\sum\limits_{\ell=1}^{N_{L}}\alpha_{i\ell}\mathbf{q}_{\ell}^{\mathrm{(L)}}+\sum\limits_{j=1}^{V_{L}}\beta_{ij}\mathbf{K}_{j},&(i=N_{L}+1,\cdots,I_{L})\end{cases}$
(52)

where the coefficients $\alpha_{i\ell}$ take values from $\\{0,\pm 1\\}$, and
$\beta_{ij}$ take values from $\\{0,+1\\}$. (See Footnote 5 for the proof.)

By our construction of the nonlocal soft limit, the first term of the above
expression, $\sum A_{ij}\mathbf{k}_{j}^{\mathrm{(L)}}$, remains finite for all
$i$ when taking the soft limit. Therefore, taking the soft limit is equivalent
to evaluating the left subgraph at the finite values of external momenta
$\mathbf{K}_{i}=\sum A_{ij}\mathbf{k}_{j}^{\mathrm{(L)}}$. According to the
first part of Theorem 1, the loop integral is analytic in the external momenta
for generic finite values in the physical region. Therefore, we conclude that
the left subgraph is analytic in $\mathbf{q}_{\ell}$ and $\mathbf{P}$ in the
simultaneous soft limit $q_{\ell}\rightarrow 0$ ($\ell=1,\cdots,D$). In this
limit, the left subgraph allows a Taylor expansion, whose zeroth order term is

$\displaystyle\mathbb{L}^{\mathrm{(L)}}\Big{(}\\{\mathbf{k}^{\mathrm{(L)}}\\}\Big{)}\equiv$
$\displaystyle~{}\mathbb{L}^{\mathrm{(L)}}\Big{(}\\{\mathbf{k}^{\mathrm{(L)}}\\};\\{\mathbf{0}\\}\Big{)}$
$\displaystyle=$
$\displaystyle\int\prod_{\ell=1}^{N_{L}}\bigg{[}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{\ell}^{\mathrm{(L)}}}\,}{(2\pi)^{3}}\bigg{]}\prod_{i=1}^{I_{L}}\bigg{|}\sum_{\ell=1}^{N_{L}}\alpha_{i\ell}\mathbf{q}_{\ell}^{\mathrm{(L)}}+\sum_{j=1}^{V_{L}}\sum_{k=1}^{B_{L}}\beta_{ij}A_{jk}\mathbf{k}_{k}^{\mathrm{(L)}}\bigg{|}^{-2s_{i\bar{i}}^{\text{(L)}}}.$
(53)

In exactly the same way, we can show that the right subgraph is also analytic
in $q_{\ell}$ in the same soft limit.

#### Factorization of loop momentum integral.

The above analysis shows that we can Taylor expand both the left and right
subgraphs around $\mathbf{q}_{\ell}=\mathbf{0}$ and $\mathbf{P}=\mathbf{0}$.
When we insert such expansions back into the full loop integral $\mathbb{L}$
in (4.2), the higher order terms in $\mathbf{q}_{\ell}$ and $\mathbf{P}$ would
eventually contribute to higher powers of $P$ when we finish the
$\mathbf{q}_{\ell}$ integral, which decreases faster in the $P\rightarrow 0$
limit. Therefore, to single out the leading term, we should keep the leading
term only, which simply means that we can directly set
$\mathbf{q}_{\ell}=\mathbf{P}=\mathbf{0}$ in both $\mathbb{L}^{\mathrm{(L)}}$
and $\mathbb{L}^{\mathrm{(R)}}$. In turn, it means that we can move
$\mathbb{L}^{\mathrm{(L)}}$ and $\mathbb{L}^{\mathrm{(R)}}$ out of the
$\mathbf{q}_{\ell}$-integral. The result is:

$\displaystyle\mathbb{L}\big{(}\\{\mathbf{k}\\}\big{)}=$
$\displaystyle~{}\mathbb{L}^{\mathrm{(L)}}\Big{(}\\{\mathbf{k}^{\mathrm{(L)}}\\}\Big{)}\mathbb{L}^{\mathrm{(R)}}\Big{(}\\{\mathbf{k}^{\mathrm{(R)}}\\}\Big{)}\mathbb{M}_{D-1}(P)\Big{[}1+\mathcal{O}({P})\Big{]},$
(54)

where $\mathbb{M}_{D-1}(P)$ is the $(D-1)$-loop melon integral in the Mellin
representation:

$\displaystyle\mathbb{M}_{D-1}(P)=$
$\displaystyle\int\prod_{\ell=1}^{D-1}\bigg{[}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{\ell}}\,}{(2\pi)^{3}}\big{|}\mathbf{q}_{\ell}\big{|}^{-2s_{\ell\bar{\ell}}}\bigg{]}\Big{|}\mathbf{P}-\sum_{i=1}^{D-1}\mathbf{q}_{i}\Big{|}^{-2s_{D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}}.$ (55)

This integral can be directly done with details shown in App. B, and the
result is:

$\displaystyle\mathbb{M}_{D-1}(P)=$
$\displaystyle~{}\displaystyle\frac{\,{P^{3(D-1)-2s_{1\bar{1}\cdots D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}}}\,}{(4\pi)^{3(D-1)/2}}\Gamma\begin{bmatrix}s_{1\bar{1}\cdots D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}-\mbox{$\frac{\,{3(D-1)}\,}{2}$}\\\
\mbox{$\frac{\,{3D}\,}{2}$}-s_{1\bar{1}\cdots D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}\end{bmatrix}\prod_{\ell=1}^{D}\Gamma\begin{bmatrix}\mbox{$\frac{\,{3}\,}{2}$}-s_{\ell\bar{\ell}}\\\
s_{\ell\bar{\ell}}\end{bmatrix}.$ (56)

It is already clear that the result of the loop momentum integral at the
leading order of $P$ factorizes into three pieces, the left subgraph
$\mathbb{L}^{\mathrm{(L)}}$, the right subgraph $\mathbb{L}^{\mathrm{(R)}}$,
and the melon integral $\mathbb{M}_{D-1}(P)$. In the $P\rightarrow 0$ limit,
both the left and right subgraphs become independent of $P$, and all the $P$
dependences, and thus all potential nonanalyticities in $P$, are from the
melon integral $\mathbb{M}_{D-1}(P)$. From (56) we see that the
(non)analyticity of $\mathbb{M}_{D-1}(P)$ in $P$ is fully controlled by the
value of the power $-2s_{1\bar{1}\cdots D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}$. After we perform the
Mellin integrals, this power will take its value from the location of poles in
all Mellin variables. Therefore, the (non)analyticity of the graph would be
controlled by the locations of these poles. A nonanalytic piece appears
whenever we take poles such that $-2s_{1\bar{1}\cdots D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}\notin\mathbb{N}$. So
the next step is to analyze the pole structure of the partial MB
representation of the graph $\mathcal{G}$ as a function of all Mellin
variables.

#### Pole structure.

A Mellin integral contour runs along a path from $-\mathrm{i}\infty$ to
$+\mathrm{i}\infty$, and in all cases of our interest, the Mellin integrand is
meromorphic. So, we should close the contour properly and pick up the residues
of appropriate poles. In our case, the power $P^{3(D-1)-2s_{1\bar{1}\cdots
D\mkern 2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}}$ in the melon
integral in (56) together with the limit $P\rightarrow 0$ shows that we should
close the MB contour from the left half plane.

Now, let us look at the pole structure of the Mellin integrand. There are
three types of poles:

  1. 1.

_Poles from time integrals._ The Mellin-space time integral
$\mathbb{T}(\\{\mathbf{k}\\})$ in (3.3) only contains right poles of all
Mellin variables. The simplest way to see this is to consider a single layer
of integral:

$\int_{-\infty}^{0}{\mathrm{d}}\tau_{i}\,(-\tau_{i})^{P_{i}-2S_{i}}e^{\mathrm{i}\mathsf{a}_{i}E_{i}\tau_{i}}$
(57)

The integral is converged when $\tau_{i}\rightarrow-\infty$ as is guaranteed
by the Bunch-Davies initial condition. Any singularity in $S_{i}$ must be from
the IR limit $\tau_{i}\rightarrow 0$. In this limit, we can expand the
exponential factor
$e^{\mathrm{i}\mathsf{a}_{i}E_{i}\tau_{i}}=\sum(\mathrm{i}\mathsf{a}_{i}E_{i}\tau_{i})^{n}/n!$,
and we see that the integral (57) is divergent if $1+P_{i}-2S_{i}+n=0$ for all
$n=0,1,2,\cdots$. They correspond to right poles of $S_{i}$. In fact, we can
do this integral explicitly, and get:

$\int_{-\infty}^{0}{\mathrm{d}}\tau_{i}\,(-\tau_{i})^{P_{i}-2S_{i}}e^{\mathrm{i}\mathsf{a}_{i}E_{i}\tau_{i}}=\displaystyle\frac{\,{1}\,}{(\mathrm{i}\mathsf{a}_{i}E_{i})^{1+P_{i}-2S_{i}}}\Gamma(1+P_{i}-2S_{i}).$
(58)

The aforementioned poles are encoded in the Euler $\Gamma$-function. The above
argument can be generalized to arbitrarily nested time integrals to show that
the full time integral, including all possible nesting functions, only
produces right poles in the Mellin variables. We refer the readers to [94] for
details. Since we only need to pick up left poles, the time-integral poles are
irrelevant to us.

  2. 2.

_Poles from the loop momentum integral._ Second, we have a set of poles from
an Euler $\Gamma$ factor in the melon integral (56). The poles from
$\Gamma(\mbox{$\frac{\,{3}\,}{2}$}-s_{\ell\bar{\ell}})$ are right poles and
can be discarded. On the other hand, there is a series of left poles from the
factor $\Gamma(s_{1\bar{1}\cdots D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}-\mbox{$\frac{\,{3(D-1)}\,}{2}$})$:

$\sum_{\ell=1}^{D}(s_{\ell}+\bar{s}_{\ell})=-n+\displaystyle\frac{\,{3(D-1)}\,}{2}.~{}~{}~{}~{}(n=0,1,2,\cdots)$
(59)

This set of poles arise due to the divergence of the melon loop integral in
the UV limit where all loop momenta goes to infinity. For this reason we call
it the UV pole. Clearly, the UV pole forces the combination
$-2s_{1\bar{1}\cdots D\mkern 2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}$ to be an integer and thus its contribution to the melon integral is
analytic in $P$ at $P=0$. So, to evaluate the nonlocal signal, we can discard
this set of poles as well.

  3. 3.

_Infrared poles from Hankel functions._ Third, we have $2D$ sets of poles from
the Euler $\Gamma$ factors in the MB representation of the $D$ cut
propagators. Assuming distinct masses $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}~{}(i=1,\cdots,D)$
for all lines, these $\Gamma$ factors are:

$\displaystyle\prod_{i=1}^{D}\Gamma\Big{[}s_{i}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},s_{i}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},\bar{s}_{i}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},\bar{s}_{i}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}}\,}{2}\Big{]}.$
(60)

So the corresponding poles are:

$\displaystyle
s_{i}=-n_{i}-\displaystyle\frac{\,{\mathsf{c}_{i}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},~{}~{}~{}~{}~{}~{}\bar{s}_{i}=-\bar{n}_{i}-\displaystyle\frac{\,{\mathsf{d}_{i}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},~{}~{}~{}~{}~{}~{}(i=1,\cdots,D)$ (61)

where $\mathsf{c}_{i},\mathsf{d}_{i}=\pm$. Note that these poles are from the
MB representation of the bulk propagators, which correspond to a late-time
expansion of the bulk propagators. The positions of these poles encode the
information of the scaling dimensions of the late-time modes, and for this
reason we shall call them infrared (IR) poles. As we shall show below, these
poles are the only sources of nonlocal signals.

For principal scalars we have $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}\in\mathbb{R}_{+}$
and thus the IR poles can generate noninteger powers for $K$ in the melon
integral (56). So we should consider all possible combinations of IR poles to
get the full nonanalytic contributions to the graph. That is, we should
consider all possible choices of $\mathsf{c}_{i}$ and $\mathsf{d}_{i}$ in
(61).

However, some combinations of $\mathsf{c}_{i}$ and $\mathsf{d}_{i}$ do not
yield nonanalytic results. There are two cases where this can happen.

First, we have a string of $\Gamma$ factors
$1/\Gamma[s_{1\bar{1}},\cdots,s_{D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}]$ in the melon loop
integral (56) which give a set of zeros of degree 1 when $s_{\ell\bar{\ell}}$
are nonpositive integers. These zeros show that we should discard all pole
combinations with $\mathsf{c}_{i}=-\mathsf{d}_{i}$, and only keep the poles
with $\mathsf{c}_{i}=\mathsf{d}_{i}$ for all $i=1,\cdots,D$. So, after this
step, the possible signal poles reduce to:

$\displaystyle
s_{i}=-n_{i}-\displaystyle\frac{\,{\mathsf{c}_{i}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},~{}~{}~{}~{}~{}~{}\bar{s}_{i}=-\bar{n}_{i}-\displaystyle\frac{\,{\mathsf{c}_{i}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},~{}~{}~{}~{}~{}~{}(i=1,\cdots,D)$ (62)

Second, it may happen that, for a particular choice of $\mathsf{c}_{i}$, the
combination $\sum\limits_{i=1}^{D}\mathsf{c}_{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}$ vanishes. For
instance, when the degree of cut $D$ is an even integer and all $D$ cut lines
share the identical mass, we can take $D/2$ of $\mathsf{c}_{i}$ to be $+1$ and
the other $D/2$ of $\mathsf{c}_{i}$ to be $-1$. The combination
$\sum\mathsf{c}_{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}$ is 0. Similar things can also happen when all masses are distinct.
For instance, we can have $D=3$ and $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}=1$, $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}=2$, and $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}=3$. Then the
choice $\mathsf{c}_{1}=\mathsf{c}_{2}=-\mathsf{c}_{3}$ clearly makes the
combination $\sum\mathsf{c}_{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}$ vanish. In such
situations, the melon integral (56) would again give a positive integer power
of $P$ and thus are analytic at $P=0$. So we should discard such combinations
as well.

