# Inflation Correlators at the One-Loop Order: Nonanalyticity, Factorization,
Cutting Rule, and OPE

Zhehan Qin111Email<EMAIL_ADDRESS>and Zhong-Zhi Xianyu222Email:
<EMAIL_ADDRESS>
_Department of Physics, Tsinghua University, Beijing 100084, China_

###### Abstract

Inflation correlators with one-loop massive exchange encode rich information
about the dynamics of the massive loop particles. Their nonanalytic behavior
in certain soft limits leads to characteristic oscillatory pattern, which is
the leading signal of many particle models of cosmological collider physics.
In this work, we investigate systematically such nonanalyticity for arbitrary
one-particle-irreducible (1PI) one-loop correlators in various soft limits.
With the partial Mellin-Barnes representation, we present and prove a
factorization theorem and a cutting rule for arbitrary 1PI one-loop inflation
correlators, which is reminiscent of the on-shell cutting rule for flat-space
scattering amplitudes. We also show how to understand this factorization
theorem from the viewpoint of operator product expansion on the future
boundary. As an application of the one-loop factorization theorem, we derive
new analytic and exact formulae for nonlocal cosmological collider signals for
massive one-loop four-point inflation correlators of all possible 1PI
topologies, including the bubble, the triangle, and the box graphs. Finally,
we show how to push the computation of nonlocal signals to higher orders in
the momentum ratio.

###### Contents

  1. 1 Introduction
  2. 2 One-Loop Factorization Theorem and Cutting Rule
    1. 2.1 Factorization theorem
    2. 2.2 Proof
    3. 2.3 Discussions
  3. 3 A Boundary OPE Perspective
    1. 3.1 Pushing the soft lines to the future boundary
    2. 3.2 Boundary OPE and factorization theorem
  4. 4 Applications: Bubble, Triangle, and Box
    1. 4.1 Bubble diagram
    2. 4.2 Triangle diagram
    3. 4.3 Box diagram
    4. 4.4 Summary: pinched operator and phases
  5. 5 Nonlocal Signal for Finite Momentum Ratios
    1. 5.1 Bubble diagram
    2. 5.2 Triangle diagram
    3. 5.3 Box diagram
  6. 6 Conclusions and Outlook
  7. A Mathematical Functions
  8. B Loop Momentum Integrals
    1. B.1 Bubble loop integral
    2. B.2 Triangle loop integral
  9. C One-Loop Four-Point Functions with Unequal Masses
    1. C.1 Bubble diagram
    2. C.2 Triangle diagram
    3. C.3 Box diagram

## 1 Introduction

It has been emphasized in recent years that the observation of large-scale
inhomogeneity and anisotropy of our universe could be a unique chance to probe
fundamental particle physics at very high energies, a program known as the
cosmological collider (CC) physics. In typical scenarios, large-scale
perturbations were generated by quantum fluctuations when the primordial
universe was inflating at a Hubble rate $H$ up to $10^{14}$GeV [1]. Such a
fast expansion, together with other field evolutions, can trigger spontaneous
production of heavy particles with masses comparable to or much greater than
$H$. The produced particles can then leave distinct signals in the scalar or
tensor modes of primordial perturbations, called CC signals. See [2, 3, 4, 5,
6, 7, 8] for earlier works, [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59] for recent theoretical progresses, and [60, 61, 62, 63] for works related
to observations.

The central objects in the study of CC physics are the correlators of
primordial perturbations, which we call _inflation correlators_. According to
the inflation theory, the inflation correlators are expectation values of
products of quantum field operators living in the inflationary universe. Since
the inflationary spacetime is approximately de Sitter (dS), the inflation
correlators can be thought of as correlation functions of bulk quantum fields
in dS, but with their endpoints pinned at the future boundary. They form the
so-called meta-observables, and are useful objects for the study of quantum
field theories (QFTs) in dS [64]. Compared to the studies of scattering
amplitudes in Minkowski spacetime or boundary correlation functions in Anti de
Sitter (AdS) spacetime, our understanding of inflation correlators, or dS
correlators, is currently at a rather primitive stage. In recent years, partly
inspired by the prospects of future cosmological observations, the study of
inflation correlators has attracted lots of attentions, and many progresses
have been made [65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 51, 78,
79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96].

Among all kinds of inflation correlators, the ones mediated by massive fields
are of particular interest to CC physics. Heavy particles with mass $m\gtrsim
H$ created during inflation quickly become nonrelativistic due to cosmic
redshift, and their wavefunction thus oscillates like $e^{\pm\mathrm{i}mt}$.
Through interactions with nearly massless and long-lived scalar modes, the
oscillation $e^{\pm\mathrm{i}mt}$ is translated to the oscillations in the
_logarithm_ of external momentum ratios in certain soft limits, which we shall
call _CC signals_. Therefore, by searching for CC signals from inflation
correlators, we will have a chance to measure many properties of the
intermediate massive particles, including their mass, spin, chemical
potential, interaction type, etc.

As often happened, it is fruitful to view the inflation correlators as
functions of _complex_ external momenta. By proper analytic continuation of
these correlators, we can often develop useful techniques and gain new
insights [80, 81, 82, 86, 87, 92]. Specialized to correlators with massive
exchanges, the CC signals manifest themselves on the complex plane of
intermediate momenta as branch cuts emanating from the origin [65]. Therefore,
by a careful study of nonanalyticity of the inflation correlators, we can
develop new understanding and find new explicit results for CC signals.

The analytic properties and CC signals of tree-level correlators with single
massive exchange have been quite well understood [9, 65]. In comparison,
massive loop correlators are much less explored. Indeed, the first complete
analytical result for arbitrary massive 1-loop exchange was obtained very
recently in [97], where 1-loop bubble graphs of 3-point and 4-point
correlators are computed. For more complex loop topologies, such as triangle
and box graphs, currently there is no explicit result available. On the other
hand, loop correlators are of great importance for CC phenomenology. In many
particle models of CC physics, large CC signals are absent at the tree level,
and show up firstly at the 1-loop level [12, 13, 22, 28, 30, 31, 32, 35, 39,
44, 48, 49]. In these models, the CC signals are typically generated in
triangle or box diagrams. Therefore, there is a practical need for computing
these diagrams, especially their CC signals. For this purpose, it is essential
to have a good understanding of the nonanalyticity in generic massive 1-loop
inflation correlators.

In this work, we initiate a systematic study of the nonanalyticity of massive
inflation correlators beyond the tree level. We begin with 1-loop processes,
and consider the most general one-particle-irreducible (1PI) graphs with
arbitrary couplings and arbitrary number of external points. In this work, by
a 1PI graph, we mean that the graph cannot be broken into disconnected parts
by removing one single _bulk_ propagator.333The reason for restricting
ourselves to 1PI graphs is that we want to focus on the nonlocal signals from
the loop propagators. For 1-loop graphs that are not 1PI, there must exist
bulk tree propagators besides the loop propagators. These bulk tree
propagators could contain additional CC signals which are unnecessary
complications to us. An example of such graphs is shown in Fig. 2. To avoid
unnecessary complications, we further take the external lines to be either
massless scalars ($m^{2}=0$) or conformal scalars ($m^{2}=2H^{2}$), while the
internal loop lines can be arbitrary massive scalars. For definiteness, we
shall only consider principal scalars with $m>3H/2$. Generalization to lighter
scalars is straightforward. It is also possible to generalize the result here
to include spinning particles with dS boost breaking dispersions, which we
shall consider in a subsequent work.

#### Outline of this work.

Below we provide a summary of the main results of this work.

1\. Nonlocal signal (Sec. 2)

We identify a particular nonanalytic behavior of a general inflation
correlator, which we call the _nonlocal signal_. In short, a nonlocal signal
refers to the existence of a branch point at the origin of a complex momentum
variable $P$ in the form of complex powers, $P^{\pm\mathrm{i}\omega}$. Here
$P\equiv|\mathbf{P}|$ is the magnitude of a 3-momentum $\mathbf{P}$,
$\mathbf{P}$ is the sum of an arbitrary subset of external 3-momenta of a
given graph, and $\omega$ is a positive real number, which determines the
frequency of the (oscillatory) nonlocal signal. Very often, this frequency is
related to the mass of the intermediate particle, and thus is a very useful
quantity for CC phenomenology.

At this point, it is worth noting that the complex function
$P^{\mathrm{i}\omega}$ with $\omega>0$ gives rise to oscillatory signal on a
logarithmic scale of $P$ for physical region $P>0$. On the other hand, when we
take analytic continuation and consider the complex $P$-plane, the function
$P^{\mathrm{i}\omega}$ develops a branch cut on the negative real axis.444Of
course, the exact location of the branch cut is of no invariant meaning. What
is important here is that the function $P^{\mathrm{i}\omega}$ does not return
to its initial value when the argument $P$ goes a full circle around $P=0$. We
illustrate this behavior in Fig. 1. Thus we see that the nonanalyticity of an
inflation correlator is intimately tied to its oscillatory CC signals.
Therefore, a study of nonanalyticity of inflation correlators is not only
interesting from a purely theoretical point of view, but also important for
phenomenological applications. We note in passing that there are other types
of nonanalyticity of an inflation correlator, such as the so-called total
energy pole (the pole that appears when the _magnitude_ sum of all external
momenta goes to zero) and partial energy poles (the poles that appear when the
_magnitude_ sum of all external momenta of a _subgraph_ goes to zero). These
poles lie deeply in the unphysical region and thus cannot be directly probed
by physically measured correlators. Finally, there is another class of
nonanalyticity which we call _local signals_. It corresponds to branch points
on the complex plane of $E$ where $E$ is the magnitude sum of all external
momenta at a vertex. A local signal can also show up in physical data as an
oscillatory pattern, but it is more subtle to analyze than the nonlocal
signal. We plan to consider the local signal in a subsequent work.555The
terminology of nonlocal signal and local signal follows from previous works on
tree-level correlators. See, e.g., [88, 51].

Figure 1: The function $P^{\mathrm{i}\omega}$ with $\omega=4$ on the complex
$P$-plane. The colors show the phase of the function while the brightness
shows the magnitude. Both the logarithmic oscillation for $P>0$ and the branch
cut for $P<0$ are clearly visible.

2\. One-loop factorization theorem (Sec. 2)

We state and prove a factorization theorem for an arbitrary 1PI 1-loop graph
with massive exchanges. With this theorem, we can find all possible nonlocal
signals in a given diagram, and get explicit and analytic expressions for all
the nonlocal signals in the corresponding soft limits, without making any
approximations. Technically, our proof of the factorization theorem is made
possible by the technique of _partial Mellin-Barnes (PMB) representation_ we
recently introduced in [51, 78].

Let us briefly explain the main content of this theorem with Fig. 2 and Fig.
3. Essentially, the theorem says that a general 1PI 1-loop graph, such as Fig.
2, exhibits a nonlocal signal, whenever the momentum injected into the graph
through a _connected_ subset of vertices goes to zero, while all other
external momenta remain fixed. (On the contrary, there is no nonlocal signal
associated with the soft limit that the momentum injected through a
_disconnected_ subset of vertices goes to zero.) For instance, Fig. 2 contains
a nonlocal signal, when
$\mathbf{P}_{N}\equiv\mathbf{K}_{1}+\cdots\mathbf{+}\mathbf{K}_{N}\rightarrow\mathbf{0}$,
in the form of $P_{N}^{3\pm 2\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1})}$. We shall
show that this nonanalytic term is contributed by the parameter region of the
loop momentum where the two blue lines in Fig. 2 are _simultaneously_ soft.
Also, the explicit expression for this nonanalytic term factorizes into a
product of three pieces: a left tree graph
$\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(L)}}$, a right tree graph
$\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(R)}}$, and a bubble signal
$\mathcal{B}_{\mathsf{c}\mathsf{d}}$. See (8) for this explicit expression and
Fig. 3 for an illustration. As shown in Fig. 3, this factorization suggests
that we can cut the graph through the two blue loop lines to construct the
nonlocal signal. For this reason, we call the two blue lines in Fig. 3 _soft
lines_ or _cut lines_.

3\. One-loop cutting rule (Sec. 2)

The cut procedure described above can be formulated more precisely as a
_cutting rule_ for computing nonlocal signals in arbitrary 1PI 1-loop graph.
In short, the cutting rule states that we can replace the propagators for the
two soft lines by their real parts. As we shall show, the real part of a bulk
propagator is independent of its Schwinger-Keldysh indices, and thus is free
from time ordering. As a result, in the computation of nonlocal signals, there
is no integral that orders the two time variables of any cut line. We shall
see that the cutting rule is a direct corollary of our factorization theorem.

4\. Boundary OPE (Sec. 3)

It is possible to understand the factorization theorem from a boundary point
of view in terms of OPEs. The key observation here is that the nonlocal signal
in a 1-loop graph is generated when the two cut lines become soft
simultaneously. For a bulk propagator in an inflating universe, the soft limit
$k\rightarrow 0$ is often equivalent to the late-time limit $\tau\rightarrow
0$, since the momentum and the time are always combined in the form of
$k\tau$. Again let us explain this point with Fig. 2. When
$\mathbf{K}_{1}+\cdots+\mathbf{K}_{N}\rightarrow\mathbf{0}$, we can find the
nonlocal signal by bringing all four modes associated with the two soft (blue)
lines to the future boundary $\tau=0$. Then, we can perform two boundary OPE,
pinching the two modes at $\tau_{1}$ and $\tau_{N}$, and pinching the other
two modes at $\tau_{N+1}$ and $\tau_{V}$. This is exactly the origin of the
bubble signal $\mathcal{B}_{\mathsf{c}\mathsf{d}}$ in the factorization
theorem.

5\. Bubble, triangle, and box (Sec. 4)

We apply the factorization theorem to construct exact and analytical
expressions for nonlocal signals in all 4-point inflaton correlators with 1PI
1-loop topologies, including the bubble, triangle, and box diagrams. According
to the factorization theorem, we need three ingredients for this computation:
the left tree graph, the right tree graph, and the bubble signal. The
expression for the bubble signal is universal for all 1-loop processes and is
explicitly given in Sec. 2, while the expressions for the left and right tree
graphs were explicitly computed in our recent work [96]. The nonlocal signal
in the 1-loop bubble graph agrees with existing results, while the nonlocal
signals in the triangle and box graphs are new. With these results, we define
a _pinched operator_ , which shows that it is possible to pinch the hard
line(s) in the triangle or box graphs to compute the nonlocal signal. See Fig.
9.

6\. Subleading corrections (Sec. 5)

We provide an algorithm that can in principle generate corrections to nonlocal
signals to arbitrary given order in the soft momentum, and thereby generalize
the factorization theorem to arbitrary momentum configurations. We apply this
method to explicitly compute the $\mathcal{O}({k_{s}^{2}})$ corrections to the
leading nonlocal signals in the $s$-channel soft limit
$k_{s}\equiv|\mathbf{k}_{1}+\mathbf{k}_{2}|\rightarrow 0$ for all the three
types of 1-loop 4-point functions, and thereby find the higher-order pinched
operators.

Above we have briefly introduced the main results of this paper. The details
will be given in corresponding sections, and we will provide further
discussions on these results in Sec. 6. There are three appendices following
the main text. In App. A, we collect several frequently used mathematical
functions and shorthand notations. In App. B, we compute explicitly the loop
momentum integrals for the bubble loop and the triangle loop with the PMB
representation. Finally, in App. C, we compute the nonlocal signals at the
leading order in the soft momentum for all 1PI 1-loop 4-point correlators with
unequal loop masses.

