# Helical Inflation Correlators:
Partial Mellin-Barnes and Bootstrap Equations

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

###### Abstract

Massive spinning particles acquire helicity-dependent chemical potentials
during the inflation from axion-type couplings. Such spinning fields can
mediate sizable inflaton correlators which we call the helical inflation
correlators. Helical inflaton correlators are approximately scale invariant,
dS boost breaking, parity-violating, and are promising observables of
cosmological collider physics. In this work, we present complete and
analytical results for 4-point helical inflation correlators with tree-level
exchanges of massive spinning particles, including both the smooth background
and the oscillatory signals. We compute the bulk Schwinger-Keldysh integrals
in two independent ways, including the partial Mellin-Barnes representation
and solving bootstrap equations. We also present new closed-form analytical
results for 3-point functions with massive scalar or helical spinning
exchanges. The analytical results allow us to concretely and efficiently
explore the phenomenological consequences of helicity-dependent chemical
potentials. In particular, we show that the chemical potential can
exponentially enhance oscillatory signals of both local and nonlocal types,
but only affects the background in a rather mild way. Our results extend the
de Sitter bootstrap program to include nonperturbative breaking of de Sitter
boosts. Our results also explicitly verify the recently proposed cutting rule
for cosmological collider signals.

###### Contents

  1. 1 Introduction
  2. 2 Spinning Fields with Helical Chemical Potential
    1. 2.1 Helical spin-1 field
    2. 2.2 Correlators and the vector seed integral
  3. 3 Inflation Correlators with Partial Mellin-Barnes Representation
    1. 3.1 Scalar exchange revisited
    2. 3.2 Partial MB representation
    3. 3.3 Summary
  4. 4 Helical Spin-1 Exchange
    1. 4.1 Transverse integral
    2. 4.2 Longitudinal integral
    3. 4.3 Summary
  5. 5 Bootstrapping Helical Correlators
    1. 5.1 Scalar
    2. 5.2 Transverse vector
    3. 5.3 Longitudinal vector
  6. 6 Variations: Higher Spins and Bispectra
  7. 7 Phenomenology
    1. 7.1 Cosmological collider signals in the squeezed limit
    2. 7.2 Background
  8. 8 Conclusions and Outlooks
  9. A Useful Formulae
  10. B General Trilinear Couplings
  11. C Fully Resolved MB Representation

## 1 Introduction

The matter distribution of our universe at large scales features an
essentially uniform background with small but important fluctuations.
According to the widely accepted inflation paradigm, the large-scale
inhomogeneities are originated from the primordial quantum fluctuations of
spacetime generated at very high-energy (small-distance) scales during the
inflation. Therefore, measurements of large-scale inhomogeneities including
cosmic microwave background (CMB) observations and large-scale structure (LSS)
surveys provide us unique windows to microscopic physical processes happening
during the inflation [1].

Mathematically, information about the primordial fluctuations is encoded in
their correlation functions, which we call _inflation correlators_ for
simplicity. We can think of inflation correlators as statistically measured
correlation function
$\langle\varphi(\mathbf{x}_{1})\cdots\varphi(\mathbf{x}_{n})\rangle$ where
$\varphi(\mathbf{x})$ is the inflaton fluctuation measured at the future
boundary of the inflation spacetime. Nontrivial correlators start from $n=2$,
which is more often called the power spectrum, while $n$-point correlators
with $n\geq 3$ are conventionally called the non-Gaussianities [2, 3, 4, 5].
The power spectrum of the primordial scalar fluctuation has been well measured
at CMB scales, providing us a wealth of information about the inflationary
universe [6]. The non-Gaussianities, on the other hand, are yet to be
discovered. The current CMB measurements only provide limits on their sizes
[7]. However, we expect non-Gaussianities to be present, since they are
signals of interactions, and interactions must be present during the
inflation.

In recent years, there have been increasing interests in the study of
inflation correlators, and significant progresses have been made in various
directions. The motivations behind this revived interest in the inflation
correlators are at least three-fold.

First, observational progresses are expected to be made in the near and far
future, allowing us to probe inflation correlators with unprecedented
precision. The expected sensitivity to 3-point functions with near-future LSS
surveys can be one order of magnitude higher than the current CMB constraints
[1, 8]. More futuristic 21cm tomography could further improve this sensitivity
by orders of magnitudes [9, 10]. These future observations would open up
entirely new parameter space with rich physics, calling for more careful and
more systematic theoretical studies of inflation correlators.

Second, the inflation correlators encode the information about the physics
around the inflation scale, which could be as high as $10^{14}$GeV [6, 11].
This is certainly among the highest energy scale that we can ever have access
to in nature. Therefore, probing the inflation correlators provides us a
unique chance to learn potential new physics at an extremely high energy
scale. This idea was already hinted at in the early studies of quasi-single-
field inflations [12, 13, 14, 15, 16, 17, 18], and was sharpened more recently
in [19] under the name of cosmological collider (CC) physics. In [19], it was
shown that the fast expansion of the inflation background can trigger
spontaneous production of heavy particles with mass up to the inflation Hubble
scale $H$. Along with their subsequent evolution, these heavy particles can
imprint characteristic oscillatory signatures on the inflation correlators, of
which a measurement could tell the mass and the spin of these heavy particles.
This makes inflation correlators directly relevant to the study of fundamental
particle physics.

Last but not least, developing computation techniques for inflation
correlators and understanding their analytical properties are interesting
theoretical problems on their own. The inflation spacetime is close to the
Poincaré patch of the de Sitter spacetime (dS), and the inflation correlators
can be viewed as correlators of bulk quantum fields in dS, with their external
points pinned to the future boundary. Inflation correlators are thus natural
analogues of scattering amplitudes in Minkowski spacetime or boundary
correlators in anti-de Sitter spacetime (AdS). However, compared to the highly
developed territory of flat-space scattering amplitudes or AdS correlators,
the study of dS correlators is still in its infancy. To give an example, the
analytical result for the simplest CC process, namely the 4-point correlator
with tree-level exchange of a massive scalar field (see Fig. 2), was found
only quite recently [20]. Full analytical results for CC processes at loop
levels are still beyond our reach at the moment. As the readers would
appreciate from the following sections, the complications of computing
inflation correlators come mainly from two sources. One is the space-time
asymmetric nature of the problem, which renders a fully covariant 4-momentum
representation inapplicable. Also, while some inflation correlators are
covariant under full dS isometries, phenomenologically more interesting
correlators typically break dS boosts. As often happens in physics, less
symmetry implies increased difficulty. The other complication is that the mode
functions of fields are often special functions due to the distortion from the
spacetime background. One has to deal with time-ordered integrals of Hankel
functions (for ordinary massive fields) or Whittaker functions (for fields
with helical chemical potentials) instead of plane waves in flat space.

Despite of the difficulties, progresses have been made in past few years. In a
recently developed cosmological bootstrap program [21], full analytical
results have been found for dS covariant 4-point and 3-point functions with
tree-level exchange of particles of arbitrary mass and (integer) spin [20, 22,
23]. Similar results were also obtained with an AdS-inspired Mellin space
approach [24, 25, 26, 27, 28]. The bootstrap program was further extended to
include dS boost breaking effects such as non-unit sound speed and
nonconvariant couplings, first to correlators of purely massless fields [29,
30, 31], and more recently to correlators containing massive fields (and thus
CC signals) [32, 33]. At the loop level, precision computations of 1-loop
correlators of both dS covariant type and boost breaking type have been done
in [34] with a full numerical approach. Analytically, an approached called
partial Mellin-Barnes (MB) representation was recently introduced in [35],
with which the complete analytical results for the nonlocal CC signals at
1-loop level were obtained, for both dS covariant case and boost breaking
case. See also [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] for recent
works on the analytical properties of inflation correlators or wavefunction
coefficients, and see [21] for a review.

From particle physics perspective, the rich CC phenomenology has also been
explored in recent years, including particle physics within and beyond the
Standard Model [48, 49, 50, 51, 52, 53, 54, 55, 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, 86, 87, 88, 89, 90], the observation strategies and forecasts
[91, 92, 93, 94]. Among the many CC models that have been considered, a
scenario with helicity-dependent chemical potentials has been emphasized owing
to its distinct and promising phenomenology, as well as its simplicity and
generality in model buildings [60, 69, 70, 74, 78, 34, 88, 35]. Such chemical
potentials arise naturally from axion-type couplings
$\partial_{\mu}J_{\text{A}}^{\mu}/\Lambda$ between the inflaton $\phi$ and
axial currents $J^{\mu}_{\text{A}}$ formed by massive spinning particles, and
thus are naturally present in various axion inflation models. Examples include
axial current $J_{\text{A}}^{\mu}=\mkern
2.0mu\overline{\mkern-2.0mu\Psi\mkern-2.0mu}\mkern
2.0mu\gamma^{\mu}\gamma^{5}\Psi$ for spin-$1/2$ fermions [60, 74] and Chern-
Simons current
$J_{\text{A}}^{\mu}=\epsilon^{\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}$ for
spin-1 bosons [78]. Generalizations to higher spins are also possible [88].
With the rolling inflaton background $\dot{\phi}_{0}\neq 0$, these operators
turn into the number density operators weighted by the helicity of the
particle, namely $(\dot{\phi}_{0}/\Lambda)J_{\text{A}}^{0}$ and thus the
combination $\mu\equiv\dot{\phi}_{0}/\Lambda$ can be viewed as a helicity-
dependent chemical potential, which we shall call _helical chemical
potential_. As a result, the spontaneous production of one helicity state will
be boosted while the other being suppressed.333The chemical potential can also
be introduced to ordinary number density operator without the helicity
dependence. However, as was shown in [74], only the helicity-dependent
chemical potential can lead to nontrivial enhancement of CC signals. See [84]
for more discussions. In [81] a helicity-independent chemical potential was
introduced for scalar fields. But this chemical potential enhance the signal
from the vertex rather than from the propagator, and thus is quite different
from the chemical potential in our current context. Cosmological consequences
of helicity-dependent chemical potential have also been extensively explored
in a wider context beyond the CC physics; See, e.g., [95, 96, 97, 98, 99, 100,
101, 102, 103, 104, 105, 106].

The chemical potential operator $(\partial_{\mu}\phi)J_{\text{A}}^{\mu}$ is
the lowest dimension operator that couples massive spinning particle and the
inflaton field and respects the shift symmetry of the inflaton
$\phi\rightarrow\phi+\text{const.}$, and thus is rather generic from the
effective field theory (EFT) viewpoint. A chemical potential comparable to or
greater than the particle mass can natually alleviate the Boltzmann
suppression factor $e^{-\pi m/H}$ that affects most of CC signals when the
mass $m$ is significantly greater than the Hubble scale $H$. In particular, in
parameter spaces that can be easily realized in particle models ($\mu\sim m\gg
H$ for spin-1 and $\mu\gg m\gg H$ for spin-1/2), the resulting CC signals
possess large amplitudes and large frequencies, while the smooth background
contribution is expected to be negligible [74, 78]. It is clear that the
helical chemical potential necessarily breaks the dS boosts as well as the
spatial parity. The resulting correlators possess characteristic angular
dependence which reflects the helicity asymmetric nature of the underlying
dynamics [68, 88]. We will call these objects _helical inflation correlators_
to highlight their characteristic dependences on the helicity of the
intermediate states.

For reliable studies of CC physics involving helical chemical potential, it is
clearly essential to develop techniques for precise and efficient computation
of helical inflation correlators. Ideally, we would like to gain full
analytical control over these objects. However, the presence of the chemical
potential makes the computation more involved. The primary reason is that the
chemical potential comes from the inflaton rolling $\dot{\phi}_{0}\neq 0$, and
thus breaks the dS boost. In particular, when the chemical potential is large,
the boost-breaking operator becomes _nonperturbative_. (See Fig. 5 and related
discussions in Sec. 5.) Therefore one must treat the chemical potential
insertions nonperturbatively, resumming all such insertions to get the
_dressed_ propagators for the massive spinning particle. This nonperturbative
boost breaking makes it hard to use the previously developed tools that rely
on full dS isometries. The result from recently developed boostless bootstrap
was not immediately applicable to this case either, as we shall further
explain in Sec. 5.

On the other hand, it is possible to complete the resummation of chemical-
potential insertions by directly solving the equation of motion, resulting in
a solution expressed by the Whittaker W function, which is technically more
complicated than the more familiar Hankel functions. (In comparison, there is
another type of strong boost-breaking effects, namely the non-unit sound speed
$c_{s}\neq 1$, for which the mode function remains to be the Hankel functions,
only with a shift in the momentum $k\rightarrow c_{s}k$.) Therefore, in order
to find analytical results for inflation correlators with helical chemical
potentials, it is essential to develop appropriate tools to deal with
Whittaker functions.

In this work, we set out to find full analytical results for a class of
helical inflation correlators at the tree level. We shall attack the problem
from two independent approaches.

In the first approach, we use the method of partial Mellin-Barnes (MB)
representation introduced in our previous work [35]. In this approach, the
complicated special functions in the intermediate massive propagators are
recast into MB integrals over power functions, which trivialize the time
integral. We then finish the MB integrals by properly closing the contours and
applying the residue theorem. In this work, we shall show how to compute both
the signal and the background part of the correlators using this method. We
will further explore the freedom of MB representation for Whittaker functions
to give two distinct derivations of the same result.

In the second approach, we derive differential equations satisfied by the
helical correlators and then solve them with appropriate boundary conditions.
We call them _bootstrap equations_ , since this approach is more in line with
the cosmological bootstrap program. Thus our bootstrap equations can be viewed
as an extension of the cosmological bootstrap program to the theories with
helical chemical potential, which is a nonperturbative source of dS boost
breaking. We also present a new way to take folded limit of the bootstrapped
series, which allows us to write down closed-form analytical results for
various 3-point functions of massive mediations without any series expansion.

The central object of this paper is a _vector seed integral_ (43) defined in
Sec. 2, from which a large number of helical correlators can be built. For
illustrating the method and for completeness, we also present a similar
computation for the simpler massive scalar exchange, by defining and computing
a _scalar seed integral_. By computing these objects with both partial MB and
bootstrap equations, we cross-check the results from the two very different
methods, and find nice agreements between them.