#### Nonlocal signal cutting rule.

A direct consequence of choosing the nonlocal poles in (62) is that we can
replace the propagators of all $D$ cut lines by their real parts, known as the
nonlocal signal cutting rule [83, 51, 92, 94]. This is a direct generalization
of the nonlocal signal cutting rule for one-loop graph proposed in [94]. To
see this, we note that the MB representation for a bulk propagator (3.3)
contains a factor $e^{\mp\mathrm{i}\pi(s_{i}-\bar{s}_{i})}$. When evaluating
this factor with the nonlocal poles (62), we get $(-1)^{n_{i}-\bar{n}_{i}}$
which is real. The whole propagator (3.3) then becomes real as well. This is
why we can replace the cut propagators by their real parts. Furthermore, after
this replacement, all four propagators $D_{\mathsf{a}\mathsf{b}}$ with
$\mathsf{a},\mathsf{b}=\pm$ become identical, and the time-ordering
$\theta$-functions in $D_{\pm\pm}$ automatically disappear. So, the time
integral factorizes into a left part and a right part.

Let us emphasize again that the cutting rule presented here is conceptually
different from the nonlocal cut introduced earlier. As we showed above, the
cutting rule
$D_{\mathsf{a}\mathsf{b}}\rightarrow\text{Re}\,D_{\mathsf{a}\mathsf{b}}$ is a
consequence of taking nonlocal signal poles (62) for all cut propagators. On
the other hand, a nonlocal cut only indicates which propagators should be made
soft in order to generate a nonlocal signal. It is not a priori clear that we
can take nonlocal signal poles to integrate out Mellin variables of all the
cut propagators. While the procedure of “taking signal poles for all cut
lines” is always possible for bulk-free graphs, we will see in the next
section that this is not the case for more general graphs.

#### Completing the Mellin integral.

Now we are ready to finish all the Mellin integrals in (26) for a bulk-free
graph and complete the proof of Theorem 2. Since we are only concerned with
the leading nonlocal signal, we only need to collect residues of leading
nonlocal poles for the Mellin variables of all cut lines, namely
$n_{i}=\bar{n}_{i}=0$ for all $i=1,\cdots,D$ in (62). Then, the melon integral
(56), together with the factors from MB representations of cut lines
(collectively denoted as $\mathbb{D}(s_{i},\bar{s}_{i})$ in (26)), gives
exactly the melon signal
$\mathfrak{M}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}(P)$ in (45). Since the time
integrals have been fully factorized into a left part and a right part,
finishing the rest of Mellin integrals simply generates the standard SK
integrals for the left subgraph and the right subgraph, as shown in (4.1) and
(4.1). In particular, the additional time-power insertion
$(-\tau_{i})^{3/2+\mathrm{i}\mathsf{c}_{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}}$ comes from the
cut propagator, and is nothing but the leading mode of a bulk propagator in
the late-time limit. Thus we see that, despite of all the complications of
loops, the cut lines can still be expanded at the late-time limit so far as
the nonlocal signal is the only concern.

###  4.3 Discussions

From the proof of Theorem 2, we see that the assumption of bulk-free graphs
brings two great simplifications. First, whenever nonlocal cuts exist, the
minimal cut is guaranteed to be unique. Second, after taking the minimal cut,
the resulting left and right subgraphs are both analytic and finite when
$P\rightarrow 0$. Importantly, the arguments leading to these simplifications
do not rely on the type of couplings and the dispersions of bulk fields. Thus,
we can immediately loosen some of the conditions of Theorem 2 as follows.

  1. 1.

As a trivial extension, we can change the external lines to be (nearly)
massless scalars (such as the inflaton fluctuations), massless spin-2
gravitons, or arbitrary mixtures of them. These are the most relevant cases
for CC applications.

  2. 2.

We can generalize the type of internal fields to include nonzero spins and
more exotic (and possibly dS boost breaking) dispersions. The tensor structure
introduced by the spinning fields does not affect the analytical properties,
and the factorization of the leading nonlocal signal into three pieces still
holds in this case. Of course, the explicit expression such as the melon
signal (45) will be changed, but it is still calculable so long as the MB
representation of bulk propagators can be analytically found. As an example,
we can consider a vector field of mass $m$ with dS-boost-breaking chemical
potential $\mu$. In terms of the vector mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu=\sqrt{m^{2}-1/4}$ and
the dimensionless chemical potential $\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu=\mu/H$, its
propagator can be found to be [92]:

$D_{-+}^{(h)}=\displaystyle\frac{\,{e^{-\pi h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2k}\mathrm{W}_{\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(2\mathrm{i}k\tau_{1})\mathrm{W}_{-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-2\mathrm{i}k\tau_{2}),$ (63)

where $\mathrm{W}_{\kappa,\mu}(z)$ is the Whittaker W function. The MB
representation of this propagator is:

$\displaystyle D_{-+}^{(h)}=$ $\displaystyle~{}e^{-\pi h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}\bar{s}}\,}{2\pi\mathrm{i}}\,e^{+\mathrm{i}\pi(s-\bar{s})/2}(2k)^{-(s+\bar{s})}(-\tau_{1})^{-s+1/2}(-\tau_{2})^{-\bar{s}+1/2}$
$\displaystyle\times{}_{2}{\mathcal{F}}_{1}\left[\begin{matrix}s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
s-\mathrm{i}h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu+\mbox{$\frac{\,{1}\,}{2}$}\end{matrix}\middle|\,\mbox{$\frac{\,{1}\,}{2}$}\right]{}_{2}{\mathcal{F}}_{1}\left[\begin{matrix}\bar{s}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\bar{s}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\bar{s}+\mathrm{i}h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu+\mbox{$\frac{\,{1}\,}{2}$}\end{matrix}\middle|\,\mbox{$\frac{\,{1}\,}{2}$}\right],$
(64)

where ${}_{2}\mathcal{F}_{1}$ is the dressed hypergeometric function, defined
in App. A. Thus we see that the pole structure of this propagator is very
similar to the case of a massive scalar. The integrand also possesses two
pairs of IR poles at $s=-n\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ and
$\bar{s}=-\bar{n}\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$.777The cutting rule,
on the other hand, is obscured for the propagators with nonzero helical
chemical potential if we use the representation (2). As shown in [51], the MB
representation is not unique, and in this case, one can use the so-called
partially resolved MB representation for the bulk propagator to make the
cutting rule manifest.

  3. 3.

We can include arbitrary dS covariant or dS breaking couplings, such as
uncontracted time derivatives or fully contracted space derivatives. A time
derivative on external lines is inconsequential, while a time derivative on an
internal line only shifts the position of right poles produced by the Mellin-
space time integral, which is again irrelevant for the computation of nonlocal
signals. On the other hand, the spatial derivatives all become factors of
momenta, which do not affect the analytic property. The only caveat is that a
spatial derivative on a cut line would make the signal decrease faster than
nonderivative couplings as $P\rightarrow 0$.

## 5 On-Shell Factorization of Arbitrary Graphs

In the previous section, we studied nonlocal signals in arbitrary bulk-free
graphs, and derived an explicit and factorized expression for the nonlocal
signal in the given nonlocal soft limit. The restriction to bulk-free graphs
implies that there are no internal vertices, and this significantly simplifies
the analysis. As a result, we are able to extend the factorization theorem to
rather general couplings and field species, as discussed above.

However, we are ultimately interested in nonlocal signals in more general
graphs, including arbitrary loop graphs with internal vertices. In this case,
the analysis can be significantly involved. Thus, in the following, we shall
first discuss several complications with loop graphs containing internal
vertices, and then focus on a particular case where the minimal cut is unique
with respect to a given bipartition to which we can directly generalize the
on-shell factorization theorem proved in the previous section for bulk-free
graphs. For graphs with multiple minimal cuts, we will see that there can be a
peculiar type of nonlocal signals, which we call hybrid signals. This hybrid
signal further complicates the analysis. Fortunately, as we shall see, such
hybrid signals are absent for graphs with complete dS covariance. Thus, we can
still formulate an on-shell factorization theorem for dS covariant graphs with
most general loop topology. Below we spell out these points in turn.

###  5.1 Complications with internal vertices

#### Leading signal.

The first complication for graphs with internal vertices is that there could
be multiple distinct irreducible cuts associated with a given nonlocal
bipartition. See Fig. 5 for an example. Then, according to Theorem 1, each of
these cuts could potentially contribute a nonlocal signal and we need to
collect all of them to get the full nonanalytic piece in the corresponding
soft limit.

This complication is partially relieved if we only consider the leading
signal. As shown in (22), each of the nonlocal signals in a given soft limit
$P\rightarrow 0$ scale as $P^{\alpha_{\ell}\pm\mathrm{i}\omega_{\ell}}$, where
both $\alpha_{\ell}$ and $\omega_{\ell}$ are positive real numbers. Thus, the
leading signal, namely the signal with the smallest $\alpha_{\ell}$, dominates
over all nonlocal signals in the soft limit.

It then remains to understand which cut gives the leading signal. Intuitively,
we expect that a nonlocal signal would be more subleading if we cut more
lines. The reason is that a nonlocal signal is generated from on-shell
resonance of heavy particles, whose number density is diluted by the expanding
physical volume $V$ according to $1/V\sim 1/a^{3}\sim(-\tau)^{3}$. From the
analysis around (11), we learn that the late-time limit $\tau\rightarrow 0$ is
translated to a squeezed limit $P\rightarrow 0$ in the correlation function.
So, by asking one more internal line to contribute the nonlocal signal, we pay
the price of an additional suppression factor of $P^{3}$. Indeed, this
intuition is very well confirmed by the explicit expression of the melon
signal (45), in which we can read the real part of the power of $P$ as
$P^{3(D-1)}$, where $D$ is the degree of the cut.

The above analysis seems to suggest that we only need to consider minimal cuts
in order to get the leading signal. However, there are further complications,
as we shall discuss below.

#### Derivative couplings.

An immediate complication comes from derivative couplings. In a general setup
where the dS boosts are explicitly broken (such as in the EFT of inflation
[116]), we can have spatial-derivative couplings without temporal-derivative
counterparts. Suppose we have spatial derivatives $-\partial_{i}^{2}$ acting
on an endpoint of a soft bulk propagator, then, in the momentum space, this
gives rise to a factor of $P^{2}$ where $P$ is the momentum of the soft
propagator. As a result, it might be possible that the signal from a minimal
cut decreases even faster than the signal from a nonminimal cut, countering
our earlier intuition.

We note that only spatial derivatives change the scaling behavior of the
signal in the soft limit. Neither the time derivatives nor the non-scale-
invariant time powers at each interaction vertex generate similar
complications. This can be very easily seen using the partial MB
representation, in which the time and momentum variables are fully separated,
and thus any change in the time integral $\mathbb{T}(\\{k\\};\\{s,\bar{s}\\})$
cannot alter the power counting of momentum partial sums in the momentum
sector $\mathbb{L}(\\{\mathbf{k}\\};\\{s,\bar{s}\\})$.

The complication of derivative couplings can be addressed by defining the
_drop_ $\Delta$ of a nonlocal cut, which measures how fast the would-be signal
decreases in the late-time/squeezed limit. In the late-time limit, a massive
propagator behaves like $k^{3}\times$nonanalytic factor. Then, when there are
spatial derivatives, we can define the total drop of a cut of Degree $D$ as:

$\displaystyle\Delta=3D+\sum_{\ell=1}^{D}\delta_{i},$ (65)

where $\delta_{i}\geq 0$ denotes the total number of spatial derivatives
acting on both sides of the $i$’th cut propagator. Then, the leading signal
would be given by the cut of minimal drop $\Delta$ rather than the minimal
degree $D$. This is indeed the case if there is a unique cut of minimal drop
$\Delta$ in a given bipartition. Due to another complication we shall discuss
below, when there are multiple cuts sharing the same minimal drop, the
situation can be more complicated.

On the other hand, we note that the complication of spatial derivatives is
always irrelevant in dS covariant graphs. The reason is simple: Whenever there
is a derivative coupling in a dS covariant graph, the spatial and temporal
derivatives must appear together. So, in the above example of
${\mathscr{L}}\supset-(\partial_{i}^{2}\sigma)\cdots$ we must have terms like
${\mathscr{L}}\supset[\sigma^{\prime\prime}-\partial_{i}^{2}\sigma]\cdots$.
While the spatial derivatives make the signal decrease faster in the soft
limit, the temporal derivatives do not. Thus, the temporal derivatives give
the leading signal, whose scalings are identical to our naive counts of number
of cuts. Thus, we conclude that, in dS covariant graphs, the leading nonlocal
signal is given by the minimal cut, at least when the minimal cut is unique.

Thus, in the following analysis, when we mention a minimal cut, we always mean
a cut of minimal degree $D$ for dS covariant graphs, or a cut of minimal drop
$\Delta$ for boost breaking graphs.

There are additional complications with graphs containing internal vertices
when the minimal cuts are not unique in a given nonlocal soft limit, to be
discussed below. For now, the analysis of the derivative couplings is
sufficient for us to formulate the on-shell factorization theorem for
arbitrary loop diagrams with unique minimal cut. Thus, we shall first
formulate this theorem in Sec. 5.2 before considering the most general case in
Sec. 5.3.

###  5.2 On-shell factorization with unique minimal cut

The above analysis shows that the complication from internal vertices can be
largely circumvented if we only consider the leading signal, and if there
exists a unique minimal cut associated with a given nonlocal bipartition.
Equipped with this knowledge, we are ready to generalize the on-shell
factorization to arbitrary loop diagrams with unique minimal cut, as
summarized in the following theorem. As we shall see, this is the most general
situation where we can formulate a theorem with certain mathematical rigor.

######  Theorem 3 (On-shell factorization with unique minimal cut)

Let $\mathcal{G}(\\{\mathbf{k}\\})$ be an arbitrary $B$-point and $L$-loop
graph with dS covariant propagators and couplings, and let
$\\{\mathbf{k}^{\mathrm{(L)}}\\}\cup\\{\mathbf{k}^{\mathrm{(R)}}\\}$ be a
nonlocal bipartition of $\mathcal{G}(\\{\mathbf{k}\\})$. Then, if there exists
a unique minimal cut of degree $D$ associated with this nonlocal bipartition,
then the nonlocal signal in the limit
$\mathbf{P}\equiv\sum\mathbf{k}^{\mathrm{(L)}}\rightarrow\mathbf{0}$
factorizes into the following three pieces at the leading order in $P$:

$\displaystyle\lim_{P\rightarrow 0}\mathcal{G}(\\{\mathbf{k}\\})=$
$\displaystyle\sum_{\begin{subarray}{c}\mathsf{c}_{1},\cdots,\mathsf{c}_{D}=\pm\\\
\text{nonlocal}\end{subarray}}\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}^{\mathrm{(L)}}\Big{(}\\{\mathbf{k}^{\mathrm{(L)}}\\}\Big{)}\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}^{\mathrm{(R)}}\Big{(}\\{\mathbf{k}^{\mathrm{(R)}}\\}\Big{)}\mathfrak{M}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}(P)$
$\displaystyle+$ $\displaystyle~{}\text{(subleading nonlocal
signals)}+\text{(terms analytic in $P$)}.$ (66)

Here the melon signal $\mathfrak{M}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}(K)$,
the left subgraph
$\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}^{\mathrm{(L)}}(\\{\mathbf{k}^{\mathrm{(L)}}\\})$,
and the right subgraph
$\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}^{\mathrm{(R)}}(\\{\mathbf{k}^{\mathrm{(R)}}\\})$
are identical to the case of Theorem 2, and are respectively given by (4.1),
(4.1), and (45).

