-volume methods in order to make use of these energies. More recently,
the same issue was pointed out in lattice calculations of $DD^{*}$ scattering,
investigating the doubly charmed tetraquark $T_{cc}(3875)^{+}$
Padmanath:2022cvl ; Du:2023hlu .

To address this issue, we have proposed an alternative approach to the
standard procedure, allowing one to extract the physical amplitude in the
region of the cut. The detailed relations presented here cover the case of a
single channel of identical scattering particles, denoted $N$, exchanging a
lighter particle, denoted $\pi$, in the $t$\- and $u$-channels.

In particular, in section 3, we have derived an alternative quantization
condition, summarized in eq. (122), which is valid both in the elastic regime,
and for subthreshold energies down to the first two-particle-exchange cut. The
quantization condition allows one to extract an intermediate infinite-volume
quantity, denoted $\overline{\mathcal{K}}^{\sf os}$, together with the
coupling governing the cut, denoted $g$. We have additionally derived integral
equations in section 4 that allow one to relate $\overline{\mathcal{K}}^{\sf
os}$ to the physical scattering amplitude. This gives a complete workflow to
extract the scattering amplitude from the finite-volume spectrum in an
extended kinematic region.

As with much of the extensive literature relating finite-volume data to
amplitudes, our derivations are based on a generic all-orders effective field
theory of nucleons and pions with interactions only constrained by symmetries.
Expressions are given for any spin of the nucleon, with emphasis on the toy
spin-zero and the physical spin-half cases. We have made an effort to detail
the steps that fail in the presence of subthreshold cuts, and to illustrate
and motivate the adaptations required to solve the issue. To summarize, two
key ingredients underpin our approach: the separation of the Bethe-Salpeter
kernel into dangerous pion exchanges and a safe remainder, and the use of
partial off-shellness as a tool for correctly representing the finite-volume
correlator, in particular the fact that it does not contain left-hand cuts.

Further developments, already underway, include the generalization of the
method to non-identical particles, non-degenerate masses and coupled channels.
The relation to the three-particle finite-volume formalism is also being
explored, e.g. whether this approach can be recovered as a limiting case of
the three-particle finite-volume relations. $DD^{*}$ scattering provides a
good testing ground for these extensions, with studies from the three-particle
perspective already ongoing.

More ambitious future work could involve the treatment of lower left-hand cuts
arising from multi-particle exchanges, such as that shown in figure 1(b),
thereby extending the range of validity of the approach to lower energies,
should these be extracted in a given system. Applications to real lattice data
are also planned. In particular, an analysis to the baryon-baryon data of ref.
Green:2021qol  is possible immediately given the results of this work.

###### Acknowledgements.

We thank Raúl Briceño, John Bulava, Zohreh Davoudi, Will Detmold, Evgeny
Eppelbaum, Andrew Jackura, Fernando Romero-López, Srijit Paul, and Steve
Sharpe for useful discussions. We especially thank Steve Sharpe for a detailed
reading of an earlier version of this manuscript. Both ABR and MTH and
supported by UK STFC grant ST/P000630/1. MTH is additionally supported by UKRI
Future Leader Fellowship MR/T019956/1.

##  Appendix A Analyticity of the Bethe-Salpeter kernel

In this appendix we give additional details concerning the conditions under
which the Bethe-Salpeter kernel is an analytic function of $E$ with
exponentially suppressed volume effects.

The discussion is broken into two subsections: First we analyze the diagram
shown in figure 7(a) from various perspectives to illustrate an important and
subtle point about diagrams with $u$-channel-like momentum routing. We show
that, in our energy range of interest:
$(2M_{N})^{2}-(2M_{\pi})^{2}<s<(2M_{N}+M_{\pi})^{2}$, this diagram has no
singularities besides the expected two-particle pole. At the same time, we
find that applying the analysis of section 3.3 to this diagram does lead to
neglected singularities inside the term denoted $\mathcal{I}^{[2]}(P)$,
together with spurious poles in the terms that are kept explicit. This is an
artifact of the partially off-shell kinematics and is removed once all
external legs of the Bethe-Salpeter kernel are required to satisfying the on-
shell condition. It must nonetheless be addressed, since the usual argument
uses properties of the partially off-shell Bethe-Salpeter kernel to reach the
final result.

After demonstrating the issue in detail for the diagram of figure 7(a), we
explain how the result generalizes to all contributions to the Bethe-Salpeter
kernel. The basic insight is that a diagram with a $u$-channel subdiagram is
invariant under the replacement $k\to P-k$, which converts it to a $t$-channel
subdiagram. In this sense, the distinction between $u$\- and $t$-channel is
artificial and relies on the particular choice of internal momentum routing.
This ultimately leads to the definition of $B^{\mathfrak{T}}$, an intermediate
quantity that resolves the issue. When the partially off-shell kinematics are
projected fully on shell, $B^{\mathfrak{T}}$ can be safely symmetrized to the
usual kernel. As a result, our final expressions are not affected by these
issues.

###  A.1 The $u$-channel loop

We begin by using time-ordered perturbation theory (TOPT) to directly identify
the singularities that arise in the full diagram shown in figure 7(a). The
diagram is understood to be evaluated in a finite volume such that all spatial
loop momenta are summed. The external kinematics are evaluated with $p$ on
shell: $p^{0}=\omega_{N}(\bm{p})$, but with
$P-p=(E-\omega_{N}(\bm{p}),\bm{P}-\bm{p})$ generally off shell.

Figure 7: (a) $u$-channel-loop diagram with momentum routing that causes
problems in the standard analysis of section 3.3; (b) the time-ordered
perturbation theory (TOPT) contributions to the diagram of (b); (c) the
$u$-channel contribution to the Bethe-Salpeter kernel in isolation; (d) TOPT
contributions corresponding to (c).

As is shown in figure 7(b), six time orderings contribute. The corresponding
singularities are given by multiplying factors of $[E_{\sf
cut}-\sum_{i}\omega_{i}]^{-1}$, one for each vertical cut, before summing over
time orderings. Here $E_{\sf cut}$ is the energy flowing across the cut and
$\sum_{i}\omega_{i}$ the sum of on-shell energies for the internal propagators
that intersect the cut. Following ref. sterman_1993 , $E_{\sf{cut}}$ is
defined as the sum of inflowing $p^{0}$ components into all vertices appearing
to the left of the cut. So, for example, when only vertex $B$ appears to the
left, then $E_{\sf{cut}}=E-\omega_{N}(\bm{p})$.