#### Comparison with existing results.

The analytical properties of dS amplidutes have been under active studies in
recent years. The discontinuity of dS amplitudes has been studied in, e.g.,
[80, 82, 81, 83, 92, 72]. Many of these results were obtained for wavefunction
coefficients instead of correlation functions. In [88], a cutting rule for
extracting CC signals (including the nonlocal and local signals) from tree-
level inflation correlators was proposed which was motivated by intuitive
arguments in the bulk dS. This cutting rule was proved rigorously in [51, 78]
with PMB method. Compared with these results, our cutting rule applies to
massive loop correlators, and can be used to compute loop signals. As all
known cutting rules in both flat space and dS, our cutting rule cuts the
integrand instead of the whole loop integral. Therefore, the cutting rules
themselves are not enough to compute the whole signals in loop processes. In
comparison, our factorization theorem provides an explicit expression for all
nonlocal signals in an arbitrary 1PI 1-loop graph, and thus is more powerful
than the cutting rules themselves. Indeed, as we have shown, the cutting rule
shows up naturally as a corollary in our proof of the factorization theorem.

The factorization found in this work is reminiscent of the factorization of
scattering amplitudes in flat space. In the latter case, when the sum of a
subset of external 4-momenta hits the physical mass of an intermediate state,
the amplitude exhibits a pole due to the on-shell propagation of the
intermediate massive particle. In our case, the existence of the branch cut
associated with nonlocal signals can also be thought of as a consequence of
on-shell production of intermediate massive particles. The difference here is
that, due to the fast expansion, an on-shell massive particle does not have a
stable wavefunction, and thus it does not produce a pole in the physical
region of the momentum space. Instead, such on-shell particles produce softer
singularities, namely branch cuts, in the unphysical regions. In the
physically accessible regions, we see nonlocal (logarithmic) oscillations
instead of discontinuities.

In the study of dS amplitudes, there is another known factorization for tree-
level correlators, which shows up when the sum of the _magnitudes_ of a subset
of external momenta goes to zero, and is known as a partial energy pole [65].
We emphasize that our factorization works in a physical limit and is
physically accessible; It is not directly related to the partial-energy-pole
factorization, which is deeply inside the unphysical region.

#### Notations and Conventions.

Most of notations and conventions of this work are in line with the ones
adopted in our previous works on similar topics [51, 78, 96]. In particular,
the spacetime metric is
${\mathrm{d}}s^{2}=a^{2}(\tau)(-{\mathrm{d}}\tau^{2}+{\mathrm{d}}\mathbf{x}^{2})$
where $a(\tau)=-1/(H\tau)$ is the scale factor for an exponentially inflating
universe, and $\tau\in(-\infty,0)$ is the conformal time. In the following, we
always take $H=1$ for simplicity. We use bold letters such as $\mathbf{k}$ to
denote 3-momenta and the corresponding italic letter $k\equiv|\mathbf{k}|$ to
denote the magnitude of a 3-momentum. Following terminologies of the
literature, we sometimes call the magnitude of a 3-momentum with the term
_energy_ , although this terminology is not very precise. For many kinds of
variables with indices, we shall use a compact notation to denote their sum.
For instance, $k_{12}\equiv k_{1}+k_{2}$, $n_{1234}=n_{1}+n_{2}+n_{3}+n_{4}$,
etc. We shall make heavy use of Mellin variables such as $s_{i}$ and
$\bar{s}_{i}$. Unbarred and barred Mellin variables are independent; they have
no relations whatsoever. When summing barred and unbarred Mellin variables, we
shall use shorthand notation such as $s_{1\bar{2}}\equiv s_{1}+\bar{s}_{2}$,
$s_{i\bar{j}k\bar{\ell}}=s_{i}+s_{\bar{j}}+s_{k}+s_{\bar{\ell}}$. With this
shorthand in mind, we shall also use $s_{\bar{k}}$ and $\bar{s}_{k}$
interchangeably.

## 2 One-Loop Factorization Theorem and Cutting Rule

In this section, we study a general inflaton correlator mediated by arbitrary
number of massive scalar fields at one-loop order. For the phenomenological
applications in CC physics, one normally consider three-point or four-point
inflaton correlators. In this section, however, we shall be slightly more
general, and consider an arbitrary $B$-point correlator with $B\geq 4$, as
shown in Fig. 2. To be concrete, we shall assume that the external lines are
either conformal scalars $\varphi_{c}$ with mass $m^{2}=2$ or (nearly)
massless scalars such as the inflaton fluctuation $\varphi$. They are coupled
via local interactions to a bunch of massive particles $\sigma_{i}$
($i=1,\cdots,V$) of mass $m_{i}$. For definiteness and also for the direct
application in CC physics, we shall assume that all $\sigma_{i}$ are principal
scalars, namely, $m_{i}>3/2$. In this case, it is more convenient to define a
real and positive _mass parameter_ $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}\equiv\sqrt{m_{i}^{2}-9/4}$. We stress that it is not necessary to
stick to principal scalars. Indeed, our results can be directly generalized to
either lighter scalars with mass $m\in(0,3H/2)$ or massive spinning particles
with dS covariant dispersion relations. The couplings between the external
modes ($\varphi_{c}$ or $\varphi$) and the internal massive modes $\sigma_{i}$
can be rather arbitrary, including arbitrary but finite integer number of
spacetime derivatives. The only technical requirement is that these couplings
should be infrared-safe, in the sense that the SK time integral does not
contain any divergence in the late-time limit.

Below, we shall first present a factorization theorem, which concerns the
nonanalytic behavior of the one-loop graph $\mathcal{T}$ in Fig. 2 when the
total momentum $\mathbf{P}_{N}$ flowing into the graph through a _connected_
subset of vertices becomes soft, see Fig. 3. In such a limit, the
factorization theorem states that the leading nonanalytic term of the graph
factorizes into a product of three terms: the left tree graph, the right tree
graph, and a bubble signal. Together with the factorization theorem, we shall
also present a cutting rule, which states that one can freely replace all the
cut loop propagators (the two blue lines in Fig. 3) by their real parts when
computing the nonanalytic term of $\mathcal{T}$.

###  2.1 Factorization theorem

#### Specification of the 1-loop graph.

The specific 1-loop graph we shall consider is shown in Fig. 2. In this graph,
we have a loop consisting of $V$ massive scalar lines with mass parameters
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}$
($i=1,\cdots,V$). They are connected at $V$ vertices with external lines. We
label each of the $V$ vertices by a number $i=1,\cdots,V$. (In this work, we
do not consider two-point mixings.) At Vertex $i$, we have $B_{i}\geq 1$
external lines, each of which carries a 3-momentum $\mathbf{k}^{(i)}_{j}$ with
$j=1,\cdots,B_{i}$. For convenience, we shall denote the total 3-momentum
flowing into the Vertex $i$ by $\mathbf{K}_{i}$, and the sum of magnitude of
3-momenta $\mathbf{k}_{j}^{(i)}$ at the Vertex $i$ by $E_{i}$. Finally, we
shall use $\mathbf{P}_{n}$ to denote the total external 3-momentum flowing
into the graph through the first $n$ vertices (namely, from Vertex $i$ with
$i=1,\cdots,n$), and we shall use $B$ to denote the total number of external
legs. We summarize our kinematic speficification of the 1-loop diagram in the
following equations:

$\displaystyle\mathbf{K}_{i}=\sum_{j=1}^{B_{i}}\mathbf{k}_{j}^{(i)},$
$\displaystyle
E_{i}\equiv\sum_{j=1}^{B_{i}}\big{|}\mathbf{k}_{j}^{(i)}\big{|},$
$\displaystyle\mathbf{P}_{n}=\sum_{j=1}^{n}\mathbf{K}_{j},$ $\displaystyle
B=\sum_{j=1}^{V}B_{j}.$ $\displaystyle(i=1,\cdots,V)$ (1)

Clearly, $\mathbf{P}_{V}=\mathbf{0}$ as a consequence of momentum
conservation.

Figure 2: A general 1-loop inflation correlator. We follow the diagrammatic
notation of [19]. In particular, all the external (black) lines are either
conformal scalars or massless scalars, while the internal (gray and blue)
lines can be either identical or distinct massive scalar fields with $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}>0$
$(i=1,\cdots,V)$. The two blue lines denote the soft lines that contribute to
the nonlocal signal when $P_{N}\rightarrow 0$.

According to the diagrammatic rule in the Schwinger-Keldysh formalism, the
graph can be written as a product of bulk propagators for all internal massive
lines and bulk-to-boundary propagators for all external lines, integrated over
all (conformal) time variables $\tau_{i}$ at all vertices, and also over the
unconstrained loop momentum $\mathbf{q}$, and summed over all SK indices
$\mathsf{a}_{i}=\pm$. We refer the readers to [19] for a pedagogical review of
the diagrammatic rule in the SK formalism. We follow most of the diagrammatic
notations and conventions in [19]. In particular, the bulk-to-boundary
propagator of a massless scalar reads:

$G_{\mathsf{a}}(k;\tau)=\displaystyle\frac{\,{1}\,}{2k^{3}}(1-\mathrm{i}\mathsf{a}k\tau)e^{\mathsf{a}\mathrm{i}k\tau},$
(2)

while the bulk-to-boundary propagator for a conformal scalar reads:

$C_{\mathsf{a}}(k;\tau)=\displaystyle\frac{\,{\tau\tau_{f}}\,}{2k}e^{\mathsf{a}\mathrm{i}k\tau},$
(3)

in which we have introduced a final time cutoff $\tau_{f}$, since a conformal
scalar mode dies away in the late-time limit as $\tau$. On the other hand, the
bulk propagator $D_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})$ of a massive scalar with mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ comes in two types.
First, we have two opposite-sign propagators, also called the homogeneous
propagators. These are solutions to homogeneous scalar equation of motion at
each of the two time variables:666See Sec. 5 of [78] for a discussion of SK
propagators as solutions to scalar equations of motion.

$\displaystyle D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(\tau_{1}\tau_{2})^{3/2}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k\tau_{1})\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}(-k\tau_{2}),$
(4) $\displaystyle D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})=$ $\displaystyle~{}\Big{[}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})\Big{]}^{*}.$ (5)

Second, we have two same-sign propagators, also called the inhomogeneous
propagators, which are solutions to inhomogeneous scalar equation of motion
with $\delta$-function source:

$\displaystyle D_{\pm\pm}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})=$ $\displaystyle~{}D_{\mp\pm}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})\theta(\tau_{1}-\tau_{2})+D_{\pm\mp}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})\theta(\tau_{2}-\tau_{1}).$ (6)

As said above, we can consider arbitrary couplings for the Vertex $i$. In
general, this coupling has the schematic form
$(\partial^{\ell}\varphi^{B_{i}})(\partial^{m}\sigma_{i})(\partial^{n}\sigma_{i+1})$
$(\ell,m,n\geq 0)$, where $\sigma_{i}$ and $\sigma_{i+1}$ are the two massive
scalars meeting at Vertex $i$, and $\varphi$ can also be replaced by
$\varphi_{c}$. Here the derivatives can be either uncontracted temporal
derivative $\partial_{\tau}$ or fully contracted spatial derivative
$\partial_{i}$. As mentioned above, the only requirement is that these
couplings do not lead to any late-time divergences in the SK time integrals
over $\tau_{i}$. (On the other hand, the early-time limit of the SK time
integral is always well behaved due to the Bunch-Davies initial condition,
which is technically implemented by appropriate $\mathrm{i}\epsilon$
prescriptions. See [19]. Also, the loop momentum integral is also free of
infrared divergences due to the finite masses in the propagators. The
potential ultraviolet divergences of the loop momentum is irrelevant so long
as we are only concerned with the nonanalytic part of the graph; More below.)

With all points clarified above, it is straightforward to write down an
expression for the 1-loop graph in Fig. 2:

$\displaystyle\mathcal{T}(\\{\mathbf{k}\\})=$
$\displaystyle\sum_{\mathsf{a}_{1},\cdots,\mathsf{a}_{V}=\pm}\int_{-\infty}^{0}\prod_{\ell=0}^{V}\bigg{[}{\mathrm{d}}\tau_{\ell}\,(\mathrm{i}\mathsf{a}_{\ell})(-\tau_{\ell})^{p_{\ell}}\prod_{i=1}^{B_{\ell}}C_{\mathsf{a}_{\ell}}\Big{(}k_{i}^{(\ell)};\tau_{\ell}\Big{)}\bigg{]}\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}D_{\mathsf{a}_{V}\mathsf{a}_{1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}\Big{(}q;\tau_{V},\tau_{1}\Big{)}$ $\displaystyle\times
D_{\mathsf{a}_{1}\mathsf{a}_{2}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}\Big{(}|\mathbf{q}+\mathbf{P}_{1}|;\tau_{1},\tau_{2}\Big{)}\cdots
D_{\mathsf{a}_{V-1}\mathsf{a}_{V}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{V})}\Big{(}|\mathbf{q}+\mathbf{P}_{V-1}|;\tau_{V-1},\tau_{V}\Big{)}.$
(7)

Here, we use $\\{\mathbf{k}\\}$ to denote the collection of all external
momenta. For simplicity, we assume the external modes to be conformal scalars,
non-derivatively coupled at each vertex, with a time-dependent coupling that
yields a factor $(-\tau_{\ell})^{p_{\ell}}$ with a generic complex power
$p_{\ell}$. We also omit all coupling constants. Generalizations to massless
and/or derivatively coupled external modes are trivial. Furthermore, when some
of the internal lines are identical particles, there could be nontrivial
symmetry factors which we do not bother to include.

#### Nonlocal signal.

Recall that the question we want to address is in which soft kinematic
configuration of the 1-loop graph in Fig. 2 can a nonlocal CC signal show up,
and, what is the frequency and the size of the nonlocal signal. Recall that
the nonlocal signal in an inflation correlator $\mathcal{T}$ means an
oscillatory behavior in the logarithm of a momentum ratio:
$\mathcal{T}\sim\cos[\omega\log(P/Q)+\varphi]$ in the limit $P\rightarrow 0$,
where $P=|\mathbf{P}|$ and $\mathbf{P}$ is the sum of some external momenta
$\mathbf{k}^{(i)}_{j}$. On the other hand, $Q$ is introduced only to make the
argument of the logarithm dimensionless, and its specific form is inessential.
The oscillatory behavior is originated from terms in $\mathcal{T}$ with
noninteger power: $P^{\mathrm{i}\omega}+\text{c.c.}$ where $\omega$ is a real
number and is usually linked to the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$. Therefore,
searching for the nonlocal signals is equivalent to searching for external
momentum sums $\mathbf{P}$ such that the correlator $\mathcal{T}$ contains a
noninteger power term $\propto P^{\pm\mathrm{i}\omega}$ in the limit
$P\rightarrow 0$.

