Boostless Cosmological Collider Bootstrap

Guilherme L. Pimentel1,2 and Dong-Gang Wang 3

1 Lorentz Institute for Theoretical Physics, Leiden University,
Leiden, 2333 CA, The Netherlands

2 Institute of Physics, University of Amsterdam,
Amsterdam, 1098 XH, The Netherlands

3 Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

Abstract

Cosmological correlation functions contain valuable information about the
primordial Universe, with possible signatures of new massive particles at very
high energies. Recent developments, including the cosmological bootstrap,
bring new perspectives and powerful tools to study these observables. In this
paper, we systematically classify inflationary three-point correlators of
scalar perturbations using the bootstrap method. For the first time, we derive
a complete set of single-exchange cosmological collider bispectra with new
shapes and potentially detectable signals. Specifically, we focus on the
primordial scalar bispectra generated from the exchange of massive particles
with all possible boost-breaking interactions during inflation. We introduce
three-point “seed” functions, from which we bootstrap the inflationary
bispectra of scalar and spinning exchanges using weight-shifting and spin-
raising operators. The computation of the seed function requires solving an
ordinary differential equation in comoving momenta, a boundary version of the
equation of motion satisfied by a propagator that linearly mixes a massive
particle with the external light scalars. The resulting correlators are
presented in analytic form, for any kinematics. These shapes are of interest
for near-future cosmological surveys, as the primordial non-Gaussianity in
boost-breaking theories can be large. We also identify new features in these
shapes, which are phenomenologically distinct from the de Sitter invariant
cases. For example, the oscillatory shapes around the squeezed limit have
different phases. Furthermore, when the massive particle has much lower speed
of sound than the inflaton, oscillatory features appear around the equilateral
configuration.

###### Contents

  1. 1 Introduction
  2. 2 The EFT of Cosmological Colliders
  3. 3 The Three-Point Scalar Seed
    1. 3.1 Free Propagators in de Sitter
    2. 3.2 A Mixed Propagator
    3. 3.3 The Primary Scalar Seed
  4. 4 More Mixed Propagators
    1. 4.1 Mixed Propagators from General Interactions
      1. 4.1.1 Building Blocks of the Mixed Propagators
      2. 4.1.2 Building Mixed Propagators
    2. 4.2 Generalized Scalar Seeds
  5. 5 Boost-Breaking Weight-Shifting Operators
    1. 5.1 From $\Delta=2$ to $\Delta=3$
    2. 5.2 Scalar Exchange Bispectra
  6. 6 Exchange of Spinning Particles
    1. 6.1 Spin-1 Exchange
      1. 6.1.1 Spin-1 Propagators
      2. 6.1.2 Spin-1 Exchange Bispectra
    2. 6.2 Spinning Propagators
    3. 6.3 Spin-$s$ Exchange Bispectra
  7. 7 Phenomenology
  8. 8 Conclusions and Outlook
  9. A More on Mixed Propagators
  10. B Details of the Generalized Scalar Seeds
    1. B.1 Explicit Solutions
    2. B.2 Singularity Structure
  11. C Comments on Diagrams beyond Single-Exchange
  12. D Spinning Fields in de Sitter

##  1 Introduction

The primordial Universe is a natural laboratory for fundamental physics, where
the laws of the microscopic world can be tested via observations on cosmic
scales. In particular, since inflation is likely to have the highest energy
densities accessible in nature, we expect that primordial correlations may
provide the ultimate test of high energy physics [1, 2]. This idea is nicely
manifested in “cosmological collider physics” [3], where the qualitative and
quantitative features of inflation are recast in terms of a giant particle
accelerator. Within this collider analogy, measuring correlation functions in
the sky corresponds to measurements of interactions of the particles
responsible for primordial fluctuations. These correlations could, for
example, be mediated by new massive particles. This scenario leads to the
natural question of studying these correlations and classifying their
distinctive observational signatures. Specifically, new particles can mediate
interactions among the curvature fluctuations, leaving their indirect imprints
in the shapes of primordial non-Gaussianity (see [4, 5, 6, 7, 8, 9, 10, 11,
12] for earlier studies, and [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] for recent
ones). It is remarkable that there is the possibility to do particle
spectroscopy in this extremely high energy environment, while having access
only to the static pattern of density fluctuations at the end of inflation. To
do so, precise predictions for the cosmological correlation functions are
needed, as well as a detailed understanding of their analytic structure.

In recent years, our theoretical understanding of the statistics of primordial
fluctuations has improved significantly. The correlation functions at the end
of inflation are now known in analytic form for a wide variety of processes.
These advances come from a new perspective toward the investigation of
cosmological correlators, following a “bootstrap” philosophy [41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63]
(also see [64] for an up-to-date review of the subject). In this new approach,
without reference to a specific model or Lagrangian, the correlators are
directly determined from a set of basic physical principles, such as locality,
unitarity and symmetry.

The cosmological bootstrap was first studied by exploiting the full de Sitter
symmetries (the de Sitter bootstrap) [41, 42, 43]. From observations, we
expect primordial fluctuations to be translation and rotation invariant, and
dilatation covariant. For inflation, there is also the possibility of
fluctuations being invariant under de Sitter boosts. In this case, the
constraints from all de Sitter isometries become very powerful. For example,
it implies that the correlators have the same kinematical symmetries of
Euclidean conformal field theories. From this perspective, we obtain analytic
control of many correlators whose computation by conventional time evolution
is rather intractable. Within the de Sitter bootstrap, it is possible to
incorporate a mild breaking of boost symmetry, and thus compute primordial
non-Gaussianity for slow-roll inflation.

From a phenomenological perspective, however, it is interesting to drop the
assumption of boost isometries. A simple way to break boosts is by giving a
subluminal speed of propagation to the scalar fluctuations [65, 66, 67].
Generically, we expect the level of non-Gaussianity to be enhanced in this
case, due to the smaller size of the sound horizon, compared to the Hubble
radius during inflation [68, 69, 70]. Technically, this happens because the
sound speed is controlled by an operator that induces strong self-interactions
of the inflaton. Theories with small speed of sound and small non-Gaussianity
typically require fine tuning. Therefore, if we detect primordial non-
Gaussianity in the near future, it is likely that primordial fluctuations
break boost symmetries. In order to have the best of both worlds, we desire
analytic control and understanding of shapes of non-Gaussianity, while
encompassing scenarios in which de Sitter boosts are strongly broken.