With full analytical results for these seed integrals, we can build more
general correlators with helical massive spinning exchanges for all helicity
states. These results also allow us to explore more reliably and more
efficiently the phenomenological consequences of these correlators. In
particular, we confirm a previous expectation that the CC signals from these
helical correlators are parametrically larger than background even for mildly
large chemical potentials. Also, we show that both the local and nonlocal CC
signals can be enhanced exponentially by the chemical potential, but the
detailed parameter dependence shows interesting difference between these two
types of signals. The fast numerical implementation of our results makes it
easier for parameter scanning in model building and also for generating
templates when confronting the model with data. Therefore, we hope that the
results presented in the work is not only of theory interest, but also of
practical use.

#### Outline of this work.

Since this work is mainly about analytical computation of helical inflation
correlators, a large portion of the main text will be rather technical. In
order to make this work more accessible, below we will give a brief summary
for each of the following sections.

In Sec. 2, after making some general remarks about the helical chemical
potentials, we present in Sec. 2.1 a detailed treatment of quantizing a
massive spin-1 field with chemical potential in dS (henceforth helical spin-1
fields), which lead to a set of _mode functions_ for the helical spin-1
fields, summarized in (22)-(24). Following a standard procedure in the
Schwinger-Keldysh (SK) formalism [57], we can construct propagators, and
eventually Feynman diagrams, from these mode functions. Then in Sec. 2.2 we
discuss the most general couplings that give rise to the 4-point functions in
Fig. 1. Then we show that the computation of this class of 4-point correlators
can be reduced to that of a vector seed integral, defined in (43).

In Sec. 3, we introduce our main tool for computing the SK integrals, namely
the partial MB representation. For illustrative purpose and also for
completeness, we use the partial MB to compute a simpler process, namely the
4-point function with tree-level exchange of a massive scalar particle. To
this end, we define a scalar seed integral in (55), and then present all the
details of computation. Readers uninterested in the technical details can
directly go to Sec. 3.3 for a summary of the result.

In Sec. 4, we present the full analytical results for the vector seed integral
defined in Sec. 2. It turns out that the correlators mediated by the two
transverse states and by the longitudinal state require separate treatments.
Therefore, we present the computation of the transverse part in Sec. 4.1, and
the longitudinal part in Sec. 4.2. In Sec. 4.1, we discuss the flexibility in
choosing the MB representation for the Whittaker function. In the same
subsection, we provide the full computation using the so-called _partially
resolved MB representation_. The computation of the same integral with an
alternative MB representation, which we call the _completely resolved MB
representation_ , will be presented in App. C. Again, readers uninterested in
the details can directly jump to Sec. 4.3 for a summary of the results in this
section. In Sec. 4.3, we also comment on the relation between our results and
a recently proposed tree-level cutting rule for CC signals [40]. Our results
explicitly verified the cutting rule for both the scalar change and the
helical spinning changes. Therefore it is possible to use our method to give a
more rigorous proof of the cutting rule stated in [40].

In Sec. 5, we turn to the method of bootstrap equations. We will first make
some general remarks about the bootstrap method, and then derive the scalar
and vector bootstrap equations in the three subsections. We then provide the
particular solutions to the inhomogeneous equations in each case in terms of
Taylor series of momentum ratios, and compare these results with the ones
obtained from partial MB representation. It turns out that the series solution
can be found for transverse correlators only if we make a suitable change of
variables, as suggested by the results from partial MB. We then solve the
homogeneous equations to get the signal part of the correlator. Different from
previous works, we impose the boundary conditions from the squeezed
configurations, which correspond to the late-time limit of the bulk integral.
Given the more free parameters in the homogeneous solutions, it turns out that
imposing the boundary condition from the late-time limit is the easiest
choice. We then check our results by sending them to the folded limit, where
all superficial divergences should cancel, as required by the Bunch-Davies
initial condition [19, 20]. We will also show how an appropriate change of
variables can help us to find neat and closed expressions for the 4-point
function in the folded limit.

In Sec. 6, we demonstrate how to extend our results to more general helical
correlators, by working out a particular example, namely a mixed 3-point
function with two scalar modes and one tensor mode, where the tensor mode is
mixed with a massive spin-2 field with helical chemical potential. This is a
process that generalizes the CC physics to tensor modes with large oscillatory
signals, and has been considered in [88]. In this section, we provide full
analytical result for this process, including the signal and the background,
in terms of the vector seed integral computed in Sec. 4. For completeness, we
will also present a corresponding closed-form expression for the scalar
3-point function with massive scalar exchange, which seems new to us.

In Sec. 7, we discuss the phenomenology of the helical 4-point correlators. In
particular, we show how the CC signals and the background depend on the mass
and the chemical potential of the intermediate particles. We also provide
representative plots for the signals and the backgrounds in Fig. 13, and also
plots for typical full correlators in Fig. 14.

Further discussions and outlooks are presented in Sec. 8. Some useful formulae
are collected in App. A. In App. B we give the results of the 4-point spin-1
correlator with the most general effective couplings in terms of the vector
seed integral. In App. C we present the result for the vector seed integral
using the completely resolved MB representation for the Whittaker function.

#### Notations and conventions.

Most of special notations used in this work will be defined in a self-
contained way when they appear. We use the mostly-minus spacetime metric. For
the most part, we use conformal time $\tau$ and comoving spatial coordinates
$\mathbf{x}$, with which the dS metric reads
${\mathrm{d}}s^{2}=a^{2}(\tau)(-{\mathrm{d}}\tau^{2}+{\mathrm{d}}\mathbf{x}^{2})$,
$a(\tau)=-1/(H\tau)$ is the scale factor and $H$ is the inflation Hubble
parameter which we will always take to be a constant. In most part of this
paper, we also take $H=1$ for notational simplicity. We occasionally use the
comoving time $t$, with which the metric reads
${\mathrm{d}}s^{2}=-{\mathrm{d}}t^{2}+a^{2}(t){\mathrm{d}}\mathbf{x}^{2}$ and
$a(t)=e^{Ht}$. Our notations and conventions for describing the 4-point
inflation correlators are mostly in line with [35], in which the readers can
find detailed discussion about the kinematics and the symmetry of the 4-point
functions. In particular, we will often use notations like $k_{12}\equiv
k_{1}+k_{2}$ (for momenta; magnitude sum rather than vector sum),
$s_{12}=s_{1}+s_{2}$ (for Mellin variables), $n_{12}=n_{1}+n_{2}$ (for
summation variables), etc. Our diagrammatic notations and computations follow
the treatment of SK formalism presented in [57], in which the detailed
diagrammatic rules about SK integrals for given diagrams have been spelled
out. We refer readers unfamiliar with the SK formalism to [57] for a pedagogic
review.

## 2 Spinning Fields with Helical Chemical Potential

The inflationary universe is very close to the Poincaré patch of the de Sitter
space, which has the largest possible number of isometries, including the
space translations and rotations, the dilatation, and also the three dS
boosts. The observed scalar power spectrum suggests that the inflation
correlators should be covariant under the space translations and rotations.
The dilatation symmetry is also realized approximately in the form of scale
invariance. On the contrary, we have no observational evidence of the three dS
boosts. From a theoretical viewpoint, the rolling background of the inflaton
itself is a source of boost breaking. Therefore, when constructing inflation
correlators, it is appropriate to consider theories respecting all dS
isometries expect the three boosts.

There are two distinct ways to introduce boost breaking dynamics to a single
propagating species in dS. One is to modify the relative sizes between the
kinetic term and the gradient term, resulting a non-unit sound speed $c_{s}$.
This is a well-known possibility and has been explored in the context of CC
physics recently in [32, 33]. The other possibility, which is realized for
spinning particles only, is the helical chemical potential and is the main
focus of this work. Usually, these two boost breaking effects can both be
obtained by evaluating some Lorentz invariant operators in an underlying
theory with the rolling inflaton background. In this respect, the chemical
potential term appears first from the dimension-5 operators, as mentioned in
the Introduction. The non-unit sound speed, on the other hand, normally
requires higher dimensional operators. Therefore, from an effective field
theory point of view, it seems that the chemical potential is more relevant
than the non-unit sound speed whenever it is present. For this reason, we
shall neglect any non-unit sound speed in the following discussions.

###  2.1 Helical spin-1 field

Helical chemical potentials can in principle be introduced to massive
particles of arbitrary nonzero spin. In this work we mainly focus on the case
of massive spin-1 particles. Generalizations to higher spins or half-integer
spins are similar, and we shall show an example of massive spin-2 in Sec. 6.
For completeness, we shall present the details of canonical quantization from
the classical Lagrangian to the mode functions and finally to the SK
propagators. Experts may wish to skip this subsection.

We can work with effective field theory of inflation to keep things general,
but we find it convenient enough to start from a (local) Lorentz invariant
Lagrangian. Therefore we introduce a real background scalar field $\phi$ which
can be either the inflaton field or a spectator field or a mixture of the
both. In either case, we require that the scalar field has a homogeneous and
isotropic background:

$\phi_{0}(t)=\langle\phi(t,\mathbf{x})\rangle,$ (1)

which is independent of space coordinate $\mathbf{x}$ but can have nontrivial
time dependence. For a wide range of phenomenological applications, we can
take the time dependence of $\phi_{0}(t)$ to be
$\phi_{0}(t)=\dot{\phi}_{0}t+\text{const.}$ where $\dot{\phi}_{0}$ is a _time-
independent_ constant speed. Such form of $\phi_{0}(t)$ is usually a good
approximation for either a rolling inflaton along its flat potential, or a
light spectator field with the mass smaller than the Hubble scale $H$. We also
note that such form of time-dependence leaves the inflation correlators scale
invariant with appropriate couplings.

#### The action and the equations of motion.

Now we can write down an action for the spin-1 field $A_{\mu}$ of mass $m$,
with a fixed spacetime background $g_{\mu\nu}$ and the scalar background
introduced above:

$S=\int{\mathrm{d}}^{4}x\,\bigg{[}\sqrt{-g}\Big{(}-\displaystyle\frac{\,{1}\,}{4}g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}-\displaystyle\frac{\,{1}\,}{2}m^{2}g^{\mu\nu}A_{\mu}A_{\nu}\Big{)}+\displaystyle\frac{\,{\phi}\,}{4\Lambda}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\bigg{]}.$
(2)

Here $\epsilon^{\mu\nu\rho\sigma}$ is the total antisymmetric tensor with
$\epsilon^{0123}=1$. We do not assume a specific origin for the mass $m$ of
the spin-1 field $A_{\mu}$. When it is originated from a Higgs mechanism, we
assume that the Higgs degree is heavy enough so that it can be decoupled from
our problem, and the above action can be regarded as written in the unitary
gauge.

The dimension-5 coupling $\phi F\mkern
2.0mu\widetilde{\mkern-2.0muF\mkern-2.0mu}\mkern 2.0mu$ is rather generic a
coupling between a CP odd scalar field $\phi$ and a spin-1 field $A_{\mu}$. It
is also the lowest-dimension coupling between the spin-1 field $A_{\mu}$ to
the scalar $\phi$ that respects both the shift symmetry of the scalar field
$\phi\rightarrow\phi+\text{const.}$ and the (albeit broken) gauge symmetry of
the spin-1 field. As mentioned in the introductory section and will be
detailed now, this coupling is also the origin of the helical chemical
potential of the massive spin-1 field once the scalar field $\phi$ acquires
its rolling background (1).

Specifically, we take the dS limit of the inflation background
${\mathrm{d}}s^{2}=a^{2}(\tau)(-{\mathrm{d}}\tau^{2}+{\mathrm{d}}\mathbf{x}^{2})$
with the scale factor $a(\tau)=-1/(H\tau)$, and also the scalar background
(1). Then the action (2) can be rewritten as:

$S=\int{\mathrm{d}}\tau{\mathrm{d}}^{3}\mathbf{x}\,{\mathscr{L}},~{}~{}~{}~{}~{}{\mathscr{L}}=\Big{(}-\displaystyle\frac{\,{1}\,}{4}F_{\mu\nu}F^{\mu\nu}-\displaystyle\frac{\,{1}\,}{2}a^{2}m^{2}A_{\mu}A^{\mu}\Big{)}+\displaystyle\frac{\,{\dot{\phi}_{0}t}\,}{4\Lambda}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$
(3)

From this equation onward, all the spacetime indices are lowered and raised by
the Minkowski metric $\eta_{\mu\nu}$ and its inverse, unless otherwise stated.
Also, in (3) we have thrown away a constant in the scalar background
$\phi_{0}(t)$ which contributes only to a boundary term that is irrelevant to
our study. We stress that it is the comoving-time derivative of the inflaton
background $\dot{\phi}_{0}={\mathrm{d}}\phi_{0}/{\mathrm{d}}t$ that is nearly
the constant. The quantity
$\phi_{0}^{\prime}\equiv{\mathrm{d}}\phi_{0}/{\mathrm{d}}\tau=a\dot{\phi}_{0}=e^{Ht}\dot{\phi}_{0}$
actually has strong time dependence. Also, the appearance of explicit time
dependence in (3) may look a bit weird, but it can be easily removed with
integration by parts, leaving a term that has an uncontracted temporal index.
Such bare temporal indices are direct consequences of broken time
diffeomorphism by the rolling scalar background [107].