_Proof._ We focus on the nonlocal signal generated from the minimal cut with
degree $D$. Since a minimal cut is necessarily irreducible, by Lemma 2, it
generates a perfect bipartition of the graph. As a result, the graph is fully
resolved into a left subgraph, a right subgraph, and $D$ cut lines. Then, the
proof goes in the same way as we did for Theorem 2 in the last section. The
difference is that the left and right subgraphs are not guaranteed to be
analytic in $q_{\ell}$ as $q_{\ell}\rightarrow 0$ where $q_{\ell}$ is the
momentum of the $\ell$’th cut line ($\ell=1,\cdots,D$). Due to the existence
of additional cuts, the left and right subgraphs can exhibit additional
nonanalytic behaviors in $q_{\ell}\rightarrow 0$. However, thanks to the
assumption of unique minimal cut, the nonlocal signals arising from those
additional nonanalytic terms would be subleading in the soft limit
$P\rightarrow 0$.

Taking account of additional nonanalytic terms in the left and right
subgraphs, we can write the soft limit $q_{\ell}\rightarrow 0$ of the Mellin-
space loop integral of the left subgraph, as,

$\displaystyle\lim_{q_{\ell}\rightarrow
0}\mathbb{L}^{\mathrm{(L)}}\Big{(}\\{\mathbf{k}^{\mathrm{(L)}}\\},\\{\mathbf{q}\\}\Big{)}=\sum_{n=0}^{\infty}\displaystyle\frac{\,{1}\,}{n!}q^{n}\bigg{[}\partial^{(n)}\mathbb{L}^{\mathrm{(L)}}\Big{(}\\{\mathbf{k}^{\mathrm{(L)}}\\},\\{\mathbf{q}\\}\Big{)}\bigg{]}_{q=0}+\text{nonanalytic
term}.$ (67)

That is, the analytic part of the left loop integral can be Taylor expanded
around $q_{\ell}\rightarrow 0$, and the leading order ($n=0$) term is
generically nonzero. The nonanalytic term, on the other hand, typically
decreases to zero in the $q_{\ell}\rightarrow 0$ limit, so long as the
interactions are IR finite. Indeed, from the expression of the melon signal in
(45), we see that a cut of degree $D$ yield a nonanalytic term (namely, a
signal) that scales like $P^{3(D-1)+\mathrm{i}\sum\mathsf{c}_{\ell}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell}}$.

So, the leading contribution from the left subgraph in the
$q_{\ell}\rightarrow 0$ limit is from the $q_{\ell}$-independent term of its
analytic part. Then, the rest of the proof goes in exactly the same way as the
last section. Now, so long as the couplings are dS covariant, the decreasing
speed of a nonlocal signal is exactly given by the degree of the corresponding
nonlocal cut. Therefore, it is guaranteed that the nonanalytic terms of left
or right subgraphs are subdominant relative to the nonlocal signal from the
minimal cut. So, the leading nonlocal signal must be from the term explicitly
shown in (67). Then, going through all similar steps as in the proof of
Theorem 2, we end up with the factorized results in (3). This completes the
proof of Theorem 3.

Clearly, the above proof can be readily generalized to the case of boost
breaking interactions and dispersions. To address the complication of spatial
derivatives, we only need to change the condition of unique minimal cut to the
unique cut of minimal drop $\Delta$. It is also straightforward to see that
the cut of minimal drop is necessarily an irreducible cut, so the perfect
bipartition (left subgraph $+$ right subgraph $+$ cut lines) still applies by
Lemma 2.

###  5.3 On-shell factorization with multiple minimal cuts

Now let us consider the most general case where there exist more than one
minimal cut in a given nonlocal bipartition. As alluded to several times,
things get even more complicated if the minimal cut is not unique for a given
bipartition. To appreciate the difficulty in this case, we need to introduce
another subtlety associated with internal vertices, namely the
$\delta$-constraint in the Mellin space.

####  $\bm{\delta}$-constraint from internal vertices.

The subtlety is this: While it may be possible to cut all the bulk lines
attached to a given internal vertex, it is impossible to get nonlocal signals
from all these lines, due to the $\delta$-constraint of an internal vertex in
the Mellin space.

To see this point more explicitly, we consider an internal vertex with the
time variable $\tau$ and let there be $N$ bulk propagators attached to this
vertex. Then, if we cut all these $N$ bulk propagators, we may want to naively
apply the cutting rule
$D_{\mathsf{a}\mathsf{b}}\rightarrow\text{Re}\,D_{\mathsf{a}\mathsf{b}}$,
which would allow us to remove all the time orderings between $\tau$ and any
other time variables. In this case, the $\tau$-integral can be isolated:

$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau(-\tau)^{p}\prod_{i=1}^{N}\,\text{Re}\,D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i})}(p_{i},\tau,\tau_{i}).$ (68)

Now, in the Mellin space, each propagator $D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i})}$ gives rise to
a factor of $(-\tau)^{-2s_{i\bar{i}}}$. Thus, the time integral yields a
$\delta$-function:

$\int_{-\infty}^{0}{\mathrm{d}}\tau(-\tau)^{p-2s_{1\bar{1}\cdots N\mkern
2.0mu\overline{\mkern-2.0muN\mkern-2.0mu}\mkern
2.0mu}}=2\pi\delta\big{[}\mathrm{i}(1+p-2s_{1\bar{1}\cdots N\mkern
2.0mu\overline{\mkern-2.0muN\mkern-2.0mu}\mkern 2.0mu})\big{]}.$ (69)

On the other hand, as we showed in the last section, for all $N$ lines to
contribute nonlocal signals, it is important to take the (IR) signal poles. In
the current situation, it means that we should choose:

$\displaystyle
s_{1}=-n_{1}+\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},\qquad\bar{s}_{1}=-\bar{n}_{1}+\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2}\cdots,$
$\displaystyle
s_{N}=-n_{N}+\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{N}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{N}}\,}{2},\qquad\bar{s}_{N}=-\bar{n}_{N}+\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{N}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{N}}\,}{2},$ (70)

where $\mathsf{c}_{1},\cdots,\mathsf{c}_{N}=\pm 1$. So, the possibility that
all these lines together contribute a nonlocal signal with
$\sum\mathrm{i}\mathsf{c}_{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}\notin\mathbb{R}$
is obviously inconsistent with the $\delta$-function constraint (69). This
gives us a sharp example showing that, while we can take a nonlocal cut by
making certain lines soft, we cannot get nonlocal signals from all these soft
lines. As a result, the cutting rule does not apply.

The $\delta$-constraints add another complication to our analysis. In the
partial MB representation, we need to finish all Mellin integrals to recover
the ordinary expressions for the graph. As shown in the proof of Theorem 2, we
only need to pick up left poles of all Mellin variables, and the only left
poles are UV poles from the loop momentum integrals, and the IR poles from the
MB representation of the bulk massive propagators. Now, the
$\delta$-constraint introduces another possibility: we can carry out
integration over some Mellin variables by the $\delta$-constraint, and the
resulting expression would flip the side of poles of other Mellin variables.
While we can postpone the use of $\delta$-constraint to deal with the side-
flipping problem, we cannot avoid the $\delta$-constraint altogether. As we
shall see below, the $\delta$-constraint may lead to peculiar signals when
there are multiple minimal cuts in a graph.

#### Hybrid signal from multiple minimal cuts.

Now we come back to the discussion of multiple minimal cuts. In this case, our
first guess may be that the leading nonlocal signal in the soft limit is the
sum of the nonlocal signals corresponding to each of the nonlocal cuts.
However, our experience with explicit examples suggests that there can be
additional contributions to the leading nonlocal signal, which can have the
same fall-off as those signals from a minimal cut, but can not be obtained by
adding nonlocal signals from all minimal cuts alone. This is closely related
to the $\delta$-constraint discussed above.

At this point, we suggest the readers have a look at the two-point mixing
example in Sec. 6.2 before reading on. To see the peculiarity of this example,
we note that, in all examples hitherto considered, the nonlocal signal was
obtained by taking signal poles for both of Mellin variables of all cut lines:

$\displaystyle s=-n\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu/2,$
$\displaystyle\bar{s}=-\bar{n}\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu/2.$ (71)

Using our intuition of on-shell resonance, we may say that the signal comes
from the resonance of both endpoints of a cut propagator. Now, we see that, in
the example of 2-point mixing graph in Fig. 11, one can generate a nonlocal
signal by asking the two modes attached to the two cubic vertices to resonate,
while the two modes at the two-point mixing vertex are “locked” by the
$\delta$-constraint. This yields a nonlocal signal that scales with the same
real power as the signal from a single soft propagator, namely $P^{0}$, but
with its imaginary power being $P^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$, which we call
a _hybrid signal_. So, in this two-point mixing example, we would expect that
the leading signals have three terms:

$\displaystyle P^{\pm 2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}},$ $\displaystyle
P^{\pm 2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}},$ $\displaystyle P^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}.$ (72)

The first two are ordinary signals, corresponding to cutting only the left
internal line or only the right internal line in Fig. 11, and requiring both
modes of a single line to resonate. The last one, namely the hybrid signal,
corresponds the peculiar situation where we cut two lines, but the resulting
signal scales as if it is from cutting one line.

We expect that this feature also appears in more general loop graphs whenever
there are multiple minimal cuts: The full leading signal is not only from the
ordinary signal of each individual minimal cut, but also include hybrid
signals from several one-side resonances. It is actually not clear if we would
get more peculiar signals in more complicated loop graphs, and this is the
major reason that we cannot get a simple and neat factorization theorem for
the most general loop diagram.

One possible way to see the appearance of the hybrid signal is to think of a
loop graph with internal vertices as a soft limit of a bulk-free graph without
any internal vertices: We can artificially insert auxiliary external legs with
energies $E_{i}$ to all internal vertices and thus turn them into external
vertices. Then, we get a bulk-free graph which appears much simpler. The
original graph can then be recovered by taking $E_{i}\rightarrow 0$ for all
the auxiliary external legs. Then, one may attempt to think of the hybrid
signal as a combination of ordinary signals in the bulk-free graph in the
corresponding soft limit. However, the real situation is not as trivial. The
reason is that, when we talk about ordinary signals, we always assume that the
momenta $P_{j}$ of the signal-generating bulk propagators are much softer than
any other external momenta, where the resonance picture is well applicable.
That is, we need $P_{j}/E_{i}\ll 1$ for all $i$ and $j$. However, when we turn
off the momenta of all auxiliary external legs, the above assumption no longer
holds. As a consequence, we need to know how to take analytical continuation
of the bulk-free graph from the region $P_{j}/E_{i}\ll 1$ to
$P_{j}/E_{i}\rightarrow\infty$. This is a nontrivial task which we leave for a
future study.

#### Absence of hybrid signals in dS covariant graphs.

Although the hybrid signals make the analysis of nonlocal signals rather
intractable, they are likely absent in dS covariant graphs. In the example of
two-point mixing graph in Sec. 6.2, the absence of the hybrid signal in the dS
covariant limit is clearly observed. There, we have a simple explanation: a
covariant mixing (such as a mass mixing) can always be rotated away by a
linear field redefinition. Of course this explanation does not apply directly
to arbitrary loop graphs, in which case we have to resort to other arguments.

Fortunately, we can find such an argument from CFT, using our interpretation
of nonlocal signals as two-point function at the future boundary. (See the end
of Sec. 2.1, and also [94] for more detailed expositions.) From our proof of
Theorem 2, we see that the leading nonlocal signal is obtained by cutting open
all soft lines while pitching all hard lines. Here the “pitching” procedure
means that we can take all hard lines out of the loop integral as we did
around (54). As a result, nonlocal signals in arbitrary loop graphs can also
be viewed as the late-time limit of a two-point function. We illustrate this
“cut and pinch” procedure in Fig. 9. Then, so long as the bulk graph is dS
covariant, the corresponding boundary two-point correlator will be conformal
covariant, and thus diagonal in the scaling dimensions:

$\lim_{\tau_{1},\tau_{2}\rightarrow
0}G(P;\tau_{1},\tau_{2})=\sum_{\Delta_{1},\Delta_{2}}G_{\Delta_{1}\Delta_{2}}(P)(-\tau_{1})^{\Delta_{1}}(-\tau_{2})^{\Delta_{2}},$
(73)

where $\Delta_{1,2}$ denote the scaling dimension of the boundary two-point
function $G_{\Delta_{1}\Delta_{2}}(P)\propto P^{\Delta_{1}+\Delta_{2}-3}$. The
crucial point is that bulk dS isometry (or equivalently, the boundary
conformal symmetry) demands that $\Delta_{1}=\Delta_{2}$ or $\Delta_{1}=\mkern
2.0mu\overline{\mkern-2.0mu\Delta\mkern-2.0mu}\mkern 2.0mu_{2}$ where $\mkern
2.0mu\overline{\mkern-2.0mu\Delta\mkern-2.0mu}\mkern 2.0mu_{2}=3-\Delta_{2}$
is the shadow counterpart of $\Delta_{2}$. The latter case with
$\Delta_{1}=\mkern 2.0mu\overline{\mkern-2.0mu\Delta\mkern-2.0mu}\mkern
2.0mu_{2}$ is irrelevant to us since it makes $G(P)$ independent of $P$ and
only produces a contact term $\propto\delta^{(3)}(\mathbf{x})$ in position
space.888However, in this part, there is still a factor
$(-\tau_{1})^{\Delta_{1}}(-\tau_{2})^{\mkern
2.0mu\overline{\mkern-2.0mu\Delta\mkern-2.0mu}\mkern 2.0mu_{2}}$ which could
be nonanalytic in the time variables. They would produce a local signal
instead of nonlocal signal. Thus the only possible nonanalytic behavior must
have $\Delta_{1}=\Delta_{2}\equiv\Delta$. Therefore the hybrid signal with
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\neq\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$ is precluded
since it corresponds to
$G(P;\tau_{1},\tau_{2})\propto(-P\tau_{1})^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}(-P\tau_{2})^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$ in the late-
time limit, inconsistent with the conformal constraint.

Figure 9: Illustration of the absence of hybrid signals in dS covariant
graphs. The leading nonlocal signal in any loop diagram can be understood as a
two-point function on the late-time boundary. In a dS covariant graph, the
resulting two-point function respects the boundary conformal symmetry, and
thus must be diagonal in the scaling dimensions at its two endpoints. This
precludes all hybrid signals.

#### On-shell factorization of most general dS covariant graphs.

Above we have shown that the main complication from internal vertices with
non-unique minimal cuts is the generation of hybrid signals, however, this
complication is absent if the graph respects the full dS isometry. Thus, if we
restrict ourselves to dS covariant processes, it would be possible to
formulate the following theorem for leading nonlocal signals in arbitrary loop
graphs:

######  Theorem 4 (Nonlocal signals in general graphs)

Let $\mathcal{G}(\\{\mathbf{k}\\})$ be an arbitrary $B$-point ($B\geq 4$) and
$L$-loop ($L\geq 0$) graph with dS covariant propagators and couplings, and
let $\\{\mathbf{k}^{\mathrm{(L)}}\\}\cup\\{\mathbf{k}^{\mathrm{(R)}}\\}$ be a
nonlocal bipartition of $\mathcal{G}(\\{\mathbf{k}\\})$. Then, the leading
nonlocal signal in the limit
$\mathbf{P}\equiv\sum\mathbf{k}^{\mathrm{(L)}}\rightarrow\mathbf{0}$ is
contributed (probably exclusively) by the sum of signals coming from all
minimal cuts:

$\displaystyle\lim_{P\rightarrow 0}\mathcal{G}(\\{\mathbf{k}\\})=$
$\displaystyle\sum_{\text{C}}\sum_{\begin{subarray}{c}\mathsf{c}_{1},\cdots,\mathsf{c}_{D}=\pm\\\
\text{nonlocal}\end{subarray}}\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}^{\mathrm{L}(C)}\Big{(}\\{\mathbf{k}^{\mathrm{(L)}}\\}\Big{)}\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}^{\mathrm{R}(C)}\Big{(}\\{\mathbf{k}^{\mathrm{(R)}}\\}\Big{)}\mathfrak{M}^{(C)}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}(K)$
$\displaystyle+$ $\displaystyle~{}\text{(other nonlocal signals)}+\text{(terms
analytic in $K$)}.$ (74)

Here $C$ labels all minimal cuts with respect to the given bipartition, and
$D$ denotes the degree of the minimal cuts. The three factors
$\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}^{\mathrm{L}(C)}(\\{\mathbf{k}^{\mathrm{(L)}}\\})$,
$\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}^{\mathrm{R}(C)}(\\{\mathbf{k}^{\mathrm{(R)}}\\})$,
$\mathfrak{M}^{(C)}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}(K)$ denote the left
subgraph, the right subgraph, and the melon signal generated by the cut $C$.
The term denoted by “other nonlocal signals” is probably subleading in the
$P\rightarrow 0$ limit.

###  5.4 Discussions

Theorem 4 shows that the leading nonlocal signal of an arbitrary (dS
covariant) graph is always generated by one (or several if the minimal cut is
not unique) $(D-1)$-loop melon subgraph with $D\geq 1$, which is formed by all
the cut lines in the corresponding minimal cut. This can be intuitively
understood as from a “cut and pinch” process: We first identify a minimal cut
that could generate a melon signal, and then pinch all the other lines and
loops. There is also a boundary OPE perspective: The cut lines are soft and we
can push them to the future boundary of dS spacetime, where the massive modes
become boundary operators with scaling dimensions $3/2\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell}$. Then we can
do the OPE to the endpoints of the cut lines, such that they are pinched
together and the soft lines become a melon graph. See [94] for more details.

It is interesting to see how we can pinch a loop. In fact, this is related to
the UV pole of the Mellin-space loop momentum integral
$\mathbb{L}(\\{\mathbf{k}\\};\\{s,\bar{s}\\})$ in (29). The simplest example
is a bubble integral (namely the 1-loop melon integral):

$\displaystyle\mathbb{M}_{1}(P)=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{1\bar{1}}}|\mathbf{P}-\mathbf{q}|^{-2s_{2\bar{2}}}=\displaystyle\frac{\,{P^{3-2s_{1\bar{1}2\bar{2}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}s_{1\bar{1}2\bar{2}}-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}}\\\
3-s_{1\bar{1}2\bar{2}},s_{1\bar{1}},s_{2\bar{2}}\end{bmatrix}.$ (75)

Clearly, the integral $\mathbb{M}_{1}(P)$ possesses a set of UV poles at
$s_{1\bar{1}2\bar{2}}=-n+3/2$ with $n=0,1,\cdots$. If we take the residue of
$\mathbb{M}_{1}(P)$ at the leading UV pole $s_{1\bar{1}2\bar{2}}=3/2$, we get:

$\underset{s_{1\bar{1}2\bar{2}}=3/2}{\text{Res}}\Big{[}\mathbb{M}_{1}(P)\Big{]}=\displaystyle\frac{\,{1}\,}{4\pi^{2}}.$
(76)

This shows that, at the leading UV pole, the residue of the bubble integral
reduces to a constant number. In this sense, the bubble integral is “pinched”
to a point.

This is in fact a more general statement. Consider a particular layer of
arbitrary loop momentum integral in the Mellin space, which corresponds to a
particular sub-loop structure in the original loop graph:

$\mathbb{L}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}\prod_{i=1}^{N}|\mathbf{q}+\mathbf{p}_{i}|^{-2s_{i\bar{i}}},$
(77)

where $\mathbf{p}_{i}$ is an arbitrary combination of external momenta and
loop momenta other than $\mathbf{q}$. In general, $\mathbb{L}$ is a function
of $\\{s_{i},\bar{s}_{i}\\}$ with some poles: When $\\{s_{i},\bar{s}_{i}\\}$
take certain values, this integral could diverge in the hard region of
$\mathbf{q}$, and the divergent part can be obtained by:

$\mathbb{L}|_{\text{divergent
part}}=\int_{\Lambda}^{\infty}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}\prod_{i=1}^{N}|\mathbf{q}+\mathbf{p}_{i}|^{-2s_{i\bar{i}}}.$
(78)

Here $\Lambda$ is a hard scale: $\Lambda\gg|\mathbf{p}_{i}|$. Thus we can then
neglect all the $\mathbf{p}_{i}$, and therefore:

$\mathbb{L}|_{\text{divergent
part}}=\int_{\Lambda}^{\infty}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}\prod_{i=1}^{N}|\mathbf{q}|^{-2s_{i\bar{i}}}=\int_{\Lambda}^{\infty}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2\sum
s_{i\bar{i}}}.$ (79)

Now notice that:

$\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2\sum
s_{i\bar{i}}}=\displaystyle\frac{\,{1}\,}{\pi}\delta\big{[}\mathrm{i}(3-2\sum
s_{i\bar{i}})\big{]},$ (80)

we can then extract the possible UV divergence of $\mathbb{L}$:

$\displaystyle\mathbb{L}|_{\text{divergent part}}=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2\sum
s_{i\bar{i}}}-\int_{0}^{\Lambda}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2\sum
s_{i\bar{i}}}$ $\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{\pi}\delta\big{[}\mathrm{i}(3-2\sum
s_{i\bar{i}})\big{]}-\displaystyle\frac{\,{1}\,}{2\pi^{2}}\displaystyle\frac{\,{\Lambda^{3-2\sum
s_{i\bar{i}}}}\,}{3-2\sum s_{i\bar{i}}}.$ (81)

Therefore, $\mathbb{L}$ indeed has a UV pole $\sum s_{i\bar{i}}=3/2$, and the
residue is:

$\underset{\sum
s_{i\bar{i}}=3/2}{\text{Res}}\,\mathbb{L}=\displaystyle\frac{\,{1}\,}{4\pi^{2}}.$
(82)

That is, when we calculate the Mellin integral, we are able to take the UV
pole of any loop integral and pinch the loop to a constant.

Now we can understand our theorem (4) with this pinch idea. All the soft lines
will be diluted due to the exponential expansion of the spacetime. Obviously
we cannot pinch all the loops, otherwise we will not obtain a signal. So we
should pinch all the lines and loops but leave one melon subgraph with minimal
degree, which corresponds to a minimal cut. If there are multiple minimal
cuts, we should sum the signals together since we should sum residues at all
the allowable pole configurations.

## 6 Examples

Below we will present some concrete examples of four-point inflation
correlators, and explicitly show how our theorems work. For definiteness, we
set all the external lines to be the conformal scalar $\varphi$ with
$m^{2}=2$, all the bulk lines to be massive scalars $\sigma_{\ell}$ in the
principal series with different mass parameters $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell}>0$. It is
assumed that all fields are coupled directly without any spatial or temporal
derivatives, but with arbitrary power dependences $(-\tau_{i})^{p_{i}}$ in the
coupling. For simplicity we also assume that $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell}$ take generic
values such that $\sum\limits_{\ell}\mathsf{c}_{\ell}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell}\neq 0$ for any
$\mathsf{c}_{\ell}\in\\{0,\pm 1\\}$ with at least one $\mathsf{c}_{\ell}\neq
0$. We will compute the Mellin-space loop integral (29) and analyze its
possible pole structures which could generate a nonlocal signal.

#### Notations.

In this section, we use solid lines to denote massless inflaton, and the
external momenta $\mathbf{k}_{1},\cdots,\mathbf{k}_{4}$ are flowing inwards.
We specify the nonlocal soft limit to be the s-channel squeezed limit, namely
$\mathbf{k}_{s}\equiv\mathbf{k}_{1}+\mathbf{k}_{2}=-\mathbf{k}_{3}-\mathbf{k}_{4}$
and $\mathbf{k}_{s}\rightarrow 0$. Each internal line $\ell$ is labeled by its
mass $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell}$.
In the Mellin space, we introduce two Mellin variables
$s_{\ell},\bar{s}_{\ell}$ to the two endpoints of the bulk line $\ell$,
respectively. We use an arrow to clarify our choice: The arrow always points
from $s_{\ell}$ to $\bar{s}_{\ell}$. The independent loop momenta
$\mathbf{q}_{i}$ will be explicitly labeled in graphs, flowing in the arrowed
direction.

###  6.1 Melon graph

As a first example, let us consider the $L$-loop melon graph as shown in Fig.
10. This graph is bulk-free, and there is only one nonlocal cut, namely
cutting all the $(L+1)$ massive propagators simultaneously.

Let us write down the expression for the melon graph. We neglect the constant
factor $-\tau_{f}/(2k)$ in the bulk-to-boundary propagator (23), and absorb
the factor $-\tau$ into the interaction $(-\tau_{i})^{p_{i}}$, then the melon
graph is expressed as the following:

$\displaystyle\mathcal{G}=-\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{+\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}\mathcal{M}^{(L)}(k_{s};\tau_{1},\tau_{2}).$
(83)

Here $\mathcal{M}^{(L)}(k_{s};\tau_{1},\tau_{2})$ is the $L$-loop integral of
the $L+1$ massive propagators:

$\displaystyle\mathcal{M}^{(L)}(k_{s};\tau_{1},\tau_{2})=$
$\displaystyle\int\prod_{i=1}^{L}\bigg{[}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{i}}\,}{(2\pi)^{3}}\bigg{]}D_{\mathsf{a}\mathsf{b}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}\Big{(}|\mathbf{q}_{1}|;\tau_{1},\tau_{2}\Big{)}D_{\mathsf{a}\mathsf{b}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}\Big{(}|\mathbf{q}_{2}-\mathbf{q}_{1}|;\tau_{1},\tau_{2}\Big{)}$
$\displaystyle\times\cdots D_{\mathsf{a}\mathsf{b}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{L})}\Big{(}|\mathbf{q}_{L}-\mathbf{q}_{L-1}|;\tau_{1},\tau_{2}\Big{)}D_{\mathsf{a}\mathsf{b}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{L+1})}\Big{(}|\mathbf{k}-\mathbf{q}_{L}|;\tau_{1},\tau_{2}\Big{)}.$
(84)

Then we apply partial MB representation to extract the nonlocal signal from
the Mellin-space loop integral (29), and the procedure is totally parallel
with the proof of Theorem 2. The result is given by (44), and the nonlocal
signal is:

$\displaystyle\mathcal{G}_{\text{NL}}=\sum_{\begin{subarray}{c}\mathsf{c}_{1},\cdots,\mathsf{c}_{L+1}=\pm\\\
\mathrm{nonlocal}\end{subarray}}\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}^{\mathrm{(L)}}(k_{12})\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}^{\mathrm{(R)}}(k_{34})\mathfrak{M}_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}(k_{s}).$
(85)

As we can see, the nonanalyticity of $k_{s}$ is fully encoded in the melon
signal $\mathfrak{M}_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}(k_{s})$, whose
explicit form of is given by (45). We also find that the signal contains
$2^{L}$ (generally different) frequencies with respect to $\log k_{s}$, given
by $|\omega_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}|$:

$\omega_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}\equiv
2\sum_{\ell=1}^{L+1}\mathsf{c}_{\ell}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell},$ (86)

where $\mathsf{c}_{1},\cdots,\mathsf{c}_{L}$ can take values from $\pm$. Each
choice gives rise to a value of the frequency, modulo the overall sign-flip of
$\mathsf{c}_{\ell}$.

Furthermore, the left and the right subgraphs in (85) can be directly
calculated by (4.1) and (4.1), and the results are:

$\displaystyle\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}^{\mathrm{(L)}}(k_{12})=$
$\displaystyle\sum_{\mathsf{a}=\pm}\int(\mathrm{i}\mathsf{a})\,{\mathrm{d}}\tau_{1}\,(-\tau_{1})^{p_{1}}e^{\mathsf{a}\mathrm{i}k_{12}\tau_{1}}{\prod_{\ell=1}^{L+1}(-\tau_{1})^{3/2+\mathsf{c}_{\ell}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell}}}$
$\displaystyle=$
$\displaystyle~{}2k_{12}^{-1-P_{1}}\cos\Big{(}\displaystyle\frac{\,{\pi}\,}{2}P_{1}\Big{)}\Gamma(1+P_{1}),$
(87)
$\displaystyle\mathcal{G}_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}^{\mathrm{(R)}}(k_{34})=$
$\displaystyle\sum_{\mathsf{b}=\pm}\int(\mathrm{i}\mathsf{b})\,{\mathrm{d}}\tau_{2}\,(-\tau_{2})^{p_{2}}e^{\mathsf{b}\mathrm{i}k_{34}\tau_{2}}{\prod_{\ell=1}^{L+1}(-\tau_{2})^{3/2+\mathsf{c}_{\ell}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{\ell}}}$
$\displaystyle=$
$\displaystyle~{}2k_{34}^{-1-P_{2}}\cos\Big{(}\displaystyle\frac{\,{\pi}\,}{2}P_{2}\Big{)}\Gamma(1+P_{2}),$
(88)

where $P_{1,2}\equiv
p_{1,2}+3(L+1)/2+\mathrm{i}\omega_{\mathsf{c}_{1}\cdots\mathsf{c}_{L+1}}/2$.

Figure 10: The 4-point $L$-loop melon graph.

###  6.2 Two-point mixing

From now on, let us consider graphs that contain internal vertices. The
simplest case is a tree-level two-point mixing, shown in Fig. 11. Given the
subtleties associated with internal vertices, we will investigate this example
more closely than the previous one. The purpose of this example is to
explicitly show the appearance of a hybrid signal for non-dS covariant
couplings and the absence of this signal for dS covariant couplings.

Since we are looking for nonlocal signals, let us assume that we can cut the
graph as usual. The expression for this graph is the following:

$\displaystyle\mathcal{G}_{\mathsf{a}\mathsf{b}\mathsf{c}}=-\mathrm{i}\mathsf{a}\mathsf{b}\mathsf{c}\int\prod_{i=1}^{3}\Big{[}{\mathrm{d}}\tau_{i}(-\tau_{i})^{p_{i}}\Big{]}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{a}\mathsf{c}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(k_{s};\tau_{1},\tau_{3})D_{\mathsf{c}\mathsf{b}}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(k_{s};\tau_{3},\tau_{2}).$ (89)

As always, we neglected unimportant prefactors from the bulk-to-boundary
propagators. We first cut the $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}$ propagator. It
is not a priori clear that we can take a symmetric cut
$D\Rightarrow\text{Re}\,D$. So, we take a safer cut:

$D_{++}^{(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(k_{s};\tau_{1},\tau_{3})\Rightarrow D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(k_{s};\tau_{1},\tau_{3})+{\color[rgb]{0,.3,1}\Big{[}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(k_{s};\tau_{1},\tau_{3})-D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(k_{s};\tau_{1},\tau_{3})\Big{]}\theta(\tau_{3}-\tau_{1})}.$ (90)

By a cut we mean to remove the blue term. Similarly,

$D_{++}^{(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(k_{s};\tau_{3},\tau_{2})\Rightarrow D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(k_{s};\tau_{3},\tau_{2})+\Big{[}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(k_{s};\tau_{3},\tau_{2})-D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(k_{s};\tau_{3},\tau_{2})\Big{]}\theta(\tau_{3}-\tau_{2}),$ (91)

but here we do not remove any term.

Figure 11: The 4-point correlator with tree-level $s$-channel exchange of two
mixed massive modes. The mixing is generally not dS covariant.

Under this cut, the integral of $\tau_{1}$ and $\tau_{2}$ is factorized. The
original integral now becomes two parts: A (fully) factorized (F) part, and a
(partially) nested (N) part which involves integral of
$\theta(\tau_{3}-\tau_{2})$. For example, we can write
$\mathcal{G}_{+\pm\pm}=\mathcal{G}_{+\pm\pm}^{\text{(F)}}+\mathcal{G}_{+\pm\pm}^{\text{(N)}}$.

Let us first look at the nested part:

$\displaystyle\mathcal{G}_{+++}^{\text{(N)}}=-\mathrm{i}\int_{\tau}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1})}(D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2})}-D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})})\theta(\tau_{3}-\tau_{2})e^{+\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}},$
(92)

where
$\int_{\tau}\cdots\equiv\int\prod\limits_{i=1}^{3}[{\mathrm{d}}\tau_{i}(-\tau_{i})^{p_{i}}]\cdots$.
We use the MB representation for the massive propagators:

$\displaystyle\mathcal{G}_{+++}^{\text{(N)}}=$
$\displaystyle~{}\displaystyle\frac{\,{-\mathrm{i}}\,}{(4\pi)^{2}}\int_{s}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{2}\Big{)}^{-2s_{1\bar{1}2\bar{2}}}e^{+\mathrm{i}\pi(s_{1}-\bar{s}_{1})}(e^{-\mathrm{i}\pi(s_{2}-\bar{s}_{2})}-e^{+\mathrm{i}\pi(s_{2}-\bar{s}_{2})})\Gamma\Big{[}\text{IR
poles}\Big{]}$
$\displaystyle\times\int_{\tau}\theta(\tau_{3}-\tau_{2})e^{+\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}}.$
(93)

Here and below, we use
$\int_{s}\cdots\equiv\int\prod\limits_{i=1}^{2}\big{(}\mbox{$\frac{\,{{\mathrm{d}}s_{i}}\,}{2\pi\mathrm{i}}$}\mbox{$\frac{\,{{\mathrm{d}}\bar{s}_{i}}\,}{2\pi\mathrm{i}}$}\big{)}$,
and,

$\Gamma\Big{[}\text{IR
poles}\Big{]}\equiv\prod_{i=1}^{2}\Gamma\Big{[}s_{i}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},s_{i}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},\bar{s}_{i}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{i}}\,}{2},\bar{s}_{i}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{i}}\,}{2}\Big{]}.$
(94)

The other four branches can be calculated similarly. It is easy to check that
the time integral will only give right poles of the Mellin variables. The next
step is to integrate out these Mellin variables. Since the integrand of (6.2)
is proportional to $k_{s}^{-2s_{1\bar{1}2\bar{2}}}$ and $k_{s}\rightarrow 0$
in the squeezed limit, we should sum residues at left poles, and the only left
poles are the IR poles. The leading poles are the following:

$s_{1}=-\mathsf{c}_{1}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2,\qquad\bar{s}_{1}=-\mathsf{c}_{2}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},\qquad
s_{2}=-\mathsf{c}_{3}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},\qquad\bar{s}_{2}=-\mathsf{c}_{4}\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}.$ (95)

However, the factor
$e^{-\mathrm{i}\pi(s_{2}-\bar{s}_{2})}-e^{+\mathrm{i}\pi(s_{2}-\bar{s}_{2})}$
in the integrand of (6.2) requires that $\mathsf{c}_{3}=-\mathsf{c}_{4}$,
otherwise $\mathcal{G}^{(\text{N})}_{+++}$ would vanish. Furthermore, to
generate a signal with nonzero frequency, we should take
$\mathsf{c}_{1}=\mathsf{c}_{2}$ such that
$\mathcal{G}^{(\text{N})}_{+++}\propto k_{s}^{\pm 2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}$. This is the
case that corresponds to a minimal cut on Line 1. Of course there is a similar
signal $\propto k_{s}^{\pm 2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$ corresponding
to a minimal cut on Line 2. These signals are ordinary signals of Fig. 11.

Now let us look at the factorized part. In this part, we have:

$\displaystyle\mathcal{G}_{+++}^{\text{(F)}}=-\mathrm{i}\int_{\tau}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1})}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}e^{+\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}},$
$\displaystyle\mathcal{G}_{++-}=+\mathrm{i}\int_{\tau}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1})}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}e^{+\mathrm{i}k_{12}\tau_{1}-\mathrm{i}k_{34}\tau_{2}},$ (96)
$\displaystyle\mathcal{G}_{+-+}^{\text{(F)}}=+\mathrm{i}\int_{\tau}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1})}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}e^{+\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}},$
$\displaystyle\mathcal{G}_{+--}^{\text{(F)}}=-\mathrm{i}\int_{\tau}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1})}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}e^{+\mathrm{i}k_{12}\tau_{1}-\mathrm{i}k_{34}\tau_{2}}.$ (97)

The other four branches can be directly obtained by taking the complex
conjugate. In total,

$\displaystyle\mathcal{G}^{\text{(F)}}\equiv$
$\displaystyle\sum_{\mathsf{a},\mathsf{b},\mathsf{c}=\pm}\mathcal{G}_{\mathsf{a}\mathsf{b}\mathsf{c}}^{\text{(F)}}=8\int_{\tau}\text{Im}\Big{[}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(k_{3};\tau_{1},\tau_{3})D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(k_{3};\tau_{3},\tau_{2})\Big{]}\sin(k_{12}\tau_{1})\sin(k_{34}\tau_{2}).$
(98)

Now, using the MB representation for the massive propagators, we get:

$\displaystyle\mathcal{G}^{\text{(F)}}=\displaystyle\frac{\,{8}\,}{(4\pi)^{2}}\int_{s}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{2}\Big{)}^{-2s_{1\bar{1}2\bar{2}}}\text{Im}\Big{[}e^{-\mathrm{i}\pi(s_{12}-s_{\bar{1}\bar{2}})}\Big{]}\Gamma\Big{[}\text{IR
poles}\Big{]}\int_{\tau}\sin(k_{12}\tau_{1})\sin(k_{34}\tau_{2}).$ (99)

The three time integrals give:

$\displaystyle\int{\mathrm{d}}\tau_{1}(-\tau_{1})^{p_{1}+3/2-2s_{1}}\sin(k_{12}\tau_{1})=-\displaystyle\frac{\,{\Gamma[p_{1}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{1}]\sin[\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{1})]}\,}{k_{12}^{p_{1}+5/2-2s_{1}}},$
(100)
$\displaystyle\int{\mathrm{d}}\tau_{2}(-\tau_{2})^{p_{2}+3/2-2s_{2}}\sin(k_{34}\tau_{2})=-\displaystyle\frac{\,{\Gamma[p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{2}]\sin[\mbox{$\frac{\,{\pi}\,}{2}$}(p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{2})]}\,}{k_{34}^{p_{2}+5/2-2s_{2}}},$
(101)
$\displaystyle\int{\mathrm{d}}\tau_{3}(-\tau_{3})^{p_{3}+3-2s_{\bar{1}\bar{2}}}=2\pi\delta\big{[}\mathrm{i}(p_{3}+4-2s_{\bar{1}\bar{2}})\big{]}.$
(102)

We can first use the $\delta$-function to remove $\bar{s}_{2}$, then we find
the integrand of (99) is proportional to $k_{s}^{-2s_{12}}$. For the other
three Mellin variables, we take the following left poles to collect leading
terms proportional to $k_{s}^{\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2})}$:

$\displaystyle s_{1}=-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2,\qquad\bar{s}_{2}=-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}/2,\qquad
s_{\bar{1}}=-\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2-n.~{}~{}~{}(n=0,1,\cdots)$ (103)

Together with the complex conjugate, the final result is:

$\displaystyle\mathcal{G}^{\text{(F)}}=$
$\displaystyle~{}\displaystyle\frac{\,{8}\,}{(4\pi)^{2}}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{2}\Big{)}^{+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}-p_{3}-4}\sin\Big{[}\displaystyle\frac{\,{\pi}\,}{2}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+p_{3}+4)\Big{]}\sum\text{Res}\Big{[}\text{IR poles}\Big{]}$
$\displaystyle\times\displaystyle\frac{\,{\Gamma[p_{1}+\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},p_{2}+\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}]\sin[\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})]\sin[\mbox{$\frac{\,{\pi}\,}{2}$}(p_{2}+\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})]}\,}{k_{12}^{p_{1}+5/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}k_{34}^{p_{2}+5/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}}.$ (104)

This is a hybrid signal as discussed in Sec. 5.3. The summation of residues at
IR poles is given by

$\displaystyle\sum\text{Res}\Big{[}\text{IR poles}\Big{]}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{\mathsf{a}=\pm}\displaystyle\frac{\,{(-1)^{n}}\,}{n!}\Gamma\Big{[}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-n,\mbox{$\frac{\,{\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})+p_{3}+4}\,}{2}$}+n,\mbox{$\frac{\,{\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})+p_{3}+4}\,}{2}$}+n\Big{]}$ $\displaystyle=$
$\displaystyle-\displaystyle\frac{\,{2\pi\Gamma[-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}]}\,}{\sinh\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,\text{Im}\,{}_{2}\mathcal{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})+4+p_{3}}\,}{2}$},\mbox{$\frac{\,{\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})+4+p_{3}}\,}{2}$}\\\ 1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\end{matrix}\middle|1\right].$ (105)

When $p_{3}=-4$ (covariant mass mixing), the above dressed hypergeometric
function is simplified to ${}_{2}\mathcal{F}_{1}[...|1]=-4/(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}^{2}-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}^{2})$ and is
manifestly real. It can be checked that the above
$\text{Im}\,{}_{2}\mathcal{F}_{1}=0$ for all integer $p_{3}\leq-4$. Therefore,
we conclude that there can be a hybrid signal for non-covariant mixing but
such a signal does not exist for covariant mass mixing, in agreement with the
general analysis in Sec. 5.3.

###  6.3 Double-bubble graph

Figure 12: The 4-point double-bubble graph.

The third example is the double-bubble graph in Fig. 12, which is like a loop
version of the tree-level 2-point mixing graph. Below we mainly focus on the
loop integral under the partial MB representation, but we also keep in mind
that the time integral at the internal vertices could give extra
$\delta$-functions. The loop integral in Mellin space is:

$\mathbb{L}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{1}}\,}{(2\pi)^{3}}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{2}}\,}{(2\pi)^{3}}|\mathbf{q}_{1}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}_{1}|^{-2s_{2\bar{2}}}|\mathbf{q}_{2}|^{-2s_{3\bar{3}}}|\mathbf{k}_{s}-\mathbf{q}_{2}|^{-2s_{4\bar{4}}}.$
(106)

The two loop momentum integrals are factorized, and the integral can be easily
computed:

$\mathbb{L}=\displaystyle\frac{\,{k_{s}^{6-2s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}}}}\,}{(4\pi)^{3}}\Gamma\begin{bmatrix}{\color[rgb]{0,.3,1}s_{1\bar{1}2\bar{2}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{0,.3,1}s_{3\bar{3}4\bar{4}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}}\\\
3-s_{1\bar{1}2\bar{2}},s_{1\bar{1}},s_{2\bar{2}},3-s_{3\bar{3}4\bar{4}},s_{3\bar{3}},s_{4\bar{4}}\end{bmatrix}.$
(107)

There are two sets of left poles from this loop integral, coming from the
factor $\Gamma(s_{1\bar{1}2\bar{2}}-3/2)$ and
$\Gamma(s_{3\bar{3}4\bar{4}}-3/2)$, respectively. In fact, they correspond to
the UV poles of the left and right loops. If we take one of the UV poles, say
$s_{3\bar{3}4\bar{4}}=3/2$, then the leading contribution becomes
$k_{s}^{3-2s_{1\bar{1}2\bar{2}}}$, and we can then set
$s_{1,\bar{1},2,\bar{2}}$ again at the IR poles:

$s_{1}=\bar{s}_{1}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2},\qquad
s_{2}=\bar{s}_{2}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}\,}{2},$ (108)

and the loop integral $\mathbb{L}$ is proportional to
$k_{s}^{3+2\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+2\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$. This
corresponds to the minimal cut of the left loop, namely cutting Line 1 and
Line 2. Similarly, we can cut the right loop, namely Line 3 and Line 4, and
take the UV pole: $s_{1\bar{1}2\bar{2}}=3/2$, then we will get a signal
proportional to $k_{s}^{3+2\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}+2\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}}$. These are all
ordinary signals. Of course, we cannot take both UV poles for the purpose of
generating a signal, since the result will simply be a constant.

The time integral of the internal vertex could give another type of signal.
For simplicity, we take a direct coupling for the internal vertex with an
arbitrary power $p_{3}$, and then the time integral gives:

$\int_{-\infty}^{0}{\mathrm{d}}\tau_{3}\,(-\tau_{3})^{p_{3}+4\times
3/2-2s_{\bar{1}\bar{2}34}}=2\pi\delta\big{[}\mathrm{i}(p_{3}+7-2s_{\bar{1}\bar{2}34})\big{]}.$
(109)

We can use this $\delta$-function to remove one Mellin variable. Then, the
integrand of $\mathbb{L}$ becomes proportional to
$k_{s}^{-1-p_{3}-2s_{12\bar{3}\bar{4}}}$. To integrate out the remaining
Mellin variables, we take residues of the integrand at the leading IR poles:

$s_{1}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2},\qquad
s_{2}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\,}{2},\qquad\bar{s}_{3}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}}\,}{2},\qquad\bar{s}_{4}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}}\,}{2}.$ (110)

we find that there is a signal of frequency $|\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}+\mathsf{c}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}|$, which is
generated by the resonances of the four modes at the two external vertices.
This is exactly a hybrid signal discussed in Sec. 5.3. However, for dS-
covariant coupling, namely $p_{3}=-4$, this signal should vanish, just like
the case of a covariant 2-point mass mixing.999One can see this point more
clearly by taking spectral decompositions of the two loops, as was done in
[95].

###  6.4 Double-triangle graph

Figure 13: The 4-point double-triangle graph.

The next example is the double-triangle graph Fig. 13. We can write down the
loop integral in the Mellin space:

$\displaystyle\mathbb{L}=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{1}}\,}{(2\pi)^{3}}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{2}}\,}{(2\pi)^{3}}|\mathbf{q}_{1}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}_{1}|^{-2s_{2\bar{2}}}|\mathbf{q}_{2}|^{-2s_{3\bar{3}}}|\mathbf{k}_{s}-\mathbf{q}_{2}|^{-2s_{4\bar{4}}}|\mathbf{q}_{1}-\mathbf{q}_{2}|^{-2s_{5\bar{5}}}$
(111)

First consider the $\mathbf{q}_{2}$ integral:

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern
2.0mu=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{2}}\,}{(2\pi)^{3}}|\mathbf{q}_{2}|^{-2s_{3\bar{3}}}|\mathbf{k}_{s}-\mathbf{q}_{2}|^{-2s_{4\bar{4}}}|\mathbf{q}_{1}-\mathbf{q}_{2}|^{-2s_{5\bar{5}}}.$
(112)

The integral allows various different series expansions, depending on the
shape of the triangle formed by vectors $\mathbf{k}_{s}$ and $\mathbf{q}_{1}$;
See App. B. Especially,

$\displaystyle\mkern 2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern
2.0mu|_{q_{1}\ll k_{s}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2s_{3\bar{3}4\bar{4}5\bar{5}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{0,.3,1}s_{3\bar{3}4\bar{4}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}5\bar{5}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}}\\\
3-s_{3\bar{3}4\bar{4}5\bar{5}},s_{3\bar{3}4\bar{4}},s_{5\bar{5}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{q_{1}^{3-2s_{3\bar{3}5\bar{5}}}k_{s}^{-2s_{4\bar{4}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{3\bar{3}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{5\bar{5}}\\\
3-s_{3\bar{3}5\bar{5}},s_{3\bar{3}},s_{5\bar{5}}\end{bmatrix},$ (113)
$\displaystyle\mkern 2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern
2.0mu|_{k_{s}\ll q_{1}}=$
$\displaystyle~{}\displaystyle\frac{\,{q_{1}^{3-2s_{3\bar{3}4\bar{4}5\bar{5}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{0,.3,1}s_{3\bar{3}4\bar{4}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}4\bar{4}},\mbox{$\frac{\,{3}\,}{2}$}-s_{5\bar{5}}\\\
3-s_{3\bar{3}4\bar{4}5\bar{5}},s_{3\bar{3}4\bar{4}},s_{5\bar{5}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{s}^{3-2s_{3\bar{3}4\bar{4}}}q_{1}^{-2s_{5\bar{5}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{3\bar{3}4\bar{4}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}}\\\
3-s_{3\bar{3}4\bar{4}},s_{3\bar{3}},s_{4\bar{4}}\end{bmatrix}.$ (114)

As we can see, the UV pole of the right loop
$s_{3\bar{3}4\bar{4}5\bar{5}}=3/2$ appears in both regions of
$\mathbf{q}_{1}$. Once we take this UV pole, we find the residue is $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern 2.0mu\rightarrow
1/(4\pi^{2})$, so the loop integral $\mathbb{L}$ is simplified to a bubble
integral:

$\displaystyle\mathbb{L}\rightarrow$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(4\pi)^{2}}\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{1}}\,}{(2\pi)^{3}}|\mathbf{q}_{1}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}_{1}|^{-2s_{2\bar{2}}}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{3-2s_{1\bar{1}2\bar{2}}}}\,}{(4\pi)^{7/2}}\Gamma\begin{bmatrix}s_{1\bar{1}2\bar{2}}-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}}\\\
3-s_{1\bar{1}2\bar{2}},s_{1\bar{1}},s_{2\bar{2}}\end{bmatrix},$ (115)

then we take leading IR poles:

$s_{1}=\bar{s}_{1}=\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2},\qquad
s_{2}=\bar{s}_{2}=\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}\,}{2},$ (116)

and we obtain an ordinary signal proportional to
$k_{s}^{3+2\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+2\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$, which
corresponds to cutting Line 1 and Line 2.

We can also take the UV pole coming from the integral of $\mathbf{q}_{1}$.
Notice that this UV pole comes from the hard region $q_{1}\gg k_{s}$, so we
can substitute $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern 2.0mu$ by the
expansion (6.4) into the loop integral $\mathbb{L}$. The first line of (6.4)
gives:

$\displaystyle\frac{\,{k_{s}^{6-2s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}5\bar{5}}}}\,}{(4\pi)^{3}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}5\bar{5}}-3},3-s_{1\bar{1}3\bar{3}4\bar{4}5\bar{5}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{0,.3,1}s_{3\bar{3}4\bar{4}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}4\bar{4}},\mbox{$\frac{\,{3}\,}{2}$}-s_{5\bar{5}}\\\
\mbox{$\frac{\,{9}\,}{2}$}-s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}5\bar{5}},s_{1\bar{1}3\bar{3}4\bar{4}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$},s_{2\bar{2}},3-s_{3\bar{3}4\bar{4}5\bar{5}},s_{3\bar{3}4\bar{4}},s_{5\bar{5}}\end{bmatrix}.$
(117)

There are two left poles from the loop integral: a UV pole for the whole graph
$s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}5\bar{5}}=3$ which gives no signal, and
the UV pole for the right loop which we have discussed before. So there is no
new leading signal. The second line of (6.4) gives:

$\displaystyle\frac{\,{k_{s}^{6-2s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}5\bar{5}}}}\,}{(4\pi)^{3}}\Gamma\begin{bmatrix}{\color[rgb]{0,.3,1}s_{1\bar{1}2\bar{2}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}5\bar{5}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{1,0,0}s_{3\bar{3}4\bar{4}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}}\\\
3-s_{1\bar{1}2\bar{2}5\bar{5}},s_{1\bar{1}5\bar{5}},s_{2\bar{2}},3-s_{3\bar{3}4\bar{4}},s_{3\bar{3}},s_{4\bar{4}}\end{bmatrix}.$
(118)

As we can see, we recover the left loop UV pole
$s_{1\bar{1}2\bar{2}5\bar{5}}=3/2$. When taking residue at this pole, we find
$\mathbb{L}\propto k_{s}^{3-2s_{3\bar{3}4\bar{4}}}$, so we can take residues
at the following leading IR poles:

$s_{3}=\bar{s}_{3}=\mp\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}}\,}{2},\qquad
s_{4}=\bar{s}_{4}=\mp\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}}\,}{2},$ (119)

and we get another ordinary signal proportional to
$k_{s}^{3+2\mathrm{i}\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}+2\mathrm{i}\mathsf{c}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}}$, corresponding
to cutting Line 3 and Line 4.

The other two left poles $s_{3\bar{3}4\bar{4}}=3/2$ and
$s_{3\bar{3}5\bar{5}}=3/2$ are more subtle. For example, we can take
$s_{3\bar{3}4\bar{4}}=3/2$, then the residue of $\mathbb{L}$ is propoertional
to $k_{s}^{3-2s_{1\bar{1}2\bar{2}5\bar{5}}}$. Naively, one may expect that we
can take the corresponding IR poles, corresponding to cutting Line 1, Line 2,
and Line 5 simultaneously, and we would obtain a signal with frequency
$|2\mathsf{c}_{1}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+2\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+2\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{5}|$. However, this
is not the case, since when Lines 1,2,5 become soft together, the integral of
$\tau_{3}$ and $\tau_{4}$ will be factorized, giving additional
$\delta$-constraints.

In fact, assuming direct couplings, the time integrals at the two internal
vertices give:

$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{3}\,(-\tau_{3})^{p_{3}+3\times
3/2-2s_{\bar{1}\bar{2}5}}=2\pi\delta\big{[}\mathrm{i}(p_{3}+11/2-2s_{\bar{1}\bar{2}5})\big{]},$
(120)
$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{4}\,(-\tau_{4})^{p_{4}+3\times
3/2-2s_{\bar{3}\bar{4}\bar{5}}}=2\pi\delta\big{[}\mathrm{i}(p_{4}+11/2-2s_{34\bar{5}})\big{]}.$
(121)

Then we see that the $k_{s}$ power becomes
$k_{s}^{-5-p_{3}-p_{4}-2s_{12\bar{3}\bar{4}}}$. This suggests that there is a
hybrid signal from the four modes attached to the “outer” two vertices, with
frequencies $|\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}+\mathsf{c}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}|$. Similar to the
previous cases, this signal should vanish for dS covariant couplings, namely
when $p_{3}=p_{4}=-4$.

###  6.5 Double-box graph

Figure 14: The 4-point (planar) double-box graph.

The case of the double-box graph Fig. 14 is very similar to the double-
triangle graph. The loop integral becomes:

$\displaystyle\mathbb{L}=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{1}}\,}{(2\pi)^{3}}|\mathbf{q}_{1}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}_{1}|^{-2s_{2\bar{2}}}|\mathbf{k}_{1}-\mathbf{q}_{1}|^{-2s_{5\bar{5}}}$
$\displaystyle\times\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{2}}\,}{(2\pi)^{3}}|\mathbf{q}_{2}|^{-2s_{3\bar{3}}}|\mathbf{k}_{s}-\mathbf{q}_{2}|^{-2s_{4\bar{4}}}|\mathbf{k}_{3}+\mathbf{q}_{2}|^{-2s_{6\bar{6}}}|\mathbf{q}_{1}-\mathbf{q}_{2}|^{-2s_{7\bar{7}}}.$
(122)

Again we first calculate the integral of $\mathbf{q}_{2}$:

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern
2.0mu=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{2}}\,}{(2\pi)^{3}}|\mathbf{q}_{2}|^{-2s_{3\bar{3}}}|\mathbf{k}_{s}-\mathbf{q}_{2}|^{-2s_{4\bar{4}}}|\mathbf{k}_{3}+\mathbf{q}_{2}|^{-2s_{6\bar{6}}}|\mathbf{q}_{1}-\mathbf{q}_{2}|^{-2s_{7\bar{7}}}.$
(123)

Depending on the shapes of vectors $\mathbf{k}_{s}$, $\mathbf{q}_{1}$ and
$\mathbf{k}_{3}$, $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern 2.0mu$ can be
expanded as different series. Here we focus on the hard region:

$\displaystyle\mkern 2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern
2.0mu|_{q_{1}\gg k_{3}\gg k_{s}}=$
$\displaystyle~{}\displaystyle\frac{\,{q_{1}^{3-2s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{0,.3,1}s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}4\bar{4}6\bar{6}},\mbox{$\frac{\,{3}\,}{2}$}-s_{7\bar{7}}\\\
3-s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}},s_{3\bar{3}4\bar{4}6\bar{6}},s_{7\bar{7}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{3}^{3-2s_{3\bar{3}4\bar{4}6\bar{6}}}q_{1}^{-2s_{7\bar{7}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{3\bar{3}4\bar{4}6\bar{6}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}4\bar{4}},\mbox{$\frac{\,{3}\,}{2}$}-s_{6\bar{6}}\\\
3-s_{3\bar{3}4\bar{4}6\bar{6}},s_{3\bar{3}4\bar{4}},s_{6\bar{6}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{s}^{3-2s_{3\bar{3}4\bar{4}}}k_{3}^{-2s_{6\bar{6}}}q_{1}^{-2s_{7\bar{7}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{3\bar{3}4\bar{4}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}}\\\
3-s_{3\bar{3}4\bar{4}},s_{3\bar{3}},s_{4\bar{4}}\end{bmatrix}.$ (124)

Plug the hard region expression into the original integral and keeping terms
nonanalytic in $k_{s}$, we obtain:

$\displaystyle~{}\mathbb{L}\Rightarrow\displaystyle\frac{\,{k_{s}^{6-2s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}6\bar{6}7\bar{7}}}k_{1}^{-2s_{5\bar{5}}}}\,}{(4\pi)^{3}}$
$\displaystyle\times\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-3},3-s_{1\bar{1}3\bar{3}4\bar{4}6\bar{6}7\bar{7}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{0,.3,1}s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}4\bar{4}6\bar{6}},\mbox{$\frac{\,{3}\,}{2}$}-s_{7\bar{7}}\\\
\mbox{$\frac{\,{9}\,}{2}$}-s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}6\bar{6}7\bar{7}},s_{1\bar{1}3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$},s_{2\bar{2}},3-s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}},s_{3\bar{3}4\bar{4}6\bar{6}},s_{7\bar{7}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{s}^{3-2s_{1\bar{1}2\bar{2}7\bar{7}}}k_{1}^{-2s_{5\bar{5}}}k_{3}^{3-2s_{3\bar{3}4\bar{4}6\bar{6}}}}\,}{(4\pi)^{3}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{1\bar{1}2\bar{2}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}7\bar{7}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{1,0,0}s_{3\bar{3}4\bar{4}6\bar{6}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}4\bar{4}},\mbox{$\frac{\,{3}\,}{2}$}-s_{6\bar{6}}\\\
3-s_{1\bar{1}2\bar{2}7\bar{7}},s_{1\bar{1}7\bar{7}},s_{2\bar{2}},3-s_{3\bar{3}4\bar{4}6\bar{6}},s_{3\bar{3}4\bar{4}},s_{6\bar{6}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{s}^{3-2s_{3\bar{3}4\bar{4}}}k_{1}^{3-2s_{1\bar{1}2\bar{2}5\bar{5}7\bar{7}}}k_{3}^{-2s_{6\bar{6}}}}\,}{(4\pi)^{3}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{1\bar{1}2\bar{2}5\bar{5}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}2\bar{2}7\bar{7}},\mbox{$\frac{\,{3}\,}{2}$}-s_{5\bar{5}},{\color[rgb]{1,0,0}s_{3\bar{3}4\bar{4}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}}\\\
3-s_{1\bar{1}2\bar{2}5\bar{5}7\bar{7}},s_{1\bar{1}2\bar{2}7\bar{7}},s_{5\bar{5}},3-s_{3\bar{3}4\bar{4}},s_{3\bar{3}},s_{4\bar{4}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{s}^{6-2s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}7\bar{7}}}k_{1}^{-2s_{5\bar{5}}}k_{3}^{-2s_{6\bar{6}}}}\,}{(4\pi)^{3}}\Gamma\begin{bmatrix}{\color[rgb]{0,.3,1}s_{1\bar{1}2\bar{2}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}7\bar{7}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{1,0,0}s_{3\bar{3}4\bar{4}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}}\\\
3-s_{1\bar{1}2\bar{2}},s_{1\bar{1}},s_{2\bar{2}},3-s_{3\bar{3}4\bar{4}},s_{3\bar{3}},s_{4\bar{4}}\end{bmatrix}.$
(125)

In the first term on the right hand side, there is a UV pole
$s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}=3/2$ for the right loop. We can take
this pole, which amounts to pinching the right loop. Then, the integrand is
proportional to $k_{s}^{3-2s_{1\bar{1}2\bar{2}}}$, and then we can take the
following IR poles:

$s_{1}=\bar{s}_{1}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2},\qquad
s_{2}=\bar{s}_{2}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}\,}{2},$ (126)

and obtain a signal of frequency $2|\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}|$, corresponding
to cutting Line 1 and Line 2. Notice that we cannot take poles
$s_{1}=-\bar{s}_{1}=-\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}/2$ to construct a
signal of frequency $2\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$, because the
factor $\Gamma(s_{1\bar{1}3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-3/2)$ in the
denominator will becomes $\Gamma(s_{1\bar{1}})$ at the UV pole
$s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}=3/2$.

In the last term on the right hand side of (6.5), there is another UV pole
$s_{1\bar{1}2\bar{2}7\bar{7}}=3/2$, at which the integrand is proportional to
$k_{s}^{3-2s_{3\bar{3}4\bar{4}}}$. Then we take IR poles:

$s_{3}=\bar{s}_{3}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}}\,}{2},\qquad
s_{4}=\bar{s}_{4}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}}\,}{2},$ (127)

which indicates another ordinary signal of frequency $2|\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}|$, corresponding
to cutting Line 3 and Line 4.

Other possible left poles lead to either no new signals, or signals
constrained by the $\delta$-functions from time integral. For instance, we can
take another left pole $s_{3\bar{3}4\bar{4}}=3/2$ in the fourth line, then the
integrand is proportional to $k_{s}^{3-2s_{1\bar{1}2\bar{2}7\bar{7}}}$.
However, if we try to cut Line 1, 2, 7 together, we will encounter the two
$\delta$-functions from the integral of $\tau_{5}$ and $\tau_{6}$:

$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{5}\,(-\tau_{5})^{p_{5}+3\times
3/2-2s_{\bar{1}37}}=2\pi\delta\big{[}\mathrm{i}(p_{5}+11/2-2s_{\bar{1}37})\big{]},$
(128)
$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}\tau_{6}\,(-\tau_{6})^{p_{6}+3\times
3/2-2s_{\bar{2}4\bar{7}}}=2\pi\delta\big{[}\mathrm{i}(p_{6}+11/2-2s_{\bar{2}4\bar{7}})\big{]},$
(129)

which make the integrand proportional to
$k_{s}^{-5-p_{5}-p_{6}-2s_{12\bar{3}\bar{4}}}$. This could generate a hybrid
signal of frequency $|\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{3}+\mathsf{c}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{4}|$ in the non-dS-
covariant case.

###  6.6 Non-planar double-box graph

Figure 15: The 4-point non-planar double-bubble graph.

Now let us consider the non-planar double box graph Fig. 15. The loop integral
is:

$\displaystyle\mathbb{L}=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{1}}\,}{(2\pi)^{3}}|\mathbf{q}_{1}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}_{1}|^{-2s_{2\bar{2}}}|\mathbf{k}_{1}-\mathbf{q}_{1}|^{-2s_{5\bar{5}}}$
$\displaystyle\times\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{2}}\,}{(2\pi)^{3}}|\mathbf{q}_{2}|^{-2s_{3\bar{3}}}|\mathbf{k}_{s}+\mathbf{k}_{3}-\mathbf{q}_{1}+\mathbf{q}_{2}|^{-2s_{4\bar{4}}}|\mathbf{k}_{3}+\mathbf{q}_{2}|^{-2s_{6\bar{6}}}|\mathbf{q}_{1}-\mathbf{q}_{2}|^{-2s_{7\bar{7}}}.$
(130)

We still consider the $\mathbf{q}_{2}$ integral $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern 2.0mu$ first, namely
the second lien of the above expression. In particular, in the region
$q_{1}\gg k_{3}\gg k_{s}$, we have:

$\displaystyle\mkern 2.0mu\widetilde{\mkern-2.0mu\mathbb{L}\mkern-2.0mu}\mkern
2.0mu|_{q_{1}\gg k_{3}\gg k_{s}}=$
$\displaystyle~{}\displaystyle\frac{\,{q_{1}^{3-2s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{0,.3,1}s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}6\bar{6}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}7\bar{7}}\\\
3-s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}},s_{3\bar{3}6\bar{6}},s_{4\bar{4}7\bar{7}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{3}^{3-2s_{3\bar{3}6\bar{6}}}q_{1}^{-2s_{4\bar{4}7\bar{7}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{3\bar{3}6\bar{6}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{6\bar{6}}\\\
3-s_{3\bar{3}6\bar{6}},s_{3\bar{3}},s_{6\bar{6}}\end{bmatrix}.$ (131)

Inserting the above expansion back into the original loop integral, and
keeping the nonanalytic terms in $k_{s}$, we obtain:

$\displaystyle\mathbb{L}\Rightarrow$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{6-2s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}6\bar{6}7\bar{7}}}k_{1}^{-2s_{5\bar{5}}}}\,}{(4\pi)^{3}}$
$\displaystyle\times\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-3},3-s_{1\bar{1}3\bar{3}4\bar{4}6\bar{6}7\bar{7}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{0,.3,1}s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}6\bar{6}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}7\bar{7}}\\\
\mbox{$\frac{\,{9}\,}{2}$}-s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}6\bar{6}7\bar{7}},s_{1\bar{1}3\bar{3}4\bar{4}6\bar{6}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$},s_{2\bar{2}},3-s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}},s_{3\bar{3}6\bar{6}},s_{4\bar{4}7\bar{7}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{s}^{3-2s_{1\bar{1}2\bar{2}4\bar{4}7\bar{7}}}k_{1}^{-2s_{5\bar{5}}}k_{3}^{-3-2s_{3\bar{3}6\bar{6}}}}\,}{(4\pi)^{3}}$
$\displaystyle\times\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{1\bar{1}2\bar{2}4\bar{4}7\bar{7}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}4\bar{4}7\bar{7}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{1,0,0}s_{3\bar{3}6\bar{6}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{3\bar{3}},\mbox{$\frac{\,{3}\,}{2}$}-s_{6\bar{6}}\\\
3-s_{1\bar{1}2\bar{2}4\bar{4}7\bar{7}},s_{1\bar{1}4\bar{4}7\bar{7}},s_{2\bar{2}},3-s_{3\bar{3}6\bar{6}},s_{3\bar{3}},s_{6\bar{6}}\end{bmatrix}.$
(132)

We can take the UV pole $s_{3\bar{3}4\bar{4}6\bar{6}7\bar{7}}$ from the right
(nonplanar) loop and thus the integrand is proportional to
$k_{s}^{3-2s_{1\bar{1}2\bar{2}}}$, and then, we can cut Line 1 and Line 2 and
take IR poles:

$s_{1}=\bar{s}_{1}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2},\qquad
s_{2}=\bar{s}_{2}=-\displaystyle\frac{\,{\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}\,}{2},$ (133)

which generates a signal of frequency $2|\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}|$. On the other
hand, we cannot take the UV pole from the $\mathbf{q}_{1}$ integral, which
will not give us any signal. Indeed, the minimal cut for this graph is unique,
which is exactly cutting Line 1 and Line 2.

###  6.7 Bubble-on-bubble graph

Figure 16: The 4-point bubble-on-bubble graph.

Our final example is the bubble-on-bubble graph Fig. 16. The loop integral can
be calculated explicitly:

$\displaystyle\mathbb{L}=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{1}}\,}{(2\pi)^{3}}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{2}}\,}{(2\pi)^{3}}|\mathbf{q}_{1}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}_{1}|^{-2s_{2\bar{2}3\bar{3}}}|\mathbf{q}_{2}|^{-2s_{4\bar{4}}}|\mathbf{k}_{s}-\mathbf{q}_{1}-\mathbf{q}_{2}|^{-2s_{5\bar{5}}}$
$\displaystyle=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{1}}\,}{(2\pi)^{3}}|\mathbf{q}_{1}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}_{1}|^{-2s_{2\bar{2}3\bar{3}}}\times\displaystyle\frac{\,{|\mathbf{k}_{s}-\mathbf{q}_{1}|^{3-2s_{4\bar{4}5\bar{5}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{0,.3,1}s_{4\bar{4}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}},\mbox{$\frac{\,{3}\,}{2}$}-s_{5\bar{5}}\\\
3-s_{4\bar{4}5\bar{5}},s_{4\bar{4}},s_{5\bar{5}}\end{bmatrix}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{k_{s}^{6-2s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}5\bar{5}}}}\,}{(4\pi)^{3}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}5\bar{5}}-3},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},3-s_{2\bar{2}3\bar{3}4\bar{4}5\bar{5}},{\color[rgb]{0,.3,1}s_{4\bar{4}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{4\bar{4}},\mbox{$\frac{\,{3}\,}{2}$}-s_{5\bar{5}}\\\
3-s_{4\bar{4}5\bar{5}},\mbox{$\frac{\,{9}\,}{2}$}-s_{1\bar{1}2\bar{2}3\bar{3}4\bar{4}5\bar{5}},s_{1\bar{1}},s_{4\bar{4}},s_{5\bar{5}},s_{2\bar{2}3\bar{3}4\bar{4}5\bar{5}}-\mbox{$\frac{\,{3}\,}{2}$}\end{bmatrix}.$
(134)

We can first take the UV pole of the small loop at $s_{4\bar{4}5\bar{5}}=3/2$,
then the integrand of $\mathbb{L}$ is proportional to
$k_{s}^{3-2s_{1\bar{1}2\bar{2}3\bar{3}}}$. Next we should specify appropriate
poles for $s_{1,\bar{1},2,\bar{2},3,\bar{3}}$. As before, the factor
$\Gamma(s_{1\bar{1}})$ in the denominator requires that we should select
leading IR poles $s_{1}=\bar{s}_{1}=\mp\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}$. Then, we can
choose leading IR poles of $s_{2,\bar{2},3,\bar{3}}$. For example, we can take
leading poles $s_{2}=\bar{s}_{2}=\mp\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$ and
$s_{3}=-\bar{s}_{3}=\mp\mathrm{i}\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}$ (notice that
there is no factor $\Gamma(s_{3\bar{3}})$ in the denominator), which gives a
signal proportional to $k_{s}^{3+2\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+2\mathrm{i}\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$. This
corresponds to cutting Line 1 and Line 2. Similarly, we can cut Line 1 and
Line 3 and obtain a signal proportional to
$k_{s}^{3+2\mathrm{i}\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+2\mathrm{i}\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}}$.

However, if we try to cut Line 2, and Line 3 simultaneously, there would be an
additional $\delta$-constraint coming from the integral of $\tau_{3}$ and
$\tau_{4}$. For example, if we consider the $\mathcal{G}_{+-+-}$ channel, then
the time integrals give:

$\displaystyle\int{\mathrm{d}}\tau_{3}\,(-\tau_{3})^{p_{3}+3\times
3/2-2s_{\bar{2}45}}=2\pi\delta\big{[}\mathrm{i}(p_{3}+11/2-2s_{\bar{2}45})\big{]},$
(135) $\displaystyle\int{\mathrm{d}}\tau_{4}\,(-\tau_{4})^{p_{4}+3\times
3/2-2s_{3\bar{4}\bar{5}}}=2\pi\delta\big{[}\mathrm{i}(p_{4}+11/2-2s_{3\bar{4}\bar{5}})\big{]},$
(136)

and thus the integrand is now proportional to
$k_{s}^{-5-p_{3}-p_{4}-2s_{1\bar{1}2\bar{3}}}$, which indicates signals of
frequencies $|2\mathsf{c}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{c}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mathsf{c}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{3}|$ in the non-dS-
covariant case, corresponding to the oscillations of the four outer modes.

## 7 Conclusion and Outlook

Inflation correlators are central observables in Cosmological Collider
physics, playing a similar role as the scattering amplitudes and $S$-matrix
for scattering experiments in Minkowski spacetime. It is thus crucial to have
a good understanding of the analytical structure of inflation correlators.
Among all kinds of singularities, the nonlocal signal produced by massive
exchanges is special: On the one hand, it gives rise to a characteristic
oscillatory signal in the physical region and thus is the main observable in
CC physics. On the other hand, it produces a branch cut starting from the
origin in the complex plane of the momentum transfer, and this feature has no
flat spacetime correspondence. For these reasons, studying the analytical
structure of the nonlocal signal is useful and important for both CC
phenomenology and general dS QFTs.

