# Closed-Form Formulae for Inflation Correlators

Zhehan Qin${}^{1\,}$111Email<EMAIL_ADDRESS>and Zhong-Zhi
Xianyu${}^{1,2\,}$222Email<EMAIL_ADDRESS>
${}^{1}\,$_Department of Physics, Tsinghua University, Beijing 100084, China_
${}^{2}\,$_Collaborative Innovation Center of Quantum Matter, Beijing 100084,
China_

###### Abstract

We derive exact and closed-form expressions for a large class of two-point and
three-point inflation correlators with the tree-level exchange of a single
massive particle. The intermediate massive particle is allowed to have
arbitrary mass, spin, chemical potential, and arbitrary nonderivative or
derivative couplings to external inflaton modes. We also allow the coupling
coefficients to have arbitrary complex power dependences on the conformal
time. Our results feature closed-form expressions involving only familiar
special functions and without any infinite sums. This is achieved by an
improved bootstrap method with a suitable change of variables. Our results
cover a wide range of cosmological collider models and can be directly used
for future phenomenological studies. Our results can also be used as basic
building blocks for constructing more complicated inflation correlators.

###### Contents

  1. 1 Introduction
  2. 2 Reducing Correlators to Seed Integrals
    1. 2.1 Seed integrals: the definition and the result
    2. 2.2 Examples
      1. 2.2.1 Scalar exchange
      2. 2.2.2 Spinning exchange
  3. 3 Bootstrapping Hankel Seed Integrals
    1. 3.1 Hankel seed integral and its bootstrap equation
    2. 3.2 Solving the bootstrap equation
    3. 3.3 Folded limit
    4. 3.4 Summary
  4. 4 Bootstrapping Whittaker Seed Integrals
    1. 4.1 Whittaker seed integral and its bootstrap equation
    2. 4.2 Solving the bootstrap equation
    3. 4.3 Folded limit
    4. 4.4 Summary
  5. 5 Conclusions and Outlooks
  6. A Useful Formulae
  7. B Squeezed limit results
    1. B.1 Hankel seed integral in squeezed limit
    2. B.2 Whittaker seed integral in squeezed limit

## 1 Introduction

Recent years have witnessed new progress in the study of particle physics
during the cosmic inflation. It was realized that current and upcoming
cosmological observations have great potential in probing the structures of
primordial scalar and tensor perturbations, and thereby could provide us a
wealth of information about the high-energy physics at the inflation scale.
Many studies in recent years have explored the possibility of probing high-
scale new physics with future data, a program sometimes called the
“cosmological collider (CC) physics”[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].

The main observables of CC physics are the $n$-point $(n\geq 2)$ correlation
functions of the primordial scalar or tensor fluctuations. We collectively
call them _inflation correlators_. They can be viewed as correlation functions
of quantum fields living in the bulk of the inflationary spacetime, which is
approximately de Sitter, but with their external legs pinned to the future
boundary of the spacetime, namely, the end of inflation. As key quantities
connecting QFT predictions with the cosmological observations, the inflation
correlators play crucial roles in CC physics, much like the Minkowskian
scattering amplitudes to the collider physics. It is thus of central
importance to have a good theoretical understanding of inflation correlators.
Although our current knowledge about inflation correlators is still very
preliminary compared to the much-developed scattering amplitudes in Minkowski
spacetime, a lot of new results have emerged in recent years in both
analytical and numerical frontiers [56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84,
85].

A particular aspect of studying inflation correlators is to find precise and
explicit results, either numerical or analytical, to a range of key processes
that were identified in the recent studies of CC phenomenologies. Until
recently, many phenomenological studies have relied on very often unjustified
approximations. In general, the CC processes involve inflation correlators
mediated by massive fields at both the tree and the loop levels, and the on-
shell production of these massive particles during inflation can leave
oscillatory signatures in the inflation correlators. While this is
phenomenologically quite appealing, the computation of these massive processes
is relatively difficult. In the diagrammatic approach in the Schwinger-Keldysh
(SK) formalism [86, 87, 88, 89, 90, 18], the computation involves multi-
layered and nested time integrals over products of special functions.

Several methods have been developed in recent years that benefit analytical
computations of inflation correlators. For example, as was pointed out in [8],
one can derive differential equations satisfied by inflation correlators, so
that one can find explicit results by solving these differential equations
with proper boundary conditions, instead of computing the SK integrals
directly. This method has been further developed in subsequent studies and was
called the cosmological bootstrap [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67]. Borrowing this terminology and following our previous work [64], we shall
call the differential equations satisfied by inflation correlators the
_bootstrap equations_.

Another recently explored method is the Mellin transform [70, 71, 72], which
exploits the dilatation symmetry of dS. The Mellin variable is essentially the
weight of dilatation eigenmodes. By using Mellin variables in place of time
variables, we can trivialize the SK time integrals. It was further shown in
[50, 64] that a more practical approach is to use Mellin variables only for
the internal propagators, while the external legs still retain their time
dependence. This approach, called partial Mellin-Barnes representation, is
convenient for computing inflation correlators, since these correlators are
defined to be equal-time correlators at the future boundary. It is thus better
to leave the time variables in external modes untransformed.

With either the bootstrap techniques or the Mellin transform, many explicit
results have been found for tree-level correlators with single massive
exchanges, including the dS covariant cases and boost-breaking cases. See,
e.g., [56, 57, 69, 62, 63, 64]. There are also new results at the 1-loop level
obtained with partial Mellin-Barnes representation [50] or dS spectral
decomposition [65]. In all these examples, when the internal massive fields
are heavy enough, the correlator contains an oscillatory/nonanalytic piece
which we call the _signal_ , and a smooth/analytic piece which we call the
_background_. The signal part typically consists of imaginary powers of
momentum ratios multiplied by a hypergeometric function, while the background
part is typically written as a double Taylor series in momentum ratios. Most
of the previous results were first obtained at the four-point level, and the
three-point correlators were obtained by taking a soft limit, although taking
this soft limit could be subtle in practice, as will be commented
below.333There are also studies that work directly at the three-point level;
See, e.g., [62]. The results for the two-point functions are even rarer.

In this work, we present new analytical results for a wide range of three-
point and two-point inflation correlators with a single massive exchange at
the tree level, shown in Fig. 1. The external modes can be either massless
scalar mode such as the inflaton fluctuation, or the closely related conformal
scalar, or the massless tensor mode. The intermediate massive particle can
have arbitrary mass, spin, and chemical potential within the physically
allowed parameter space. We allow very general couplings between the external
mode and the intermediate massive particle, including both the nonderivative
and derivative couplings, and the coupling coefficient is allowed to have very
general complex power dependence on the conformal time $\tau$. Thus our result
covers models with time-dependent couplings, especially the oscillatory
couplings that are present in models with oscillating background fields, e.g.,
[51]. The couplings with arbitrary complex power dependence on the conformal
time are also useful when we treat our correlators as subgraphs of more
complicated correlators.

Our results feature exact and closed-form formulae. In contrast to previously
obtained results, our expressions for the three-point and two-point
correlators do not contain any Taylor series. Instead, all the momentum
dependences are fully captured by familiar special functions whose analytical
properties are well understood. This makes our result particularly suitable
for understanding the analytic properties of inflation correlators. Also, our
compact results can also be used as handy building blocks when constructing
more complicated inflation correlators, a topic we shall explore in a separate
work [91].

The strategy we adopt in this paper is similar in spirit to many previous
works on cosmological bootstraps. That is, we begin with a tree-level four-
point correlator
$\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\varphi_{\mathbf{k}_{4}}\rangle^{\prime}$
with a single massive exchange in the $s$-channel, with momentum
$\mathbf{k}_{s}\equiv\mathbf{k}_{1}+\mathbf{k}_{2}$. The result for the three-
point and two-point correlators can be obtained by taking soft limits
$\mathbf{k}_{4}\rightarrow 0$ and $\mathbf{k}_{2},\mathbf{k}_{4}\rightarrow
0$, which we call the single folded limit and the double folded limit,
respectively. (See Fig. 2.) The regularity of folded limits of four-point
functions follows from the Bunch-Davies initial condition for all the fields
involved [56]. So, there is no conceptual difficulty in taking the folded
limits. In practice, however, the four-point functions usually contain terms
singular in folded limits, and these singularities must cancel out in the full
expression. The matter is further complicated by the fact that the double
folded limit is usually at the boundary of convergent regions of the Taylor
series for the background. All these complications make the folded limit less
trivial than it seems.

In this work, we circumvent these complications by making a proper change of
variables. The advantage of adopting a new set of variables for the bootstrap
equations has already been observed in [64]. Here we shall make fuller use of
it. The key idea is simple: after stripping off trivial external momentum
factors, an $s$-channel four-point correlator depends on various momenta only
through two independent momentum ratios. In previous works, the two
independent momentum ratios are normally taken as $r_{1}\equiv
k_{s}/(k_{1}+k_{2})$ and $r_{2}\equiv k_{s}/(k_{3}+k_{4})$, where
$k_{i}\equiv|\mathbf{k}_{i}|$. Our new observation is that it is more
advantageous to use $u_{i}\equiv 2r_{i}/(1+r_{i})$ $(i=1,2)$ instead of
$r_{1,2}$. For physical parameters $0\leq r_{1,2}\leq 1$, the new variable
$u_{i}$ is a monotonic function of $r_{i}$ and also takes its value from
$[0,1]$. The key advantage of $u$ variables is that the inhomogeneous
bootstrap equation can be solved by an ansatz of single-layer Taylor series in
the single folded limit $u_{2}\rightarrow 1$, and it turns out that this
Taylor series can be easily summed to give a closed-form special function,
which in our case is always a generalized hypergeometric function
${}_{3}\text{F}_{2}$. With the closed-form expression for the three-point
functions, the double folded limit $u_{1,2}\rightarrow 1$ is also easily
taken, and the result is again a closed-form expression.

Now we give an outline for the rest of this work. In Sec. 2, we introduce the
correlators to be computed in this work, and explain that all these
correlators can be easily reduced to simple linear combinations of several
_seed integrals_ , and thereby reduce the computation of the correlators to
that of the seed integrals. Depending on whether there is a nonzero chemical
potential for the intermediate massive particle, we need two types of seed
integrals: For massive fields of arbitrary mass and spin but without chemical
potential, the corresponding internal propagator can always be expressed in
terms of Hankel functions. For such correlators, we define a _Hankel seed
integral_. On the other hand, for an intermediate field with a chemical
potential, the transversely polarized internal propagator is expressed in
terms of Whittaker W functions. For such cases, we define a _Whittaker seed
integral_. We then give various explicit examples, showing how to reduce the
correlators to seed integrals. Then, we present all the details of solving the
bootstrap equations for the Hankel seed integral in Sec. 3, and for the
Whittaker seed integral in Sec. 4. We conclude in Sec. 5 with further
discussions. Some useful formulae are summarized in App. A, and a computation
of the seed integrals in the squeezed limit is presented in App. B.

Readers uninterested in the technical details of bootstrapping seed integrals
can skip most of Sec. 3 and Sec. 4. For readers who only want to use the
results, the shortest route is to glance through Sec. 2 to get an idea of how
to reduce a correlator to the seed integral, and then jump directly to the
results. A quick summary of our results is as follows. The Hankel seed
integral $\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},u_{2})$ is defined in (10).
The closed-form expression for $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},1)$ is given in (3.4) and
(90), while the closed-form expression for $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(1,1)$ is given in (3.4) and (93).
The Whittaker seed integral $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{(h)p_{1}p_{2}}_{\mathsf{a}\mathsf{b}}(u_{1},u_{2})$ is defined in (11).
The closed-form expression for its single folded limit $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(h)p_{1}p_{2}}(u,1)$ is given in (4.4) and
(4.4), and the closed-form expression for the double folded limit $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(h)p_{1}p_{2}}(1,1)$ is given in (130) and
(4.4).

Explicit analytical results already exist in the literature for many special
cases considered in this work, but almost always expressed in terms of Taylor
series.444In the appendix of [56] it was shown that such Taylor series in $r$
variables may be written in terms of the Kampé de Fériet function, which is a
very general form of hypergeometric function of two variables. However, it
seems that the numerical implementation of this function is not readily
available in Mathematica. For these known cases, our results are new in that
they are expressed in compact and closed form in terms of familiar special
functions, instead of Taylor series. Our results also include some completely
new cases that were not treated in the literature, including most of two-point
functions, the three-point function with (complex) time-dependent couplings,
and also the chemical-potential-boosted three-point function with arbitrary
couplings.

##### Notations and conventions.

We use mostly plus signature for the spacetime metric and we fix the
background geometry to be the inflationary patch of dS. With the conformal
time $\tau\in(-\infty,0)$ and comoving spatial coordinates
$\mathbf{x}\in\mathbb{R}^{3}$, the metric reads
${\mathrm{d}}s^{2}=a^{2}(\tau)(-{\mathrm{d}}\tau^{2}+{\mathrm{d}}\mathbf{x}^{2})$
and $a(\tau)=-1/(H\tau)$, where $H$ is the inflation Hubble parameter.
Throughout this work, we take $H=1$ for simplicity. We follow the diagrammatic
notations and conventions in [18]. Frequently used shorthand notations include
$k_{ij}\equiv k_{i}+k_{j}$ $(i,j=1,2,3,4)$, $k_{123}\equiv k_{1}+k_{2}+k_{3}$,
$p_{12}\equiv p_{1}+p_{2}$, $\bar{p}_{12}\equiv p_{1}-p_{2}$. Indices sans
serif such as $\mathsf{a},\mathsf{b}$ are often but not always labels for SK
branches, but they always take values in $\pm 1$. In most places, we use the
mass parameter $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu$ instead of the mass $m$ for the intermediate massive field. The two are
related by $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu=\sqrt{m^{2}-9/4}$ for scalars and $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu=\sqrt{m^{2}-(s-1/2)^{2}}$ for fields of spin $s\neq 0$. In all mid-
steps, we treat $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu$ as a positive real parameter. The results for complementary fields
($\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ being
purely imaginary) can be obtained in the final results by analytic
continuation. For spinning fields, the chemical potential is denoted by
$\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu$. We use
$\langle\varphi_{\mathbf{k}_{1}}\cdots\varphi_{\mathbf{k}_{n}}\rangle^{\prime}$
to denote correlators of 3-momentum modes $\varphi_{\mathbf{k}}$, and the
prime in $\langle\cdots\rangle^{\prime}$ means that the momentum-conserving
$\delta$-function is removed. Other variables and parameters will be defined
in the main text when they appear.