From the action (3) it is straightforward to get the equation of motion for
the spin-1 field $A_{\mu}$:

$\displaystyle\partial_{\mu}F^{\mu\nu}-a^{2}m^{2}A^{\nu}-2a\mu\epsilon^{0\nu\rho\sigma}F_{\rho\sigma}=0.$
(4)

Here the dimension-1 parameter $\mu\equiv\dot{\phi}_{0}/\Lambda$ is nothing
but the helical chemical potential. Acting $\partial_{\nu}$ on the equation of
motion (4), we get a constraint:

$2aHA^{0}+\partial_{\nu}A^{\nu}=0,$ (5)

which can be written in a more covariant form
$g^{\mu\nu}\nabla_{\mu}A_{\nu}=0$ although we prefer not to do so.444Our
treatment here also applies to a Higgsed $U(1)$ gauge theory so long as one
chooses (5) as the gauge condition. However, in the case of a Higgsed gauge
theory, other choices of gauge conditions are also possible, which may lead to
expressions superficially different from what we are presenting here.
Substituting the constraint (5) back into the equation (4), we get two
equations for the temporal and spatial components of $A_{\mu}=(A_{0},A_{i})$,
respectively:

$\displaystyle
A_{0}^{\prime\prime}-\partial_{j}^{2}A_{0}+(2aHA_{0})^{\prime}+a^{2}m^{2}A_{0}=0,$
(6) $\displaystyle
A_{i}^{\prime\prime}-\partial_{j}^{2}A_{i}+2aH\partial_{i}A_{0}+a^{2}m^{2}A_{i}+2a\mu\epsilon_{ijk}\partial^{j}A^{k}=0.$
(7)

Although we have 4 equations here, they are not independent due to the
constraint (5). As is well known, a massive spin-1 field as discussed here
gives rise to 3 independent propagating degrees of freedom.555This applies to
all mass $0<m<H/2$ and $m>H/2$. Although the spin-1 representations of dS
isometry group belong to different categories for $0<m<H/2$ and $m>H/2$, this
peculiarity of dS representation does not affect the degree counting. In
particular, the possibility of partially massless states kicks in only for
spin $s>2$ [108].  It is convenient to choose helicity eigenstates with
$h=0,\pm 1$ to describe them. The helicity
$h=\mathbf{s}\cdot\mathbf{k}/|\mathbf{k}|$ is the angular momentum of the mode
along the direction of propagation. In a Lorentz invariant theory, the
helicity is frame dependent, and thus is not an intrinsic property for massive
spinning particles. More explicitly, one can always boost a helicity
eigenstate to flip the direction of propagation while keeping the direction of
angular momentum invariant, and thereby flip the sign of the helicity.
However, in our case, the dS boosts (the dS counterparts of the Lorentz
boosts) are spontaneously broken by the scalar background $\phi_{0}(t)$. As a
result, the scalar background no longer takes the space-independent form
$\phi_{0}(t)$ for an observer boosted relative to the local comoving frame.
Therefore, the background scalar $\phi_{0}(t)$ unambiguously picks out a
comoving frame in which the universe looks isotropic. It can be easily seen
that the helicity is invariant under all other unbroken symmetries, including
the space translations and rotations, and the dilatation. The dilatation is of
course broken by the scalar background $\dot{\phi}_{0}$ as well. However, this
is inconsequential so long as other fields couple only derivatively to the
scalar and so long as $\dot{\phi}_{0}$ is constant.

#### Canonical quantization.

Now we review the canonical quantization of the massive spin-1 field in dS in
the presence of a helical chemical potential. Given the symmetry of the
problem, it is convenient to go to the Fourier space for the 3-dimensional
space and consider a specific Fourier mode with comoving momentum
$\mathbf{k}$. Then, the Fourier modes $A_{\mu,\mathbf{k}}(\tau)$ can be
written as linear superposition of annihilation and creation operators,
$a_{\mathbf{k}}^{(h)}$ and $a_{\mathbf{k}}^{(h){\dagger}}$, for each helicity
eigenstate:

$\displaystyle
A_{\mu}(\tau,\mathbf{x})=\int\displaystyle\frac{\,{{\mathrm{d}}^{3}\mathbf{k}}\,}{(2\pi)^{3}}e^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}A_{\mu,\mathbf{k}}(\tau),$
$\displaystyle
A_{\mu,\mathbf{k}}(\tau)=\sum_{h=-1}^{+1}\Big{[}A^{(h)}_{\mu,\mathbf{k}}(\tau)a_{\mathbf{k}}^{(h)}+A^{(h)*}_{\mu,-\mathbf{k}}(\tau)a_{-\mathbf{k}}^{(h){\dagger}}\Big{]},$
(8)

where $A_{\mu,\mathbf{k}}^{(h)}(\tau)$ are the helicity eigenstates, with
$h=0$ denoting the longitudinal mode and $h=\pm 1$ denoting the two transverse
modes. In the flat spacetime, each helicity eigenstate can be further written
as a product of a time-dependent mode function
$e^{-\mathrm{i}E_{\mathbf{k}}t}$ and a time-independent polarization vector
$e_{\mathbf{k}}^{(h)}$. Things are a little more complicated in dS for the
longitudinal mode: As a consequence of nontrivial time dependence in the mode
function, together with the constraint equation (5), the longitudinal mode is
no longer a simple product of a time-dependent mode function and a time-
independent polarization vector. To take care of this complication, we take
$k_{i}=(0,0,k)^{T}$ without loss of generality. Then we can define four
polarization vectors $e^{(\lambda)}$ with $\lambda=T,+,-,L$ as below (Note
that $\lambda$ is _not_ a label of helicity):

$\displaystyle e_{\mu,\mathbf{k}}^{(T)}=\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix},$ $\displaystyle
e_{\mu,\mathbf{k}}^{(\pm)}=\displaystyle\frac{\,{1}\,}{\sqrt{2}}\begin{pmatrix}0\\\
1\\\ \pm\mathrm{i}\\\ 0\end{pmatrix},$ $\displaystyle
e_{\mu,\mathbf{k}}^{(L)}=\begin{pmatrix}0\\\ 0\\\ 0\\\
1\end{pmatrix},\text{~{}~{}~{}with~{}~{}~{}}\mathbf{k}=\begin{pmatrix}0\\\
0\\\ k\end{pmatrix}.$ (9)

For general $\mathbf{k}$ not in the 3-direction, the corresponding
polarization vectors can be obtained by the usual rule of 3-dimensional
rotations. That is, $e^{(T)}_{\mu}$ will remain intact as a scalar under
3-rotations, while $e_{\mu,\mathbf{k}}^{(L)}=\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern 2.0mu$ and
$e_{\mu,\mathbf{k}}^{(\pm)}=(\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{u}\mkern-2.0mu}\mkern
2.0mu\pm\mathrm{i}\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{v}\mkern-2.0mu}\mkern 2.0mu)/\sqrt{2}$ up to
arbitrary complex phase factors, where $(\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{u}\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{v}\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern 2.0mu)$ form a right-
handed frame of the 3-dimensional space. We note in particular that these
expressions are valid only in the comoving frame, and we are not allowed to
perform dS boost due to the boost breaking nature of the problem. We will not
need expressions for these polarization vectors in boosted frames.

With the aid of the (physically redundant) set of polarization basis (9), we
can write the helicity eigenstates $A_{\mu,\mathbf{k}}^{(h)}(\tau)$ in terms
of a set of mode functions $B^{(\lambda)}$ as below:

$\displaystyle A_{\mu,\mathbf{k}}^{(\pm
1)}(\tau)=B^{(\pm)}(k,\tau)e_{\mu,\mathbf{k}}^{(\pm)},$ (10) $\displaystyle
A_{0,\mathbf{k}}^{(0)}(\tau)=B^{(T)}(k,\tau)e_{\mu,\mathbf{k}}^{(T)}+B^{(L)}(k,\tau)e_{\mu,\mathbf{k}}^{(L)},$
(11)

The equations of motion for the field $A_{\mu}$ then give rise to a set of
equations for the mode functions. The temporal equation (6) gives:

$\displaystyle
B^{(T)}{}^{\prime\prime}+2aHB^{(T)}{}^{\prime}+\big{[}k^{2}+a^{2}(m^{2}+2H^{2})\big{]}B^{(T)}=0,$
(12)

while the spatial equation (7) mixes different mode functions. To decouple the
spatial equation, we contract it with the polarization vectors in (9).
Contracting (7) with $e_{\mu,\mathbf{k}}^{(L)}$ gives the longitudinal
equation:

$\displaystyle
B^{(L)}{}^{\prime\prime}+(k^{2}+a^{2}m^{2})B^{(L)}+2\mathrm{i}aHkB^{(T)}=0,$
(13)

and contracting (7) with $e_{\mu,\mathbf{k}}^{(\mp)}$ gives the transverse
equations:

$\displaystyle B^{(\pm)}{}^{\prime\prime}+(k^{2}\pm 2a\mu
k+a^{2}m^{2})B^{(\pm)}=0.$ (14)

It can be readily checked that the longitudinal equation and the temporal
equations are compatible with the constraint (5) and therefore are not
independent. We can conveniently choose (12) and (14) as an independent and
complete set of equations, and the longitudinal mode function $B^{(L)}$ can be
found from $B^{(T)}$ by solving the constraint (5):

$B^{(L)}=\displaystyle\frac{\,{1}\,}{\mathrm{i}k}\big{(}B^{(T)}{}^{\prime}+2aHB^{(T)}\big{)}.$
(15)

Now it is straightforward to solve the mode equations up to overall
normalization constants:

$\displaystyle B^{(\pm)}(k,\tau)=$
$\displaystyle~{}\mathcal{N}^{(\pm)}\displaystyle\frac{\,{e^{\mp\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu/2}}\,}{\sqrt{2k}}\mathrm{W}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(2\mathrm{i}k\tau),$
(16) $\displaystyle B^{(T)}(k,\tau)=$
$\displaystyle~{}\mathcal{N}^{(T)}\displaystyle\frac{\,{\sqrt{\pi}}\,}{2}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}H(-\tau)^{3/2}\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k\tau).$ (17)

Here $\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu\equiv\mu/H$ and $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\equiv\sqrt{(m/H)^{2}-1/4}$, $\mathrm{W}_{\kappa,\mu}(z)$ is the
Whittaker W function and $\mathrm{H}_{\mu}^{(1)}(z)$ is the Hankel function of
the first kind. We shall call $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ the mass parameter
and we add a tilde to distinguish it from the more conventionally defined
parameter $\nu=\sqrt{1/4-m^{2}/H^{2}}$. In this paper, we shall exclusively
consider massive fields in the principal series, meaning that $\nu$ is purely
imaginary, and $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu$ is positive real. Note also that the relation between $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ and $m$ depends on
the spin: For spin-$s$ $(s\neq 0)$ fields, $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu=\sqrt{(m/H)^{2}-(s-1/2)^{2}}$, and for scalars $(s=0)$, $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu=\sqrt{(m/H)^{2}-9/4}$.

With the application of CC physics in mind, we always assume $m>H/2$, the so-
called principal series. The generalization to complementary series $0<m<H/2$
is straightforward.

The normalization constants $\mathcal{N}^{(\pm)}$ and $\mathcal{N}^{(T)}$ are
determined by the canonical commutators. On the one hand, we have the
canonical equal-time commutator of the field variable $A_{i}(\tau,\mathbf{x})$
with its canonical conjugate momentum
$\Pi^{i}(\tau,\mathbf{x})\equiv\partial{\mathscr{L}}/\partial A_{i}^{\prime}$
where ${\mathscr{L}}$ is given in (3):

$\displaystyle[A_{i}(\tau,\mathbf{x}),\Pi^{j}(\tau,\mathbf{y})]=\mathrm{i}\,\delta_{i}^{j}\delta^{(3)}(\mathbf{x}-\mathbf{y}),$
(18)

and all other brackets vanish. On the other hand, we have the canonical
commutators of annihilation and creation operators
$[a_{\mathbf{k}}^{(h)},a_{\mathbf{q}}^{(h^{\prime}){\dagger}}]=(2\pi)^{3}\delta^{hh^{\prime}}\delta^{(3)}(\mathbf{k}-\mathbf{q})$
and all other commutators vanish. These two sets of commutators, together with
the mode decomposition, fix the normalization of the mode functions. More
explicitly, we can again take $k_{i}=(0,0,k)$ without loss of generality.
Then, the transverse field commutators
$[A_{i}(\tau,\mathbf{x}),A_{j}^{\prime}(\tau,\mathbf{y})]=\mathrm{i}\delta_{ij}\delta^{(3)}(\mathbf{x}-\mathbf{y})$
with $i=1,2$ lead to the normalization condition for the transverse mode
functions:

$\mathrm{i}=\mathcal{W}\Big{(}B^{(\pm)},B^{(\pm)*}\Big{)}=\big{|}\mathcal{N}^{(\pm)}\big{|}^{2}.$
(19)

where $\mathcal{W}(f,g)=fg^{\prime}-f^{\prime}g$ is the Wronskian of two
functions $f$ and $g$. Thus the coefficients of the transverse modes are fixed
to be $\mathcal{N}^{(\pm)}=1$ up to irrelevant phases.

For the longitudinal mode $A_{\mu,\mathbf{k}}^{(0)}$, we have
$\Pi_{3}=A_{3}^{\prime}-\partial_{3}A_{0}$, and thus the field commutator
reads
$\big{[}A_{3}(\tau,\mathbf{x}),A_{3}^{\prime}(\tau,\mathbf{y})-\partial_{3}A_{0}(\tau,\mathbf{y})\big{]}=\mathrm{i}\delta^{(3)}(\mathbf{x}-\mathbf{y})$.
Together with
$[a_{\mathbf{p}}^{(0)},a_{-\mathbf{q}}^{(0){\dagger}}]=(2\pi)^{3}\delta^{(3)}(\mathbf{p}+\mathbf{q})$,
it gives rise to the following condition:

$\displaystyle\Big{[}B^{(L)}(k,\tau)B^{(L)*}{}^{\prime}(k,\tau)-B^{(L)*}(k,\tau)B^{(L)}{}^{\prime}(k,\tau)\Big{]}$
$\displaystyle-\mathrm{i}k\Big{[}B^{(L)}(k,\tau)B^{(T)*}(k,\tau)-B^{(L)*}(k,\tau)B^{(T)}(k,\tau)\Big{]}=\mathrm{i}$
(20)

Using (15) to remove $B^{(L)}$ in the second line, we get a somewhat unusual
Wronskian condition:

$\mathrm{i}=\mathcal{W}\Big{(}B^{(L)}(k,\tau),B^{(L)*}(k,\tau)\Big{)}-\mathcal{W}\Big{(}B^{(T)}(k,\tau),B^{(T)*}(k,\tau)\Big{)}=\mathrm{i}\big{|}\mathcal{N}^{(T)}\big{|}^{2}\displaystyle\frac{\,{m^{2}}\,}{k^{2}}.$
(21)

This fixes the normalization coefficient $\mathcal{N}^{(T)}=k/m$ up to an
irrelevant phase.