With the above explanations, now, we are at the position to state the main
theorem of this section, which we call the one-loop factorization theorem:

#### One-loop factorization theorem.

Consider the 1-loop graph shown in Fig. 2, and let $\mathbf{P}_{n}$ be as
defined in (1). Then, for each of $N=1,\cdots,V-1$,777When $N=1$ or $N={V-1}$,
it is required that $B_{1}>1$ or $B_{V}>1$, respectively, for our theorem to
hold. More explanations below. the 1-loop graph is nonanalytic in $P_{N}$ at
$P_{N}=0$ on the complex $P_{N}$-plane, in the form of noninteger powers
$P_{N}^{\pm 2\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1})}$ (when
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\neq\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}$) or
$P_{N}^{\pm 4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}$ (when $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}=\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}$). In the limit
$P_{N}\rightarrow 0$, each nonanalytic term with a given noninteger power
factorizes into a product of a _left tree correlator_
$\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(L)}}$, a _right tree correlator_
$\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(L)}}$, and a _bubble signal_
$\mathcal{B}_{\mathsf{c}\mathsf{d}}(P_{N})$:

$\displaystyle\lim_{P_{N}\rightarrow
0}\mathcal{T}\big{(}\\{\mathbf{k}\\}\big{)}=\sum_{\mathsf{c},\mathsf{d}=\pm}\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(L)}}\Big{(}\\{\mathbf{k}^{\text{(L)}}\\}\Big{)}\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(R)}}\Big{(}\\{\mathbf{k}^{\text{(R)}}\\}\Big{)}\mathcal{B}_{\mathsf{c}\mathsf{d}}(P_{N})+\text{terms
analytic in $P_{N}$}.$ (8)

where the summation variables $\mathsf{c},\mathsf{d}$ are subject to an
additional constraint $\mathsf{c}=\mathsf{d}$ if $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}=\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}$. In (8), the
left tree graph $\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(L)}}$ and the
right tree graph $\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(R)}}$ are
respectively given by:

$\displaystyle\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(L)}}\Big{(}\\{\mathbf{k}^{\text{(L)}}\\}\Big{)}=$
$\displaystyle\sum_{\mathsf{a}_{1},\cdots,\mathsf{a}_{N}=\pm}\int_{-\infty}^{0}\prod_{\ell={1}}^{N}\bigg{[}{\mathrm{d}}\tau_{\ell}\,(\mathrm{i}\mathsf{a}_{\ell})(-\tau_{\ell})^{p_{\ell}}\prod_{i=1}^{B_{\ell}}C_{\mathsf{a}_{\ell}}\Big{(}k_{i}^{(\ell)};\tau_{\ell}\Big{)}\bigg{]}$
$\displaystyle\times\prod_{n={1}}^{N-1}\Big{[}D_{\mathsf{a}_{n}\mathsf{a}_{n+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{n+1})}\big{(}P_{n};\tau_{n},\tau_{n+1}\big{)}\Big{]}(-\tau_{1})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}(-\tau_{N})^{3/2+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}},$ (9)
$\displaystyle\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(R)}}\Big{(}\\{\mathbf{k}^{\text{(R)}}\\}\Big{)}=$
$\displaystyle\sum_{\mathsf{a}_{N+1},\cdots,\mathsf{a}_{V}=\pm}\int_{-\infty}^{0}\prod_{\ell={N+1}}^{V}\bigg{[}{\mathrm{d}}\tau_{\ell}\,(\mathrm{i}\mathsf{a}_{\ell})(-\tau_{\ell})^{p_{\ell}}\prod_{i=1}^{B_{\ell}}C_{\mathsf{a}_{\ell}}\Big{(}k_{i}^{(\ell)};\tau_{\ell}\Big{)}\bigg{]}$
$\displaystyle\times\prod_{n={N+1}}^{V-1}\Big{[}D_{\mathsf{a}_{n}\mathsf{a}_{n+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{n+1})}\big{(}P_{n};\tau_{n},\tau_{n+1}\big{)}\Big{]}(-\tau_{N+1})^{3/2+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1}}(-\tau_{V})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{V}},$ (10)

where $\\{\mathbf{k}^{\text{(L)}}\\}$ denotes the set of external momenta
$\mathbf{k}_{j}^{(i)}$ with $j=1,\cdots,B_{i}$ and $i=1,\cdots,N$, and
$\\{\mathbf{k}^{\text{(R)}}\\}$ denotes the set of external momenta
$\mathbf{k}_{j}^{(i)}$ with $j=1,\cdots,B_{i}$ and $i={N+1},\cdots,V$. Note
that the two groups of momenta are separately conserved, i.e.,
$\sum\\{\mathbf{k}^{\text{(L)}}\\}=\sum\\{\mathbf{k}^{\text{(R)}}\\}=\mathbf{0}$,
as a consequence of the total momentum conservation plus the condition
$P_{N}\rightarrow 0$. The bubble signal
$\mathcal{B}_{\mathsf{c}\mathsf{d}}(P_{N})$ encodes all the nonanalytic
behavior of $P_{N}$ as $P_{N}\rightarrow 0$, and is given by:

$\displaystyle\mathcal{B}_{\mathsf{c}\mathsf{d}}(P_{N})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(4\pi)^{7/2}}\Gamma\begin{bmatrix}-\mbox{$\frac{\,{3}\,}{2}$}-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1},\mbox{$\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}$}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1},-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}\\\
3+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}\end{bmatrix}$
$\displaystyle\times
P_{N}^{3}\Big{(}\displaystyle\frac{\,{P_{N}}\,}{2}\Big{)}^{2\mathrm{i}(\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathsf{d}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1})}.$ (11)

Here $\Gamma[\cdots]$ is our shorthand notation for products of Euler $\Gamma$
functions. See App. A for the definition.

Figure 3: An illustration of the 1-loop factorization theorem. On the left
hand side, the two blue lines denote the soft lines, and the two cuts (dashed
orange lines) show the cutting rule: one can replace the two soft lines by
their real parts. On the right hand side, the three graphs denote the left
tree graph, the bubble signal, and the right tree graph, respectively. The
four hatched larger blobs on the right hand side denote the insertions with
complex powers of conformal time variables.

#### One-loop nonlocal cutting rule.

As a side product of our derivation of the factorization theorem, we also get
a cutting rule for obtaining the nonlocal signal in arbitrary 1-loop graph.
That is, in the computation of the nonanalytic terms of the one-loop graph
$\mathcal{T}$ to any orders in $P_{N}$ as $P_{N}\rightarrow 0$, one can cut
the two “soft” loop propagators, $D_{\mathsf{a}_{V}\mathsf{a}_{1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(q;\tau_{V},\tau_{1})$ and
$D_{\mathsf{a}_{N}\mathsf{a}_{N+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1})}(q;\tau_{N},\tau_{N+1})$, by replacing them with their real
parts:

$\displaystyle D_{\mathsf{a}_{V}\mathsf{a}_{1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}\Big{(}q;\tau_{V},\tau_{1}\Big{)}\Rightarrow$
$\displaystyle~{}\text{Re}\,D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}\Big{(}q;\tau_{V},\tau_{1}\Big{)};$ (12) $\displaystyle
D_{\mathsf{a}_{N}\mathsf{a}_{N+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1})}\Big{(}q;\tau_{N},\tau_{N+1}\Big{)}\Rightarrow$
$\displaystyle~{}\text{Re}\,D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1})}\Big{(}q;\tau_{N},\tau_{N+1}\Big{)}.$ (13)

Here, we deliberately remove the SK indices on the right hand sides of these
equations, to highlight the fact that the real part of a bulk massive
propagator is independent of SK labels. As a consequence, the time integrals
nesting over the two cut lines are automatically removed in the computation of
the nonanalytic terms, which justifies the name “cutting rule.” We emphasize
that, as will be made explicit in the following proof, the cutting rule
applies to the nonlocal CC signals to all orders in the expansion of the soft
momenta $P_{N}$.

###  2.2 Proof

In this subsection, we shall prove the theorem stated above. To help the
readers to keep track of what is going on, we first outline the main steps of
the proof before going into details. Readers uninterested in technical details
may only read the outline or skip the entire subsection.

#### Outline.

The proof comes in the following several steps.

  1. 1.

_Partial MB representation_ : We use the MB representation for all massive
loop lines [51, 78]. The MB representation turns all the momentum and time
dependences in the Hankel functions into powers, which greatly simplifies the
time integrals and the loop momentum integral.

  2. 2.

_Nested time integrals_ : We then consider the time integrals. Without
performing these integrations explicitly, we prove that all (nested) time
integrals only produce right poles in all Mellin variables.888Right poles mean
a series of poles at $s=an+b$ with $a>0$, $b\in\mathbb{C}$, and
$n\in\mathbb{N}$. Our prescription is that all Mellin integral contours should
be deformed to the left side of all right poles, even when $\text{Re}\,b<0$.
Likewise, left poles mean a semi-infinite series of poles emanating to the
left half of the complex $s$ plane, and the Mellin integral contour should be
deformed to the right side of all left poles.

  3. 3.

_Loop momentum integral_ : We then consider the loop integral over the
momentum $\mathbf{q}$. After finishing the loop integral, the result is fully
expressed in terms of external momenta $\mathbf{P}_{\ell}$. We show that any
potential nonanalytic terms in $P_{N}\rightarrow 0$ must come from a region
where both $\mathbf{q}$ and $\mathbf{q}+\mathbf{P}_{N}$ become soft, and this
region gives rise to a term $\propto P_{N}^{3-2s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}}$. Since we are
looking at $P_{N}\rightarrow 0$ limit, this term tells us two things: 1) We
should pick up “left poles” for the Mellin integrals over
$s_{V},\bar{s}_{1},s_{N},\bar{s}_{N+1}$; 2) The location of the left poles of
these four Mellin variables should be such that $2s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}$ takes noninteger
values. Only in this case the term $P_{N}^{3-2s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}}$ is truly
nonanalytic in $P_{N}$ at $P_{N}=0$. Coupled with Step 2, we see that all
poles from the time integrals can be discarded since they are right poles.

  4. 4.

_Nonlocal signal poles_ : We then collect all possible left poles with
$2s_{V\bar{1}N(\mkern 2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}\notin\mathbb{Z}$. We call them nonlocal signal poles as explained in
Step 3. The nonlocal signal poles are entirely from the MB representations of
the Hankel functions in the two soft propagators
$D_{\mathsf{a}_{V}\mathsf{a}_{1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(q;\tau_{V},\tau_{1})$ and
$D_{\mathsf{a}_{N}\mathsf{a}_{N+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1})}(q;\tau_{N},\tau_{N+1})$. Both the loop momentum integral and the
time integrals do not contribute to nonlocal signal poles.

  5. 5.

_Cutting rule_ : We then prove that, at the nonlocal signal poles identified
in Step 4, the two soft propagators $D_{\mathsf{a}_{V}\mathsf{a}_{1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(q;\tau_{V},\tau_{1})$ and
$D_{\mathsf{a}_{N}\mathsf{a}_{N+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1})}(q;\tau_{N},\tau_{N+1})$ are independent of any SK indices. They
can be freely replaced by their real parts. In particular, the time-ordering
$\theta$ functions become irrelevant in these real parts. Consequently, the
nested time integrals over $\tau_{V}$ and $\tau_{1}$, and over $\tau_{N}$ and
$\tau_{N+1}$, are automatically factorized.

  6. 6.

_Collecting residues_ : We finally collect the residues of the Mellin
integrand for the four “soft Mellin variables”
$(s_{V},\bar{s}_{1},s_{N},\bar{s}_{N+1})$ at the nonlocal signal poles at the
leading order in $P_{N}$, and finish the Mellin integral over all other Mellin
variables. The result just gives the leading nonanalytic contribution to the
1-loop graph $\mathcal{T}$ in the limit $P_{N}\rightarrow 0$, which has a
factorized expression as given in (8).

Below we carry out these steps in details.

#### Partial MB representation.

First, we use the partial MB representation of the 1-loop graph
$\mathcal{T}(\\{\mathbf{k}\\})$ in (2.1). See Sec. 3 of [78] for a more
comprehensive introduction to the partial MB representation. In short, we take
the MB representation of all internal massive propagators, with all external
lines unchanged. The MB representations for the two opposite-sign propagators
$D_{\pm\mp}^{(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})$ in (4) and (5) are given by:

$\displaystyle D_{\pm\mp}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi}\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,e^{\mp\mathrm{i}\pi(s_{1}-s_{2})}\Big{(}\displaystyle\frac{\,{k}\,}{2}\Big{)}^{-2s_{12}}(-\tau_{1})^{-2s_{1}+3/2}(-\tau_{2})^{-2s_{2}+3/2}$
$\displaystyle\times\Gamma\Big{[}s_{1}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{1}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{2}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{2}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}\Big{]},$ (14)

while the same-sign propagators are still given by (6) with the opposite-sign
propagators in (6) replaced by their MB representation in (2.2).

In the 1-loop graph $\mathcal{T}(\\{\mathbf{k}\\})$ in (2.1), for the massive
propagator $D_{\mathsf{a}_{i}\mathsf{a}_{j}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{j})}(p;\tau_{i},\tau_{j})$ connecting the two time variables $\tau_{i}$
and $\tau_{j}$, we designate two Mellin variables, $s_{i}$ for the mode at
$\tau_{i}$, and $\bar{s}_{j}$ at $\tau_{j}$, as shown in Fig. 4. With these
Mellin variables, we rewrite all $V$ internal propagators as in (2.2).
Therefore, we have a total of $2V$ Mellin integrals over
$(s_{1},\cdots,s_{V},\bar{s}_{1},\cdots,\bar{s}_{V})$. Note that $s_{\ell}$
and $\bar{s}_{\ell}$ are independent variables.

Figure 4: The Mellin variables for the loop modes.

#### Poles of time integrals.

After taking the partial MB representation, let us look at the integrals over
the time variables $\tau_{1},\cdots,\tau_{V}$ in (2.1). Using the explicit
form the of the bulk-to-boundary propagator for conformal scalars (3),
together with the MB representation for the loop massive propagator (2.2), we
observe that the time integrals have the following schematic form:

$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}\cdots{\mathrm{d}}\tau_{V}\,(-\tau_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}}\cdots(-\tau_{V})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern 2.0mu_{V}-2s_{V\mkern
2.0mu\overline{\mkern-2.0muV\mkern-2.0mu}\mkern
2.0mu}}e^{\mathrm{i}(\mathsf{a}_{1}E_{1}\tau_{1}+\cdots+\mathsf{a}_{V}E_{V}\tau_{V})}$
$\displaystyle\times\mathcal{N}(\tau_{1},\tau_{2})\mathcal{N}(\tau_{2},\tau_{3})\cdots\mathcal{N}(\tau_{V}-1,\tau_{V})\mathcal{N}(\tau_{V},\tau_{1}),$
(15)

with $\mkern 2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern 2.0mu_{\ell}\equiv
p_{\ell}+B_{\ell}+3/2$. Here $p_{\ell}$ comes from the explicit time
dependence in the time integrals in (2.1), $B_{\ell}$ counts the total number
of external conformal scalar modes at Vertex $\ell$, and $3/2$ comes from the
MB representation of the loop propagators. Here the “nesting” function
$\mathcal{N}(\tau_{i},\tau_{j})$ can take any of the three possible choices:
$1$, $\theta(\tau_{i}-\tau_{j})$, and $\theta(\tau_{j}-\tau_{i})$, depending
on the choices of SK indices at all vertices. By using relations such as
$\theta(\tau_{i}-\tau_{j})+\theta(\tau_{j}-\tau_{i})=1$, it is easy to see
that the time integral (2.2) can be organized into a linear combination of
many terms, and each term is a product of several monotonically ordered time
integrals. For example, a typical term could look like:

$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}\cdots{\mathrm{d}}\tau_{V}\,(-\tau_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}}\cdots(-\tau_{V})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern 2.0mu_{V}-2s_{V\mkern
2.0mu\overline{\mkern-2.0muV\mkern-2.0mu}\mkern
2.0mu}}e^{\mathrm{i}(\mathsf{a}_{1}E_{1}\tau_{1}+\cdots+\mathsf{a}_{V}E_{V}\tau_{V})}$
$\displaystyle\times\Big{[}\theta(\tau_{1}-\tau_{2})\cdots\theta(\tau_{j}-\tau_{j+1})\Big{]}\Big{[}\theta(\tau_{k}-\tau_{k+1})\cdots\theta(\tau_{\ell}-\tau_{\ell+1})\Big{]},$
(16)

where $1\leq j$, $j+1<k$, $k+1<\ell$, and $\ell+1\leq V$. In this particular
term, we identify two nested factors, one of which orders the time variables
such that $\tau_{1}>\tau_{2}\cdots>\tau_{j}+1$, and the other gives
$\tau_{k}>\cdots>\tau_{\ell+1}$. Since each nested factor can be integrated
separately, we only need to consider one nested integral at a time. In
particular, an integral over a time variable without any $\theta$ functions
can be regarded as a singly nested time integral.