In this paper, we make progress in developing the boostless bootstrap for the
primordial bispectrum—the three-point function of density perturbations.
Despite having less symmetries at our disposal, the bispectrum is a simple
observable, in which kinematics is tight enough that it is still possible to
run the bootstrap. Recently, the boostless bootstrap was successfully applied
to classify three- and four- points of the massless fields in de Sitter by
leveraging the remaining symmetries and locality constraints [51, 52, 53, 54,
55]. This general approach provides a complete set of correlators from single
field inflation including all the boost-breaking interactions. Our focus here
is the bispectrum due to the presence of mediator massive particles during
inflation. In other words, we are interested in developing a bootstrap for the
cosmological collider with potentially large non-Gaussianity, which is the
most interesting case for upcoming observations. It is perhaps surprising that
the bootstrap approach is applicable to the exchange bispectrum even in the
boost-breaking scenario. We suspect that this is because the bispectrum is
only sensitive to the longitudinal modes of the massive particles.

In the de Sitter bootstrap, we first compute de Sitter-invariant four-point
functions, and then deform them to obtain a minimal level of boost breaking
[41]. In this paper, as boosts can be strongly broken, we compute the
bispectrum using simpler building blocks, without reference to four-point
functions (see [71] for an alternative approach). The starting point is the
correlator of two conformally coupled scalars and a massless scalar which
linearly mixes with a scalar particle of arbitrary mass, as shown in Figure 1.
The mixed propagator satisfies an interesting differential equation in time
that internally “collapses” the massive particle, producing the massless bulk-
to-boundary propagator for a massless scalar. Then, we show that this “scalar
seed” three-point correlator satisfies an inhomogeneous differential equation,
and proceed to solve this equation analytically. More surprisingly, from this
scalar seed, we are able to bootstrap the inflaton correlators exchanging a
particle of arbitrary mass and spin, as well as arbitrary vertices (both for
quadratic and cubic interactions). Like in the de Sitter bootstrap, all
possible boost-breaking interactions are derived from hitting the seed
diagrams with “weight-shifting” operators. Similarly, we generate the spin-
exchange bispectra from the scalar seeds by using “spin-raising” operators.

The resulting shapes share some similarities with their de Sitter symmetric
counterparts, having features due to the mass and spin of the exchanged
particles, but they also have new properties that are unique to the
possibility of subluminal sound speeds111More precisely, the ratio of sound
speeds between the inflaton fluctuations and the exchanged particles.. The
oscillatory phases are now different with the ones predicted by de Sitter
invariant interactions. Moreover, the oscillations which are prominent around
the squeezed limit in de Sitter invariant theories can also appear close to
equilateral configurations. This is only possible in a scenario where the
sound horizon of the mediator field is much smaller than the one of the
inflaton. In that case, the exchange correlator can probe multiple sound
horizon crossings for the massive particle before it decays into the inflaton.
Even when all inflatons have similar wavelengths, the linear mixing leg
provides a different clock during inflation, thus modulating the resulting
non-Gaussian signal.

Figure 1: The structure of this paper. We compute the bispectrum of
conformally coupled scalars with a massless scalar that linearly mixes with a
massive scalar field. From this building block, and various “weight-shifting”
and “spin-raising” operations, we generate a wide variety of inflationary
bispectra of phenomenological interest. As we relax the requirement of boost
symmetry, the signals can be large enough to be detectable in near-future
cosmological surveys. The procedure to obtain the scalar seed, as well as the
various weight-shifting moves, are presented in the sections indicated above.

##### Outline

We start in Section 2 reviewing the effective field theory of inflation
applied to the cosmological collider scenario. We highlight how large
couplings are achieved for boost-breaking interactions in this setup, which
illustrates why the boostless bootstrap is interesting. The rest of the paper
is organized as shown in Figure 1. In Section 3, we present the propagator of
a massive scalar $\sigma$ linearly mixing with a massless field (the inflaton)
$\phi$. Next, we apply this mixed propagator to compute the three-point
function of two conformally coupled scalars $\varphi$ with an inflaton. This
correlator serves as a scalar seed of the bootstrap. In Section 4, we consider
the most general quadratic interactions between $\phi$ and $\sigma$, and
compute the resulting (generalized) scalar seeds. The effects of different
sound speeds between $\phi$ and $\sigma$ are taken into account here. These
seed functions are related to the one in Section 3 through recursive
relations, whose explicit forms are presented in Appendix B. In Section 5, we
introduce the boost-breaking weight-shifting operators, which map the seed
functions with conformally coupled scalars to the three-point correlators of
massless external fields. In Section 6 we analyze spinning particle exchanges.
As only the longitudinal mode of the particle propagates, we find that
relatively simple spin-raising operators relate the spinning-exchange
bispectra to the generalized scalar seeds. We discuss the phenomenology of the
new shapes in Section 7.

The appendices contain various technical details of the computations used
throughout the main text. In Appendix A we present the asymptotic behaviour of
the mixed propagators. In Appendix B we provide the solutions and singularity
analysis of the generalized scalar seeds. In Appendix C, we briefly comment on
the double-exchange and triple-exchange diagrams. In Appendix D, we review the
theory of free spinning particles in de Sitter space. Throughout the paper we
take the convention of natural units $c=\hbar=1$, the reduced Planck mass
$M_{\rm pl}^{2}=1/8\pi G$, and the metric signature $(-,+,+,+)$.

##  2 The EFT of Cosmological Colliders

In this section, we briefly review the effective field theory (EFT) of
inflation, with a single clock picking a foliation of spacetime, and also
additional massive fields beyond the inflaton. See the original papers [70,
13, 23] for more details about the construction of the EFT. We illustrate the
basic idea and collect the relevant results for the rest of the paper, showing
the most relevant interaction vertices for large non-Gaussianities.

The key idea of the EFT is to separate the background dynamics from the
dynamics of the quantum fluctuations. The background provides a natural
foliation of spacetime, and dictates the allowed symmetries and interactions
of the fluctuations. In this framework, many models of inflation lead to the
same EFT. A specific model gives predictions for the EFT coefficients. More
importantly, by being agnostic about the origin of the background dynamics,
the EFT provides a framework in which there is perturbative control of the
fluctuations, even when the background dynamics is UV-sensitive.