To summarize, we have obtained the mode functions for the massive spin-1
fields in dS with helical chemical potential as follows:

$\displaystyle B^{(\pm)}(k,\tau)=$
$\displaystyle~{}\displaystyle\frac{\,{e^{\mp\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu/2}}\,}{\sqrt{2k}}\mathrm{W}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(2\mathrm{i}k\tau),$
(22) $\displaystyle B^{(T)}(k,\tau)=$
$\displaystyle~{}\displaystyle\frac{\,{\sqrt{\pi}k}\,}{2m}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}H(-\tau)^{3/2}\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k\tau),$ (23)
$\displaystyle B^{(L)}(k,\tau)=$
$\displaystyle-\displaystyle\frac{\,{\mathrm{i}\sqrt{\pi}}\,}{4m}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu/2}H(-\tau)^{1/2}\Big{[}\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau)-k\tau\Big{(}\mathrm{H}^{(1)}_{1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau)-\mathrm{H}^{(1)}_{-1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau)\Big{)}\Big{]}.$ (24)

#### SK propagators.

With the mode functions obtained above, it is straightforward to construct the
propagators of the massive spinning fields. Owing to the aforementioned
complication of the longitudinal mode, namely the temporal component $B^{(T)}$
not proportional to the longitudinal component $B^{(L)}$, we find it more
useful to define the propagators with respect to the redundant polarization
basis $e_{\mu,\mathbf{k}}^{(\lambda)}$ given in (9). Following the standard SK
path integral formalism [57], the tree-level propagators can be obtained as:

$\displaystyle D_{++}^{(\lambda)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}e_{\mathbf{k}}^{(\lambda)\mu*}e_{-\mathbf{k}}^{(\lambda)\nu*}\big{\langle}0|\text{T}\\{A_{\mu,\mathbf{k}}(\tau_{1})A_{\nu,-\mathbf{k}}(\tau_{2})\\}|0\big{\rangle}^{\prime},$
(25) $\displaystyle D_{--}^{(\lambda)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}e_{\mathbf{k}}^{(\lambda)\mu*}e_{-\mathbf{k}}^{(\lambda)\nu*}\big{\langle}0|\mkern
2.0mu\overline{\mkern-2.0mu\text{T}\mkern-2.0mu}\mkern
2.0mu\\{A_{\mu,\mathbf{k}}(\tau_{1})A_{\nu,-\mathbf{k}}(\tau_{2})\\}|0\big{\rangle}^{\prime},$
(26) $\displaystyle D_{+-}^{(\lambda)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}e_{\mathbf{k}}^{(\lambda)\mu*}e_{-\mathbf{k}}^{(\lambda)\nu*}\big{\langle}0|A_{\nu,-\mathbf{k}}(\tau_{2})A_{\mu,\mathbf{k}}(\tau_{1})|0\big{\rangle}^{\prime},$
(27) $\displaystyle D_{-+}^{(\lambda)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}e_{\mathbf{k}}^{(\lambda)\mu*}e_{-\mathbf{k}}^{(\lambda)\nu*}\big{\langle}0|A_{\mu,\mathbf{k}}(\tau_{1})A_{\nu,-\mathbf{k}}(\tau_{2})|0\big{\rangle}^{\prime},$
(28)

where $|0\rangle$ is the initial state of the problem which we will take as
the usual Bunch-Davies vacuum, T and $\mkern
2.0mu\overline{\mkern-2.0mu\text{T}\mkern-2.0mu}\mkern 2.0mu$ denote the time
ordering and anti time ordering of operators, respectively, and
$\langle\cdots\rangle^{\prime}$ denotes a correlator with momentum-conserving
$\delta$-function removed. Then, with the equations (8), (10), (11), and
(22)-(24), one can find the explicit expressions of the above propagators in
terms of mode functions. All these propagators can be expressed in terms of
the Wightman functions, namely $D_{>}^{(\lambda)}(k;\tau_{1},\tau_{2})$ and
its complex conjugate $D_{<}^{(\lambda)}(k;\tau_{1},\tau_{2})\equiv
D_{>}^{(\lambda)*}(k;\tau_{1},\tau_{2})$:

$D_{>}^{(\lambda)}(k;\tau_{1},\tau_{2})\equiv
B^{(\lambda)}(k,\tau_{1})B^{(\lambda)*}(k,\tau_{2}).$ (29)

Then, we have:

$\displaystyle
D_{++}^{(\lambda)}(k;\tau_{1},\tau_{2})=D_{>}^{(\lambda)}(k;\tau_{1},\tau_{2})\theta(\tau_{1}-\tau_{2})+D_{<}^{(\lambda)}(k;\tau_{1},\tau_{2})\theta(\tau_{2}-\tau_{1}),$
(30) $\displaystyle
D_{--}^{(\lambda)}(k;\tau_{1},\tau_{2})=D_{<}^{(\lambda)}(k;\tau_{1},\tau_{2})\theta(\tau_{1}-\tau_{2})+D_{<}^{(\lambda)}(k;\tau_{1},\tau_{2})\theta(\tau_{2}-\tau_{1}),$
(31) $\displaystyle
D_{+-}^{(\lambda)}(k;\tau_{1},\tau_{2})=D_{<}^{(\lambda)}(k;\tau_{1},\tau_{2}),$
(32) $\displaystyle
D_{-+}^{(\lambda)}(k;\tau_{1},\tau_{2})=D_{>}^{(\lambda)}(k;\tau_{1},\tau_{2}),$
(33)

One can show, as we will do in Sec. 5, that the two opposite-sign propagators
$D_{\pm\mp}^{(\lambda)}$ satisfies the sourceless (and thus homogenous)
equation of motion for the vector field $A_{\mu}$, while the same-sign
propagators satisfy the same equation but with a nonzero source (and thus
inhomogeneous). For this reason we will also call $D^{(\lambda)}_{\pm\pm}$ the
inhomogeneous propagators and $D^{(\lambda)}_{\pm\mp}$ the homogeneous
propagators.

For later use, we also present the explicit expressions for the Wightman
function $D_{>}^{(\lambda)}$ for various polarizations:

$\displaystyle D^{(\pm)}_{>}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}B^{(\pm)}(k,\tau_{1})B^{(\pm)*}(k,\tau_{2})=\displaystyle\frac{\,{e^{\mp\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2k}\mathrm{W}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(2\mathrm{i}k\tau_{1})\mathrm{W}_{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-2\mathrm{i}k\tau_{2}),$ (34) $\displaystyle
D^{(T)}_{>}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}B^{(T)}(k,\tau_{1})B^{(T)*}(k,\tau_{2})=\displaystyle\frac{\,{\pi
H^{2}k^{2}}\,}{4m^{2}}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(\tau_{1}\tau_{2})^{3/2}\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau_{1})\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k\tau_{2}),$ (35)
$\displaystyle D^{(L)}_{>}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{k^{2}}\bigg{(}\partial_{\tau_{1}}-\displaystyle\frac{\,{2}\,}{\tau_{1}}\bigg{)}\bigg{(}\partial_{\tau_{2}}-\displaystyle\frac{\,{2}\,}{\tau_{2}}\bigg{)}D^{(T)}_{>}(k;\tau_{1},\tau_{2}).$
(36)

One caveat is that the relation between the longitudinal propagator and the
temporal propagator as written in (36) applies only to homogeneous type
propagators, namely $D^{(L)}_{\gtrless}$ or $D^{(L)}_{\pm\mp}$. Owing to the
presence of Heaviside $\theta$-functions, the two inhomogeneous propagators
$D_{\pm\pm}^{(L)}$ and $D_{\pm\pm}^{(T)}$ are not related by simply acting the
differential operators $\partial_{\tau_{1,2}}-2/\tau_{1,2}$ as in (36). There
will be additional contact term generated involved in such a relation. This
point will be relevant for our computation of longitudinal correlators below,
especially when using the bootstrap equations in Sec. 5.

###  2.2 Correlators and the vector seed integral

From this point onward, we shall take $H=1$ to simplify expressions.

With the massive spin-1 fields properly quantized and their propagators
obtained, we are now ready to construct helical inflation correlators. In this
work, we focus on the tree-level exchange of massive spinning particles mainly
in 4-point correlators, namely the trispectrum. The 3-point correlators,
namely the bispectrum, will be considered in Sec. 6 as soft limits of 4-point
correlators.

Figure 1: The 4-point correlators mediated by a massive spin-1 field in the
$s$-channel at the tree level. The left diagram shows the correlator of four
identical external scalars
$\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\varphi_{\mathbf{k}_{4}}\rangle^{\prime}$,
which corresponds to the amplitude in (42). The right diagram corresponds to
correlators with two distinct external scalars
$\langle\varphi_{1,\mathbf{k}_{1}}\varphi_{2,\mathbf{k}_{2}}\varphi_{1,\mathbf{k}_{3}}\varphi_{2,\mathbf{k}_{4}}\rangle^{\prime}$,
which corresponds to the amplitude in (2.2).

The 4-point correlators we will be considering are shown in Fig. 1. In this
figure, we only draw the $s$-channel exchange. There are two other similar
processes with $t$-channel and $u$-channel exchanges that can be obtained by
simple permutations $\mathbf{k}_{1}\leftrightarrow\mathbf{k}_{4}$ and
$\mathbf{k}_{1}\leftrightarrow\mathbf{k}_{3}$, respectively. To form a
correlator as in Fig. 1, we need a trilinear interaction vertex with two
external scalar fields and one internal spinning field, to be introduced
below.

#### Trilinear couplings.

From the previous subsection, we see that the helical chemical potential
affects only the transverse components. Therefore, the trilinear coupling must
see the spatial components $A_{i}$ of the spin-1 field in order to see the
effect of the chemical potential. For this reason, we shall only consider
couplings involving $A_{i}$ and neglect couplings to $A_{0}$. Also, for
convenience, we shall consider a process with two distinct external massless
scalars $\varphi_{1}$ and $\varphi_{2}$, shown by the right diagram in Fig. 1.
(The result for four identical external scalars, namely the left diagram of
Fig. 1, can be obtained by permuting the momenta. See below.) With the
symmetry of the problem given, we can write down the following most general
trilinear interaction vertex among $\varphi_{1}$, $\varphi_{2}$, and $A_{i}$:

$\mathcal{O}_{\mathbb{P}}=\lambda_{\mathbb{P}}a^{2-(J+K)-2(M+N+R)}\Big{[}\partial_{\tau}^{J}\partial_{i_{1}}\cdots\partial_{i_{M}}(\partial_{j}\partial^{j})^{N}\varphi_{1}\Big{]}\Big{[}\partial_{\tau}^{K}\partial^{i}\partial^{i_{1}}\cdots\partial^{i_{M}}(\partial_{k}\partial^{k})^{R}\varphi_{2}\Big{]}A_{i}.$
(37)

Here the subscript $\mathbb{P}=(J,K,M,N,R)$ is a set of nonnegative integers
specifying the number of various temporal and spatial derivatives, and it also
serves as a label for the operator $\mathcal{O}_{\mathbb{P}}$ and the coupling
strength $\lambda_{\mathbb{P}}$. The power of the scale factor $a$ is uniquely
fixed by the scale invariance. It is of course possible to act temporal and
spatial derivatives on $A_{i}$, but these derivatives can always be moved to
$\varphi_{1}$ and $\varphi_{2}$ using integration by parts. Therefore, any
operators involving derivatives of $A_{i}$ can be written as linear
combinations of operators of the above form up to irrelevant boundary terms.

It might be useful to find the result for the correlator with couplings
written in the above very general form. However, it is clearly rather
cumbersome to stay in the most general case throughout our presentation. Also,
the final result will be rather complicated and uninspiring. Therefore we
shall present the result for the 4-point correlator with the most general
coupling (37) in App. B. Here, for the sake of clarity, we will set $J=1$ and
$K=M=N=R=0$ in the operator $\mathcal{O}_{\mathbb{P}}$ in (37), and we simply
call the resulting operator $\mathcal{O}$ :

$\mathcal{O}=\lambda a\varphi_{1}^{\prime}(\partial^{i}\varphi_{2})A_{i}.$
(38)

This could well be the simplest choice for the trilinear coupling, and is the
lowest dimension coupling one can write down for the three fields
$\varphi_{1}$, $\varphi_{2}$, and $A_{i}$ that respects the shift symmetry of
the scalar fields. Note in particular that the shift symmetry requires us to
retain at least one derivative to each of the two scalar fields.

#### SK integral.

Now we are ready to write down an expression for the 4-point correlator shown
in the right diagram of Fig. 1, with the two vertices given by the simple
operator $\mathcal{O}$ in (38), following the diagrammatic rule summarized in
[57]. More explicitly, we use $G_{\mathsf{a}}(k,\tau)$ to denote the bulk-to-
boundary propagator for the external scalar fields:

$G_{\mathsf{a}}(k,\tau)=\displaystyle\frac{\,{1}\,}{2k^{3}}(1-\mathrm{i}\mathsf{a}k\tau)e^{\mathrm{i}\mathsf{a}k\tau}.$
(39)

Since we are assuming both $\varphi_{1}$ and $\varphi_{2}$ massless, we do not
use different letters to distinguish their propagators. Then, we can write
down the amplitude mediated by each of the four polarizations of the spin-1
field:

$\displaystyle\mathcal{T}_{\text{2}}^{(\lambda)}=$
$\displaystyle-\lambda^{2}(\mathrm{i}\mathbf{k}_{2}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\lambda)})(\mathrm{i}\mathbf{k}_{4}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\lambda)*})\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{ab}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{-\tau_{1}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{-\tau_{2}}$
$\displaystyle\times\big{[}\partial_{\tau_{1}}G_{\mathsf{a}}(k_{1},\tau_{1})\big{]}G_{\mathsf{a}}(k_{2},\tau_{1})\big{[}\partial_{\tau_{2}}G_{\mathsf{b}}(k_{3},\tau_{2})\big{]}G_{\mathsf{b}}(k_{4},\tau_{2})D_{\mathsf{ab}}^{(\lambda)}(k_{s};\tau_{1},\tau_{2}).$
(40)

Then, use (39), we find

$\displaystyle\mathcal{T}_{\text{2}}^{(\lambda)}=$
$\displaystyle~{}\lambda^{2}(\mathbf{k}_{2}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\lambda)})(\mathbf{k}_{4}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\lambda)*})\displaystyle\frac{\,{1}\,}{16k_{1}k_{2}^{3}k_{3}k_{4}^{3}}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}$
$\displaystyle~{}\times\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}(1-\mathrm{i}\mathsf{a}k_{2}\tau_{1})(1-\mathrm{i}\mathsf{b}k_{4}\tau_{2})e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{a}\mathsf{b}}^{(\lambda)}(k_{s};\tau_{1},\tau_{2}).$
(41)

The process with four identical scalars, namely the left diagram of Fig. 1,
can be obtained directly by permutation of the four external momenta:

$\mathcal{T}_{\text{1}}^{(\lambda)}(k_{I})=\displaystyle\frac{\,{1}\,}{4}\bigg{[}\mathcal{T}_{\text{2}}^{(\lambda)}(k_{I})+(k_{1}\leftrightarrow
k_{2})+(k_{3}\leftrightarrow k_{4})+\binom{k_{1}\leftrightarrow
k_{2}}{k_{3}\leftrightarrow k_{4}}\bigg{]}.$ (42)

There are some cancellations in this permutation in the leading order CC
signals in the squeezed limit. However, the CC signals are still present in
the subleading terms, albeit with a different angular dependence. See [35] for
more discussions.

