2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p_{1},3+p_{2}\end{matrix}\right]$
$\displaystyle-\displaystyle\frac{\,{e^{\mp\mathrm{i}p_{12}\pi/2}}\,}{2^{5+p_{12}}}\Gamma\Big{[}5+p_{12},\mbox{$\frac{\,{5}\,}{2}$}+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}{}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{2}\left[\begin{matrix}5+p_{12},\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,1\\\
\mbox{$\frac{\,{7}\,}{2}$}+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{7}\,}{2}$}+p_{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|1\right],$ (85) $\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu|\pm\mp}^{p_{1}p_{2}}(1,1)=\displaystyle\frac{\,{e^{\mp\mathrm{i}\bar{p}_{12}\pi/2}}\,}{2^{5+p_{12}}}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p_{1},3+p_{2}\end{matrix}\right].$ (86)

With these explicit expressions, the summation in (84) can be directly done,
and the result is:

$\mathcal{P}^{(2)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)=-\displaystyle\frac{\,{8(2+\mathrm{i}\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma(2+2\mathrm{i}\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\sin(\pi\mathrm{i}\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{3+2\mathrm{i}\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}.$ (87)

Now we put together the left graph $\mathcal{T}_{\mathsf{c}}^{(L)}$ in (83),
the right graph $\mathcal{T}_{\mathsf{c}}^{(R)}$ in (80), and the bubble
signal $\mathcal{B}_{\mathsf{c}}$ in (72), as illustrated in Fig. 7, and get
the expression for the nonlocal CC signal in the triangle graph in the
squeezed limit $k_{s}\rightarrow 0$:

$\displaystyle\lim_{k_{s}\rightarrow
0}\Big{[}\mathcal{T}_{\text{triangle}}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3}}\,}{(4\pi)^{7/2}k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{4k_{12}k_{34}}\Big{)}^{2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\sinh^{2}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$
$\displaystyle\times\Gamma\Big{[}2+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-\displaystyle\frac{\,{3}\,}{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}\Gamma^{2}\Big{[}\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}+\text{c.c.}.$
(88)

We note that this result holds for all $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu>0$.

###  4.3 Box diagram

Figure 8: The nonlocal signal in the 1-loop box graph.

Finally, we consider the box diagram with the following interaction, as shown
in Fig. 8:

$\Delta{\mathscr{L}}=\displaystyle\frac{\,{1}\,}{2}a^{3}\varphi^{\prime}\sigma^{2}.$
(89)

This time, both the left and the right tree graphs have exactly the same
structure with the left tree graph of the triangle graph computed in the last
subsection. Therefore, we can directly quote the previous result and write:

$\displaystyle\mathcal{T}_{\mathsf{c}}^{(\text{L})}(k_{1},k_{2})=\mathcal{P}^{(2)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{12}^{-4-2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{4k_{1}k_{2}},$
$\displaystyle\mathcal{T}_{\mathsf{c}}^{(\text{R})}(k_{3},k_{4})=\mathcal{P}^{(2)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{34}^{-4-2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{4k_{3}k_{4}},$
(90)

where $\mathcal{P}_{\mathsf{c}}^{(2)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ is given in (87).
Then, we can directly substitute this result into the general expression (71).
Together with the bubble signal $\mathcal{B}_{\mathsf{c}}$ in (72), we get the
nonlocal signal of the box diagram in the limit $k_{s}\rightarrow 0$:

$\displaystyle\lim_{k_{s}\rightarrow
0}\Big{[}\mathcal{T}_{\text{box}}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=$
$\displaystyle-\displaystyle\frac{\,{k_{s}^{3}}\,}{(4\pi)^{7/2}k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{4k_{12}k_{34}}\Big{)}^{2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\displaystyle\frac{\,{(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)^{4}}\,}{(3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)^{2}}\sinh^{2}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$
$\displaystyle\times\Gamma\Big{[}3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-\displaystyle\frac{\,{3}\,}{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}\Gamma^{2}\Big{[}\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}+\text{c.c.}.$
(91)

###  4.4 Summary: pinched operator and phases

In this section, we have computed the nonlocal signals from all 1PI 1-loop
4-point graphs, including the bubble, the triangle, and the box diagrams. By
applying the 1-loop factorization theorem, we have obtained the exact and
analytical expressions for the nonlocal CC signals in the $s$-channel squeezed
limit $k_{s}\rightarrow 0$ for all these graphs.

$\parbox{195.12767pt}{\includegraphics[width=195.12767pt]{fd_triangle}}\Rightarrow\parbox{195.12767pt}{\includegraphics[width=195.12767pt]{fd_pinched_triangle}}$
Figure 9: The triangle diagram and its pinched version.

#### Pinched operator.

Our result shows that, so far as the nonlocal CC signal is concerned, it is
legitimate to pinch the hard lines in the loop graph, so that the oscillatory
signals are essentially generated by an effective bubble diagram. We
illustrate this procedure for the triangle diagram in Fig. 9. We emphasize
that this pinch procedure does not require the pinched mode to be heavy,
although we do expect that our result recovers the EFT intuition when $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\gg 1$. That is, the
coupling constant of the pinched operator $\mathcal{O}_{\pm}$ should scale as
$1/\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}$. To
explain this point more explicitly, we introduce the following effective
_pinched operators_ :

$\mathcal{O}_{\pm}=\mathcal{O}_{\pm}^{(0)}+\mathcal{O}_{\pm}^{(1)}+\cdots.$
(92)

This pinched operator $\mathcal{O}_{\pm}$ is defined such that the bubble
graph with one insertion of $\mathcal{O}_{\pm}$, as shown in the right graph
of Fig. 9, gives rise to a nonlocal signal in the $s$-channel soft limit
$k_{s}\rightarrow 0$ identical to the signal from the triangle graph, namely
the left graph of Fig. 9. For later convenience, we have rewritten the pinched
operator $\mathcal{O}_{\pm}$ as a summation of $n$-th order pinched operator
$\mathcal{O}_{\pm}^{(n)}$ with $n=0,1,\cdots$. An $n$’th order pinched
operator generates a nonlocal signal starting from order $k_{s}^{3+2n}$. We
will have more to say about the higher order couplings in Sec. 5.2. For now,
let us focus on the leading operator $\mathcal{O}_{\pm}^{(0)}$. By comparing
the two pinch coefficients defined in (76) and (87), we can identify the
leading pinched operator $\mathcal{O}_{\pm}^{(0)}$ to be:

$\mathcal{O}_{\pm}^{(0)}=\displaystyle\frac{\,{1}\,}{4}\mathcal{C}^{(0)}_{\pm}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)a^{2}\varphi^{\prime
2}\sigma^{2},$ (93)

where the pinched coupling constant $\mathcal{C}_{\pm}^{(0)}$ is given by:

$\displaystyle\mathcal{C}_{\pm}^{(0)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\equiv\displaystyle\frac{\,{2\mathcal{P}^{(2)}_{\pm}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{\mathcal{P}^{(1)}_{\pm}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}=-\displaystyle\frac{\,{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+2}\,}{(\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+3/2)^{2}(\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+1)}.$ (94)

Here we include a factor of $2$, since the symmetry factors for the triangle
and the bubble diagrams are $1$ and $2$, respectively. As we can see, the
pinched operator provides a convenient way to generate nonlocal signals in
diagrams with more complicated topologies. For instance, the nonlocal signal
in the box graph can be obtained by computing a bubble graph with both of its
vertices replaced by the pinched operator.121212For the nonlocal signal of the
box diagram in Fig. 8, one can start from the nonlocal signal of the bubble
diagram (4.1), replace the two $\mathcal{P}_{\pm}^{(1)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ by
$\mathcal{P}_{\pm}^{(2)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$, and add a factor
$2$ since the symmetry factor for box diagram is $1$, and the result will be
(4.3). Furthermore, there is another box graph with
($\mathbf{k}_{3}\leftrightarrow\mathbf{k}_{4}$) giving identical nonlocal
signal in the $s$-channel squeezed limit. As a result, the total $s$-channel
nonlocal signal is given by (4.3) multiplied by 2, and it can be equivalently
obtained by computing a bubble graph with two pinched operators (94). In this
section, we are only working at the leading order in the $k_{s}$ expansion,
and we only need a single pinched operator $\mathcal{O}_{\pm}^{(0)}$. At
higher orders in $k_{s}$, the situation can be more complicated and the choice
of pinched operators is not unique. We can simply make convenience choices at
each order, as we shall show in the next section.

Figure 10: The size of the effective coupling from pinching the hard line as a
function of the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ of the loop mode.
The magenta curve shows the exact result in (94), and the black dashed curve
shows the EFT approximation.

#### Signal phase.

A peculiar feature of the effective pinched coupling $\mathcal{C}_{\pm}^{(0)}$
in (94) is that it is complex, and is dependent on the choices of the two
scaling dimensions $2\Delta_{\pm}=3\pm 2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ of the soft loop.
This is quite natural from the boundary viewpoint, since we have two sets of
boundary operators $\sigma_{\pm}$ with scaling dimensions
$\Delta_{\pm}=3/2\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$, and each set
independently generates its own nonlocal signal. Phenomenologically, the
complex coupling implies that there will be nontrivial phase shift in the
nonlocal CC signals in the triangle or box graph relative to the bubble graph.
This phase shift is physical and measurable, as has been emphasized in [51].
Therefore, the phase shift gives us a chance to tell apart the bubble,
triangle, and box diagrams even in the deep squeezed limit.

Specifically, we can rewrite the nonlocal CC signals in all three diagrams as:

$\displaystyle\lim_{k_{s}\rightarrow
0}\Big{[}\mathcal{T}_{I}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=$
$\displaystyle~{}\mathcal{A}_{I}^{(0)}\displaystyle\frac{\,{k_{s}^{3}}\,}{k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}^{2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.}$
$\displaystyle=$
$\displaystyle~{}|\mathcal{A}^{(0)}_{I}|\displaystyle\frac{\,{k_{s}^{3}}\,}{k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\cos\bigg{[}2\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\log\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}+\vartheta_{I}^{(0)}\bigg{]}$
(95)

where $I\in\\{\text{bubble},\text{triangle},\text{box}\\}$,
$\mathcal{A}_{I}^{(0)}$ denotes the (complex) amplitude of the signal, and
$\vartheta^{(0)}_{I}\equiv\text{Arg}\mathcal{A}_{I}^{(0)}$ denotes the phase
shift of the signal relative to the folded point $k_{s}=k_{12}=k_{34}$. Here
we use a superscript $(0)$ to highlight that this is the leading result in the
squeezed limit $k_{s}\rightarrow 0$. Corrections of higher orders in $k_{s}$
will be considered in the next section. Then, from the three expressions for
the nonlocal signals, (4.1), (4.2), and (4.3), we get the phases for the
nonlocal signals from these three graphs:

$\displaystyle\vartheta^{(0)}_{\text{bubble}}=$
$\displaystyle~{}\text{Arg}\bigg{\\{}-4^{-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma\Big{[}4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-\displaystyle\frac{\,{3}\,}{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}\Gamma^{2}\Big{[}\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}\bigg{\\}},$
(96) $\displaystyle\vartheta^{(0)}_{\text{triangle}}=$
$\displaystyle~{}\text{Arg}\bigg{\\{}4^{-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma\Big{[}2+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-\displaystyle\frac{\,{3}\,}{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}\Gamma^{2}\Big{[}\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}\bigg{\\}},$
(97) $\displaystyle\vartheta^{(0)}_{\text{box}}=$
$\displaystyle~{}\text{Arg}\bigg{\\{}-4^{-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\displaystyle\frac{\,{(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)^{4}}\,}{(3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)^{2}}\Gamma\Big{[}3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-\displaystyle\frac{\,{3}\,}{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}\Gamma^{2}\Big{[}\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}\bigg{\\}}.$
(98)

We plot these phases as functions of the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ in Fig. 11. The
differences among these phases are evident from the plot, and thus one can in
principle use the phase information to distinguish the signals from all
different 1-loop diagrams. We should note that the phase also depends on
particle species and interaction types in a complicated way. Here we are only
considering a particular type of coupling with a particular choice of loop
particles, but our method here directly applies to more general cases. Further
discussions about the use of signal phases can be found in [51].