## 2 Reducing Correlators to Seed Integrals

Figure 1: Three-point correlator
$\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle^{\prime}$
and two-point correlator
$\langle\varphi_{\mathbf{k}}\varphi_{-\mathbf{k}}\rangle^{\prime}$ mediated by
a single massive field $\sigma$ at the tree level. Here $\varphi$ can be
either the inflaton which is a massless scalar, or the graviton which is a
massless spin-2 particle. Our diagrammatic notation follows [18]. In
particular, the shaded circles represent vertices of either SK branch
$\mathsf{a}=\pm$ which is summed over in the final result, and the white
squares denote the boundary points at $\tau=0$.

In this section, we first introduce the objects to be computed in this work,
including the three-point and two-point correlators mediated by a single
massive field at the tree level. Next, we introduce the two types of seed
integrals, the Hankel seed integrals and the Whittaker seed integrals, which
are built from the SK integrals in a standard diagrammatic calculation in the
bulk. Then, we provide a number of examples showing how to reduce SK integrals
to a simple linear combination of seed integrals, including both scalar and
spinning exchanges with several types of couplings.

###  2.1 Seed integrals: the definition and the result

##### Ingredients: propagators and vertices.

The inflation correlators considered in this work have two types of topologies
shown in Fig. 1. That is, we consider correlators of the external field
$\varphi$ mediated by a single massive field $\sigma$ at the three-point and
two-point levels. The external field $\varphi$ can be a massless inflaton, a
conformal scalar ($m^{2}=2$), or even a massless spin-2 graviton, since all
these cases share very similar bulk-to-boundary propagators. Therefore, for
the clarity of presentation, we will often take $\varphi$ as the massless
inflaton field below, with a notable exception when we discuss the example of
a massive spinning exchange with nonzero chemical potential.

In the standard SK formalism, the bulk-to-boundary propagator of the inflaton
field $\varphi$, represented by all the black external lines in Fig. 1, is
given by [18]:

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

Here $k$ is the magnitude of the momentum flowing in the propagator. The
conformal time $\tau$ and the SK index $\mathsf{a}=\pm$ are both for the bulk
endpoint. See [18] for more detailed discussions.

On the other hand, the blue internal lines in Fig. 1 represent the bulk
propagator of the massive field $\sigma$. In this work, the massive field is
allowed to have either dS covariant or dS boost-breaking dispersion relation.
The dS covariant case is technically simpler thanks to the presence of the
full dS isometries, and is thus interesting on its own. The dS boost breaking
case is more relevant to CC physics, since the rolling of the inflaton
background necessarily breaks the dS boosts. More importantly, boost breaking
processes typically lead to larger observable signals, and thus are more
interesting for phenomenological model buildings [21, 29, 28, 34, 38, 48, 62,
63, 64, 55, 54].

Similar to the Minkowskian QFT, free massive particles with dS invariant
dispersion relation are classified by their mass $m$ and spin $s$. The spin
can take any nonnegative integer or half-integer value, although we shall only
consider integer spin in this work, since only the integer-spin field can
contribute to tree-level inflaton correlators. For the mass $m$, as mentioned
at the end of the introduction, we shall always use the _mass parameter_
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ in place of
the mass $m$. The two are related by $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu=\sqrt{m^{2}-9/4}$ for
a scalar field and $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu=\sqrt{m^{2}-(s-1/2)^{2}}$ for a field of spin $s\neq 0$. For simplicity,
in the calculations performed in this work, we always assume $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ to be positive real.
This is the situation most relevant to CC physics. However, our final analytic
results are also applicable to lighter particles by analytic continuation, in
which cases the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ takes purely
imaginary value.

It is useful to present some basic results about a massive scalar particle.
Given a massive scalar field $\sigma(\tau,\mathbf{x})$ with mass parameter
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu>0$, we can
canonically quantize it in the usual way, by writing it as a linear
combination of the creation operator $a_{-\mathbf{k}}^{\dagger}$ and
annihilation operator $a_{\mathbf{k}}$ in the 3-momentum space:

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

where the time-dependent coefficient $\sigma(k,\tau)$ is called the mode
function, and is determined by the Klein-Gordon equation in dS together with
the Bunch-Davies initial condition. Explicitly:

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

where $\mathrm{H}_{\nu}^{(1)}(z)$ is the Hankel function of first kind. Using
the mode function, one can construct the bulk propagator
$D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ for $\sigma$ field, in which
$\mathsf{a},\mathsf{b}=\pm$ and $\tau_{1,2}$ are SK indices and conformal time
variables at the two bulk endpoints of the propagator. According to the
standard procedure in the SK formalism, as reviewed in [18], one first
constructs the two Wightman functions $D_{>}(k;\tau_{1},\tau_{2})$ and
$D_{<}(k;\tau_{1},\tau_{2})$. The “greater” Wightman function $D_{>}$ is
given, in terms of the mode function $\sigma(k,\tau)$, by:

$D_{>}(k;\tau_{1},\tau_{2})=\sigma(k,\tau_{1})\sigma^{*}(k,\tau_{2})=\displaystyle\frac{\,{\pi
e^{-\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{4}(\tau_{1}\tau_{2})^{3/2}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k\tau_{1})\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}(-k\tau_{2}),$
(4)

while the “less” Wightman function
$D_{<}(k;\tau_{1},\tau_{2})=D_{>}^{*}(k;\tau_{1},\tau_{2})$. Then, the four SK
propagators $D_{\mathsf{a}\mathsf{b}}$ with $\mathsf{a},\mathsf{b}=\pm$ are
given as:

$\displaystyle
D_{\pm\pm}(k;\tau_{1},\tau_{2})=D_{\gtrless}(k;\tau_{1},\tau_{2})\theta(\tau_{1}-\tau_{2})+D_{\lessgtr}(k;\tau_{1},\tau_{2})\theta(\tau_{2}-\tau_{1}),$
(5) $\displaystyle
D_{\pm\mp}(k;\tau_{1},\tau_{2})=D_{\lessgtr}(k;\tau_{1},\tau_{2}).$ (6)

On the other hand, it is also important to consider the boost-breaking
dispersion relations for the massive particle, as mentioned above. In this
case, one needs more parameters to characterize the particle. If we break the
three dS boost symmetries but retain the dS dilatation, then there are two
additional parameters one can include in the quadratic Lagrangian of a massive
field: First, one can have a non-unit sound speed $c_{s}$ which characterizes
the relative sizes between the kinetic energy $\sigma^{\prime 2}$ and the
gradient energy $(\partial_{i}\sigma)^{2}$. The non-unit sound speed can be
introduced in our computation by replacing all momentum $k$ of the massive
field by $c_{s}k$. The problem of including non-unit sound speed has been
treated in detail in [62, 63], and we do not consider them from now on.

Second, when the dS boost is broken by, say, a rolling inflaton background,
the helicity of a massive spinning particle
$h\equiv\mathbf{s}\cdot\mathbf{k}/|\mathbf{k}|$ becomes unambiguously defined,
since we are no longer allowed to boost particles into different helicity
states. Therefore, one can include a helicity-weighted particle number
operator in the Hamiltonian, whose coefficient is a helicity-dependent
chemical potential $\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu$ [34], which we shall simply call chemical potential for short.555The
chemical potential $\mu$ is a dim-1 parameter, and it is usually convenient to
define a dimensionless chemical potential $\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu=\mu/H$. Since we take
$H=1$ in this work, the two become the same.  Technically, the chemical
potential modifies the mode function of a massive spinning particle from the
original Hankel-type functions to the Whittaker W functions. For example, a
massive spin-1 particle of mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ and chemical
potential $\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu$
has the following mode function $B^{(h)}(k,\tau)$ for its two transverse
polarizations $h=\pm 1$:

$B^{(h)}(k,\tau)=\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu/2}}\,}{\sqrt{2k}}\mathrm{W}_{\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(2\mathrm{i}k\tau),$
(7)

where $\text{W}_{\kappa,\nu}(z)$ is the Whittaker W function. The bulk
propagators for particles of nonzero chemical potential can then be
constructed from their mode functions in the same way as shown above for a
massive scalar. It is then clear that the cases with nonzero chemical
potential require a separate treatment. Therefore, we classify the correlators
considered in this work into two categories, one corresponds to the Hankel-
type correlators, and includes all massive fields with dS covariant
dispersion; the other is the Whittaker-type correlators, which covers the
cases of massive spinning fields with nonzero chemical potential.

With all bulk-to-boundary propagators and bulk propagators specified in Fig.
1, it remains to include the appropriate couplings for all the internal
vertices. In this work, we allow the massive field $\sigma$ to have very
general couplings to the external inflaton lines. The couplings can be either
nonderivative or derivative, and they can contain an arbitrary number of
uncontracted time derivatives or contracted spatial derivatives. Normally, if
one imposes scale invariance, the time dependence in the coupling coefficient
is fully determined by the number of fields and derivatives. However, we do
not have to make the assumption of scale invariance, which is broken at least
by the inflaton potential. Indeed, it is even possible to include complicated
time dependences in the coupling coefficients, such as a component oscillatory
in physical time, $c(\tau)\supset(-\tau)^{p}\cos\omega
t\sim(-\tau)^{p\pm\mathrm{i}\omega}$, which may appear in some resonant
models.

##### Seed integrals: definitions and results.

With all ingredients introduced above, we are now in a position to construct
correlators and to compute them. The general strategy of this work is that we
compute two types of _seed integrals_ , one for Hankel-type correlators, and
one for Whittaker-type correlators. These seed integrals are essentially the
SK integrals for the four-point correlators with $s$-channel exchange of a
massive field, shown in Fig. 2. As four-point functions with $s$-channel
mediation, the seed integrals depend on the magnitudes of the four external
momenta, $k_{i}\equiv|\mathbf{k}_{i}|$ $(i=1,2,3,4)$, as well as the
magnitudes of the $s$-channel momentum $k_{s}\equiv|\mathbf{k}_{s}|$, with
$\mathbf{k}_{s}\equiv\mathbf{k}_{1}+\mathbf{k}_{2}$. We also use shorthand
notations such as $k_{12}\equiv k_{1}+k_{2}$, $k_{34}\equiv k_{3}+k_{4}$, and
$k_{123}\equiv k_{1}+k_{2}+k_{3}$.

$\xrightarrow[(u_{2}\rightarrow 1)]{\mathbf{k}_{4}\rightarrow 0}$
$\xrightarrow[(u_{1}\rightarrow 1)]{\mathbf{k}_{2}\rightarrow 0}$

Figure 2: Three-point and two-point correlators as successive folded limits of
a four-point correlator.

We define the seed integrals such that they depend on these momenta only
through two independent ratios. A widely used pair of independent momentum
ratios are the following:

$\displaystyle r_{1}\equiv\displaystyle\frac{\,{k_{s}}\,}{k_{12}},$
$\displaystyle r_{2}\equiv\displaystyle\frac{\,{k_{s}}\,}{k_{34}}.$ (8)

However, a key point in this work is that it is more advantageous to use the
following two variables $u_{1,2}$ instead of $r_{1,2}$, as was observed in
[64]:

$\displaystyle u_{1}\equiv\displaystyle\frac{\,{2k_{s}}\,}{k_{12}+k_{s}},$
$\displaystyle u_{2}\equiv\displaystyle\frac{\,{2k_{s}}\,}{k_{34}+k_{s}}.$ (9)

As we shall see in later sections, it is the use of the $u$-variables that
enables us to conveniently take the folded limits of the four-point function,
and to get the closed-form analytical formulae for the three-point and two-
point functions.

Now we are ready to introduce the seed integrals, to which a large class of
correlators can be reduced, as will be shown later in this section. As
explained above, we introduce two types of seed integrals. One is the _Hankel
seed integral_ $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},u_{2})$, which is defined by:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},u_{2})\equiv(-\mathsf{a}\mathsf{b})k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{a}\mathsf{b}}(k_{s};\tau_{1},\tau_{2}).$
(10)

Here $\mathsf{a},\mathsf{b}=\pm 1$ are SK indices as required by the
diagrammatic rule in the SK formalism, and $(p_{1},p_{2})$ are a pair of
numbers that can take most _complex_ values, except that we require
$\text{Re}~{}p_{1,2}>-5/2$ to make the seed integral well defined.
Furthermore, $D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ is the bulk SK
propagator for a massive scalar field with mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$, whose explicit form
has been given in (4), (5), and (6). With these explicit expressions, one can
see that the seed integral defined above depends on various momenta only
through two independent ratios, which we choose to be $u_{1}$ and $u_{2}$.

The Hankel seed integral will be computed explicitly in Sec. 3 with one or
both of $u_{1}$ and $u_{2}$ taken to be 1, so that one can directly read the
three-point and two-point correlators from them. The result for the three-
point limit $\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},1)$ is summarized in (3.4) and
(90), and the result for the two-point limit $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(1,1)$ is given in (3.4) and (93).

Likewise, we introduce the following _Whittaker seed integral_ for the
Whittaker-type correlators:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{(h)p_{1}p_{2}}_{\mathsf{a}\mathsf{b}}(u_{1},u_{2})\equiv(-\mathsf{a}\mathsf{b})k_{s}^{3+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{a}\mathsf{b}}^{(h)}(k_{s};\tau_{1},\tau_{2}).$
(11)

Everything here is in parallel with the Hankel seed integral, except that here
we are including the bulk SK propagator
$D_{\mathsf{a}\mathsf{b}}^{(h)}(k_{s};\tau_{1},\tau_{2})$ for the helicity-$h$
$(h=\pm 1)$ component of a massive spin-1 field, constructed from the mode
function (7). Therefore, we add a superscript $(h)$ in the seed integral to
distinguish it from the Hankel seed integral. The computation of the Whittaker
seed integral will be given in Sec. 4. The result for the three-point limit
$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(h)p_{1}p_{2}}(u,1)$ is given in (4.4) and
(4.4), and the result for the two-point limit $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(h)p_{1}p_{2}}(1,1)$ is given in (130) and
(4.4).

One notable difference in our definition of seed integrals from the previously
defined seed integrals (including our earlier work [64]) is that the SK
indices $\mathsf{a},\mathsf{b}$ are left unsummed. As we shall see later, when
writing correlators as linear combinations of seed integrals, the coefficients
of the linear combination usually involve additional SK indices, and this fact
necessitates the results of the seed integrals for each individual SK branch.
In previous works, this complication was usually overcome by acting
appropriate differential operators on a fully summed seed integral. Acting on
such differential operators has the effect of raising the powers $p_{1,2}$ by
integer units. However, when we want to change $p_{1,2}$ continuously on their
complex planes, the method of acting differential operators no longer works.
Also, for the three-point and two-point functions, there is simply no
sufficient momentum dependence to act the differential operator. For example,
to raise the power of $p_{2}$ in a four-point seed integral, one needs to act
a differential operator that contains $\partial_{u_{2}}$. However, for the
three-point function, $u_{2}$ has been set to a constant, so the differential
operator is no longer applicable. It is for these two reasons that we choose
to keep the SK indices explicit in the definition of the seed integrals. We
shall explain this point more clearly with an example below.

###  2.2 Examples

In this subsection, we provide various examples, showing how to reduce the
correlators to seed integrals. We choose a range of examples that often appear
in models of CC physics. We also provide sufficient details and mid-steps to
illustrate the procedure.

####  2.2.1 Scalar exchange

The first group of examples all involve tree-level exchange of a massive
scalar field $\sigma$, but the massive scalar can couple to the external
inflaton $\varphi$ in many different ways. The first three examples below
consider three special choices of couplings, all coming from CC model
buildings. The fourth example considers the most general coupling one can
write down between $\sigma$ and $\varphi$.

##### Time-derivative coupling.

The presence of a rolling inflaton background breaks the time diffeomorphism,
and thus allows us to introduce uncontracted temporal indices when writing the
effective Lagrangian for the fluctuating fields. In particular, one can
consider couplings between $\sigma$ and the conformal time derivative of
$\varphi^{\prime}\equiv{\mathrm{d}}\varphi/{\mathrm{d}}\tau$. Therefore, for
the trilinear and bilinear couplings shown in Fig. 1, we can choose:

$\displaystyle\mathcal{O}_{\varphi\varphi\sigma}=\displaystyle\frac{\,{1}\,}{2}(-\tau)^{-2}(\varphi^{\prime})^{2}\sigma,$
$\displaystyle\mathcal{O}_{\varphi\sigma}=(-\tau)^{-3}\varphi^{\prime}\sigma.$
(12)

The explicit $\tau$ dependence in the couplings is dictated by the scale
invariance, and we do not include the constant coefficients for clarity. It is
trivial to put them back in the final expression. The above couplings are
perhaps the simplest choice from a purely technical point of view, owing to
the fact that the bulk-to-boundary propagator $G_{\mathsf{a}}(k,\tau)$ in (1)
becomes simpler when acting on a conformal time derivative:
$G_{\mathsf{a}}^{\prime}(k,\tau)=\tau e^{+\mathrm{i}\mathsf{a}k\tau}/(2k)$.
The SK integral thus takes the following form: (See [18] for the diagrammatic
rules in the SK formalism.)

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma_{\mathbf{k}_{3}}}^{\prime}=$
$\displaystyle-\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{(-\tau_{1})^{2}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{(-\tau_{2})^{3}}G_{\mathsf{a}}^{\prime}(k_{1};\tau_{1})G_{\mathsf{a}}^{\prime}(k_{2};\tau_{1})G_{\mathsf{b}}^{\prime}(k_{3};\tau_{2})D_{\mathsf{a}\mathsf{b}}(k_{3};\tau_{1},\tau_{2})$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{8k_{1}k_{2}k_{3}}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int{\mathrm{d}}\tau_{1}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{(-\tau_{2})^{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{3}\tau_{2}}D_{\mathsf{a}\mathsf{b}}(k_{3};\tau_{1},\tau_{2}).$
(13)

Here the notation
$\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma_{\mathbf{k}_{3}}}^{\prime}$
means that we have only included the graph with the massive propagator
carrying the momentum $\mathbf{k}_{3}$. There are also other two permutations
to be included in the full result
$\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma}^{\prime}$.
Now, comparing the above expression with the Hankel seed integral defined in
(10), we immediately find:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma}^{\prime}=\displaystyle\frac{\,{-1}\,}{8k_{1}k_{2}k_{3}^{4}}\sum_{\mathsf{a},\mathsf{b}=\pm}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{0,-2}(u,1)+(\text{2 perms}),$ (14)

where $u=2k_{3}/k_{123}$. In a similar way, one can compute the correction to
the two-point function
$\delta\langle\varphi_{\mathbf{k}}\varphi_{-\mathbf{k}}\rangle_{\sigma}^{\prime}$
with two insertions of bilinear couplings $\mathcal{O}_{\varphi\sigma}$
introduced above:

$\displaystyle\delta\langle\varphi_{\mathbf{k}}\varphi_{-\mathbf{k}}\rangle_{\sigma}^{\prime}=$
$\displaystyle-\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{(-\tau_{1})^{3}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{(-\tau_{2})^{3}}G_{\mathsf{a}}^{\prime}(k;\tau_{1})G_{\mathsf{b}}^{\prime}(k;\tau_{2})D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4k^{3}}\sum_{\mathsf{a},\mathsf{b}=\pm}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{-2,-2}(1,1).$ (15)

In this simple example, we have only one seed integral with its SK indices
directly summed, thanks to the simple form of the bulk-to-boundary propagator
with one time derivative. We will see more complicated combinations in the
following examples where the combination coefficients themselves contain SK
indices.

##### dS-invariant coupling.

Our second example is a slight modification of the first example. Instead of
time-derivative coupling for the trilinear coupling, we make the following dS
invariant choice:

$\displaystyle\mathcal{O}_{\varphi\varphi\sigma}=\displaystyle\frac{\,{1}\,}{2}(-\tau)^{-2}\eta^{\mu\nu}(\partial_{\mu}\varphi)(\partial_{\nu}\varphi)\sigma,$
$\displaystyle\mathcal{O}_{\varphi\sigma}=(-\tau)^{-3}\varphi^{\prime}\sigma.$
(16)

At the same time, we still keep the time derivative coupling for the bilinear
vertex. We note in passing that dS invariant couplings for the bilinear
vertex, such as $\partial_{\mu}\sigma\partial^{\mu}\varphi$ and
$\sigma\varphi$, are in a sense trivial, since such couplings can always be
rotated away by a simple linear field redefinition. Consequently, dS invariant
two-point mixings normally do not lead to physically interesting effects in CC
physics.

It is straightforward to find the following SK integral for the three-point
correlator constructed with the above two couplings:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma_{\mathbf{k}_{3}}}^{\prime}=$
$\displaystyle~{}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{(-\tau_{1})^{2}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{(-\tau_{2})^{3}}G_{\mathsf{b}}^{\prime}(k_{3};\tau_{2})D_{\mathsf{a}\mathsf{b}}(k_{3};\tau_{1},\tau_{2})$
$\displaystyle~{}\times\Big{[}G_{\mathsf{a}}^{\prime}(k_{1};\tau_{1})G_{\mathsf{a}}^{\prime}(k_{2};\tau_{1})+\mathbf{k}_{1}\cdot\mathbf{k}_{2}G_{\mathsf{a}}(k_{1};\tau_{1})G_{\mathsf{a}}(k_{2};\tau_{1})\Big{]}.$
(17)

Using the explicit expression for the bulk-to-boundary propagator
$G_{\mathsf{a}}(k;\tau)$ in (1) and then comparing the result with the
definition of the seed integral (10), we have:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma}^{\prime}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{16(k_{1}k_{2}k_{3})^{2}}\sum_{\mathsf{a},\mathsf{b}=\pm}\Big{[}(\varrho_{12}^{2}-1)\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{0,-2}(u,1)+\mathrm{i}\mathsf{a}\displaystyle\frac{\,{\varrho_{12}(1-\varrho_{1}^{2}-\varrho_{2}^{2})}\,}{\varrho_{1}\varrho_{2}}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{-1,-2}(u,1)$
$\displaystyle~{}+\displaystyle\frac{\,{1-\varrho_{1}^{2}-\varrho_{2}^{2}}\,}{\varrho_{1}\varrho_{2}}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{-2,-2}(u,1)\Big{]}+(\text{2 perms}),$ (18)

where $u=2k_{3}/k_{123}$ as before, and we have defined $\varrho_{1}\equiv
k_{1}/k_{3}$, $\varrho_{2}\equiv k_{2}/k_{3}$, $\varrho_{12}\equiv
k_{12}/k_{3}$ for simplicity. Also, we have used the momentum conservation to
write
$\mathbf{k}_{1}\cdot\mathbf{k}_{2}=\frac{1}{2}(k_{3}^{2}-k_{1}^{2}-k_{2}^{2})$.
As mentioned before, the coefficients in front of some seed integrals have
explicit dependence on SK indices. In this particular example, one can get rid
of the SK-indices-dependent coefficients by acting appropriate differential
operators. One can directly check that the following equation holds:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma}^{\prime}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{8(k_{1}k_{2}k_{3})^{2}}\Big{[}-k_{1}k_{2}\partial_{k_{12}}^{2}+(\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern 2.0mu_{1}\cdot\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern
2.0mu_{2})(1-k_{1}\partial_{k_{12}})(1-k_{2}\partial_{k_{12}})\Big{]}\sum_{\mathsf{a},\mathsf{b}=\pm}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{-2,-2}(u,1)$ $\displaystyle~{}+(\text{2
perms}).$ (19)

When acting the differential operator $\partial_{k_{12}}$, the $u$ parameter
in $\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{-2,-2}(u,1)$ should be understood as
$u=2k_{3}/(k_{12}+k_{3})$. Thus we see that it is sometimes possible to avoid
SK-index-dependent coefficients by acting appropriate differential operators
on the fully summed seed integrals. But this is not always possible, as we
shall see in a later example with the most general couplings.

##### Oscillating coupling.

Now we consider an example with explicitly time-dependent couplings. In
particular, we consider a trilinear coupling that is oscillatory in _physical_
time $t$.

$\displaystyle\mathcal{O}_{\varphi\varphi\sigma}=\displaystyle\frac{\,{1}\,}{2}(-\tau)^{-2}\big{[}1+\lambda(-\tau)^{p}\cos(\omega
t+\delta)\big{]}(\varphi^{\prime})^{2}\sigma,$
$\displaystyle\mathcal{O}_{\varphi\sigma}=(-\tau)^{-3}\varphi^{\prime}\sigma.$
(20)

Here $\lambda$ is a small dimensionless parameter, $\delta$ is a phase, and
the real exponent $p$ describes how the oscillation amplitude changes with
time. Such a trilinear coupling can appear when there is a background field
oscillating with a physical frequency $\omega$.

Now let us focus on the three-point correlator contributed by the oscillatory
term in $\mathcal{O}_{\varphi\varphi\sigma}$, namely, the term proportional to
$\lambda$. With the same procedure, we write down the SK integral and compare
it with the Hankel seed integral, which shows that the
$\mathcal{O}({\lambda})$ part of the correlator can be written in the
following way:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\lambda}^{\prime}=$
$\displaystyle~{}\displaystyle\frac{\,{-\lambda}\,}{16k_{1}k_{2}k_{3}^{4}}\sum_{\mathsf{a},\mathsf{b}=\pm}\Big{[}e^{-\mathrm{i}\delta}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{\,p+\mathrm{i}\omega,-2}(u,1)+e^{+\mathrm{i}\delta}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{\,p-\mathrm{i}\omega,-2}(u,1)\Big{]}+(\text{2
perms}).$ (21)

Here we have used the fact that the physical time is related to the conformal
time via $\tau=-e^{-t}$.

##### General couplings.

Finally we consider the most general coupling between the massive scalar field
$\sigma$ and the external inflaton mode $\varphi$ respecting the spatial
translation and rotation. Using integration by parts, we can move all the
derivatives on $\sigma$ to $\varphi$. After doing this, the most general
trilinear and bilinear couplings take the following form:

$\displaystyle\mathcal{O}_{\varphi\varphi\sigma}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2}(-\tau)^{R}\Big{[}\partial_{\tau}^{J}\partial_{i_{1}}\cdots\partial_{i_{M}}(\partial_{j}\partial^{j})^{N}\varphi\Big{]}\Big{[}\partial_{\tau}^{K}\partial^{i_{1}}\cdots\partial^{i_{M}}(\partial_{k}\partial^{k})^{L}\varphi\Big{]}\sigma,$
$\displaystyle\mathcal{O}_{\varphi\sigma}=$
$\displaystyle~{}(-\tau)^{S}\Big{[}\partial_{\tau}^{P}(\partial_{i}\partial^{i})^{Q}\varphi\Big{]}\sigma.$
(22)

Here $R$ and $S$ can be any complex numbers with $\text{Re}\,R,S>-5/2$, while
$J,K,M,N,L,P,Q$ can take any nonnegative integer values. If we further assume
scale symmetry, then the powers of time $R$ and $S$ are fixed to be
$R=-4+J+K+2(M+N+L)$ and $S=-4+P+2Q$, but we do not have to make this
assumption. Then, applying the diagrammatic rule in Schwinger-Keldysh
formalism, one can write down the SK integral for the three-point correlator:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma_{\mathbf{k}_{3}}}^{\prime}=-\displaystyle\frac{\,{1}\,}{2}(-\mathbf{k}_{1}\cdot\mathbf{k}_{2})^{M}(-k_{1}^{2})^{N}(-k_{2}^{2})^{L}(-k_{3}^{2})^{Q}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}(-\tau_{1})^{R}(-\tau_{2})^{S}$
$\displaystyle~{}\times\Big{[}\partial_{\tau_{1}}^{J}G_{\mathsf{a}}(k_{1};\tau_{1})\Big{]}\Big{[}\partial_{\tau_{1}}^{K}G_{\mathsf{a}}(k_{2};\tau_{1})\Big{]}\Big{[}\partial_{\tau_{2}}^{P}G_{\mathsf{b}}(k_{3};\tau_{2})\Big{]}D_{\mathsf{a}\mathsf{b}}(k_{s};\tau_{1},\tau_{2})+(\mathbf{k}_{1}\leftrightarrow\mathbf{k}_{2}).$
(23)

To compare this with the Hankel seed integral (10), it is useful to note the
following expression for the bulk-to-boundary propagator with $J$ time
derivatives:

$\partial_{\tau}^{J}G_{\mathsf{a}}(k_{1};\tau_{1})=\displaystyle\frac{\,{1}\,}{2k^{3}}(\mathrm{i}\mathsf{a}k)^{J}(1-J-\mathrm{i}\mathsf{a}k\tau)e^{+\mathrm{i}\mathsf{a}k\tau}.$
(24)

Then it is straightforward to find the following expression:

$\displaystyle~{}\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\sigma}^{\prime}=\displaystyle\frac{\,{(-1)^{M+N+L+Q}(\mathbf{k}_{1}\cdot\mathbf{k}_{2})^{M}k_{1}^{2N+J-1}k_{2}^{2L+K-1}k_{3}^{2Q+P-R-S-6}}\,}{16k_{1}^{2}k_{2}^{2}k_{3}^{2}}\sum_{\mathsf{a},\mathsf{b}=\pm}(\mathrm{i}\mathsf{a})^{J}(\mathrm{i}\mathsf{a})^{K}(\mathrm{i}\mathsf{b})^{P}$
$\displaystyle~{}\times\bigg{[}(1-J)(1-K)(1-P)\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{R,S}(u,1)-\mathrm{i}\mathsf{a}\big{(}(1-K)\varrho_{1}+(1-J)\varrho_{2}\big{)}(1-P)\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{R+1,S}(u,1)$
$\displaystyle~{}-(1-P)\varrho_{1}\varrho_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{R+2,S}(u,1)-\mathrm{i}\mathsf{b}(1-J)(1-K)\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{R,S+1}(u,1)$
$\displaystyle~{}-\mathsf{a}\mathsf{b}\big{(}(1-K)\varrho_{1}+(1-J)\varrho_{2}\big{)}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{R+1,S+1}(u,1)-\mathrm{i}\mathsf{b}\varrho_{1}\varrho_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{R+2,S+1}(u,1)\bigg{]}+(\text{5 perms}).$ (25)

Again, this expression contains coefficients that depend on SK indices. In
this example, it is no longer possible to eliminate this dependence by acting
on differential operators on a summed seed integral, in particular because the
$u_{2}$ variable has been set to 1. Therefore it seems most convenient to
leave the SK indices unsummed in the definition of the seed integral, in order
to accommodate the most general couplings as presented here.

####  2.2.2 Spinning exchange

Next we consider the exchange of massive particles with nonzero spin $s$. The
angular momentum conservation at each of the two-point mixing vertices in Fig.
1 implies that only the longitudinal component of the spinning field can mix
with the external scalar mode, and that only the helicity-2 component of the
spinning field can mix with the external graviton. Therefore, in the following
examples, we will only consider longitudinal polarizations for the three-point
and two-point scalar correlators with spinning exchange. In the last example,
we shall also consider an example of graviton correlators, in which we will
have a helicity-2 exchange, possibly with a nonzero chemical potential.

The main purpose of the following examples is to show how to reduce the SK
integrals with spinning field into seed integrals defined with spin-0 or
spin-1 massive propagators. Therefore we will no longer consider the most
general couplings. Instead, we will focus on the problems such as how to
relate the longitudinal mode function of a spinning particle with the mode
function of a massive scalar field. We shall begin with the simplest example,
namely the spin-1 case, and then consider the general spin-$s$ field.

##### Spin-1.

The three-point function with longitudinal spin-1 exchange has been
extensively studied in [64], and here we show the main results together with
some important mid-steps for illustration. A massive spin-1 particle can be
obtained after canonically quantizing a vector field $A_{\mu}$ with mass
parameter $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$.
For this field, we choose the following simple couplings to the external
inflaton modes:

$\displaystyle\mathcal{O}_{\varphi\varphi
A}=\displaystyle\frac{\,{1}\,}{2}(-\tau)^{-1}\varphi^{\prime}(\partial_{i}\varphi)A_{i},$
$\displaystyle\mathcal{O}_{\varphi
A}=(-\tau)^{-1}(\partial_{i}\varphi^{\prime})A_{i}$ (26)

Then, the SK integral for the three-point function mediated by the
longitudinal mode of $A$ reads:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{A}^{\prime}=$
$\displaystyle~{}\bigg{\\{}\displaystyle\frac{\,{1}\,}{2}\Big{[}\mathbf{k}_{2}\cdot\bm{\epsilon}^{(L)}(\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern
2.0mu_{3})\Big{]}\Big{[}\mathbf{k}_{3}\cdot\bm{\epsilon}^{(L)*}(\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern
2.0mu_{3})\Big{]}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{-\tau_{1}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{-\tau_{2}}$
$\displaystyle~{}\times
G_{\mathsf{a}}^{\prime}(k_{1},\tau_{1})G_{\mathsf{a}}(k_{2},\tau_{1})G_{\mathsf{b}}^{\prime}(k_{3},\tau_{2})D^{(L)}_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})\bigg{\\}}+(\text{5
perms}).$ (27)

Here we have introduced the longitudinal polarization vector
$\bm{\epsilon}^{(L)}(\mkern 2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern
2.0mu)=\mkern 2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern
2.0mu\equiv\mathbf{k}/k$, as well as the longitudinal bulk propagator
$D_{\mathsf{a}\mathsf{b}}^{(L)}(k_{3};\tau_{1},\tau_{2})$. This bulk
propagator is built from the longitudinal mode function $B^{(L)}(k,\tau)$ in
(7) following the standard procedure. Now, by a careful examination of the
equation of motion of $A_{\mu}$ as well as the constraint
$\nabla^{\mu}A_{\mu}=0$, one can show that the longitudinal mode function
$B^{(L)}(k,\tau)$ can be built from a massive scalar mode function of Hankel-
type [13, 64]. Here we directly quote the result. It is convenient to define
the following differential operator $\mathcal{L}_{\tau}$:

$\displaystyle\mathcal{L}_{\tau}=\partial_{\tau}-\displaystyle\frac{\,{2}\,}{\tau}.$
(28)

Then, one can show that the longitudinal component in $A_{i}$ is related to
the temporal polarization $B^{(L)}_{0}$ in $A_{0}$ via an action of
$\mathcal{L}_{\tau}$ operator. In addition, the mode function for the temporal
polarization $B^{(L)}_{0}$ is identical to that of the massive scalar
$\sigma(k,\tau)$ up to a constant factor:

$\displaystyle
B^{(L)}(k,\tau)=\displaystyle\frac{\,{1}\,}{\mathrm{i}k}\mathcal{L}_{\tau}B_{0}^{(L)}(k,\tau),$
(29) $\displaystyle
B^{(L)}_{0}(k,\tau)=\displaystyle\frac{\,{\sqrt{\pi}k}\,}{2m}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}(-\tau)^{3/2}\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau)=\displaystyle\frac{\,{k}\,}{m}\sigma(k,\tau).$ (30)

From the above relations, we can immediately find a relation between the
longitudinal propagator $D_{\mathsf{a}\mathsf{b}}^{(L)}(k;\tau_{1},\tau_{2})$
and the massive scalar propagator
$D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$:

$\displaystyle
D_{\mathsf{a}\mathsf{b}}^{(L)}(k;\tau_{1},\tau_{2})=\displaystyle\frac{\,{1}\,}{m^{2}}\Big{[}\mathcal{L}_{\tau_{1}}\mathcal{L}_{\tau_{2}}D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})-\mathrm{i}\mathsf{a}\tau_{1}\tau_{2}\delta_{\mathsf{a}\mathsf{b}}\delta(\tau_{1}-\tau_{2})\Big{]}.$
(31)

There is a notable contact term proportional to
$\delta_{\mathsf{a}\mathsf{b}}\delta(\tau_{1}-\tau_{2})$ in this expression.
It is nonzero only for the two same-sign propagators $D_{\pm\pm}^{(L)}$. This
term arises when we commute the $\mathcal{L}_{\tau}$ operator, originally
acting on the mode function, with the step function in $D_{\pm\pm}^{(L)}$; See
(5).

Now we substitute (31) in (2.2.2), which then leads to the following SK
integral involving the massive scalar propagator
$D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{A}^{\prime}=$
$\displaystyle~{}\displaystyle\frac{\,{\mathbf{k}_{2}\cdot\mathbf{k}_{3}}\,}{2m^{2}}\bigg{\\{}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{-\tau_{1}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{-\tau_{2}}G_{\mathsf{a}}^{\prime}(k_{1};\tau_{1})G_{\mathsf{a}}(k_{2};\tau_{1})G_{\mathsf{b}}^{\prime}(k_{3};\tau_{2})\mathcal{L}_{\tau_{1}}\mathcal{L}_{\tau_{2}}D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$
$\displaystyle-\sum_{\mathsf{a}=\pm}\mathrm{i}\mathsf{a}\int{\mathrm{d}}\tau_{1}\,G_{\mathsf{a}}^{\prime}(k_{1};\tau_{1})G_{\mathsf{a}}(k_{2};\tau_{1})G_{\mathsf{a}}^{\prime}(k_{3};\tau_{1})\bigg{\\}}+(\text{5
perms}).$ (32)

The first term in the second line comes from the contact term in (31). Then,
we perform integration by parts, to move $\mathcal{L}_{\tau_{1}}$ and
$\mathcal{L}_{\tau_{2}}$ to external modes. There is no boundary term since
the integrand vanishes at both $\tau=-\infty$ (by $\mathrm{i}\epsilon$
prescription enforced by the Bunch-Davies initial condition) and $\tau=0$ (by
explicitly counting the powers of $\tau_{1,2}$ in the integrand). After this
is done, we are ready to write the above SK integral as a linear combination
of the Hankel seed integral:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{A}^{\prime}=$
$\displaystyle-\displaystyle\frac{\,{1}\,}{m^{2}}\displaystyle\frac{\,{\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern 2.0mu_{2}\cdot\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern
2.0mu_{3}}\,}{16(k_{1}k_{2}k_{3})^{2}}\bigg{\\{}\sum_{\mathsf{a},\mathsf{b}=\pm}\bigg{[}4\varrho_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{-1,-1}(u,1)-2\mathrm{i}\mathsf{b}\varrho_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{-1,0}(u,1)$
$\displaystyle-2\mathrm{i}\mathsf{a}\varrho_{1}(\varrho_{1}-2\varrho_{2})\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{0,-1}(u,1)-\mathsf{a}\mathsf{b}\varrho_{1}(\varrho_{1}-2\varrho_{2})\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{0,0}(u,1)+2\varrho_{1}\varrho_{2}\varrho_{12}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{1,-1}(u,1)$
$\displaystyle-\mathrm{i}\mathsf{b}\varrho_{1}\varrho_{2}\varrho_{12}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{1,0}(u,1)\bigg{]}+\displaystyle\frac{\,{4\varrho_{1}\varrho_{3}^{2}(\varrho_{123}+3\varrho_{2})}\,}{\varrho_{123}^{4}}\bigg{\\}}+(\text{5
perms}).$ (33)

Here again $u=2k_{3}/k_{123}$, and we define the same momentum ratios
$\varrho_{i}$ as in (2.2.1). Therefore, we see that the computation of three-
point correlators with spin-1 exchange can also be reduced to that of the
Hankel seed integral.

##### General integer spin.

The procedure described above for spin-1 field can be readily applied to
fields with general integer spin $s$, although the algebra can be quite
tedious. A spin-$s$ particle can be described by a transverse, traceless, and
totally symmetric tensor $\Psi_{\mu_{1}\cdots\mu_{s}}$ of rank-$s$. There are
again a lot of possible choices for the trilinear and bilinear vertices in
Fig. 1. We do not pursue the most general possibilities, but only illustrate
the method with the following simple example:

$\displaystyle\mathcal{O}_{\varphi\varphi\sigma}=(-\tau)^{-3+2s}\varphi^{\prime}(\partial_{i_{1}}\cdots\partial_{i_{s}}\varphi)\Psi_{i_{1}\cdots
i_{s}},$
$\displaystyle\mathcal{O}_{\varphi\sigma}=(-\tau)^{-3+2s}(\partial_{i_{1}}\cdots\partial_{i_{s}}\varphi^{\prime})\Psi_{i_{1}\cdots
i_{s}},$ (34)

where the explicit time dependence is again fixed by the scale symmetry. With
the couplings given, the SK integral for the three-point correlator with
$\Psi$-exchange reads:

$\displaystyle\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{\Psi}^{\prime}=$
$\displaystyle~{}\bigg{\\{}\displaystyle\frac{\,{(-1)^{s+1}}\,}{2}\Big{[}k_{2}^{i_{1}}\cdots
k_{2}^{i_{s}}{\epsilon}_{i_{1}\cdots i_{s}}^{(L)}({\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern
2.0mu}_{3})\Big{]}\Big{[}k_{3}^{j_{1}}\cdots
k_{3}^{j_{s}}{\epsilon}_{j_{1}\cdots j_{s}}^{(L)*}({\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern
2.0mu}_{3})\Big{]}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{(-\tau_{1})^{3-2s}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{(-\tau_{2})^{3-2s}}$
$\displaystyle~{}\times
G_{\mathsf{a}}^{\prime}(k_{1},\tau_{1})G_{\mathsf{a}}(k_{2},\tau_{1})G_{\mathsf{b}}^{\prime}(k_{3},\tau_{2})D^{(s,L)}_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})\bigg{\\}}+(\text{5
perms}).$ (35)

Here $\epsilon_{i_{1}\cdots i_{s}}^{(L)}(\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern 2.0mu)$ is the longitudinal
polarization tensor. Up to a normalization factor, $\epsilon_{i_{1}\cdots
i_{s}}^{(L)}(\mkern 2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern 2.0mu)$ is
the unique combination of $\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern 2.0mu_{i}$ and $\delta_{ij}$
with the property of being transverse, traceless, and totally symmetric with
all indices. Therefore, in the above expression, the factor
$k_{3}^{j_{1}}\cdots k_{3}^{j_{s}}{\epsilon}_{j_{1}\cdots
j_{s}}^{(L)*}({\mkern 2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern
2.0mu}_{3})$ contributes a constant factor, while $k_{2}^{i_{1}}\cdots
k_{2}^{i_{s}}{\epsilon}_{i_{1}\cdots i_{s}}^{(L)}({\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern 2.0mu}_{3})$ contributes to a
angular dependent factor $\propto\mathrm{P}_{s}(\cos\theta_{23})$ where
$\mathrm{P}_{n}(z)$ is the Legendre polynomial and $\theta_{23}$ is the angle
between $\mathbf{k}_{2}$ and $\mathbf{k}_{3}$. The factor $\mathrm{P}_{n}(z)$
is well known in CC physics as the signal of a spin-$s$ exchange. However, let
us emphasize that this factor also depends on the form of the couplings. Had
we coupled the spin-$s$ in a different way to the external mode, this factor
will change accordingly.