Figure 2: A diagrammatic illustration of how large boost-breaking interactions
arise for the exchange bispectrum. In Feynman diagrams of cosmological
correlators, the breaking of the boost symmetry is normally associated with
evaluating external legs of the massless scalar to the background (the dashed
lines with $\dot{\Phi}$). The (boostless) EFT diagram on the left encompasses
all the higher-derivative contributions on the right-hand side systematically.

An example to keep in mind is that of a dynamical scalar field acting as the
“clock” of the background evolution (e.g. the inflaton). The time-dependence
of this clock field $\Phi(t)$ is usually the source of the boost symmetry
breaking in cosmology. If we look at its kinetic term $(\partial\Phi)^{2}$ and
expand around the background solution, we obtain $\dot{\Phi}^{2}g^{00}$. The
metric component $g^{00}$ is the fluctuating degree of freedom in the EFT. If
we look at higher-derivative operators involving $\Phi$, they would generate
other operators for $g^{00}$, etc. The EFT packages all of these contributions
in a background-agnostic fashion, meaning that operators that would be higher
derivative, and thus suppressed, for the background dynamics, only appear as
Wilson coefficients in the Lagrangian of the fluctuations, where the de Sitter
boosts can be strongly broken. This is demonstrated by using Feynman diagrams
in Figure 2, where the EFT diagram with boost-breaking interactions is a sum
of all the diagrams on the right hand side, with various legs being put to the
background.

Within this framework, it is easy to show that non-Gaussian signals can be
computed reliably as long as perturbation theory doesn’t break down, namely
$f_{\rm NL}\lesssim O(A_{\zeta}^{-1/2})$. For means of comparison, theories in
which the background dynamics is weakly coupled typically predict a much lower
$f_{\rm NL}\sim O(1)$. To test primordial non-Gaussianity in the near future,
we’d like to have a framework that allows us to systematically classify non-
Gaussianity for which $f_{\rm NL}\sim O(10)$, which are the current
experimental bounds. As we will show below, the EFT provides us such a
framework. Moreover, it strongly suggests that large non-Gaussianities are
only achievable in models where the boost symmetries are broken [72], both in
the form of the interaction vertices, and in the dispersion relation of the
fluctuations.

In order to study scalar fluctuations, it is often useful to consider the
decoupling limit. In this limit, the metric has a scalar longitudinal mode,
which is (by a gauge transformation) related to curvature perturbations. This
is the Goldstone boson $\pi$ associated with the breaking of the time-
translation invariance in an expanding spacetime. For most practical purposes,
it can be treated as a massless field in the quasi-de Sitter background of
inflation, but perhaps with a non-relativistic dispersion relation. In
particular, we are interested in its couplings to other massive fields during
inflation. In general, these new particles can be massive scalars or spinning
fields, and their interactions with the Goldstone might not be covariant.

First, let’s briefly review the single-clock EFT focusing on self-interactions
of the $\pi$ field. Then we present the relevant results for its couplings
with extra massive particles. At leading order in derivatives, the action of
the single-clock EFT is

$\displaystyle S=\int{\rm d}^{4}x\sqrt{-g}\left[\frac{1}{2}M_{\rm
Pl}^{2}R+M_{\rm Pl}^{2}\dot{H}g^{00}-M_{\rm
Pl}^{2}(3H^{2}+\dot{H})+\sum_{n=2}^{\infty}\frac{M_{n}^{4}}{n!}(\delta
g^{00})^{n}+\cdots\right],$ (2.1)

where $\delta g^{00}\equiv g^{00}+1$. The coefficients of the operators $1$
and $g^{00}$ are adjusted to ensure that the background cosmology has Hubble
rate $H$. Then the action starts quadratic in fluctuations. For $M_{n}\to 0$
we recover slow-roll inflation. To see the fluctuating scalar degree of
freedom, we introduce the “pion” via a time reparametrization
$t\to\tilde{t}=t+\pi(t,{\bf x})$. The metric transforms as

$\displaystyle g^{00}$ $\displaystyle\to g^{00}+2\partial_{\mu}\pi
g^{0\mu}+\partial_{\mu}\pi\partial_{\nu}\pi g^{\mu\nu}\,.$ (2.2)

Substituting this into (2.1) gives the action for the Goldstone boson. In
general, this action contains a complicated mixing between the Goldstone mode
and metric fluctuations. We are interested in the decoupling limit, where the
gravitational interactions are neglected [70]. In this case, the
transformation (2.2) is
$g^{00}\to-1-2\dot{\pi}-\dot{\pi}^{2}+a^{-2}(\partial_{i}\pi)^{2}$, and the
Goldstone Lagrangian becomes

${\cal L}_{\pi}=M_{\rm
Pl}^{2}\dot{H}(\partial_{\mu}\pi)^{2}+2M_{2}^{4}\left[\dot{\pi}^{2}-\frac{\dot{\pi}(\partial_{i}\pi)^{2}}{a^{2}}\right]+\left(2M_{2}^{4}-\frac{4}{3}M_{3}^{4}\right)\dot{\pi}^{3}+\cdots\,.$
(2.3)

We see that $M_{2}\neq 0$ induces a nontrivial sound speed for the Goldstone
boson,

$c_{s}^{2}\equiv\frac{M_{\rm Pl}^{2}\dot{H}}{M_{\rm
Pl}^{2}\dot{H}-2M_{2}^{4}}\,.$ (2.4)

A small value of $c_{s}$ (large value of $M_{2}$) is correlated with an
enhanced cubic interaction $\dot{\pi}(\partial_{i}\pi)^{2}$ and large
equilateral non-Gaussianity. This is partly why the boost-breaking scenario is
phenomenologically important. The resulting bispectra and trispectra of the
self-interacting pion were computed in [51, 52, 53] via the “boostless
bootstrap” approach.