Figure 11: The phases of nonlocal CC signals in the bubble, triangle, and box
diagrams as functions of the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$.

## 5 Nonlocal Signal for Finite Momentum Ratios

Up to this point in this work, we have only considered the explicit
expressions for the 1-loop nonlocal signals in a given soft limit, such as in
the factorization theorem stated in (8). For practical applications of CC
physics, we have to measure the CC signals for a range of configurations where
$P_{N}$ is nonzero but is parametrically smaller than all other independent
external momenta. For such configurations, the result of (8) will be corrected
by a factor of $[1+\mathcal{O}({P_{N}^{2}})]$. Among other things, such
corrections introduce a momentum-ratio dependence in the signal phase
$\vartheta^{(0)}$ defined in (4.4). As emphasized in [51], such running of
signal phase can provide important physical information about the particles
and interactions in a loop process. Therefore, it is phenomenologically useful
to push our factorization results to higher orders in the soft momentum.

In this section, we demonstrate how to go beyond the leading order results of
nonlocal CC signals by reconsidering all 1PI diagrams studied in Sec. 4. We
will calculate explicitly the $\mathcal{O}({k_{s}^{2}})$ corrections to the
nonlocal CC signals of 1-loop bubble, triangle, and box diagrams with four
external points. To achieve this goal, we can no longer quote the
factorization theorem (8) directly, since it is a statement in the leading
order in the squeezed limit. However, the proof in Sec. 2.2 can be easily
generalized to higher orders. So, before working on specific examples, we
describe how to obtain the higher order corrections in the general settings of
Sec. 2.

Recall that the nonlocal signals in the $P_{N}\rightarrow 0$ limit are
contributed by the signal poles (36). The choice
$n_{V}=\bar{n}_{1}=n_{N}=\bar{n}_{N+1}=0$ gives the leading result, while
other choices give corrections of higher orders in $P_{N}$. As emphasized in
Sec. 2, the cutting rule presented in (12) holds to all orders in the
expansion of $P_{N}$. Therefore, we can still separate the 1-loop nonlocal
signal into three parts at higher orders of $P_{N}$. With these points in
mind, let us summarize the sources of corrections to the leading-order result
(8):

  1. 1.

The contributions of higher order positive/negative poles in (36). From the
boundary OPE point of view, these contributions correspond to the higher order
terms in the late-time expansion (55)-(58) of the four mode functions in the
two soft lines.

  2. 2.

The higher-order terms in the loop momentum integral expansion, namely the
$\mathcal{O}({P_{N}})$ part in the first line of (2.2). From the boundary
viewpoint, they correspond to the higher terms of the OPEs (62) and (63).

  3. 3.

The last one is a little subtle. In the left tree graph (2.1) and the right
tree graph (2.1), the sum of all external momentum should be
$\sum\\{\mathbf{k}^{(\text{L})}\\}=-\sum\\{\mathbf{k}^{(\text{R})}\\}=\mathbf{P}_{N}$.
At the leading order, we can set $\mathbf{P}_{N}=\mathbf{0}$ and thus turn off
the momentum flow $\mathbf{P}_{N}$ carried by the bubble signal. At higher
orders, the soft yet nonzero momentum $\mathbf{P}_{N}$ could introduce more
complicated kinematic dependence in the nonlocal signal. Take the 4-point
triangle diagram in Fig. 7 as an example. In this case, we have $N=2$ and
$\mathbf{k}_{1}+\mathbf{k}_{2}\equiv\mathbf{k}_{s}\rightarrow\mathbf{0}$. At
the leading order, the left tree graph depends on only one scalar variable
$k_{12}$ since the constraint $\mathbf{k}_{1}+\mathbf{k}_{2}=\mathbf{0}$
implies $k_{1}=k_{2}=k_{12}/2$. However, for nonzero $\mathbf{k}_{s}$, the
left tree graph can depend on three independent variable $k_{12}$,
$\bar{k}_{12}\equiv k_{1}-k_{2}$, and $k_{s}$. Now, the triangle inequality
requires $|\bar{k}_{12}|\leq k_{s}$. Therefore, the new dependences on $k_{s}$
and $\bar{k}_{12}$ must be at least of order $\mathcal{O}({k_{s}})$.

Taking all above contributions into account, we can write down an expression
which generalizes the factorization theorem (8) to all orders in $P_{N}$:

$\displaystyle\mathcal{T}\big{(}\\{\mathbf{k}\\}\big{)}=$
$\displaystyle\sum_{m}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{V}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}\bar{s}_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{N}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}\bar{s}_{N+1}}\,}{2\pi\mathrm{i}}\mathcal{T}_{m}^{\text{(L)}}\Big{(}\\{\mathbf{k}^{\text{(L)}}\\},\\{s\\}\Big{)}\mathcal{T}_{m}^{\text{(R)}}\Big{(}\\{\mathbf{k}^{\text{(R)}}\\},\\{s\\}\Big{)}\mathcal{B}_{m}\Big{(}P_{N},\\{s\\}\Big{)}$
$\displaystyle+\text{terms analytic in $P_{N}$ as $P_{N}\rightarrow 0$},$ (99)

where $\\{s\\}\equiv\\{s_{V},\bar{s}_{1},s_{N},\bar{s}_{N+1}\\}$. Here the
subscript $m$ denotes the each term in the Taylor expansion of the loop
integral, and the integral over Mellin variables should be finished by summing
residues at all nonlocal poles (36), which are poles of the factor (2.2)
contained in the $m$-th order “bubble signal” $\mathcal{B}_{m}$.

Specialized to the 1PI 1-loop 4-point functions studied in Sec. 4, we again
consider the $s$-channel soft limit $k_{s}\rightarrow 0$ and parameterize the
full nonlocal signal in the following form:

$\Big{[}\mathcal{T}_{I}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=\displaystyle\frac{\,{k_{s}^{3}}\,}{k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\bigg{[}\mathcal{A}_{I}(\\{\mathbf{k}\\})\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}^{2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+\text{c.c.}\bigg{]},$ (100)

where again $I\in\\{\text{bubble},\text{triangle},\text{box}\\}$.
$\mathcal{A}_{I}$ is a dimensionless function of external momenta and can be
expanded around the squeezed limit $k_{s}\rightarrow 0$, whose leading term
reproduces the expression (4.4):

$\lim_{k_{s}\rightarrow
0}\mathcal{A}_{I}(\\{\mathbf{k}\\})=\mathcal{A}_{I}^{(0)}.$ (101)

We find it is convenient to choose the following set of independent momentum
magnitudes to express $\mathcal{A}_{I}$:

$\mathcal{A}_{I}(\\{\mathbf{k}\\})=\mathcal{A}_{I}(k_{12},k_{34},\bar{k}_{12},\bar{k}_{34},k_{s},k_{t}),$
(102)

where $\bar{k}_{12}\equiv k_{1}-k_{2}$ and $\bar{k}_{34}\equiv k_{3}-k_{4}$.
For the particles and couplings we are choosing in this work, there is no
$k_{t}$ dependence in the $s$-channel exchange diagrams. Also, the triangle
inequalities from the momentum conservations require that
$|\bar{k}_{12}|,|\bar{k}_{34}|\leq k_{s}$. Thus, when we treat $k_{s}$ as a
small expansion parameter, $\bar{k}_{12}$ and $\bar{k}_{34}$ are automatically
small as well. Furthermore, because of the exchange symmetries
$k_{1}\leftrightarrow k_{2}$ and $k_{3}\leftrightarrow k_{4}$, the complex
amplitude $\mathcal{A}_{I}$ depends on $\bar{k}_{12}$ and $\bar{k}_{34}$ only
through even powers such as $\bar{k}_{12}^{2}$ and $\bar{k}_{34}^{2}$. With
all these remarks made, in the following, we shall set $k_{1}=k_{2}$ and
$k_{3}=k_{4}$ to remove the dependences on $\bar{k}_{12}$ and $\bar{k}_{34}$,
just to keep the presentation simple.

###  5.1 Bubble diagram

Higher order corrections to the bubble diagram in Fig. 6 are simple: We did
not assume $\mathbf{k}_{s}=\mathbf{0}$ when calculating left and right tree
graphs in (74) and (75). Also, there is no $\mathcal{O}(k_{s})$ terms in the
loop integral (2.2), so the higher order corrections come only from the higher
order signal poles in (36). Therefore, we only need to sum over residues at
all the positive and negative poles in (36). The final result is a series of
momentum ratios $k_{s}/k_{12}$ and $k_{s}/k_{34}$, and has been worked out in
our previous work [51], which is equivalent to (4.1). Here we simply quote the
first correction, which can be obtained from (4.1) by keeping $n=0$ term in
the summation and expanding the two ${}_{2}\text{F}_{1}$ functions to linear
order in $r_{1}^{2}$ and $r_{2}^{2}$:

$\mathcal{A}_{\text{bubble}}=\mathcal{A}_{\text{bubble}}^{(0)}\bigg{[}1+\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{12}}+\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{34}^{2}}+\mathcal{O}(k_{s}^{4})\bigg{]},$
(103)

where $\mathcal{A}_{\text{bubble}}^{(0)}$ is the complex amplitude of the
leading-order nonlocal signal of the bubble diagram, which is given in (4.1),
and the coefficient $\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ is defined as:

$\alpha(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\equiv\displaystyle\frac{\,{(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(5+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{5+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}.$ (104)

###  5.2 Triangle diagram

Unlike the bubble diagram, all three types of corrections contribute to the
higher order terms of triangle diagram. In this subsection, we will study the
higher order terms of the triangle diagram in details, and calculate the
leading corrections to the squeezed limit result. In principle, we can treat
the three types of corrections separately. However, we will take another
approach, since the loop integral is simple enough (though more complicated
than the bubble diagram) such that we can derive the full nonlocal signal
associated with $k_{s}$.