So, without loss of generality, let us consider a nested integral of the
following form:

$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}\cdots{\mathrm{d}}\tau_{\ell}\,(-\tau_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}}\cdots(-\tau_{\ell})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{\ell}-2s_{\ell\bar{\ell}}}e^{\mathrm{i}\mathsf{a}(E_{1}\tau_{1}+\cdots+E_{\ell}\tau_{\ell})}\theta(\tau_{1}-\tau_{2})\cdots\theta(\tau_{\ell-1}-\tau_{\ell}).$
(17)

Notice that we have set all SK indices equal:
$\mathsf{a}_{1}=\cdots=\mathsf{a}_{\ell}\equiv\mathsf{a}$, since the nested
time integral arises only in this case. We would like to identify its pole
structure in all Mellin variables
$(s_{1},\cdots,s_{\ell},\bar{s}_{1},\cdots,\bar{s}_{\ell})$. Any pole of the
above integral must come from a divergence of the integral, so we are led to
ask when this integral diverges. First, the integrand is well behaved for any
finite value of $\tau_{i}>0~{}(i=1,\cdots,\ell)$, and therefore, the potential
divergences must come from two endpoints of the integral. Second, the integral
is well behaved at $\tau_{i}\rightarrow-\infty$ for physical values of
momenta, as a result of Bunch-Davies initial condition. Therefore, any pole of
the above integral must come from the limit when one or several of $\tau_{i}$
goes to 0.

So, let us focus on the integral in the limit $\tau_{i}\rightarrow 0$ by
introducing a small cutoff
$0<\epsilon\ll\min\\{E_{1}^{-1},\cdots,E_{\ell}^{-1}\\}$:

$\displaystyle\int_{-\epsilon}^{0}{\mathrm{d}}\tau_{1}\cdots{\mathrm{d}}\tau_{\ell}\,(-\tau_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}}\cdots(-\tau_{\ell})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{\ell}-2s_{\ell\bar{\ell}}}e^{\mathrm{i}\mathsf{a}(E_{1}\tau_{1}+\cdots+E_{\ell}\tau_{\ell})}\theta(\tau_{1}-\tau_{2})\cdots\theta(\tau_{\ell-1}-\tau_{\ell}).$
(18)

In this restricted region, we can Taylor expand all exponential factors, and
get

$\displaystyle\sum_{n_{1},\cdots,n_{\ell}=0}^{\infty}\displaystyle\frac{\,{(-\mathrm{i}\mathsf{a})^{n_{1}+\cdots+n_{\ell}}E_{1}^{n_{1}}\cdots
E_{\ell}^{n_{\ell}}}\,}{n_{1}!\cdots
n_{\ell}!}\int_{-\epsilon}^{0}{\mathrm{d}}\tau_{1}\cdots{\mathrm{d}}\tau_{\ell}\,(-\tau_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}+n_{1}}\cdots(-\tau_{\ell})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{\ell}-2s_{\ell\bar{\ell}}+n_{\ell}}$
$\displaystyle\times\theta(\tau_{1}-\tau_{2})\cdots\theta(\tau_{\ell-1}-\tau_{\ell}).$
(19)

Then the integral over $\tau_{1}$ can be done:

$\displaystyle\int_{-\epsilon}^{0}{\mathrm{d}}\tau_{1}\,(-\tau_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}+n_{1}}\theta(\tau_{1}-\tau_{2})=\displaystyle\frac{\,{(-\tau_{2})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}+n_{1}+1}}\,}{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}+n_{1}+1}.$ (20)

This is divergent when $\mkern 2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}+n_{1}=-1$. Therefore, we identify a set of poles for
$s_{1\bar{1}}$:

$s_{1\bar{1}}=\displaystyle\frac{\,{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}+n_{1}+1}\,}{2}.~{}~{}~{}(n_{1}=0,1,2,\cdots)$ (21)

These are right poles. Next, we insert the $\tau_{2}$ factor in the above
result back to the integrand and finish the $\tau_{2}$ integral:

$\displaystyle\int_{-\epsilon}^{0}{\mathrm{d}}\tau_{2}\,(-\tau_{2})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{12}-2s_{1\bar{1}2\bar{2}}+n_{12}+1}\theta(\tau_{2}-\tau_{3})=\displaystyle\frac{\,{(-\tau_{3})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{12}-2s_{1\bar{1}2\bar{2}}+n_{12}+2}}\,}{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{12}-2s_{1\bar{1}2\bar{2}}+n_{12}+2}.$ (22)

Likewise, we identify a second set of poles:

$s_{1\bar{1}2\bar{2}}=\displaystyle\frac{\,{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{12}+n_{12}+2}\,}{2}.~{}~{}~{}(n_{1},n_{2}=0,1,2,\cdots)$ (23)

These are again right poles.

Now it becomes clear that the above procedure can be carried through all
nested time variables, and the poles of the original nested time integral are
given by:

$\displaystyle s_{1\bar{1}}=\displaystyle\frac{\,{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}+n_{1}+1}\,}{2},~{}~{}s_{1\bar{1}2\bar{2}}=\displaystyle\frac{\,{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{12}+n_{12}+2}\,}{2},~{}~{}\cdots~{}~{}s_{1\bar{1}\cdots\ell\bar{\ell}}=\displaystyle\frac{\,{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1\dots\ell}+n_{1\cdots\ell}+\ell}\,}{2},$ (24)

where $n_{1},\cdots,n_{\ell}=0,1,2,\cdots$. Therefore we have proved that the
nested time integrals only give rise to right poles.

It is useful to check the above result for special cases with $\ell=1$ and $2$
in (17). For $\ell=1$,

$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}(-\tau_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}}e^{\mathrm{i}\mathsf{a}E_{1}\tau_{1}}=\displaystyle\frac{\,{1}\,}{(\mathrm{i}\mathsf{a}E_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}+1}}\Gamma(\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern 2.0mu_{1}-2s_{1\bar{1}}+1).$
(25)

Clearly, the poles are given by $s_{1\bar{1}}=(\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern 2.0mu_{1}+n_{1}+1)/2$, in
agreement with the above analysis.

Then, for the case of $\ell=2$,

$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}}(-\tau_{2})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{2}-2s_{2\bar{2}}}e^{\mathrm{i}\mathsf{a}(E_{1}\tau_{1}+E_{2}\tau_{2})}\theta(\tau_{1}-\tau_{2})$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(\mathrm{i}\mathsf{a}E_{2})^{\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{12}-2s_{1\bar{1}2\bar{2}}+2}}\times{}_{2}\mathcal{F}_{1}\left[\begin{matrix}\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}+1,\mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{12}-2s_{1\bar{1}2\bar{2}}+2\\\ \mkern
2.0mu\widehat{\mkern-2.0mup\mkern-2.0mu}\mkern
2.0mu_{1}-2s_{1\bar{1}}+2\end{matrix}\middle|-\displaystyle\frac{\,{E_{1}}\,}{E_{2}}\right],$
(26)

and the poles are entirely from the two upper $\Gamma$ factors in the dressed
hypergeometric function ${}_{2}\mathcal{F}_{1}$ (defined in App. A), and these
poles are also identical to what we found above.

The upshot of this part is that the time integrals, however nested, only
produce right poles for all Mellin variables.

#### Momentum integral.

Next let us look at the loop momentum integral over $\mathbf{q}$ in (2.1).
After the partial MB representation, the loop momentum reads:

$\displaystyle\mathcal{L}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}\mathcal{P};$
$\displaystyle\mathcal{P}\equiv|\mathbf{q}|^{-2s_{V\bar{1}}}|\mathbf{q}+\mathbf{P}_{1}|^{-2s_{1\bar{2}}}|\mathbf{q}+\mathbf{P}_{2}|^{-2s_{2\bar{3}}}\cdots|\mathbf{q}+\mathbf{P}_{V-1}|^{-2s_{(V-1)\mkern
2.0mu\overline{\mkern-2.0muV\mkern-2.0mu}\mkern 2.0mu}}.$ (27)

Remember that we are interested in the limit $P_{N}\rightarrow 0$ with all
other momenta appearing in (2.1) kept finite, where $N$ can be any integer
from 1 to $V-1$. In this limit, we can always pick up a value $Q$ such that
$P_{N}<Q<\text{min}\\{P_{1},\cdots,P_{N-1},P_{N+1},\cdots,P_{V-1}\\}$, and
then separate the momentum integral into a soft-momentum integral
$\mathcal{L}_{\text{S}}$ and a hard-momentum integral
$\mathcal{L}_{\text{H}}$:

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{S}}+\mathcal{L}_{\text{H}};$
$\displaystyle\mathcal{L}_{\text{S}}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}\mathcal{P}\,\theta(Q-|\mathbf{q}|),$
$\displaystyle\mathcal{L}_{\text{H}}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}\mathcal{P}\,\theta(|\mathbf{q}|-Q).$
(28)

For the soft-momentum integral, we can Taylor expand all but the two factors
$|\mathbf{q}|^{-2s_{V\bar{1}}}$ and
$|\mathbf{q}+\mathbf{P}_{N}|^{-2s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}}$ around
$\mathbf{q}=\mathbf{0}$. The Taylor series is guaranteed to be convergent for
the entire range of the soft-momentum integral:

$\displaystyle|\mathbf{q}+\mathbf{P}_{n}|=\sum_{\ell=0}^{\infty}\displaystyle\frac{\,{1}\,}{\ell!}q^{i_{1}}\cdots
q^{i_{\ell}}\bigg{[}\displaystyle\frac{\,{\partial^{\ell}|\mathbf{q}+\mathbf{P}_{n}|}\,}{\partial
q^{i_{1}}\cdots\partial
q^{i_{\ell}}}\bigg{]}_{q=0}.~{}~{}~{}(n=1,\cdots,N-1,N+1,\cdots,{V-1})$ (29)

In particular, when $P_{N}\rightarrow 0$, the leading term
$|\mathbf{q}+\mathbf{P}_{n}|=P_{n}+\mathcal{O}({q})$ for all $n\neq N$ will
contribute to the leading term in $P_{N}$, while higher order terms in $q$
lead to higher order terms in $P_{N}$. With these considerations, the soft-
momentum integral $\mathcal{L}_{\text{S}}$ is reduced to:

$\displaystyle\mathcal{L}_{\text{S}}=$
$\displaystyle~{}\Big{(}P_{1}^{-2s_{1\bar{2}}}\cdots P_{N-1}^{-2s_{(N-1)\mkern
2.0mu\overline{\mkern-2.0muN\mkern-2.0mu}\mkern
2.0mu}}\Big{)}\Big{(}P_{N+1}^{-2s_{(N+1)(\mkern
2.0mu\overline{\mkern-2.0muN+2\mkern-2.0mu}\mkern 2.0mu)}}\cdots
P_{V-1}^{-2s_{(V-1)\bar{V}}}\Big{)}$
$\displaystyle\times\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{V\bar{1}}}|\mathbf{q}+\mathbf{P}_{N}|^{-2s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}}\theta\Big{(}Q-|\mathbf{q}|\Big{)}\Big{[}1+\mathcal{O}({P_{N}})\Big{]}.$
(30)

On the other hand, the hard-momentum integral $\mathcal{L}_{\text{H}}$ is
analytic in $P_{N}$, as one can see by Taylor expanding the integrand of
$\mathcal{L}_{\text{H}}$ around $P_{N}=0$, which is guaranteed to be a
convergent series since the integral variable $q>Q>P_{N}$. This tells us that
any possible nonanalytic behavior in $P_{N}$ at $P_{N}=0$ must come from the
soft-momentum integral (2.2). In particular, so far as nonanalyticity in
$P_{N}$ is concerned, we are free to extend the upper limit of the integral
(2.2) to $\infty$, since the additional terms introduced by this extension is
again analytic in $P_{N}$. Therefore, we have:

$\displaystyle\mathcal{L}=$
$\displaystyle~{}\Big{(}P_{1}^{-2s_{1\bar{2}}}\cdots P_{N-1}^{-2s_{(N-1)\mkern
2.0mu\overline{\mkern-2.0muN\mkern-2.0mu}\mkern
2.0mu}}\Big{)}\Big{(}P_{N+1}^{-2s_{(N+1)(\mkern
2.0mu\overline{\mkern-2.0muN+2\mkern-2.0mu}\mkern 2.0mu)}}\cdots
P_{V-1}^{-2s_{(V-1)\bar{V}}}\Big{)}$
$\displaystyle\times\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{V\bar{1}}}|\mathbf{q}+\mathbf{P}_{N}|^{-2s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}}\Big{[}1+\mathcal{O}({P_{N}})\Big{]}+\text{terms analytic in
$P_{N}$}.$ (31)

The remaining integral is nothing but the bubble 1-loop integral, and can be
directly done, as shown in App. B.1:

$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{V\bar{1}}}|\mathbf{q}+\mathbf{P}_{N}|^{-2s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}}$ $\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{P_{N}^{3-2s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}-s_{V\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}\\\
3-s_{V\bar{1}N(\mkern 2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)},s_{V\bar{1}},s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}\end{bmatrix}.$ (32)

Therefore we finally get:

$\displaystyle\mathcal{L}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}-s_{V\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}\\\
3-s_{V\bar{1}N(\mkern 2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)},s_{V\bar{1}},s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}\end{bmatrix}P_{N}^{3-2s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}}\Big{[}1+\mathcal{O}({P_{N}})\Big{]}$
$\displaystyle\times\Big{(}P_{1}^{-2s_{1\bar{2}}}\cdots
P_{N-1}^{-2s_{(N-1)\mkern 2.0mu\overline{\mkern-2.0muN\mkern-2.0mu}\mkern
2.0mu}}\Big{)}\Big{(}P_{N+1}^{-2s_{(N+1)(\mkern
2.0mu\overline{\mkern-2.0muN+2\mkern-2.0mu}\mkern 2.0mu)}}\cdots
P_{V-1}^{-2s_{(V-1)\bar{V}}}\Big{)}+\text{terms analytic in $P_{N}$}$ (33)

We see that the nonanalytic behavior of $P_{N}$ is controlled by the power
$P_{N}^{3-2s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}}$. Nonanalyticity
requires that $2s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}$ takes noninteger
values. This value is provided by the location of poles of the four variables
$(s_{V},\bar{s}_{1},s_{N},\bar{s}_{N+1})$. Also, this nonanalytic power must
be combined into the power of a momentum ratio in the final result. Now, since
we are taking $P_{N}\rightarrow 0$, the convergence of the resulting series
requires that we pick up left poles for all these variables.

#### Left pole structure.

From the analysis of the momentum integral, we have seen that the
nonanalyticity of the graph at $P_{N}=0$ is contributed by left poles of
Mellin variables. Now we take a closer look at these poles.

Above we have shown that the time integral does not yield any left poles. The
momentum integral, on the other hand, has one set of left poles:

$s_{V\bar{1}N\mkern 2.0mu\overline{\mkern-2.0mu(N+1)\mkern-2.0mu}\mkern
2.0mu}=-n+\displaystyle\frac{\,{3}\,}{2}.~{}~{}~{}~{}~{}(n=0,1,\cdots)$ (34)

But these poles only contribute to integer powers of $P_{N}$, which are
analytic at $P_{N}=0$, and therefore we can discard them.