#### Helical Vector Seed Integral.

One can well follow the standard procedure and substitute the explicit
expressions of the spin-1 propagators (34) and (36) into (2.2). However, the
computation of the two time integrals is not that trivial. Therefore, we will
isolate this integral from the rest of the expression, by defining the
following _helical vector seed integral_ :

$\displaystyle\mathcal{I}^{(\lambda)p_{1}p_{2}}_{\mathsf{ab}}\equiv$
$\displaystyle-\mathsf{ab}\,k_{s}^{3+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{ab}}^{(\lambda)}(k_{s};\tau_{1},\tau_{2}).$
(43)

Let us make some remarks about this vector seed integral before going on to
consider the 4-point correlator $\mathcal{T}_{\text{2}}^{(\lambda)}$.

First, the minus sign and also the factor $\mathsf{a}\mathsf{b}$ in front of
everything in (43) are from the two vertices, which give
$\mathrm{i}\mathsf{a}$ and $\mathrm{i}\mathsf{b}$ as the diagrammatic rule
required. Second, we have introduced two arbitrary powers of time
$(-\tau_{1})^{p_{1}}$ and $(-\tau_{2})^{p_{2}}$, and it turns out that this
slight generalization suffices to accommodate a wide range of diagrams,
including the diagrams with the most general coupling (37). It is for this
reason that we call $\mathcal{I}^{(\lambda)p_{1}p_{2}}_{\mathsf{ab}}$ a seed
integral. Third, we have also introduced the factor $k_{s}^{3+p_{1}+p_{2}}$ to
balance the powers of time inside the integral, so that the whole expression
$\mathcal{I}^{(\lambda)p_{1}p_{2}}_{\mathsf{ab}}$ is dimensionless and scale
invariant, and must be expressible in terms of momentum ratios.666This is not
the unique choice. We can also balance the $\tau$-powers by including a factor
of $k_{12}^{3/2+p_{1}}k_{34}^{3/2+p_{2}}$. Our choice $k_{s}^{3+p_{1}+p_{2}}$
slightly simplifies some derivations and the final expressions. It turns out
that the integral (43) depends on two independent momentum ratios, reminiscent
of the behavior of a 4-point correlator in ordinary conformal field theories.
As we shall see below, it is convenient to define the following momentum
ratios:

$\displaystyle r_{1}\equiv\displaystyle\frac{\,{k_{s}}\,}{k_{12}},$
$\displaystyle r_{2}\equiv\displaystyle\frac{\,{k_{s}}\,}{k_{34}}.$
$\displaystyle
u_{1}\equiv\displaystyle\frac{\,{2k_{s}}\,}{k_{12}+k_{s}}=\displaystyle\frac{\,{2r_{1}}\,}{1+r_{1}},$
$\displaystyle
u_{2}\equiv\displaystyle\frac{\,{2k_{s}}\,}{k_{34}+k_{s}}=\displaystyle\frac{\,{2r_{2}}\,}{1+r_{2}}.$
(44)

Clearly, the momentum conservation at each vertex, namely
$\mathbf{k}_{1}+\mathbf{k}_{2}=\mathbf{k}_{s}=-(\mathbf{k}_{3}+\mathbf{k}_{4})$,
requires that $r_{1},r_{2},u_{1},u_{2}\in[0,1]$. As we shall see in Sec. 4, it
turns out more convenient to use $(r_{1},r_{2})$ as independent variables for
the longitudinal integral $\mathcal{I}^{(L)p_{1}p_{2}}_{\mathsf{ab}}$ while
$(u_{1},u_{2})$ is a better choice for the transverse integral
$\mathcal{I}^{(\pm)p_{1}p_{2}}_{\mathsf{ab}}$.

#### From the seed integral to the correlator.

Now, in terms of the vector seed integral (43), we can write the 4-point
correlator $\mathcal{T}_{\text{2}}^{(\lambda)}$ in two different ways. First,
the correlator can be written as a linear combination of seed integrals with
different SK indices:

$\displaystyle\mathcal{T}_{\text{2}}^{(\lambda)}=$
$\displaystyle~{}\displaystyle\frac{\,{-\lambda^{2}}\,}{16k_{1}k_{2}k_{3}k_{4}k_{s}^{3}}\displaystyle\frac{\,{(\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern
2.0mu_{2}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\lambda)})(\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern
2.0mu_{4}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\lambda)*})}\,}{k_{2}k_{4}}$
$\displaystyle~{}\times\sum_{\mathsf{a},\mathsf{b}=\pm}\Big{[}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\lambda)0,0}-\mathrm{i}\mathsf{a}\displaystyle\frac{\,{k_{2}}\,}{k_{s}}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\lambda)1,0}-\mathrm{i}\mathsf{b}\displaystyle\frac{\,{k_{4}}\,}{k_{s}}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\lambda)0,1}-\mathsf{a}\mathsf{b}\displaystyle\frac{\,{k_{2}k_{4}}\,}{k_{s}^{2}}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\lambda)1,1}\Big{]}.$
(45)

The advantage of the representation (2.2) is that one only need to form simple
linear combination once the result for the vector seed integral is known.
However, an unsatisfactory aspect of this representation is that the
coefficients of some terms contain the SK indices $\mathsf{a},\mathsf{b}$,
which means that we have to know the result of
$\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}$ for each
$\mathsf{a},\mathsf{b}=\pm$. While there is no problem in obtaining the
results for each individual integral, it is somewhat redundant to present the
result for each $\mathsf{a},\mathsf{b}$ separately. In fact, the results for
different $\mathsf{a},\mathsf{b}=\pm$ are intimately related, not least
because the bulk propagator $D_{\mathsf{a}\mathsf{b}}$ can all be constructed
from a single Wightman function $D_{>}$.

There is an alternative way to relate the vector seed integral and the
correlator, which is more in line with the spirit of the recent bootstrap
papers [20, 22, 32, 33], where the correlators of more complicated processes
are all constructed from a seed correlator by acting appropriate differential
or integral operators. For our example of
$\mathcal{T}_{\text{2}}^{(\lambda)}$, one can easily verify the following
relation:

$\displaystyle\mathcal{T}_{\text{2}}^{(\lambda)}=\displaystyle\frac{\,{-\lambda^{2}}\,}{16k_{1}k_{2}k_{3}k_{4}k_{s}^{3}}\displaystyle\frac{\,{(\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern
2.0mu_{2}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\lambda)})(\mkern
2.0mu\widehat{\mkern-2.0mu\mathbf{k}\mkern-2.0mu}\mkern
2.0mu_{4}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\lambda)*})}\,}{k_{2}k_{4}}\times(1-k_{2}\partial_{k_{12}})(1-k_{4}\partial_{k_{34}})\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\lambda)0,0}.$
(46)

In this representation, we only need the summed vector seed integral
$\sum_{\mathsf{a},\mathsf{b}}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\lambda)p_{1}p_{2}}$.
However, we need to act differential operators to the summed integral, which
is itself a very complicated expression. In practical calculations, we do not
have to stick to one representation, but can freely exploit both
representations in order to simplify the calculation. We will illustrate this
point in the subsequent sections. Of course, whatever approach taken, the
final answer should be the same, in part because acting differential or
integral operators on the seed correlator is usually equivalent to changing
some parameters in it.

To summarize, in this section, we have reduced the computation of the helical
4-point correlators to the computation of a vector seed integral (43). In Sec.
4, we will carry out this computation and present an exact and analytical
expression for it. We will also provide independent derivations for the seed
integrals for particular choices of $p_{1,2}$ in Sec. 5, using a bootstrap
approach.

## 3 Inflation Correlators with Partial Mellin-Barnes Representation

From the previous section, we see that the major task in computing the
correlator is to evaluate the SK integrals, preferably in an exact and
analytic form. For a wide range of applications involving helicity dependent
chemical potentials, it suffices to compute a single vector seed integral,
defined in (43). The seed integral is a two-layer integral over products of
special functions with time orderings, which make the computation difficult.

The approach we shall take to compute these integrals is the so-called partial
Mellin-Barnes representation that we introduced in [35]. Before applying this
method to the vector seed integral, however, we shall demonstrate the use of
partial Mellin transformation with a simpler example, namely the 4-point
correlator mediated by a massive scalar field. The mode function associated
with massive scalars involves the Hankel functions, which is somewhat simpler
than the Whittaker function in the spin-1 mode function. Also, the result
obtained in this section can be recycled when we consider the longitudinal
spin-1 exchange. Finally, we shall present a new expression for the
“background” part of the correlator, which is expressed as a fast converging
series of $k_{s}/k_{12}$ alone, while the dependence on $k_{12}/k_{34}$ is
fully resummed into a hypergeometric function, as opposed to the double-series
expressions given in previous works. A detailed comparison between these two
types of results will be given in Sec. 5.

#### Mellin transformation.

The Mellin transformation of a function $f(x)$ is defined by

$F(s)=\int_{0}^{\infty}{\mathrm{d}}x\,x^{s-1}f(x).$ (47)

The inverse Mellin transformation, also known as the Mellin-Barnes (MB)
representation, is given by

$f(x)=\int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\,x^{-s}F(s),$
(48)

where $c\in(\alpha,\beta)$ is a real number and $(\alpha,\beta)$ is an open
interval in the real axis that includes 0. Very often, we can simply take
$c=0$.

Like the more familiar Fourier transformation, the Mellin transformation is a
linear transformation of functions, but with power function $x^{s}$ instead of
exponential function $e^{\mathrm{i}kx}$, as the transformation kernel. In dS,
the background geometry possesses a dilatation symmetry generated by
$\tau\partial_{\tau}$, with power functions $(-\tau)^{\Delta}$ as
eigenfunctions. (In AdS, it is the radial direction that has a similar
symmetry.) Therefore, the Mellin transformation in dS is really a natural
analogy of Fourier transformation of time in Minkowski space. One can thus
imagine that the Feynman rules and computations for dS correlators can be
drastically simplified if we Mellin transform the time direction, in addition
to the usual spatial Fourier transformation. For instance, the difficult time
integral for each interaction vertex is replaced by a $\delta$-function of
Mellin variables, much like a distorted form of energy conservation in curved
space. Indeed, the advantages of Mellin amplitudes in the context of inflation
correlators have been investigated in [24, 25, 26, 27, 28].

On the other hand, the inflation correlators are boundary correlators of
operators with fixed time $\tau=0$, and therefore, to obtain these correlators
from the Mellin amplitudes, we need to perform inverse Mellin transformation
separately at _each_ external points, which is likely not a trivial step, and
this limits the direct use of Mellin amplitudes for practical study of
inflation correlators.

To take the advantage of Mellin transformation in a more practical way, we
adopted the _partial Mellin-Barnes representation_ introduced in [35]. The
essential idea is simple: We use MB representations only for all massive
internal modes. The MB representation resolves various special functions
associated with these massive modes into powers of time variable $\tau$, and
thus trivializes the time integrals. Then, the MB integral can be carried out
by closing the contour appropriately and picking up residues of enclosed
singularities.

###  3.1 Scalar exchange revisited

Given that the partial MB representation has not been widely adopted in the
study of inflation correlators, in this section, we shall walk the readers
through the main steps of this method with a simpler example, namely the tree-
level exchange of a massive real scalar field in 4-point correlator. The
complete analytical result for this process has been worked out in [20]. In
addition to providing independent checks, our final expression features an
improved series expression for the background piece of the correlator, which
resums the powers of $r_{1}/r_{2}$ to all orders into tractable Gauss
hypergeometric functions.

Figure 2: The 4-point inflation correlator with tree-level exchange of a
massive scalar field $\sigma$ in the $s$-channel.

To be concrete, we consider the 4-point correlator shown in Fig. 2. The
external legs represent bulk-to-boundary propagators of $\varphi$ field with
momenta $\mathbf{k}_{i}$ ($i=1,\cdots,4$). In the study of realistic inflation
correlators, $\varphi$ is usually taken to be the nearly massless inflaton
fluctuation. Its bulk-to-boundary propagator is given in (39).

It is sometimes convenient to take the external legs to be conformal scalars
$\phi_{c}$ with mass $m^{2}=2H^{2}$. These scalars do not propagate to the
future boundary $\tau=0$ of dS; their mode functions are proportional to
$-\tau$ and vanish when $\tau=0$. Therefore, to write their bulk-to-boundary
propagator $C_{\mathsf{a}}(k,\tau)$, we introduce a final-time cutoff
$\tau=\tau_{f}$:

$C_{\mathsf{a}}(k,\tau)=\displaystyle\frac{\,{\tau\tau_{f}}\,}{2k}e^{\mathsf{a}\mathrm{i}k\tau}.$
(49)

The internal (blue) line in Fig. 2 represents the bulk propagator
$D_{\mathsf{a}\mathsf{b}}(k,\tau_{1},\tau_{2})$ of a massive scalar field
$\sigma$. The four bulk propagators are again related to the Wightman
functions $D_{>}$ and its complex conjugate $D_{<}$ in a way similar to the
vector case shown in (30)-(33). (One simply removes the superscript
$(\lambda)$ in those expressions.) For a massive scalar, the Wightman function
$D_{>}$ is given by:

$D_{>}(k;\tau_{1},\tau_{2})=\displaystyle\frac{\,{\pi}\,}{4}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}H^{2}(\tau_{1}\tau_{2})^{3/2}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k\tau_{1})\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}(-k\tau_{2}),$
(50)

where $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\equiv\sqrt{m^{2}/H^{2}-9/4}$ is the mass parameter.