#### Loop integral

Let us take a closer look at the loop momentum integral (27) for the triangle
diagram:

$\mathcal{L}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{3\bar{1}}}|\mathbf{q}+\mathbf{k}_{1}|^{-2s_{1\bar{2}}}|\mathbf{q}+\mathbf{k}_{s}|^{-2s_{2\bar{3}}}.$
(105)

This integral is calculated in App. B.2, and the nonanalytic part in $k_{s}$
can be expanded as a double series in $k_{s}/k_{1}$ and $1-k_{2}/k_{1}$ in the
squeezed limit $k_{s}\rightarrow 0$:

$\mathcal{L}=\displaystyle\frac{\,{1}\,}{(4\pi)^{3/2}}k_{s}^{3-2s_{3\bar{1}2\bar{3}}}k_{2}^{-2s_{1\bar{2}}}\sum_{m,\ell=0}^{\infty}\mathbf{P}_{m}^{\ell}(\\{s_{i}\\})\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{2m}\Big{(}1-\displaystyle\frac{\,{k_{2}}\,}{k_{1}}\Big{)}^{\ell}+\text{terms
analytic in $k_{s}$}.$ (106)

We can easily rewrite the above expression in terms of $k_{12}$,
$\bar{k}_{12}$ and $k_{s}$. However, if we focus on the symmetric
configuration $k_{1}=k_{2}$, only terms with $\ell=0$ survive and the loop
integral is simplified to:

$\mathcal{L}|_{k_{1}=k_{2}}=\displaystyle\frac{\,{1}\,}{(4\pi)^{3/2}}k_{s}^{3-2s_{3\bar{1}2\bar{3}}}k_{1}^{-2s_{1\bar{2}}}\sum_{m=0}^{\infty}\mathbf{P}_{m}(\\{s_{i}\\})\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{2m}+\text{terms
analytic in $k_{s}$},$ (107)

where

$\mathbf{P}_{m}(\\{s_{i}\\})=f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})\times\displaystyle\frac{\,{(-1)^{m}}\,}{m!}\Gamma\left[\begin{matrix}-m-\mbox{$\frac{\,{3}\,}{2}$}+s_{3\bar{1}2\bar{3}},m+\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{1}},m+\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{3}}\\\
2m+3-s_{3\bar{1}2\bar{3}},s_{3\bar{1}},s_{2\bar{3}}\end{matrix}\right],$ (108)

and $f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$ is a polynomial:

$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})=(s_{1\bar{2}})_{m}(3-s_{3\bar{1}2\bar{3}}-s_{1\bar{2}})_{m}.$
(109)

Similar to the case of bubble loop integral (2.2), the only left poles in
$\mathbf{P}_{m}(\\{s_{i}\\})$ integral come from the factor
$\Gamma(-m-3/2-s_{3\bar{1}2\bar{3}})$. In addition, the factors
$1/\Gamma(s_{3\bar{1}})$ and $1/\Gamma(s_{2\bar{3}})$ in (108) imply that the
cutting rule (12) holds for the two soft propagators at each order, therefore
we can simply sum over residues at nonlocal poles (36) to complete the
integral over $s_{3,\bar{1},2,\bar{3}}$. At the leading order in $k_{s}$, this
is exactly our factorization theorem. We note that, although
$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$ depends on these 4 Mellin
variables, this polynomial does not affect the pole structure, so the nonlocal
poles in this case are still given by (36). In fact, given that
$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$ is dependent on $s_{1\bar{2}}$, we
find it more proper to absorb the factor
$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$ into the definition of the left
tree graph. We will come back to this point below. With this
$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$ factor removed, we define the
$m$-th order bubble signal as:

$\displaystyle\mathcal{B}_{m}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{(-1)^{m}k_{s}^{3}}\,}{m!(4\pi)^{7/2}}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{2m}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{3}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{\bar{1}}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{\bar{3}}}\,}{2\pi\mathrm{i}}\,e^{\mathrm{i}\pi(s_{\bar{1}2}-s_{\bar{3}3})}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{2}\Big{)}^{-2s_{3\bar{1}2\bar{3}}}$
$\displaystyle\times\Gamma\left[\begin{matrix}-m-\mbox{$\frac{\,{3}\,}{2}$}+s_{3\bar{1}2\bar{3}},m+\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{1}},m+\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{3}}\\\
2m+3-s_{3\bar{1}2\bar{3}},s_{3\bar{1}},s_{2\bar{3}}\end{matrix}\right]\prod_{i=3,\bar{1},2,\bar{3}}\Gamma\Big{[}s_{i}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{i}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}\Big{]}.$ (110)

We also note that the leading term with $m=0$ recovers the expression (72).

#### Left and right tree graphs.

Next let us look at the two $m$’th order tree graphs,
$\mathcal{T}_{m}^{\text{(L)}}$ and $\mathcal{T}_{m}^{\text{(R)}}$, given in
(5). For the triangle loop, the corresponding two tree graphs are shown on the
right hand side of Fig. 7.

Similar to the leading result in Sec. 4.2, the right tree graph only receives
a polynomial of conformal time $(-\tau_{3})^{-2s_{\bar{3}3}}$:

$\displaystyle\mathcal{T}^{(\text{R})}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{3}k_{4}}\sum_{\mathsf{a}_{3}=\pm}(\mathrm{i}\mathsf{a}_{3})\int_{-\infty}^{0}{\mathrm{d}}\tau_{3}\,e^{\mathsf{a}_{3}\mathrm{i}k_{34}\tau_{3}}\times(-\tau_{3})^{3/2-2s_{\bar{3}}}(-\tau_{3})^{3/2-2s_{3}}$
$\displaystyle=$ $\displaystyle-\displaystyle\frac{\,{\sin(\pi
s_{\bar{3}3})}\,}{2k_{3}k_{4}}k_{34}^{-4+2s_{\bar{3}3}}\Gamma(4-2s_{\bar{3}3}),$
(111)

which is in fact independent of $m$, so we omit the subscript.

On the other hand, the left tree graph $\mathcal{T}_{m}^{\text{(L)}}$, is much
less trivial. As we mentioned above, here we choose to include the polynomial
$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$ in the definition of the left tree
graph. We remind the reader that this polynomial comes from the loop momentum
integral (108). Due to this extra polynomial factor, in general, the integral
over $s_{1}$ and $\bar{s}_{2}$ will not directly recover the propagator
$D^{(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k;\tau_{1},\tau_{2})$. Explicitly, the
expression for the $m$’th order left tree graph now reads:

$\displaystyle\mathcal{T}^{(\text{L})}_{m}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}^{2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}(-\mathsf{a}_{1}\mathsf{a}_{2})\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{-1/2-2s_{\bar{1}}}(-\tau_{2})^{-1/2-2s_{2}}e^{\mathsf{a}_{1}\mathrm{i}k_{1}\tau_{1}+\mathsf{a}_{2}\mathrm{i}k_{1}\tau_{2}}$
$\displaystyle\times\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{\bar{2}}}\,}{2\pi\mathrm{i}}\,f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k_{1};\tau_{1},\tau_{2};s_{1},s_{\bar{2}}).$
(112)

where $\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}$ denotes the Mellin integrand of the
PMB representation of the propagator
$D_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}$. The explicit
expressions for $\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}$ can be easily extracted from the PMB
representation (2.2):

$\displaystyle\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\pm\mp}(k;\tau_{1},\tau_{2};s_{1},s_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi}e^{\mp\mathrm{i}\pi(s_{1}-s_{2})}\Big{(}\displaystyle\frac{\,{k}\,}{2}\Big{)}^{-2s_{12}}(-\tau_{1})^{-2s_{1}+3/2}(-\tau_{2})^{-2s_{2}+3/2}$
$\displaystyle\times\Gamma\Big{[}s_{1}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{1}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{2}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{2}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}\Big{]},$ (113)
$\displaystyle\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\pm\pm}(k;\tau_{1},\tau_{2};s_{1},s_{2})=$
$\displaystyle~{}\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mp\pm}(k;\tau_{1},\tau_{2};s_{1},s_{2})\theta(\tau_{1}-\tau_{2})+\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\pm\mp}(k;\tau_{1},\tau_{2};s_{1},s_{2})\theta(\tau_{2}-\tau_{1}).$
(114)

Thus, in the absence of the polynomial
$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$, the Mellin integral in the second
line of (5.2) can be directly done and gives nothing but the massive
propagator $D_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k_{1};\tau_{1},\tau_{2})$. Now, with the presence of
$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$, we make use of the following
relation, which can be easily verified with the expressions in (5.2) and
(114):

$s_{1\bar{2}}\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k_{1};\tau_{1},\tau_{2};s_{1},s_{\bar{2}})=\mathcal{D}_{\tau_{1},\tau_{2}}\mathfrak{D}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k_{1};\tau_{1},\tau_{2};s_{1},s_{\bar{2}}),\quad\mathcal{D}_{\tau_{1},\tau_{2}}\equiv\displaystyle\frac{\,{1}\,}{2}(3-\tau_{1}\partial_{\tau_{1}}-\tau_{2}\partial_{\tau_{2}}).$
(115)

Consequently, the polynomial $f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})$ in the
integral (5.2) can be replaced by a differential operator
$f_{m}(\mathcal{D}_{\tau_{1},\tau_{2}},s_{3\bar{1}2\bar{3}})$. Then, we can
commute this operator with the integral over $s_{1},s_{\bar{2}}$. The integral
can then be directly done, yielding the original propagator
$D_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}$:

$\displaystyle\mathcal{T}^{(\text{L})}_{m}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}^{2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}(-\mathsf{a}_{1}\mathsf{a}_{2})\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{-1/2-2s_{\bar{1}}}(-\tau_{2})^{-1/2-2s_{2}}e^{\mathrm{i}\mathsf{a}_{1}k_{1}\tau_{1}+\mathrm{i}\mathsf{a}_{2}k_{1}\tau_{2}}$
$\displaystyle\times
f_{m}(\mathcal{D}_{\tau_{1},\tau_{2}},s_{3\bar{1}2\bar{3}})D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k_{1};\tau_{1},\tau_{2}).$ (116)

Having finished the Mellin integral, the last step is to perform an
integration by parts and move the $\tau_{1,2}$-derivatives in
$\mathcal{D}_{\tau_{1},\tau_{2}}$ to the front of exponential factor
$e^{\mathrm{i}\mathsf{a}_{1}k_{1}\tau_{1}+\mathrm{i}\mathsf{a}_{2}k_{1}\tau_{2}}$.
The resulting expression can then be expressed as a combination of seed
integrals $\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu|\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}$ defined in (82).

Now, with the expressions for the $m$’th order bubble signal $\mathcal{B}_{m}$
in (5.2), the right tree graph $\mathcal{T}^{\text{(R)}}$ in (5.2), and the
left tree graph $\mathcal{T}^{(\text{L})}_{m}$ in (5.2), we can calculate the
nonlocal signals in (5) to all orders in the $s$-channel momentum $k_{s}$.
Here we are only considering the symmetric configuration $k_{1}=k_{2}$. For
the asymmetric case, we can start from (106) and follow the same procedure.

Below, we do this calculation explicitly to find the first nontrivial
correction to the leading order nonlocal signal in the triangle diagram in
Fig. 7. As was made clear in the discussion above (5), these corrections come
from three different sources. First, we have the contributions from the higher
order nonlocal poles beyond $n_{3}=\bar{n}_{1}=n_{2}=\bar{n}_{3}=0$. Second,
we have the contributions of higher powers of $k_{s}$ from the loop momentum
integral, which correspond to $m=1$ terms in (5). Third, there are
contributions from asymmetric configuration which are of order
$\bar{k}_{12}^{2}$. For simplicity, we shall take the freedom of choosing
external momenta and set $k_{1}=k_{2}$, so that the third contributions of
$\mathcal{O}({\bar{k}_{12}^{2}})$ do not appear.