Then the remaining left poles are all contributed by the $\Gamma$ functions
from the MB representation of Hankel functions in the massive propagators, as
shown in (2.2). For the four relevant variables
$(s_{V},\bar{s}_{1},s_{N},\bar{s}_{N+1})$, these $\Gamma$ factors are:

$\displaystyle~{}\Gamma\begin{bmatrix}s_{V}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},s_{V}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},\bar{s}_{1}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},\bar{s}_{1}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2}\end{bmatrix}$ $\displaystyle\times$
$\displaystyle~{}\Gamma\begin{bmatrix}s_{N}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1}}\,}{2},s_{N}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1}}\,}{2},\bar{s}_{N+1}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1}}\,}{2},\bar{s}_{N+1}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1}}\,}{2}\end{bmatrix}.$ (35)

We have 4 Mellin variables, each of which has 2 groups of left poles. Say,
$s_{V}=-n_{V}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}/2$. For
convenience, we call $s_{V}=-n_{V}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}/2$ the negative
poles, and $s_{V}=-n_{V}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}/2$ the positive
poles. Now, in the result of loop momentum integral (2.2), the factor
$1/\Gamma(s_{V\bar{1}})$ implies that the poles of $s_{V}$ and $\bar{s}_{1}$
must be both positive or both negative. Similarly, the factor
$1/\Gamma(s_{N(\mkern 2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)})$ in (2.2) implies that the poles of $s_{N}$ and $\bar{s}_{N+1}$ must
be both positive or both negative. Then we have the following possibilities:

$\displaystyle s_{V}=-n_{V}-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},~{}\bar{s}_{1}=-\bar{n}_{1}-\mathsf{c}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},~{}s_{N}=-n_{N}-\mathsf{d}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1}}\,}{2},~{}\bar{s}_{N+1}=-\bar{n}_{N+1}-\mathsf{d}\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}}\,}{2},$ (36)

where $\mathsf{c}=\pm 1$ and $\mathsf{d}=\pm 1$. In the degenerate case
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}=\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}$, we should
also set $\mathsf{c}=\mathsf{d}$, in order that $2s_{V\bar{1}N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}\notin\mathbb{Z}$.
These are the only poles that can contribute to the nonlocal signal in
$P_{N}$. We call them signal poles.

#### Cutting the nested time integral.

Above we have identified all possible poles that could contribute to the
nonlocal signal in the limit $P_{N}\rightarrow 0$, as summarized in (36). An
immediate yet important consequence of this result is that the nested time
integrals across the cut are automatically factorized. That is, the time
integrals nesting the two sides of the cut neither contribute left poles, nor
contribute any finite residues at the signal poles in (36). The first half of
this statement (the absence of left poles) has been proved above. Now we show
more explicitly that these nested time integrals vanish at all signal poles in
(36).

There are two possible nested time integrals across the cut, one nesting
$\tau_{V}$ and $\tau_{1}$, the other nesting $\tau_{N}$ and $\tau_{N+1}$. The
treatments for both integrals are identical so we only consider the former
case.

The integrals over $\tau_{V}$ and $\tau_{1}$ are nested only when
$\mathsf{a}_{V}=\mathsf{a}_{1}\equiv\mathsf{a}$, due to the $\theta$ functions
in the same-sign propagator $D_{\pm\pm}$, as shown in (6). It proves
convenient to rewrite $D_{\pm\pm}$ as:

$\displaystyle D_{\pm\pm}(q;\tau_{V},\tau_{1})=$
$\displaystyle~{}\text{Re}\,D_{\pm\pm}(q;\tau_{V},\tau_{1})+\mathrm{i}\,\text{Im}\,D_{\pm\pm}(q;\tau_{V},\tau_{1});$
(37) $\displaystyle\text{Re}\,D_{\pm\pm}(q;\tau_{V},\tau_{1})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2}\Big{[}D_{\mp\pm}(q;\tau_{V},\tau_{1})+D_{\pm\mp}(q;\tau_{V},\tau_{1})\Big{]}$
(38) $\displaystyle\mathrm{i}\,\text{Im}\,D_{\pm\pm}(q;\tau_{V},\tau_{1})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2}\Big{[}D_{\mp\pm}(q;\tau_{V},\tau_{1})-D_{\pm\mp}(q;\tau_{V},\tau_{1})\Big{]}\text{sgn}(\tau_{V}-\tau_{1}),$
(39)

where $\text{sgn}(z)\equiv z/|z|$ extracts the sign of a real number $z\neq
0$. Thus we see that the real part of $D_{\pm\pm}$ is automatically
factorized, while the imaginary part of $D_{\pm\pm}$ is nested.

With the above representation for $D_{\pm\pm}$, we can express all four
$D_{\mathsf{a}_{V}\mathsf{a}_{1}}$ in terms of opposite-sign propagators
$D_{\mp\pm}$. Now let us see what $D_{\mp\pm}$ contributes inside the Mellin
integrand. By examining the PMB representation of the propagator
$D_{\mathsf{a}_{V}\mathsf{a}_{1}}$ in (2.2), we see that the SK indices affect
the integrand only through the factor
$e^{\mp\mathrm{i}\pi(s_{V}-\bar{s}_{1})}$. Now, at the poles given in (36),
this factor evaluates to $(-1)^{n_{V}-\bar{n}_{1}}$, which is independent of
the two index choices of $D_{\mp\pm}$. In particular, this implies that:

  1. 1.

The real parts of all four propagators, namely
$\text{Re}\,D_{\mathsf{a}_{V}\mathsf{a}_{1}}$ with $\mathsf{a}_{V}=\pm 1$ and
$\mathsf{a}_{1}=\pm 1$, all contribute the same factor
$(-1)^{n_{V}-\bar{n}_{1}}$ to the Mellin integrand at the poles in (36).

  2. 2.

The imaginary parts of all four propagators,
$\text{Im}\,D_{\mathsf{a}_{V}\mathsf{a}_{1}}$, vanish at the poles (36).

The same argument also applies to all integrals nesting $\tau_{N}$ and
$\tau_{N+1}$. Therefore, we conclude that we can freely replace the two cut
propagators, $D_{\mathsf{a}_{V}\mathsf{a}_{1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(q;\tau_{V},\tau_{1})$ and
$D_{\mathsf{a}_{N}\mathsf{a}_{N+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1})}(q;\tau_{N},\tau_{N+1})$, by their real parts, when computing the
nonanalytic part of the correlator as $P_{N}\rightarrow 0$, as summarized in
(12). In particular, this replacement means that the time integrals nesting
over the two cut lines are automatically factorized at the signal poles.
Therefore, we can think of this replacement as a cutting rule for extracting
nonlocal signals of one-loop inflation correlators for a given soft momentum
$P_{N}$. We stress that this analysis is done for all signal poles, and thus
the cutting rule holds to all orders in the expansion of $P_{N}$.

#### Residues in the soft limit.

Finally, we evaluate the whole Mellin integrals at the poles in (36) with
$n_{V}=\bar{n}_{1}=n_{N}=\bar{n}_{N+1}=0$, which we call the _leading poles_
since they give the leading terms of the nonlocal signal in the soft limit
$P_{N}\rightarrow 0$. There are four types of contributions to these residues.
First, we notice that, at the leading poles in (36), four of the $\Gamma$
factors in (2.2) are singular with residue $1$, while the other four are
regular. Together with two prefactors $1/(4\pi)$ and the factor
$2^{2s_{V\bar{1}N(\mkern 2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern
2.0mu)}}$ in (2.2), they give:

$\displaystyle\displaystyle\frac{\,{2^{-2\mathrm{i}(\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathsf{d}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1})}}\,}{(4\pi)^{2}}\Gamma\begin{bmatrix}-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1},-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}\end{bmatrix}.$
(40)

Second, we evaluate the loop momentum integral at the leading poles in (36)
and isolate the result in the first line of (2.2):

$\displaystyle\mathcal{L}\Rightarrow$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}-\mbox{$\frac{\,{3}\,}{2}$}-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1},\mbox{$\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}$}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}\\\
3+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1},-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1}\end{bmatrix}P_{N}^{3+2\mathrm{i}(\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathsf{d}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1})}.$ (41)

Let us put these two pieces together, and define the result to be
$\mathcal{B}_{\mathsf{c}\mathsf{d}}(P_{N})$ which we call the bubble signal:

$\displaystyle\mathcal{B}_{\mathsf{c}\mathsf{d}}(P_{N})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(4\pi)^{7/2}}\Gamma\begin{bmatrix}-\mbox{$\frac{\,{3}\,}{2}$}-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1},\mbox{$\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}$}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1},-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}\\\
3+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{d}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}\end{bmatrix}$
$\displaystyle\times
P_{N}^{3}\Big{(}\displaystyle\frac{\,{P_{N}}\,}{2}\Big{)}^{2\mathrm{i}(\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathsf{d}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1})}.$ (42)

It is called the bubble signal since this part is essentially the same as what
we would get from computing a bubble loop; See Sec. 4.1. The bubble signal
encodes all the nonanalyticity of $P_{N}$ in the limit $P_{N}\rightarrow 0$.

Now, we have two momentum factors remained, coming from the second line of the
loop momentum integral in (2.2):

$\Big{(}P_{1}^{-2s_{1\bar{2}}}\cdots P_{N-1}^{-2s_{(N-1)\mkern
2.0mu\overline{\mkern-2.0muN\mkern-2.0mu}\mkern
2.0mu}}\Big{)}\Big{(}P_{N+1}^{-2s_{(N+1)(\mkern
2.0mu\overline{\mkern-2.0muN+2\mkern-2.0mu}\mkern 2.0mu)}}\cdots
P_{V-1}^{-2s_{(V-1)\bar{V}}}\Big{)}$ (43)

Comparing them with the original momentum integral (27), we see that this
result amounts to replacing $|\mathbf{q}+\mathbf{P}_{\ell}|$ factors by
$P_{\ell}$ for all $\ell=1,\cdots,N-1,N+1,\cdots,V-1$. Remember that we have
only evaluated four Mellin integrals with
$(s_{V},\bar{s}_{1},s_{N},\bar{s}_{N+1})$. All other Mellin integrals are left
undone at the moment. Now, we can put the factor $P_{1}^{-2s_{1\bar{2}}}\cdots
P_{N-1}^{-2s_{(N-1)\mkern 2.0mu\overline{\mkern-2.0muN\mkern-2.0mu}\mkern
2.0mu}}$ back, and finish the Mellin integrals over
$(s_{1},\cdots,s_{N-1},\bar{s}_{2},\cdots,\bar{s}_{N})$. This gives us nothing
but the expression for the “left” tree graph
$\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(L)}}(\\{\mathbf{k}^{\text{(L)}}\\})$
after the cut, as given in (2.1), and is the third piece of contribution to
the residues. Likewise, finishing the Mellin integrals over
$(s_{N+1},\cdots,s_{V-1},\bar{s}_{N+2},\cdots,\bar{s}_{V})$, we get the
“right” tree graph
$\mathcal{T}_{\mathsf{c}\mathsf{d}}^{\text{(R)}}(\\{\mathbf{k}^{\text{(R)}}\\})$
after the cut, as given in (2.1), which makes the fourth contribution to the
residues at the signal poles.

Putting all these pieces together, we finally reach a main result of this
section, namely (8). This completes our proof of the factorization theorem.

###  2.3 Discussions

The factorization theorem stated and proved in this section not only tells us
when do we expect to see a nonlocal CC signal in a generic 1-loop graph, but
also gives the explicit expression for the leading nonlocal CC signal in terms
of the two tree graphs $\mathcal{T}^{\text{L}}_{\mathsf{c}\mathsf{d}}$ and
$\mathcal{T}^{\text{R}}_{\mathsf{c}\mathsf{d}}$. Therefore, this theorem can
be directly put in use to compute more complicated 1-loop graphs than the
simplest bubble graph. We shall pursue this direction in Sec. 4 and derive the
most general nonlocal signals for all 1PI 1-loop graphs of the 4-point
inflation correlators. On the other hand, there are several propositions one
can directly extract from our proof of the factorization theorem. In the rest
of this section, we shall briefly discuss these points, and we plan to expand
on some of them in greater details in a future work.

#### Adjacent cut lines.

First, in the figures and also in the proof, we have considered the cases
where the two cut lines do not share any common vertex. That is, we assume
$P_{N}\rightarrow 0$ with $1<N<V-1$. However, the theorem still holds for
$N=1$ or $N=V-1$, namely, when the two cut lines share one common vertex, but
with an important additional condition, namely that there are at least two
external lines connected to the common vertex (Vertex 1 or Vertex $V$ for
$N=1$ or $N={V-1}$, respectively). The reason for this additional condition is
that we require enough freedom in the kinematics to independently take the
$P_{N}\rightarrow 0$ limit. When there is only one external line connected to,
say, Vertex 1, we have $B_{1}=1$. (Recall the definitions in (1).) Then, the
“total external energy” $E_{1}\equiv|\mathbf{K}_{1}|$ at this vertex is
identical to the magnitude of $\mathbf{P}_{1}$. In this case it is impossible
to take $P_{1}\rightarrow 0$ with $E_{1}$ fixed, and there could be additional
nonanalytic behavior in $E_{1}$ which becomes degenerate with the
nonanalyticity in $P_{1}$.

#### Subgraph.

Second, although we have presented our theorem for 1-loop boundary correlators
assuming all bulk-to-boundary propagators being massive or conformal, our
proof does not really rely on this assumption. The only occasion that we quote
the explicit form of the bulk-to-boundary propagators is when we consider the
pole structure of the time integrals. (See the discussion starting from
(2.2).) Even at this place, what we really used is the regularity of the
exponential factor $e^{\mathrm{i}\mathsf{a}_{\ell}E_{\ell}\tau_{\ell}}$ in the
$\tau_{\ell}\rightarrow 0$ limit. This regularity is still true when we
replace these conformal or massless bulk-to-boundary propagators by arbitrary
massive bulk propagators. Therefore, our theorem also holds as a statement for
a 1-loop bulk subgraph, _provided that all the external momenta
$\\{\mathbf{k}\\}$ in this subgraph can be held fixed_. That is, we require
that those external momenta are not part of other loop integrals, so that it
is possible to take $\mathbf{P}_{N}\rightarrow 0$ limit with all other momenta
held at fixed finite values.

#### No nonlocal signal in a disconnected soft limit.

In the factorization theorem and its proof, we have only considered the limit
$P_{N}\rightarrow 0$ with $N=1,\cdots,V-1$. Recall that
$\mathbf{P}_{N}=\mathbf{K}_{1}+\cdots+\mathbf{K}_{N}$ is the total energy
flowing into the graph through the first $N$ vertices, that is, through a
_connected_ subset of vertices. One may naturally ask what happens if we take
the soft limit of the total momentum flowing into the graph through a
_disconnected_ subset of vertices, with all other combinations of momenta
fixed at finite values, which we call a “disconnected soft limit.” Is there
any nonlocal CC signal associated with this disconnected soft limit? The
answer to this question is no. To see this, let us have a second look at the
integrand of the loop momentum integral (27):

$\mathcal{P}=|\mathbf{q}|^{-2s_{V\bar{1}}}|\mathbf{q}+\mathbf{P}_{1}|^{-2s_{1\bar{2}}}|\mathbf{q}+\mathbf{P}_{2}|^{-2s_{2\bar{3}}}\cdots|\mathbf{q}+\mathbf{P}_{V-1}|^{-2s_{(V-1)\mkern
2.0mu\overline{\mkern-2.0muV\mkern-2.0mu}\mkern 2.0mu}}.$ (44)

There are a total of $V$ factors here. When we take a connected soft limit
(say $\mathbf{P}_{N}\rightarrow 0$), there are always a pair of factors in
(44) sharing almost the same momentum. (In the case of $P_{N}\rightarrow 0$,
the pair is given by $|\mathbf{q}|^{-2_{V\bar{1}}}$ and
$|\mathbf{q}+\mathbf{P}_{N}|^{-2s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}}$.) As shown in the
proof, in this case, we can “pinch” the rest $V-2$ loop lines by setting
$\mathbf{q}\rightarrow 0$ in them, and move them out of the loop momentum
integral. What remain in the integrand are just the two factors
$|\mathbf{q}|^{-2_{V\bar{1}}}$ and $|\mathbf{q}+\mathbf{P}_{N}|^{-2s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}}$. As shown in
(2.2), integrating these two factors gives rise to the desired nonanalyticity
when $P_{N}\rightarrow 0$.