There are many possible choices for the two vertices in Fig. 2. Since we are
considering the scalar correlator only for illustrative purpose, we do not aim
at the most general coupling. Instead, we will only mention several frequently
encountered examples.

In the case of the massless inflaton fluctuation being the external modes, the
simplest choice is the following one:

$\Delta{\mathscr{L}}=\displaystyle\frac{\,{1}\,}{2}\lambda
a^{2}\varphi^{\prime 2}\sigma.$ (51)

We choose the derivative coupling to keep the (approximate) shift symmetry of
the inflaton field.

For the conformal scalar $\phi_{c}$, one can simply choose a direct coupling:

$\Delta{\mathscr{L}}=\displaystyle\frac{\,{1}\,}{2}\lambda_{c}a^{4}\phi_{c}^{2}\sigma.$
(52)

With all the propagators and vertices given, it is straightforward to write
down the expression for the correlator in Fig. 2 following the standard
procedure [57]. For instance, the inflaton correlator
$\langle\varphi^{4}\rangle$ with the interaction (51) is given by

$\displaystyle\mathcal{T}_{\varphi}\equiv$
$\displaystyle~{}\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\varphi_{\mathbf{k}_{4}}\rangle_{s}^{\prime}$
$\displaystyle=$
$\displaystyle-\lambda^{2}\sum_{\mathsf{a,b}=\pm}\mathsf{ab}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{(-\tau_{1})^{2}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{(-\tau_{2})^{2}}\,G_{\mathsf{a}}^{\prime}(k_{1},\tau_{1})G_{\mathsf{a}}^{\prime}(k_{2},\tau_{1})G_{\mathsf{b}}^{\prime}(k_{3},\tau_{2})G_{\mathsf{b}}^{\prime}(k_{4},\tau_{2})D_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2})$
$\displaystyle=$
$\displaystyle-\displaystyle\frac{\,{\lambda^{2}}\,}{16k_{1}k_{2}k_{3}k_{4}}\sum_{\mathsf{a,b}=\pm}\mathsf{ab}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2}).$
(53)

The conformal scalar correlator $\langle\phi_{c}^{4}\rangle$, on the other
hand, is given by:

$\displaystyle\mathcal{T}_{\phi_{c}}\equiv$
$\displaystyle~{}\langle\phi_{c,\mathbf{k}_{1}}\phi_{c,\mathbf{k}_{2}}\phi_{c,\mathbf{k}_{3}}\phi_{c,\mathbf{k}_{4}}\rangle_{s}^{\prime}$
$\displaystyle=$
$\displaystyle-\displaystyle\frac{\,{\lambda_{c}^{2}\tau_{f}^{4}}\,}{16k_{1}k_{2}k_{3}k_{4}}\sum_{\mathsf{a,b}=\pm}\mathsf{ab}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(\tau_{1}\tau_{2})^{-2}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2}).$
(54)

By examining these and other more general examples, it turns out useful to
define the following _scalar seed integral_ :

$\displaystyle\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(r_{1},r_{2})\equiv-\mathsf{ab}\,k_{s}^{5+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2}).$
(55)

As in the vector seed integral, here we have kept the powers of $\tau_{1}$ and
$\tau_{2}$ general. Also, the integral
$\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(r_{1},r_{2})$ thus defined is
dimensionless and scale invariant, and depends on various momenta only through
two ratios $r_{1}=k_{s}/k_{12}$ and $r_{2}=k_{s}/k_{34}$.

It is now trivial to see that our previous two examples can be conveniently
expressed as:

$\mathcal{T}_{\varphi}=\displaystyle\frac{\,{\lambda^{2}}\,}{16k_{1}k_{2}k_{3}k_{4}k_{s}^{5}}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}^{00}_{\mathsf{a}\mathsf{b}}(r_{1},r_{2}),$
(56)

and,

$\mathcal{T}_{\phi_{c}}=\displaystyle\frac{\,{\lambda_{c}^{2}\tau_{f}^{4}}\,}{16k_{1}k_{2}k_{3}k_{4}k_{s}}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}^{-2,-2}_{\mathsf{a}\mathsf{b}}(r_{1},r_{2}).$
(57)

Figure 3: The 3-point correlator of the inflaton fluctuation $\varphi$
mediated by a massive scalar field $\sigma$ at the tree level.

The scalar seed integral thus defined can be used to compute more general
inflation correlators. A phenomenologically important example is the 3-point
inflaton correlator $\langle\varphi^{3}\rangle$, shown in Fig. 3. In this
figure, the left vertex is again given by (51), while the right two-point
mixing vertex is given by the interaction
$\Delta{\mathscr{L}}=\lambda_{2}a^{3}\varphi^{\prime}\sigma$. Then the 3-point
function can be expressed as

$\displaystyle\mathcal{B}_{\phi}=\langle\varphi_{\mathbf{k}_{1}}\varphi_{\mathbf{k}_{2}}\varphi_{\mathbf{k}_{3}}\rangle_{3}^{\prime}=\displaystyle\frac{\,{\lambda\lambda_{2}}\,}{8k_{1}k_{2}k_{3}^{4}}\lim_{r_{2}\rightarrow
1^{-}}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{0,-2}(r_{1},r_{2}).$
(58)

Here we add a subscript 3 to highlight the factor that we are including only
the figure with the massive propagator carrying momentum $\mathbf{k}_{3}$. The
full result should also include two permutations, which we shall always
neglect for simplicity. Incidentally, we note that setting directly $r_{1}=1$
or $r_{2}=1$ in $\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(r_{1},r_{2})$
may give rise to superficially divergent expressions, which are nevertheless
canceled among different SK branches. We will come back to this point in Sec.
5. An explicit closed-form analytical expression for (58) will be given in
(6).

Similar expressions can be obtained for more general examples including
spatial derivative couplings. Therefore, the computation of scalar-exchange
correlators has been largely reduced to the computation of scalar seed
integral, which we shall elaborate in the rest of this section.

###  3.2 Partial MB representation

In this subsection we are going to use the partial MB representation to
compute the scalar seed integral (55). This subsection is somewhat technical.
Readers uninterested in the details can jump directly to Sec. 3.3 for a
summary of the result.

The main strategy is to use the MB representation to resolve the bulk massive
propagators into powers of time. In the scalar seed integral, the bulk
propagator (50) includes a pair of Hankel functions. The MB representation of
the Hankel function is given by [109]:

$\text{H}_{\nu}^{(j)}(az)=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(az/2)^{-2s}}\,}{\pi}e^{(-1)^{j+1}(2s-\nu-1)\pi\mathrm{i}/2}\Gamma\Big{[}s-\displaystyle\frac{\,{\nu}\,}{2},s+\displaystyle\frac{\,{\nu}\,}{2}\Big{]}.~{}~{}~{}~{}(j=1,2)$
(59)

It is then straightforward to get the MB representation for the four bulk
propagators:

$\displaystyle D_{\pm\mp}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,e^{\mp\mathrm{i}\pi(s_{1}-s_{2})}\Big{(}\displaystyle\frac{\,{k}\,}{2}\Big{)}^{-2s_{12}}(-\tau_{1})^{-2s_{1}+3/2}(-\tau_{2})^{-2s_{2}+3/2}$
$\displaystyle\times\Gamma\Big{[}s_{1}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{1}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{2}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{2}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}\Big{]},$ (60)
$\displaystyle D_{\pm\pm}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}D_{\mp\pm}(k;\tau_{1},\tau_{2})\theta(\tau_{1}-\tau_{2})+D_{\pm\mp}(k;\tau_{1},\tau_{2})\theta(\tau_{2}-\tau_{1}).$
(61)

Here and below, we use shorthand $s_{12}=s_{1}+s_{2}$. Substituting the MB
bulk propagators (3.2) and (61) into the scalar seed integral (55), we resolve
all time dependences into powers, and thus simplify the time integral. This is
particularly true for the opposite-sign integrals
$\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}(r_{1},r_{2})$, in which the time integrals
yield new factors of Euler $\Gamma$ functions:

$\displaystyle\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi}e^{\mp\mathrm{i}\pi(p_{1}-p_{2})/2}r_{1}^{5/2+p_{1}}r_{2}^{5/2+p_{2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-2s_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-2s_{2}}$
$\displaystyle\times\Gamma\Big{[}p_{1}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{1},p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{2},s_{1}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{1}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\Big{]}.$
(62)

The same-sign integrals, however, are a bit more complicated owing to the
Heaviside $\theta$ functions for the (anti-)time-orderings. There are multiple
ways to deal with this time ordering. It turns that different treatments will
yields results with different convergent speeds, and the convergence speed
also depends on whether $k_{12}>k_{34}$ or $k_{12}<k_{34}$.

Assuming $k_{12}>k_{34}$ (and thus $r_{1}<r_{2}$) for the moment. It turns out
useful to adopt the following rearrangement of the bulk propagator
$D_{\pm\pm}$:

$\displaystyle D_{\pm\pm}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}D_{\gtrless}(k;\tau_{1},\tau_{2})+\Big{[}D_{\lessgtr}(k;\tau_{1},\tau_{2})-D_{\gtrless}(k;\tau_{1},\tau_{2})\Big{]}\theta(\tau_{2}-\tau_{1}).$
(63)

Correspondingly, we rewrite the seed integral
$\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}(r_{1},r_{2})$ as:

$\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}(r_{1},r_{2})=\mathcal{I}_{\pm\pm,\text{F},>}^{p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\pm\pm,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2}).~{}~{}~{}~{}(r_{1}<r_{2})$
(64)

Here the factorized (F) integral
$\mathcal{I}_{\pm\pm,\text{F},>}^{p_{1}p_{2}}(r_{1},r_{2})$ and the time-
ordered (TO) integral
$\mathcal{I}_{\pm\pm,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2})$ are respectively
defined as: (The subscript $>$ associated with these integrals serves as a
reminder that we are focusing on the region $k_{12}>k_{34}$.)

$\displaystyle\mathcal{I}_{\pm\pm,\text{F},>}^{p_{1}p_{2}}(r_{1},r_{2})\equiv$
$\displaystyle-
k_{s}^{5+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\pm\mathrm{i}(k_{12}\tau_{1}+k_{34}\tau_{2})}D_{\gtrless}(k_{s};\tau_{1},\tau_{2}),$
(65)
$\displaystyle\mathcal{I}_{\pm\pm,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2})\equiv$
$\displaystyle-
k_{s}^{5+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{2}\int_{-\infty}^{\tau_{2}}{\mathrm{d}}\tau_{1}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\pm\mathrm{i}(k_{12}\tau_{1}+k_{34}\tau_{2})}$
$\displaystyle~{}\times\Big{[}D_{\lessgtr}(k;\tau_{1},\tau_{2})-D_{\gtrless}(k;\tau_{1},\tau_{2})\Big{]}.$
(66)

Then, the factorized time integral, very similar to the opposite-sign
integrals (62), can be carried out directly:

$\displaystyle{\mathcal{I}}_{\pm\pm,\text{F},>}^{p_{1}p_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi}e^{\mp\mathrm{i}\pi(p_{1}+p_{2})/2}r_{1}^{5/2+p_{1}}r_{2}^{5/2+p_{2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,(\pm\mathrm{i}e^{\pm
2\mathrm{i}\pi
s_{1}})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-2s_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-2s_{2}}$
$\displaystyle\times\Gamma\Big{[}p_{1}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{1},p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{2},s_{1}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{1}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\Big{]},$
(67)

The time-ordered integral can also be carried out with the aid of (A):

$\displaystyle{\mathcal{I}}_{\pm\pm,\text{TO},>}^{p_{1}p_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi}e^{\mp\mathrm{i}\pi(p_{1}+p_{2})/2}r_{1}^{5+p_{1}+p_{2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,(\mp\mathrm{i}e^{\pm
2\mathrm{i}\pi s_{1}}\pm\mathrm{i}e^{\pm 2\mathrm{i}\pi
s_{2}})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-2s_{12}}$
$\displaystyle\times\Gamma\Big{[}p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{2},p_{1}+p_{2}+5-2s_{12},s_{1}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{1}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\Big{]}$
$\displaystyle\times{}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}\left[\begin{matrix}p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{2},p_{1}+p_{2}+5-2s_{12}\\\
p_{2}+\mbox{$\frac{\,{7}\,}{2}$}-2s_{2}\end{matrix}\middle|\,-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\right].$
(68)

Here ${}_{2}\mkern 2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}$ is the regularized hypergeometric function. See (276) for the
definition.

It turns out that the above results obtained with the assumption of
$k_{12}>k_{34}$ are also valid when $k_{12}<k_{34}$. We suspect the reason is
that our result for the background has resummed all powers of $k_{34}/k_{12}$
into a well behaved function which is analytic at $k_{12}=k_{34}$, so that the
series of $r_{1}/r_{2}$ originally defined in the range of $k_{12}>k_{34}$ is
analytically continued to the opposite side $k_{12}<k_{34}$. However, we also
observe that the convergence of the $r_{1}$ series in (68) becomes slow as
$r_{1}\rightarrow 1$.