Studying 7(b), we note that only two of the six time orderings have the
$E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k})$ singularity leading to the
relevant two-particle pole. These are the first two diagrams, labeled $ABC$
and $BAC$, respectively. The full singularity structure of the two orderings
is

$\displaystyle[ABC]$
$\displaystyle=\frac{1}{\omega_{N}(\bm{p})-\omega_{N}(\bm{P}-\bm{k})-\omega_{\pi}(\bm{P}-\bm{p}-\bm{k}-\bm{\ell})-\omega_{\pi}(\bm{\ell})}\frac{1}{E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k})}\,,$
$\displaystyle[BAC]$
$\displaystyle=\frac{1}{E-\omega_{N}(\bm{p})-\omega_{N}(\bm{k})-\omega_{\pi}(\bm{P}-\bm{p}-\bm{k}-\bm{\ell})-\omega_{\pi}(\bm{\ell})}\frac{1}{E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k})}\,,$

where we have introduced $[ABC]$, etc. as a shorthand, representing the poles
generated by a given vertex ordering.

Crucially, both the $[ABC]$ and $[BAC]$ have no singularities in the relevant
kinematic range, besides that at
$E=\omega_{N}(\bm{k})+\omega_{N}(\bm{P}-\bm{k})$. However, if we consider the
$u$-channel loop of figure 7(c) and the corresponding time orderings of figure
7(d) in isolation, we instead get the following contributions:

$\displaystyle[AB]$
$\displaystyle=\frac{1}{\omega_{N}(\bm{p})-E+\omega_{N}(\bm{k})-\omega_{\pi}(\bm{P}-\bm{p}-\bm{k}-\bm{\ell})-\omega_{\pi}(\bm{\ell})}\,,$
$\displaystyle[BA]$
$\displaystyle=\frac{1}{E-\omega_{N}(\bm{p})-\omega_{N}(\bm{k})-\omega_{\pi}(\bm{P}-\bm{p}-\bm{k}-\bm{\ell})-\omega_{\pi}(\bm{\ell})}\,.$

Note that the notion of external momentum has changed. For example, vertex $A$
now has the external momenta $p-P+k$ flowing into it. As a result, the cut in
$[AB]$ has energy flow $E_{\sf cut}=\omega_{N}(\bm{p})-E+\omega_{N}(\bm{k})$.
We see that, while $[BA]$ exactly matches the factor appearing within $[BAC]$,
the $[AB]$ contribution differs. The reason this matters is that the $[AB]$
form is what appears after $k^{0}$ integration is applied to the general two-
particle loop in section 3.3. See, in particular, eq. (74), which is repeated
in the paragraph after next.

The first factor in $[ABC]$ can be converted to $[AB]$ via the replacement
$\omega_{N}(\bm{P}-\bm{k})\to E-\omega_{N}(\bm{k})$. Thus, the difference
between the two is something that vanishes whenever
$E=\omega_{N}(\bm{k})+\omega_{N}(\bm{P}-\bm{k})$. Such replacements are often
valid in deriving finite-volume relations, since they amount to neglecting a
difference with exponentially suppressed $L$ dependence. In this case the
replacement is invalid, since the form of $[AB]$ in isolation has unphysical
singularities. For example, in the case where $\bm{P}=\bm{0}$ and
$\bm{p}=-\bm{k}$, the $[AB]$ factor has a pole at
$E=2\omega_{N}(\bm{p})-2\omega_{\pi}(\bm{\ell})$. There is no pole in the
original diagram at this location; the singularity is only an artifact of
setting the Bethe-Salpeter kernel to partially on-shell kinematics.

An alternative way to identify the same complication is to track what goes
wrong in the derivation of section 3.3, for this type of diagram. The problem
is the step going from eq. (73),

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

to eq. (74),

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

(both repeated here for convenience). As we now show, the $k^{0}$ integration
has contributions other than the $k^{0}=\omega_{N}(\bm{k})$ part and these
generate singularities in $\mathcal{I}^{[2]}(P)$, which thus carries power-
like $L$ dependence. Simultaneously, singularities and associated power-like
$L$-dependence are hidden within $\mathcal{L}(P,k)$ in eq. (74). The poles
cancel and are also both removed when all external legs are set on shell.

To avoid clutter of notation, we demonstrate the problem explicitly for
slightly simplified three-vector kinematics. In particular, we take
$\bm{P}=\bm{0}$ and $\bm{p}=\bm{k}$ (aligned rather than anti-aligned as
above). For these values, the pion energies are distinct:
$\omega_{\pi}(\bm{\ell})$ and $\omega_{\pi}(2\bm{p}+\bm{\ell})$ while all on-
shell nucleons carry $\omega_{N}(\bm{p})$. The relevant singularities are then
captured by the function

$\displaystyle\mathcal{F}(\bm{p},\bm{\ell})$
$\displaystyle\equiv\int\\!\frac{dk^{0}}{2\pi}\int\\!\frac{d\ell^{0}}{2\pi}\frac{1}{\ell^{0}-\omega_{\pi}(\bm{\ell})+i\epsilon}\frac{1}{\ell^{0}+\omega_{\pi}(\bm{\ell})-i\epsilon}$
$\displaystyle\frac{1}{k^{0}+\ell^{0}+\omega_{N}(\bm{p})-E-\omega_{\pi}(2\bm{p}+\bm{\ell})+i\epsilon}\frac{1}{k^{0}+\ell^{0}+\omega_{N}(\bm{p})-E+\omega_{\pi}(2\bm{p}+\bm{\ell})-i\epsilon}$
$\displaystyle\frac{1}{k^{0}-\omega_{N}(\bm{p})+i\epsilon}\frac{1}{k^{0}+\omega_{N}(\bm{p})-i\epsilon}\frac{1}{k^{0}-E-\omega_{N}(\bm{p})+i\epsilon}\frac{1}{k^{0}-E+\omega_{N}(\bm{p})-i\epsilon}\,,$

which follows from simply factorizing the four covariant propagators.