Now we consider how the additional fields are coupled to the Goldstone in the
EFT framework. We include both scalars and spinning fields in our
discussion.222For the EFT with spinning fields, here we follow the
construction in [13] which assumes the full dS isometries for spinning
particles. A different approach is presented in [23] where a dS-invariant UV
completion is not required. However, as only the helicity-0 longitudinal mode
contributes to the cosmological collider bispectrum, final results from these
two approaches are expected to be the same. We leave more detailed discussion
in Section 6. For a spin-$s$ field $\sigma^{\mu_{1}\cdots\mu_{s}}$, the basic
building blocks in the EFT are $\sigma^{0\cdots 0}$ and all Lorentz-invariant
self-interactions, e.g.
$\sigma^{\mu_{1}\cdots\mu_{s}}\sigma_{\mu_{1}\cdots\mu_{s}}$. The latter are
diff-invariant and will not induce couplings to the Goldstone in the
decoupling limit. We may also have contractions with curvature tensors, which
are higher order in derivatives. As this work focuses on single-exchange
diagrams, we are interested in quadratic and cubic vertices with one massive
field leg. For this type of interactions, in order of increasing spin, we
obtain:

  * •

Spin-0 Since the massive scalars do not respect shift symmetry, the lowest
derivative interactions with the Goldstone are simply given by

$\displaystyle{\cal L}_{\rm int}^{(0)}$ $\displaystyle=\omega_{0}^{3}\hskip
1.0pt\delta g^{00}\sigma+\tilde{\omega}_{0}^{3}\hskip 1.0pt(\delta
g^{00})^{2}\sigma\,.$ (2.5)

In the decoupling limit, the mixing Lagrangian becomes

$\displaystyle{\cal L}^{(0)}_{\rm int}=\rho_{0}\hskip
1.0pt\dot{\pi}_{c}\sigma+\frac{1}{\Lambda_{0}}\frac{(\partial_{i}\pi_{c})^{2}\sigma}{a^{2}}+\frac{1}{\tilde{\Lambda}_{0}}\dot{\pi}_{c}^{2}\sigma,$
(2.6)

where $\pi_{c}\equiv f_{\pi}^{2}\pi$ is the canonically normalized Goldstone,
with $f_{\pi}^{4}=2M_{pl}^{2}|\dot{H}|c_{s}$ being the symmetry breaking scale
[73]. The coupling constants are given by

$\rho_{0}=-\frac{2\omega_{0}^{3}}{f_{\pi}^{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\Lambda_{0}=-2\frac{f_{\pi}^{2}}{\rho_{0}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{\Lambda}_{0}=\frac{f_{\pi}^{4}}{4\tilde{\omega}_{0}^{3}-\omega_{0}^{3}}\leavevmode\nobreak\
.$ (2.7)

Here we see that, as a consequence of the nonlinearly realized time
translation symmetry, the couplings $\rho_{0}$ and $\Lambda_{0}$ are
correlated, though the $\tilde{\Lambda}_{0}$ coupling is independent.

  * •

Spin-1 For spin-1, the operators of the effective action involve $g^{00}$ and
$\sigma^{0}$. Taking into account the tadpole constraints, the mixing
Lagrangian at leading order in derivatives is

$\displaystyle{\cal L}_{\pi\sigma}^{(1)}$ $\displaystyle=\omega_{1}^{3}\hskip
1.0pt\delta g^{00}\sigma^{0}+\tilde{\omega}_{1}^{3}\hskip 1.0pt(\delta
g^{00})^{2}\sigma^{0}\,.$ (2.8)

which, in the decoupling limit, gives

$\displaystyle{\cal L}_{\pi\sigma}^{(1)}$ $\displaystyle=\omega_{1}^{3}\hskip
1.0pta^{-2}\big{(}2\partial_{i}\pi\sigma_{i}-(\partial_{i}\pi)^{2}\sigma_{0}-2\dot{\pi}\partial_{i}\pi\sigma_{i}\big{)}+(3\omega_{1}^{3}+4\tilde{\omega}_{1}^{3})\hskip
1.0pt\dot{\pi}^{2}\sigma_{0}+\cdots\,.$ (2.9)

Only the cubic mixing $\dot{\pi}\partial_{i}\pi\sigma_{i}$ will lead to the
characteristic angular structure from spin exchange. Therefore, it is
interesting to consider the bispectrum from the interaction vertices
$\dot{\pi}\partial_{i}\pi\sigma_{i}$ and $\partial_{i}\pi\sigma_{i}$. Again,
due to the nonlinearly realized symmetry, a single parameter $\omega_{1}$
controls the size of these two interactions. Combining the above, we can write

$\displaystyle{\cal L}^{(1)}_{\rm mix}=\frac{1}{a^{2}}\left(\rho_{1}\hskip
1.0pt\partial_{i}\pi_{c}\sigma_{i}+\frac{1}{\Lambda_{1}}\dot{\pi}_{c}\partial_{i}\pi_{c}\sigma_{i}\right),$
(2.10)

with two correlated couplings

$\displaystyle\rho_{1}\equiv\frac{2\omega_{1}^{3}}{f_{\pi}^{2}}\,,\quad\Lambda_{1}\equiv-\frac{f_{\pi}^{2}}{\rho_{1}}\,\,.$
(2.11)

  * •

Spin-2 and higher For the interactions between a massive spin-2 field and the
Goldstone boson, the steps are similar. Focusing on the cubic operator which
produces a characteristic angular dependence from spin exchange, we obtain

$\displaystyle{\cal L}^{(2)}_{\rm mix}=\frac{1}{a^{4}}\left(\rho_{2}\hskip
1.0pt\partial_{i}\partial_{j}\pi_{c}\hat{\sigma}_{ij}+\frac{1}{\Lambda^{2}_{2}}\,\dot{\pi}_{c}\partial_{i}\partial_{j}\pi_{c}\hat{\sigma}_{ij}\right),$
(2.12)

This time the $\rho_{2}$ and $\Lambda_{2}$ parameters are independent. A
similar structure persists at higher spin $s>2$; we find the following mixing
Lagrangian

$\displaystyle{\cal L}^{(s)}_{\rm mix}=\frac{1}{a^{2s}}\left(\rho_{s}\hskip
1.0pt\partial_{i_{1}\cdots i_{s}}\pi_{c}\hat{\sigma}_{i_{1}\cdots
i_{s}}+\frac{1}{\Lambda_{s}^{s}}\dot{\pi}_{c}\partial_{i_{1}\cdots
i_{s}}\pi_{c}\hat{\sigma}_{i_{1}\cdots i_{s}}\right),$ (2.13)

where $\partial_{i_{1}\cdots
i_{s}}\equiv\partial_{i_{1}}\cdots\partial_{i_{s}}$ and $\rho_{s}$,
$\Lambda_{s}$ are independent parameters.