Alternatively, we can also perform the calculation again with the assumption
$k_{12}<k_{34}$ from the very beginning. For instance, we can also define the
factorized integral
$\mathcal{I}_{\pm\pm,\text{F},<}^{p_{1}p_{2}}(r_{1},r_{2})$ and the time-
ordered integral $\mathcal{I}_{\pm\pm,\text{TO},<}^{p_{1}p_{2}}(r_{1},r_{2})$.
The results can be simply obtained by switching the variables
$(r_{1},p_{1})\leftrightarrow(r_{2},p_{2})$. In practice, it turns out that
the integrals expressed in the following way would have fast convergence speed
for all physical range of parameters:

$\displaystyle\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{I}_{\pm\pm,\text{F}}^{p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\pm\pm,\text{TO}}^{p_{1}p_{2}}(r_{1},r_{2}),$
(69) $\displaystyle\mathcal{I}_{\pm\pm,\text{F}}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{I}_{\pm\pm,\text{F},>}^{p_{1}p_{2}}(r_{1},r_{2})\theta(r_{2}-r_{1})+\mathcal{I}_{\pm\pm,\text{F},>}^{{p_{2}p_{1}}}(r_{2},r_{1})\theta(r_{1}-r_{2}),$
(70) $\displaystyle\mathcal{I}_{\pm\pm,\text{TO}}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{I}_{\pm\pm,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2})\theta(r_{2}-r_{1})+\mathcal{I}_{\pm\pm,\text{TO},>}^{{p_{2}p_{1}}}(r_{2},r_{1})\theta(r_{1}-r_{2}).$
(71)

#### Signal and background.

We are now ready to carry out the MB integrals in (62), (67) and (68), by
closing the contours and picking up the enclosed poles. Again let us focus on
the case of $k_{12}>k_{34}$. In this region we have $0<r_{1}<r_{2}<1$, which
shows that we should close the contour from left for each of the integrals.
Then, all the enclosed poles are from the $\Gamma$ functions of the form
$\Gamma(s_{1,2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu/2)$, which are called
“left poles.” These poles can be divided into two sets:

$\displaystyle s_{1}=-n_{1}\mp\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2},\quad
s_{2}=-n_{2}\pm\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2},$ (72)
$\displaystyle s_{1}=-n_{1}\mp\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2},\quad
s_{2}=-n_{2}\mp\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}.$ (73)

Then using the residue theorem, we can complete the integrals. The integrals
(62) and (67) can be combined together:

$\displaystyle{\mathcal{I}}_{\text{S},>}^{p_{1}p_{2}}(r_{1},r_{2})\equiv$
$\displaystyle~{}\mathcal{I}_{+-}^{p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{++,\text{F},>}^{p_{1}p_{2}}(r_{1},r_{2})+\text{c.c.}.$
(74)

More explicitly,

$\displaystyle~{}{\mathcal{I}}_{\text{S},>}^{p_{1}p_{2}}(r_{1},r_{2})$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi}r_{1}^{5/2+p_{1}}r_{2}^{5/2+p_{2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,\Big{[}e^{-\mathrm{i}\pi(p_{1}-p_{2})/2}+\mathrm{i}e^{-\mathrm{i}\pi(p_{1}+p_{2})/2}e^{2\mathrm{i}\pi
s_{1}}\Big{]}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-2s_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-2s_{2}}$
$\displaystyle\times\Gamma\Big{[}p_{1}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{1},p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-2s_{2},s_{1}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{1}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}\Big{]}+\text{c.c.}.$ (75)

The result of this integral is obtained by summing over the residues of all
the poles listed in (72) and (73). It is clear that the first set of poles
(72) contribute to a result proportional to
$(r_{1}/r_{2})^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, which is the local
(L) signal, while the second set (73) leads to the nonlocal (NL) signal of the
form $(r_{1}r_{2})^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$. For positive real
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$, it is clear
that the local and nonlocal signals exhibit oscillatory behavior in
$\log(r_{1}/r_{2})$ and $\log(r_{1}r_{2})$, respectively, and this is why they
are called “signals.” For phenomenological applications, it would be useful to
spell out these signals more clearly. Therefore, we make the following
rearrangement:

${\mathcal{I}}_{\text{S},>}^{p_{1}p_{2}}(r_{1},r_{2})={\mathcal{I}}_{\text{L},>}^{p_{1}p_{2}}(r_{1},r_{2})+{\mathcal{I}}_{\text{NL},>}^{p_{1}p_{2}}(r_{1},r_{2}),$
(76)

in which the local-signal integral is:

$\displaystyle~{}{\mathcal{I}}_{\text{L},>}^{p_{1}p_{2}}(r_{1},r_{2})$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi}r_{1}^{5/2+p_{1}}r_{2}^{5/2+p_{2}}\sum_{n_{1},n_{2}=0}^{\infty}\bigg{\\{}\displaystyle\frac{\,{(-1)^{n_{12}}}\,}{n_{1}!n_{2}!}\Big{[}e^{-\mathrm{i}\pi(p_{1}-p_{2})/2}+\mathrm{i}e^{-\mathrm{i}\pi(p_{1}+p_{2})/2}e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{]}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\Big{[}2n_{1}+p_{1}+\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,2n_{2}+p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigg{\\}}+\text{c.c.}$ $\displaystyle=$
$\displaystyle~{}\mathcal{C}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}}(r_{1})\mathbf{F}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{2}}(r_{2})+\text{c.c.}.$ (77)

Here we have defined

$\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p}(r)\equiv
r^{5/2+p}\Gamma\Big{[}\displaystyle\frac{\,{5}\,}{2}+p+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{5}\,}{4}$}+\mbox{$\frac{\,{p}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{7}\,}{4}$}+\mbox{$\frac{\,{p}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,r^{2}\right],$ (78)

and

$\mathcal{C}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}\equiv\displaystyle\frac{\,{1}\,}{2\pi}\Big{[}\cos\big{(}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}-p_{2})\big{)}+\sin\big{(}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+p_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\big{)}\Big{]}.$ (79)

Similarly, the nonlocal signal can be worked out as

$\displaystyle{\mathcal{I}}_{\text{NL},>}^{p_{1}p_{2}}=$
$\displaystyle~{}\mathcal{C}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}}(r_{1})\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{2}}(r_{2})+\text{c.c.}.$ (80)

It then remains to compute the time-ordered integral (68). It can be seen that
the second set of poles (73) do not contribute to the result, so that we can
consider the first set of poles (72) only. (The regularized hypergeometric
function is regular with respect to all the three orders and thus contribute
no poles.) The result is thus analytic in $r_{1}$ and $r_{2}$ at
$r_{1}=r_{2}=0$, and corresponds to the “EFT” part, or the “background” (BG)
part, of the correlator. (Again, see [35] for explanations of the background
part.) Therefore, we can write:

$\displaystyle{\mathcal{I}}_{\text{BG},>}^{p_{1}p_{2}}(r_{1},r_{2})\equiv{\mathcal{I}}_{++,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2})+{\mathcal{I}}_{--,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2}),$
(81)

and more explicitly,

$\displaystyle{\mathcal{I}}_{\text{BG},>}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\cos\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+p_{2})\big{]}r_{1}^{5+p_{1}+p_{2}}\sum_{n_{1},n_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{12}}}\,}{n_{1}!n_{2}!}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{12}}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{1}}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times{\mathcal{F}}\left[\begin{matrix}2n_{2}+p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,2n_{12}+p_{1}+p_{2}+5\\\
2n_{2}+p_{2}+\mbox{$\frac{\,{7}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\right]+\text{c.c.},$
(82)

where we have used the Pochhammer symbol $(z)_{n}\equiv\Gamma(z+n)/\Gamma(z)$
and the “dressed” hypergeometric function $\mathcal{F}$. See (277) for the
definition. Thus we have finished the computation of the scalar seed integral
$\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(r_{1},r_{2})$ for
$r_{1}<r_{2}$. The result for $r_{1}>r_{2}$ is obtained by switching
$(r_{1},p_{1})\leftrightarrow(r_{2},p_{2})$ in all the results above. We note
that the nonlocal signal (80) is symmetric with respect to
$(r_{1},p_{1})\leftrightarrow(r_{2},p_{2})$, so that we can simply drop the
subscript “$>$”:

${\mathcal{I}}_{\text{NL}}^{p_{1}p_{2}}(r_{1},r_{2})={\mathcal{I}}_{\text{NL},>}^{p_{1}p_{2}}(r_{1},r_{2}).$
(83)

On the other hand, the local signal (77) and the background (82) are
asymmetric.

###  3.3 Summary

Now we summarize the result for the scalar seed integral obtained above. As we
have seen, the scalar seed integral can be expressed as a function of two
independent momentum ratios $r_{1}=k_{s}/k_{12}$ and $r_{2}=k_{s}/k_{34}$. The
following expressions apply to the case of $r_{1}<r_{2}$. The results for
$r_{1}>r_{2}$ can be obtained from the following expressions simply by
switching the variables $(r_{1},p_{1})\leftrightarrow(r_{2},p_{2})$.

For convenience, we only present the scalar seed integral with all SK indices
summed. According to the analytic properties at $r_{1}=0$ and $r_{2}=0$, the
summed integral can be put into the following three terms, respectively called
the background (BG), the local signal (L), and the nonlocal signal (NL):

$\displaystyle\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(r_{1},r_{2})=\mathcal{I}_{\text{BG},>}^{p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\text{L},>}^{p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\text{NL}}^{p_{1}p_{2}}(r_{1},r_{2})$
(84)

The background piece $\mathcal{I}_{\text{BG},>}^{p_{1}p_{2}}$ is analytic in
both $r_{1}$ and $r_{2}$ as $r_{1},r_{2}\rightarrow 0$, and therefore
possesses a power-series representation. The local signal
$\mathcal{I}_{\text{L},>}^{p_{1}p_{2}}$ is nonanalytic in $r_{1}/r_{2}$ as the
ratio reaches 0, but is analytic in $k_{s}$ as $k_{s}\rightarrow 0$. Finally,
the nonlocal signal $\mathcal{I}_{\text{NL},>}^{p_{1}p_{2}}$ is nonanalytic in
$r_{1}r_{2}$ as $r_{1}r_{2}\rightarrow 0$, and thus is also nonanalytic in
$k_{s}$ as $k_{s}\rightarrow 0$. The explicit expressions for these integrals
are summarized as follows:

$\displaystyle\mathcal{I}_{\text{BG},>}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\cos\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+p_{2})\big{]}r_{1}^{5+p_{1}+p_{2}}\sum_{n_{1},n_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{12}}}\,}{n_{1}!n_{2}!}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{12}}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{1}}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times{\mathcal{F}}\left[\begin{matrix}2n_{2}+p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,2n_{12}+p_{1}+p_{2}+5\\\
2n_{2}+p_{2}+\mbox{$\frac{\,{7}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\right]+\text{c.c.}$
(85) $\displaystyle\mathcal{I}_{\text{L},>}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{C}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}}(r_{1})\mathbf{F}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{2}}(r_{2})+\text{c.c.},$ (86)
$\displaystyle\mathcal{I}_{\text{NL}}^{p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{C}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}}(r_{1})\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{2}}(r_{2})+\text{c.c.}.$ (87)

Here $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu=\sqrt{m^{2}/H^{2}-9/4}$ is the mass parameter of the intermediate
massive scalar, which we always assume real. $(z)_{n}$ is the Pochhammer
symbol, defined in (274), $\mathcal{F}$ is the dressed hypergeometric
function, defined in (277). The function $\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p}$ is defined in
(78) and the coefficient $\mathcal{C}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}$ is
defined in (79).

## 4 Helical Spin-1 Exchange

In this section we compute the vector seed integral (43) using the partial MB
representation, which leads to the main result of this paper, namely an exact
and analytical expression for the helical 4-point correlators. This section
will be rather technical. Readers uninterested in the computation details are
invited to go directly to Sec. 4.3 for a summary of the result.

The vector seed integral (43) is defined for each intermediate polarization,
labeled by $\lambda=\pm,L$. As one can see from (22) and (24), the two
transverse polarizations with $\lambda=\pm$ are quite different from the
longitudinal polarization with $\lambda=L$. Therefore we shall treat these two
cases separately, with the transverse integral
$\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\pm)p_{1}p_{2}}$ computed in Sec. 4.1
and the longitudinal integral
$\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(L)p_{1}p_{2}}$ in Sec. 4.2.

###  4.1 Transverse integral

We now use the partial Mellin transformation to carry out the transverse
integral $\mathcal{I}_{\mathsf{a}\mathsf{b}}^{({\pm})p_{1}p_{2}}$. As one can
see from (10), for the two transverse components, the helicity basis is in
one-to-one relationship with the polarization basis, and therefore we are free
to use the helicity label $(h)$ to replace the polarization label $(\lambda)$
in various expressions.

To perform the partial Mellin transformation on the transverse mode function
(22), we could use the following Mellin-Barnes representation for the
Whittaker function, which we call the completely resolved representation
[109]:

$\displaystyle\mathrm{W}_{\kappa,\nu}(az)=\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\,\mathcal{F}\left[\begin{matrix}s-\nu,s+\nu\\\
s-\kappa+\mbox{$\frac{\,{1}\,}{2}$}\end{matrix}\middle|\mbox{$\frac{\,{1}\,}{2}$}\right](az)^{-s+1/2},$
(88)

where $\mathcal{F}$ is the dressed hypergeometric function defined in (277).
The advantage of this representation is that all $z$-dependences in the
Whittaker function have been completely resolved into power functions, which
trivializes the time integral. The drawback, however, is that the integrand
involves a hypergeometric function. Although this posts no essential
difficulty for us to perform the contour integral, the resulting expression
has a somewhat more complicated form. We will present the result derived from
this completely resolved representation in App. C. Here, instead, we make use
of the fact that the Whittaker function, when multiplied by an appropriate
exponential factor, has a far simpler Mellin-Barnes representation. There are
two ways to add this exponential factor, which lead to the following two
alternative results, which we call partially resolved representations [109]:

$\displaystyle\mathrm{W}_{\kappa,\nu}(az)=$
$\displaystyle~{}e^{az/2}\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\,\Gamma\left[\begin{matrix}s-\nu,s+\nu\\\
s-\kappa+\mbox{$\frac{\,{1}\,}{2}$}\end{matrix}\right](az)^{-s+1/2}.$ (89)
$\displaystyle\mathrm{W}_{\kappa,\nu}(az)=$
$\displaystyle~{}e^{-az/2}\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\,\Gamma\left[\begin{matrix}s-\nu,s+\nu,-s-\kappa+\mbox{$\frac{\,{1}\,}{2}$}\\\
\mbox{$\frac{\,{1}\,}{2}$}-\kappa-\nu,\mbox{$\frac{\,{1}\,}{2}$}-\kappa+\nu\end{matrix}\right](az)^{-s+1/2}.$
(90)

We have the freedom to use either of them when computing the seed integral so
long as all parameters stay in the validity ranges of the above two
representations. (See [109] for detailed specification of validity ranges of
all MB representations quoted here.)