Now, if we take a disconnected soft limit, say,
$\mathbf{K}_{2}+\mathbf{K}_{V}\rightarrow 0$ while keeping all other momentum
combinations finite, then it is clear that no two momenta in the $V$ factors
of (44) can become close to each other. Consequently, in all parameter space
of the loop momentum $\mathbf{q}$, there can be at most one soft momentum at a
time, and when a given momentum become soft (say $\mathbf{q}$), all the other
$V-1$ lines can be “pinched” and pulled out of the integral. What remains
inside the loop integral would be a single factor
$|\mathbf{q}|^{-2s_{V\bar{1}}}$. Clearly, this amounts to setting $s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)}=0$ in (2.2), and it
gives us a vanishing result due to the $\Gamma(s_{N(\mkern
2.0mu\overline{\mkern-2.0muN+1\mkern-2.0mu}\mkern 2.0mu)})$ factor in the
denominator on the right hand side of (2.2). So, we conclude that a
disconnected soft limit does not generate any nonlocal CC signal.

An immediate corollary of this discussion is that it is impossible to get
nonlocal signal by “cutting” a single loop propagator; One has to cut at least
two lines at a time. On the other hand, one may also ask if we could
simultaneously see the nonlocal CC signals from more than two loop lines. The
answer is yes, and one can achieve this by taking several successive cuts. We
shall expand on all these points more systematically in a future work.

## 3 A Boundary OPE Perspective

It is usually helpful to look at inflation correlators from a boundary point
of view. In particular, the bulk isometries of dS spacetime are mapped to the
boundary conformal symmetries, which can be powerful tool to constrain the
form of inflation correlators already at the purely kinematic level. In our
current discussion, the 1-loop factorization theorem in (8) from the last
section states that we can cut the two soft lines and pinch all remaining hard
lines to get the nonanalytic piece of a 1-loop graph in a soft limit. This is
very similar to an OPE procedure on the future boundary: From the position-
space point of view, the two tree graphs,
$\mathcal{T}_{\mathsf{c}\mathsf{d}}^{(L)}$ in (2.1) and
$\mathcal{T}_{\mathsf{c}\mathsf{d}}^{(R)}$ in (2.1), are very far from each
other, compared with the typical separation of points within each graph.
Therefore, to compute the full loop graph at the leading order in the momentum
transfer $P_{N}$, it seems reasonable to perform an OPE for operators within
each subgraph, which simply pinches each subgraph to a common point in the
position space at the leading order.

However, this naive way of applying OPE is not quite correct. The reason is
that all massive lines are attached to bulk vertices, and each bulk vertex
carries a time variable $\tau_{j}$ that can take any value in $(-\infty,0)$;
See (2.1). _A priori_ , the SK integral receives contributions from all
possible values of $\tau_{j}$, and it is not clear that one can simply treat
these bulk vertices as boundary insertions.

The way to make progress is to make the following two key observations: First,
in the bulk propagators (and also in the bulk mode functions), the time
variables $\tau$ always appear within the combination $k\tau$ where $k$ is the
momenta carried by the propagator (or by the mode function). As a result, a
bulk mode carrying very soft momenta $k\rightarrow 0$ will have
$k\tau\rightarrow 0$, which means that this bulk mode is effectively living on
the boundary $\tau\rightarrow 0$. Second, although the loop propagators can _a
priori_ carry arbitrarily large (loop) momentum, the nonlocal signal is always
contributed by the region of the loop integral where the two “cut” propagators
(the two blue lines in Fig. 2) carry very small momenta, as we have made clear
in the proof of the factorization theorem.

Combining the above two observations, we see that it is possible to treat the
four bulk modes in the two cut propagators as boundary modes. We can then
apply the boundary OPE to these four boundary modes, and this effectively
pinches an arbitrary 1-loop configuration to a bubble. Thus it is indeed
possible to reinterpret the 1-loop factorization theorem by a boundary OPE.
Below, we shall expand on these points and show more explicitly how the
factorization emerges from a boundary OPE viewpoint. We should emphasize at
this point that the this OPE interpretation should not be treated as an
independent proof of the factorization theorem. It is in any case less
rigorous than the bulk proof in the last section. Still, this OPE
interpretation may provide useful physical insights into what is happening in
the generation of nonlocal CC signals.

###  3.1 Pushing the soft lines to the future boundary

The picture of the OPE is most transparent in the position space. Therefore,
we shall begin by rewriting the 1-loop graph $\mathcal{T}(\\{\mathbf{k}\\})$,
as shown in Fig. 2, in terms of the expectation value of a string of bulk
operator product in the position space. For this purpose, we introduce $V$
distinct massive scalar fields $\sigma^{(j)}(x)$ with mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{j}$
$(j=1,\cdots,V)$. Then, we have:

$\displaystyle\mathcal{T}(\\{\mathbf{k}\\})(2\pi)^{3}\delta(\mathbf{P}_{V})=$
$\displaystyle\sum_{\mathsf{a}_{1},\cdots,\mathsf{a}_{V}=\pm}\int_{-\infty}^{0}\prod_{\ell=1}^{V}\bigg{[}{\mathrm{d}}\tau_{\ell}\,\mathrm{i}\mathsf{a}_{\ell}(-\tau_{\ell})^{p_{\ell}}\prod_{i=1}^{B_{\ell}}C_{\mathsf{a}_{\ell}}\Big{(}k_{i}^{(\ell)};\tau_{\ell}\Big{)}\bigg{]}\int\prod_{n=1}^{V}\Big{[}{\mathrm{d}}^{3}\mathbf{x}_{n}\,e^{-\mathrm{i}\mathbf{K}_{n}\cdot\mathbf{x}_{n}}\Big{]}$
$\displaystyle\times\Big{\langle}\big{[}\sigma^{(1)}\sigma^{(2)}(x_{1})\big{]}_{\mathsf{a}_{1}}\big{[}\sigma^{(2)}\sigma^{(3)}(x_{2})\big{]}_{\mathsf{a}_{2}}\cdots\big{[}\sigma^{(V)}\sigma^{(1)}(x_{V})\big{]}_{\mathsf{a}_{V}}\Big{\rangle},$
(45)

where $x_{\ell}\equiv(\tau_{\ell},\mathbf{x}_{\ell})$. The factor
$(2\pi)^{3}\delta(\mathbf{P}_{V})$ on the left hand side comes from the total
3-momentum conservation produced by the Fourier transform of the position-
space correlator
$\langle[\sigma^{(1)}\sigma^{(2)}]_{\mathsf{a}_{1}}\cdots\rangle$ in the last
line. This position-space correlator can be thought of as the generating
function $Z[J]$ of a bulk theory of $V$ free massive scalars
$\sigma^{(\ell)}$, in the presence of external source terms
$\int{\mathrm{d}}\tau{\mathrm{d}}^{3}\mathbf{x}\,J_{\ell}(\tau,\mathbf{x})\sigma^{(\ell)}\sigma^{(\ell+1)}$,
where $J_{\ell}(x)=(-\tau)^{p_{\ell}}\phi_{c}^{B_{\ell}}(x)$ and $\phi_{c}$ is
the external conformal scalar. Furthermore, the subscript of the correlator
$\langle\cdots\rangle_{\mathsf{a}_{1}\cdots\mathsf{a}_{V}}$ denotes the
appropriate SK branch. More explicitly:

$\displaystyle\big{\langle}\mathcal{O}_{\mathsf{a}_{1}}(x_{1})\cdots\mathcal{O}_{\mathsf{a}_{n}}(x_{n})\big{\rangle}=\bigg{\langle}\mkern
2.0mu\overline{\mkern-2.0mu\text{T}\mkern-2.0mu}\mkern
2.0mu\Big{[}\prod_{\\{i|\mathsf{a}_{i}=-\\}}\mathcal{O}(x_{i})\Big{]}\text{T}\Big{[}\prod_{\\{j|\mathsf{a}_{j}=+\\}}\mathcal{O}(x_{j})\Big{]}\bigg{\rangle},$
(46)

where T denotes the time ordering and $\mkern
2.0mu\overline{\mkern-2.0mu\text{T}\mkern-2.0mu}\mkern 2.0mu$ denotes the
anti-time ordering. More details of these SK correlators can be found in [19].

It is straightforward to see that the expression (3.1) is equivalent to the
original expression (2.1). In fact, at the leading order in perturbation
theory, the bulk correlator is reduced to a product of propagators after Wick
contractions. Since we are assuming $V$ distinct massive scalars in this
process, the Wick contraction is unique, and the result is:

$\Big{\langle}\big{[}\sigma^{(1)}\sigma^{(2)}(x_{1})\big{]}_{\mathsf{a}_{1}}\cdots\big{[}\sigma^{(V)}\sigma^{(1)}(x_{V})\big{]}_{\mathsf{a}_{V}}\Big{\rangle}\sim
D^{(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}_{\mathsf{a}_{V}\mathsf{a}1}(x_{V},x_{1})\cdots D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{V})}_{\mathsf{a}_{V-1}\mathsf{a}_{V}}(x_{V-1},x_{V}).$ (47)

where $D_{\mathsf{a}_{i}\mathsf{a}_{j}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{j})}(x_{i},x_{j})$
is the position-space bulk propagator for the field $\sigma^{(j)}$, and is
related to the previously used momentum-space propagator via:

$D_{\mathsf{a}_{i}\mathsf{a}_{j}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{j})}(x_{i},x_{j})=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{k}}\,}{(2\pi)^{3}}D_{\mathsf{a}_{i}\mathsf{a}_{j}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{j})}(k;\tau_{i},\tau_{j})e^{\mathrm{i}\mathbf{k}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})}.$
(48)

If we insert (47) and (48) into the right hand side of (3.1), and finish the
Fourier transformation, we recover nothing but (2.1) with an extra factor
$(2\pi)^{3}\delta(\mathbf{P}_{V})$ denoting the momentum conservation.

Now let us consider the soft limit $P_{N}\rightarrow 0$ and the corresponding
nonlocal signal in (3.1). Recall that in momentum space, the nonlocal signal
associated with $P_{N}$ comes totally from the configuration where the
$\sigma^{(1)}$ propagator and the $\sigma^{(N+1)}$ propagator carry soft
momenta, while all other loop lines carry hard momenta. This particular
configuration in the momentum space has two implications for the corresponding
position-space picture:

  1. 1.

On a fixed equal-time slice, the bulk operators on the right hand side of
(3.1) are divided into two sets far away from each other:
$\\{\sigma^{(j)}(x_{i})|1\leq i\leq N\\}$ and $\\{\sigma^{(j)}(x_{i})|N+1\leq
i\leq V\\}$. These two sets of operators are connected by the two “soft”
propagators $D_{\mathsf{a}_{V}\mathsf{a}_{1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1})}(x_{V},x_{1})$
and $D_{\mathsf{a}_{N}\mathsf{a}_{N+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1})}(x_{N},x_{N+1})$. The softness in momentum space implies large
distance in position space. On the other hand, operators among each set are
close to each other in 3-dimension equal-time slice in the bulk, since all the
other momenta are “hard” compared to the two soft momenta.

  2. 2.

In the time direction, the four bulk modes associated with the two soft
propagators are close to the late-time boundary. To see this, let us consider
the time integral over $\tau_{i}$ ($i=V,1,N,N+1$) more carefully. Due to the
exponential factor $e^{\pm\mathrm{i}E_{i}\tau_{i}}$ in (3.1), the time
integral receives most contribution from the region $-1/E_{i}<\tau_{i}<0$. For
this range of $\tau_{i}$, we always have $|q\tau_{1}|\ll 1$, $|q\tau_{V}|\ll
1$, $|(\mathbf{q}+\mathbf{P}_{N})\tau_{N}|\ll 1$, and
$|(\mathbf{q}+\mathbf{P}_{N})\tau_{N+1}|\ll 1$. The soft modes with momenta
$|\mathbf{q}|,|\mathbf{q}+\mathbf{P}_{N}|\ll E_{i}$ crossed the horizon at the
time $\tau_{i}^{*}\sim-1/|\mathbf{q}|$ or $-1/|\mathbf{q}+\mathbf{P}_{N}|$.
These conditions allow us to expand the operator $\sigma^{(j)}(x_{i})$ around
$\tau_{i}\sim 0$.

#### More on the late-time expansion

Above we have shown that it is legitimate to push the four soft modes,
$\sigma^{(1)}(x_{1})$, $\sigma^{(1)}(x_{V})$, $\sigma^{(N+1)}(x_{N})$, and
$\sigma^{(N+1)}(x_{N+1})$, to the future boundary. That is, we can perform
small $\tau$ expansion for them. Now, let us pause and have a closer look at
the late-time expansion of a bulk massive field $\sigma(\tau,\mathbf{x})$:

$\sigma(x)=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{k}}\,}{(2\pi)^{3}}\,\Big{[}u(k,\tau)a_{\mathbf{k}}+u^{*}(k,\tau)a_{-\mathbf{k}}^{\dagger}\Big{]}e^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}.$
(49)

Here $k=|\mathbf{k}|$ denotes the magnitude of the 3-momentum.
$a_{\mathbf{k}}$ and $a^{\dagger}_{\mathbf{k}}$ are the annihilation and the
creation operators, respectively, with standard commutation relation
$[a_{\mathbf{p}},a_{\mathbf{q}}^{\dagger}]=(2\pi)^{3}\delta(\mathbf{p}-\mathbf{q})$.
$u(k,\tau)$ is the mode function satisfying the equation of motion:

$u(k,\tau)=\displaystyle\frac{\,{\sqrt{\pi}}\,}{2}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}(-\tau)^{3/2}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}(-k\tau).$ (50)

Now that the time dependence in the operator $\sigma(x)$ is all in the mode
function, so we can expand the mode function around $\tau\sim 0$:

$\lim_{\tau\rightarrow
0}u(k,\tau)=-\mathrm{i}\sqrt{\displaystyle\frac{\,{2}\,}{\pi
k^{3}}}\bigg{[}e^{\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}\Gamma(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{(}-\displaystyle\frac{\,{k\tau}\,}{2}\Big{)}^{3/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}\Gamma(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{(}-\displaystyle\frac{\,{k\tau}\,}{2}\Big{)}^{3/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\bigg{]}.$ (51)

Insert (51) into (49), we obtain the late-time expansion for the scalar field:

$\lim_{\tau\rightarrow
0}\sigma(\tau,\mathbf{x})=\sigma_{+}(\mathbf{x})(-\tau)^{3/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+\sigma_{-}(\mathbf{x})(-\tau)^{3/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu},$ (52)

where $\sigma_{\pm}$ are boundary operators:

$\sigma_{\pm}(\mathbf{x})=\displaystyle\frac{\,{-\mathrm{i}}\,}{\sqrt{4\pi}}\Gamma(\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{k}}\,}{(2\pi)^{3}}\,\Big{(}\displaystyle\frac{\,{k}\,}{2}\Big{)}^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{[}e^{\pm\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}a_{\mathbf{k}}-e^{\mp\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}a^{\dagger}_{-\mathbf{k}}\Big{]}e^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}.$
(53)

As mentioned before, the bulk isometries of dS induce conformal symmetries on
the future boundary, and it can be shown that the two boundary fields
$\sigma_{\pm}(\mathbf{x})$ thus defined transform as primary scalars with
conformal weights $\Delta_{\pm}=3/2\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$.

The fields $\sigma_{\pm}(\mathbf{x})$ are the leading two boundary operators
in the late-time expansion (52). One can also consider higher order terms:

$\sigma(\tau,\mathbf{x})=\sum_{\Delta\in\\{\Delta_{\pm}\\}}\sigma_{\Delta}(\mathbf{x})(-\tau)^{\Delta},\qquad\\{\Delta_{\pm}\\}=\Big{\\{}2n+\displaystyle\frac{\,{3}\,}{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{|}n=0,1,2\cdots\Big{\\}},$ (54)

The higher order operators $\sigma_{\Delta}$ with $n\neq 0$ in (54) are
descendants of the corresponding primary fields $\sigma_{\pm}$, and their
effects are suppressed by at least $\mathcal{O}(\tau^{2})$ in the late-time
limit, and thus can be neglected when we consider the leading nonlocal signal
in the squeezed limit. We also point out that each term in the expansion (54)
is in one-to-one correspondence with the contribution of residues at nonlocal
poles (36) of the $\Gamma$ factors $\Gamma(s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2)\Gamma(s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu/2)$ in the MB
representation (2.2).

###  3.2 Boundary OPE and factorization theorem

With all preparation made above, we can now perform the late-time expansion
(52) to the four soft modes in the position-space correlator
$\langle[\sigma^{(1)}\sigma^{(2)}]_{\mathsf{a}_{1}}\cdots\rangle$ in (3.1):

$\displaystyle\sigma^{(1)}(x_{V})\sim$
$\displaystyle~{}\sum_{\mathsf{c}_{1}=\pm}\sigma^{(1)}_{\mathsf{c}_{1}}(\mathbf{x}_{V})(-\tau_{V})^{3/2+\mathsf{c}_{1}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}},$ (55)
$\displaystyle\sigma^{(1)}(x_{1})\sim$
$\displaystyle~{}\sum_{\mathsf{c}_{2}=\pm}\sigma^{(1)}_{\mathsf{c}_{2}}(\mathbf{x}_{1})(-\tau_{1})^{3/2+\mathsf{c}_{2}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}},$ (56)
$\displaystyle\sigma^{(N+1)}(x_{N})\sim$
$\displaystyle~{}\sum_{\mathsf{d}_{1}=\pm}\sigma^{(N+1)}_{\mathsf{d}_{1}}(\mathbf{x}_{N})(-\tau_{N})^{3/2+\mathsf{d}_{1}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}},$ (57)
$\displaystyle\sigma^{(N+1)}(x_{N+1})\sim$
$\displaystyle~{}\sum_{\mathsf{d}_{2}=\pm}\sigma^{(N+1)}_{\mathsf{d}_{2}}(\mathbf{x}_{N+1})(-\tau_{N+1})^{3/2+\mathsf{d}_{2}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}},$ (58)

where we only keep the leading terms. Inserting (55)-(58) into the position-
space correlator in (3.1), we get:

$\displaystyle~{}\Big{\langle}\big{[}\sigma^{(1)}\sigma^{(2)}(x_{1})\big{]}_{\mathsf{a}_{1}}\cdots\big{[}\sigma^{(V)}\sigma^{(1)}(x_{V})\big{]}_{\mathsf{a}_{V}}\Big{\rangle}\sim\mathcal{M}_{\mathsf{a}_{1}\cdots\mathsf{a}_{V}}^{\text{Bulk}}\times\sum_{\mathsf{c}_{1},\mathsf{c}_{2},\mathsf{d}_{1},\mathsf{d}_{2}}\mathcal{M}_{\mathsf{c}_{1}\mathsf{c}_{2}\mathsf{d}_{1}\mathsf{d}_{2}}^{\text{Boundary}}$
$\displaystyle\times(-\tau_{V})^{3/2+\mathsf{c}_{1}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}(-\tau_{1})^{3/2+\mathsf{c}_{2}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}(-\tau_{N})^{3/2+\mathsf{d}_{1}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N+1}}(-\tau_{N+1})^{3/2+\mathsf{d}_{2}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}},$ (59)

where

$\displaystyle\mathcal{M}_{\mathsf{a}_{1}\cdots\mathsf{a}_{V}}^{\text{Bulk}}\equiv$
$\displaystyle~{}\prod_{i=1}^{N-1}D_{\mathsf{a}_{i}\mathsf{a}_{i+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i+1})}(x_{i},x_{i+1})\prod_{j=N+1}^{V-1}D_{\mathsf{a}_{j}\mathsf{a}_{j+1}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{j+1})}(x_{i},x_{j+1})$ (60)
$\displaystyle\mathcal{M}_{\mathsf{c}_{1}\mathsf{c}_{2}\mathsf{d}_{1}\mathsf{d}_{2}}^{\text{Boundary}}\equiv$
$\displaystyle~{}\Big{\langle}\sigma^{(1)}_{\mathsf{c}_{1}}(\mathbf{x}_{V})\sigma^{(1)}_{\mathsf{c}_{2}}(\mathbf{x}_{1})\sigma^{({N+1})}_{\mathsf{d}_{1}}(\mathbf{x}_{N})\sigma^{({N+1})}_{\mathsf{d}_{2}}(\mathbf{x}_{N+1})\Big{\rangle}.$
(61)

Now we focus on the boundary correlator (61). The four boundary operators in
the bra-ket are divided into two groups:
$\\{\sigma^{(1)}_{\mathsf{c}_{2}}(\mathbf{x}_{1}),\sigma^{({N+1})}_{\mathsf{d}_{1}}(\mathbf{x}_{N})\\}$
and
$\\{\sigma^{({N+1})}_{\mathsf{d}_{2}}(\mathbf{x}_{N+1}),\sigma^{(1)}_{\mathsf{c}_{1}}(\mathbf{x}_{V})\\}$.
The two groups are far from each other, while the two operators in each group
are close to each other. This configuration allows us to do the OPE at the
boundary:999The OPEs in (62) and (63) correspond to a free theory. If we turn
on interactions among these boundary fields, there will be other terms
appearing on the right hand side, which may have lower scaling dimension than
the operators shown here. In particular, for the product of two identical
massive scalars $\sigma^{(i)}(\mathbf{x})\sigma^{(i)}(\mathbf{y})$, the right
hand side also includes the identity operator. These operators contribute only
when we include more interaction insertions. The identity operator does not
contribute to the connected diagram.

$\displaystyle\sigma_{\mathsf{c}_{2}}^{(1)}(\mathbf{x}_{1})\sigma_{\mathsf{d}_{1}}^{({N+1})}(\mathbf{x}_{N})=$
$\displaystyle~{}\Big{[}\sigma_{\mathsf{c}_{2}}^{(1)}(\mathbf{x}_{N})+(\mathbf{x}_{1}-\mathbf{x}_{N})\cdot\nabla\sigma_{\mathsf{c}_{2}}^{(1)}(\mathbf{x}_{N})+\cdots\Big{]}\sigma_{\mathsf{d}_{1}}^{({N+1})}(\mathbf{x}_{N}),$
(62)
$\displaystyle\sigma_{\mathsf{d}_{2}}^{({N+1})}(\mathbf{x}_{N+1})\sigma_{\mathsf{c}_{1}}^{(1)}(\mathbf{x}_{V})=$
$\displaystyle~{}\Big{[}\sigma_{\mathsf{d}_{2}}^{({N+1})}(\mathbf{x}_{V})+(\mathbf{x}_{N+1}-\mathbf{x}_{V})\cdot\nabla\sigma_{\mathsf{d}_{2}}^{({N+1})}(\mathbf{x}_{V})\cdots\Big{]}\sigma_{\mathsf{c}_{1}}^{(1)}(\mathbf{x}_{V}).$
(63)

For the soft configuration where the two groups of modes
$\\{\sigma^{(j)}(x_{i})|1\leq i\leq N\\}$ and $\\{\sigma^{(j)}(x_{i})|N+1\leq
i\leq V\\}$ are far from each other, the behavior of the boundary correlator
(61) is dominated by the leading terms, namely:

$\displaystyle\sigma_{\mathsf{c}_{2}}^{(1)}(\mathbf{x}_{1})\sigma_{\mathsf{d}_{1}}^{({N+1})}(\mathbf{x}_{N})\sim$
$\displaystyle~{}\Big{[}\sigma_{\mathsf{c}_{2}}^{(1)}\sigma_{\mathsf{d}_{1}}^{({N+1})}\Big{]}(\mathbf{x}_{N}),$
(64)
$\displaystyle\sigma_{\mathsf{d}_{2}}^{({N+1})}(\mathbf{x}_{N+1})\sigma_{\mathsf{c}_{1}}^{(1)}(\mathbf{x}_{V})\sim$
$\displaystyle~{}\Big{[}\sigma_{\mathsf{d}_{2}}^{({N+1})}\sigma_{\mathsf{c}_{1}}^{(1)}\Big{]}(\mathbf{x}_{V}).$
(65)

Now we can calculate the right hand side of (3.1) and check that the leading
nonlocal piece does give the factorization theorem (8). We rewrite the
integral (3.1) with the leading late-time expansion (55)-(58) and the leading
OPE (64), (65), together with the following rearrangement of the Fourier
kernel:

$\displaystyle~{}\exp\Big{[}-\mathrm{i}\sum_{\ell=1}^{V}\mathbf{K}_{\ell}\cdot\mathbf{x}_{\ell}\Big{]}=\exp\Big{[}-\mathrm{i}\mathbf{P}_{V}\cdot\mathbf{x}_{V}\Big{]}\exp\Big{[}-\mathrm{i}\mathbf{P}_{N}\cdot(\mathbf{x}_{N}-\mathbf{x}_{V})\Big{]}$
$\displaystyle\times\exp\Big{[}-\mathrm{i}\sum_{\ell=1}^{N-1}\mathbf{K}_{\ell}\cdot\mathbf{x}_{\ell}+\mathrm{i}\mathbf{P}_{N-1}\cdot\mathbf{x}_{N}\Big{]}\exp\Big{[}-\mathrm{i}\sum_{\ell=N+1}^{V-1}\mathbf{K}_{\ell}\cdot\mathbf{x}_{\ell}+\mathrm{i}(\mathbf{P}_{V-1}-\mathbf{P}_{N})\cdot\mathbf{x}_{V}\Big{]}.$
(66)

We can further change the measurement for later convenience:

${\mathrm{d}}^{3}\mathbf{x}_{1}\cdots{\mathrm{d}}^{3}\mathbf{x}_{V}={\mathrm{d}}^{3}\mathbf{x}_{1}\cdots{\mathrm{d}}^{3}\mathbf{x}_{N-1}{\mathrm{d}}^{3}\mathbf{y}_{N}{\mathrm{d}}^{3}\mathbf{x}_{N+1}\cdots{\mathrm{d}}^{3}\mathbf{x}_{V},\qquad\mathbf{y}_{N}\equiv\mathbf{x}_{N}-\mathbf{x}_{V}.$
(67)

The integral (3.1) is thus divided into four parts:

  1. 1.

The first part involves the boundary correlator (61):

$\displaystyle\int{\mathrm{d}}^{3}\mathbf{y}_{N}\,e^{-\mathrm{i}\mathbf{P}_{N}\cdot(\mathbf{x}_{N}-\mathbf{x}_{V})}\Big{\langle}\sigma^{(1)}_{\mathsf{c}_{1}}(\mathbf{x}_{V})\sigma^{(1)}_{\mathsf{c}_{2}}(\mathbf{x}_{N})\sigma^{({N+1})}_{\mathsf{d}_{1}}(\mathbf{x}_{V})\sigma^{({N+1})}_{\mathsf{d}_{2}}(\mathbf{x}_{N})\Big{\rangle}$
$\displaystyle=$
$\displaystyle\int{\mathrm{d}}^{3}\mathbf{y}_{N}\,e^{-\mathrm{i}\mathbf{P}_{N}\cdot\mathbf{y}_{N}}\Big{\langle}\sigma^{(1)}_{\mathsf{c}_{1}}(\mathbf{0})\sigma^{(1)}_{\mathsf{c}_{2}}(\mathbf{y}_{N})\sigma^{({N+1})}_{\mathsf{d}_{1}}(\mathbf{0})\sigma^{({N+1})}_{\mathsf{d}_{2}}(\mathbf{y}_{N})\Big{\rangle},$
(68)

where we have made used of the spatial translation symmetry. Again, in the
free-theory limit, this four-point correlation function breaks into a product
of two two-point functions
$\langle\sigma^{(1)}_{\mathsf{c}_{1}}(\mathbf{0})\sigma^{(1)}_{\mathsf{c}_{2}}(\mathbf{y}_{N})\rangle$
and
$\langle\sigma^{({N+1})}_{\mathsf{d}_{1}}(\mathbf{0})\sigma^{({N+1})}_{\mathsf{d}_{2}}(\mathbf{y}_{N})\rangle$.
Since $\sigma^{(1)}_{\mathsf{c}}$ and $\sigma^{(N+1)}_{\mathsf{d}}$ are
primary scalars on the boundary, when
$\mathsf{c}_{1}=\mathsf{c}_{2}\equiv\mathsf{c}$ and
$\mathsf{d}_{1}=\mathsf{d}_{2}\equiv\mathsf{d}$, we have
$\langle\sigma^{(1)}_{\mathsf{c}}(\mathbf{0})\sigma^{(1)}_{\mathsf{c}}(\mathbf{y}_{N})\rangle\sim
y_{N}^{-3\mp\mathrm{i}\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}$ and
$\langle\sigma^{({N+1})}_{\mathsf{d}_{1}}(\mathbf{0})\sigma^{({N+1})}_{\mathsf{d}_{2}}(\mathbf{y}_{N})\rangle\sim
y_{N}^{-3\mp\mathrm{i}\mathsf{d}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N+1}}$. We see that
the equal-sign choices $\mathsf{c}_{1}=\mathsf{c}_{2}\equiv\mathsf{c}$ and
$\mathsf{d}_{1}=\mathsf{d}_{2}\equiv\mathsf{d}$ just give rise to the bubble
signal in (2.1).101010Naively, we would expect that the opposite-sign choice
$\mathsf{c}_{1}=-\mathsf{c}_{2}$ makes the two-point correlator
$\langle\sigma^{(1)}_{\mathsf{c}_{1}}(\mathbf{0})\sigma^{(1)}_{\mathsf{c}_{2}}(\mathbf{y}_{N})\rangle$
vanish, since the two fields $\sigma^{(1)}_{\mathsf{c}_{1}}$ and
$\sigma^{(1)}_{\mathsf{c}_{2}}$ now have different scaling dimensions.
However, $\sigma^{(1)}_{\mathsf{c}_{1}}$ and $\sigma^{(1)}_{\mathsf{c}_{2}}$
does not really vanish, but is proportional to $\delta^{(3)}(\mathbf{y}_{N})$,
since the two representations with $\Delta_{\pm}=3/2\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ are related by a
shadow transformation, and thus are isomorphic to each other. With this said,
the result
$\langle\sigma^{(1)}_{\mathsf{c}_{1}}(\mathbf{0})\sigma^{(1)}_{\mathsf{c}_{2}}(\mathbf{y}_{N})\rangle\sim\delta^{(3)}(\mathbf{y}_{N})$
shows that we should discard all opposite-sign combinations, since they do not
produce any complex power dependences on the coordinate difference
$\mathbf{y}_{N}$, and so they do not contribute to nonlocal signals. Similar
analysis can also be done for $\mathsf{d}_{1}=-\mathsf{d}_{2}$. The net result
is that we only need to keep the same-sign choice for both soft lines.

  2. 2.