The couplings are free parameters, but must satisfy some bounds to keep the
effective theory under theoretical control. The necessary requirements are
that the interactions must be treated perturbatively, that the fluctuations
propagate subluminally, and that the couplings are technically natural, in the
sense of being robust to radiative corrections. A detailed analysis implies
the following bounds [13]

$\displaystyle\frac{\rho^{2}_{s}}{m^{2}}\,\leq\,\,\frac{1-c_{s}^{2}}{c_{s}^{3}}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left(\frac{H}{\Lambda_{s}}\right)^{s}\ \lesssim\ \left[\frac{\big{(}2\pi
A_{\zeta}^{1/2}\big{)}^{s+1}}{c_{s}^{5}}\right]^{1/2}\,,$ (2.14)

where $m$ is the mass of the additional field, and $A_{\zeta}$ is the
amplitude of the curvature perturbation power spectrum. While sizes of some
interactions can be strongly constrained, in general large non-Gaussianity
signals are still allowed.

##### From EFT to Bootstrap

The discussion above shows that the couplings can be large within the EFT, in
particular when de Sitter boosts are broken. We may take another look at
Figure 2. For a weakly coupled theory, the bispectrum is dominated by the
first diagram on the right, which is the case studied in the de Sitter
bootstrap [41]. There the symmetry breaking is mild because of the slow-roll
condition. However, for theories with strongly coupled dynamics, all the
diagrams on the right may contribute, and thus it is possible to have sizable
breaking of the boost symmetry. The upshot of the EFT analysis is that for
small sound speed and boost-breaking interactions, the primordial bispectra
for masses $m\sim H$ are potentially detectable, with strengths that could be
as large as the currently allowed bounds for equilateral non-Gaussianity.

With this general picture and motivation in mind, in the next sections we will
develop the boostless bootstrap of cosmological colliders. Our goal is a
precise determination of all primordial bispectrum shapes due to the exchange
of a massive, scalar or spinning particle, in theories that break boost
symmetry. In practice, we ignore all the coupling constants above, but keep
the forms of the boost-breaking interactions as specific examples. As we will
show, from the bootstrap, we will systematically obtain a complete set of
these correlators, with complete analytic understanding of their shape
functions.

##  3 The Three-Point Scalar Seed

Cosmological correlators at the reheating surface are evaluated at late times
in a quasi-de Sitter space, well approximated by its asymptotic spacelike
boundary. A key insight of the cosmological bootstrap is that the time
evolution and interactions of particles during inflation are encoded in the
momentum dependence of cosmological correlators on the boundary. The local,
causal and unitary bulk evolution imply that the cosmological correlators
satisfy an interesting set of differential equations. While this idea was
first realized by exploiting all the de Sitter isometries [41], we expect that
similar equations exist for theories in which de Sitter boosts are broken.
Indeed, we will derive the differential equations below, from the known bulk
time integrals for the correlators in the case of broken boosts.

In this section, we will derive and solve the differential equations for a
“seed” cosmological correlator. We begin by introducing a linear mixing bulk-
to-boundary propagator, show that it satisfies a differential equation of its
own, and use that observation to construct the primary scalar seed of three-
point functions. This is the correlator of two conformally coupled scalars and
one inflaton exchanging a scalar particle of arbitrary mass. The seed function
derived here provides a benchmark example of a cosmological collider
correlator with broken boosts, and will serve as the building block for the
general bispectra of inflation.

###  3.1 Free Propagators in de Sitter

We begin with a brief review of free propagators of scalar fields during
inflation and Feynman rules for computing cosmological correlators. Expert
readers may skip this part and move on to Section 3.2 directly.

The background geometry of the inflationary universe can be well approximated
by de Sitter (dS) space, with line element

$ds^{2}=a(\eta)^{2}{(-d\eta^{2}+d{\bf x}^{2})}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
a(\eta)=-\frac{1}{H\eta}$ (3.1)

where $H$ is the Hubble scale and $\eta$ is the conformal time. In the
following we consider quantum fields propagating on this fixed background.
Instead of restricting to de Sitter invariant theories, our analysis shall
incorporate the cases with broken boost symmetries, while keeping the
dilations, spatial translations and rotations intact. This means that not all
interactions are built out of contractions of the background metric with
spacetime derivatives. Sometimes they will involve contractions with the space
components or the time component of the metric only. In general, free scalars
are described by the action

$S_{2}=\int d\eta d^{3}{\bf
x}\frac{1}{H^{2}\eta^{2}}\left[\frac{1}{2}(\partial_{\eta}\sigma)^{2}-\frac{1}{2}c_{\sigma}^{2}(\partial_{i}\sigma)^{2}-\frac{1}{2}m^{2}\sigma^{2}\right]\leavevmode\nobreak\
,$ (3.2)

where $m$ and $c_{\sigma}$ are the mass and the sound speed of the field
$\sigma$ respectively. At this level, the breaking of the dS boosts is
associated with $c_{\sigma}\neq 1$. In the Fourier space, we decompose the
field operator as $\sigma({\bf k},\eta)=\sigma_{k}(\eta)a({\bf k})+h.c.$.
Since we may absorb $c_{\sigma}$ in the momentum ${\bf k}$ by redefining
$c_{\sigma}{\bf k}\rightarrow{\bf k}$, without losing generality we shall set
$c_{\sigma}=1$ in the following analysis. Then the mode function
$\sigma_{k}(\eta)$ satisfies the equation of motion

$\left(\mathcal{O}_{\eta}+{m^{2}}/{H^{2}}\right)\sigma_{k}(\eta)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm with}\leavevmode\nobreak\
\leavevmode\nobreak\
\mathcal{O}_{\eta}\equiv\eta^{2}\partial^{2}_{\eta}-2\eta\partial_{\eta}+k^{2}\eta^{2}\leavevmode\nobreak\
.$ (3.3)

Assuming Bunch-Davies vacuum at early times, the $\sigma$ mode function is
explicitly given by

$\sigma_{k}(\eta)=-i\frac{H\sqrt{\pi}}{2}e^{i\pi/4}e^{-\pi\mu/2}(-\eta)^{3/2}H^{(1)}_{i\mu}(-k\eta)\leavevmode\nobreak\
\xrightarrow{\eta\rightarrow-\infty}\leavevmode\nobreak\
iH\eta\frac{e^{-ik\eta}}{\sqrt{2k}}$ (3.4)

where

$\mu=\sqrt{\frac{m^{2}}{H^{2}}-\frac{9}{4}}$ (3.5)

On the late-time boundary $\eta\rightarrow 0$, the massive scalar behaves as

$\lim_{\eta\rightarrow 0}\sigma({\bf k},\eta)=O^{+}({\bf
k})\eta^{\Delta^{+}}+O^{-}({\bf k})\eta^{\Delta^{-}}$ (3.6)

where the conformal dimensions are $\Delta^{\pm}=\frac{3}{2}\pm i\mu$. Two
particular cases that we will be interested in are the massless scalar $\phi$
(with $m^{2}=0$) and the conformally coupled scalar $\varphi$ (with
$m^{2}=2H^{2}$). For later convenience, we restore the sound speeds of these
two fields and set them as $c_{s}$, and then their mode functions and the
corresponding scaling dimensions $\Delta$ (in the sense of the power law
behaviour of the decaying mode) are given by

$\displaystyle\phi_{k}(\eta)$ $\displaystyle=$
$\displaystyle\frac{H}{\sqrt{2c_{s}^{3}k^{3}}}(1+ic_{s}k\eta)e^{-ic_{s}k\eta}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Delta=3$ (3.7)
$\displaystyle\varphi_{k}(\eta)$ $\displaystyle=$ $\displaystyle
i\frac{H\eta}{\sqrt{2c_{s}k}}e^{-ic_{s}k\eta}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Delta=2$ (3.8)

The scalar curvature fluctuations are well approximated by those of the
massless scalar $\phi$.333Explicitly, $\phi$ is related to the canonically
normalized Goldstone $\pi_{c}$ in the previous section via
$\phi=\pi_{c}/\sqrt{c_{s}^{3}}$. In this paper, we shall focus on the three-
point functions of these two types of scalars. They will be external lines in
Feynman diagrams, while the $\sigma$ field with general mass corresponds to
the exchanged massive particle.

From the bulk perspective, the standard approach to compute cosmological
correlators is the Schwinger-Keldysh or in-in formalism [74, 75]. Here we give
a very brief introduction to the method, and refer to recent reviews in [76,
77] for more details. First, it is convenient to introduce two types of
propagators for the quantum fields, bulk-to-bulk and bulk-to-boundary
propagators. As the massive field $\sigma$ appears in the internal lines, we
are interested in its bulk-to-bulk propagators

$\displaystyle G_{++}^{\sigma}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\sigma_{k}(\eta)\sigma^{*}_{k}(\eta^{\prime})\Theta(\eta-\eta^{\prime})+\sigma^{*}_{k}(\eta)\sigma_{k}(\eta^{\prime})\Theta(\eta^{\prime}-\eta)$
$\displaystyle G_{+-}^{\sigma}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\sigma^{*}_{k}(\eta)\sigma_{k}(\eta^{\prime})$ $\displaystyle
G_{-+}^{\sigma}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\sigma_{k}(\eta)\sigma^{*}_{k}(\eta^{\prime})$ $\displaystyle
G_{--}^{\sigma}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\sigma_{k}(\eta)\sigma^{*}_{k}(\eta^{\prime})\Theta(\eta^{\prime}-\eta)+\sigma^{*}_{k}(\eta)\sigma_{k}(\eta^{\prime})\Theta(\eta-\eta^{\prime})\leavevmode\nobreak\
,$ (3.9)

where $\Theta$ is the Heaviside function and we use $+$ and $-$ to represent
the time-ordered and anti-time-ordered pieces in the in-in integration
contour. These propagators describe the motion of a comoving mode $\sigma_{k}$
from one bulk time $\eta$ to another time $\eta^{\prime}$. By using the
equation of motion (3.3), the $G_{\pm\pm}^{\sigma}(k,\eta,\eta^{\prime})$
propagators satisfy the following inhomogeneous equation

$\left(\mathcal{O}_{\eta}+{m^{2}}/{H^{2}}\right)G_{\pm\pm}^{\sigma}(k,\eta,\eta^{\prime})=\mp
iH^{2}\eta^{2}\eta^{\prime 2}\delta(\eta-\eta^{\prime})\leavevmode\nobreak\ ,$
(3.10)

while $G_{\pm\mp}^{\sigma}(k,\eta,\eta^{\prime})$ satisfies the homogeneous
equation correspondingly. These propagators are related to each other by
complex conjugation, $G_{++}=G_{--}^{*}$ and $G_{+-}=G_{-+}^{*}$.

For fields associated with the external lines, we introduce the bulk-to-
boundary propagators, which describe the propagation from some bulk time
$\eta$ to the late-time boundary of de Sitter $\eta_{0}\rightarrow 0$. For the
massless scalar $\phi$, they are given by

$K_{+}(k,\eta)=\phi_{k}(\eta_{0})\phi^{*}_{k}(\eta)\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
K_{-}(k,\eta)=\phi^{*}_{k}(\eta_{0})\phi_{k}(\eta)\leavevmode\nobreak\ ,$
(3.11)

while the ones for the conformally coupled scalar $K_{\pm}^{\varphi}(k,\eta)$
are given by the same form but with the $\varphi_{k}(\eta)$ mode function. It
is clear that the bulk-to-boundary propagators satisfy the homogeneous
equation $\left(\mathcal{O}_{\eta}+{m^{2}}/{H^{2}}\right)K=0$, and they have
$K_{+}=K^{*}_{-}$.

Using this set of propagators, it becomes straightforward to derive the
Feynman rules for computing boundary correlators in interaction theories. For
contact diagrams, only the bulk-to-boundary propagators are needed, and we
have one time integral from $\eta=-\infty$ to $\eta\rightarrow 0$ to capture
the field interactions in the bulk. The computation becomes more complicated
when we study exchange diagrams from this bulk perspective. The internal lines
are associated with bulk-to-bulk propagators which lead to multiple nested
time integrals in the in-in formalism. In general, it is very difficult to
find analytical expressions, and thus one has to resort to numerical methods
for solving these integrals.

###  3.2 A Mixed Propagator

To simplify the computation of three-point correlators from exchange
processes, we first consider the two-point function
$\mathcal{K}_{\pm}\equiv\langle\sigma_{\pm}(\eta)\phi(\eta_{0})\rangle$. This
object is a bulk-to-boundary propagator generated by quadratic interactions.
Physically, it describes the conversion process from a massive field $\sigma$
at the bulk time $\eta$ to an inflaton, which then freely propagates to the
boundary. In this section let us focus on the simplest quadratic interaction
$\dot{\phi}\sigma$. Setting the coupling constant to unity, we express this
mixed propagator as

$\mathcal{K}_{\pm}(k,\eta)=\pm
i\int_{-\infty}^{0}d\eta^{\prime}a(\eta^{\prime})^{3}\left[G_{\pm\pm}^{\sigma}(k,\eta,\eta^{\prime})\partial_{\eta^{\prime}}K_{\pm}(k,\eta^{\prime})-G_{\pm\mp}^{\sigma}(k,\eta,\eta^{\prime})\partial_{\eta^{\prime}}K_{\mp}(k,\eta^{\prime})\right]\leavevmode\nobreak\
.$ (3.12)

Note that in (3.12), we make the simplifying assumption that $\phi$ and
$\sigma$ have the same sound speed, which is not generally true for boost-
breaking theories. We consider the case of different sound speeds and other
mixing interactions in Section 4.

This linear mixing is ubiquitous in cosmological backgrounds when multiple
fields are present. In maximally symmetric spacetimes, one can always
diagonalize the field basis and remove such mixings. During inflation, the de
Sitter boosts are broken by the time-dependent inflaton profile $\Phi(t)$,
which generically leads to quadratic interactions.444One simple example is to
consider the shift-symmetric coupling $(\partial\Phi)^{2}\sigma$. As shown in
Figure 2, expanding $\Phi(t,{\bf x})=\Phi(t)+\phi(t,{\bf x})$ with
$\dot{\Phi}(t)\neq 0$, we obtain the linear mixing
$\sim\dot{\Phi}\dot{\phi}\sigma$. In the EFT analysis, it is given by the
first term in (2.6). In general, the $\dot{\phi}\sigma$ interaction will arise
when the inflaton trajectory deviates from the geodesics in the multi-
dimensional field manifold [78]. The mixed propagator generates new shapes of
cosmological correlators, beyond those of self-interactions of the inflaton,
as we shall see.

The explicit form of this simplest mixed propagator has been studied in [79].
Here instead of computing the $\eta^{\prime}$ integral directly, we are
interested in deriving a differential equation for $\mathcal{K}$. From the
equation for $G$-propagators in (3.10), it is easy to see that the evolution
of the mixed propagator satisfies the following inhomogeneous equation

$\left(\mathcal{O}_{\eta}+{m^{2}}/{H^{2}}\right)\mathcal{K}_{\pm}(k,\eta)=-{H}\frac{\eta^{2}}{2k}e^{\pm
ik\eta}.$ (3.13)

Notice that the free bulk-to-boundary propagator $K_{\pm}$ satisfies a similar
equation, but without the source term. As a consequence of the
$\dot{\phi}\sigma$ interaction, this nonzero source marks the main feature of
the mixed propagator. Next we introduce the dimensionless mixed propagator
$\hat{\mathcal{K}}_{\pm}(k\eta)\equiv(k^{3}/H)\mathcal{K}_{\pm}(k,\eta)$,
which is a function of the combination $k\eta$ only. Therefore, we are allowed
to trade $\eta$-derivatives with $k$-derivatives on $\hat{\mathcal{K}}$, and
(3.13) is equivalent to

$\left(\mathcal{O}_{k}+{m^{2}}/{H^{2}}\right)\hat{\mathcal{K}}_{\pm}(k\eta)=-\frac{1}{2}k^{2}\eta^{2}e^{\pm
ik\eta}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm with}\leavevmode\nobreak\ \mathcal{O}_{k}\equiv
k^{2}\partial_{k}^{2}-2k\partial_{k}+k^{2}\eta^{2}.$ (3.14)

To better understand the evolution behaviour of the mixed propagator, it is
useful to look at its early-time and late-time limits, where the time
integrals can be performed. At early times, or equivalently the short-
wavelength limit $k\gg-1/\eta$, we have

$\displaystyle\lim_{\eta\rightarrow-\infty}\hat{\mathcal{K}}_{+}(k\eta)$
$\displaystyle=$ $\displaystyle\frac{i}{4}k\eta
e^{ik\eta}\log(-2k\eta)\leavevmode\nobreak\ .$ (3.15)

Thus the free Bunch-Davies vacuum has been dressed by the mixing, but still
only positive-frequency mode appears. The late-time behaviour corresponds to
the soft limit

$\lim_{k\rightarrow
0}\hat{\mathcal{K}}_{+}(k\eta)=\sum_{\pm}A_{\pm}\left(\frac{-k\eta}{2}\right)^{\frac{3}{2}\pm
i\mu}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm with}\leavevmode\nobreak\
A_{\pm}=\frac{\pi^{3/2}e^{\frac{\pi\mu}{2}\mp\frac{i\pi}{4}}(1\mp
ie^{-\pi\mu})}{\sinh(2\pi\mu)\Gamma(1\pm i\mu)}\leavevmode\nobreak\ ,$ (3.16)

which encodes the scaling of $\sigma$ on the boundary in (3.6). This
interesting behaviour of the mixed propagator is crucial for the new features
in the cosmological correlators.

###  3.3 The Primary Scalar Seed

In this section, we compute the three point function of two conformally
coupled scalars exchanging a massive particle with a massless scalar. We do
the computation for the cubic vertex $\varphi^{2}\sigma$ and set the coupling
constants to be unity in our analysis. This correlator will serve as a seed to
build all the cosmological collider bispectra later.

With the help of the mixed propagator, the exchange interaction can be
simplified into a “contact-like” form. Explicitly the three-point function is
given by

$\displaystyle\langle\varphi_{{\bf k}_{1}}\varphi_{{\bf k}_{2}}\phi_{{\bf
k}_{3}}\rangle^{\prime}$ $\displaystyle=$ $\displaystyle i\int d\eta
a(\eta)^{4}\left[K^{\varphi}_{+}(k_{1},\eta)K^{\varphi}_{+}(k_{2},\eta)\mathcal{K}_{+}(k_{3},\eta)-c.c.\right]+{\rm
perms.}$ (3.17) $\displaystyle=$
$\displaystyle\frac{iH\eta_{0}^{2}}{4k_{1}k_{2}k_{3}^{2}}\mathcal{\hat{I}}(k_{12},k_{3})+{\rm
perms.}\leavevmode\nobreak\ ,$

where $k_{12}\equiv k_{1}+k_{2}$ and the prime on the correlator means that
the momentum-conserving delta function has been stripped. Meanwhile, we have
defined the primary scalar seed 555The name primary is used because later we
will need other scalar seeds.