#### Opposite-sign integrals.

First consider the opposite-sign integrals
$\mathcal{I}_{\pm\mp}^{(h)p_{1}p_{2}}$:

$\mathcal{I}^{(h)p_{1}p_{2}}_{\pm\mp}(r_{1},r_{2})=k_{s}^{3+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\pm\mathrm{i}k_{12}\tau_{1}\mp\mathrm{i}k_{34}\tau_{2}}D_{\pm\mp}^{(h)}(k_{s};\tau_{1},\tau_{2}).$
(91)

The two time integrals in these expressions are already factorized and so the
integration is straightforward. For a reason that will be clear soon, we
choose (90) for the both Whittaker functions in $D_{\pm\mp}^{(h)}$:

$\displaystyle D_{\pm\mp}^{(h)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu}}\,}{2\pi^{2}}(\cosh
2\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh
2\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)e^{\pm\mathrm{i}k(\tau_{1}-\tau_{2})}$
$\displaystyle\times\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,e^{\mp\mathrm{i}\pi(s_{1}-s_{2})/2}(2k_{s})^{-s_{12}}(-\tau_{1})^{-s_{1}+1/2}(-\tau_{2})^{-s_{2}+1/2}$
$\displaystyle\times\Gamma\Big{[}-s_{1}+\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-s_{2}+\mbox{$\frac{\,{1}\,}{2}$}\mp\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,s_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}.$ (92)

Inserting the MB representation into (91) and completing the time integrals,
we obtain:

$\displaystyle{\mathcal{I}}_{\pm\mp}^{(h)p_{1}p_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu}}\,}{2\pi^{2}}(\cosh
2\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh
2\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\big{(}\displaystyle\frac{\,{u_{1}}\,}{2}\big{)}^{3/2+p_{1}}\big{(}\displaystyle\frac{\,{u_{2}}\,}{2}\big{)}^{3/2+p_{2}}e^{\mp\mathrm{i}(p_{1}-p_{2})\pi/2}$
$\displaystyle\times\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,u_{1}^{-s_{1}}u_{2}^{-s_{2}}\Gamma\big{[}p_{1}+\mbox{$\frac{\,{3}\,}{2}$}-s_{1},p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-s_{2}\big{]}$
$\displaystyle\times\Gamma\big{[}-s_{1}+\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-s_{2}+\mbox{$\frac{\,{1}\,}{2}$}\mp\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,s_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{]},$ (93)

where we have used the previously defined momentum ratios
$u_{1}=2r_{1}/(1+r_{1})$ and $u_{2}=2r_{2}/(1+r_{2})$.

At this point we can explain why we choose (90) to represent the propagator:
Had we chosen the other representation (89), the integral (4.1), and also all
the subsequent results, would be expressed in terms of
$w_{1}=2r_{1}/(1-r_{1})$ and $w_{2}=2r_{2}/(1-r_{2})$ instead of $u_{1,2}$. As
we shall see below, the result of the vector seed integral involves the
hypergeometric function ${}_{2}\mathrm{F}_{1}$ of the momentum ratio. We know
that ${}_{2}\mathrm{F}_{1}(a,b;c;z)$ for generic orders $(a,b,c)$ is singular
at $z=1$. Thus, had we use (89) which gives rise to results in terms of
$w_{1,2}$, we would encounter spurious divergences at $w_{1,2}=1$, namely
$r_{1,2}=1/3$, a configuration of no special physical significance. Of course
this superficial divergence must get canceled in the final result, but its
presence would make the final expression rather ugly, and also make the
numerical implementation hard. Therefore, here and below, we will always make
choices from (89) and (90) such that we always get $u_{1,2}$ instead of
$w_{1,2}$ in the final expressions.777Of course, our choice to represent the
result in terms of $u_{1,2}$ does not remove the singularity of the
hypergeometric function at $u_{1,2}=1$. However, $u_{1,2}=1$, namely
$r_{1,2}=1$, correspond to the folded limit of the momentum configuration,
which does have a physical significance. Requiring no divergences at the
folded limit correspond to choosing the Bunch-Davies vacuum as the initial
states for all the quantum fields.

Given $0<u_{1,2}<1$, we should close the contours of MB integrals in (4.1)
from left with a large semicircle. Applying the residue theorem, we pick up
the following left poles of the integrand:

$\displaystyle s_{1}=-n_{1}\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,\quad
s_{2}=-n_{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,$ (94) $\displaystyle
s_{1}=-n_{1}\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,\quad
s_{2}=-n_{2}\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu.$ (95)

The result for the MB integral is thus obtained by summing over the residues
of all the above left poles:

$\displaystyle{\mathcal{I}}_{\pm\mp}^{(h)p_{1}p_{2}}=\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu}}\,}{2\pi^{2}}(\cosh
2\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh
2\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\big{(}\displaystyle\frac{\,{u_{1}}\,}{2}\big{)}^{3/2+p_{1}}\big{(}\displaystyle\frac{\,{u_{2}}\,}{2}\big{)}^{3/2+p_{2}}e^{\mp\mathrm{i}(p_{1}-p_{2})\pi/2}$
$\displaystyle\times\sum_{n_{1}=0}^{\infty}\bigg{\\{}\displaystyle\frac{\,{(-1)^{n_{1}}}\,}{n_{1}!}u_{1}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n_{1}}\Gamma\big{[}n_{1}+p_{1}+\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-n_{1}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\bigg{\\}}$
$\displaystyle\times\sum_{n_{2}=0}^{\infty}\bigg{\\{}\displaystyle\frac{\,{(-1)^{n_{2}}}\,}{n_{2}!}u_{2}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+n_{2}}\Gamma\big{[}n_{2}+p_{2}+\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\mbox{$\frac{\,{1}\,}{2}$}\mp\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\bigg{\\}}.$ (96)

The summation can be done in closed form, and the result can be expressed as:

$\displaystyle{\mathcal{I}}_{\pm\mp}^{(h)p_{1}p_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu}}\,}{2\pi^{2}}(\cosh
2\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh
2\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)e^{\mp\mathrm{i}(p_{1}-p_{2})\pi/2}$
$\displaystyle\times\Big{[}\mathbf{G}^{p_{1}}_{\pm h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})u_{1}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+\mathbf{G}^{p_{1}}_{\pm h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})u_{1}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{]}\Big{[}\mathbf{G}^{p_{2}}_{\mp h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{2})u_{2}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+\mathbf{G}^{p_{2}}_{\mp h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{2})u_{2}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\Big{]},$ (97)

where we have defined

$\displaystyle\mathbf{G}^{p}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u)\equiv\mathrm{i}\pi\text{csch}(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{(}\displaystyle\frac{\,{u}\,}{2}\Big{)}^{3/2+p}\mathcal{F}\left[\begin{matrix}\mbox{$\frac{\,{3}\,}{2}$}+p+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|u\right].$ (98)

#### Same-sign integrals: factorized part.

We focus on the double-plus integral $\mathcal{I}^{(h)p_{1}p_{2}}_{++}$, and
still considering the case of $k_{12}>k_{34}$. The result for $k_{12}<k_{34}$
can be obtained by switching the variables
$(u_{1},p_{1})\leftrightarrow(u_{2},p_{2})$ as before, and the result for
$\mathcal{I}^{(h)p_{1}p_{2}}_{--}$ can be obtained by taking complex
conjugation.

Similar to the case of the scalar integral discussed in the previous section,
the same-sign integral can be separated into a factorized part and a time-
ordered part:

$\mathcal{I}_{\pm\pm}^{(h)p_{1}p_{2}}(r_{1},r_{2})=\mathcal{I}_{\pm\pm,\text{F},>}^{(h)p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\pm\pm,\text{TO},>}^{(h)p_{1}p_{2}}(r_{1},r_{2}).~{}~{}~{}~{}(r_{1}<r_{2})$
(99)

The factorized integral is given by:

$\mathcal{I}^{(h)p_{1}p_{2}}_{++,\text{F},>}(r_{1},r_{2})=-k_{s}^{3+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{+\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}}D_{>}^{(h)}(k_{s};\tau_{1},\tau_{2}),$
(100)

The computation of this integral is much similar to the opposite-sign
integral, except that we should now use a different MB representation for the
Wightman function $D_{>}$, since the exponential factor in (100) is
$e^{+\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}}$:

$\displaystyle D_{>}^{(h)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{\pi\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu]}e^{+\mathrm{i}k(\tau_{1}+\tau_{2})}$
$\displaystyle\times\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,e^{\mathrm{i}\pi(s_{1}-s_{2})/2}\cos\pi(s_{1}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)(2k_{s})^{-s_{12}}(-\tau_{1})^{-s_{1}+1/2}(-\tau_{2})^{-s_{2}+1/2}$
$\displaystyle\times\Gamma\Big{[}-s_{1}+\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-s_{2}+\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,s_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}.$ (101)

Substituting this representation into (100) and carrying out the time
integral, we get:

$\displaystyle{\mathcal{I}}^{(h)p_{1}p_{2}}_{++,\text{F},>}=$
$\displaystyle-\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{\pi\Gamma\big{[}\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu\big{]}}\big{(}\displaystyle\frac{\,{u_{1}}\,}{2}\big{)}^{3/2+p_{1}}\big{(}\displaystyle\frac{\,{u_{2}}\,}{2}\big{)}^{3/2+p_{2}}e^{-\mathrm{i}\pi(p_{1}+p_{2})/2}$
$\displaystyle\times\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,e^{+\mathrm{i}\pi(s_{1}+1/2)}\cos\pi(s_{1}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)u_{1}^{-s_{1}}u_{2}^{-s_{2}}\Gamma\big{[}p_{1}+\mbox{$\frac{\,{3}\,}{2}$}-s_{1},p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-s_{2}\big{]}$
$\displaystyle\times\Gamma\big{[}-s_{1}+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-s_{2}+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,s_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{]}.$ (102)

Then, following the same procedure as before, we close the contour from the
left, pick up all enclosed (left) poles, namely (94) and (95), and sum over
the residues at these poles, to get the final result:

$\displaystyle{\mathcal{I}}^{(h)p_{1}p_{2}}_{++,\text{F},>}=$
$\displaystyle~{}\displaystyle\frac{\,{-\mathrm{i}e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{\pi\Gamma\big{[}\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu\big{]}}e^{-\mathrm{i}\pi(p_{1}+p_{2})/2}$
$\displaystyle\times\Big{[}e^{+\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(\nu+h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)]\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})u_{1}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\cosh[\pi(-\nu+h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)]\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})u_{1}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\Big{]}$
$\displaystyle\times\Big{[}\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{2})u_{2}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{2})u_{2}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\Big{]},$ (103)

where $\mathbf{G}_{h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p}(u)$
is the same function as is defined in (98).

#### Local and nonlocal signals.

It is not surprising that the factorized part of the same-sign integral has
very similar structure as the opposite-sign integrals. In particular, both
results are nonanalytic in $u_{1}$ and $u_{2}$ as $u_{1,2}\rightarrow 0$.
Similar to the case of scalar integral, these two parts contribute to the
local and nonlocal signals:

$\displaystyle\mathcal{I}_{++,\text{F},>}^{(h)p_{1}p_{2}}+\mathcal{I}_{--,\text{F},>}^{(h)p_{1}p_{2}}+\mathcal{I}_{+-}^{(h)p_{1}p_{2}}+\mathcal{I}_{-+}^{(h)p_{1}p_{2}}=\mathcal{I}_{\text{L},>}^{(h)p_{1}p_{2}}+\mathcal{I}_{\text{NL}}^{(h)p_{1}p_{2}},$
(104)

with the local signal $\mathcal{I}_{\text{L},>}^{(h)p_{1}p_{2}}$ and the
nonlocal signal $\mathcal{I}_{\text{NL}}^{(h)p_{1}p_{2}}$ given by:

$\displaystyle\mathcal{I}_{\text{L},>}^{(h)p_{1}p_{2}}(u_{1},u_{2})=$
$\displaystyle~{}\mathcal{G}^{(h)p_{1}p_{2}}_{\text{L}}(u_{1},u_{2})\Big{(}\displaystyle\frac{\,{u_{1}}\,}{u_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.}$
$\displaystyle\mathcal{I}_{\text{NL}}^{(h)p_{1}p_{2}}(u_{1},u_{2})=$
$\displaystyle~{}\mathcal{G}^{(h)p_{1}p_{2}}_{\text{NL}}(u_{1},u_{2})(u_{1}u_{2})^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.}$ (105)

where we have defined:

$\displaystyle\mathcal{G}_{\text{L}}^{(h)p_{1}p_{2}}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-\pi h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2\pi^{2}}\bigg{\\{}\big{(}\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\big{)}\Big{[}e^{-\mathrm{i}(p_{1}-p_{2})\pi/2}\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})$
$\displaystyle+e^{+\mathrm{i}(p_{1}-p_{2})\pi/2}\mathbf{G}^{p_{1}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})\Big{]}$
$\displaystyle~{}+\displaystyle\frac{\,{-2\mathrm{i}\pi e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu]}e^{-\mathrm{i}(p_{1}+p_{2})\pi/2}\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})$
$\displaystyle~{}+\displaystyle\frac{\,{2\mathrm{i}\pi e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\Gamma[\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu]}e^{+\mathrm{i}(p_{1}+p_{2})\pi/2}\mathbf{G}^{p_{1}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})\bigg{\\}},$
(106) $\displaystyle\mathcal{G}_{\text{NL}}^{(h)p_{1}p_{2}}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-\pi h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2\pi^{2}}\bigg{\\{}\big{(}\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\big{)}\Big{[}e^{-\mathrm{i}(p_{1}-p_{2})\pi/2}\mathbf{G}^{p_{1}}_{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}(u_{1})\mathbf{G}^{p_{2}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern