# Dispersive Bootstrap of Massive Inflation Correlators

Haoyuan Liu${}^{\,a\,}$111Email<EMAIL_ADDRESS>Zhehan
Qin${}^{\,a\,}$222Email<EMAIL_ADDRESS>Zhong-Zhi
Xianyu${}^{\,a,b\,}$333Email<EMAIL_ADDRESS>
${}^{a}\,$_Department of Physics, Tsinghua University, Beijing 100084, China_
${}^{b}\,$_Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026,
China_

###### Abstract

Inflation correlators with massive exchanges are central observables of
cosmological collider physics, and are also important theoretical data for us
to better understand quantum field theories in dS. However, they are difficult
to compute directly due to many technical complications of the Schwinger-
Keldysh integral. In this work, we initiate a new bootstrap program for
massive inflation correlators with dispersion relations on complex momentum
planes. We classify kinematic variables of a correlator into vertex energies
and line energies, and develop two distinct types of dispersion relations for
both of them, respectively called vertex dispersion and line dispersion
relations. These dispersion methods allow us to obtain full analytical results
of massive correlators from a knowledge of their oscillatory signals alone,
while the oscillatory signal at the tree level can be related to simpler
subgraphs via the cutting rule. We further apply this method to massive loop
correlators, and obtain new analytical expressions for loop diagrams much
simpler than existing results from spectral decomposition. In particular, we
show that the analyticity demands the existence of an “irreducible background”
in the loop correlator, which is unambiguously defined, free of UV divergence,
and independent of renormalization schemes.

`USTC-ICTS/PCFT-24-23`

###### Contents

  1. 1 Introduction
  2. 2 Analytical Structure on a Complex Vertex-Energy Plane
    1. 2.1 Inflation correlators
    2. 2.2 Dispersion relations
    3. 2.3 Analytical continuation by contour deformation
    4. 2.4 Vertex dispersion relation of the seed integral
  3. 3 Bootstrapping Correlators with Vertex Dispersion Relation
    1. 3.1 Three-point single-exchange graph
    2. 3.2 Three-point one-loop bubble graph
    3. 3.3 Four-point single-exchange graph
  4. 4 Analytical Structure on the Complex Line-Energy Plane
  5. 5 Bootstrapping Correlators with Line Dispersion Relation
  6. 6 Conclusions and Outlooks
  7. A Notations
  8. B Useful Functions and Properties
  9. C Vertex Dispersion Integral with PMB Representation
  10. D Line Dispersion Integral with PMB Representation
  11. E Dispersion Integral for a Minkowski One-Loop Correlator

## 1 Introduction

There have been active and ongoing efforts in the study of $n$-point
correlation functions of primordial curvature fluctuations in recent years [1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71]. These functions are, on the one
hand, observables extracted from cosmic microwave background (CMB) or large-
scale structure (LSS) data, and, on the other hand, generated by quantum
process of particle productions and interactions during the cosmic inflation.
Therefore, these correlation functions, subsequently called _inflation
correlators_ , are the central object that bridge the observational data with
quantum field theory in inflationary spacetime.

A particular class of correlation functions mediated by massive particles have
attracted many attentions in recent years [72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114,
115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129,
130, 131, 132, 133, 134, 135]. A propagating massive particle during inflation
could impact the inflaton fluctuations through a resonant process, and leaves
a distinct pattern in the inflation correlators as logarithmic oscillations in
momentum ratios. The logarithmic nature is a consequence of exponential
expansion of the inflating universe [136, 80, 94, 137], while the oscillations
encode rich physical information about the massive particles. For these
reasons, the logarithmic oscillations have been dubbed “clock signals” and
“cosmological collider (CC) signals.”

The phenomenological studies of CC physics have identified many scenarios
producing large CC signals [73, 86, 93, 104, 101, 102, 96, 103, 105, 109,
111], which are promising targets for the current and upcoming CMB and LSS
observations [138, 139, 140, 141, 142, 143, 144, 145, 146]. To connect theory
predictions to observational data, it is crucial to perform efficient and
accurate computations of inflation correlators. It’s not surprising that
progress from analytical studies can facilitate this process. Theory-wise,
inflation correlators encode important data of quantum field theories in the
bulk de Sitter (dS), and are interesting objects in their own rights. Given
the great success of amplitude program in other spacetime backgrounds such as
Minkowski and AdS, we are now increasingly motivated in developing amplitude
techniques in dS, which are more relevant to our very own universe.

Many progresses have been made recently in the study of dS correlators or
cosmological correlators in general. Relevant to this work is the analytical
structure of massive inflation correlators in momentum space, which have been
explored in recent years from different angles, e.g., [7, 22, 21, 26, 28, 33,
44, 49, 50, 68, 79]. To explain this analytical structure, it is convenient to
start from a soft limit where the momentum $K$ of a bulk massive propagator
goes to zero [79, 7, 33, 49, 50]. As will be detailed below, a general graph
in this limit can be separated into three pieces: a _nonlocal signal_ which is
in nonanalytic in the soft momentum $K$ in the form of a branch cut; a _local
signal_ which is analytic in $K$, but nonanalytic in the energy ratios also in
the form of a branch cut; and finally, a _background_ which is analytic in
both momentum $K$ and other energy variables.

Although we use the analytical property to classify the signals and the
background, this classification has a practical consequence when doing real
computations. To explain this point, we note that a bulk computation of a
given graph involves a time integral at each bulk vertex and a momentum
integral for each independent loop [85]. In particular, the bulk propagators
contain a part that depends on the ordering of its two time variables, and
this makes the bulk time integral heavily nested. Therefore, a direct
integration is typically difficult.444See, however, a recently proposed method
to compute arbitrary nested time integrals [56, 67]. However, a curious
observation is that the computation of signals (both nonlocal and local) is
generally simpler than the background. The reason is that, to get the signals,
one can execute appropriate cuts of the graph to remove certain nested time
integrals. The simplicity of signals also shows up in final results:
Typically, both the signal and the background are (generalized) hypergeometric
functions of momentum ratios, but the background is of higher “transcendental
weight”555Here we are using the term “transcendental weight” to characterize
the complexity of hypergeometric series arising in inflation correlators. Very
loosely, an irreducible hypergeometric function of $n$-variables can be
thought of as having weight $n$. This meaning can be made precise by the
family-chain decomposition, as explained in [67]. than the signal [46, 56, 67,
135].

In addition, a closer inspection shows that the computation of nonlocal signal
is simpler than that of local signal. To get the nonlocal signal, one can take
a simpler _nonlocal cut_ of the graph, which replaces the cut propagator by
its real part [49]. The nonlocal signal also obeys the on-shell factorization
at arbitrary loop orders [50]. In comparison, the computation of local signals
requires a subtle and asymmetric cut, which depends on external kinematics and
also retains the imaginary part of the propagator [33]. Besides, it remains
challenging to identify local signals at arbitrary loop orders although some
progress is ongoing.

To recapitulate, our past experience shows that there is a “hierarchy” in the
complexity and also the difficulty of computing the three parts of a given
graph: In descending order, we have background $>$ local signal $>$ nonlocal
signal. Thus, it is tempting to ask if we can bootstrap the full result of a
given graph starting from its signal part alone, or better, if we can
bootstrap the full shape with the knowledge of the nonlocal signal only.

To answer these questions, in this work, we initiate a “dispersive” bootstrap
program for massive inflation correlators, with the dispersion relation as a
key ingredient. The dispersion relation is a very well studied technique,
tailored to recover the full function from knowledge of its discontinuities.
As the first step, we apply the dispersion relations and get full analytical
expressions for a range of massive inflation correlators at both tree and
1-loop levels. The ingredient of the dispersion integral can be either the
full signal (both local and nonlocal) or the nonlocal signal alone.
Technically, these ingredients can be obtained by computing factorized time
integrals, which correspond to simpler subgraphs at the tree level. The
essential idea of this method is schematically illustrated in Fig. 1.

Figure 1: A schematic illustration of the dispersive bootstrap of inflation
correlators with a massive exchange. On the left hand side, we have an 4-point
boundary correlator of inflatons (black lines) mediated by a massive bulk
propagator (cyan wiggly line) characterized by a (properly rescaled) momentum
variable $r$. On the right hand side, this correlator is expressed as a
dispersion integral along its branch cut, whose discontinuity can be obtained
by cutting the graph open and computing the two simpler subgraphs.

The dispersion relation is an old tool. It has played a central role in the
flat-space S-matrix bootstrap program [147, 148, 149, 150, 151, 152, 153, 154,
155, 156]. There have also been many studies on the cutting rule and
dispersion relations in CFT [157, 158, 159, 160, 161]. Given many types of
cutting rules for inflation correlators proposed recently [22, 20, 21, 33, 49,
50], it is a natural next step to try to “glue” those cut subgraphs back
together. While there are many discussions on dispersion relations at a
conceptual level, we are not aware of any previous study using dispersion
relations to explicitly bootstrap massive inflation correlators. We fill this
gap by providing explicit calculations with dispersion relations for a few
typical examples.

Our results at the tree level are not new; All the tree correlators considered
in this work have been worked out using other methods, and our method here is
by no means “simpler” than existing methods such as cosmological bootstrap [7,
8, 46] or partial Mellin-Barnes representation [37, 39]. Rather, we use these
known examples as tests of principle for the dispersive bootstrap method. We
expect that one can use this method to “glue” more subgraphs and get full
results for more complicated graphs, either analytically or numerically, where
other methods may not be immediately applicable.

On the other hand, at the 1-loop level, we do obtain new analytical
expressions for a class of 1-loop 3-point functions. Our expressions are
substantially simpler than known results obtained with spectral decomposition
[42], and are far easier to implement numerically. This result shows that the
dispersive bootstrap can be a promising way to compute inflation correlators
with massive loops, which we will further develop in a future study.

An appealing feature of our dispersion technique at the 1-loop level is that
it is insensitive to the renormalization ambiguities, because the UV sensitive
part of the 1-loop correlator can always be subtracted by a local counterterm
and thus is local and analytic. In a sense, the background part of the 1-loop
diagram obtained by the dispersion relation can be viewed as an “irreducible”
companion of the signals, whose existence is enforced by the correct
analytical behavior of the full correlator.

#### Outline of this work

At the heart of our dispersive bootstrap is a detailed understanding of the
analytical structure of a specific graph contribution to an inflation
correlator. In general, after properly removing all tensor structures, a tree-
graph contribution to the inflation correlator is a scalar function of two
types of kinematic variables: the _vertex energies_ and the _line energies_.
The vertex energy is the magnitude sum of momenta of all external lines at a
vertex, while a line energy is the magnitude of the momentum flowing in an
internal line.

For physically reachable kinematical configurations (henceforth _physical
regions_), vertex and line energies are necessarily positive real. However, to
develop dispersion relations, we need to study a graph as a function of
complex energies. Our strategy is to consider only one variable being complex
at a time, with all other variables staying in their physical regions. We can
complexify either a vertex energy or a line energy. In both cases, a massive
inflaton correlator develops branch points on the corresponding complex plane,
connected by branch cuts. With these branch cuts, we can build corresponding
dispersion integrals which compute the full correlator. Thus, we have two
distinct types of dispersion relations: the _vertex dispersion relation_ built
on a vertex energy complex plane, and the _line dispersion relation_ built on
a line energy complex plane. As we shall see, for a four-point correlator with
single massive exchange, the vertex dispersion relation computes the whole
graph from its signal, both local and nonlocal. On the other hand, the line
dispersion relation computes the whole graph from its nonlocal signal only.

While the vertex and line dispersion relations can be constructed for very
general tree graphs, in this work, for definiteness, we will focus on 4-point
correlators with $s$-channel massive exchange (Fig. 2) and the related 3-point
single-exchange correlators (Fig. 5), the only exception being the 3-point
1-loop bubble graph (Fig. 7), which is related to tree graphs via spectral
decomposition.

In Sec. 2, we begin with a brief review of inflation correlators and the
dispersion relation. In particular, we introduce the four-point seed integral
$\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ in (11) which is the central
object to be studied in this work. Here $k_{i}\equiv|\bm{k}_{i}|$
($i=1,\cdots,4,s$) are magnitudes of momenta (also called energies) shown in
Fig. 2, and $k_{ij}\equiv k_{i}+k_{j}$. A very important technical step is the
analytical continuation of inflation correlators on the complex energy plane.
Thus, in Sec. 2.3, we use a few toy examples to explain how to take analytical
continuation by contour deformation of an integral expression as a function of
its (unintegrated) parameters. Then, we put this method in use in Sec. 2.4 and
identify the branch cut of the seed integral
$\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ on the complex $k_{12}$ plane.
With this method, we can compute the discontinuity of the seed integral across
this branch cut without computing the integral itself, as summarized in (55),
which is the main result of this section.

Then, in Sec. 3, we use the vertex dispersion relation to bootstrap a few
3-point and 4-point correlators. For the 3-point correlator, we also consider
a one-loop example, where we make use of the loop signal computed via spectral
decomposition and dispersively bootstrap the full loop correlator. While our
computation of tree graphs recovers previously known results, we get a new
analytical expression for the 3-point 1-loop correlator substantially simpler
than the existing result.

In Sec. 4, we switch to a different perspective and consider the seed integral
$\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ on the complex $k_{s}$ plane.
We show that the seed integral also possesses a few branch points on $k_{s}$
plane which are connected by branch cuts. The discontinuities of these branch
cuts are again computable. Remarkably, all the discontinuities in this case
can be related to the discontinuity of the nonlocal signal alone, as shown in
(127). So, we can build up a line dispersion relation connecting the whole
seed integral with its nonlocal signal.

Then, in Sec. 5, we use the line dispersion to recover the full seed integral
from the nonlocal signal. This calculation has the advantage that it uses a
minimal set of data to bootstrap the full shape, but the drawback that the
computation is complicated. It is nevertheless a useful proof of concept and
points to possibilities of (analytical or semi-analytical) computation of more
complicated correlators from their readily available nonlocal signal alone. We
provide further discussions and outlooks in Sec. 6. In the first two
appendices, we collect a few frequently used notations (App. A) and special
functions, together with their useful properties (App. B). We collect the
details of analytical evaluations of vertex and line dispersion integrals in
App. C and App. D, respectively. Finally, in App. E, we use a simple 1-loop
correlator in Minkowski spacetime to demonstrate the relation between the
dispersive method and a conventional calculation with dimensional
regularization.