#### Corrections from higher order nonlocal poles.

To compute the leading corrections from the higher order nonlocal poles, we
can set $m=0$ in (5). Then, from (109), we get
$f_{m}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})=1$. In this case, the left tree
graph is simplified to:

$\displaystyle\mathcal{T}^{(\text{L})}_{0}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}^{2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}(-\mathsf{a}_{1}\mathsf{a}_{2})\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{-1/2-2s_{\bar{1}}}(-\tau_{2})^{-1/2-2s_{2}}e^{\mathrm{i}\mathsf{a}_{1}k_{1}\tau_{1}+\mathrm{i}\mathsf{a}_{2}k_{1}\tau_{2}}$
$\displaystyle\times D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k_{1};\tau_{1},\tau_{2})$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}k_{2}}\Big{(}\displaystyle\frac{\,{k_{12}}\,}{2}\Big{)}^{-4+2s_{\bar{1}2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}\mathcal{I}_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)-1/2-2s_{\bar{1}},-1/2-2s_{2}}(1,1),$ (117)

where the seed integral $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu|\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}$ is defined in (82), and we have
symmetrized expression with respect to $k_{1}\leftrightarrow k_{2}$. On the
other hand, the bubble signal (5.2) with $m=0$ is:

$\displaystyle\mathcal{B}_{0}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3}}\,}{(4\pi)^{7/2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{3}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{\bar{1}}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{\bar{3}}}\,}{2\pi\mathrm{i}}\,e^{\mathrm{i}\pi(s_{\bar{1}2}-s_{\bar{3}3})}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{2}\Big{)}^{-2s_{3\bar{1}2\bar{3}}}$
$\displaystyle\times\Gamma\left[\begin{matrix}-\mbox{$\frac{\,{3}\,}{2}$}+s_{3\bar{1}2\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{3}}\\\
3-s_{3\bar{1}2\bar{3}},s_{3\bar{1}},s_{2\bar{3}}\end{matrix}\right]\times\prod_{i=3,\bar{1},2,\bar{3}}\Gamma\Big{[}s_{i}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{i}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}\Big{]}.$ (118)

Recall that the nonlocal poles for the triangle diagram are:

$s_{3}=-n_{3}-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},\quad\bar{s}_{1}=-\bar{n}_{1}-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2},\quad
s_{2}=-n_{2}-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},\quad\bar{s}_{3}=-\bar{n}_{3}-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}.$ (119)

For the leading correction we can set one of the four integers $n_{3}$,
$\bar{n}_{1}$, $n_{2}$, $\bar{n}_{3}$ to be $1$, and the other three to be
$0$. Summing up the four choices, the result is:

$\mathcal{A}_{\text{triangle}}^{(0)}\bigg{[}1+\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{12}}+\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{34}^{2}}\bigg{]}+\text{c.c.},$
(120)

where $\mathcal{A}_{\text{triangle}}^{(0)}$ is the complex amplitude of the
nonlocal signal in the triangle 1-loop graph, which we calculated in Sec. 4.2
with the result given in (4.2). The coefficient $\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ was defined in
(104). In the current example, the piece proportional to $\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ is contributed by
the sum of the residues of two choices of signal poles: either $n_{3}=1$ with
all other $n$’s being 0, or $\bar{n}_{3}=1$ with all other $n$’s being 0.
Likewise, the coefficient $\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ is defined as:

$\displaystyle\beta(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)=$
$\displaystyle-\displaystyle\frac{\,{\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}[\mathcal{I}_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)3/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(1,1)+\mathcal{I}_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,3/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(1,1)]}\,}{(5+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}\mathcal{I}_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(1,1)}$
$\displaystyle=$ $\displaystyle~{}\displaystyle\frac{\,{(1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(3+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(5+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)},$ (121)

and it is contributed by the signal poles with either $\bar{n}_{1}=1$ with all
other $n$’s being zero, or $n_{2}=1$ with all other $n$’s being zero.

The result in (120) summarizes the contributions from the higher order
nonlocal poles of $m=0$ term in (5). As explained, this part should be
combined with the contributions from the leading poles (with all $n$’s being
0) in the $m=1$ term of (5), which we shall compute next.

#### Correction from the higher order terms in the loop momentum integral.

Now, let us consider the contributions from $m=1$ term in (5) specialized to
the triangle diagram. In this case, the bubble signal (5.2) becomes:

$\displaystyle\mathcal{B}_{1}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{(-1)k_{s}^{3}}\,}{(4\pi)^{7/2}}\Big{(}\displaystyle\frac{\,{2k_{s}}\,}{k_{12}}\Big{)}^{2}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{3}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{\bar{1}}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{\bar{3}}}\,}{2\pi\mathrm{i}}\,e^{\mathrm{i}\pi(s_{\bar{1}2}-s_{\bar{3}3})}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{2}\Big{)}^{-2s_{3\bar{1}2\bar{3}}}$
$\displaystyle\times\Gamma\left[\begin{matrix}-\mbox{$\frac{\,{5}\,}{2}$}+s_{3\bar{1}2\bar{3}},\mbox{$\frac{\,{5}\,}{2}$}-s_{3\bar{1}},\mbox{$\frac{\,{5}\,}{2}$}-s_{2\bar{3}}\\\
5-s_{3\bar{1}2\bar{3}},s_{3\bar{1}},s_{2\bar{3}}\end{matrix}\right]\times\prod_{i=3,\bar{1},2,\bar{3}}\Gamma\Big{[}s_{i}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{i}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}\Big{]},$ (122)

where we have substituted $k_{s}/k_{1}$ by $2k_{s}/k_{12}$.

On the other hand, taking $m=1$ in (109), we get
$f_{1}(s_{1\bar{2}},s_{3\bar{1}2\bar{3}})=(s_{1\bar{2}})(3-s_{3\bar{1}2\bar{3}}-s_{1\bar{2}})$.
Therefore, the left tree graph $\mathcal{T}^{(\text{L})}_{1}$ with symmetric
configuration $k_{1}=k_{2}$ becomes:

$\displaystyle\mathcal{T}^{(\text{L})}_{1}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}^{2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}(-\mathsf{a}_{1}\mathsf{a}_{2})\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{-1/2-2s_{\bar{1}}}(-\tau_{2})^{-1/2-2s_{2}}e^{\mathrm{i}\mathsf{a}_{1}k_{1}\tau_{1}+\mathrm{i}\mathsf{a}_{2}k_{1}\tau_{2}}$
$\displaystyle\times
f_{1}(\mathcal{D}_{\tau_{1},\tau_{2}},s_{3\bar{1}2\bar{3}})D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k_{1};\tau_{1},\tau_{2}).$ (123)

After integrating by parts, we obtain:

$\displaystyle\mathcal{T}_{1}^{(\text{L})}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}k_{2}}\Big{(}\displaystyle\frac{\,{k_{12}}\,}{2}\Big{)}^{-4+2s_{\bar{1}2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}\bigg{\\{}(2-s_{\bar{1}2})(1-s_{\bar{3}3})\mathcal{I}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)-1/2-2s_{\bar{1}},-1/2-2s_{2}}_{\mathsf{a}_{1}\mathsf{a}_{2}}(1,1)$
$\displaystyle+\displaystyle\frac{\,{\mathrm{i}}\,}{4}(3-2s_{\bar{1}2}+2s_{\bar{3}3})\Big{[}\mathsf{a}_{1}\mathcal{I}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)1/2-2s_{\bar{1}},-1/2-2s_{2}}_{\mathsf{a}_{1}\mathsf{a}_{2}}(1,1)+\mathsf{a}_{2}\mathcal{I}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)-1/2-2s_{\bar{1}},1/2-2s_{2}}_{\mathsf{a}_{1}\mathsf{a}_{2}}(1,1)\Big{]}$
$\displaystyle+\displaystyle\frac{\,{1}\,}{4}\Big{[}\mathcal{I}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)3/2-2s_{\bar{1}},-1/2-2s_{2}}_{\mathsf{a}_{1}\mathsf{a}_{2}}(1,1)+\mathcal{I}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)-1/2-2s_{\bar{1}},3/2-2s_{2}}_{\mathsf{a}_{1}\mathsf{a}_{2}}(1,1)\Big{]}$
$\displaystyle+\displaystyle\frac{\,{1}\,}{2}\mathsf{a}_{1}\mathsf{a}_{2}\mathcal{I}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)1/2-2s_{\bar{1}},1/2-2s_{2}}_{\mathsf{a}_{1}\mathsf{a}_{2}}(1,1)\bigg{\\}}.$
(124)

Now we can finish the four Mellin integrals over
$(s_{3},\bar{s}_{1},s_{2},\bar{s}_{3})$ by taking residues at the leading
poles, namely

$s_{3}=-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},\quad\bar{s}_{1}=-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2},\quad
s_{2}=-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},\quad\bar{s}_{3}=-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2},$ (125)

and the resulting correction is:

$\mathcal{A}_{\text{triangle}}^{(0)}\bigg{[}1+\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{12}}\bigg{]}+\text{c.c.},$
(126)

where the coefficient $\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ is defined as:

$\displaystyle\gamma(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)=$ $\displaystyle~{}\displaystyle\frac{\,{3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(5+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\times\displaystyle\frac{\,{\mathcal{T}_{1}^{(\text{L})}|_{k_{1}=k_{2}}}\,}{\mathcal{T}_{0}^{(\text{L})}|_{k_{1}=k_{2}}}\Big{|}_{s_{3}=\bar{s}_{1}=s_{2}=\bar{s}_{3}=-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu/2}$ $\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{2(1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(3+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)^{2}(3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)^{2}(5+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(5+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}.$ (127)

On the right hand side of the first equality, the first factor comes from the
change of the bubble signal $\mathcal{B}_{1}/\mathcal{B}_{0}$ at leading
positive poles. Combining the corrections (120) and (5.2), we obtain the
leading correction:

$\mathcal{A}_{\text{triangle}}|_{k_{1}=k_{2}}=\mathcal{A}_{\text{triangle}}^{(0)}\bigg{[}1+\Big{(}\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)+\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{)}\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{12}}+\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{34}}+\mathcal{O}(k_{s}^{4})\bigg{]}.$
(128)

As explained before, when $k_{1}\neq k_{2}$, there will be another correction
of order $\bar{k}_{12}^{2}$. Therefore, we can write down the correction for
general configuration as:

$\mathcal{A}_{\text{triangle}}=\mathcal{A}_{\text{triangle}}^{(0)}\bigg{[}1+\Big{(}\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)+\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{)}\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{12}}+\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{34}}+\mathcal{O}(\bar{k}_{12}^{2},k_{s}^{4})\bigg{]}.$
(129)

#### Pinched operator to the next order.

Now, we use the $\mathcal{O}({k_{s}^{2}})$ correction of the nonlocal signal
(128) from the triangle diagram to find the first order pinched operator
$\mathcal{O}_{\pm}^{(1)}$ defined in (92). For this purpose, let us consider
the pinched diagram on the right hand side of Fig. 9. If we use the pinched
operator at the leading order $\mathcal{O}_{\pm}^{(0)}$ for the left vertex,
use the original four-point vertex, namely the second term in (79), for the
right vertex, and compute the nonlocal signal to the second order in $k_{s}$,
we will get the following result for the complex amplitude:

$\mathcal{A}_{\text{triangle}}^{(0)}\bigg{[}1+\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{12}}+\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{34}}+\mathcal{O}(k_{s}^{4})\bigg{]},$
(130)

This differs from the triangle nonlocal signal at $\mathcal{O}({k_{s}^{2}})$,
showing that we should include a first order pinched operator
$\mathcal{O}_{\pm}^{(1)}$ which takes account of the difference between (130)
and (128):

$\mathcal{A}_{\text{triangle}}^{(0)}\Big{[}\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)+\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)-\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{]}\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{12}}.$ (131)

Since the only requirement is that $\mathcal{O}_{\pm}^{(1)}$ generates a
nonlocal signal as above, the form of $\mathcal{O}_{\pm}^{(1)}$ is not
uniquely fixed. For example, we can make the following choice, which turns out
to be convenient:

$\mathcal{O}_{\pm}^{(1)}=\displaystyle\frac{\,{1}\,}{4}\mathcal{C}_{\pm}^{(1)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\varphi^{\prime
2}\partial_{i}^{2}\sigma^{2},$ (132)

where the two additional derivatives on $\sigma$ just produces the right
scaling of the signal. Now, to find the coupling constant
$\mathcal{C}^{(1)}_{\pm}$, we should calculate the bubble diagram generated by
the operator (132) at the left vertex (together with the original operator
$a^{2}\varphi^{\prime 2}\sigma^{2}/4$ at the right vertex), to the leading
order. The calculation is very similar to the bubble case in Sec. 4.1: The
$\partial_{i}^{2}$ gives $-k_{s}^{2}$, and there is an extra $(-\tau_{1})^{2}$
in the left tree graph integral (74), which is fixed by the dilatation
symmetry of the problem. As a result, the factor
$\Gamma(4+2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ in (76) should be
replaced by $-\Gamma(6+2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)k_{12}^{-2}$.
Therefore, the operator $\mathcal{O}_{\pm}^{(1)}$ generates the following
nonlocal signal:

$\mathcal{C}_{+}^{(1)}\mathcal{A}^{(0)}_{\text{bubble}}\displaystyle\frac{\,{\Gamma(6+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{\Gamma(4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}^{2}}+\mathcal{O}(k_{s}^{4})=\displaystyle\frac{\,{\mathcal{C}_{+}^{(1)}}\,}{\mathcal{C}_{+}^{(0)}}\mathcal{A}^{(0)}_{\text{triangle}}\times(4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(5+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}^{2}}+\mathcal{O}(k_{s}^{4}).$
(133)

Now, by comparing (133) and (131), we can determine the coupling
$\mathcal{C}^{(1)}_{\pm}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$:

$\mathcal{C}^{(1)}_{+}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)=\displaystyle\frac{\,{\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)+\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)-\alpha(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{(4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(5+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\mathcal{C}_{+}^{(0)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu),\qquad\mathcal{C}^{(1)}_{-}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)=\mathcal{C}^{(1)}_{+}(-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu),$ (134)

and the effective pinched operator to the next order:

$\mathcal{O}_{\pm}=\displaystyle\frac{\,{1}\,}{4}\mathcal{C}_{\pm}^{(0)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)a^{2}\varphi^{\prime
2}\sigma^{2}+\displaystyle\frac{\,{1}\,}{4}\mathcal{C}_{\pm}^{(1)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\varphi^{\prime
2}\partial_{i}^{2}\sigma^{2}+\cdots.$ (135)

###  5.3 Box diagram

Given all the results for the triangle diagram, we do not have to compute the
box signal from scratch. Thanks to the pinch procedure discussed previously,
we can generate the nonlocal signal of a box diagram by the nonlocal signal of
a bubble diagram, with two pinched operators
$\mathcal{O}_{\pm}\sim\mathcal{O}_{\pm}^{(0)}+\mathcal{O}_{\pm}^{(1)}$
inserted, and the correction to the box diagram can be simply read out from
(129) as the following,

$\mathcal{A}_{\text{box}}=\mathcal{A}_{\text{box}}^{(0)}\bigg{[}1+\Big{(}\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)+\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{)}\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{12}}+\Big{(}\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)+\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{)}\displaystyle\frac{\,{k_{s}^{2}}\,}{k^{2}_{34}}+\mathcal{O}(\bar{k}_{12}^{2},\bar{k}_{34}^{2},k_{s}^{4})\bigg{]}.$
(136)

where the coefficients $\beta(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ and $\gamma(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ are given in (5.2)
and (5.2), respectively.

## 6 Conclusions and Outlook

Inflation correlators are on the one hand observables in Cosmological Collider
physics, and on the other hand dS counterparts of scattering amplitudes in
Minkowski spacetime. A better understanding of these objects is thus crucial
for both CC phenomenology and theoretical study of dS QFTs. A special example
in this regard is the nonlocal signal in massive inflation correlators. They
generate characteristic oscillatory signals in CC observables, and also
produce branch cut on the complex plane of external momenta. For this reason,
the theoretical question of the analytic structure of massive inflation
correlators is closely related to the phenomenological question of identifying
all nonlocal signals in a given process.

In this work, we make a first step towards understanding the nonlocal signal
in general inflation correlators. By working with arbitrary 1PI 1-loop
processes, we formulated a factorization theorem (8), and proved it using the
method of partial Mellin-Barnes representation. With the factorization
theorem, we can not only identify all the nonlocal signals in a given 1PI
1-loop graph, but also get its explicit expression. The theorem shows that, at
the leading order in the soft momentum, this expression factorizes into three
parts: the left tree graph (2.1), the right tree graph (2.1), and the bubble
signal (2.1). Our proof of the theorem made it clear that the nonlocal signal
could only appear in the configuration where two of the internal momenta
become soft simultaneously. As a byproduct of the proof, we found that the
time orderings between the endpoints of the two soft lines do not contribute
to the nonlocal signal, so we can simply “cut” these two soft lines when
computing the nonlocal signal, which is a natural generalization of the known
tree-level cutting rules to the 1-loop order. Furthermore, we can also “pinch”
an endpoint of a soft propagator with an endpoint of the other soft
propagator, to get an effective pinched operator. We also show that the
factorization theorem cam be intuitively understood from a boundary OPE point
of view: The two soft lines are effectively closed to the future boundary of
dS, so that the corresponding four massive bulk operators can be pushed to the
boundary. As a result, the two soft propagators are automatically cut, and we
can do the OPE and pinch the remaining bulk operators.

With the help of the factorization theorem, we calculated the leading nonlocal
signals of all possible 1-loop 4-point correlators, including the bubble,
triangle, and box diagrams. As guaranteed by the theorem, the loop diagrams
are factorized to simpler tree graphs, together with a universal bubble signal
that contains the nonanalyticity. For 4-point correlators we considered, the
tree graphs from the factorizations are 2-point functions of inflaton, either
contact or with an exchange of a massive scalar. The contact 2-point
correlators are easy to calculate, and the closed-form expressions for the
single-exchange case have been derived in our earlier work [96]. It is then
straightforward to construct nonlocal signals for loop diagrams by these
building blocks. Our results are intuitive: The coupling constant in the
effective pinched operator scales as $1/m^{2}$ at large mass, which meets the
expectation from the EFT viewpoint. We also emphasize that both the
factorization theorem and the cutting rule still hold at higher orders, and we
carefully calculated the leading corrections to those nonlocal signals for
illustration. These corrections can be also regarded as the contributions of
subleading effective pinched operators.

There are many open questions to be explored along this direction.

First, in this work, we have only considered the simple case with massive
scalar fields running in the loop. It would be useful to generalize our
analysis to include spinning fields and also to include dS boost breaking
effects. This generalization is particularly relevant to CC phenomenology. The
reason is that the nonlocal signal generated by a single dS covariant scalar
of mass $m\gtrsim H$ suffers from the Boltzmann suppression $e^{-\pi m}$. For
a 1-loop process where we have two soft lines contributing nonlocal signals
simultaneously, there would be double suppression $\propto e^{-2\pi m}$.
Therefore, seeing dS covariant loop signal is challenging for cosmological
observations. While this problem can be partially circumvented in a dS
covariant way by considering nonminimal scenarios such as curvaton mechanism
or modulated reheating [28, 33], a natural solution within the minimal
inflation setup is to consider dS boost breaking dispersions and interactions.
As has been emphasized in the literature [22, 30, 31, 35, 39, 79, 49, 51], in
this case, one can make use of the boost-breaking helical chemical potential
to remove the Boltzmann suppression. As a result, the nonlocal signals in such
scenarios can be naturally large even for loop process. Although, due to the
loss of full dS isometry, the calculation of inflation correlators or just
(local or nonlocal) signals becomes much more difficult in this case, our main
tool used in this work, namely the PMB representation, does not explicitly
rely on the existence of full dS isometries. Our previous works [51, 78] also
showed that the PMB representation is particularly useful to study dS boost
breaking correlators. Therefore, we expect that our results in this paper can
be directly generalized to those boost breaking cases.

Second, as we mentioned in the Introduction, besides the nonlocal signal,
there is another class of nonanalyticity of inflation correlators, called the
local signal, which shows up as branch cuts in the soft limit of partial
energy sum instead of partial momentum sum. The local signal is also an
observable of CC physics. It would be interesting to do a similar analysis to
identify all local signals in a 1-loop graph and find their explicit
expressions in the corresponding soft limits. It would also be interesting to
explore the loop counterparts of other types of nonanalyticities in a tree
correlator, such as the total energy pole and the partial energy poles.

Finally, it would be interesting to generalize our result to arbitrary loop
order. We leave all these open questions for future studies.

#### Acknowledgments.

We thank Hayden Lee and Yi Wang for useful discussions. This work is supported
by the National Key R&D Program of China (2021YFC2203100), NSFC under Grant
No. 12275146, an Open Research Fund of the Key Laboratory of Particle
Astrophysics and Cosmology, Ministry of Education of China, and a Tsinghua
University Initiative Scientific Research Program.

## Appendix

##  Appendix A Mathematical Functions

Here we collect some mathematical functions and compact notations frequently
used in this work.

First, we denote the products of Euler $\Gamma$ functions in the following
compact way:

$\displaystyle\Gamma\left[z_{1},\cdots,z_{m}\right]\equiv$
$\displaystyle~{}\Gamma(z_{1})\cdots\Gamma(z_{m}),$ (137)
$\displaystyle\Gamma\left[\begin{matrix}z_{1},\cdots,z_{m}\\\
w_{1},\cdots,w_{n}\end{matrix}\right]\equiv$
$\displaystyle~{}\displaystyle\frac{\,{\Gamma(z_{1})\cdots\Gamma(z_{m})}\,}{\Gamma(w_{1})\cdots\Gamma(w_{n})}.$
(138)

Second, the Pochhammer symbol is often used:

$\displaystyle(z)_{n}\equiv\Gamma\left[\begin{matrix}z+n\\\
z\end{matrix}\right].$ (139)

Third, the generalized hypergeometric function ${}_{p}\mathrm{F}_{q}$ is
defined by the following series when the series is convergent, and is defined
by analytic continuation outside the convergence reigion:

$\displaystyle{}_{p}\mathrm{F}_{q}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{matrix}\middle|z\right]=\sum_{n=0}^{\infty}\displaystyle\frac{\,{(a_{1})_{n}\cdots(a_{p})_{n}}\,}{(b_{1})_{n}\cdots(b_{q})_{n}}\displaystyle\frac{\,{z^{n}}\,}{n!}.$
(140)

We sometimes also use the following dressed hypergeometric function:

${}_{p}\mathcal{F}_{q}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{matrix}\middle|z\right]=\Gamma\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{matrix}\right]{}_{p}\mathrm{F}_{q}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{matrix}\middle|z\right].$ (141)

Finally, the PMB representation of the inflation correlator in this work draws
heavily use of the Mellin-Barnes representation of the Hankel functions:

$\text{H}_{\nu}^{(j)}(az)=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(az/2)^{-2s}}\,}{\pi}e^{(-1)^{j+1}(2s-\nu-1)\pi\mathrm{i}/2}\Gamma\Big{[}s-\displaystyle\frac{\,{\nu}\,}{2},s+\displaystyle\frac{\,{\nu}\,}{2}\Big{]}.~{}~{}~{}~{}(j=1,2)$
(142)

##  Appendix B Loop Momentum Integrals

Under the partial MB representation, the loop integral is resolved to the form
of (27):

$\mathcal{L}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{V\bar{1}}}|\mathbf{q}+\mathbf{P}_{1}|^{-2s_{1\bar{2}}}|\mathbf{q}+\mathbf{P}_{2}|^{-2s_{2\bar{3}}}\cdots|\mathbf{q}+\mathbf{P}_{V-1}|^{-2s_{(V-1)\mkern
2.0mu\overline{\mkern-2.0muV\mkern-2.0mu}\mkern 2.0mu}}.$ (143)

If we stick to the leading order in the squeezed limit $P_{N}\rightarrow 0$,
this integral is simplified to the bubble integral, as shown in Sec. 2.2.
However, if we want to consider the higher order corrections, which is the
case in Sec. 5, we shall treat this loop integral more carefully, taking
higher order corrections into account. In some simple cases, including the
bubble and the triangle diagrams, we can explicitly calculate (the nonlocal
part of) the loop integral to arbitrary order, as we will show below.

###  B.1 Bubble loop integral

In the calculation of the bubble signal in Sec. 2.2, we encounter the bubble
loop integral in the form of (2.2):

$\mathcal{L}_{\text{bubble}}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}\,|\mathbf{q}|^{-2a}|\mathbf{q}+\mathbf{k}_{s}|^{-2b}.$
(144)

To compute this integral, we use the standard method of Feynman
parameterization:

$|\mathbf{q}|^{-2a}|\mathbf{q}+\mathbf{k}_{s}|^{-2b}=\Gamma\left[\begin{matrix}a+b\\\
a,b\end{matrix}\right]\int_{0}^{1}{\mathrm{d}}\xi_{1}{\mathrm{d}}\xi_{2}\,\delta(1-\xi_{1}-\xi_{2})\displaystyle\frac{\,{\xi_{1}^{a-1}\xi_{2}^{b-1}}\,}{(\xi_{1}|\mathbf{q}|^{2}+\xi_{2}|\mathbf{q}+\mathbf{k}_{s}|^{2})^{a+b}}.$
(145)

Now we can make a shift to the variable $\mathbf{q}$ by introducing new
variable $\mathbf{p}$ such that:

$\mathbf{q}=\mathbf{p}-\xi_{2}\mathbf{k}_{s}.$ (146)

Together with the constraint $\xi_{1}+\xi_{2}=1$, the denominator of the
integrand in (145) becomes:

$(\xi_{1}|\mathbf{q}|^{2}+\xi_{2}|\mathbf{q}+\mathbf{k}_{s}|^{2})^{a+b}=(|\mathbf{p}|^{2}+\Delta)^{a+b},\qquad\Delta=\xi_{1}\xi_{2}k_{s}^{2},$
(147)

so we can easily perform the integral over $\mathbf{p}$:

$\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{p}}\,}{(2\pi)^{3}}\displaystyle\frac{\,{1}\,}{(|\mathbf{p}|^{2}+\Delta)^{a+b}}=\displaystyle\frac{\,{1}\,}{(4\pi)^{3/2}}\Gamma\left[\begin{matrix}a+b-\mbox{$\frac{\,{3}\,}{2}$}\\\
a+b\end{matrix}\right]\Delta^{3/2-a-b}.$ (148)

Then insert (145), (147) and (148) into the original integral (144), we
obtain:

$\mathcal{L}_{\text{bubble}}=\displaystyle\frac{\,{k_{s}^{3-2a-2b}}\,}{(4\pi)^{3/2}}\Gamma\left[\begin{matrix}a+b-\mbox{$\frac{\,{3}\,}{2}$}\\\
a,b\end{matrix}\right]\int_{0}^{1}{\mathrm{d}}\xi_{1}{\mathrm{d}}\xi_{2}\,\delta(1-\xi_{1}-\xi_{2})\xi_{1}^{1/2-b}\xi_{2}^{1/2-a}.$
(149)

The integral over Feynman parameters can be calculated explicitly:

$\int_{0}^{1}{\mathrm{d}}\xi_{1}{\mathrm{d}}\xi_{2}\,\delta(1-\xi_{1}-\xi_{2})\xi_{1}^{\alpha}\xi_{2}^{\beta}=\Gamma\left[\begin{matrix}1+\alpha,1+\beta\\\
2+\alpha+\beta\end{matrix}\right],$ (150)

so finally we obtain:

$\mathcal{L}_{\text{bubble}}=\displaystyle\frac{\,{k_{s}^{3-2a-2b}}\,}{(4\pi)^{3/2}}\Gamma\left[\begin{matrix}a+b-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}-a,\mbox{$\frac{\,{3}\,}{2}$}-b\\\
3-a-b,a,b\end{matrix}\right].$ (151)

###  B.2 Triangle loop integral

Next we consider a more nontrivial case, namely the loop integral of a
triangle diagram (105) which we considered in Sec. 5.2:

$\mathcal{L}_{\text{triangle}}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2a}|\mathbf{q}+\mathbf{k}_{s}|^{-2b}|\mathbf{q}+\mathbf{k}_{1}|^{-2c}.$
(152)

Similarly, we apply Feynman parameterization:

$\displaystyle|\mathbf{q}|^{-2a}|\mathbf{q}+\mathbf{k}_{s}|^{-2b}|\mathbf{q}+\mathbf{k}_{1}|^{-2c}=$
$\displaystyle~{}\Gamma\left[\begin{matrix}a+b+c\\\
a,b,c\end{matrix}\right]\int_{0}^{1}{\mathrm{d}}\xi_{1}{\mathrm{d}}\xi_{2}{\mathrm{d}}\xi_{3}\,\delta(1-\xi_{1}-\xi_{2}-\xi_{3})$
$\displaystyle\times\displaystyle\frac{\,{\xi_{1}^{a-1}\xi_{2}^{b-1}\xi_{3}^{c-1}}\,}{(\xi_{1}|\mathbf{q}|^{2}+\xi_{2}|\mathbf{q}+\mathbf{k}_{s}|^{2}+\xi_{3}|\mathbf{q}+\mathbf{k}_{1}|^{2})^{a+b+c}}.$
(153)

After the following change of variable:

$\mathbf{q}=\mathbf{p}-\xi_{2}\mathbf{k}_{s}-\xi_{3}\mathbf{k}_{1},$ (154)

the denominator of the integrand in (B.2) becomes:

$(\xi_{1}|\mathbf{q}|^{2}+\xi_{2}|\mathbf{q}+\mathbf{k}_{s}|^{2}+\xi_{3}|\mathbf{q}+\mathbf{k}_{1}|^{2})^{a+b+c}=|\mathbf{p}|^{2}+\Delta,\qquad\Delta=\xi_{1}\xi_{2}k_{s}^{2}+\xi_{1}\xi_{3}k_{1}^{2}+\xi_{2}\xi_{3}k_{2}^{2},$
(155)

where we have used the constraints $\xi_{1}+\xi_{2}+\xi_{3}=1$ and
$\mathbf{k}_{1}\cdot\mathbf{k}_{s}=(k_{s}^{2}+k_{1}^{2}-k_{2}^{2})/2$. Then
the integral over $\mathbf{p}$ can be again calculated:

$\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{p}}\,}{(2\pi)^{3}}\displaystyle\frac{\,{1}\,}{(|\mathbf{p}|^{2}+\Delta)^{a+b+c}}=\displaystyle\frac{\,{1}\,}{(4\pi)^{3/2}}\Gamma\left[\begin{matrix}a+b+c-\mbox{$\frac{\,{3}\,}{2}$}\\\
a+b+c\end{matrix}\right]\Delta^{3/2-a-b-c}.$ (156)

Insert (B.2), (155) and (156) into (152), we obtain:

$\mathcal{L}_{\text{triangle}}=\displaystyle\frac{\,{1}\,}{(4\pi)^{3/2}}\Gamma\left[\begin{matrix}a+b+c-\mbox{$\frac{\,{3}\,}{2}$}\\\
a,b,c\end{matrix}\right]\int_{0}^{1}{\mathrm{d}}\xi_{1}{\mathrm{d}}\xi_{2}{\mathrm{d}}\xi_{3}\,\delta(1-\xi_{1}-\xi_{2}-\xi_{3})\xi_{1}^{a-1}\xi_{2}^{b-1}\xi_{3}^{c-1}\Delta^{3/2-a-b-c}.$
(157)

#### Symmetric configuration

As we can see, the triangle loop integral depends on the shape of the triangle
formed by the three momenta $\mathbf{k}_{1}$, $\mathbf{k}_{2}$ and
$\mathbf{k}_{s}$, and this dependence is manifest in the expression of
$\Delta$ in (155). Let us consider a special configuration, namely we fix the
shape of the triangle by by setting $k_{1}=k_{2}$. The result, though much
simpler than the general case, gives the leading correction of the integral
(152) in arbitrary shapes (so long as $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$
are not anti-parallel).

Now that $k_{1}=k_{2}$, we obtain

$\Delta|_{k_{1}=k_{2}}=\xi_{1}\xi_{2}k_{s}^{2}+\xi_{3}(1-\xi_{3})k_{1}^{2}.$
(158)

The integral (157) is still difficult. However, we only focus on the nonlocal
part of this integral, so we can make use of the MB representation to resolve
the $k_{s}$-dependence in the factor $\Delta^{3/2-a-b-c}$. With the help of
the following identity:

$\displaystyle\frac{\,{1}\,}{(X_{0}+X_{1}+\cdots+X_{n})^{\lambda}}=\displaystyle\frac{\,{1}\,}{\Gamma(\lambda)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\prod_{\ell=1}^{n}\Big{[}\displaystyle\frac{\,{{\mathrm{d}}z_{\ell}}\,}{2\pi\mathrm{i}}X_{\ell}^{z_{\ell}}\Big{]}X_{0}^{-\lambda-
z_{1}-\cdots-
z_{n}}\Gamma(\lambda+z_{1}+\cdots+z_{n})\prod_{\ell=1}^{n}\Gamma(-z_{\ell}),$
(159)

we can rewrite $\Delta^{3/2-a-b-c}$ as:

$\Delta^{3/2-a-b-c}|_{k_{1}=k_{2}}=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}z}\,}{2\pi\mathrm{i}}\,[\xi_{3}(1-\xi_{3})k_{1}^{2}]^{z}[\xi_{1}\xi_{2}k_{s}^{2}]^{3/2-a-b-
c-z}\times\Gamma\left[\begin{matrix}a+b+c-\mbox{$\frac{\,{3}\,}{2}$}+z,-z\\\
a+b+c-\mbox{$\frac{\,{3}\,}{2}$}\end{matrix}\right],$ (160)

and therefore,

$\displaystyle\mathcal{L}_{\text{triangle}}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2a-2b-2c}}\,}{(4\pi)^{3/2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}z}\,}{2\pi\mathrm{i}}\,\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{-2z}\Gamma\left[\begin{matrix}a+b+c-\mbox{$\frac{\,{3}\,}{2}$}+z,-z\\\
a,b,c\end{matrix}\right]$
$\displaystyle\times\int_{0}^{1}{\mathrm{d}}\xi_{1}{\mathrm{d}}\xi_{2}{\mathrm{d}}\xi_{3}\,\delta(1-\xi_{1}-\xi_{2}-\xi_{3})\xi_{1}^{1/2-b-c-z}\xi_{2}^{1/2-a-c-z}\xi_{3}^{c-1+z}(1-\xi_{3})^{z}.$
(161)

The integral over Feynman parameters in the second line can be calculated
explicitly:

$\int_{0}^{1}{\mathrm{d}}\xi_{1}{\mathrm{d}}\xi_{2}{\mathrm{d}}\xi_{3}\,\delta(1-\xi_{1}-\xi_{2}-\xi_{3})\xi_{1}^{\alpha}\xi_{2}^{\beta}\xi_{3}^{\gamma}(1-\xi_{3})^{\delta}=\Gamma\begin{bmatrix}1+\alpha,1+\beta,1+\gamma,2+\alpha+\beta+\delta\\\
2+\alpha+\beta,3+\alpha+\beta+\gamma+\delta\end{bmatrix},$ (162)

so we obtain:

$\displaystyle\mathcal{L}_{\text{triangle}}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2a-2b-2c}}\,}{(4\pi)^{3/2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}z}\,}{2\pi\mathrm{i}}\,\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{-2z}\Gamma\left[\begin{matrix}a+b+c-\mbox{$\frac{\,{3}\,}{2}$}+z,-z\\\
a,b,c\end{matrix}\right]$
$\displaystyle\times\begin{bmatrix}\mbox{$\frac{\,{3}\,}{2}$}-a-c-z,\mbox{$\frac{\,{3}\,}{2}$}-b-c-z,c+z,3-a-b-2c-z\\\
3-a-b-2c-2z,3-a-b-c\end{bmatrix}.$ (163)

The last step is to integrate over Mellin variable $z$, using the residue
theorem. Since $k_{s}<k_{1}$, we should close the contour from left and sum
over residues at the following two sets of left poles:

$z=-m+\displaystyle\frac{\,{3}\,}{2}-a-b-c,\qquad z=-m-c,$ (164)

which come from the factors $\Gamma(a+b+c-3/2+z)$ and $\Gamma(c+z)$,
respectively. Furthermore, we find that the first set of poles contributes to
residues proportional to $k_{s}^{2m}$, which is analytic as $k_{s}\rightarrow
0$. Therefore, we are only concerned with contribution of the second set of
poles:

$\displaystyle\mathcal{L}_{\text{triangle}}|_{k_{1}=k_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2a-2b}k_{1}^{-2c}}\,}{(4\pi)^{3/2}}\sum_{m=0}^{\infty}\displaystyle\frac{\,{(-1)^{m}}\,}{m!}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{2m}\Gamma\left[\begin{matrix}-m+a+b-\mbox{$\frac{\,{3}\,}{2}$},m+c\\\
a,b,c\end{matrix}\right]$
$\displaystyle\times\begin{bmatrix}m+\mbox{$\frac{\,{3}\,}{2}$}-a,m+\mbox{$\frac{\,{3}\,}{2}$}-b,m+3-a-b-c\\\
2m+3-a-b,3-a-b-c\end{bmatrix}+\text{terms analytic in $k_{s}$}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2a-2b}k_{1}^{-2c}}\,}{(4\pi)^{3/2}}\sum_{m=0}^{\infty}\displaystyle\frac{\,{(-1)^{m}}\,}{m!}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{2m}(3-a-b-c)_{m}(c)_{m}$
$\displaystyle\times\begin{bmatrix}-m+a+b-\mbox{$\frac{\,{3}\,}{2}$},m+\mbox{$\frac{\,{3}\,}{2}$}-a,m+\mbox{$\frac{\,{3}\,}{2}$}-b\\\
2m+3-a-b,a,b\end{bmatrix}+\text{terms analytic in $k_{s}$}.$ (165)

#### General configuration

Now we consider the most general case, with $\Delta$ given in (155). Again we
can use the MB representation (159) and express $\Delta^{3/2-a-b-c}$ in (157)
as the following:

$\displaystyle\Delta^{3/2-a-b-c}=$
$\displaystyle\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}z_{2}}\,}{2\pi\mathrm{i}}\,(\xi_{1}\xi_{3}k_{1}^{2})^{z_{1}}(\xi_{2}\xi_{3}k_{2}^{2})^{z_{2}}(\xi_{1}\xi_{2}k_{s}^{2})^{3/2-a-b-
c-z_{1}-z_{2}}$
$\displaystyle\times\Gamma\begin{bmatrix}a+b+c-\mbox{$\frac{\,{3}\,}{2}$}+z_{1}+z_{2},-z_{1},-z_{2}\\\
a+b+c-\mbox{$\frac{\,{3}\,}{2}$}\end{bmatrix}.$ (166)

Then we can integrate out Feynman parameters using:

$\int_{0}^{1}{\mathrm{d}}\xi_{1}{\mathrm{d}}\xi_{2}{\mathrm{d}}\xi_{3}\,\delta(1-\xi_{1}-\xi_{2}-\xi_{3})\xi_{1}^{\alpha}\xi_{2}^{\beta}\xi_{3}^{\gamma}=\Gamma\begin{bmatrix}1+\alpha,1+\beta,1+\gamma\\\
3+\alpha+\beta+\gamma\end{bmatrix},$ (167)

and the integral (157) becomes:

$\displaystyle\mathcal{L}_{\text{triangle}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2a-2b-2c}}\,}{(4\pi)^{3/2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}z_{2}}\,}{2\pi\mathrm{i}}\,\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{-2z_{1}}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{2}}\Big{)}^{-2z_{2}}\Gamma\begin{bmatrix}a+b+c-\mbox{$\frac{\,{3}\,}{2}$}+z_{1}+z_{2},-z_{1},-z_{2}\\\
a,b,c\end{bmatrix}$
$\displaystyle\times\Gamma\begin{bmatrix}\mbox{$\frac{\,{3}\,}{2}$}-a-c-
z_{1},\mbox{$\frac{\,{3}\,}{2}$}-b-c-z_{2},c+z_{1}+z_{2}\\\
3-a-b-c\end{bmatrix}.$ (168)

Now we should integrate out $z_{1}$ and $z_{2}$, respectively. Assume
$k_{1}>k_{2}$, and let us integrate out $z_{2}$ first. Since $k_{2}<k_{s}$, we
should again close the contour from left and sum over residues at left poles:

$z_{2}=-m_{2}-a-b-c+\displaystyle\frac{\,{3}\,}{2}-z_{1},\qquad
z_{2}=-m_{2}-c-z_{1},$ (169)

where the first set contributes to analytic terms proportional to
$k_{s}^{2m_{2}}$, so we only consider the second set, which gives:

$\displaystyle\mathcal{L}_{\text{triangle}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2a-2b}k_{2}^{-2c}}\,}{(4\pi)^{3/2}}\sum_{m_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{m_{2}}}\,}{m_{2}!}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{2}}\Big{)}^{2m_{2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{2\pi\mathrm{i}}\,\Big{(}\displaystyle\frac{\,{k_{2}}\,}{k_{1}}\Big{)}^{-2z_{1}}$
$\displaystyle\times\Gamma\begin{bmatrix}-m_{2}+a+b-\mbox{$\frac{\,{3}\,}{2}$},-z_{1},m_{2}+c+z_{1}\\\
a,b,c\end{bmatrix}\times\Gamma\begin{bmatrix}\mbox{$\frac{\,{3}\,}{2}$}-a-c-
z_{1},m_{2}+\mbox{$\frac{\,{3}\,}{2}$}-b+z_{1}\\\ 3-a-b-c\end{bmatrix}$
$\displaystyle+\text{terms analytic in $k_{s}$}.$ (170)

Next we should integrate out $z_{1}$. Since we have assumed $k_{1}>k_{2}$, we
should still sum over residues at left poles:

$z_{1}=-m_{1}-m_{2}-c,\qquad
z_{1}=-m_{1}-m_{2}-\displaystyle\frac{\,{3}\,}{2}+b.$ (171)

This time we can complete the summation of $m_{1}$ to hypergeometric functions
with argument $k_{2}^{2}/k_{1}^{2}$:

$\displaystyle\mathcal{L}_{\text{triangle}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2a-2b}k_{2}^{-2c}}\,}{(4\pi)^{3/2}}\sum_{m_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{m_{2}}}\,}{m_{2}!}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{2m_{2}}$
$\displaystyle\times\bigg{\\{}\Big{(}\displaystyle\frac{\,{k_{2}}\,}{k_{1}}\Big{)}^{2c}\Gamma\left[\begin{matrix}-m_{2}-\mbox{$\frac{\,{3}\,}{2}$}+a+b,m_{2}+c,m_{2}+\mbox{$\frac{\,{3}\,}{2}$}-a,\mbox{$\frac{\,{3}\,}{2}$}-b-c\\\
a,b,c,3-a-b-c\end{matrix}\right]$
$\displaystyle\times{}_{2}\mathrm{F}_{1}\left[\begin{matrix}m_{2}+c,m_{2}+\mbox{$\frac{\,{3}\,}{2}$}-a\\\
-\mbox{$\frac{\,{1}\,}{2}$}+b+c\end{matrix}\middle|\displaystyle\frac{\,{k_{2}^{2}}\,}{k_{1}^{2}}\right]$
$\displaystyle+\Big{(}\displaystyle\frac{\,{k_{2}}\,}{k_{1}}\Big{)}^{3-2b}\Gamma\left[\begin{matrix}-m_{2}-\mbox{$\frac{\,{3}\,}{2}$}+a+b,m_{2}+\mbox{$\frac{\,{3}\,}{2}$}-b,-\mbox{$\frac{\,{3}\,}{2}$}+b+c,m_{2}+3-a-b-c\\\
a,b,c,3-a-b-c\end{matrix}\right]$
$\displaystyle\times{}_{2}\mathrm{F}_{1}\left[\begin{matrix}m_{2}+\mbox{$\frac{\,{3}\,}{2}$}-b,m_{2}+3-a-b-c\\\
\mbox{$\frac{\,{5}\,}{2}$}-b-c\end{matrix}\middle|\displaystyle\frac{\,{k_{2}^{2}}\,}{k_{1}^{2}}\right]\bigg{\\}}+\text{terms
analytic in $k_{s}$}.$ (172)

Finally, we can expand the above the result around $k_{2}\sim k_{1}$. Let us
write:

$\mathcal{L}_{\text{triangle}}=\displaystyle\frac{\,{k_{s}^{3-2a-2b}k_{2}^{-2c}}\,}{(4\pi)^{3/2}}\sum_{m,\ell=0}^{\infty}\mathbf{P}_{m}^{\ell}(a,b,c)\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}^{2m}\Big{(}1-\displaystyle\frac{\,{k_{2}}\,}{k_{1}}\Big{)}^{\ell}+\text{terms
analytic in $k_{s}$}.$ (173)

This is exactly the right hand side of (106). The coefficients
$\mathbf{P}_{m}^{\ell}$ can be determined by matching each term in the Taylor
expansion of (B.2). Below we list the expressions of $\mathbf{P}_{m}^{\ell}$
for $\ell=0,1,2$ and arbitrary $m$:

$\displaystyle\mathbf{P}_{m}^{0}(a,b,c)=$
$\displaystyle~{}\displaystyle\frac{\,{(-1)^{m}}\,}{m!}(c)_{m}(3-a-b-c)_{m}\Gamma\left[\begin{matrix}-m+a+b-\mbox{$\frac{\,{3}\,}{2}$},m+\mbox{$\frac{\,{3}\,}{2}$}-a,m+\mbox{$\frac{\,{3}\,}{2}$}-b\\\
2m+3-a-b,a,b\end{matrix}\right],$ (174)
$\displaystyle\mathbf{P}_{m}^{1}(a,b,c)=$
$\displaystyle~{}\displaystyle\frac{\,{m(2m+3-2a)-(2m+3-2b)c}\,}{2m+3-a-b}\times\mathbf{P}_{m}^{0}(a,b,c),$
(175) $\displaystyle\mathbf{P}_{m}^{2}(a,b,c)=$
$\displaystyle~{}\Big{[}m(2m+3-2a)\big{(}2m^{2}+(5-2a)m+1-a+b\big{)}$
$\displaystyle-(2m+3-2b)\big{(}4m^{2}+(10-4a)m+7-3a-b\big{)}c$
$\displaystyle+(2m+3-2b)(2m+5-2b)c^{2}\Big{]}\Big{/}\Big{[}2(2m+3-a-b)_{2}\Big{]}\times\mathbf{P}_{m}^{0}(a,b,c).$
(176)

One can also check that terms with $\ell=0$ recover the result (B.2) in the
symmetric case.

##  Appendix C One-Loop Four-Point Functions with Unequal Masses

In this appendix, we consider the leading results of all possible 1-loop
4-point functions, which has been considered in Sec. 4, but now with unequal
masses for all the loop modes. For convenience, we always set the mass
parameters of the two soft lines to be $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}$ and $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$, respectively.

###  C.1 Bubble diagram

First we consider the bubble diagram with the following interaction:

$\Delta{\mathscr{L}}=\displaystyle\frac{\,{1}\,}{2}a^{2}\varphi^{\prime
2}\sigma^{(1)}\sigma^{(2)},$ (177)

where $\sigma^{(1)}$ and $\sigma^{(2)}$ are distinct massive scalar fields,
with mass parameters $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}$ and $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$, respectively.
Following our factorization theorem (8), the bubble signal is given by (2.1):

$\mathcal{B}_{\mathsf{c}\mathsf{d}}=\displaystyle\frac{\,{k_{s}^{3}}\,}{(4\pi)^{7/2}}\Gamma\begin{bmatrix}-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},\mbox{$\frac{\,{3}\,}{2}$}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\\\
3+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}\end{bmatrix}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{2}\Big{)}^{2\mathrm{i}(\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathsf{d}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2})}.$ (178)

The left and the right tree graphs are simple:

$\displaystyle\mathcal{T}_{\mathsf{c}\mathsf{d}}^{(\text{L})}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}k_{2}}\sum_{\mathsf{a}_{1}=\pm}(\mathrm{i}\mathsf{a}_{1})\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}\,e^{\mathsf{a}_{1}\mathrm{i}k_{12}\tau_{1}}(-\tau_{1})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}(-\tau_{1})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$
$\displaystyle=$
$\displaystyle~{}\mathcal{P}^{(1)}_{\mathsf{c}\mathsf{d}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})\displaystyle\frac{\,{k_{12}^{-4-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}}\,}{4k_{1}k_{2}},$ (179)
$\displaystyle\mathcal{T}_{\mathsf{c}\mathsf{d}}^{(\text{R})}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{3}k_{4}}\sum_{\mathsf{a}_{2}=\pm}(\mathrm{i}\mathsf{a}_{2})\int_{-\infty}^{0}{\mathrm{d}}\tau_{2}\,e^{\mathsf{a}_{2}\mathrm{i}k_{34}\tau_{2}}(-\tau_{2})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}(-\tau_{2})^{3/2+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$
$\displaystyle=$
$\displaystyle~{}\mathcal{P}^{(1)}_{\mathsf{c}\mathsf{d}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})\displaystyle\frac{\,{k_{34}^{-4-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}}\,}{4k_{3}k_{4}},$ (180)

where the first order pinch coefficient now becomes:

$\mathcal{P}^{(1)}_{\mathsf{c}\mathsf{d}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2})\equiv
2\sin\Big{[}\mathrm{i}\displaystyle\frac{\,{\pi}\,}{2}(\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathsf{d}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})\Big{]}\Gamma(4+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}).$ (181)

Therefore, the leading nonlocal signal is:

$\lim_{k_{s}\rightarrow
0}\Big{[}\mathcal{T}_{\text{bubble}}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=\displaystyle\frac{\,{k_{s}^{3}}\,}{k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\bigg{[}\mathcal{A}_{\text{bubble},+}^{(0)}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{+}}+\mathcal{A}_{\text{bubble},-}^{(0)}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{-}}\bigg{]}+\text{c.c.},$ (182)

where

$\mathcal{A}_{\text{bubble},\pm}^{(0)}=-\displaystyle\frac{\,{4^{-1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm}}}\,}{(4\pi)^{7/2}}\sinh^{2}\Big{(}\displaystyle\frac{\,{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm}}\,}{2}\Big{)}(3+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm})\Gamma\Big{[}4+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm},\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},\displaystyle\frac{\,{3}\,}{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-\displaystyle\frac{\,{3}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\pm}\Big{]},$ (183)

and

$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm}\equiv\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\pm\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},$ (184)

are the positive and negative frequency of the signal, respectively.

###  C.2 Triangle diagram

As the second example, we consider the triangle diagram with the interaction:

$\Delta{\mathscr{L}}=a^{3}\varphi^{\prime}\sigma^{(1)}\sigma^{(3)}+a^{3}\varphi^{\prime}\sigma^{(2)}\sigma^{(3)}+\displaystyle\frac{\,{1}\,}{2}a^{2}\varphi^{\prime
2}\sigma^{(1)}\sigma^{(2)}.$ (185)

The bubble signal and the right tree graph are the same as the bubble case,
given by (178) and (C.1), respectively. The left tree graph is:

$\displaystyle\mathcal{T}_{\mathsf{c}\mathsf{d}}^{(\text{L})}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}k_{2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}(-\mathsf{a}_{1}\mathsf{a}_{2})\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{\tau_{1}^{2}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{\tau_{2}^{2}}e^{\mathsf{a}_{1}\mathrm{i}k_{1}\tau_{1}+\mathsf{a}_{2}\mathrm{i}k_{2}\tau_{2}}D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3})}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k_{1};\tau_{1},\tau_{2})$
$\displaystyle\times(-\tau_{1})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}(-\tau_{2})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$
$\displaystyle=$
$\displaystyle~{}\mathcal{P}^{(2)}_{\mathsf{c}\mathsf{d}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2};\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3})\displaystyle\frac{\,{k_{12}^{-4-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}}\,}{4k_{1}k_{2}},$ (186)

where the second order pinch coefficient can be expressed by the following
combination of seed integrals:

$\mathcal{P}^{(2)}_{\mathsf{c}\mathsf{d}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2};\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3})\equiv
2^{4+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}|\mathsf{a}_{1}\mathsf{a}_{2}}^{-1/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-1/2+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}(1,1).$ (187)

Therefore, the leading nonlocal signal for the triangle diagram is:

$\lim_{k_{s}\rightarrow
0}\Big{[}\mathcal{T}_{\text{triangle}}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=\displaystyle\frac{\,{k_{s}^{3}}\,}{k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\bigg{[}\mathcal{A}_{\text{triangle},+}^{(0)}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{+}}+\mathcal{A}_{\text{triangle},-}^{(0)}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{-}}\bigg{]}+\text{c.c.},$ (188)

where

$\displaystyle\mathcal{A}_{\text{triangle},\pm}^{(0)}=$
$\displaystyle~{}\displaystyle\frac{\,{\mathrm{i}2^{1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm}}}\,}{(4\pi)^{7/2}}\sinh\Big{(}\displaystyle\frac{\,{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm}}\,}{2}\Big{)}(3+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm})\Gamma\Big{[}\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},\displaystyle\frac{\,{3}\,}{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-\displaystyle\frac{\,{3}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\pm}\Big{]}$
$\displaystyle\times\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}|\mathsf{a}_{1}\mathsf{a}_{2}}^{-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-1/2\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}(1,1).$ (189)

We can also define the effective pinched operator:

$\mathcal{O}_{\mathsf{c}\mathsf{d}}=\mathcal{O}^{(0)}_{\mathsf{c}\mathsf{d}}+\cdots,\qquad\mathcal{O}_{\mathsf{c}\mathsf{d}}^{(0)}=\displaystyle\frac{\,{1}\,}{2}\mathcal{C}_{\mathsf{c}\mathsf{d}}^{(0)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2};\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3})a^{2}\varphi^{\prime 2}\sigma^{(1)}\sigma^{(2)},$ (190)

and the pinched coupling is given by:

$\mathcal{C}_{\mathsf{c}\mathsf{d}}^{(0)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2};\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3})\equiv\displaystyle\frac{\,{\mathcal{P}^{(2)}_{\mathsf{c}\mathsf{d}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2};\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3})}\,}{\mathcal{P}^{(1)}_{\mathsf{c}\mathsf{d}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2};\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3})}.$ (191)

Although the expression for $\mathcal{C}_{\mathsf{c}\mathsf{d}}^{(0)}$
involves generalized hypergeometric functions and thus is complicated, we find
it scales as $1/\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}^{2}$ at large $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}$ by fitting the
asymptotic behavior when $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}\rightarrow\infty$
and $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1,2}$
fixed. This behavior meets our expectation:

$\lim_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}\rightarrow\infty}\mathcal{A}_{\text{triangle},\pm}^{(0)}\sim\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}^{-2}\mathcal{A}_{\text{bubble},\pm}^{(0)}.$ (192)

###  C.3 Box diagram

Finally we consider the box diagram with the interaction:

$\Delta{\mathscr{L}}=a^{3}\varphi^{\prime}\sigma^{(1)}\sigma^{(3)}+a^{3}\varphi^{\prime}\sigma^{(2)}\sigma^{(3)}+a^{3}\varphi^{\prime}\sigma^{(1)}\sigma^{(4)}+a^{3}\varphi^{\prime}\sigma^{(2)}\sigma^{(4)}.$
(193)

Now the bubble signal is still (178). The left tree graph is given in (C.2),
and the right tree graph is similar, given by:

$\displaystyle\mathcal{T}_{\mathsf{c}\mathsf{d}}^{(\text{R})}=$
$\displaystyle~{}\mathcal{P}^{(2)}_{\mathsf{c}\mathsf{d}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2};\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{4})\displaystyle\frac{\,{k_{34}^{-4-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}}\,}{4k_{3}k_{4}},$ (194)

and thus the nonlocal signal is:

$\lim_{k_{s}\rightarrow
0}\Big{[}\mathcal{T}_{\text{box}}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=\displaystyle\frac{\,{k_{s}^{3}}\,}{k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\bigg{[}\mathcal{A}_{\text{box},+}^{(0)}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{+}}+\mathcal{A}_{\text{box},-}^{(0)}\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{k_{12}k_{34}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{-}}\bigg{]}+\text{c.c.},$ (195)

where

$\displaystyle\mathcal{A}_{\text{box},\pm}^{(0)}=$
$\displaystyle~{}\displaystyle\frac{\,{16}\,}{(4\pi)^{7/2}}\Gamma\begin{bmatrix}\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},\displaystyle\frac{\,{3}\,}{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-\displaystyle\frac{\,{3}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\pm}\\\
3+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{\pm}\end{bmatrix}$
$\displaystyle\times\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}|\mathsf{a}_{1}\mathsf{a}_{2}}^{-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-1/2\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}(1,1)\times\sum_{\mathsf{a}_{3},\mathsf{a}_{4}=\pm}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{4}|\mathsf{a}_{3}\mathsf{a}_{4}}^{-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-1/2\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}(1,1),$ (196)

and the large mass behaviors are:

$\displaystyle\lim_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{4}\rightarrow\infty}\mathcal{A}_{\text{box},\pm}^{(0)}\sim$
$\displaystyle~{}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{4}^{-2}\mathcal{A}_{\text{triangle},\pm}^{(0)},$ (197)
$\displaystyle\lim_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3},\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{4}\rightarrow\infty}\mathcal{A}_{\text{box},\pm}^{(0)}\sim$
$\displaystyle~{}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}^{-2}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{4}^{-2}\mathcal{A}_{\text{bubble},\pm}^{(0)}.$ (198)

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