The propagator $D_{\mathsf{a}\mathsf{b}}^{(s,L)}$ corresponding to the
longitudinal polarization tensor $\epsilon_{i_{1}\cdots i_{s}}^{(L)}$ with $s$
spatial indices is again built from the mode function for the helicity-0
component of the spin-$s$ field with all-spatial indices $\Psi_{i_{1}\cdots
i_{s}}$. Similar to the previous spin-1 case, the mode function
$\Psi^{(s,L)}_{s}(k,\tau)$ can also be constructed from the scalar mode
function $\sigma(k,\tau)$ of the same mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ by the action of a
differential operator. For spin-$s$, this differential operator is defined
recursively. Here we quote the result, and the details can be found in, e.g.,
App. A of [13]. First, similar to spin-1 case, the temporal mode function
$\Psi^{(s,L)}_{0}(k,\tau)$ is identical to the scalar mode function up to a
constant factor:

$\displaystyle\Psi^{(s,L)}_{0}(k,\tau)=\displaystyle\frac{\,{\pi^{3/4}\text{sech}^{1/2}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2^{s/2}}\Gamma^{1/2}\begin{bmatrix}1+s\\\
\mbox{$\frac{\,{1}\,}{2}$}+s,\mbox{$\frac{\,{1}\,}{2}$}+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{bmatrix}k^{s}\sigma(k,\tau),$ (36)

where we have used the compact notation for the Euler-$\Gamma$ product, as
defined in (133). Then, the longitudinal mode function
$\Psi^{(s,L)}_{s}(k,\tau)$ with respect to the longitudinal polarization of
$s$ spatial indices is constructed recursively from the following relation:

$\displaystyle\Psi^{(s,L)}_{n}(k,\tau)=\displaystyle\frac{\,{1}\,}{\mathrm{i}k}\mathcal{L}_{\tau}\Psi^{(s,L)}_{n-1}(k,\tau)-\sqrt{\pi}\sum_{m=0}^{n-1}\Gamma\begin{bmatrix}1+n,\mbox{$\frac{\,{1+m+n}\,}{2}$}\\\
1+m,1-m+n,\mbox{$\frac{\,{1}\,}{2}$}+n,\mbox{$\frac{\,{1+m-n}\,}{2}$}\end{bmatrix}\Psi^{(s,L)}_{m}(k,\tau).$
(37)

For example, the all-spatial longitudinal mode functions,
$\Psi_{2}^{(2,L)}(k,\tau)$ for a spin-2 particle and
$\Psi_{3}^{(3,L)}(k,\tau)$ for a spin-3 particle, are respectively given by:

$\displaystyle\Psi_{2}^{(2,L)}(k,\tau)=$
$\displaystyle~{}\sqrt{\displaystyle\frac{\,{2}\,}{3(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{9}\,}{4}$})}}\bigg{(}\mathcal{L}_{\tau}^{2}-\displaystyle\frac{\,{k^{2}}\,}{3}\bigg{)}\sigma(k,\tau),$
(38) $\displaystyle\Psi_{3}^{(3,L)}(k,\tau)=$
$\displaystyle~{}\sqrt{\displaystyle\frac{\,{2}\,}{5(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{9}\,}{4}$})(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{25}\,}{4}$})}}\bigg{(}\mathcal{L}_{\tau}^{3}-\displaystyle\frac{\,{14k^{2}}\,}{15}\mathcal{L}_{\tau}\bigg{)}\sigma(k,\tau),$
(39)

where we have dropped unimportant overall phases. Then, in parallel with the
previous spin-1 example, we can move the differential operator on
$\sigma(k,\tau)$ to the external modes by using the integration by parts, and
thereby reduce the correlator of spin-$s$ exchange to a linear combination of
the Hankel seed integral.

##### Tensor and mixed correlators.

In all examples considered above, the correlators are reduced to linear
combinations of Hankel seed integral. The Whittaker seed integral has not been
used until now. The reason is that the Whittaker mode function appears only
for spinning field with helicity-dependent chemical potentials. Such chemical
potentials usually vanish for the longitudinal polarization, and therefore it
does not play a role in the scalar two-point and three-point correlators at
the tree level. (However, the chemical-potential-enhanced mode does play a
role in scalar three-point function at the one-loop level or the scalar four-
point function at the tree level.) To see the effect of the chemical potential
in tree-level three-point functions, we should consider spinning external
states, among which the graviton correlators are most relevant to realistic CC
physics. Indeed, it was shown in [48] that a helical chemical potential can be
introduced to a massive spin-2 particle $\Sigma$. When $\Sigma$ linearly mixes
with the massless graviton mode $\gamma_{\mathbf{k}}^{(\pm 2)}$, it can
mediate a mixed three-point function
$\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\gamma_{\mathbf{k}_{3}}^{(\pm
2)}\rangle^{\prime}$, described again by the left diagram of Fig. 1, but with
the right external mode replaced by $\gamma_{\mathbf{k}_{3}}$. There is also a
two-point function of graviton modes $\langle\gamma_{\mathbf{k}}^{(\pm
2)}\gamma_{-\mathbf{k}}^{(\pm 2)}\rangle^{\prime}$ contributed by the massive
spin-2 mode, similar to the right diagram of Fig. 1. The corresponding
couplings can be chosen as:

$\displaystyle\mathcal{O}_{\varphi\varphi\Sigma}=\displaystyle\frac{\,{1}\,}{2}(\partial_{i}\varphi)(\partial_{j}\varphi)\sigma_{ij},$
$\displaystyle\mathcal{O}_{\gamma\Sigma}=(-\tau)^{-1}\gamma_{ij}^{\prime}\sigma_{ij}.$
(40)

To write down the SK integral, we also need the bulk-to-boundary propagator
$T_{\mathsf{a}}(k;\tau)$ for the external graviton, as well as the mode
function $\Sigma^{(\pm 2)}(k,\tau)$ for the helicity-2 component of the
massive spin-2 field $\Sigma$. They are respectively given by:

$\displaystyle
T_{\mathsf{a}}(k,\tau)=\displaystyle\frac{\,{1}\,}{4k^{3}}(1-\mathrm{i}\mathsf{a}k\tau)e^{\mathrm{i}\mathsf{a}k\tau},$
(41) $\displaystyle\Sigma^{(\pm
2)}(k,\tau)=-\displaystyle\frac{\,{e^{\mp\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu/2}}\,}{2\sqrt{k}\tau}\mathrm{W}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(2\mathrm{i}k\tau)=-\displaystyle\frac{\,{1}\,}{\sqrt{2}\tau}B^{(\pm
1)}(k,\tau),$ (42)

where $B^{(\pm 1)}(k,\tau)$ is the mode function for the massive spin-1 field
with helicity $h=\pm 1$, given in (7). From this one can build the SK
integral, and then rewrite it as a linear combination of Whittaker seed
integrals. The details have been spelled out in [64]. Here we show the final
result:

$\displaystyle\langle\varphi(\mathbf{k}_{1})\varphi(\mathbf{k}_{2})\gamma^{(\pm
2)}(\mathbf{k}_{3})\rangle^{\prime}$ $\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern 2.0mu_{1i}\mkern
2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern 2.0mu_{2j}\epsilon_{ij}^{(\pm
2)}(\mkern 2.0mu\widehat{\mkern-2.0muk\mkern-2.0mu}\mkern
2.0mu_{3})}\,}{32(k_{1}k_{2}k_{3})^{2}}\sum_{\mathsf{a},\mathsf{b}=\pm}\Big{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(\pm)-1,-1}(u,1)+\mathrm{i}\mathsf{a}\varrho_{12}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(\pm)0,-1}(u,1)-\varrho_{1}\varrho_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(\pm)1,-1}(u,1)\Big{]},$ (43)

where, again, $u=2k_{3}/k_{123}$, and the various momentum ratios
$\varrho_{i}$ are defined in the same way as in (2.2.1). It turns out that
this correlator can also be obtained by acting an appropriate differential
operator on a fully summed SK integral. Interested readers can find more
details in [64].

It seems that the Whittaker seed integral is of limited use in representing
inflation correlators. As we shall show in a separate work [91], this seed
integral will be more useful when we treat the two-point or three-point
functions considered here as subgraphs of more complicated diagrams. In that
case, the Whittaker seed integral will be a very convenient building block for
constructing more complicated processes at the loop level.

## 3 Bootstrapping Hankel Seed Integrals

In this section, we provide the details of computing the three-point and two-
point functions with Hankel-type tree-level mediation. We will derive the
bootstrap equations satisfied by the seed integrals, first in $r$-variables,
and then in $u$-variables. We then solve the bootstrap equations and obtain
both the particular solution to the inhomogeneous equation and the general
solution to the homogeneous equation. The final answer is determined by proper
boundary conditions, which we choose to impose from the squeezed limit. Then,
we take the single and double folded limit, to get the desired result for the
seed integrals with one or both of $u_{1}$ and $u_{2}$ taken to 1. Readers
uninterested in the technical details can directly go to Sec. 3.4 for the
final results.

###  3.1 Hankel seed integral and its bootstrap equation

As mentioned above, our starting point is the Hankel seed integral
$\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},u_{2})$ defined in
(10). Here we shall first write the seed integrals as functions of
$r_{1}=k_{s}/k_{12}$ and $r_{2}=k_{s}/k_{34}$, and transform to
$(u_{1},u_{2})$ later:

$\displaystyle\mathcal{I}^{p_{1}p_{2}}_{\mathsf{a}\mathsf{b}}(r_{1},r_{2})\equiv-\mathsf{a}\mathsf{b}\,k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{a}\mathsf{b}}(k_{s};\tau_{1},\tau_{2}).$
(44)

where $D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ is the bulk propagator
for a massive scalar field, given in (5) and (6). An important and well-known
property of the Schwinger-Keldysh propagators is that they satisfy the
equation of motion for the massive scalar field with or without a
$\delta$-source:

$\displaystyle(\tau_{1}^{2}\partial_{\tau_{1}}^{2}-2\tau_{1}\partial_{\tau_{1}}+k_{s}^{2}\tau_{1}^{2}+m^{2})D_{\pm\mp}(k_{s};\tau_{1},\tau_{2})=0,$
(45)
$\displaystyle(\tau_{1}^{2}\partial_{\tau_{1}}^{2}-2\tau_{1}\partial_{\tau_{1}}+k_{s}^{2}\tau_{1}^{2}+m^{2})D_{\pm\pm}(k_{s};\tau_{1},\tau_{2})=\mp\mathrm{i}\tau_{1}^{2}\tau_{2}^{2}\delta(\tau_{1}-\tau_{2}).$
(46)

This will be the key property we shall make use of when deriving the bootstrap
equation for the Hankel seed integral.

Now, we start to derive the bootstrap equation. It turns out convenient to
redefine several variables. First, we shall use $z_{1}\equiv-k_{12}\tau_{1}$
and $z_{2}\equiv-k_{34}\tau_{2}$ instead of the conformal time $\tau_{1}$ and
$\tau_{2}$. Notice that the negative sign in the definition of $z_{1,2}$ makes
them real and positive: $z_{1,2}\in(0,+\infty)$ in the physical regime.
Second, we shall define a “hatted” massive propagator $\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}(z_{1},z_{2})$:

$\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}(z_{1},z_{2})=k^{3}D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})=\displaystyle\frac{\,{\pi
e^{-\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{4}(z_{1}z_{2})^{3/2}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(z_{1})\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}(z_{2}).$ (47)

The nice thing about the hatted propagator is that it depends on the time and
the momentum only through the combination $k\tau_{1}$ and $k\tau_{2}$, and
therefore, we can rewrite it as a function of two independent variables
$z_{1}$ and $z_{2}$. With these redefinitions, we can rewrite the Hankel seed
integral (44) as

$\displaystyle\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}(-\mathsf{a}\mathsf{b})r_{1}^{1+p_{1}}r_{2}^{1+p_{2}}\int_{0}^{\infty}{\mathrm{d}}z_{1}{\mathrm{d}}z_{2}\,z_{1}^{p_{1}}z_{2}^{p_{2}}e^{-\mathrm{i}\mathsf{a}z_{1}-\mathrm{i}\mathsf{b}z_{2}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}(r_{1}z_{1},r_{2}z_{2}).$ (48)

It now becomes manifest that the Hankel seed integral depends on external
momenta only through $r_{1}$ and $r_{2}$.

Recall that the massive propagator $D_{\mathsf{a}\mathsf{b}}$ satisfies the
equation of motion (45) and (46). This pair of equations can be rewritten in
terms of the hatted propagator as

$\displaystyle(z_{1}^{2}\partial_{z_{1}}^{2}-2z_{1}\partial_{z_{1}}+z_{1}^{2}+m^{2})\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu_{\pm\mp}(z_{1},z_{2})=0,$
(49)
$\displaystyle(z_{1}^{2}\partial_{z_{1}}^{2}-2z_{1}\partial_{z_{1}}+z_{1}^{2}+m^{2})\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}(z_{1},z_{2})=\mp\mathrm{i}z_{1}^{2}z_{2}^{2}\delta(z_{1}-z_{2}).$
(50)

For the hatted propagator $\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}(r_{1}z_{1},r_{2}z_{2})$ in (48), the two
variables $z_{1}$ and $z_{2}$ are multiplied by the momentum ratios $r_{1}$
and $r_{2}$, respectively. Therefore, for this particular propagator, we can
change the variables in the equations (49) and (50), and rewrite them as
differential equations with respect to $r_{1}$ and $r_{2}$:

$\displaystyle(r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2})\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}(r_{1}z_{1},r_{2}z_{2})=0,$ (51)
$\displaystyle(r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2})\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}(r_{1}z_{1},r_{2}z_{2})=\mp\mathrm{i}r_{1}^{2}z_{1}^{2}r_{2}^{2}z_{2}^{2}\delta(r_{1}z_{1}-r_{2}z_{2}),$
(52)

Now, we can insert the differential operator
$(r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2})$
in front of the hatted propagator in (48), which reduces the massive
propagator into 0 or a term proportional to $\delta$ function. Either way, the
integral is trivialized, and this will become the right-hand side of the final
bootstrap equation.

To derive the left-hand side of the bootstrap equation, we commute the
differential operator
$(r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2})$
with the $z$-integrals. It is clear that the only obstacle in our attempt of
pulling the differential operator to the left of the integral sign is the
$r_{1}^{2}z_{1}^{2}$ term, which depends on the integration variable $z_{1}$.
This obstacle can be removed by integration by parts. That is, for any well-
behaved function $f(rz)$, we have:

$0=\int_{0}^{\infty}{\mathrm{d}}z\,\partial_{z}\big{[}z^{p+1}e^{-\mathrm{i}\mathsf{a}z}f(rz)\big{]}=\int_{0}^{\infty}{\mathrm{d}}z\,z^{p}e^{-\mathrm{i}\mathsf{a}z}(p+1-\mathrm{i}\mathsf{a}z+r\partial_{r})f(rz),$
(53)

which then gives

$\int_{0}^{\infty}{\mathrm{d}}z\,z^{p}e^{-\mathrm{i}\mathsf{a}z}zf(rz)=-\mathrm{i}\mathsf{a}(r\partial_{r}+p+1)\int_{0}^{\infty}{\mathrm{d}}z\,z^{p}e^{-\mathrm{i}\mathsf{a}z}f(rz).$
(54)

A repetition of the same procedure then gives:

$\displaystyle\int_{0}^{\infty}{\mathrm{d}}z\,z^{p}e^{-\mathrm{i}\mathsf{a}z}z^{2}f(rz)=-(r\partial_{r}+p+2)(r\partial_{r}+p+1)\int_{0}^{\infty}{\mathrm{d}}z\,z^{p}e^{-\mathrm{i}\mathsf{a}z}f(rz),$
(55)

which shows that the $z^{2}$ term can be as well transformed into a
differential operator with respect to $r$. Then, it is straightforward to get
the following differential equations:

$\displaystyle\Big{[}(r_{1}^{2}-r_{1}^{4})\partial_{r_{1}}^{2}-\big{(}2r_{1}+(4+2p_{1})r_{1}^{3}\big{)}\partial_{r_{1}}+\big{(}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{9}\,}{4}$})-(p+1)(p+2)r_{1}^{2}\big{)}\Big{]}$
$\displaystyle\times\big{[}r_{1}^{-1-p_{1}}r_{2}^{-1-p_{2}}\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}(r_{1},r_{2})\big{]}=0,$
(56)
$\displaystyle\Big{[}(r_{1}^{2}-r_{1}^{4})\partial_{r_{1}}^{2}-\big{(}2r_{1}+(4+2p_{1})r_{1}^{3}\big{)}\partial_{r_{1}}+\big{(}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{9}\,}{4}$})-(p+1)(p+2)r_{1}^{2}\big{)}\Big{]}$
$\displaystyle\times\big{[}r_{1}^{-1-p_{1}}r_{2}^{-1-p_{2}}\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}(r_{1},r_{2})\big{]}=e^{\mp\mathrm{i}p_{12}\pi/2}\displaystyle\frac{\,{r_{1}^{4+p_{2}}r_{2}^{4+p_{1}}}\,}{(r_{1}+r_{2})^{5+p_{12}}}\Gamma(5+p_{12}).$
(57)

After slight simplifications, we arrived at the bootstrap equations for the
Hankel seed integral in terms of $r$-variables:

$\displaystyle\mathcal{D}_{r_{1}}^{p_{1}}\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}(r_{1},r_{2})=0,$
(58)
$\displaystyle\mathcal{D}_{r_{1}}^{p_{1}}\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}(r_{1},r_{2})=e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(5+p_{12})\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{r_{1}+r_{2}}\Big{)}^{5+p_{12}};$
(59)
$\displaystyle\mathcal{D}_{r}^{p}\equiv(r^{2}-r^{4})\partial_{r}^{2}-\big{[}(4+2p)r+2r^{3}\big{]}\partial_{r}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{(5+2p)^{2}}\,}{4}.$ (60)

One can well work with this set of equations and solve for the Hankel seed
integral $\mathcal{I}_{\mathsf{a}\mathsf{b}}$, as in most previous works on
this topic. However, a crucial observation made in [64] is that it is much
easier to take the three-point and two-point limits if we choose in instead to
work with the following new variables:

$u_{i}\equiv\displaystyle\frac{\,{2r_{i}}\,}{1+r_{i}},\quad i=1,2.$ (61)

Correspondingly, we rewrite the Hankel seed integral as functions of
$u_{1,2}$, namely, we define:

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},u_{2})\equiv\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}\big{(}r_{1}(u_{1}),r_{2}(u_{2})\big{)}.$
(62)

Then, the bootstrap equations for $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u_{1},u_{2})$ are:

$\displaystyle\mathcal{D}_{u_{1}}^{p_{1}}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(u_{1},u_{2})=0,$ (63)
$\displaystyle\mathcal{D}_{u_{1}}^{p_{1}}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{p_{1}p_{2}}(u_{1},u_{2})=e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(5+p_{12})\Big{(}\displaystyle\frac{\,{u_{1}u_{2}}\,}{2(u_{1}+u_{2}-u_{1}u_{2})}\Big{)}^{5+p_{12}};$
(64)
$\displaystyle\mathcal{D}_{u}^{p}\equiv(u^{2}-u^{3})\partial_{u}^{2}-\Big{[}(4+2p)u-(1+p)u^{2}\Big{]}\partial_{u}+\Big{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\big{(}p+\mbox{$\frac{\,{5}\,}{2}$}\big{)}^{2}\Big{]}.$ (65)

This completes the derivation of the bootstrap equations for the Hankel seed
integral in $u$-variables. Below we are going to solve these equations to get
the four-point function and especially its folded limit.

###  3.2 Solving the bootstrap equation

Now we start to solve the bootstrap equations (63) and (64) for the Hankel
seed integrals.

For the homogeneous equation (63), the solution is a proper linear combination
of two independent solutions:

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(u_{1},u_{2})=\sum_{\mathsf{a},\mathsf{b}=\pm}\alpha_{\pm\mp|\mathsf{a}\mathsf{b}}\mathcal{Y}_{\mathsf{a}}^{p_{1}}(u_{1})\mathcal{Y}_{\mathsf{b}}^{p_{2}}(u_{2}).$
(66)

Here $\mathcal{Y}_{\pm}^{p}(u)$ is a pair of linearly independent solutions to
the homogeneous equation (63), which we choose to be:

$\mathcal{Y}_{\pm}^{p_{1}}(u_{1})=2^{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}\displaystyle\frac{\,{u_{1}}\,}{2}\Big{)}^{5/2+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma\Big{[}\displaystyle\frac{\,{5}\,}{2}+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\ 1\pm
2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|u_{1}\right].$ (67)

Here we have included some numerical factors in
$\mathcal{Y}_{\pm}^{p_{1}}(u_{1})$ for later convenience. Note that we have
used the fact that the seed integral $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(u_{1},u_{2})$ has a symmetry with respect to the
exchange $u_{1}\leftrightarrow u_{2}$, and that the seed integrals satisfy an
identical set of bootstrap equations with respect to $u_{2}$. Therefore, we
expect that the final result for $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(u_{1},u_{2})$ should be bilinear in
$\mathcal{Y}_{\mathsf{a}}^{p_{1}}(u_{1})$ and
$\mathcal{Y}_{\mathsf{b}}^{p_{2}}(u_{2})$. The coefficients
$\alpha_{\pm\mp|\mathsf{a}\mathsf{b}}$ should be determined by imposing a
proper boundary condition. In this work, we impose the boundary condition by
matching the result to an explicit calculation in the squeezed limit $u_{1}\ll
u_{2}\ll 1$, as will be elaborated below.

For the inhomogeneous equation (64), the solution is the sum of two parts: one
particular solution $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{p_{1}p_{2}}(u_{1},u_{2})$ with the presence of the
source term, and one homogeneous solution which is again a linear combination
of two independent solutions to the homogeneous equation:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{p_{1}p_{2}}(u_{1},u_{2})=\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{p_{1}p_{2}}(u_{1},u_{2})+\sum_{\mathsf{a},\mathsf{b}=\pm}\alpha_{\pm\pm|\mathsf{a}\mathsf{b}}\mathcal{Y}_{\mathsf{a}}^{p_{1}}(u_{1})\mathcal{Y}_{\mathsf{b}}^{p_{2}}(u_{2}).$
(68)

The coefficients $\alpha_{\pm\pm;\mathsf{a}\mathsf{b}}$ are again determined
by matching the result to an explicit calculation in the squeezed limit.

Below, we shall first work out the particular solution $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{p_{1}p_{2}}(u_{1},u_{2})$, and then construct the
homogeneous solution. To find the particular solution, we do not work with
arbitrary momentum configuration; Instead, we shall directly work with the
three-point limit by taking $u_{2}\rightarrow 1$, since we are concerned only
with the three-point and two-point functions in this work, and the single
folded limit $u_{2}\rightarrow 1$ is regular for the particular solution. The
general four-point correlator for arbitrary momentum configuration can be
calculated following the same procedure in our previous work [64].

On the other hand, to find the homogeneous solution, we still work with the
4-point function, and impose the boundary conditions in the squeezed limit
$u_{1,2}\rightarrow 0$. It turns out that this limit is enough to fix all the
coefficients in the homogeneous solution. Only after this is done, we take the
single folded limit $u_{2}\rightarrow 1$ to get the result for the three-point
function. The reason that we work with the four-point function for the
homogeneous solution is that the single folded limit is singular for each term
in the homogeneous solution, and the regular three-point limit is achieved
only when we take the “correct” linear combination of all these terms.

##### Particular solution.

As said, we solve the inhomogeneous equation (64) in the single folded limit
$u_{2}=1$. Also writing $u_{1}=u$, the equation becomes:

$\mathcal{D}_{u}^{p_{1}}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{p_{1}p_{2}}(u,1)=e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(5+p_{12})\Big{(}\displaystyle\frac{\,{u}\,}{2}\Big{)}^{5+p_{12}}.$
(69)

Note that $\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{++,\text{Inh}}^{p_{1}p_{2}}$ and $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{--,\text{Inh}}^{p_{1}p_{2}}$ satisfy the equation with the same
differential operator on the left-hand side but with the source term differed
by a phase.

We use the following Taylor series as an ansatz for the solution:

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{p_{1}p_{2}}(u,1)=2^{-5-p_{12}}e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(5+p_{12})\sum_{n=0}^{\infty}\mathcal{X}_{n}u^{5+p_{12}+n}.$
(70)

Then, the equation (69) gives rise to the following recursion relation for the
Taylor coefficients $\mathcal{X}_{n}$:

$\mathcal{X}_{0}=\displaystyle\frac{\,{1}\,}{(\mbox{$\frac{\,{5}\,}{2}$}+p_{2})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}},\qquad\mathcal{X}_{n+1}=\displaystyle\frac{\,{(n+3+p_{2})(n+5+p_{12})}\,}{(n+\mbox{$\frac{\,{7}\,}{2}$}+p_{2})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}}\mathcal{X}_{n}.$
(71)

Substituting this back into the ansatz (70), we see that the summation is a
standard (generalized) hypergeometric series, which can be directly finished,
and we get the following result for the particular solution:

$\mathcal{X}_{n}=\displaystyle\frac{\,{(3+p_{2})_{n}(5+p_{12})_{n}}\,}{(p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n+1}(p_{2}+\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{n+1}},$ (72)

where $(a)_{n}\equiv\Gamma(a+n)/\Gamma(a)$ is the Pochhammer symbol. Thus

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{p_{1}p_{2}}(u,1)=\displaystyle\frac{\,{e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(5+p_{12})u^{5+p_{12}}}\,}{2^{5+p_{12}}[(\mbox{$\frac{\,{5}\,}{2}$}+p_{2})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}]}{}_{3}\mathrm{F}_{2}\left[\begin{matrix}1,3+p_{2},5+p_{12}\\\
\mbox{$\frac{\,{7}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{7}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|u\right].$ (73)

This completes our derivation for the particular solution to the bootstrap
equation.

##### Homogeneous solutions.

As mentioned above, the problem of finding the homogeneous solution boils down
to a determination of coefficients $\alpha_{\pm\mp|\mathsf{a}\mathsf{b}}$ and
$\alpha_{\pm\pm|\mathsf{a}\mathsf{b}}$, as given in (66) and (68).

We impose the boundary condition in the hierarchical squeezed limit $u_{1}\ll
u_{2}\ll 1$. In this limit, the tree-seed integral can be directly done. We
put this calculation in App. B.1, and here we quote the result:

$\displaystyle\lim_{u_{1}\ll u_{2}\ll 1}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{p_{1}p_{2}}_{\pm\mp}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{\mp\mathrm{i}\bar{p}_{12}\pi/2}}\,}{4\pi}\Big{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{+}^{p_{1}}(u_{1})+\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{-}^{p_{1}}(u_{1})\Big{]}\Big{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{+}^{p_{2}}(u_{2})+\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{-}^{p_{2}}(u_{2})\Big{]},$ (74) $\displaystyle\lim_{u_{1}\ll u_{2}\ll
1}\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{p_{1}p_{2}}_{\pm\pm}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{\pm\mathrm{i}e^{\mp\mathrm{i}p_{12}\pi/2}}\,}{4\pi}\Big{[}e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{\pm}^{p_{1}}(u_{1})+e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{\mp}^{p_{1}}(u_{1})\Big{]}\Big{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{+}^{p_{2}}(u_{2})+\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{-}^{p_{2}}(u_{2})\Big{]}.$ (75)

where $\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{\pm}^{p}(u)$ is the two independent solutions
$\mathcal{Y}_{\pm}^{p}(u)$ given in (67) in the squeezed limit $u\rightarrow
0$:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{\pm}^{p}(u)=2^{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}\displaystyle\frac{\,{u}\,}{2}\Big{)}^{5/2+p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma\Big{[}\displaystyle\frac{\,{5}\,}{2}+p+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}.$ (76)

Given the form of the squeezed-limit solution, we do not have to write down
all the coefficients $\alpha_{\mathsf{cd}|\mathsf{a}\mathsf{b}}$ explicitly.
Rather, we only need to make the replacement $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{Y}\mkern-2.0mu}\mkern
2.0mu_{\pm}^{p}(u)\rightarrow\mathcal{Y}_{\pm}^{p}(u)$ in (74) and (75), and
the results are guaranteed to be the correct homogeneous solutions:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{p_{1}p_{2}}_{\pm\mp}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{\mp\mathrm{i}\bar{p}_{12}\pi/2}}\,}{4\pi}\Big{[}{\mathcal{Y}}_{+}^{p_{1}}(u_{1})+{\mathcal{Y}}_{-}^{p_{1}}(u_{1})\Big{]}\Big{[}{\mathcal{Y}}_{+}^{p_{2}}(u_{2})+{\mathcal{Y}}_{-}^{p_{2}}(u_{2})\Big{]},$
(77) $\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{p_{1}p_{2}}_{\pm\pm}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{\pm\mathrm{i}e^{\mp\mathrm{i}p_{12}\pi/2}}\,}{4\pi}\Big{[}e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathcal{Y}}_{\pm}^{p_{1}}(u_{1})+e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathcal{Y}}_{\mp}^{p_{1}}(u_{1})\Big{]}\Big{[}{\mathcal{Y}}_{+}^{p_{2}}(u_{2})+{\mathcal{Y}}_{-}^{p_{2}}(u_{2})\Big{]}.$
(78)

At this point, an important observation is that the squeezed-limit results
(74) and (75) scale with the momentum ratio $u_{1}$ as
$u_{1}^{5/2+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, which is identical
to the scaling of the general homogeneous solution (67). On the contrary, the
particular solution (73) scales as $u_{1}^{5+p_{12}}$ in the squeezed limit
$u_{1}\ll 1$. [Note that $u_{1}=u$ in (73).] Therefore, for positive real
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$, as long as
$p_{2}>-5/2$, the particular solution (73) is always subdominant, and the
general solutions in (66) and (68) can be matched to the squeezed-limit result
(74) and (75), respectively. On the other hand, when $p_{2}\leq-5/2$, the
particular solution will dominate the squeezed limit, and there is no way to
match the squeezed-limit result. In fact, the Hankel seed integral
$\mathcal{I}^{p_{1}p_{2}}$ is divergent when $p_{1,2}\leq-5/2$, and the
divergence does not cancel out among different SK branches. Physically, the
powers $p_{1,2}\leq-5/2$ appear because of too nonlocal interactions, due to
which the perturbation theory actually breaks down. Therefore, we shall always
assume $p_{1,2}>-5/2$ in this work when $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ is positive real.

###  3.3 Folded limit

##### Single folded limit.

Now we consider the single folded limit $u_{2}\rightarrow 1$. The general
solution (67) can become singular in this limit through the hypergeometric
factor ${}_{2}\text{F}_{1}$, and the singular terms must cancel themselves in
(77) and (78), so that only the finite part of the hypergeometric function
contributes to the folded limit. Therefore, we only keep the finite part of
${}_{2}\text{F}_{1}$ function. In the folded limit $u\rightarrow 1$,

$\text{Fin}\bigg{\\{}\lim_{u\rightarrow
1}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}a,b\\\
c\end{matrix}\middle|u\right]\bigg{\\}}=\Gamma\left[\begin{matrix}c,s\\\
a+s,b+s\end{matrix}\right],~{}~{}~{}~{}~{}~{}(s\equiv c-a-b\notin\mathbb{Z})$
(79)

Here $\text{Fin}\\{\cdots\\}$ means the finite part of the limit. As
indicated, this formula holds only when the balance $s=c-a-b$ is not an
integer. The case of $s\in\mathbb{Z}$ can be dealt with by taking the limit.
Then, the finite part of the general solution $\mathcal{Y}_{\pm}^{p}(u)$ in
(67) as $u\rightarrow 1$ is:

$\text{Fin}\Big{\\{}\lim_{u\rightarrow
1}\mathcal{Y}_{\pm}^{p}(u)\Big{\\}}=\pm\mathrm{i}2^{-5/2-p}\pi^{1/2}\text{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma\left[\begin{matrix}-2-p,\mbox{$\frac{\,{5}\,}{2}$}+p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
-\mbox{$\frac{\,{3}\,}{2}$}-p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\end{matrix}\right],$
(80)

Now, all the general solutions $\mathcal{Y}_{\pm}^{p_{2}}(u_{2})$ in (77) and
(78) come in the combination
$\mathcal{Y}_{+}^{p_{2}}(u_{2})+\mathcal{Y}_{-}^{p_{2}}(u_{2})$. It can be
shown that the divergences in the limit $u_{2}\rightarrow 1$ cancel out in
this combination, and we have the following regular limit:

$\lim_{u\rightarrow
1}\Big{[}\mathcal{Y}_{+}^{p}(u)+\mathcal{Y}_{-}^{p}(u)\Big{]}=2^{-3/2-p}\pi^{1/2}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p\end{matrix}\right].$ (81)

Substituting this result back into (77) and (78), we find the correct
homogeneous solution in the single folded limit. The explicit results are
summarized in (3.4) and (90) at the end of this section.

##### Double-folded limit.

Now we consider the double folded limit, in order to get an expression for the
two-point correlator. The double folded limit is obtained by further taking
$u\rightarrow 1$ in $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u,1)$. Remarkably, all of the four
seed integral $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u,1)$ are individually regular in
the limit of $u\rightarrow 1$. First, the regularity of $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(u,1)$ is again due to the combination
${\mathcal{Y}}_{+}^{p_{1}}(u)+{\mathcal{Y}}_{-}^{p_{1}}(u)$, which we have
shown to be regular in the limit of $u\rightarrow 1$ in the previous
subsection. In fact, by direct substitution of (81) in $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(u,1)$ in (90), we immediately get:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(1,1)=$
$\displaystyle~{}\displaystyle\frac{\,{e^{\mp\mathrm{i}\bar{p}_{12}\pi/2}}\,}{2^{5+p_{12}}}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p_{1},3+p_{2}\end{matrix}\right].$ (82)

On the other hand, in $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{p_{1}p_{2}}(u,1)$ in (3.4), the combination $e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathcal{Y}}_{+}^{p_{1}}(u)+e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathcal{Y}}_{-}^{p_{1}}(u)$ is divergent when $u\rightarrow 1$, and
this divergence is to be canceled by another divergent piece from the
${}_{3}\text{F}_{2}$ function in (3.4), as a consequence of the Bunch-Davies
initial condition for all the mode functions. Therefore, we shall only retain
the finite part of each of these two pieces, and add them together to get the
result in the double folded limit.

The finite part of the combination $e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathcal{Y}}_{+}^{p_{1}}(u)+e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathcal{Y}}_{-}^{p_{1}}(u)$ can again be obtained by using the formula
in (79). To obtain the finite part of the ${}_{3}\text{F}_{2}$ function, we
use the following formula [92]:

$\text{Fin}\bigg{\\{}\lim_{u\rightarrow
1}{}_{3}\mathrm{F}_{2}\left[\begin{matrix}a,b,c\\\
d,e\end{matrix}\middle|u\right]\bigg{\\}}=\Gamma\left[\begin{matrix}d,e,s\\\
c,a+s,b+s\end{matrix}\right]{}_{3}\mathrm{F}_{2}\left[\begin{matrix}d-c,e-c,s\\\
a+s,b+s\end{matrix}\middle|1\right],$ (83)

where $s\equiv d+e-a-b-c$ is the balance of the hypergeometric function. Once
again, the above formula holds when $s$ is not an integer, and the integer
case can be dealt with by taking the limit. Therefore, for the
${}_{3}\text{F}_{2}$ function in the inhomogeneous solution (73), we can make
the following assignment of parameters,

$a=5+p_{12},\quad b=3+p_{2},\quad c=1,\quad
d=\displaystyle\frac{\,{7}\,}{2}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,\quad
e=\displaystyle\frac{\,{7}\,}{2}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,\quad s=-2-p_{1}.$
(84)

Combined with the finite part of the homogeneous solution as shown in (68), we
obtain the double-folded limit of the $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{p_{1}p_{2}}$ as follows:

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

There are a large number of relations among hypergeometric functions of
argument unity. With these relations we can recast the result into a somewhat
simpler form. For example, we can use the following relation (Eq. 4.3.4.4 of
[93]):

$\displaystyle~{}{}_{3}\mathrm{F}_{2}\left[\begin{matrix}a,b,d+e-a-b-1\\\
d,e\end{matrix}\middle|1\right]=\Gamma\left[\begin{matrix}d,e,d-a-b,e-a-b\\\
d-a,d-b,e-a,e-b\end{matrix}\right]$
$\displaystyle+\displaystyle\frac{\,{1}\,}{a+b-d}\Gamma\left[\begin{matrix}d,e\\\
a,b,d+e-a-b\end{matrix}\right]{}_{3}\mathrm{F}_{2}\left[\begin{matrix}d-a,d-b,1\\\
1+d-a-b,d+e-a-b\end{matrix}\middle|1\right].$ (86)

Then, with the following assignment of parameters:

$a=\displaystyle\frac{\,{5}\,}{2}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,\quad
b=-2-p_{1},\quad d=3+p_{2},\quad e=1-p_{1}+p_{2},$ (87)

we obtain:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{p_{1}p_{2}}(1,1)=\displaystyle\frac{\,{\pm\mathrm{i}e^{\mp\mathrm{i}{p}_{12}\pi/2}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{2^{5+p_{12}}}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p_{1},3+p_{2}\end{matrix}\right]$
$\displaystyle-\displaystyle\frac{\,{e^{\mp\mathrm{i}p_{12}\pi/2}}\,}{2^{5+p_{12}}}\Gamma\Big{[}5+p_{12},\mbox{$\frac{\,{5}\,}{2}$}+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}{}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{2}\left[\begin{matrix}5+p_{12},\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,1\\\
\mbox{$\frac{\,{7}\,}{2}$}+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{7}\,}{2}$}+p_{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|1\right].$ (88)

This result has the advantage that it is manifestly symmetric in $p_{1}$ and
$p_{2}$, and is regular when $p_{1}$ and $p_{2}$ take generic integer values.

###  3.4 Summary

At this point, we have finished the computation of the Hankel seed integral
$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(u,1)$ in the single folded limit. We
present the full result below.

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{p_{1}p_{2}}(u,1)=$
$\displaystyle~{}\displaystyle\frac{\,{\pm\mathrm{i}e^{\mp\mathrm{i}{p}_{12}\pi/2}}\,}{2^{7/2+p_{2}}\pi^{1/2}}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p_{2}\end{matrix}\right]\Big{[}e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathcal{Y}}_{\pm}^{p_{1}}(u)+e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathcal{Y}}_{\mp}^{p_{1}}(u)\Big{]}$
$\displaystyle+\displaystyle\frac{\,{e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(5+p_{12})u^{5+p_{12}}}\,}{2^{5+p_{12}}[(\mbox{$\frac{\,{5}\,}{2}$}+p_{2})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}]}{}_{3}\mathrm{F}_{2}\left[\begin{matrix}1,3+p_{2},5+p_{12}\\\
\mbox{$\frac{\,{7}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{7}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|u\right],$ (89) $\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(u,1)=$
$\displaystyle~{}\displaystyle\frac{\,{e^{\mp\mathrm{i}\bar{p}_{12}\pi/2}}\,}{2^{7/2+p_{2}}\pi^{1/2}}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p_{2}\end{matrix}\right]\Big{[}{\mathcal{Y}}_{+}^{p_{1}}(u)+{\mathcal{Y}}_{-}^{p_{1}}(u)\Big{]},$
(90)

where $\mathcal{Y}_{\pm}^{p}(u)$ is a pair of independent solutions to the
homogeneous bootstrap equation:

$\mathcal{Y}_{\pm}^{p}(u)=2^{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}\displaystyle\frac{\,{u}\,}{2}\Big{)}^{5/2+p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma\Big{[}\displaystyle\frac{\,{5}\,}{2}+p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\ 1\pm
2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|u\right].$ (91)

With these results, one can easily generate explicit analytical expressions
for a large class of three-point correlators, as discussed in Sec. 2.

The result for the two-point function can be read from the Hankel seed
integral in the double folded limit. The results are:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{p_{1}p_{2}}(1,1)=\displaystyle\frac{\,{\pm\mathrm{i}e^{\mp\mathrm{i}{p}_{12}\pi/2}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{2^{5+p_{12}}}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p_{1},3+p_{2}\end{matrix}\right]$
$\displaystyle-\displaystyle\frac{\,{e^{\mp\mathrm{i}p_{12}\pi/2}}\,}{2^{5+p_{12}}}\Gamma\Big{[}5+p_{12},\mbox{$\frac{\,{5}\,}{2}$}+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}{}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{2}\left[\begin{matrix}5+p_{12},\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,1\\\
\mbox{$\frac{\,{7}\,}{2}$}+p_{1}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{7}\,}{2}$}+p_{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|1\right],$ (92) $\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{p_{1}p_{2}}(1,1)=\displaystyle\frac{\,{e^{\mp\mathrm{i}\bar{p}_{12}\pi/2}}\,}{2^{5+p_{12}}}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{2}$}+p_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3+p_{1},3+p_{2}\end{matrix}\right].$ (93)

## 4 Bootstrapping Whittaker Seed Integrals

Now we consider the case where the intermediate massive particle has a boost-
breaking chemical potential. This scenario is particularly interesting for CC
phenomenology due to its exponentially enhanced signal and also its parity-
violating nature. Mathematically, the mode functions for such fields involve
the Whittaker W function, which needs a separate treatment from the previously
considered Hankel type. In this section, we shall consider the Whittaker case
and go through the same procedure as the last section. We shall be briefer in
this section since most of the intermediate steps are in parallel with the
previous section.