#### Comparison with previous works

The dispersion relation is a topic with rich history. It is not surprising
that this relation, together with several closely related concepts such as
discontinuities, the optical theorem, cutting rules, has been explored in the
context of cosmological correlators (and, relatedly, the wavefunction
coefficients) from various different angles [14, 20, 21, 33, 26, 22, 49, 50,
53]. There are a few similarities and differences between the discontinuities
studied in the previous works and the current work, on which we very briefly
comment here.

In previous works such as [162, 21, 20], the discontinuity of an amplitude
(typically a wavefunction coefficient) is normally defined to be the
difference between the amplitude and its complex conjugate with one or several
energies’ signs flipped. In this combination, one can replace one or a product
of several propagators by the real part. (It was the imaginary part in [21,
20] due to a different convention.) Since the real part of a bulk propagator
is always factorized, the discontinuity of an amplitude defined in this way
possesses a cutting rule. The nice thing about this definition is that it has
a natural origin from the the unitarity of the theory, and therefore, one can
use this discontinuity to formulate an optical theorem for cosmological
amplitudes [18]. Generalized to the loop level, such a discontinuity can be
expressed as momentum integrals of products of (discontinuity of) tree sub-
diagrams [53]. The dispersion relations for wavefunctions were used to
construct wavefunction coefficients with massless scalars in [26]. Similar
dispersion relations in full Mellin space were discussed in [14].

In comparison, the discontinuity we are going to use is defined with respect
to a correlator alone, without invoking its complex conjugate. More
importantly, for the dispersion relation to work as a bootstrap tool, we need
to identify all branch cuts of a correlator on the entire complex plane of an
energy, where the energy can take arbitrary unphysical value. To extract this
information, it is essential to take analytical continuation of a correlator
beyond its physical domain, which is not a trivial task as we shall show.

Furthermore, our starting point is the correlators rather than the bulk
propagators, so our dispersive bootstrap can be used to directly construct the
full correlators, for both tree and loop diagrams, rather than the integrand
as in [53].

With that said, there is certainly a connection between our definition of
discontinuity of a correlator and the discontinuity defined in previous works.
For instance, we find that the discontinuity of a tree diagram is also
factorized, and expressible in terms of factorized part of propagators. Also,
in the line dispersion relation introduced in this work, the discontinuity in
the squeezed limit corresponds exactly to the nonlocal signal, so the
discontinuity also obeys the nonlocal cutting rule and the factorization
theorem as the nonlocal signal [49, 50]. It would be interesting to explore
the deeper connections between this work and previous works such as [18, 21,
22, 53] where basic properties of amplitudes such as unitarity and locality
are manifest. We leave this to future exploration.

#### Notations and conventions

We work in the slow-roll limit of the inflation where the spacetime is
described by the inflation patch of the dS spacetime, and the spacetime metric
reads
${\mathrm{d}}s^{2}=a^{2}(\tau)(-{\mathrm{d}}\tau^{2}+{\mathrm{d}}\bm{x}^{2})$.
Here $\bm{x}\in\mathbb{R}^{3}$ is the spatial comoving coordinate,
$\tau\in(-\infty,0)$ is the conformal time, and $a(\tau)=-1/(H\tau)$ is the
scale factor with $H$ being the constant Hubble parameter. We take the energy
unit $H=1$ throughout this work. We use bold italic letters such as $\bm{k}$
to denote 3-momenta and the corresponding italic letter $k\equiv|\bm{k}|$ to
denote its magnitude, which is also called an energy. For sums of several
indexed quantities, we use a shorthand notation such as $k_{12}\equiv
k_{1}+k_{2}$. Other frequently used variables are collected in App. A.
Finally, we make heavy use of the discontinuity of a complex function across
its branch cut and it is useful to fix our convention from the very beginning.
In this work, the branch cut of a function $f(z)$ appears almost always on the
real axis of $z$. Therefore, we define the discontinuity of a function $f(z)$
for such a branch cut as:

$\mathop{\mathrm{Disc}}_{z}f(z)\equiv\lim_{\epsilon\rightarrow
0^{+}}\Bigl{[}f(z+\mathrm{i}\epsilon)-f(z-\mathrm{i}\epsilon)\Bigr{]}.~{}~{}~{}~{}(z\in\mathbb{R})$
(1)

## 2 Analytical Structure on a Complex Vertex-Energy Plane

###  2.1 Inflation correlators

In this subsection, we set the stage by reviewing the basic kinematic
structure of the correlation functions to be studied in this work. We consider
generic boundary correlators of a massless or conformal scalar field, with
arbitrary massive bulk exchanges. Apart from a three-point example in the next
section, we will mostly consider tree-level diagrams. Also, we assume all bulk
fields are directly coupled, i.e., without derivatives acting on them.
Generalizations to derivative couplings or spinning exchanges are
straightforward by including appropriate tensor structures.

#### Vertex energies and line energies

Using the standard diagrammatic rule in the Schwinger-Keldysh (SK) formalism
[85], it is straightforward to write down an integral expression for any tree-
level correlation function. For definiteness, let us consider a scalar theory
with a conformal scalar field $\phi_{c}$ and a collection of $N_{F}$ massive
fields $\sigma_{A}$ $(A=1,\cdots,N_{F})$. In dS, a conformal scalar field
$\phi_{c}$ has an effective mass $m^{2}=2$, while the masses of $\sigma_{A}$
can be arbitrary. We assume these fields are coupled directly via polynomial
interactions with (possibly) power time dependences. Then, the SK integral for
a generic tree-level correlator of $\phi_{c}$ takes the following form:

$\displaystyle\mathcal{G}(\bm{k}_{1},\cdots,\bm{k}_{N})=\sum_{\mathsf{a}_{1},\cdots,\mathsf{a}_{V}=\pm}\int_{-\infty}^{0}\prod_{\ell=1}^{V}\Big{[}{\mathrm{d}}\tau_{\ell}(-\mathrm{i}\mathsf{a}_{\ell})(-\tau_{\ell})^{p_{\ell}}\Big{]}\prod_{i=1}^{N}C_{\mathsf{a}_{i}}(k_{i},\tau_{i})\prod_{j=1}^{I}D_{\mathsf{a}_{j}\mathsf{b}_{j}}(K_{j};\tau_{j},\tau_{j}^{\prime}).$
(2)

This is an integral of $V$ time variables $\tau_{\ell}$ for all $V$ vertices,
with the integrand being products of time-dependent coupling factors
$(-\tau_{\ell})^{p_{\ell}}$ and two types of propagators. We assume the powers
$p_{\ell}$ are not too negative such that the graph remains perturbative in
the $\tau\rightarrow 0$ limit. The bulk-to-boundary propagator
$C_{\mathsf{a}}(k;\tau)$ is constructed from a conformal scalar field
$\phi_{c}$ with mass $m^{2}=2$:

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

Here $|\tau_{f}|\ll 1$ is a final time cutoff, and is introduced to
characterize the leading fall-off behavior of a conformal scalar as
$\tau\rightarrow 0$. In physical situations with external modes being massless
scalars or tensors, this cutoff is unnecessary.666Also, the case of external
massless mode can be conveniently obtained from the conformal scalar case here
by acting appropriate differential operators of kinematic variables [79, 7,
8]. Moreover, $D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ is the bulk
propagator for the massive scalar field $\sigma$ with mass $m$:

$\displaystyle D_{-+}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(\tau_{1}\tau_{2})^{3/2}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k\tau_{1})\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}(-k\tau_{2}),$
(4) $\displaystyle D_{+-}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(\tau_{1}\tau_{2})^{3/2}\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(-k\tau_{1})\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}(-k\tau_{2}),$
(5) $\displaystyle D_{\pm\pm}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}D_{\mp\pm}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})\theta(\tau_{1}-\tau_{2})+D_{\pm\mp}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}(k;\tau_{1},\tau_{2})\theta(\tau_{2}-\tau_{1}),$ (6)

where $\mathrm{H}_{\nu}^{(j)}(z)$ $(j=1,2)$ is the Hankel function of $j$’th
type. In this work, we choose $\sigma$ to be in the principal series, namely,
$m>3/2$, so that the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\equiv\sqrt{m^{2}-9/4}$ is positive, and we get oscillatory signals from
$\sigma$. Generalization to complementary scalar with $0<m<3/2$ is completely
straightforward.

In (2), we have summations over all SK indices $\mathsf{a}_{\ell}=\pm$ for all
$V$ vertices. When doing so, we require each of the SK indices appearing in
the subscript of propagators to be identified with the corresponding index on
the vertex to which the propagator attach.

It is trivial to see that the conformal scalar bulk-to-boundary propagator (3)
satisfies the relation $C_{\mathsf{a}}(k_{1};\tau)\cdots
C_{\mathsf{a}}(k_{n};\tau)=C_{\mathsf{a}}(k_{1}+\cdots+k_{n};\tau)$ up to
multiplications of prefactors $\tau_{\ell}\tau_{f}/(2k_{\ell})$. As a result,
the graph $\mathcal{G}(\\{\bm{k}_{i}\\})$ depends on all spatial vector
momenta $\\{\bm{k}_{i}\\}$ only through two particular classes of scalar
variables, the _vertex energies_ $E_{\ell}$ $(\ell=1,\cdots,V)$ and the _line
energies_ $K_{j}$ ($j=1,\cdots,I$): A vertex energy is assigned to each vertex
of the tree diagram, and equals to the _magnitude sum_ of the momenta of all
_external lines_ (bulk-to-boundary propagators) attached to the vertex. A line
energy, on the other hand, is assigned to each internal line (bulk propagator)
of the tree diagram, and equals to the magnitude of the momentum flowing
through this bulk line. Clearly, by momentum conservation, a line energy can
always be expressed as the magnitude of a _vector sum_ of the momenta of all
external lines at either side of the bulk line.

Following the above analysis, we can always write the graph as:

$\mathcal{G}(\bm{k}_{1},\cdots,\bm{k}_{N})=\prod_{i=1}^{N}\Big{(}\displaystyle\frac{\,{\tau_{f}}\,}{2k_{i}}\Big{)}\times\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{G}\mkern-2.0mu}\mkern
2.0mu(E_{1},\cdots,E_{V};K_{1},\cdots,K_{I}).$ (7)

We emphasize that this dependence works only for a particular diagram. Since
we will develop dispersion relations at the diagrammatic level, this set of
variables suit our purpose well. Explicitly, we have:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{G}\mkern-2.0mu}\mkern
2.0mu\big{(}\\{E_{\ell}\\};\\{K_{j}\\}\big{)}=\sum_{\mathsf{a}_{1},\cdots,\mathsf{a}_{V}=\pm}\int_{-\infty}^{0}\prod_{\ell=1}^{V}\Big{[}{\mathrm{d}}\tau_{\ell}(-\mathrm{i}\mathsf{a}_{\ell})(-\tau_{\ell})^{p_{\ell}}e^{\mathrm{i}\mathsf{a}_{\ell}E_{\ell}\tau_{\ell}}\Big{]}\prod_{j=1}^{I}D_{\mathsf{a}_{j}\mathsf{b}_{j}}(K_{j};\tau_{j},\tau_{j}^{\prime}).$
(8)

The dispersion relations always involve analytical continuation of the
correlator in the complex plane of some variables. Typically, we consider the
complex plane of only one variable at a time, and keep all other variables
fixed in their physical region. For the tree diagram $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{G}\mkern-2.0mu}\mkern
2.0mu(\\{E_{\ell}\\},\\{K_{j}\\})$, we can choose to analytically continue a
vertex energy $E_{\ell}$ or a line energy $K_{j}$. With these two choices, we
can respectively develop a _vertex dispersion relation_ , and a _line
dispersion relation_. Each of them has its own merits and drawbacks.

Figure 2: The 4-point correlator of conformal scalars $\phi_{c}$ with a single
massive scalar exchange $\sigma$ in $s$-channel.

#### Four-point seed integral

To be concrete, we will derive explicit dispersion relations for a tree-level
four-point function of a conformal scalar $\phi_{c}$ with single exchange of a
massive scalar $\sigma$ in the $s$-channel, shown in Fig. 2. Dispersion
relations for more general correlation functions have similar structures and
will be presented in a future work. Assuming a direct coupling
$-\mbox{$\frac{\,{1}\,}{2}$}\sqrt{-g}\lambda\phi_{c}^{2}\sigma$, the integral
expression for this graph reads:

$\displaystyle\mathcal{G}_{s}(\bm{k}_{1},\cdots,\bm{k}_{4})=$
$\displaystyle-\lambda^{2}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{(-\tau_{1})^{4}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{(-\tau_{2})^{4}}$
$\displaystyle\times
C_{\mathsf{a}}(k_{1};\tau_{1})C_{\mathsf{a}}(k_{2};\tau_{1})C_{\mathsf{b}}(k_{3};\tau_{2})C_{\mathsf{b}}(k_{4};\tau_{2})D_{\mathsf{a}\mathsf{b}}(k_{s};\tau_{1},\tau_{2}).$
(9)

In light of the explicit expression for the conformal propagator (3), it is
useful to define the following dimensionless _seed integral_ , as introduced
in [39], which enables direct generalization to arbitrary interactions and
massless scalar/tensor external modes:777Note that our choice of arguments of
the seed integral $\mathcal{I}^{p_{1}p_{2}}$ is different from previous papers
including [39], where the seed integral is defined to be a function of two
dimensionless momentum ratios, often chosen as $r_{1}=k_{s}/k_{12}$ and
$r_{2}=k_{s}/k_{34}$. Here, we prefer to explicitly retain the dependence on
the three energies $k_{12}$, $k_{34}$, and $k_{s}$, since it is more
transparent to consider the analytical property of the seed integral on the
complex plane of an energy variable instead of a momentum ratio.

$\displaystyle\mathcal{I}^{p_{1}p_{2}}_{\mathsf{ab}}(k_{12},k_{34},k_{s})=-\mathsf{ab}\
k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2});$
(10)
$\displaystyle\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\equiv\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}^{p_{1}p_{2}}_{\mathsf{a}\mathsf{b}}(k_{12},k_{34},k_{s}).$
(11)

The introduction of arbitrary power factors $(-\tau_{i})^{p_{i}}$ $(i=1,2)$ is
to take account of various interaction types and external mode functions. The
exponents $p_{1,2}$ can in general take complex values (as in models with
resonant background). However, we will take $p_{1,2}$ to be real purely to
reduce the complication of the analysis. The generalization to complex
$p_{1,2}$ is straightforward.