In this work, we generalize our previous result in [94] to all loop orders:
With the help of partial Mellin-Barnes representation, we state and prove a
factorization theorem (4), which can be used to detect and identify all
possible nonlocal signal in an arbitrary graph. At 1-loop order, a nonlocal
signal could appear when two internal lines become soft simultaneously, and we
can cut these two soft lines, pinch their endpoints and get a bubble signal.
Similarly, the nonlocal signal of an arbitrary graph is associated with a
nonlocal cut, and the leading signal in the nonlocal soft limit
$\mathbf{P}\rightarrow 0$ corresponds to the minimal cut. The signal appears
when all these lines become soft simultaneously, and we can pinch their
endpoints and get a melon subgraph, and the nonanalyticity of $P$ is manifest
in its melon signal.

Partial Mellin-Barnes representation has been proven suitable and useful when
studying inflation correlators, and there are still many interesting targets
to explore with this tool. Under this representation, the graph breaks into
different pieces in (26), and both the time and loop momentum integrals are
much simplified. Moreover, the external momenta only appear in the loop
integral $\mathbb{L}$, while the external energies only appear in the time
integral $\mathbb{T}$ (if there is no degeneracy between momenta and energies,
of course). Therefore, it seems that considering the loop integral
$\mathbb{L}$ is enough to analyze the nonlocal signals. However, this is not
exactly the case, since time integral $\mathbb{T}$ could give some
$\delta$-functions of Mellin variables, which can result in the hybrid signals
briefly discussed in Sec. 5.3 and Sec. 6.2. Although such signals are absent
in bulk-free graphs and forbidden by the conformal symmetry in graphs with dS-
covariant couplings, it will be interesting to have a detailed study of such
hybrid signals.

Also, as mentioned in the Introduction, there are other kinds of
nonanalyticity, including the local signal associated with the partial
external energy sum. Since the energy only appears in time integrals, we can
derive similar detection rules for such local signals using partial MB
representation. This requires a careful study of the Mellin-space time
integral $\mathbb{T}$ in (3.3), and we will pursue this topic in another work.

Branch cuts associated with CC signals are in a sense unique to dS amplitudes,
which reflect the particle production and resonances in an inflationary
universe. So they are absent in the corresponding flat-space amplitudes. It is
well known that we have full control of a meromorphic function if we know its
behavior at all the singularities (for example, residues at simple poles). The
case of functions with branch cuts is similar: once we know the discontinuity
along every branch cut, we can restore the function using the dispersion
relation. We have seen that the massive inflation correlators have a branch
cut starting from the origin of the complex plane of certain momentum
transfers, and we are able to calculate the nonanalyticity using our
factorization theorem. It is thus natural to ask whether we can reconstruct
the full correlator using dispersion relation. This is not as trivial as it
sounds, since our result is only valid around the squeezed limit. However, to
do a dispersion integral along the full branch cut, we need a more
comprehensive understanding of the nonanalyticity away from the origin point.
Meanwhile, we should also identify all the other singular structures on the
complex plane for the dispersion integrals to work. We also leave this
question to a future work.

#### Acknowledgments.

We thank Clifford Cheung, Bing-Chu Fan, Xi Tong, Yi Wang, and Yuhang Zhu for
useful discussions. ZX thanks the HKUST Jockey Club Institute for Advanced
Study for hospitality during the completion of this work. This work is
supported by the National Key R&D Program of China (2021YFC2203100), NSFC
under Grant No. 12275146, an Open Research Fund of the Key Laboratory of
Particle Astrophysics and Cosmology, Ministry of Education of China, and a
Tsinghua University Initiative Scientific Research Program.

##  Appendix A Notations

In this work, we draw heavily use of following shorthands for products and
fractions of Euler $\Gamma$ functions:

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

We also use the dressed version of generalized hypergeometric functions,
defined in the following way when the series converges:

$\displaystyle{}_{p}\mathcal{F}_{q}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{matrix}\middle|z\right]=\sum_{n=0}^{\infty}\Gamma\left[\begin{matrix}a_{1}+n,\cdots,a_{p}+n\\\
b_{1}+n,\cdots,b_{q}+n\end{matrix}\right]\displaystyle\frac{\,{z^{n}}\,}{n!}.$
(139)

The definition can be extended beyond the radius of convergences by analytical
continuation.

A central formula for our partial MB representation for massive inflation
correlators is the following MB representation of the Hankel functions:

$\text{H}_{\nu}^{(j)}(az)=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(az/2)^{-2s}}\,}{\pi}e^{(-1)^{j+1}(2s-\nu-1)\pi\mathrm{i}/2}\Gamma\Big{[}s-\displaystyle\frac{\,{\nu}\,}{2},s+\displaystyle\frac{\,{\nu}\,}{2}\Big{]}.~{}~{}~{}~{}(j=1,2)$
(140)

Finally, we collect frequently used symbols in Table. 1.

Table 1: List of symbols Symbol |  | Equation
---|---|---
$\mathbf{k}_{i}$ $(i=1,\cdots,B)$ | Momentum of an external line |
$\mathbf{k}_{i}^{\mathrm{(L)}}$ $(i=1,\cdots,B_{L})$ | Momentum of an external line in the left sugraph |
$\mathbf{k}_{i}^{\mathrm{(R)}}$ $(i=1,\cdots,B_{R})$ | Momentum of an external line in the right sugraph |
$\mathbf{p}_{i}$ $(i=1,\cdots,I)$ | Momentum of an internal line |
$\mathbf{q}_{i}$ $(i=1,\cdots,L)$ | Loop momentum variable |
$\mathbf{q}_{i}^{\mathrm{(L)}}$ $(i=1,\cdots,N_{L})$ | Loop momentum variable of the left subgraph |
$\mathbf{q}_{i}^{\mathrm{(R)}}$ $(i=1,\cdots,N_{R})$ | Loop momentum variable of the right subgraph |
$\mathbf{K}_{i}$ $(i=,1\cdots,V)$ | Total external momentum flowing into the $i$’th vertex |
$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathbf{P}\mkern-2.0mu}\mkern 2.0mu_{i}$ | Partial sum of external momenta |
$\mathbf{P}\equiv\sum\mathbf{k}^{\mathrm{(L)}}$ | Total external momentum of the left subgraph |
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\equiv\sqrt{m^{2}-9/4}$ | Mass parameter of a principal scalar of mass $m$ |
$\mathcal{G}(\\{\mathbf{k}\\})$ | (Amplitude of) a graph | (3.3)
$C_{\mathsf{a}}(k;\tau)$ | Bulk-to-boundary propagator of a conformal scalar | (23)
$D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ | Bulk propagator of a massive scalar | (13)-(15)
$\mathsf{a},\mathsf{b},\mathsf{c},\cdots$ | SK indices or other indices taking values from $\pm 1$ |
$s_{i},\bar{s}_{i}$ | Mellin variable |
$\mathbb{T}(\\{\mathbf{k}\\};\\{s,\bar{s}\\})$ | Mellin-space time integral | (3.3)
$\mathbb{L}(\\{\mathbf{k}\\};\\{s,\bar{s}\\})$ | Mellin-space loop momentum integral | (29)
$\mathbb{M}_{L}(P)$ |  $L$-loop melon integral in Mellin space | (56)
$\mathfrak{M}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}(P)$ | Melon signal | (45)

##  Appendix B Useful Integrals

#### Melon integral.

To derive the melon signal
$\mathfrak{M}_{\mathsf{c}_{1}\cdots\mathsf{c}_{D}}(P)$ (45), we need to
compute the Mellin-space melon integral (55), and the result is given in (56).
Below we will show the calculation explicitly. The basic tool is the bubble
integral:

$\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{1\bar{1}}}|\mathbf{P}-\mathbf{q}|^{-2s_{2\bar{2}}}=\displaystyle\frac{\,{P^{3-2s_{1\bar{1}2\bar{2}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}s_{1\bar{1}2\bar{2}}-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}}\\\
3-s_{1\bar{1}2\bar{2}},s_{1\bar{1}},s_{2\bar{2}}\end{bmatrix}.$ (141)

We start with the definition:

$\displaystyle\mathbb{M}_{D-1}(P)=$
$\displaystyle\int\prod_{\ell=1}^{D-1}\bigg{[}\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{\ell}}\,}{(2\pi)^{3}}\big{|}\mathbf{q}_{\ell}\big{|}^{-2s_{\ell\bar{\ell}}}\bigg{]}\Big{|}\mathbf{P}-\sum_{i=1}^{D-1}\mathbf{q}_{i}\Big{|}^{-2s_{D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}}.$ (142)

We first consider the integral of $\mathbf{q}_{1}$, which is a bubble
integral:

$\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{1}}\,}{(2\pi)^{3}}\big{|}\mathbf{q}_{1}\big{|}^{-2s_{1\bar{1}}}\Big{|}\mathbf{P}-\sum_{i=1}^{D-1}\mathbf{q}_{i}\Big{|}^{-2s_{D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}}=\displaystyle\frac{\,{|\mathbf{P}-\sum_{i=2}^{D-1}\mathbf{q}_{i}|^{3-2s_{1\bar{1}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}{\color[rgb]{1,0,0}s_{1\bar{1}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}-\mbox{$\frac{\,{3}\,}{2}$}},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}\\\
{\color[rgb]{0,.3,1}3-s_{1\bar{1}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}},s_{1\bar{1}},s_{D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}\end{bmatrix}.$ (143)

Then we can finish the integral of $\mathbf{q}_{2}$, which is also a bubble
integral:

$\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}_{2}}\,}{(2\pi)^{3}}\big{|}\mathbf{q}_{2}\big{|}^{-2s_{2\bar{2}}}\Big{|}\mathbf{P}-\sum_{i=2}^{D-1}\mathbf{q}_{i}\Big{|}^{3-2s_{1\bar{1}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}}=\displaystyle\frac{\,{|\mathbf{P}-\sum_{i=3}^{D-1}\mathbf{q}_{i}|^{6-2s_{1\bar{1}2\bar{2}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}s_{1\bar{1}2\bar{2}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}-3,\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}},{\color[rgb]{0,.3,1}3-s_{1\bar{1}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu}}\\\
\mbox{$\frac{\,{9}\,}{2}$}-s_{1\bar{1}2\bar{2}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu},s_{2\bar{2}},{\color[rgb]{1,0,0}s_{1\bar{1}D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}-\mbox{$\frac{\,{3}\,}{2}$}}\end{bmatrix}.$ (144)

We see that the two $\Gamma$ functions in red are canceled between (143) and
(144), and so are those in blue. We can then repeat the procedure and finish
the full integral, and the result is exactly (56):

$\mathbb{M}_{D-1}(P)=\displaystyle\frac{\,{P^{3(D-1)-2s_{1\bar{1}\cdots
D\mkern 2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}}}\,}{(4\pi)^{3(D-1)/2}}\Gamma\begin{bmatrix}s_{1\bar{1}\cdots D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}-\mbox{$\frac{\,{3(D-1)}\,}{2}$}\\\
\mbox{$\frac{\,{3D}\,}{2}$}-s_{1\bar{1}\cdots D\mkern
2.0mu\overline{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu}\end{bmatrix}\prod_{\ell=1}^{D}\Gamma\begin{bmatrix}\mbox{$\frac{\,{3}\,}{2}$}-s_{\ell\bar{\ell}}\\\
s_{\ell\bar{\ell}}\end{bmatrix}.$ (145)

#### Expansion of loop integrals

For more complicated Mellin-space loop momentum integrals, it is very
difficult to compute the closed-form result. However, we can calculate the
leading contribution in special hierachical limits of the momentum
configuration. For instance, we consider the triangle integral:

$\mathbb{L}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}|^{-2s_{2\bar{2}}}|\mathbf{k}_{1}-\mathbf{q}|^{-2s_{3\bar{3}}}.$
(146)

This is the loop integral of a triangle with external momenta
$\mathbf{k}_{s}$, $\mathbf{k}_{1}$ and
$\mathbf{k}_{2}=-\mathbf{k}_{1}-\mathbf{k}_{s}$. Furthermore, we can consider
the squeezed limit $k_{1}\gg k_{s}$. Then the leading result is:

$\displaystyle\mathbb{L}|_{k_{1}\gg k_{s}}=$
$\displaystyle~{}\displaystyle\frac{\,{k_{1}^{3-2s_{1\bar{1}2\bar{2}3\bar{3}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}s_{1\bar{1}2\bar{2}3\bar{3}}-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}3\bar{3}}\\\
3-s_{1\bar{1}2\bar{2}3\bar{3}},s_{1\bar{1}},s_{2\bar{2}3\bar{3}}\end{bmatrix}$
$\displaystyle+\displaystyle\frac{\,{k_{s}^{3-2s_{1\bar{1}2\bar{2}}}k_{1}^{-2s_{3\bar{3}}}}\,}{(4\pi)^{3/2}}\Gamma\begin{bmatrix}s_{1\bar{1}2\bar{2}}-\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}-s_{1\bar{1}},\mbox{$\frac{\,{3}\,}{2}$}-s_{2\bar{2}}\\\
3-s_{1\bar{1}2\bar{2}},s_{1\bar{1}},s_{2\bar{2}}\end{bmatrix}+\mathcal{O}\Big{(}\displaystyle\frac{\,{k_{s}}\,}{k_{1}}\Big{)}.$
(147)

The detailed derivation can be found in the Appendix B of [94]. Here we
provide a much simpler proof which could be applied to general situations,
including the 2-loop examples in Sec. 6.

In the limit $k_{s}\rightarrow 0$, there could be two contributions. One piece
is analytic in $k_{s}$, then we can set $k_{s}\rightarrow 0$ in the integrand
of (146). Equivalently, we can expand the integrand and take the leading
contribution in the region $q\gg k_{1}$:

$\mathbb{L}|_{q\gg
k_{1}}=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{1\bar{1}2\bar{2}}}|\mathbf{k}_{1}-\mathbf{q}|^{-2s_{3\bar{3}}},$
(148)

which becomes a bubble integral and gives the first line of (B). Another piece
is nonanalytic in $k_{s}$. This piece appears in the region $q\ll k_{1}$, so
the factor $|k_{1}-q|^{-2s_{3\bar{3}}}$ becomes $|k_{1}|^{-2s_{3\bar{3}}}$,
and we obtain:

$\mathbb{L}|_{q\ll
k_{1}}=|k_{1}|^{-2s_{3\bar{3}}}\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{q}}\,}{(2\pi)^{3}}|\mathbf{q}|^{-2s_{1\bar{1}}}|\mathbf{k}_{s}-\mathbf{q}|^{-2s_{2\bar{2}}},$
(149)

which is again a bubble integral and gives the second line of (B).

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