The second part is the integral of the first product $\prod_{i=1}^{N-1}D$ in
the bulk correlator
$\mathcal{M}^{\text{(Bulk)}}_{\mathsf{a}_{1}\cdots\mathsf{a}_{V}}$ in (60)
with the third factor on the right hand side of (3.2), over the spatial
coordinates $\mathbf{x}_{1}\cdots\mathbf{x}_{N-1}$ and temporal coordinates
$\tau_{1}\cdots\tau_{N}$, with SK indices
$\mathsf{a}_{1},\cdots,\mathsf{a}_{N}$ summed. This is nothing but the left
tree graph $\mathcal{T}_{\mathsf{c}\mathsf{d}}^{(\text{L})}$ (2.1). We note
that, here and also in the next entry, one should set
$\mathbf{P}_{N}=\mathbf{0}$ for the leading result in the soft limit.

  3. 3.

Similarly, the third part is the integral of the second product
$\prod_{i={N+1}}^{V-1}D$ in the bulk correlator (60) with the fourth factor on
the right hand side of (3.2), over the spatial coordinates
$\mathbf{x}_{N+1}\cdots\mathbf{x}_{V-1}$ and temporal coordinates
$\tau_{N+1}\cdots\tau_{V}$, with SK indices
$\mathsf{a}_{N+1},\cdots,\mathsf{a}_{V}$ summed over. This is nothing but the
right tree graph $\mathcal{T}_{\mathsf{c}\mathsf{d}}^{(\text{R})}$ (2.1).

  4. 4.

Finally, the last part is simply:

$\int{\mathrm{d}}^{3}\mathbf{x}_{V}\,e^{-\mathrm{i}\mathbf{P}_{V}\cdot\mathbf{x}_{V}}=(2\pi)^{3}\delta(\mathbf{P}_{V}),$
(69)

which is the extra momentum conservation constraint in (3.1).

Combining the four parts together, we recover the same result of our
factorization theorem (8). Thus we have shown that the “cut and pinch” process
of the factorization theorem does have a boundary interpretation in terms of
OPE.

## 4 Applications: Bubble, Triangle, and Box

Figure 5: All possible 4-point inflation correlators with nontrivial 1PI
1-loop topologies. The external (black) lines denote bulk-to-boundary
propagators of massless field $\varphi$, while the internal (gray and blue)
lines denote bulk propagator of the massive scalar $\sigma$. The blue lines
represent the soft lines that contribute to the nonlocal signal in the
$s$-channel soft limit $k_{s}\rightarrow 0$.

In this section, we apply the factorization theorem (8) in Sec. 2 to the case
of 4-point inflation correlators of all possible 1PI topologies, including the
bubble, the triangle, and the box diagrams. See Fig. 5. (The case of seagull
diagram is also 1PI but is trivial.) These three 1-loop diagrams have been
very well studied in flat spacetime, and it has long been known that they form
a linearly independent basis for expressing more general 1-loop diagrams via
Passarino-Veltman reduction [98, 99]. In dS, the boundary correlators
corresponding to these three diagrams are much less understood. The complete
analytical result for the 1-loop bubble was obtained only very recently [97],
and the result for the triangle and box diagrams are still unavailable at the
moment. Our new analytical results in this section thus mark a step towards a
full analytical understanding of these graphs in dS.

With the application to CC physics in mind, we consider the case where the
four external lines are time-derivatively coupled massless scalars $\varphi$.
The corresponding bulk-to-boundary propagator is:

$G_{\mathsf{a}}^{\prime}(k;\tau)=\displaystyle\frac{\,{1}\,}{2k}\tau
e^{\mathsf{a}\mathrm{i}k\tau}.$ (70)

We shall consider the $s$-channel squeezed limit, i.e., we send
$k_{s}\equiv|\mathbf{k}_{1}+\mathbf{k}_{2}|\rightarrow 0$ and keep all other
independent magnitudes of momenta
($k_{1},\cdots,k_{4},k_{t}\equiv|\mathbf{k}_{1}+\mathbf{k}_{4}|$) finite.
Also, for simplicity, we shall assume that all massive modes in the loop are
identical, with a common mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$. This is the most
commonly appeared case in CC models. The more general situation with different
masses in the loop will be presented in App. C.

Specialized to the 4-point 1-loop diagrams, the factorization theorem (8) can
be rewritten as:

$\displaystyle\lim_{k_{s}\rightarrow
0}\mathcal{T}\big{(}\\{\mathbf{k}\\}\big{)}=\sum_{\mathsf{c}=\pm}\mathcal{T}_{\mathsf{c}}^{\text{(L)}}\big{(}\mathbf{k}_{1},\mathbf{k}_{2}\big{)}\mathcal{T}_{\mathsf{c}}^{\text{(R)}}\big{(}\mathbf{k}_{3},\mathbf{k}_{4}\big{)}\mathcal{B}_{\mathsf{c}}(k_{s})+\text{terms
analytic in $k_{s}$},$ (71)

where
$\mathcal{T}_{\mathsf{c}}^{(\text{L,R})}\equiv\mathcal{T}_{\mathsf{c}\mathsf{c}}^{(\text{L,R})}$.
The bubble signal (2.1) is reduced to:

$\displaystyle\mathcal{B}_{\mathsf{c}}(k_{s})=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3}}\,}{(4\pi)^{7/2}}\Gamma\begin{bmatrix}-2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu-\mbox{$\frac{\,{3}\,}{2}$}\\\ 3+2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{bmatrix}\Gamma^{2}\Big{[}\mbox{$\frac{\,{3}\,}{2}$}+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{2}\Big{)}^{4\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}.$ (72)

The computation of the nonlocal signals in the 4-point functions are then
reduced to a computation of the left and the right tree graphs, which we shall
discuss for all three cases in turn.

###  4.1 Bubble diagram

Figure 6: The nonlocal signal in the 1-loop bubble graph. On the right hand
side, the two hatched larger blobs denote the insertions with complex powers
of conformal time variables.

We first consider the bubble diagram, for which we choose the following
coupling between the external massless mode $\varphi$ and the internal massive
mode $\sigma$:

$\Delta{\mathscr{L}}=\displaystyle\frac{\,{1}\,}{4}a^{2}\varphi^{\prime
2}\sigma^{2}.$ (73)

Again, the massive field $\sigma$ has the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$.

In this simple case, the computation of the left and right tree graphs is
pretty straightforward:

$\displaystyle\mathcal{T}_{\mathsf{c}}^{(\text{L})}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}k_{2}}\sum_{\mathsf{a}_{1}=\pm}(\mathrm{i}\mathsf{a}_{1})\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}\,e^{\mathsf{a}_{1}\mathrm{i}k_{12}\tau_{1}}(-\tau_{1})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-\tau_{1})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}=\mathcal{P}^{(1)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{12}^{-4-2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{4k_{1}k_{2}},$
(74) $\displaystyle\mathcal{T}_{\mathsf{c}}^{(\text{R})}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{3}k_{4}}\sum_{\mathsf{a}_{2}=\pm}(\mathrm{i}\mathsf{a}_{2})\int_{-\infty}^{0}{\mathrm{d}}\tau_{2}\,e^{\mathsf{a}_{2}\mathrm{i}k_{34}\tau_{2}}(-\tau_{2})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-\tau_{2})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}=\mathcal{P}^{(1)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{34}^{-4-2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{4k_{3}k_{4}},$
(75)

where we have introduced the _pinch coefficient of first order_
$\mathcal{P}_{\mathsf{c}}^{(1)}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$, which is defined
by:

$\mathcal{P}^{(1)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\equiv
2\sin(\pi\mathrm{i}\mathsf{c}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma(4+2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (76)

Now we can directly substitute the two amplitudes
$\mathcal{T}^{\text{(L)}}_{\mathsf{c}}$,
$\mathcal{T}^{\text{(R)}}_{\mathsf{c}}$ obtained here, and the bubble signal
$\mathcal{B}_{\mathsf{c}}$ in (72), into the factorization equation (71), to
obtain the leading nonlocal signal in the bubble diagram. In addition, the
bubble diagram with identical particle has an extra symmetry factor $1/2$.
Altogether, the final result is:

$\displaystyle\lim_{k_{s}\rightarrow
0}\Big{[}\mathcal{T}_{\text{bubble}}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=$
$\displaystyle-\displaystyle\frac{\,{k_{s}^{3}}\,}{8(4\pi)^{7/2}k_{1}k_{2}k_{3}k_{4}k_{12}^{4}k_{34}^{4}}\sinh^{2}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{(}\displaystyle\frac{\,{k_{s}^{2}}\,}{4k_{12}k_{34}}\Big{)}^{2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times(3+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma\Big{[}4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-\displaystyle\frac{\,{3}\,}{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}\Gamma^{2}\Big{[}\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}+\text{c.c.}.$
(77)

We illustrate this result in Fig. 6.

On the other hand, the complete analytical result for this diagram has been
worked out in [97]. In particular, there is a “nonlocal signal” piece in the
full result. This nonlocal signal is the only piece nonanalytic in $k_{s}$ as
$k_{s}\rightarrow 0$ and we quote it here:

$\displaystyle~{}\Big{[}\mathcal{T}_{\text{bubble}}(\\{\mathbf{k}\\})\Big{]}_{\text{NL}}=\displaystyle\frac{\,{(r_{1}r_{2})^{3/2+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{8k_{1}k_{2}k_{3}k_{4}(k_{12}k_{34})^{5/2}}\frac{}{}\sum_{n=0}^{\infty}\displaystyle\frac{\,{(1+n)_{\frac{1}{2}}\big{[}(1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n)_{\frac{1}{2}}\big{]}^{2}(1+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n)_{\frac{1}{2}}}\,}{\pi^{2}\cos(2\pi\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(1+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+2n)_{2}}(\mbox{$\frac{\,{3}\,}{2}$}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+2n)$
$\displaystyle~{}\times{}_{2}\mathcal{F}_{1}\left[\begin{matrix}2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n,\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+n\\\
\mbox{$\frac{\,{5}\,}{2}$}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+2n\end{matrix}\middle|r_{1}^{2}\right]{}_{2}\mathcal{F}_{1}\left[\begin{matrix}2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n,\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+n\\\
\mbox{$\frac{\,{5}\,}{2}$}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+2n\end{matrix}\middle|r_{2}^{2}\right](r_{1}r_{2})^{2n}+\text{c.c.}.$
(78)

Here $r_{1}=k_{s}/k_{12}$ and $r_{2}=k_{s}/k_{34}$. Note that this result
holds to all orders in the powers of $k_{s}$. To compare this result with what
we obtained in (4.1), we take the $k_{s}\rightarrow 0$ limit, which amounts to
keep the $n=0$ term only. It is straightforward to check that the $n=0$ term
of the above result agrees exactly with our result (4.1). Note that the full
result (4.1) was obtained via the spectral decomposition method. It is thus
good to see the agreement between the two results from very different methods.

###  4.2 Triangle diagram

Next, we consider the triangle diagram. Contrary to the bubble diagram, there
is no available analytical result for the triangle 1-loop diagram. Even worse,
it was not clear how we can reliably compute the nonlocal signal for the
triangle loop in the squeezed limit. For the CC application, it was usually
assumed that one of the three loop lines can be pinched so that one can
effectively compute the triangle nonlocal signal with a bubble graph. While
such a pinch could be intuitive, the coupling strength of the resulting
effective operator after the pinch was undetermined. In the literature, it was
normally assumed that this effective coupling strength scales as $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{-2}$, where $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ is the mass
parameter of the _pinched_ line. This behavior may be well justified by the
EFT argument when $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\gg 1$. When $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\sim 1$ or $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu<1$, the situation is much less clear.

Figure 7: The nonlocal signal in the 1-loop triangle graph.

Below, we use the factorization theorem to solve this problem. Coupled with
our previous results for the partial amplitude which corresponds to the left
tree graph, we obtain the triangle nonlocal signal in the $s$-channel squeezed
limit valid for all $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu>0$.

The triangle 1-loop for the 4-point function requires a four-point vertex and
a three-point vertex, which we choose to be:

$\Delta{\mathscr{L}}=\displaystyle\frac{\,{1}\,}{2}a^{3}\varphi^{\prime}\sigma^{2}+\displaystyle\frac{\,{1}\,}{4}a^{2}\varphi^{\prime
2}\sigma^{2},$ (79)

where the massive field $\sigma$ has mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu>0$. Our starting
point is again (71), and we need to compute the left and right tree graphs for
the triangle graph. See Fig. 7. Clearly, the right tree graph is identical to
the case of bubble graph computed in (75):

$\mathcal{T}_{\mathsf{c}}^{(\text{R})}=\mathcal{P}^{(1)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{34}^{-4-2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{4k_{3}k_{4}}.$
(80)

The left tree graph $\mathcal{T}_{\mathsf{c}}^{\text{(L)}}$ is less trivial,
which, according to the general expression

$\displaystyle\mathcal{T}_{\mathsf{c}}^{(\text{L})}(k_{1},k_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k_{1}k_{2}}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}(-\mathsf{a}_{1}\mathsf{a}_{2})\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{\tau_{1}^{2}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{\tau_{2}^{2}}e^{\mathsf{a}_{1}\mathrm{i}k_{1}\tau_{1}+\mathsf{a}_{2}\mathrm{i}k_{2}\tau_{2}}D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{a}_{1}\mathsf{a}_{2}}(k_{1};\tau_{1},\tau_{2})$
$\displaystyle\times(-\tau_{1})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-\tau_{2})^{3/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}.$ (81)

At the leading order of $k_{s}$ in the squeezed limit $k_{s}\rightarrow 0$, we
should set $k_{1}=k_{2}$. The resulting left tree graph
$\mathcal{T}_{\mathsf{c}}^{(\text{L})}(k_{1},k_{1})$ is nothing but a two-
point correlator mediated by a single massive scalar $\sigma$ at the tree
level, but its two couplings have unusual complex power dependences on the two
time variables. This amplitude has been computed in [96] in terms of a two-
point seed integral $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu|\mathsf{a}_{1}\mathsf{a}_{2}}^{p_{1}p_{2}}(1,1)$. In [96], the seed
integral $\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\nu|\mathsf{a}_{1}\mathsf{a}_{2}}^{p_{1}p_{2}}(u_{1},u_{2})$ is defined
as:111111See Eq. (10) of [96]. For applications to cases involving multiple
masses, we include an additional subscript $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ in $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\nu|\mathsf{a}_{1}\mathsf{a}_{2}}^{p_{1}p_{2}}(u_{1},u_{2})$ to
highlight the mass parameter of the field mediating the two-point correlator.

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu|\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},u_{2})\equiv(-\mathsf{a}\mathsf{b})k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{a}\mathsf{b}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k_{s};\tau_{1},\tau_{2}).$ (82)

where $u_{1}\equiv 2k_{s}/(k_{12}+k_{s})$ and $u_{2}\equiv
2k_{s}/(k_{34}+k_{s})$. Comparing (4.2) with (82), we get:

$\mathcal{T}_{\mathsf{c}}^{(\text{L})}(k_{1},k_{2})=\mathcal{P}^{(2)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{k_{12}^{-4-2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{4k_{1}k_{2}},$
(83)

where we have introduced the _pinch coefficient of second order_
$\mathcal{P}^{(2)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$, which is defined
by:

$\mathcal{P}^{(2)}_{\mathsf{c}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\equiv
2^{4+2\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu|\mathsf{a}_{1}\mathsf{a}_{2}}^{-1/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-1/2+\mathsf{c}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(1,1).$ (84)

The seed integral $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu|\mathsf{a}_{1}\mathsf{a}_{2}}^{p_{1}p_{2}}(1,1)$ with both arguments
being unity was computed in [96] with the following results:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu|\pm\pm}^{p_{1}p_{2}}(1,1)=\displaystyle\frac{\,{\pm\mathrm{i}e^{\mp\mathrm{i}{p}_{12}\pi/2}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{2^{5+p_{12}}}\Gamma\left[\begin{matrix}\mbox{$\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