By construction, it is evident that the seed integral
$\mathcal{I}^{p_{1}p_{2}}_{\mathsf{ab}}(k_{12},k_{34},k_{s})$ is
dimensionless, and thus can be expressed as a function of dimensionless
momentum ratios. We will exploit this fact when doing explicit computations.
Also, the graph ${\mathcal{G}}_{s}(\bm{k}_{1},\cdots,\bm{k}_{4})$ is
expressible in terms of the seed integral as:

$\displaystyle\mathcal{G}_{s}(\bm{k}_{1},\cdots,\bm{k}_{4})=\displaystyle\frac{\,{\lambda^{2}\tau_{f}^{4}}\,}{16k_{1}k_{2}k_{3}k_{4}k_{s}}\mathcal{I}^{-2,-2}(k_{12},k_{34},k_{s}).$
(12)

Thus, we have reduced the whole problem to an analysis of the seed integral.
It is certainly possible to compute the entire seed integral by other methods
such as partial Mellin-Barnes representation [39] or bootstrap equations [7,
39]. However, to be in accordance with the spirit of the dispersive bootstrap,
we avoid such a direct computation, but pay more attention to the analytical
structure of the seed integral itself. Finally, it is worth noting that the
physical regions of the energies $(k_{12},k_{34},k_{s})$ are given by $0\leq
k_{s}\leq k_{12}$ and $0\leq k_{s}\leq k_{34}$ due to the triangle
inequalities from momentum conservation.

###  2.2 Dispersion relations

In the current and next subsections, we make some mathematical preparations
for deriving the vertex dispersion relation in Sec. 2.4. In this subsection,
we very briefly explain what a dispersion relation is for nonexperts. Readers
familiar with this topic are free to skip this entire subsection.

At the mathematical level, a dispersion relation is nothing but a clever
manipulation of the contour integral on a complex plane. As a simple but very
typical example, suppose we have $f(r)$ as a function of complex variable $r$,
which possesses a branch cut along the negative real axis $r<0$, but is
otherwise analytic everywhere. Furthermore, it is convenient (but not
necessary) to assume that $f(r)$ decreases fast enough as
$|r|\rightarrow\infty$. Suppose that all quantitative information we have
about $f(r)$ is its _discontinuity_ along the branch cut:

$\mathop{\text{Disc}}_{r}f(r)\equiv\lim_{\epsilon\rightarrow
0^{+}}\Big{[}f(r+\mathrm{i}\epsilon)-f(r-\mathrm{i}\epsilon)\Big{]}.~{}~{}~{}~{}~{}(r\in\mathbb{R})$
(13)

Then, a dispersion relation makes use of this quantitative information to
recover the original function $f(r)$ for an arbitrary given point $r$ on the
complex plane. As shown in the left panel of Fig. 3, we enclose the given
point $r$ by a small contour $\mathcal{C}$. Then, we have the following
equality by virtue of the residue theorem:

$f(r)=\int_{\mathcal{C}}\displaystyle\frac{\,{{\mathrm{d}}r^{\prime}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{f(r^{\prime})}\,}{r^{\prime}-r}.$
(14)

Now, as shown in the right panel of Fig. 3, we can deform the contour
$\mathcal{C}$ to a big circle $\mathcal{C}^{\prime}$ without changing the
answer of the integration. The new contour $\mathcal{C}^{\prime}$ is chosen
with radius $|r^{\prime}|\rightarrow\infty$ except on the negative real axis,
to which the contour approaches from both sides. By our assumption of the
analytical property of $f(r)$, the integration of
$f(r^{\prime})/(r-r^{\prime})$ along the big circle at
$|r^{\prime}|\rightarrow\infty$ vanishes. Then, we get:

$f(r)=\int_{\mathcal{C}^{\prime}}\displaystyle\frac{\,{{\mathrm{d}}r^{\prime}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{f(r^{\prime})}\,}{r^{\prime}-r}=\int_{-\infty}^{+\infty}\displaystyle\frac{\,{{\mathrm{d}}r^{\prime}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{\mathop{\mathrm{Disc}}_{r^{\prime}}f(r^{\prime})}\,}{r^{\prime}-r}.$
(15)

Thus, by performing an integration along the branch cut, we recover the value
of $f(r)$ at any point $r$.

The requirement that $f(r^{\prime})$ decreases faster enough when
$|r^{\prime}|\rightarrow\infty$ is to make sure that the integration over
$f(r^{\prime})/(r-r^{\prime})$ vanishes along the large circle at infinity.
This requirement can be loosen: So long as $f(r^{\prime})$ is bounded by a
power function of finite order, namely $|f(r^{\prime})/r^{\prime
n}|\rightarrow 0$ as $|r^{\prime}|\rightarrow\infty$ for some
$n\in\mathbb{Z}_{+}$, we can consider the following new function
$g(r_{1},\cdots,r_{n};r^{\prime})$:

$g(r_{1},\cdots;r_{n};r^{\prime})\equiv\displaystyle\frac{\,{f(r^{\prime})}\,}{(r^{\prime}-r_{1})\cdots(r^{\prime}-r_{n})},$
(16)

where $r_{1}\cdots r_{n}$ are $n$ arbitrarily chosen points. Then it is clear
that $g(r_{1},\cdots,r_{n};r^{\prime})$ decreases fast enough at infinity. So,
we can use $g$ in place of $f$ to do dispersion integral, at the expense that
we need the values of $f(r^{\prime})$ at $n$ discrete points
$r^{\prime}=r_{1},\cdots,r_{n}$. This way of dealing with large-circle
divergence is called _subtraction_ , and the number $n$ is called the _order_
of the subtraction.

$\Rightarrow$

Figure 3: An illustration of the dispersion integral for the function $f(r)$
with a branch cut shown by the magenta wiggly line. (One can think of $f(r)$
as the correlator shown in Fig. 1.) So long as the function $f(r^{\prime})$ in
question decreases fast enough as $|r^{\prime}|\rightarrow\infty$, one can
deform the small circle $\mathcal{C}$ enclosing the point $r^{\prime}=r$ to a
larger contour $\mathcal{C}^{\prime}$, and thus relate the value $f(r)$ with
an integral along the branch cut.

It is worth mentioning that the study of dispersion relations has a long
history in physics, with the Kramers-Krönig relation in classical
electrodynamics as a notable early example [163, 164]. In the S-matrix
bootstrap program for relativistic field theories, the dispersion relations
played a central role [147, 148, 149, 150, 151, 152, 153, 154, 155, 156]. In
these examples, the desired analytical property of the scattering amplitude is
closely related to causality [147, 148, 149, 154, 156]. At the perturbative
level, the analytical properties can also be diagnosed by methods such as
Landau analysis [147, 44, 154]. At a fixed order in perturbation theory, the
dispersion relation relates loop amplitudes with tree amplitudes, and in well-
situated cases, it allows one to reconstruct loop amplitudes from simpler tree
amplitudes. More remarkably, one can exploit the dispersion relation beyond
the perturbation theory [147, 154]. This has been shown useful in the study of
hadron physics, e.g., [155, 156]. Also, one can use the dispersion relation to
connect UV and IR parts of a theory and derive nontrivial positivity bounds
for low-energy effective theories [165, 166, 167, 168].

###  2.3 Analytical continuation by contour deformation

To derive a dispersion relation for the seed integral in (10), we need to
understand its analytical property as a function of _complex_ energies. Now we
face an obvious problem: While the original seed integral is well defined for
energies taking physical values, it is not for arbitrary complex energies.
Therefore, we need to redefine the seed integral so that it also applies to
complex energies. We want to do it without evaluating the full integral. To
see how this works, we demonstrate our method with three toy examples, before
considering the full seed integral in the next subsection.

#### One-fold integral

First, let us consider a complex function $I(z)$ for $z\in\mathbb{C}$ defined
by the following integral:

$\displaystyle I(z)\equiv\int_{0}^{\infty}{\mathrm{d}}w\,e^{\mathrm{i}zw},$
(17)

where the integral contour is chosen to be the positive real axis. The
integral is convergent when $z\neq 0$ and $\text{Arg}\,z\in(0,\pi)$, and is
integrated to the following result:

$\displaystyle I(z)=-\displaystyle\frac{\,{1}\,}{\mathrm{i}z}.$ (18)

Clearly, this expression is analytic everywhere in $z$ except when $z=0$. In
particular, it is analytic for $\text{Arg}\,z\notin(0,\pi)$, where the
original integral (17) is no longer well defined. So the question is how we
can modify the original integral so that it is well defined for arbitrary
$z\neq 0$. The answer for this example is simple enough. Indeed, let us
consider the following integral:

$\tilde{I}_{\delta}(z)\equiv\int_{0}^{z^{-1}e^{\mathrm{i}\delta}\infty}{\mathrm{d}}w\,e^{\mathrm{i}zw},$
(19)

where $\delta$ is a small positive real number. That is, the contour is
deformed to approach $w=\infty$ from the direction
$\text{Arg}\,w=-\text{Arg}\,z+\delta$. This integral is convergent for any
$z\neq 0$, and, in the mean time, we have:

$\lim_{\delta\rightarrow
0^{+}}\tilde{I}_{\delta}(z)=I(z).~{}~{}~{}~{}(\text{Im}\,z>0)$ (20)

Therefore, we can take $\tilde{I}_{\delta\rightarrow 0^{+}}(z)$ as the
analytical continuation of the original integral $I(z)$ to any $z\neq 0$. The
lesson here is that, when $z$ takes a value at which the original integral
$I(z)$ is not convergent (at infinity), we can deform the contour properly so
that the integral is convergent again.

#### One-fold integral with a branch cut

Taking analytical continuation by deforming the integral contour may have
obstructions when the integrand contains branch cuts. To see this, consider
the following example:

$J(z)\equiv\int_{0}^{\infty}{\mathrm{d}}w\,e^{\mathrm{i}zw}\sqrt{w},$ (21)

The situation is similar: The integral is well defined when
$\text{Arg}\,z\in(0,\pi)$, but the integrated result is analytic in a larger
region:

$J(z)=\displaystyle\frac{\,{\sqrt{\pi}e^{3\mathrm{i}\pi/4}}\,}{2z^{3/2}}.~{}~{}~{}~{}~{}(0<\text{Arg}\,z<\pi)$
(22)

This time we can consider the following integral:

$\tilde{J}_{\delta}(z)=\int_{0}^{z^{-1}e^{\mathrm{i}\delta}\infty}{\mathrm{d}}w\,e^{\mathrm{i}zw}\sqrt{w}.$
(23)

We still have $\tilde{J}_{\delta\rightarrow 0^{+}}(z)=J(z)$ when
$\text{Im}\,z>0$. However, the new phenomenon here is that the integral
possesses a branch point at $z=0$ due to the factor $\sqrt{w}$. For
definiteness, we can take the branch cut to be along the negative real axis
$w\in(-\infty,0)$. Then we see that this branch cut implies the existence of a
branch cut of the integral $\tilde{J}_{\delta}(z)$ along $z\in(-\infty,0)$
when $\delta\rightarrow 0^{+}$. Let us compute the discontinuity of this
branch cut:

$\displaystyle\mathop{\text{Disc}}_{z}\tilde{J}_{\delta}(z)=$
$\displaystyle~{}\tilde{J}_{\delta}(ze^{-\mathrm{i}\epsilon})-\tilde{J}_{\delta}(ze^{+\mathrm{i}\epsilon})$
$\displaystyle=$
$\displaystyle\int_{0}^{z^{-1}e^{+\mathrm{i}\epsilon}e^{\mathrm{i}\delta}\infty}{\mathrm{d}}w\,e^{\mathrm{i}zw}\sqrt{w}-\int_{0}^{z^{-1}e^{-\mathrm{i}\epsilon}e^{\mathrm{i}\delta}\infty}{\mathrm{d}}w\,e^{\mathrm{i}zw}\sqrt{w}$
$\displaystyle=$
$\displaystyle-\int_{0}^{-\infty}{\mathrm{d}}u\,\mathop{\text{Disc}}_{u}\Big{[}e^{-\mathrm{i}|z|ue^{\mathrm{i}\delta}}\sqrt{u}\Big{]}=2\mathrm{i}\int_{0}^{\infty}{\mathrm{d}}u\,e^{+\mathrm{i}|z|ue^{\mathrm{i}\delta}}\sqrt{u}.$
(24)

Then we let $\delta\rightarrow 0$, and get:

$\displaystyle\mathop{\text{Disc}}_{z}J(z)=2\mathrm{i}\lim_{\delta\rightarrow
0^{+}}\int_{0}^{\infty}{\mathrm{d}}u\,e^{+\mathrm{i}|z|ue^{\mathrm{i}\delta}}\sqrt{u}=-\displaystyle\frac{\,{\sqrt{\pi}e^{+\mathrm{i}\pi/4}}\,}{|z|^{3/2}}.$
(25)

Therefore, we have found a relation between the discontinuity of the integral
$J(z)$ and the integrand. The lesson from this example is that deforming the
contour to approach the branch cut of the integrand from two different
directions will lead to a discontinuity of the integral itself, and this
contour deformation procedure provides us a way to relate the discontinuities
of the integral and the integrand.

#### Two-fold integral

We will have to deal with time orderings when studying the seed integral. So,
our third example will be a two-fold time-ordered integral:

$K(z_{1},z_{2})\equiv\int_{0}^{\infty}{\mathrm{d}}w_{1}{\mathrm{d}}w_{2}\,e^{\mathrm{i}z_{1}w_{1}+\mathrm{i}z_{2}w_{2}}\theta(w_{1}-w_{2}).$
(26)

Again, when $\text{Im}\,z_{1}>0$ and $\text{Im}\,z_{2}>0$ hold at the same
time, the integral is well defined, and can be directly integrated to:

$K(z_{1},z_{2})=-\displaystyle\frac{\,{1}\,}{z_{1}(z_{1}+z_{2})}.~{}~{}~{}~{}~{}(\text{Im}\,z_{1}>0\text{~{}and~{}}\text{Im}\,z_{2}>0)$
(27)

Now we want to analyze the above integral for more general choice for $z_{1}$
and $z_{2}$. In particular, we assume that $z_{2}>0$ stays in the positive
real axis, while $z_{1}\in\mathbb{C}$ can take arbitrary complex values. Then,
we can first rewrite the original integral as an iterated integral:

$K(z_{1},z_{2})\equiv\int_{0}^{\infty}{\mathrm{d}}w_{1}\int_{0}^{w_{1}}{\mathrm{d}}w_{2}\,e^{\mathrm{i}z_{1}w_{1}+\mathrm{i}z_{2}w_{2}}=\displaystyle\frac{\,{1}\,}{\mathrm{i}z_{2}}\int_{0}^{\infty}{\mathrm{d}}w_{1}\,\Big{[}e^{\mathrm{i}(z_{1}+z_{2})w_{1}}-e^{\mathrm{i}z_{1}w_{1}}\Big{]}.$
(28)

As shown above, the inner-layer integral is trivially convergent, and we only
need to deal with the $w_{1}$-integral, which may be divergent. Now, we want
to deform the contour of $w_{1}$-integral to make it convergent for any
$z_{1}\neq 0$ and $z_{1}+z_{2}\neq 0$. As a consistent deformation of the
original integral, we should use the same contour for both terms. Then, we
need a judicious choice for the direction along which the contour goes to
infinity. That is, we want to deform the integral contour in the following
way:

$\displaystyle\frac{\,{1}\,}{\mathrm{i}z_{2}}\bigg{[}\int_{0}^{(z_{1}+z_{2})^{-1}e^{\mathrm{i}\delta_{1}}\infty}{\mathrm{d}}w_{1}\,e^{\mathrm{i}(z_{1}+z_{2})w_{1}}-\int_{0}^{z_{1}^{-1}e^{\mathrm{i}\delta_{2}}\infty}{\mathrm{d}}w_{1}\,e^{\mathrm{i}z_{1}w_{1}}\bigg{]},$
(29)

such that two conditions hold at the same time: 1)
$0<\delta_{1},\delta_{2}<\pi$ so that both integrals converge; 2)
$\delta_{1}-\delta_{2}=\text{Arg}\,(z_{1}+z_{2})-\text{Arg}\,z_{2}~{}\text{mod}\,2\pi$,
so that both integrals share the same contour. Clearly, the two conditions can
always be satisfied simultaneously, except when
$\text{Arg}\,(z_{1}+z_{2})-\text{Arg}\,z_{1}=\pi$, in which case no contour
deformation works. For $z_{2}>0$, this corresponds to $z_{1}>-z_{2}$. So, we
conclude that, the above contour deformation always works well for any
$z_{1}\neq-z_{2}$ and $z_{2}>0$ so long as $z_{1}$ is away from an interval on
the negative real axis $(-z_{2},0)$.

How to deal with this interval? The solution is to rewrite the original
integral in a different way:

$K(z_{1},z_{2})=\int_{0}^{\infty}{\mathrm{d}}w_{1}{\mathrm{d}}w_{2}\,e^{\mathrm{i}z_{1}w_{1}+\mathrm{i}z_{2}w_{2}}\Big{[}1-\theta(w_{2}-w_{1})\Big{]}.$
(30)

Then, the first term is factorized and thus is trivial, and the second term is
again a nested integral but with the role of $w_{1}$ and $w_{2}$ switched.
Thus, all above analysis still applies to this nested integral, and the
contour deformation trick applies for all $z_{1}\neq-z_{2}$ except in the
interval $z_{1}\in(-\infty,-z_{2})$.

So the lesson is that, when we try to take the analytical continuation of a
nested integral by deforming the contour, the two cases of $z_{1}<-z_{2}$ and
$-z_{2}<z_{1}<0$ should be treated separately.

###  2.4 Vertex dispersion relation of the seed integral

Now we come back to the seed integral (10). We observe that the integrands of
$\mathcal{I}_{\mathsf{ab}}^{p_{1}p_{2}}$ contain exponential functions, power
functions and Hankel functions. Moreover, the opposite-sign integrals
$\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}$ are factorized, meaning that the
integrands are of product form $f(\tau_{1})g(\tau_{2})$. On the contrary, the
same-sign integrals $\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}$ are nested, due to the
time-ordered factor $\theta(\tau_{1}-\tau_{2})$ or
$\theta(\tau_{2}-\tau_{1})$. Although the seed integrals are much more
complicated than the toy examples considered above, they share some common
features. In particular, the integrand of a seed integral is regular along the
integral path, so that any potential singular behavior must be from a
diveregence in the early time limit. (The integral is always convergent in the
late-time limit by our assumption of IR regularity.) Therefore, let us
consider the asymptotic behavior of Hankel functions in the early time limit
$\tau\rightarrow-\infty$:

$\displaystyle\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau)\sim\displaystyle\frac{\,{C_{1}}\,}{\sqrt{-k\tau}}e^{-\mathrm{i}k\tau},$
$\displaystyle\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau)\sim\displaystyle\frac{\,{C_{2}}\,}{\sqrt{-k\tau}}e^{+\mathrm{i}k\tau},$
(31)

where $C_{1}$ and $C_{2}$ are $k\tau$-independent constants. So, we see that,
although the integrand of the seed integral is complicated, its behavior at
the early time limit is simple, and is controlled by exponential functions,
much like the toy examples considered above. Then, the previous discussion
shows that, if we allow $\\{k_{12},k_{34},k_{s}\\}$ to be outside the physical
region, the seed integral in its original form (10) can not always be
convergent. To make sense of the seed integral for arbitrary
$\\{k_{12},k_{34},k_{s}\\}$, we need analytical continuation. Below, we carry
out this analytical continuation on the complex plane of the vertex energy
$k_{12}$ while $k_{34}$ and $k_{s}$ are fixed within their physical regions.
From this result, we will derive the vertex dispersion relation.

#### Factorized integrals

We start from the factorized integrals $\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}$
which are free of time orderings and thus simpler. Without loss of generality,
we focus on $\mathcal{I}^{p_{1}p_{2}}_{+-}$, and the treatment for
$\mathcal{I}^{p_{1}p_{2}}_{-+}$ is very similar. Below we rewrite
$\mathcal{I}^{p_{1}p_{2}}_{+-}$ in an explicitly factorized form:

$\displaystyle\mathcal{I}^{p_{1}p_{2}}_{\mathsf{+-}}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathcal{U}^{p_{1}}_{+}(k_{12},k_{s})\mathcal{U}^{p_{2}}_{-}(k_{34},k_{s}).$
(32) $\displaystyle\mathcal{U}^{p_{1}}_{+}(k_{12},k_{s})=$
$\displaystyle~{}k_{s}^{5/2+p_{1}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}\,(-\tau_{1})^{3/2+p_{1}}e^{+\mathrm{i}k_{12}\tau_{1}}\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(-k_{s}\tau_{1}),$ (33)
$\displaystyle\mathcal{U}^{p_{2}}_{-}(k_{34},k_{s})=$
$\displaystyle~{}k_{s}^{5/2+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{2}\,(-\tau_{2})^{3/2+p_{2}}e^{-\mathrm{i}k_{34}\tau_{2}}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k_{s}\tau_{2}).$ (34)

To analyze the behavior of $\mathcal{I}^{p_{1}p_{2}}_{+-}$ on the complex
$k_{12}$ plane, it suffices to consider
$\mathcal{U}^{p_{1}}_{+}(k_{12},k_{s})$ alone. Of course, the integrals
$\mathcal{U}^{p_{1}}_{\pm}(k_{12},k_{s})$ are simple enough to be done
directly. However, we prefer to analyze their analytical structure without
really evaluating them. Thus we will defer the direct integration until next
section, where one can find the explicit results of
$\mathcal{U}^{p_{1}}_{\pm}(k_{12},k_{s})$ in (62).

For convenience, let us fix $k_{s}$ in the physical region $k_{s}>0$. (We can
also fix $k_{34}$ in the physical region $k_{34}>k_{s}$ although this is
irrelevant for the analysis of $\mathcal{U}^{p_{1}}_{+}(k_{12},k_{s})$.) From
(31), we know that, at the early time limit, the integrand of
$\mathcal{U}^{p_{1}}_{+}(k_{12},k_{s})$ is controlled by the exponential
factor:

$\displaystyle e^{\mathrm{i}(k_{12}+k_{s})\tau_{1}}.$ (35)

Therefore, for fixed integral contour $\tau_{1}\in(-\infty,0)$, the phase of
$k_{12}+k_{s}$ controls the convergence of
$\mathcal{U}^{p_{1}}_{+}(k_{12},k_{s})$ when $\tau_{1}\rightarrow-\infty$. For
example, if $\mathrm{Im}[k_{12}+k_{s}]>0$, the integral in (33) will diverge,
although the function $\mathcal{U}^{p_{1}}_{+}(k_{12},k_{s})$ can actually be
analytically continued to this region. In order to make this analytical
continuation, we improve the definition of $\mathcal{U}_{+}^{p_{1}}$ in (33)
by deforming the integration contour of
$\mathcal{U}^{p_{1}}_{+}(k_{12},k_{s})$ in the following way, similar to what
we did for the first two toy examples before:

$\displaystyle{\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})\equiv
k_{s}^{5/2+p_{1}}\int_{-(1-\mathrm{i}0^{+})(k_{12}+k_{s})^{-1}\infty}^{0}{\mathrm{d}}\tau_{1}\,(-\tau_{1})^{3/2+p_{1}}e^{\mathrm{i}k_{12}\tau_{1}}\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(-k_{s}\tau_{1}).$ (36)

Clearly, this new definition agrees with (33) for all $k_{12}$ and $k_{s}$ in
the interior of the physical region. On the other hand, the change of
integration path is continuous in $k_{12}$, so is the integral
${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$ for generic values of $k_{12}$. (One
can see this point more explicitly by taking derivative of
${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$ with respect to $k_{12}$.) However,
like the second toy example (21), the integrand of
${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$ contains a branch point at
$\tau_{1}=0$ due to the Hankel function and the power factor. The branch point
emanates a branch cut which we take to be on the positive real axis
$\tau_{1}\in(0,+\infty)$, and this branch cut can be an obstacle for contour
deformation. As a result, when the integration contour is brought to the
vicinity of the branch cut of the integrand, a discontinuity may occur and
lead to a branch cut for ${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$ with
respect to $k_{12}$.

Since the branch cut of the integrand in (36) is on the positive real axis of
$\tau_{1}$, the integral contour of ${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$
has a chance to approach this branch cut if $(k_{12}+k_{s})$ has a phase close
to $\pm\pi$. For $k_{s}>0$, this corresponds to $k_{12}\in(-\infty,-k_{s})$.
Thus we conclude that the only possible branch cut of
${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$ on the complex $k_{12}$ plane for
fixed $k_{s}>0$ is in the interval $k_{12}\in(-\infty,-k_{s})$, with the two
branch points $k_{12}=-\infty$ and $k_{12}=-k_{s}$. (In fact, the point
$k_{12}=-k_{s}$ is often divergent, since the integral
${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$ at this point is typically divergent
in the early time limit no matter how we deform the contour. This is called a
partial energy singularity in the literature.) Apart from this integral as
well as the two endpoints, the function
${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$ is analytical in $k_{12}$ everywhere
else.

Now let us determine the discontinuity across the branch cut of
${\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})$ at $k_{12}\in(-\infty,-k_{s})$.
Using the method identical to (2.3), we have:

$\displaystyle\mathop{\mathrm{Disc}}_{k_{12}}{\mathcal{U}}_{+}^{p_{1}}(k_{12},k_{s})=k_{s}^{5/2+p_{1}}\int_{\infty}^{0}{\mathrm{d}}\tau_{1}\,\mathop{\mathrm{Disc}}_{\tau_{1}}\Bigl{[}e^{+\mathrm{i}k_{12}\tau_{1}}(-\tau_{1})^{3/2+p_{1}}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\Bigr{]}.~{}~{}~{}~{}(k_{12}<-k_{s})$ (37)

Then, using the known discontinuity of the power function and the Hankel
function as given in (160) and (157), we get:

$\displaystyle\mathop{\mathrm{Disc}}_{\tau_{1}}\Bigl{[}e^{+\mathrm{i}k_{12}\tau_{1}}(-\tau_{1})^{3/2+p_{1}}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\Bigr{]}=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}}e^{+\mathrm{i}k_{12}\tau_{1}}\tau_{1}^{3/2+p_{1}}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(k_{s}\tau_{1})\theta(\tau_{1}).$ (38)

Therefore,

$\displaystyle\mathop{\mathrm{Disc}}_{k_{12}}\mathcal{U}_{+}^{p_{1}}(k_{12},k_{s})=$
$\displaystyle~{}2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}}k_{s}^{5/2+p_{1}}\int_{\infty}^{0}{\mathrm{d}}\tau_{1}\
e^{+\mathrm{i}k_{12}\tau_{1}}\tau_{1}^{3/2+p_{1}}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(k_{s}\tau_{1})$
$\displaystyle=$ $\displaystyle~{}2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}+1}k_{s}^{5/2+p_{1}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}\
e^{-\mathrm{i}k_{12}\tau_{1}}(-\tau_{1})^{3/2+p_{1}}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k_{s}\tau_{1})$
$\displaystyle=$ $\displaystyle~{}2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}+1}\mathcal{U}_{+}^{p_{1}}(-k_{12},k_{s}).$ (39)

Now it is trivial to put back all factors independent of $k_{12}$ in (32), and
get the discontinuity for $\mathcal{I}_{+-}^{p_{1}p_{2}}$. The discontinuity
of the other factorized seed integral $\mathcal{I}_{-+}^{p_{1}p_{2}}$ can be
analyzed in the same way and the result is very similar. So we summarize the
results for both factorized seed integrals together:

$\displaystyle\mathop{\mathrm{Disc}}_{k_{12}}\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}+1}\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}(-k_{12},k_{34},k_{s})\theta(-k_{12}-k_{s}).$
(40)

#### Nested integrals

Now we move on to the nested seed integrals
$\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}$. We will focus on
$\mathcal{I}_{++}^{p_{1}p_{2}}$ and the treatment of
$\mathcal{I}_{--}^{p_{1}p_{2}}$ is similar. Substituting the same-sign type
propagators (6) in (10), we get an explicit expression for the integral
$\mathcal{I}_{++}^{p_{1}p_{2}}$:

$\displaystyle\mathcal{I}_{++}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=-\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}e^{\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}}$
$\displaystyle\times\Big{[}\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})\theta(\tau_{1}-\tau_{2})+\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})\theta(\tau_{2}-\tau_{1})\Big{]}.$ (41)

As before, we fix $k_{s}$ and $k_{34}$ to be in the interior of the physical
region, i.e., $k_{34}>k_{s}>0$, and analyze the integral
$\mathcal{I}_{++}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ on the complex $k_{12}$
plane, where we need to perform analytical continuation by contour
deformation.

The way to deform the contour has been indicated in the third toy example
(26). In particular, for fixed values of $k_{34}>k_{s}>0$ and for arbitrary
real $k_{12}$, we need to consider separately two cases: $k_{12}<-k_{34}$ and
$-k_{34}<k_{12}<k_{s}$, both in the unphysical region. In each case, we need
to pick up a specific ordering for the two time variables.

Let us first analyze the case of $k_{12}<-k_{34}$, for which we choose to
rewrite the integral as:

$\displaystyle\mathcal{I}_{++}^{p_{1}p_{2}}=$
$\displaystyle~{}\mathcal{I}_{++,\mathrm{F},>}^{p_{1}p_{2}}+\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}},$
(42) $\displaystyle\mathcal{I}_{++,\mathrm{F},>}^{p_{1}p_{2}}\equiv$
$\displaystyle-\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,e^{\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}}$
$\displaystyle~{}\times(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k_{s}\tau_{2}),$
(43) $\displaystyle\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}\equiv$
$\displaystyle-\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,e^{\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}}(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}$
$\displaystyle\times\Big{[}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})-\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})\Big{]}\theta(\tau_{2}-\tau_{1}).$ (44)

The subscript $>$ means that we are working with the condition
$|k_{12}|>|k_{34}|$. This notation is in line with the one taken in [39]. The
analysis for the factorized integral
$\mathcal{I}_{++,\mathrm{F},>}^{p_{1}p_{2}}$ is identical to that for
$\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}$, and we have:

$\displaystyle\mathop{\mathrm{Disc}}_{k_{12}}\mathcal{I}_{++,\mathrm{F},>}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=$
$\displaystyle~{}2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}+1}\mathcal{I}_{++,\mathrm{F},>}^{p_{1}p_{2}}(-k_{12},k_{34},k_{s})\theta(-k_{12}-k_{34}).$
(45)

On the other hand, given the asymptotic behavior of the Hankel functions (31),
the analysis for the iterated integral
$\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}$ is in parallel with the one for
our third toy example (26). In particular, when
$\tau_{1},\tau_{2}\rightarrow-\infty$, the integrand of
$\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}$ behaves, up to unimportant power
functions (denoted as $\\#$), like:

$\\#e^{\mathrm{i}(k_{12}+k_{s})\tau_{1}}e^{\mathrm{i}(k_{34}-k_{s})\tau_{2}}-\\#e^{\mathrm{i}(k_{12}-k_{s})\tau_{1}}e^{\mathrm{i}(k_{34}+k_{s})\tau_{2}}.$
(46)

Thus, after finishing the inner-layer integral over $\tau_{2}$, we get four
terms which behave in the $\tau_{1}\rightarrow-\infty$ limit like (again, up
to unimportant power functions and constant coefficients):

$\displaystyle e^{\mathrm{i}(k_{12}+k_{34})\tau_{1}},$ $\displaystyle
e^{\mathrm{i}(k_{12}+k_{s})\tau_{1}},$ $\displaystyle
e^{\mathrm{i}(k_{12}+k_{34})\tau_{1}},$ $\displaystyle
e^{\mathrm{i}(k_{12}-k_{s})\tau_{1}}.$ (47)

Therefore, for fixed $k_{34}>k_{s}>0$, one can deform the integration contour
on the $\tau_{1}$ plane to make all above four terms convergent, if $k_{12}$
is away from the interval $(-k_{34},0)$. This is exactly the condition
$k_{12}<-k_{34}$ that we imposed from the very beginning. Then, we see that
the integral $\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}$ is analytic
everywhere in $k_{12}$ when $k_{12}$ is away from the negative real axis. On
the negative real axis, the interval $(-k_{34},0)$ is not covered by the
current case, while the interval $(-\infty,-k_{34})$ may contain a branch cut
due to the potential discontinuities of the integrand of
$\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}$.

However, by a direct computation, we can show that
$\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}$ is in fact free of branch cut
even in $(-\infty,-k_{34})$. Explicitly:

$\displaystyle\mathop{\mathrm{Disc}}_{k_{12}}\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$
$\displaystyle=$ $\displaystyle-\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}k_{s}^{5+p_{12}}\lim_{\epsilon\rightarrow 0^{+}}\bigg{\\{}\int_{\infty
e^{\mathrm{i}\epsilon}}^{0}{\mathrm{d}}\tau_{1}\int_{\tau_{1}}^{0}{\mathrm{d}}\tau_{2}\
e^{\mathrm{i}(-k_{12}+\mathrm{i}\epsilon)\tau_{1}+\mathrm{i}k_{34}\tau_{2}}(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}$
$\displaystyle\times\Big{[}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})-\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})\Big{]}-(\epsilon\rightarrow-\epsilon)\bigg{\\}}.$ (48)

We can reparameterize the two time variables in (2.4) so that the two nested
integrals in the curly brackets can be combined:

$\displaystyle-\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}k_{s}^{5+p_{12}}\int_{\infty}^{0}{\mathrm{d}}\tau_{1}\int_{\tau_{1}}^{0}{\mathrm{d}}\tau_{2}\bigg{\\{}e^{\mathrm{i}(-k_{12}+\mathrm{i}\epsilon)\tau_{1}^{+}}e^{\mathrm{i}k_{34}\tau_{2}^{+}}(-\tau_{1}^{+})^{3/2+p_{1}}(-\tau_{2}^{+})^{3/2+p_{2}}$
$\displaystyle\times\Big{[}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1}^{+})\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2}^{+})-\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1}^{+})\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2}^{+})\Big{]}-(\epsilon\rightarrow-\epsilon)\bigg{\\}},$
(49)

where $\tau_{1,2}^{+}\equiv\tau_{1,2}e^{\mathrm{i}\epsilon}$. Thus, (2.4) says
that we can find the discontinuity of the nested integral
$\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}$ by computing a “discontinuity” of
its integrand. Then, using the known discontinuities of the Hankel and power
functions on their branch cuts, collected in (160) and (157), it is
straightforward to show that the integrand of (2.4) actually vanishes. Thus we
conclude that $\mathcal{I}_{++,\mathrm{N},>}^{p_{1}p_{2}}$ has no branch cut
in $k_{12}$ when $k_{12}\in(-\infty,-k_{34})$.

The other case with $-k_{34}<k_{12}<k_{s}$ always satisfies
$|k_{12}|<|k_{34}|$, and therefore we separate the integral
$\mathcal{I}_{++}^{p_{1}p_{2}}$ in a different way:

$\displaystyle\mathcal{I}_{++}^{p_{1}p_{2}}=$
$\displaystyle~{}\mathcal{I}_{++,\mathrm{F},<}^{p_{1}p_{2}}+\mathcal{I}_{++,\mathrm{N},<}^{p_{1}p_{2}},$
(50) $\displaystyle\mathcal{I}_{++,\mathrm{F},<}^{p_{1}p_{2}}\equiv$
$\displaystyle-\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\
e^{\mathrm{i}k_{12}\tau_{1}}e^{\mathrm{i}k_{34}\tau_{2}}$
$\displaystyle\times(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k_{s}\tau_{2}),$
(51) $\displaystyle\mathcal{I}_{++,\mathrm{N},<}^{p_{1}p_{2}}\equiv$
$\displaystyle-\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\
e^{\mathrm{i}k_{12}\tau_{1}}e^{\mathrm{i}k_{34}\tau_{2}}(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}$
$\displaystyle\times\bigg{[}\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})-\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})\bigg{]}\theta(\tau_{1}-\tau_{2}).$ (52)

Like before, the subscript “$<$” here means that the way we split the integral
works when $|k_{12}|<|k_{34}|$. Then, in complete parallel with the previous
case, we can show that the the factorized part
$\mathcal{I}_{++,\mathrm{F},<}^{p_{1}p_{2}}$ has a branch cut in the interval
$-k_{34}<k_{12}<-k_{s}$, whose discontinuity is proportional to the factorized
integral $\mathcal{I}_{++,\mathrm{F},<}^{p_{1}p_{2}}$ itself:

$\displaystyle\mathop{\mathrm{Disc}}_{k_{12}}\mathcal{I}^{p_{1}p_{2}}_{++,\mathrm{F},<}(k_{12},k_{34},k_{s})$
$\displaystyle=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}+1}\mathcal{I}^{p_{1}p_{2}}_{\pm\pm,\mathrm{F},<}(-k_{12},k_{34},k_{s})\theta(k_{12}+k_{34})\theta(-k_{12}-k_{s}).$
(53)

On the other hand, the nested part
$\mathcal{I}_{++,\mathrm{N},<}^{p_{1}p_{2}}$ does not have any branch cut in
the region where it is defined (namely, $|k_{12}|<|k_{34}|$). Therefore, the
discontinuity in this case is also fully from the factorized integral.

Above we have present a detailed analysis for the integral
$\mathcal{I}_{++}^{p_{1}p_{2}}$. The treatment for
$\mathcal{I}_{--}^{p_{1}p_{2}}$ is completely the same. In particular, one can
separate $\mathcal{I}_{--}^{p_{1}p_{2}}$ into
$\mathcal{I}_{{--},\text{F},\gtrless}^{p_{1}p_{2}}$ and
$\mathcal{I}_{--,\text{N},\gtrless}^{p_{1}p_{2}}$ when
$|k_{12}|\gtrless|k_{34}|$. So, we can summarize our result for both same-sign
seed integrals as follows:

$\displaystyle\mathop{\mathrm{Disc}}_{k_{12}}\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}+1}\mathcal{I}_{\pm\pm,\text{F},>}^{p_{1}p_{2}}(-k_{12},k_{34},k_{s})\theta(-k_{12}-k_{34})$
$\displaystyle+2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}+1}\mathcal{I}^{p_{1}p_{2}}_{\pm\pm,\mathrm{F},<}(-k_{12},k_{34},k_{s})\theta(k_{12}+k_{34})\theta(-k_{12}-k_{s}).$
(54)

Figure 4: The analytical structure of the seed integral
$\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ in (11) on the complex
$k_{12}$ plane, with $k_{34}$ and $k_{s}$ staying in their physical regions
$k_{34}>k_{s}>0$. The physical region for $k_{12}$ ($k_{12}\geq k_{s}$) is
marked in green. The seed integral is analytic in $k_{12}$ everywhere except
when $k_{12}\in(-\infty,-k_{s}]$. In this interval, we have three branch
points: a partial energy (PE) pole at $k_{12}=-k_{s}$, the total energy (TE)
pole at $k_{12}=-k_{34}$. The other partial energy pole $k_{34}=-k_{s}$ is
never met for finite $k_{12}$, and can be thought of as sitting at
$k_{12}=-\infty$ due to the scale invariance of the seed integral. The branch
cut can be put in the entire interval $k_{12}\in(-\infty,-k_{s}]$, with the
discontinuity itself being discontinuous at $k_{12}=-k_{34}$.

#### Summary

Now we have completed the analysis for the seed integral
$\mathcal{I}^{p_{1}p_{2}}_{\mathsf{a}\mathsf{b}}$ on the complex $k_{12}$
plane, with $k_{34}$ and $k_{s}$ fixed in the interior of their physical
region $k_{34}\geq k_{s}\geq 0$. The discontinuities of all four SK branches
are given in (40) and (2.4), respectively.

When performing the dispersion integrals, we don’t have to separate the seed
integral according to their SK branches. Therefore, it is useful to sum over
SK indices $\mathsf{a},\mathsf{b}=\pm$ and to get the analytical structure for
the full seed integral $\mathcal{I}^{p_{1}p_{2}}$ in (11):

$\displaystyle\mathop{\text{Disc}}_{k_{12}}\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-1)^{p_{1}+1}\mathcal{I}^{p_{1}p_{2}}_{\text{S}}(-k_{12},k_{34},k_{s})\theta(-k_{12}-k_{s}).$
(55)

In this expression, we have defined the _signal_ part of the seed integral as:

$\displaystyle\mathcal{I}_{\text{S}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=$
$\displaystyle~{}\Bigl{(}\mathcal{I}_{++,\text{F},>}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})+\mathcal{I}_{--,\text{F},>}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\Bigr{)}\theta\Big{(}|k_{12}|-|k_{34}|\Big{)}$
$\displaystyle+\Bigl{(}\mathcal{I}_{++,\text{F},<}^{p_{1}p_{2}}(k_{34},k_{12},k_{s})+\mathcal{I}_{--,\text{F},<}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\Bigr{)}\theta\Big{(}|k_{34}|-|k_{12}|\Big{)}$
$\displaystyle+\mathcal{I}_{+-}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})+\mathcal{I}_{-+}^{p_{1}p_{2}}(k_{12},k_{34},k_{s}).$
(56)

Eqs. (55) and (2.4) are the main results of this section. They form the basis
for the vertex dispersion relation, detailed in the next section.

Note that the “signal” defined in (2.4) is nothing but all factorized pieces
in (40) and (2.4) summed, and it is this signal piece that is responsible for
all discontinuities of the seed integral on $k_{12}$ plane. On the other hand,
it does agree with the signal defined in previous works through the analytical
properties in $k_{s}/k_{12}$ and $k_{s}/k_{34}$ as $k_{s}\rightarrow 0$ [33,
37, 39]. Thus the results (55) and (2.4) make precise our intuition that the
CC signal corresponds to the nonanalyticity of the correlator.

## 3 Bootstrapping Correlators with Vertex Dispersion Relation

In this section, we will put the vertex dispersion relation in use, to
bootstrap a few 3-point and 4-point correlators with massive exchanges. We
begin with the simplest example, the 3-point tree correlator with single
massive exchange in Sec. 3.1. The dispersive bootstrap yields a closed-form
analytical expression for this example, identical to the one found with
improved bootstrap equation in [46]. Then, in Sec. 3.2, we bootstrap the
3-point correlator mediated by two massive fields via a bubble loop. We will
show that, with an additional input of spectral decomposition explored in a
previous work [42], the vertex dispersion relation can be generalized to loop
processes, leading to analytical expressions much simpler than the one found
with pure spectral method in [42]. In particular, our one-loop result here
features a neat separation of the renormalization-dependent local part and the
renormalization-independent nonlocal part, thus allows for unambiguous
extraction of on-shell effects from loop process. Finally, in Sec. 3.3, we
bootstrap the 4-point correlator with a single massive exchange in the
$s$-channel. This is a well studied example, and we use it to demonstrate the
use of vertex dispersion relation for kinematics more complicated than 3-point
examples.

###  3.1 Three-point single-exchange graph

We begin with the simplest nontrivial example, namely a 3-point correlator
with a single massive exchange. To be specific, we will consider the single
massive exchange from the following interactions:

$\displaystyle\Delta{\mathscr{L}}=\lambda_{2}a^{3}\varphi^{\prime}\sigma+\displaystyle\frac{\,{1}\,}{2}\lambda_{3}a^{2}\varphi^{\prime\,2}\sigma,$
(57)

where $\varphi$ is a massless scalar field (typically the inflaton fluctuation
in the context of CC physics), and $\sigma$ is a real massive scalar field of
mass $m$. For convenience, we take $m>3/2$ so that the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ is a positive real,
although generalization to light mass $0<m<3/2$ is straightforward. Also,
$\lambda_{3}$ and $\lambda_{2}$ in (57) are coupling constants, and the powers
of the scale factor $a=-1/\tau$ are inserted to ensure the scale invariance.
Then, there is a single independent tree diagram, shown in Fig. 5 that
contributes to the 3-point correlator $\langle\varphi\varphi\varphi\rangle$ at
the leading order $\mathcal{O}({\lambda_{2}\lambda_{3}})$, together with other
two obtained by trivial momentum permutations. This process appears in a
simple realization of the original quasi-single-field inflation with dim-6
inflaton-spectator coupling $(\partial_{\mu}\phi)^{2}\sigma^{2}$ [105], and
turns out to be the leading signal in this model with comparable signal
strength with double-massive-exchange and triple-massive-exchange graphs.

The time integral for the diagram in Fig. 5 can be expressed in terms of the
4-point seed integral in (11) as:

$\langle\varphi_{\bm{k}_{1}}\varphi_{\bm{k}_{2}}\varphi_{\bm{k}_{3}}\rangle^{\prime}=\displaystyle\frac{\,{\lambda_{2}\lambda_{3}}\,}{k_{1}k_{2}k_{3}^{4}}\bigg{[}\mathcal{I}^{0,-2}(k_{12},k_{3},k_{3})+\text{2
perms}\bigg{]},$ (58)

Therefore, technically, the 3-point function we are going to compute can be
viewed as a limiting case of a 4-point correlator with $k_{4}\rightarrow
0^{+}$.

Figure 5: The 3-point correlator of a massless scalar $\varphi$ with a single
massive scalar exchange $\sigma$.

Then, the problem reduces to the computation of
$\mathcal{I}^{0,-2}(k_{12},k_{3},k_{3})$. For this particular integral, it
turns out useful to use a new variable $u\equiv 2k_{3}/k_{123}$. With this
definition, the physical region $0\leq k_{3}\leq k_{12}$ can be written as
$u\in[0,1]$. It is known that this variable is useful for obtaining closed-
form analytical expressions for many 3-point functions [46]. From the
perspective of dispersion integral, the simplification can be observed from
the fact that the partial-energy limit and the total-energy limit merge into a
single limit $u\rightarrow-\infty$, while the branch point $k_{12}=\infty$
corresponds to $u=0$. Thus, the branch cut of the full 3-point function
extends from $0$ to $-\infty$ on the entire negative real axis on the complex
$u$ plane, which makes the dispersion integral simpler.

To avoid potential confusions, we use a new notation $\mathcal{X}(u)$ to
denote the dimensionless three-point seed integral as a function of
$u=2k_{3}/k_{123}$:

$\mathcal{X}_{\mathsf{a}\mathsf{b}}(u=2k_{3}/k_{123})\equiv\mathcal{I}^{0,-2}_{\mathsf{a}\mathsf{b}}(k_{12},k_{3},k_{3}).$
(59)

Then, we can rewrite the full 3-point seed integral as:

$\displaystyle\mathcal{X}(u)\equiv\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{X}_{\mathsf{a}\mathsf{b}}(u)=\Big{[}\mathcal{X}_{++,\text{N}}(u)+\mathcal{X}_{++,\text{F}}(u)+\mathcal{X}_{+-}(u)\Big{]}+\text{c.c.},$
(60)

where $\mathcal{X}_{\text{++,\text{N}}}$ and
$\mathcal{X}_{\text{++,\text{F}}}$ are nested and factorized part of
$\mathcal{X}_{++}$, defined from the corresponding seed integrals as in (59);
See (2.4) and (2.4).

Our dispersive bootstrap of $\mathcal{X}(u)$ comes naturally with two steps
following the result in (55): First, we will compute the signal part
$\mathcal{X}(u)$ and its discontinuity across the branch cut. Second, we will
perform the dispersion integral along the branch cut to get the full result.
Below we carry out these two steps in turn.

#### Computing the signal

From the analysis of the previous section, we know that the discontinuity of
$\mathcal{X}(u)$ on the negative real axis is fully from
$\mathcal{X}_{\pm\pm,\text{F}}(u)$ and $\mathcal{X}_{\pm\mp}(u)$, which can be
combined together into the signal $\mathcal{X}_{\mathrm{S}}(u)$:

$\displaystyle\mathcal{X}_{\text{S}}(u)\equiv$
$\displaystyle~{}\mathcal{X}_{++,\text{F}}(u)+\mathcal{X}_{+-}(u)+\text{c.c.}$
$\displaystyle=$
$\displaystyle~{}\mathcal{I}^{0,-2}_{++,\mathrm{F},>}(k_{12},k_{3},k_{3})+\mathcal{I}^{0,-2}_{+-}(k_{12},k_{3},k_{3})+\mathrm{c.c.}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{[}-{\mathcal{U}}_{-}^{0}(e^{\mathrm{i}\pi}k_{12},k_{3}){\mathcal{U}}_{+}^{-2}(k_{3},k_{3})+{\mathcal{U}}_{+}^{0}(k_{12},k_{3}){\mathcal{U}}_{-}^{-2}(k_{3},k_{3})\Big{]}+\text{c.c.}.$
(61)

In writing this expression, we have removed the $\theta$ functions in the
original expression (2.4), because the relation $|k_{12}|>|k_{34}|$ always
holds true in the regions of our interest, including the region
$u\in(-\infty,0)$ where the branch cut lies, and the physical region
$u\in(0,1)$. Also, we note that, in the final expression, we have a factor
${\mathcal{U}}^{0}_{-}(e^{\mathrm{i}\pi}k_{12},k_{3})$, in which the first
argument $k_{12}$ should be analytically continued by a rotation of
$e^{\mathrm{i}\pi}$. One can readily check that this way of analytical
continuation brings ${\mathcal{U}}^{p_{1}}_{-}(k_{12},k_{3})$ to the
corresponding $\tau_{1}$-integral appears in
$\mathcal{I}_{++,F,>}^{p_{1}p_{2}}$.

As mentioned before, the single-layer integrals $\mathcal{U}$ can be directly
done, and the results are:

$\displaystyle{\mathcal{U}}_{\pm}^{p}(K_{1},K_{2})=\displaystyle\frac{\,{e^{\mp\mbox{$\frac{\,{\mathrm{i}\pi}\,}{4}$}(3+2p)+\mbox{$\frac{\,{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}}}\,}{\pi}\Bigl{(}\displaystyle\frac{\,{K_{2}}\,}{K_{1}}\Bigr{)}^{5/2+p}\biggl{[}\Bigl{(}\displaystyle\frac{\,{K_{2}}\,}{K_{1}}\Bigr{)}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{F}^{p}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(K_{2}/K_{1})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\biggr{]},$ (62)

where we have defined a function $\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p}(z)$ for later
convenience:

$\displaystyle\mathbf{F}^{p}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z)\equiv-\mathrm{i}\pi^{1/2}2^{3/2+p}\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu){}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{5}\,}{4}$}+\mbox{$\frac{\,{p}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{7}\,}{4}$}+\mbox{$\frac{\,{p}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}z^{2}\biggr{]}.$ (63)

Here ${}_{2}\mathcal{F}_{1}$ is the dressed version of Gauss’s hypergeometric
function, whose definition is collected in App. B.

It is straightforward to insert (62) into (3.1) to get an expression for the
signal $\mathcal{X}_{\text{S}}$. However, there are two small technical points
worth mentioning. First, we want to write $\mathcal{X}_{\text{S}}$ as a
function of $u=2k_{3}/k_{123}$, and this can be easily done by using the
following identity of the hypergeometric function:

$\displaystyle{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,b\\\
2b\end{matrix}\bigg{|}\displaystyle\frac{\,{2r}\,}{1+r}\biggr{]}=(1+r)^{a}{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{a}\,}{2}$},\mbox{$\frac{\,{a+1}\,}{2}$}\\\
b+\frac{1}{2}\end{matrix}\bigg{|}r^{2}\biggr{]}.$ (64)

Second, in (3.1) we have a factor ${\mathcal{U}}_{+}^{-2}(k_{3},k_{3})$, which
involves the cancellation of divergence between the two functions
$\mathbf{F}^{-2}_{\pm\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(k_{3}/k_{3})$:

$\displaystyle\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(k_{3}/k_{3})+\mathbf{F}^{-2}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(k_{3}/k_{3})=(2\pi^{3})^{1/2}\mathrm{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (65)

With these two points clarified, we obtain the signal part of 3-point
correlator:

$\displaystyle\mathcal{X}_{\text{S}}(u)=\displaystyle\frac{\,{\pi\bigl{(}\mathrm{i}+\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}}\,}{4\sinh(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u\biggr{]}u^{5/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (66)

Now, we can quote our previous result (55) to get the discontinuity of
$\mathcal{X}(u)$. After switching to $u=2k_{3}/k_{123}$ as the argument, the
result reads:

$\displaystyle\mathop{\mathrm{Disc}}_{u}{\mathcal{X}}(u)$
$\displaystyle=-\mathop{\mathrm{Disc}}_{k_{12}}\mathcal{I}^{0,-2}(k_{12},k_{3},k_{3})$
$\displaystyle=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathcal{I}^{0,-2}_{\mathrm{S}}(-k_{12},k_{3},k_{3})\theta(-k_{12}-k_{s})$
$\displaystyle=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathcal{X}_{\mathrm{S}}\Bigl{(}\displaystyle\frac{\,{u}\,}{u-1}\Bigr{)}\theta(-u),$
(67)

in which the minus sign in the first line follows from the relation
$u=2k_{3}/k_{123}$. Then, from (66), we find the discontinuity of the full
3-point correlator as:

$\displaystyle\mathop{\mathrm{Disc}}_{u}\mathcal{X}(u)=\displaystyle\frac{\,{\pi}\,}{4}\bigl{(}\mathrm{i}-\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}\,{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u\biggr{]}(-u)^{5/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (68)

Here we have used the following identity to simplify the expression:

$\displaystyle{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,b\\\
c\end{matrix}\bigg{|}u\biggr{]}=(1-u)^{-a}{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,c-b\\\
c\end{matrix}\bigg{|}\displaystyle\frac{\,{u}\,}{u-1}\biggr{]}.$ (69)

Of course, the discontinuity in (68) can be directly read from the analytical
expression for $\mathcal{X}_{\text{S}}$ in (66) by using the known analytical
properties of hypergeometric function ${}_{2}\mathcal{F}_{1}[\cdots|u]$ and
the power function $u^{5/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$. However, from
(3.1), we can check that the discontinuity from the hypergeometric functions
get canceled. The net discontinuity (68) is fully from the power factor
$u^{5/2-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}$, and this will be the key ingredient for our computation of dispersion
integral in the next part.

#### Dispersion integral

With the discontinuity of the function $\mathcal{X}(u)$ known, we are ready to
form a dispersion integral, which computes the full correlator from its
(factorized) discontinuity.

$\displaystyle\mathcal{X}(u)=\displaystyle\frac{\,{u^{3}}\,}{2\pi\mathrm{i}}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}u^{\prime}}\,}{u^{\prime\,3}(u^{\prime}-u)}\mathop{\mathrm{Disc}}_{u^{\prime}}\mathcal{X}(u^{\prime}).$
(70)

Here we have introduced a third-order subtraction ($u^{\prime 3}$) to make
sure that contour integral vanishes on the large circle. To understand this
choice, we note that the large $u$ limit of $\mathcal{X}(u)$ corresponds to
the total-energy limit $k_{123}\rightarrow 0$. By power counting of time, one
can see that the seed integral behaves like $\mathcal{X}(u)\sim u^{2}$ as
$|u|\rightarrow\infty$. Thus, a third-order subtraction suffices to make the
dispersion integral well defined.

(70) is a well documented integral and can be directly done by `Mathematica`.
However, it is instructive to compute this integral more explicitly, by using
the partial Mellin-Barnes (PMB) representation [37, 39]. This method will be
useful for more complicated integrals in the following sections where we do
not have readily available integral formulae. Also, as we shall see, there is
a nice correspondence between the pole structure of the Mellin integral and
the analytical property of the final result. (See Fig. 6).

$\Rightarrow$

Figure 6: The dispersion integrals for the 3-point correlator. The left panel
shows the integral (70) on the complex $u^{\prime}$ plane, where we have a
branch cut of the 3-point correlator in the unphysical region
$u^{\prime}\in(-\infty,0)$, and an inserted pole at the physical point
$u^{\prime}=u$ with $0<u<1$. On the right panel, we show the corresponding
Mellin integral (3.1) on the complex plane of Mellin variable $s$. The blue
and red poles on the left side of the integration contour contribute to the
signal and the background of the full 3-point correlator, respectively.

To apply the PMB technique, we use the following MB representation for the
hypergeometric function:888Generally, there is certain flexibility to deform
the integral contour, so long as all poles coming from “$\Gamma(+as+\cdots)$
are to the left of the contour, and those poles from “$\Gamma(-bs+\cdots)$”
are to the right. (Here $a,b\in\mathbb{R}_{+}$.) For convenience, here we just
label the lower/upper bound of the integral as $\mp\mathrm{i}\infty$.