###  4.1 Whittaker seed integral and its bootstrap equation

As introduced in Sec. 2, the Whittaker seed integral (11) is defined based on
the SK integral for the tree-level four-point correlator with the exchange of
a spin-1 particle of mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ and helicity-
dependent chemical potential $\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu$. In parallel with
the previous section, we first derive the bootstrap equations for this seed
integral with $r$-variables, and then change to $u$-variables. Thus we rewrite
the Whittaker seed integral (11) in the following way:

$\displaystyle\mathcal{I}^{(h)p_{1}p_{2}}_{\mathsf{a}\mathsf{b}}(r_{1},r_{2})=-\mathsf{a}\mathsf{b}\,r_{1}^{1+p_{1}}r_{2}^{1+p_{2}}\int_{0}^{\infty}{\mathrm{d}}z_{1}{\mathrm{d}}z_{2}\,z_{1}^{p_{1}}z_{2}^{p_{2}}e^{-\mathrm{i}\mathsf{a}z_{1}-\mathrm{i}\mathsf{b}z_{2}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(h)}(r_{1}z_{1},r_{2}z_{2}).$ (94)

In the seed integral above, $\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(h)}(z_{1},z_{2})$ is again a hatted propagator,
this time built from the helicity-$h$ component ($h=\pm 1$) of the massive
spin-1 propagator $D_{\mathsf{a}\mathsf{b}}^{(h)}(k;\tau_{1},\tau_{2})$:

$\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(h)}(z_{1},z_{2})=kD_{\mathsf{a}\mathsf{b}}^{(h)}(k;\tau_{1},\tau_{2}).$
(95)

Again, we use the dimensionless and positive variables $z_{i}=-k\tau_{i}$
$(i=1,2)$. Here the four Schwinger-Keldysh propagators
$D_{\mathsf{a}\mathsf{b}}^{(h)}(k;\tau_{1},\tau_{2})$ are again related to the
Wightman functions $D_{>}^{(h)}$ and $D_{<}^{(h)}=D_{>}^{(h)*}$, and the
“greater” Wightman function is given by

$\displaystyle D^{(h)}_{>}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-\pi h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2k}\mathrm{W}_{\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(2\mathrm{i}k\tau_{1})\mathrm{W}_{-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-2\mathrm{i}k\tau_{2}).$ (96)

Let us emphasize that, although we are building the Whittaker seed integral
from the spin-1 propagators, this seed integral can be used to compute
correlators involving higher spin states. Very often, the boost-breaking
chemical potential changes the dispersion of a spin-$s$ field in such a way
that its highest or lowest helicity component ($h=\pm s$) is most enhanced.
For such states, the mode function is identical to the $h=\pm 1$ states of the
spin-1 field up to a prefactor, and thus above seed integral is perfectly
applicable to these cases. One only needs to be careful that the mass
parameter $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ is
related to the mass $m$ of a spin-$s$ state via $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu=\sqrt{m^{2}-(s-1/2)^{2}}$.

The equations of motion satisfied by the propagator
$D_{\mathsf{a}\mathsf{b}}^{(h)}(k;\tau_{1},\tau_{2})$ then lead to the
following set of equations satisfied by the hatted propagators:

$\displaystyle(r_{1}^{2}\partial_{r_{1}}^{2}+r_{1}^{2}z_{1}^{2}+2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mur_{1}z_{1}+m^{2})\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{(h)}(r_{1}z_{1},r_{2}z_{2})=0,$ (97)
$\displaystyle(r_{1}^{2}\partial_{r_{1}}^{2}+r_{1}^{2}z_{1}^{2}+2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mur_{1}z_{1}+m^{2})\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{(h)}(r_{1}z_{1},r_{2}z_{2})=\mp\mathrm{i}r_{1}r_{2}z_{1}z_{2}\delta(r_{1}z_{1}-r_{2}z_{2}).$
(98)

Therefore, we insert the differential operator
$r_{1}^{2}\partial_{r_{1}}^{2}+r_{1}^{2}z_{1}^{2}+2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mur_{1}z_{1}+m^{2}$ in
front of the hatted propagator in the Whittaker seed integral (94), to derive
the bootstrap equation. On one hand, the differential operator reduces the
integral to either 0 or a local term, which becomes the right-hand side of the
bootstrap equation. On the other hand, we commute the differential operator
with the integrals over $z_{1,2}$ with the help of (54) and (55), and end up
with a new differential operator acting on the whole seed integral, which is
the left-hand side of the bootstrap equation. The result is:

$\displaystyle\mathcal{D}_{\pm,r_{1}}^{p_{1}}\mathcal{I}_{\pm\mp}^{(h)}(r_{1},r_{2})=0,$
(99)
$\displaystyle\mathcal{D}_{\pm,r_{1}}^{p_{1}}\mathcal{I}_{\pm\pm}^{(h)}(r_{1},r_{2})=-e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(3+p_{12})\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{r_{1}+r_{2}}\Big{)}^{3+p_{12}};$
(100)
$\displaystyle\mathcal{D}_{\pm,r}^{p}\equiv(r^{2}-r^{4})\partial_{r}^{2}-\Big{[}2(1+p)r\pm
2\mathrm{i}h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mur^{2}+2r^{3}\Big{]}\partial_{r}+\Big{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{(3+2p)^{2}}\,}{4}$}\Big{]}.$ (101)

Next we use the new variables $u_{i}=(2r_{i})/(1+r_{i})$ in place of $r_{i}$
$(i=1,2)$, in terms of which the seed integral is rewritten as

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}^{(h)p_{1}p_{2}}(u_{1},u_{2})\equiv\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(h)p_{1}p_{2}}\big{(}r_{1}(u_{1}),r_{2}(u_{2})\big{)}$
(102)

Then we can derive the following set of equations with respect to the
$u$-variable:

$\displaystyle\mathcal{D}_{\pm,u_{1}}^{p_{1}}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{(h)}(u_{1},u_{2})=0,$ (103)
$\displaystyle\mathcal{D}_{\pm,u_{1}}^{p_{1}}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{(h)}(u_{1},u_{2})=-e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(3+p_{12})\Big{(}\displaystyle\frac{\,{u_{1}u_{2}}\,}{2(u_{1}+u_{2}-u_{1}u_{2})}\Big{)}^{3+p_{12}};$
(104)
$\displaystyle\mathcal{D}_{\pm,u}^{p}=(u^{2}-u^{3})\partial_{u}^{2}-\Big{[}(2+2p)u+(\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu-p)u^{2}\Big{]}\partial_{u}+\Big{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\big{(}p+\mbox{$\frac{\,{3}\,}{2}$}\big{)}^{2}\Big{]}.$ (105)

These are the bootstrap equations for the Whittaker seed integral, from which
we shall bootstrap the three-point and two-point functions in the following
subsections.

###  4.2 Solving the bootstrap equation

In parallel with the procedure in the last section, we now solve the bootstrap
equations (103) and (104). Again, the solution to the homogeneous equation
(103) can be written as a linear combination of the two independent solutions

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{(h)p_{1}p_{2}}(u_{1},u_{2})=\sum_{\mathsf{a},\mathsf{b}=\pm}\beta_{\pm\mp|\mathsf{a}\mathsf{b}}\,\mathcal{U}_{\pm|\mathsf{a}}^{p_{1}}(u_{1})\,\mathcal{U}_{\mp|\mathsf{b}}^{p_{2}}(u_{2}),$
(106)

with $\mathcal{U}^{p}_{\mathsf{a}|\mathsf{b}}(u)$ ($\mathsf{b}=\pm 1$) the two
independent solutions to the following equations:

$\displaystyle\mathcal{D}_{\mathsf{a},u}^{p}\mathcal{U}^{p}_{\mathsf{a}|\mathsf{b}}(u)=0.$
(107)

Here $\mathcal{D}_{\mathsf{a},u}^{p}$ are the differential operators in (105).
Note that we have two distinct operators $\mathcal{D}_{\mathsf{a},u}^{p}$,
labeled by $\mathsf{a}=\pm$. For each fixed choice of $\mathsf{a}=\pm$, there
are a pair of independent solutions, labeled by $\mathsf{b}=\pm$, hence the
notation $\mathcal{U}^{p}_{\mathsf{a}|\mathsf{b}}$ for the solutions.
Explicitly, we can choose $\mathcal{U}^{p}_{\mathsf{a}|\mathsf{b}}$ to be:

$\mathcal{U}_{\mathsf{a}|\mathsf{b}}^{p}(u)=\mathrm{i}\,\mathsf{a}\mathsf{b}\,2^{\mathrm{i}\mathsf{a}\mathsf{b}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\pi\text{csch}(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{(}\displaystyle\frac{\,{u}\,}{2}\Big{)}^{3/2+p+\mathrm{i}\mathsf{a}\mathsf{b}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{}_{2}\mathcal{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{3}\,}{2}$}+p+\mathrm{i}\mathsf{a}\mathsf{b}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mathsf{a}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu+\mathrm{i}\mathsf{a}\mathsf{b}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+2\mathrm{i}\mathsf{a}\mathsf{b}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|u\right].$ (108)

Here, again, we have included some numerical factors for later convenience.
The coefficients $\beta_{\pm\mp|\mathsf{a}\mathsf{b}}$ in (106) should be
determined by the boundary conditions in the squeezed limit, as will be
detailed below.

For the inhomogeneous equation (104), the solution is a sum of a particular
solution $\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{(h)p_{1}p_{2}}(u_{1},u_{2})$ and a homogeneous
solution which is again a linear combination of independent solutions:

$\displaystyle\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{(h)p_{1}p_{2}}(u_{1},u_{2})=\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{(h)p_{1}p_{2}}(u_{1},u_{2})+\sum_{\mathsf{a},\mathsf{b}=\pm}\beta_{\pm\pm|\mathsf{a}\mathsf{b}}\,\mathcal{U}^{p_{1}}_{\pm|\mathsf{a}}(u_{1})\,\mathcal{U}^{p_{2}}_{\pm|\mathsf{b}}(u_{2}).$
(109)

Below we will first find the particular solution $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm,\text{Inh}}^{(h)p_{1}p_{2}}(u_{1},u_{2})$, and then determine
the homogeneous solutions by matching the results in the squeezed limit.

##### Particular solution.

Similar to the previous section, we do not pursue the most general result for
the particular solution. Rather, we shall set $u_{2}=1$ and write $u_{1}=u$ in
the inhomogeneous equation (104):

$\displaystyle\mathcal{D}_{\pm,u}^{p_{1}}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{(h)p_{1}p_{2}}_{\pm\pm,\text{Inh}}(u,1)=-e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(3+p_{12})\Big{(}\displaystyle\frac{\,{u}\,}{2}\Big{)}^{3+p_{12}}.$
(110)

This equation can again be solved by trying the following series ansatz:

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{(h)p_{1}p_{2}}_{\pm\pm,\text{Inh}}(u,1)=-2^{-3-p_{12}}e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(3+p_{12})\sum_{n=0}^{\infty}\mathcal{V}_{n,\pm}u^{3+p_{12}+n}.$
(111)

The differential equation then leads to the following recursion relations for
the series coefficients:

$\mathcal{V}_{0,\pm}=\displaystyle\frac{\,{1}\,}{(\mbox{$\frac{\,{3}\,}{2}$}+p_{2})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}},\qquad\mathcal{V}_{n+1,\pm}=\displaystyle\frac{\,{(n+2+p_{2}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)(n+3+p_{12})}\,}{(n+\mbox{$\frac{\,{5}\,}{2}$}+p_{2})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}\mathcal{V}_{n,\pm},$ (112)

and the recursion relation can be directly solved to get a general term
formula:

$\mathcal{V}_{n,\pm}=\displaystyle\frac{\,{(2+p_{2}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)_{n}(3+p_{12})_{n}}\,}{(p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n+1}(p_{2}+\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{n+1}}.$ (113)

This shows that the series ansatz is again a standard hypergeometric series.
Finishing the summation, we get:

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu^{(h)p_{1}p_{2}}_{\pm\pm,\text{Inh}}(u,1)=-\displaystyle\frac{\,{e^{\mp\mathrm{i}p_{12}\pi/2}\Gamma(3+p_{12})u^{3+p_{12}}}\,}{2^{3+p_{12}}[(\mbox{$\frac{\,{3}\,}{2}$}+p_{2})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}]}{}_{3}\mathrm{F}_{2}\left[\begin{matrix}1,2+p_{2}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,3+p_{12}\\\
\mbox{$\frac{\,{5}\,}{2}$}+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|u\right].$ (114)

This completes our computation of the particular solution.

##### Homogeneous solutions.

To determine the homogeneous solutions in both $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}^{(h)p_{1}p_{2}}(u_{1},u_{2})$ and $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{I}\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}^{(h)p_{1}p_{2}}(u_{1},u_{2})$, we again compute these integrals
in the squeezed limit $u_{1}\ll u_{2}\ll 1$. Some details are given in App.
B.2, and the results are shown in (158). By matching the results in the
squeezed limit, we can determine the coefficients
$\beta_{\mathsf{a}\mathsf{b}|\mathsf{c}\mathsf{d}}$:

$\displaystyle\beta_{\pm\mp|++}=\beta_{\pm\mp|--}=\beta_{\pm\mp|+-}=\beta_{\pm\mp|-+}=\displaystyle\frac{\,{e^{\mp\mathrm{i}\bar{p}_{12}\pi/2}e^{-\pi
h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2\pi^{2}}(\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu),$ (115)
$\displaystyle\beta_{\pm\pm|++}=\beta_{\pm\pm|+-}=\displaystyle\frac{\,{\mp\mathrm{i}e^{\mp\mathrm{i}p_{12}\pi/2}e^{-\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\pi\Gamma\big{[}\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{]}},$ (116)
$\displaystyle\beta_{\pm\pm|-+}=\beta_{\pm\pm|--}=\displaystyle\frac{\,{\mp\mathrm{i}e^{\mp\mathrm{i}p_{12}\pi/2}e^{-\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\pi\Gamma\big{[}\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{]}}.$ (117)

###  4.3 Folded limit

##### Single-folded limit.

Now we consider the folded limits of the Whittaker seed integrals. First, to
obtain the three-point function, we take the single-folded limit
$u_{2}\rightarrow 1$. As in the previous section, each of the solutions
$\mathcal{U}_{\mathsf{a}|\mathsf{b}}^{p}(u)$ is singular when $u\rightarrow
1$. However, the singular terms must cancel out in the final expression, as a
result of choosing the Bunch-Davies initial condition. Therefore, we only need
to retain the finite parts of the solutions
$\mathcal{U}_{\mathsf{a}|\mathsf{b}}^{p}(u)$ in the folded limit $u\rightarrow
1$. For notational simplicity, we define the finite part of
$\mathcal{U}_{\mathsf{a}|\mathsf{b}}^{p}(1)$ as:

$U_{\mathsf{a}|\mathsf{b}}^{p}\equiv\text{Fin}\Big{\\{}\mathcal{U}_{\mathsf{a}|\mathsf{b}}^{p}(1)\Big{\\}}=\displaystyle\frac{\,{\mathrm{i}\mathsf{a}\mathsf{b}\pi}\,}{2^{3/2+p}\sinh(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\left[\begin{matrix}\mbox{$\frac{\,{3}\,}{2}$}+p+\mathrm{i}\mathsf{a}\mathsf{b}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mathsf{a}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu+\mathrm{i}\mathsf{a}\mathsf{b}\mkern