Beginning with the $\ell^{0}$ integral (which is internal to
$\mathcal{L}(P,k)$ in the general construction), and closing the contour in
the lower half of the complex $\ell^{0}$ plane, we note that two terms arise
from encircling the poles at $\ell^{0}=\omega_{\pi}(\bm{\ell})-i\epsilon$ and
$\ell^{0}=-k^{0}-\omega_{N}(\bm{p})+E+\omega_{\pi}(2\bm{p}+\bm{\ell})-i\epsilon$.
The result reads

$\displaystyle\mathcal{F}(\bm{p},\bm{\ell})$
$\displaystyle\equiv(-i)\frac{1}{2\omega_{\pi}(\bm{\ell})}\int\\!\frac{dk^{0}}{2\pi}\frac{1}{k^{0}+\omega_{\pi}(\bm{\ell})+\omega_{N}(\bm{p})-E-\omega_{\pi}(2\bm{p}+\bm{\ell})+i\epsilon}$
$\displaystyle\hskip
100.0pt\frac{1}{k^{0}+\omega_{\pi}(\bm{\ell})+\omega_{N}(\bm{p})-E+\omega_{\pi}(2\bm{p}+\bm{\ell})-i\epsilon}$
$\displaystyle\hskip
100.0pt\frac{1}{k^{0}-\omega_{N}(\bm{p})+i\epsilon}\frac{1}{k^{0}+\omega_{N}(\bm{p})-i\epsilon}$
$\displaystyle\hskip
100.0pt\frac{1}{k^{0}-E-\omega_{N}(\bm{p})+i\epsilon}\frac{1}{k^{0}-E+\omega_{N}(\bm{p})-i\epsilon}$
$\displaystyle+(-i)\frac{1}{2\omega_{\pi}(2\bm{p}+\bm{\ell})}\int\\!\frac{dk^{0}}{2\pi}\frac{1}{-k^{0}-\omega_{N}(\bm{p})+E+\omega_{\pi}(2\bm{p}+\bm{\ell})-\omega_{\pi}(\bm{\ell})+i\epsilon}$
$\displaystyle\hskip
100.0pt\frac{1}{-k^{0}-\omega_{N}(\bm{p})+E+\omega_{\pi}(2\bm{p}+\bm{\ell})+\omega_{\pi}(\bm{\ell})-i\epsilon}$
$\displaystyle\hskip
100.0pt\frac{1}{k^{0}-\omega_{N}(\bm{p})+i\epsilon}\frac{1}{k^{0}+\omega_{N}(\bm{p})-i\epsilon}$
$\displaystyle\hskip
100.0pt\frac{1}{k^{0}-E-\omega_{N}(\bm{p})+i\epsilon}\frac{1}{k^{0}-E+\omega_{N}(\bm{p})-i\epsilon}\,.$

When we then subsequently evaluate the $k^{0}$ integral, each of these two
terms generates three, for a total of six. But the paradigm established in the
main text only keeps the $k^{0}=\omega_{N}(\bm{k})$ contributions explicit
with all others buried inside $\mathcal{I}^{[2]}(P)$. However, if we consider
the
$k^{0}=-\omega_{N}(\bm{p})+E+\omega_{\pi}(2\bm{p}+\bm{\ell})+\omega_{\pi}(\bm{\ell})-i\epsilon$
contribution within the second term above, we identify a term of the form

$\displaystyle\mathcal{F}(\bm{p},\bm{\ell})$
$\displaystyle\supset-(-i)^{2}\frac{1}{2\omega_{\pi}(\bm{\ell})}\frac{1}{2\omega_{\pi}(2\bm{p}+\bm{\ell})}\bigg{[}\frac{1}{E-2\omega_{N}(\bm{p})+\omega_{\pi}(2\bm{p}+\bm{\ell})+\omega_{\pi}(\bm{\ell})}$
$\displaystyle\hskip
150.0pt\frac{1}{E+\omega_{\pi}(2\bm{p}+\bm{\ell})+\omega_{\pi}(\bm{\ell})}$
$\displaystyle\hskip
150.0pt\frac{1}{\omega_{\pi}(2\bm{p}+\bm{\ell})+\omega_{\pi}(\bm{\ell})-2\omega_{N}(\bm{p})}\frac{1}{\omega_{\pi}(2\bm{p}+\bm{\ell})+\omega_{\pi}(\bm{\ell})}\bigg{]}\,,$

which includes a singularity at
$E=2\omega_{N}(\bm{p})-\omega_{\pi}(\bm{\ell})-\omega_{\pi}(2\bm{p}+\bm{\ell})$.

In the construction of section 3.3, this singularity is included in
$\mathcal{I}^{[2]}(P)$. Similarly, keeping the $k^{0}=\omega_{N}(\bm{p})$
result leads to a contribution to $\mathcal{L}(P,k)$ with the same pole, the
one identified in the TOPT analysis of figure 7(c) above. These poles cancel
in the full expression for the diagram, but they formally disrupt the
smoothness assumptions for both $\mathcal{I}^{[2]}(P)$ and $\mathcal{L}(P,k)$.

###  A.2 Shuffling $t$\- and $u$-type subdiagrams

Having discussed the problem of $u$-channel-like Bethe-Salpeter contributions
in the context of a particular example, we now turn to the solution, first for
the specific case considered and then for all contributions to the Bethe-
Salpeter kernel.

For the example of the single $u$-channel loop, the problem is solved by
simply replacing $\mathcal{L}(P,k)\to\mathcal{L}(P,P-k)$ before $k^{0}$
integration. Since the expression is invariant under $k\to P-k$, this does not
change the value of the full diagram, but it does change the definition (and
indeed the basic properties) of particular contributions such as
$\mathcal{I}^{[2]}(P)$. In particular, the TOPT contributions identified above
are transformed to

$\displaystyle[ABC]$
$\displaystyle=\frac{1}{\omega_{N}(\bm{p})-\omega_{N}(\bm{k})-\omega_{\pi}(\bm{k}-\bm{p}-\bm{\ell})-\omega_{\pi}(\bm{\ell})}\frac{1}{E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k})}\,,$
$\displaystyle[BAC]$
$\displaystyle=\frac{1}{E-\omega_{N}(\bm{p})-\omega_{N}(\bm{P}-\bm{k})-\omega_{\pi}(\bm{k}-\bm{p}-\bm{\ell})-\omega_{\pi}(\bm{\ell})}\frac{1}{E-\omega_{N}(\bm{k})-\omega_{N}(\bm{P}-\bm{k})}\,,$
$\displaystyle[AB]$
$\displaystyle=\frac{1}{\omega_{N}(\bm{p})-\omega_{N}(\bm{k})-\omega_{\pi}(\bm{k}-\bm{p}-\bm{\ell})-\omega_{\pi}(\bm{\ell})}\,,$
$\displaystyle[BA]$
$\displaystyle=\frac{1}{-\omega_{N}(\bm{p})+\omega_{N}(\bm{k})-\omega_{\pi}(\bm{k}-\bm{p}-\bm{\ell})-\omega_{\pi}(\bm{\ell})}\,.$

In this case we find that the $[AB]$ and $[BA]$ terms faithfully represent the
true singularities, without adding any spurious poles. In the case of $[AB]$,
the expression for the sub-diagram matches the full diagram exactly, while for
$[BA]$ it is related by a $E-\omega_{N}(\bm{P}-k)\to\omega_{N}(\bm{k})$
replacement. In contrast to the situation of the preceding subsection, here
the replacement does not generate spurious singularities and in fact leads to
an $E$-independent expression for the sub-diagram. (To see that $[AB]$ and
$[BA]$ are safe note that either denominator vanishing would correspond to the
kinematics of an on-shell $N\to N\pi\pi$ decay, which is not possible.)

It remains to show how the replacement $\mathcal{L}(P,k)\to\mathcal{L}(P,P-k)$
is generalized to all diagrams. The guiding principle is to replace
$u$-channel-like momentum routing with $t$-channel-like routing. However, this
is ambiguous since certain diagrams, like those shown in figure 8(a), have
both $u$\- and $t$-channel like behavior. Such ambiguities are resolved by
performing all internal $\ell^{0}$ integrals in a given Bethe-Salpeter
contribution before applying the momentum reassignments.

This leads to the following diagrammatic definition of the modified Bethe-
Salpeter kernel $B^{\mathfrak{T}}(P,p,p^{\prime})$:

  1. 1.

For a generic diagram contributing to the Bethe-Salpeter kernel, label the
four vertices connected to propagators in the adjacent two-particle loops as
$A_{\sf L}$, $B_{\sf L}$, $A_{\sf R}$, and $B_{\sf R}$, where ${\sf L}$ and
${\sf R}$ stand for left and right, respectively. [See figure 8(b).]

  2. 2.

If $A_{\sf L}$ and $B_{\sf L}$ or $A_{\sf R}$ and $B_{\sf R}$ are the same
vertex, (if [$(A_{\sf L}=B_{\sf L})$ or $(A_{\sf R}=B_{\sf R})$] is true),
then the diagram is a contribution to $B^{\mathfrak{T}}(P,p,p^{\prime})$
without any further modification.

  3. 3.

If $A_{\sf L}$ and $B_{\sf L}$ are distinct from each other, and also $A_{\sf
R}$ and $B_{\sf R}$ are distinct, then evaluate the TOPT contributions to the
diagram by enumerating all orderings of $A_{\sf L}$, $B_{\sf L}$, $A_{\sf R}$,
and $B_{\sf R}$, as well as the set of all vertices not attached to
neighboring loops. Note, it can still be the case that $A_{\sf L}=A_{\sf R}$
or $B_{\sf L}=B_{\sf R}$ (or else $A_{\sf L}=B_{\sf R}$ or $B_{\sf L}=A_{\sf
R}$) as with the single $u$-channel loop.

  4. 4.

For concreteness, define the momentum flowing into each vertex from the
neighboring loops as follows: $k$ into $A_{\sf L}$, $P-k$ into $B_{\sf L}$,
$k^{\prime}$ into $A_{\sf R}$, $P-k^{\prime}$ into $B_{\sf R}$. [Again see
figure 8(b).] Then all time orderings contribute to
$B^{\mathfrak{T}}(P,p,p^{\prime})$ without modification, except for the
following orderings of the external vertices, also shown in figure 8(c):

$A_{\sf L}\cdots B_{\sf R}\cdots A_{\sf R}\cdots B_{\sf L}$,

$A_{\sf L}\cdots B_{\sf R}\cdots B_{\sf L}\cdots A_{\sf R}$,

$B_{\sf R}\cdots A_{\sf L}\cdots A_{\sf R}\cdots B_{\sf L}$,

$B_{\sf R}\cdots A_{\sf L}\cdots B_{\sf L}\cdots A_{\sf R}$,

where the ellipses represent other internal vertices and cuts that can appear
in the ordering. (For our $u$-channel loop, both orderings are of this type.)

  5. 5.

For all orderings given above, dangerous cuts appear between the middle two
vertices (i.e. in the central ellipses). The energy flowing across these cuts
is $u$-channel like, of the form
$\omega_{N}(\bm{k})-E+\omega_{N}(\bm{k}^{\prime})$, and this leads to the
spurious singularities discussed above. Therefore, for all such orderings we
replace $\bm{k}$ with $\bm{P}-\bm{k}$ and $\omega_{N}(\bm{k})$ with
$E-\omega_{N}(\bm{k})$. This replaced definition yields the contribution of
this diagram to $B^{\mathfrak{T}}(P,p,p^{\prime})$. For all other time
orderings the contribution to $B^{\mathfrak{T}}(P,p,p^{\prime})$ is
unmodified. This completes our construction of the modified Bethe-Salpeter
kernel.

Figure 8: (a) Examples of diagrams for which the notion of $u$-channel-like
and $t$-channel like is ambiguous. (b) Scheme for labeling vertices in a
generic diagram. (c) Summary of orderings for which a momentum rerouting is
required.

This modified kernel is useful because it allows us to follow the procedure of
section 3.3 without encountering spurious singularities. In particular, the
$k^{0}$ integration of eq. (73) is now valid. The ordering of vertices
guarantees that the only energies that can flow across the cuts in the TOPT
approach are $E$ and $\pm[\omega_{N}(\bm{k})-\omega_{N}(\bm{k}^{\prime})]$.
Thus, in the case where no single meson exchange occurs,
$B^{\mathfrak{T}}(P,p,p^{\prime})$ is analytic in $s$ and has exponentially
suppressed volume effects for
$4M_{N}^{2}-4M_{\pi}^{2}<s<(2M_{N}+M_{\pi})^{2}$. This is the same range of
validity holds for the subtracted kernel
$\overline{B}^{\mathfrak{T}}(P,p,p^{\prime})$ in theories with a single meson
exchange.