$\displaystyle{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\ 1\mp
2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u^{\prime}\biggr{]}=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\
(-u^{\prime})^{-s}\Gamma\biggl{[}\begin{matrix}s,\mbox{$\frac{\,{1}\,}{2}$}-s\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}-s\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\ 1-s\mp
2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}.$ (71)

We see that the MB representation effectively turns a complicated dependence
on $u^{\prime}$ into a simple power function $(-u^{\prime})^{-s}$. As a
result, the dispersion integral over $u^{\prime}$ in (70) can now be trivially
carried out as:

$\displaystyle\frac{\,{u^{3}}\,}{2\pi\mathrm{i}}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}u^{\prime}\,(-u^{\prime})^{5/2\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu-s}}\,}{u^{\prime\,3}(u^{\prime}-u)}=\displaystyle\frac{\,{\sec\big{[}\pi(s\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\big{]}}\,}{2\mathrm{i}}u^{5/2-s\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}.$ (72)

It then remains to finish the Mellin integral over $s$:

$\displaystyle\mathcal{X}(u)=$ $\displaystyle\
\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}{u^{5/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\sec\big{[}{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\pi(s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\big{]}\Gamma\biggl{[}\begin{matrix}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}s},\mbox{$\frac{\,{1}\,}{2}$}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1-s-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}$
$\displaystyle\times\displaystyle\frac{\,{\pi}\,}{8}\bigl{(}1+\mathrm{i}\
\mathrm{csch}(\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}+(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\rightarrow-\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu).$ (73)

Since we have taken the variable $u$ in the physical region, i.e.,
$u\in(0,1)$, we can perform the above Mellin integral by closing the contour
from the left side. The integrand decreases fast enough when $s$ goes to
infinity in the left plane, so that the integral over the large semi-circle on
the left plane vanishes, and we can finish the integral by collecting the
residues of all poles to the left side of the original integration contour.
From (3.1), we see that there are two sets of left poles contributing to the
final results, whose origins are highlighted in red and blue colors:999When
computing integrals via PMB representation, if a Gamma function contributes
poles, then all of its poles need to be collected. For example, here there are
a set of poles from $\Gamma[s]$, then we need to pick up the whole set of
these poles, i.e., $s=-n$ where $n=0,1,2,\cdots$. The case where poles come
from $\sec[\pi(s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)]$ is a little
subtler: If we change the upper bound of the integral over $u^{\prime}$ from
$0$ to $-\epsilon$ where $\epsilon$ is a small positive real number, one can
find we will get $\Gamma[1/2+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$ instead of
$\sec[\pi(s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)]$. It is only in the
limit $\epsilon\rightarrow 0$ that the Gamma function
$\Gamma[1/2+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$ will meet another
Gamma function $\Gamma[1/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$ and give rise to
$\sec[\pi(s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)]$. This implies when
considering poles from $\sec[\pi(s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)]$ we actually need
to collect all poles from $\Gamma[1/2+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$, while poles from
$\Gamma[1/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$ should be omitted.
This analysis gives us another set of poles, i.e., $s=-1/2-n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ where
$n=0,1,2,\cdots$.

$\left\\{\begin{aligned}
&{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}s=-\mbox{$\frac{\,{1}\,}{2}$}-n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu};\\\
&{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}s=-n}.~{}~{}~{}~{}(n=0,1,2,\cdots)\end{aligned}\right.$
(74)

We also show these poles in the right panel of Fig. 6. Clearly, from the
factor $u^{5/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ in the integrand in
(3.1), we see that the poles $s=-1/2-n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ correspond to the
background, whose residues sum to:

$\displaystyle\mathcal{X}_{\text{B}}(u)$
$\displaystyle=\sum_{n=0}^{\infty}\bigl{(}1+\mathrm{i}\
\mathrm{csch}(\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}\displaystyle\frac{\,{(-1)^{n}}\,}{8}u^{3+n}\Gamma\biggl{[}\begin{matrix}1+n,3+n,-\frac{1}{2}-n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\frac{3}{2}+n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$
$\displaystyle=-\displaystyle\frac{\,{2u^{3}}\,}{1+4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}1,1,3\\\
\frac{3}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\frac{3}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u\biggr{]}.$ (75)

On the other hand, the poles at $s=-n$ in the integrand of (3.1) give rise to
the signal:

$\displaystyle\mathcal{X}_{\text{S}}(u)$
$\displaystyle=\sum_{n=0}^{\infty}\mathrm{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\bigl{(}1+\mathrm{i}\
\mathrm{csch}(\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}\displaystyle\frac{\,{\pi}\,}{8}u^{5/2+n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma\biggl{[}\begin{matrix}\frac{1}{2}+n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\frac{5}{2}+n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$
$\displaystyle=\displaystyle\frac{\,{\pi\bigl{(}\mathrm{i}+\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}}\,}{4\sinh(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u\biggr{]}u^{5/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (76)

Thus, the whole three-point correlator ${\mathcal{X}}(u)$ is neatly expressed
as a sum of the signal and the background:

$\displaystyle\mathcal{X}(u)=$
$\displaystyle~{}\bigg{\\{}\displaystyle\frac{\,{\pi\bigl{(}\mathrm{i}+\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}}\,}{4\sinh(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u\biggr{]}u^{5/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\bigg{\\}}$
$\displaystyle~{}-\displaystyle\frac{\,{2u^{3}}\,}{1+4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}\,{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}1,1,3\\\
\frac{3}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\frac{3}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u\biggr{]}.$ (77)

This agrees with the results found previously using a different method [46].

To recapitulate our strategy, the PMB representation converts special
functions into simple power functions, making the dispersion integrals easier
to compute. Thereafter, the integration over Mellin variables can be directly
computed via residue theorem. Therefore, the PMB representation provides a
convenient way to calculate dispersion integrals analytically. For inflation
correlators more complicated than the one considered here, the PMB
representation remains useful, and will be shown below.

###  3.2 Three-point one-loop bubble graph

Figure 7: The 3-point correlator with 1-loop massive scalar exchange. For
simplicity, we take the two bulk-to-bulk propagators to have the same mass
parameter $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$.

Although most of the discussions of this work focus on tree-level processes,
the dispersion technique can also be applied to loop processes. In this
subsection, we will explore a simple 1-loop diagram with dispersion relations,
with the help of the technique of spectral decomposition [42]. Our example
comes from the following interactions between the massless scalar $\varphi$
and the principal scalar $\sigma$:

$\displaystyle\Delta{\mathscr{L}}=\displaystyle\frac{\,{\kappa_{4}}\,}{4}a^{2}\varphi^{\prime
2}\sigma^{2}+\displaystyle\frac{\,{\kappa_{3}}\,}{2}a^{3}\varphi^{\prime}\sigma^{2}.$
(78)

Then, at $\mathcal{O}({\kappa_{4}\kappa_{3}})$, there is a unique diagram (up
to trivial permutations) contributing to the 3-point correlator of $\varphi$
with a bubble loop formed by $\sigma$; See Fig. 7.

Similar to the tree-level case, we can extract a dimensionless seed integral
from the correlator:

$\displaystyle\langle\varphi_{\bm{k}_{1}}\varphi_{\bm{k}_{2}}\varphi_{\bm{k}_{3}}\rangle^{\prime}_{\text{1-loop}}=\displaystyle\frac{\,{\kappa_{3}\kappa_{4}}\,}{8k_{1}k_{2}k_{3}^{4}}\bigg{[}{\mathcal{J}}^{0,-2}\Big{(}\displaystyle\frac{\,{2k_{3}}\,}{k_{123}}\Big{)}+\text{2
perms}\bigg{]}.$ (79)

Here $\mathcal{J}^{p_{1}p_{2}}$ is the corresponding seed integral, defined as
a function of the momentum ratio $u=2k_{3}/k_{123}$:

$\displaystyle\mathcal{J}^{p_{1}p_{2}}(u)\equiv-\displaystyle\frac{\,{1}\,}{2}\sum_{\mathsf{a,b}=\pm}\mathsf{ab}\
k_{3}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{3}\tau_{2}}\mathbf{Q}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{ab}}(k_{3};\tau_{1},\tau_{2}).$ (80)

Here, $\mathbf{Q}^{(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{ab}}$ denotes the 3-momentum loop integral:

$\displaystyle\mathbf{Q}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{ab}}(k_{3};\tau_{1},\tau_{2})\equiv\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\bm{q}}\,}{(2\pi)^{3}}D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{ab}}\Bigl{(}q;\tau_{1},\tau_{2}\Bigr{)}D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{ab}}\Bigl{(}|\bm{k}_{3}-\bm{q}|;\tau_{1},\tau_{2}\Bigr{)}.$
(81)

Here we mark out the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ of propagators as
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ is important
in the following analysis.

As explained in previous works [169, 42, 52], the loop integral (81) can be
recast as a (continuous) linear superposition of massive propagators
${D}^{(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime})}_{\mathsf{ab}}$ with different values of $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime}$, weighted
by a spectral function $\rho^{\mathrm{dS}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime})$:

$\displaystyle\mathbf{Q}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}_{\mathsf{ab}}(k_{3};\tau_{1},\tau_{2})=\int_{-\infty-\mathrm{i}\epsilon}^{+\infty+\mathrm{i}\epsilon}{\mathrm{d}}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}\displaystyle\frac{\,{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}}\,}{\pi\mathrm{i}}\rho^{\mathrm{dS}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime})D^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime})}_{\mathsf{ab}}(k_{3};\tau_{1},\tau_{2}).$ (82)

With the assumption that both the time integrals in (80) and the spectral
integral in (82) are convergent,101010The convergence of the spectral integral
(82) requires a proper regularization procedure, such as dimensional
regularization, to make the spectral function $\rho^{\mathrm{dS}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime})$ finite in
the first place. However, as we will see below, our treatment of the loop
process is completely independent of the regularization, and we can safely
stay in $d=3$ throughout the discussion. we can switch the order of two
integrals, and write the loop correlator $\mathcal{J}^{p_{1}p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ as a spectral
integral over tree correlator $\mathcal{I}^{p_{1}p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime}}$:

$\displaystyle\mathcal{J}^{p_{1}p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u)=\int_{-\infty-\mathrm{i}\epsilon}^{+\infty+\mathrm{i}\epsilon}{\mathrm{d}}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}\displaystyle\frac{\,{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}}\,}{2\pi\mathrm{i}}\rho^{\mathrm{dS}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime})\mathcal{I}^{p_{1}p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}}(k_{12},k_{3},k_{3}).$ (83)

Now we specialize to the case of $(p_{1},p_{2})=(0,-2)$ as indicated in (79),
and form a dispersion integral for $\mathcal{J}^{0,-2}$. Such a dispersion
integral is possible, because all the 3-point tree-level correlators
$\mathcal{I}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}}^{0,-2}$ with different mass parameters $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime}$ satisfy the
same dispersion relation (70). Therefore, their linear superposition in (83)
should satisfies a similar dispersion integral. However, we should expect that
the subtraction order for the loop correlator differs from the tree due to the
different UV behavior. Therefore, let us write down the dispersion integral
for the loop seed integral $\mathcal{J}^{0,-2}(u)$ on $u$ plane in the
following way:

$\displaystyle{\mathcal{J}}^{0,-2}(u)=\displaystyle\frac{\,{u^{m}}\,}{2\pi\mathrm{i}}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}u^{\prime}}\,}{u^{\prime
m}(u^{\prime}-u)}\mathop{\mathrm{Disc}}_{u^{\prime}}{\mathcal{J}}_{\mathrm{S}}^{0,-2}(u^{\prime}).$
(84)

Here we leave the subtraction order $m$ arbitrary, and we will determine it
later.

As mentioned above, the loop seed integral $\mathcal{J}^{p_{1}p_{2}}$ has been
computed purely from spectral decomposition in [42]. However, the result in
[42] shows a significant hierarchy in the degree of complication between the
signal and the background: The signal part of the loop diagram is a discrete
sum of tree signals weighted by a simple coefficients, which can be understood
as summing over all quasinormal modes of the loop. On the other hand, the
background part is quite complicated, which, after regularization and
renormalization, contains a highly intricate special function in the
renormalized spectral function. Below, we shall exploit this hierarchy, using
the signal computed via the spectral decomposition to bootstrap the full
correlator, and thus bypassing any complications of regularization and
renormalization.

Thus, our starting point will be the signal part of the loop seed integral
computed via the spectral decomposition [42]:

$\displaystyle\mathcal{J}^{0,-2}_{\text{S}}(u)=$ $\displaystyle\
\displaystyle\frac{\,{u^{4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{128\pi\sin(-2\pi\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\sum_{n=0}^{\infty}\displaystyle\frac{\,{(3+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+4n)(1+n)_{\frac{1}{2}}(1+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n)_{\frac{1}{2}}}\,}{(\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n)_{\frac{1}{2}}(\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+n)_{\frac{1}{2}}}$
$\displaystyle\times{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}2+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+2n,4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+2n\\\
4+4\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+4n\end{matrix}\bigg{|}u\biggr{]}u^{2n}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (85)

We then need to get the discontinuity of the signal along the branch cut. For
the signal (3.2), its discontinuity along the branch cut $u\in(-\infty,0)$ is
simply contributed by the $u^{\pm 2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ factor, which is
similar to the tree-level case. The result is

$\displaystyle\mathop{\mathrm{Disc}}_{u}{\mathcal{J}}^{0,-2}_{\mathrm{S}}(u)=$
$\displaystyle\ \displaystyle\frac{\,{(-u)^{4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{64\pi\mathrm{i}}\sum_{n=0}^{\infty}\displaystyle\frac{\,{(3+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+4n)(1+n)_{\frac{1}{2}}(1+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n)_{\frac{1}{2}}}\,}{(\frac{1}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n)_{\frac{1}{2}}(\frac{3}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+n)_{\frac{1}{2}}}$
$\displaystyle\times{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}2+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+2n,4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+2n\\\
4+4\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+4n\end{matrix}\bigg{|}u\biggr{]}u^{2n}\theta(-u)+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (86)

Now we are ready to use (84) and (3.2) to compute the full correlator.
However, at this point, we need to choose a subtraction (namely, to choose a
value of $m$ in (84)) to make sure that the integral (84) converges when
$u\rightarrow 0$ and $u\rightarrow-\infty$. Examining the behavior of the
integrand in these two limits, we see that the convergence as $u\rightarrow 0$
requires $m\leq 4$ while the convergence as $u\rightarrow-\infty$ prefers a
large $m$. So, $m=4$ is an optimal choice.

Similar to the 3-point tree-level case, for every term in (3.2), the
dispersion integral can be done either by `Mathematica` directly or by PMB
representation. The final result for the loop seed integral
$\mathcal{J}^{0,-2}(u)$ is again the sum of the signal
$\mathcal{J}_{\text{S}}^{0,-2}(u)$ and the background
$\mathcal{J}^{0,-2}_{\text{BG}}(u)$. The signal is already given in (3.2), and
the background is given by:

$\displaystyle{\mathcal{J}}^{0,-2}_{\mathrm{BG}}(u)=\ $
$\displaystyle\displaystyle\frac{\,{u^{4}}\,}{128\pi\sin(2\pi\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\sum_{n=0}^{\infty}\displaystyle\frac{\,{(3+4n+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(1+n)_{\frac{1}{2}}(1+n+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{\frac{1}{2}}}\,}{(\mbox{$\frac{\,{1}\,}{2}$}+n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{\frac{1}{2}}(\mbox{$\frac{\,{3}\,}{2}$}+n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{\frac{1}{2}}}$
$\displaystyle\times{}_{3}{\mathcal{F}}_{2}\biggl{[}\begin{matrix}1,2,4\\\
1-2n-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,4+2n+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u\biggr{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (87)

Here ${}_{3}\mathcal{F}_{2}$ is the dressed version of the generalized
hypergeometric function, whose definition is collected in App. B.

Some readers may find it mysterious that no UV divergence ever shows up in our
calculation. The reason is in fact clear: The UV divergence in this 1-loop
correlator can be fully subtracted by a local counterterm
${\mathscr{L}}\supset\delta_{\lambda}a\varphi^{\prime 3}$ with divergent
coefficient $\delta_{\lambda}$. At the correlator level, this counterterm
produces a contact diagram $\propto\delta_{\lambda}u^{3}$, and thus is
analytical on the entire $u$ plane. If we follow the standard loop
calculation, we would find a divergent part proportional to $u^{3}$, plus a
finite part with more complicated $u$ dependence. Then we can use any
convenient regularization method to remove the divergence, and use any proper
renormalization scheme to determine the finite coefficient of the $u^{3}$
term. The arbitrariness of the coefficient of the $u^{3}$ term is an intrinsic
uncertainty of the loop calculation.

We think that this is an important lesson, especially for readers not very
familiar with loop calculations, so let us reiterate it: When computing a
superficially UV divergent loop correlator, the UV divergence is simply an
artifact of our computation method and unphysical. Therefore, we may find a
method so that UV divergences never appear and we never need to do UV
regularization. Indeed, our dispersion method here is such an example where
the regularization is never needed.

On the contrary, when computing a 1-loop correlator with whatever methods, the
result may contain a finite number of terms (in our case, the number is 1),
whose kinematic dependence is totally fixed but coefficient undetermined.
Indeed, the kinematic dependences of these terms are simply given by the
corresponding tree graphs from the local counterterm in ordinary calculations,
while the coefficients of these terms are never fixed by computation only;
Instead, they should be determined by a renormalization condition, or, in a
loose sense, by experimental data. Thus, to summarize: in a UV-divergent loop
correlator, the UV divergence may be avoidable, but the renormalization
ambiguity is not avoidable.

For readers familiar with flat-space loop calculations with dimensional
regularization, in App. E, we provide a direct comparison between our
dispersive calculation and the more conventional computation for a Minkowski
1-loop correlator, where one can see explicitly that the dispersion integral
itself is free of any UV divergence or renormalization dependence, and that
all renormalization-dependent information is fully encoded in the subtraction
point.

Back to our dispersion method, it is now clear that the renormalization
ambiguity cannot be probed by the nonanalyticity of the correlator, and
therefore we are not going to recover them from a dispersion integral. What we
did recover in (3.2), therefore, is a background free of any UV ambiguity,
whose existence is demanded by analyticity of the correlator. For this reason,
we call it the _irreducible background_.

The physical meaning of this irreducible background is clear: For the loop
diagram in question, we can imagine to integrate out all loop modes and get
infinitely many effective 3-point self-interaction vertices of the external
mode, with increasing number of derivatives. These derivative couplings
contribute to the 3-point function in the form of a Taylor expansion of $u$,
starting from $u^{3}$. Except the renormalization-dependent term $\propto
u^{3}$, all terms starting from $\mathcal{O}({u^{4}})$ are UV free and
unambiguously determined by the loop computation. They can still be treated as
from local (albeit derivative) interactions, but the coefficients of these
interactions are unambiguous prediction of the model. Our result for the
background (3.2) precisely recovers these terms.

With the above remark on renormalization ambiguity in mind, we can summarize
our result for the loop seed integral as:

$\displaystyle{\mathcal{J}}^{0,-2}(u)=Cu^{3}-\displaystyle\frac{\,{u^{4}}\,}{128\pi\sin(2\pi\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\sum_{n=0}^{\infty}\displaystyle\frac{\,{(3+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+4n)(1+n)_{\frac{1}{2}}(1+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n)_{\frac{1}{2}}}\,}{(\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n)_{\frac{1}{2}}(\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+n)_{\frac{1}{2}}}$
$\displaystyle\times\bigg{\\{}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}2+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+2n,4+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+2n\\\
4+4\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+4n\end{matrix}\bigg{|}u\biggr{]}u^{2n+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}-{}_{3}{\mathcal{F}}_{2}\biggl{[}\begin{matrix}1,2,4\\\
1-2n-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,4+2n+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}u\biggr{]}\bigg{\\}}$ $\displaystyle+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (88)

Here the first term $Cu^{3}$ is a local term, whose coefficient $C$ is to be
determined by a renormalization scheme. The rest of terms, including the
signal and the irreducible background, are free from renormalization
ambiguities. They are both organized as an infinite summation over quasi-
normal modes of the bubble loop.

Although it is difficult to analytically compare our result (3.2) with the
known background obtained from the spectral decomposition in [42] , we find
that their numerical results only differ by a $u^{3}$-term, which is exactly
the undetermined local part $Cu^{3}$ in (3.2). Given the very complicated form
of the background in [42], we consider this agreement a rather nontrivial
check of both methods.

###  3.3 Four-point single-exchange graph

Figure 8: The analytical structure of the 4-point seed integral
$\mathcal{Y}(r_{1}^{\prime},r_{2})$ on the complex $r_{1}^{\prime}$ plane,
with $r_{2}\in(0,1)$ fixed in the interior of its physical region. There are
three branch points at $r_{1}^{\prime}=0$ (signal branch point),
$r_{1}^{\prime}=-r_{2}$ (total-energy pole), and $r_{1}^{\prime}=-1$ (partial-
energy pole). with a branch cut connecting them in the interval
$r_{1}^{\prime}\in(-1,0)$. In this plot, we also show the insertion of a pole
at $r_{1}^{\prime}=r_{1}$ at which we compute the seed integral via the
dispersion integral (3.3), and the blue circle surrounding the branch cut is
the contour for the dispersion integral.

As our last application of the vertex dispersion relation, we return to the
4-point seed integral (11). Once again, we work with a particular choice of
the exponents $(p_{1},p_{2})=(-2,-2)$. As explained in (12), this corresponds
to the case of nonderivative coupling $\phi_{c}^{2}\sigma$ between the
conformal scalar $\phi_{c}$ in the external legs and a general principal
massive scalar $\sigma$ in the bulk line.

Similar to the previous 3-point examples, we want to exploit the scale
invariance of the process, which implies that the the seed integral
$\mathcal{I}^{-2,-2}(k_{12},k_{34},k_{s})$ depends on three energy variables
only through two independent momentum ratios. For the 4-point case, it is
convenient to choose the following pair of ratios:

$\displaystyle r_{1}\equiv\displaystyle\frac{\,{k_{s}}\,}{k_{12}},$
$\displaystyle r_{2}\equiv\displaystyle\frac{\,{k_{s}}\,}{k_{34}}.$ (89)

The physical region $0\leq k_{s}\leq\text{min}\\{k_{12},k_{34}\\}$ then
corresponds to $r_{1,2}\in[0,1]$. We then translate the analytical structure
of the seed integral on the complex $k_{12}$ plane (Fig. 4) to the complex
$r_{1}$ plane, keeping $r_{2}\in(0,1)$ staying in the interior of the physical
region. We show the result in Fig. 8, where the total-energy pole
$k_{12}=-k_{34}$, the partial energy pole $k_{12}=-k_{s}$, and the signal
branch point $k_{12}\rightarrow-\infty$ correspond to $r_{1}=-r_{2}$,
$r_{1}=-1$, and $r_{1}=0$, respectively. Also, the branch cut is now entirely
moved to the interval $r_{1}\in(-1,0)$.

To highlight that we are working with $r_{1}$ and $r_{2}$ as arguments of the
seed integral, we use a new notation $\mathcal{Y}(r_{1},r_{2})$ for the
4-point seed integral:

$\mathcal{Y}(r_{1}=k_{s}/k_{12},r_{2}=k_{s}/k_{34})\equiv\mathcal{I}^{-2,-2}(k_{12},k_{34},k_{s}).$
(90)

Then, from (2.4), we can read the signal
$\mathcal{Y}_{\text{S}}(r_{1},r_{2})\equiv\mathcal{I}_{\text{S}}^{-2,-2}(k_{12},k_{34},k_{s})$
of the seed integral, which is responsible for all the discontinuities:

$\displaystyle\mathcal{Y}_{\text{S}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{Y}_{\text{S},>}(r_{1},r_{2})\theta(r_{2}-r_{1})+\mathcal{Y}_{\text{S},<}(r_{1},r_{2})\theta(r_{1}-r_{2});$
(91) $\displaystyle\mathcal{Y}_{\text{S},>}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{I}^{-2,-2}_{+-}(k_{12},k_{34},k_{s})+\mathcal{I}^{-2,-2}_{++,\mathrm{F},>}(k_{12},k_{34},k_{s})+\mathrm{c.c.}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{[}{\mathcal{U}}_{+}^{-2}(k_{12},k_{s}){\mathcal{U}}_{-}^{-2}(k_{34},k_{s})-{\mathcal{U}}_{-}^{-2}(e^{\mathrm{i}\pi}k_{12},k_{s}){\mathcal{U}}_{+}^{-2}(k_{34},k_{s})\Big{]}+\text{c.c.};$
(92) $\displaystyle\mathcal{Y}_{\mathrm{S},<}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{Y}_{\mathrm{S},>}(r_{2},r_{1}).$ (93)

Using the expressions for $\mathcal{U}^{p}_{\pm}$ in (62), we can find the
explicit result for the signal:

$\displaystyle\mathcal{Y}_{\mathrm{S},>}(r_{1},r_{2})=\Bigl{(}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi}r_{1}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Bigr{)}\Bigl{(}r_{2}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{2})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\Bigr{)},$ (94)

where $\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p}$ is defined in (63). Then, with $r_{2}\in(0,1)$ fixed in the
interior of the physical region, the discontinuity of the seed function
$\mathcal{Y}(r_{1},r_{2})$ on the real axis of $r_{1}$ is itself a piecewise
function of $r_{1}$:

$\displaystyle\mathop{\mathrm{Disc}}_{r_{1}}\mathcal{Y}(r_{1},r_{2})=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathcal{Y}_{\mathrm{S}}(-r_{1},r_{2})\theta(-r_{1})\theta(r_{1}+1)$
$\displaystyle=2\mathrm{i}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Bigl{[}\mathcal{Y}_{\mathrm{S},>}(-r_{1},r_{2})\theta(r_{1}+r_{2})\theta(-r_{1})+\mathcal{Y}_{\mathrm{S},<}(-r_{1},r_{2})\theta(r_{1}+1)\theta(-r_{1}-r_{2})\Bigr{]}.$
(95)

This result is derived directly from (55), although there is a sign difference
in $\mathop{\mathrm{Disc}}\limits_{r_{1}}\mathcal{Y}(r_{1},r_{2})$ and
$\mathop{\mathrm{Disc}}\limits_{k_{12}}\mathcal{I}^{-2,-2}(k_{12},k_{34},k_{s})$
due to the relation $r_{1}=k_{s}/k_{12}$. Since the seed integral
$\mathcal{Y}(r_{1},r_{2})$ is regular when $|r_{1}|\rightarrow\infty$ and
$r_{2}$ fixed at a finite point, we can directly construct a dispersion
integral for $\mathcal{Y}(r_{1},r_{2})$ from (3.3), with a first-order
subtraction to ensure the vanishing integral along the large circle:

$\displaystyle\mathcal{Y}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{r_{1}}\,}{2\pi\mathrm{i}}\int_{-1}^{0}\displaystyle\frac{\,{{\mathrm{d}}r}\,}{r(r-r_{1})}\mathop{\mathrm{Disc}}_{r}\mathcal{Y}(r,r_{2})$
$\displaystyle=$ $\displaystyle~{}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\displaystyle\frac{\,{r_{1}}\,}{\pi}\bigg{[}\int_{-r_{2}}^{0}{\mathrm{d}}r\displaystyle\frac{\,{\mathcal{Y}_{\mathrm{S},>}(-r,r_{2})}\,}{r(r-r_{1})}+\int_{-1}^{-r_{2}}{\mathrm{d}}r\displaystyle\frac{\,{\mathcal{Y}_{\mathrm{S},<}(-r,r_{2})}\,}{r(r-r_{1})}\bigg{]},$
(96)

With the explicit expressions for the signal $\mathcal{Y}_{\text{S}}$ in (94)
and (93), the dispersion integral (3.3) can be rewritten as:

$\displaystyle\mathcal{Y}(r_{1},r_{2})=\bigg{[}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi^{2}}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu){r_{1}}\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},r_{2})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigg{]}\bigg{[}r_{2}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{2})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\bigg{]}$
$\displaystyle+\bigg{[}{r_{1}}\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},r_{2})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigg{]}\bigg{[}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi^{2}}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{2}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{2})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\bigg{]},$ (97)

where $\mathbf{I}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}$ and $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$ are the two
integrals that are derived from the vertex dispersion relation:

$\displaystyle\mathbf{I}_{\mkern