##  Appendix B Manipulating the finite-volume $S$ function

In this appendix, we describe the steps required to go from eq. (74) to eqs.
(79) and (80) of the main text. Recalling the definitions, the task is to show
that

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

can be rewritten as

$\mathcal{L}\,\circ_{\sf fv}\,\mathcal{R}^{\dagger}=\mathcal{I}^{[3]}(P)\\\
+\widetilde{\mathcal{L}}_{\bm{k}^{\star}\ell
m}(P)\,\frac{1}{2L^{3}}\,\frac{i4\pi\,Y_{\ell
m}(\hat{\bm{k}}^{\star})\,Y^{*}_{\ell^{\prime}m^{\prime}}(\hat{\bm{k}}^{\star})\,\delta_{\bm{k}^{\star}\bm{k}^{\prime\star}}\,|\bm{k}^{\star}|^{\ell+\ell^{\prime}}\,H(\bm{k}^{\star})}{4\omega_{N}(\bm{k})\,\big{[}(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}\big{]}}\,\widetilde{\mathcal{R}}^{*}_{\bm{k}^{\prime\star}\ell^{\prime}m^{\prime}}(P)\,,$
(195)

where we have combined eqs. (79) and (80) to reach eq. (195).

Taking the difference of these two results and using the definitions of
$\widetilde{\mathcal{L}}_{\ell m}(P,|\bm{k}^{\star}|)$ and
$\widetilde{\mathcal{R}}_{\ell m}(P,|\bm{k}^{\star}|)$, eqs. (75) and (76), it
remains to show

$\mathcal{I}^{[2]}(P)-\mathcal{I}^{[3]}(P)=\frac{i}{2L^{3}}\sum_{\bm{k}}\frac{1}{2\omega_{N}(\bm{k})}\mathcal{L}(P,k)\,\mathcal{R}^{*}(P,k)\\\
\times\bigg{[}\frac{1}{(E-\omega_{N}(\bm{k}))^{2}-\omega_{N}(\bm{P}-\bm{k})^{2}}-\frac{H(\bm{k}^{\star})/2}{(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}}\bigg{]}\bigg{|}_{k^{0}=\omega_{N}(\bm{k})}\,,$
(196)

with the key claim being that $\mathcal{I}^{[2]}(P)-\mathcal{I}^{[3]}(P)$ only
has exponentially suppressed $L$-dependence. This follows from the Poisson
summation formula, provided that the summand on the right-hand side has a
horizontal strip of analyticity in the complex-$|\bm{k}|$ plane, which
includes the real axis. Here we abuse notation by thinking of $z=|\bm{k}|$ as
a complex variable.

For $P^{2}=s<(2M_{N}+M_{\pi})^{2}$, the analyticity holds for
$\mathcal{L}(P,k)\,\mathcal{R}^{*}(P,k)$ by construction, and also for
$\omega_{N}(\bm{k})$, which is analytic for $|\text{Im}[|\bm{k}|]|<M_{N}$. The
non-trivial step is to show the same for the difference of the two terms in
square brackets. To demonstrate this we define the quantity in question via

$Q(P,\bm{k})=\frac{1}{(E-\omega_{N}(\bm{k}))^{2}-\omega_{N}(\bm{P}-\bm{k})^{2}}-\frac{H(\bm{k}^{\star})/2}{(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}}\,.$ (197)

We then observe that

$(E-\omega_{N}(\bm{k}))^{2}-\omega_{N}(\bm{P}-\bm{k})^{2}=(P-k)^{2}-M_{N}^{2}=(E^{\star}-\omega_{N}(\bm{k}^{\star}))^{2}-\omega_{N}(\bm{k}^{\star})^{2}\,,$
(198)

where we have used $P^{\mu}=(E,\bm{P})^{\mu}$ and
$k^{\mu}=(\omega_{N}(\bm{k}),\bm{k})^{\mu}$ in the middle step. This makes
manifest that the combination is a Lorentz scalar and can be written in the CM
frame, as we have done.

Further algebraic manipulations then yield

$(E-\omega_{N}(\bm{k}))^{2}-\omega_{N}(\bm{P}-\bm{k})^{2}=\frac{E^{\star}}{E^{\star}+2\omega_{N}(\bm{k}^{\star})}\big{[}E^{\star
2}-4\omega_{N}(\bm{k}^{\star})^{2}\big{]}\,.$ (199)

Substituting $(k_{\sf os}^{\star})^{2}=E^{\star 2}/4-M_{N}^{2}$ and then
substituting back for $Q(P,\bm{k})$, we finally deduce

$Q(P,\bm{k})=\bigg{[}\frac{E^{\star}+2\omega_{N}(\bm{k}^{\star})}{4E^{\star}}-\frac{H(\bm{k}^{\star})}{2}\bigg{]}\,\frac{1}{(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}}\,.$ (200)

Note that the factor in square brackets scales as $[(k_{\sf
os}^{\star})^{2}-(\bm{k}^{\star})^{2}]$ as $(\bm{k}^{\star})^{2}\to(k_{\sf
os}^{\star})^{2}$, and therefore cancels the pole. This holds because

$\bigg{[}\frac{E^{\star}+2\omega_{N}(\bm{k}^{\star})}{4E^{\star}}-\frac{H(\bm{k}^{\star})}{2}\bigg{]}\bigg{|}_{(\bm{k}^{\star})^{2}=(k_{\sf
os}^{\star})^{2}}=0\,,$ (201)

together with the fact that all functions are analytic for
$(\bm{k}^{\star})^{2}>-M_{N}^{2}$. We thus conclude a non-zero strip of
analyticity for $Q(P,\bm{k})$

##  Appendix C Details of the derivation

In this appendix we discuss the manipulations required to go from eq. (112):

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

to eq. (120):

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

where we have repeated both results here for convenience.

The basic idea is that the initial four infinite-volume quantities
($\mathcal{I}^{[1]}_{C}(P)$, $\overline{\mathcal{K}}^{[1]}(P)$, $A^{[1]}(P)$,
and $A^{[1]\dagger}(P)$) are modified to alternatives ($\mathcal{I}_{C}(P)$,
$\overline{\mathcal{K}}^{\sf os}(P)$, $A^{\sf os}(P)$, and $A^{\sf
os}(P)^{\dagger}$) with $\ell m$ indices in place of $\bm{k}^{\star}\ell m$
everywhere, i.e. the $\bm{k}^{\star}$ index is removed in quantities labelled
by ${\sf os}$, which stands for _on shell_. This is possible because, if both
nucleons in a given two-nucleon state are on shell, then the state is
completely specified by energy and angular momentum and the $\bm{k}^{\star}$
index is redundant. This is not true for the quantities labeled by $[1]$ as
these are off-shell and the magnitude of $\bm{k}^{\star}$ can be freely
varied.

The simplest instance of replacing off-shell with on-shell quantities is given
by the relation

$\mathcal{I}^{[1]}_{C}(P)+A^{[1]}(P)iS(P,L)A^{[1]\dagger}(P)=\widetilde{\mathcal{I}}^{[1]}_{C}(P)+A^{[1]{\sf
os}}(P)\,\xi\,iS(P,L)\,\xi^{\dagger}\,A^{[1]{\sf os}}(P)^{\dagger}\,,$ (202)

which trivially follows from definition

$\displaystyle\widetilde{\mathcal{I}}^{[1]}_{C}(P)$
$\displaystyle=\mathcal{I}^{[1]}_{C}(P)+A^{[1]}(P)iS(P,L)A^{[1]\dagger}(P)-A^{[1]{\sf
os}}(P)\,\xi\,iS(P,L)\,\xi^{\dagger}\,A^{[1]{\sf os}}(P)^{\dagger}\,,$ (203)

where $A_{\ell m}^{[1]{\sf os}}(P)=A_{\bm{k}^{\star}\ell
m}^{[1]}(P)\Big{|}_{|\bm{k}^{\star}|=k^{\star}_{\sf os}}$ and similar for the
conjugated factor. This replacement is valid because $A_{\ell m}^{[1]{\sf
os}}(P)$ does not have a left-hand cut and the singularity of $S(P,L)$ cancels
between the last two terms of eq. (203). Thus the sum, implicit in the second
two terms, can be replaced with an integral up to exponentially suppressed
terms that we neglect.

In this appendix we will describe a series of such replacements that have the
effect of converting eq. (112) to eq. (120). The guiding principles of the
replacements are as follows:

  1. 1.

In quantities without a left-hand cut in the region that we control
($4M_{N}^{2}-4M_{\pi}^{2}<s$), one can safely set $|\bm{k}^{\star}|$ to
$k_{\sf os}^{\star}$ up to differences that cancel neighboring $S(P,L)$ poles.

  2. 2.

The cancellation of $S(P,L)$ poles, in turn, allows one to replace sums with
integrals and the integrated terms are absorbed in redefinitions of infinite-
volume quantities, such as in eq. (203).

  3. 3.

An iterative process then emerges. This is because the resulting integrals
create new quantities for which the replacement $|\bm{k}^{\star}|\to k_{\sf
os}^{\star}$ can safely be performed.

To explain in detail, it will be useful to introduce a more compact notation.
We use bold symbols to represent the building blocks compactly and suppress
all arguments, factors of $i$ and $g^{2}$, and overlines. We then define the
$(n+1)$th or $(n^{\prime}+1)$th term in our starting and target sums,
respectively, as

$\displaystyle C^{[1],(n)}_{L}(P)$
$\displaystyle={\textbf{A}}^{\\![1]}\,{\textbf{S}}\,\Big{[}\big{(}{\textbf{K}}^{[1]}+{\textbf{T}}\big{)}\,{\textbf{S}}\Big{]}^{n-1}\,{{\textbf{A}}}^{[1]\dagger}\,,$
(204) $\displaystyle C^{[{\sf os}],(n^{\prime})}_{L}(P)$
$\displaystyle={\textbf{A}}^{\\!\sf
os}\,\xi\,{\textbf{S}}\left[\left(\xi^{\dagger}\,{\textbf{K}}^{\sf
os}\,\xi+{\textbf{T}}\right){\textbf{S}}\right]^{n^{\prime}-1}\xi^{\dagger}\,{\textbf{A}}^{\\!\sf
os\dagger}\,,$ (205)

with $n=0$ corresponding to the S-independent quantity in each case, i.e.
$\mathcal{I}^{[1]}_{C}(P)$ or $\mathcal{I}_{C}(P)$.

The key claim is that, for any non-negative integer $n$, $C_{L}^{[1],(n)}$
generates a series of contributions to $C^{[{\sf os}],(n^{\prime})}_{L}(P)$
for all $n^{\prime}\leq n$. We now sketch the argument for $n=0,1,2,3$, since
new features emerge at each of these orders. We then give the all orders
construction.

For $n=0$ it is straightforward; we simply note that
$\mathcal{I}^{[1]}_{C}(P)$ is one of the terms entering the definition of
$\mathcal{I}_{C}(P)$. In addition, we have already addressed $n=1$ above in
our motivating example. In our more compact notation we begin with

$C^{[1],(1)}_{L}(P)={\textbf{A}}^{\\![1]}\,{\textbf{S}}\,{{\textbf{A}}}^{[1]\dagger}\,.$
(206)

The on-shell projection is then exactly shown in eq. (202). We generate
contributions to $n^{\prime}=0$ and $n^{\prime}=1$, i.e. to

$C^{[{\sf os}],(0)}_{L}(P)=\mathcal{I}_{C}(P)\,,\qquad C^{[{\sf
os}],(1)}_{L}(P)={\textbf{A}}^{\\!\sf
os}\,\xi\,{\textbf{S}}\,\xi^{\dagger}\,{\textbf{A}}^{\\!\sf os\dagger}\,.$
(207)

Before moving to higher $n$, we introduce some additional notation and show
how this applies to the $n=1$ case. In particular, we define

$\delta^{\dagger}{{\textbf{A}}}^{\\![1]\,{\sf
os}\,\dagger}={{\textbf{A}}}^{\\![1]\,\dagger}-\xi^{\dagger}{\textbf{A}}^{\\![1]\,\sf
os\,\dagger}\,,\qquad{\textbf{A}}^{\\![1]\,{\sf
os}}\delta={\textbf{A}}^{\\![1]}-{\textbf{A}}^{\\![1]\,\sf os}\,\xi\,,$ (208)

and write

$C^{[1],(1)}_{L}(P)={\textbf{A}}^{\\![1]}\,{\textbf{S}}\,{{\textbf{A}}}^{[1]\dagger}={\textbf{A}}^{\\![1]\,{\sf
os}}\,(\xi+\delta)\,{\textbf{S}}\,(\xi^{\dagger}+\delta^{\dagger})\,{\textbf{A}}^{\\![1]\,{\sf
os}\,\dagger}\,.$ (209)

Note that the quantities with a $\delta$ are not truly on shell as the label
implies. We nonetheless use this notation so that $\xi$ and $\delta$ can be
compactly grouped as shown. We next use the rule that, whenever one or more
$\delta$ factors appears next to an S, the singularity cancels and the sum is
replaced with an integral.101010It should be emphasized that similar tricks
were used heavily in addressing singularities and redefinitions for the three-
to-three finite-volume formalism Hansen:2014eka  and in the study of two-to-
two processes with an external current Briceno:2015tza . In this way, of the
four terms generated by multiplying out the $(\xi+\delta)$ and
$(\xi^{\dagger}+\delta^{\dagger})$ binomials, three contribute to $C^{[{\sf
os}],(0)}_{L}(P)$, as claimed.

We are now ready to turn to $n=2$, where a new and subtle aspect of the
derivation arises. We write out the expression as

$\displaystyle C^{[1],(2)}_{L}(P)$
$\displaystyle={\textbf{A}}^{\\![1]}\,{\textbf{S}}\,{\textbf{K}}^{[1]}\,{\textbf{S}}\,{{\textbf{A}}}^{[1]\dagger}+{\textbf{A}}^{\\![1]}\,{\textbf{S}}\,{\textbf{T}}\,{\textbf{S}}\,{{\textbf{A}}}^{[1]\dagger}\,,$
(210) $\displaystyle\begin{split}&={\textbf{A}}^{\\![1]\,{\sf
os}}\,(\xi+\delta)\,{\textbf{S}}\,(\xi^{\dagger}+\delta^{\dagger})\,{\textbf{K}}^{[1]\,{\sf
os}}\,(\xi+\delta)\,{\textbf{S}}\,(\xi^{\dagger}+\delta^{\dagger})\,{{\textbf{A}}}^{\\![1]\,{\sf
os}\,\dagger}\\\ &\hskip 150.0pt+{\textbf{A}}^{\\![1]\,{\sf
os}}\,(\xi+\delta)\,{\textbf{S}}\,{\textbf{T}}\,{\textbf{S}}\,(\xi^{\dagger}+\delta^{\dagger})\,{{\textbf{A}}}^{\\![1]\,{\sf
os}\,\dagger}\,,\end{split}$ (211)

where in the second equality we have applied the $\xi+\delta$ decomposition to
both $\textbf{A}^{\\![1]}$ and $\textbf{K}^{[1]}$.

First consider the $2^{4}=16$ terms generated by expanding all binomials in
the first term of eq. (211). Using the usual rule that a $\delta$ always
annihilates a neighboring S we see that all terms can be identified as
contributions to $C_{L}^{[\sf os],(n^{\prime})}(P)$ with $n^{\prime}\leq 2$.

More effort is required for the second term of (211). Of the $2^{2}=4$ terms,
the term with $\xi$ and $\xi^{\dagger}$ has a straightforward match in the
${\textbf{A}}^{\\!\sf
os}\,\xi\,{\textbf{S}}\,{\textbf{T}}\,{\textbf{S}}\,\xi^{\dagger}\,{\textbf{A}}^{\\!\sf
os\dagger}$ part of $C^{[{\sf os}],(1)}_{L}(P)$. Also, the term with $\delta$
and $\delta^{\dagger}$ is clear, since all singularities are cancelled and one
reaches a fully integrated contribution to $C^{[{\sf os}],(0)}_{L}(P)$. The
challenge is the remaining contributions with a single factor of either
$\delta$ or $\delta^{\dagger}$.

Consider, for concreteness, ${\textbf{A}}^{\\![1]\,{\sf
os}}\,\delta\,{\textbf{S}}\,{\textbf{T}}\,{\textbf{S}}\,\xi^{\dagger}\,{{\textbf{A}}}^{\\![1]\,{\sf
os}\,\dagger}$. The issue is that T cannot be placed on shell but, at the same
time, the $\delta$ cancelling the pole in S will lead to an integral combining
the off-shell T into the left endcap. We require a final piece of notation to
reflect this:

${\textbf{A}}^{\\![k+1]}={\textbf{A}}^{\\![k]\,{\sf
os}}\,\delta\,{\textbf{S}}\,{\textbf{T}}\,.$ (212)

Of which the special case relevant here is simply
${\textbf{A}}^{\\![2]}={\textbf{A}}^{\\![1]\,{\sf
os}}\,\delta\,{\textbf{S}}\,{\textbf{T}}$. So we find that T can convert an
on-shell contribution back to an off-shell contribution. In other words,
${\textbf{A}}^{\\![2]}$ has dependence on $\bm{k}^{\star}$ within
$\bm{k}^{\star}\ell m$, and is thus not defined on the desired index space.

Having articulated the extra complication, the resolution follows from an
imitation of what we have already done. The terms that we have to address are
of the form
${\textbf{A}}^{\\![2]}\,{\textbf{S}}\,\xi^{\dagger}\,{{\textbf{A}}}^{\\![1]\,{\sf
os}\,\dagger}$ and ${\textbf{A}}^{\\![1]\,{\sf
os}}\,\xi\,{\textbf{S}}\,{{\textbf{A}}}^{\\![2]\,\dagger}$. But, now that the
T factors within ${\textbf{A}}^{\\![2]}$ have integrated momenta, these no
longer have cuts when set on shell. It follows that the on-shell projections
of ${\textbf{A}}^{\\![2]}$ and ${{\textbf{A}}}^{\\![2]\,\dagger}$ are safe and
we can write

$\displaystyle{\textbf{A}}^{\\![2]}\,{\textbf{S}}\,\xi^{\dagger}\,{{\textbf{A}}}^{\\![1]\,{\sf
os}\,\dagger}$ $\displaystyle={\textbf{A}}^{\\![2]\,{\sf
os}}\,(\xi+\delta)\,{\textbf{S}}\,\xi^{\dagger}\,{{\textbf{A}}}^{\\![1]\,{\sf
os}\,\dagger}\,,$ (213) $\displaystyle{\textbf{A}}^{\\![1]\,{\sf
os}}\,\xi\,{\textbf{S}}\,{{\textbf{A}}}^{\\![2]\,\dagger}$
$\displaystyle={\textbf{A}}^{\\![1]\,{\sf
os}}\,\xi\,{\textbf{S}}\,(\xi^{\dagger}+\delta^{\dagger})\,{{\textbf{A}}}^{\\![2]\,{\sf
os}\,\dagger}\,.$ (214)

At this stage, the $\delta$-dependent terms are absorbed into $C_{L}^{[\sf
os],(0)}(P)$ and the remaining terms are absorbed into the part of
$C_{L}^{[\sf os],(1)}(P)$ that is linear in S.

The last claim above relies on the definition for the full on-shell matrix
element

${\textbf{A}}^{\\!\sf os}=\sum_{k=1}^{\infty}{\textbf{A}}^{\\![k]\,\sf
os}\,,\qquad{\textbf{A}}^{\\!{\sf
os}\,\dagger}=\sum_{k=1}^{\infty}{\textbf{A}}^{\\![k]\,{\sf os}\,\dagger}\,.$
(215)

In words, these definitions indicate that the final endcap factors are built
from an infinite set of T factors, attached with integrals to a neighboring
factor of S in which the pole has been cancelled. In all such cases, the
factor of T is not evaluated at $|\bm{k}^{\star}|=k^{\star}_{\sf os}$ until
after integration. In this way, one avoids introducing spurious left-hand cuts
in the finite-volume correlator.

A final new feature arises within $C_{L}^{[1],(3)}$, defined as

$C^{[1],(3)}_{L}(P)={\textbf{A}}^{\\![1]}\,{\textbf{S}}\,\big{(}{\textbf{K}}^{[1]}+{\textbf{T}}\big{)}\,{\textbf{S}}\,\big{(}{\textbf{K}}^{[1]}+{\textbf{T}}\big{)}\,{\textbf{S}}\,{{\textbf{A}}}^{[1]\dagger}\,,$
(216)

namely that factors of T must also be absorbed into a redefinition of the K
matrix.

At this stage, we think it more pedagogical to explain this partly in words,
without introducing additional heavy notation to represent the absorptions.
When one inserts the $(\xi+\delta)$ decomposition for all
${\textbf{A}}^{\\![1]}$, ${{\textbf{A}}}^{[1]\dagger}$ and
${\textbf{K}}^{[1]}$ in eq. (216), a term arises with the left
${\textbf{K}}^{[1]}$ replaced by $\xi^{\dagger}{\textbf{K}}^{[1]\,{\sf
os}}\delta$, leading to combinations such as the following:

$C^{[1],(3)}_{L}(P)\supset{\textbf{A}}^{\\![1]\,{\sf
os}}\,\xi\,{\textbf{S}}\,\xi^{\dagger}\Big{(}{\textbf{K}}^{[1]\,{\sf
os}}\,\delta\,{\textbf{S}}\,{\textbf{T}}\Big{)}{\textbf{S}}\,\xi^{\dagger}\,{{\textbf{A}}}^{\\![1]\,{\sf
os}\,\dagger}\,,$ (217)

where the $\supset$ symbol indicates we have only kept one term. Observe that
the factor in parentheses here is analogous to ${\textbf{A}}^{\\![2]}$ defined
in eq. (212) above. Any number of T factors can be attached on either side of
K in this way. After such an attachment, the function no longer has a cut in
the region of interest and can be set on shell. The sum over all such on-
shell, T-absorbing factors defines the final K-matrix denoted by
${\textbf{K}}^{\sf os}$.

Fortunately we do not have to make use of this definition, since the relation
of ${\textbf{K}}^{\sf os}$ to the finite-volume correlation function provides
an alternative definition in terms of the two-to-two scattering amplitude.
Here the situation is analogous to the three-particle formalism of refs.
Hansen:2014eka . Also in that work a very ugly iterative definition of a
K-matrix was provided but never used, favoring instead an alternative
derivation of a more direct relation to the scattering amplitude
Hansen:2015zga .

At this stage we have presented all features that arise in expressing
$C_{L}^{[1],(n)}(P)$ in terms of $C_{L}^{[{\sf os}],(n^{\prime})}(P)$ for
$n^{\prime}\leq n$. We summarize the general construction as follows:

  1. 1.

For a given $C_{L}^{[1],(n)}(P)$ insert all allowed factors of $\xi+\delta$
and $\xi^{\dagger}+\delta^{\dagger}$. A total of $2n$ such binomials arise, 2
from the endcaps and the remaining $(2n-2)$ from the $n-1$ insertions of
$\textbf{K}^{[1]}$.

  2. 2.

Multiplying out the terms yields $2^{2n}$ combinations, with various sequences
of $\xi$, $\xi^{\dagger}$, $\delta$ and $\delta^{\dagger}$.

  3. 3.

All contributions are immediately identified within $C_{L}^{[{\sf
os}],(n^{\prime})}(P)$ with the exception of those for which an off-shell T is
attached to either an endcap or a K-matrix.

  4. 4.

These remaining terms correspond to a contribution within
$C_{L}^{[1],(n-j)}(P)$ for some $j\geq 1$. That is, they match a factor with
fewer explicit Ts, since some are absorbed into $\textbf{A}^{[k]}$,
$\textbf{A}^{[k]\dagger}$ or the corresponding K matrix.

  5. 5.

For these remaining terms, new $\xi+\delta$ insertions arise and are processed
as above.

  6. 6.

The iterative procedure always either generates terms that match those in
$C_{L}^{[{\sf os}],(n^{\prime})}(P)$ or else reduces the explicit T factors in
the remaining, unmatched terms.

  7. 7.

It follows that, after $n$ such iterations on $C_{L}^{[1],(n)}(P)$, the
process terminates and the matching to the desired expression is complete.

Thus, we have achieved our aim to show the equivalence of eqs. (112) and (120)
(the first two equations of this appendix). We emphasize that the result
requires a sequence of redefinitions that is analogous to the three-particle
finite-volume scattering formalism of refs. Hansen:2014eka ; Hansen:2015zga .
These parallels are expected since the same underlying $NN\pi$ state is
leading to the left-hand cut addressed here.

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