$\mathcal{\hat{I}}(k_{12},k_{3})\equiv\frac{1}{k_{3}}\int_{-\infty}^{0}\frac{d\eta}{\eta^{2}}\left[e^{ik_{12}\eta}\hat{\mathcal{K}}_{+}(k_{3}\eta)-e^{-ik_{12}\eta}\hat{\mathcal{K}}_{-}(k_{3}\eta)\right],$
(3.18)

which will be used as a building block to construct more general three-point
functions. Notice that $\mathcal{\hat{I}}$ is dimensionless and depends only
on the ratio $k_{3}/k_{12}$. The direct integration of (3.18) is difficult.
Instead, using (3.14), we find

$\left[(k_{12}^{2}-k_{3}^{2})\partial_{k_{12}}^{2}+2k_{12}\partial_{k_{12}}+\frac{m^{2}}{H^{2}}-2\right]\mathcal{\hat{I}}(k_{12},k_{3})=i\frac{k_{3}}{k_{T}}\leavevmode\nobreak\
,$ (3.19)

where $k_{T}\equiv k_{1}+k_{2}+k_{3}$. In terms of $u\equiv{k_{3}}/{k_{12}}$,
this equation becomes

$\left[\Delta_{u}+\left(\mu^{2}+\frac{1}{4}\right)\right]\mathcal{\hat{I}}(u)={i}\frac{u}{1+u},$
(3.20)

where we have defined the differential operator

$\Delta_{u}\equiv
u^{2}(1-u^{2})\partial_{u}^{2}-2u^{3}\partial_{u}\leavevmode\nobreak\ .$
(3.21)

This equation has a hidden conformal symmetry, which is closely related to the
differential equation of the four-point scalar seed in de Sitter bootstrap.
Meanwhile the boost-breaking effect of the linear mixing is manifested in the
source term. Next, we will first derive its analytical solution explicitly,
and then discuss the connection with the scalar seed in de Sitter bootstrap.

##### Analytical solution

The equation (3.20) is a second order ordinary differential equation with
three singular points $u=0,\pm 1$. To find its solution, we separate
$\mathcal{\hat{I}}(u)$ into a homogeneous part $\mathcal{\hat{H}}(u)$ and a
particular part $\mathcal{\hat{S}}(u)$. For the particular solution with
$u\in[0,1]$, we use the following series expansion around the regular singular
point $u=0$

$\mathcal{\hat{S}}(u)=i\sum_{n=0}^{\infty}c_{n}u^{n+1}.$ (3.22)

Substituting this ansatz into (3.20), we find the recursive relation of the
series coefficients

$c_{0}=\frac{1}{\frac{1}{4}+\mu^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ c_{1}=\frac{-1}{\frac{9}{4}+\mu^{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
c_{n}=\frac{(-1)^{n}+n(n-1)c_{n-2}}{(n+\frac{1}{2})^{2}+\mu^{2}}\leavevmode\nobreak\
,$ (3.23)

which can be solved as

$c_{n}=\sum_{m=0}^{\lfloor
n/2\rfloor}\frac{(-1)^{n}n!/(n-2m)!}{\left[\left(n+\frac{1}{2}\right)^{2}+\mu^{2}\right]\left[\left(n-\frac{3}{2}\right)^{2}+\mu^{2}\right]...\left[\left(n+\frac{1}{2}-2m\right)^{2}+\mu^{2}\right]}.$
(3.24)

This series solution is regular at $u=0$, but has singular behaviour when
$u\rightarrow\pm 1$.

Next, we derive the general solution $\mathcal{\hat{H}}(u)$, which can be
written as

$\mathcal{\hat{H}}(u)=-\frac{i}{2}\sum_{\pm}C_{\pm}\left(\frac{iu}{2\mu}\right)^{\frac{1}{2}\pm
i\mu}{}_{2}F_{1}\left[\frac{1}{4}\pm\frac{i\mu}{2},\frac{3}{4}\pm\frac{i\mu}{2};1\pm{i\mu};u^{2}\right]\leavevmode\nobreak\
,$ (3.25)

with two free coefficients $C_{\pm}$. To fix them, we impose the non-analytic
behaviour of the primary scalar seed at $u\rightarrow 0$. By using the soft
limit of the mixed propagator in (3.16), the integral in $\mathcal{\hat{I}}$
can be explicitly solved as

$\mathcal{\hat{I}}(u\rightarrow
0)=-\frac{i}{2}\sum_{\pm}B_{\pm}\left(\frac{u}{2}\right)^{\frac{1}{2}\pm
i\mu},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm with}\leavevmode\nobreak\ \leavevmode\nobreak\
B_{\pm}=\frac{\pi^{3/2}}{\cosh\pi\mu}\left(1\mp\frac{i}{\sinh\pi\mu}\right)\frac{\Gamma(\frac{1}{2}\pm
i\mu)}{\Gamma(1\pm i\mu)}\leavevmode\nobreak\ .$ (3.26)

This non-analytic soft limit can only be present in the homogeneous solution,
as $\mathcal{\hat{S}}(u\rightarrow 0)$ is a rational function. Thus by
matching the coefficients in this limit, we fix
$C_{\pm}=(-i\mu)^{\frac{1}{2}\pm i\mu}B_{\pm}$. It is interesting that this
limit automatically fixes the two free coefficients of the differential
equation, so no additional boundary condition is necessary.

Figure 3: The analytical solution of the primary scalar seed with $\mu=3$. For
comparison, the blue dots are numerical results from the direct integration of
(3.18).

The form of the solution is demonstrated in Figure 3. As we can see, the
homogeneous part contains the non-analytic oscillations in the $u\rightarrow
0$ limit; while the series solution is convergent as long as $u$ is not too
close to 1. The full expression
$\mathcal{\hat{I}}(u)=\mathcal{\hat{S}}(u)+\mathcal{\hat{H}}(u)$ matches well
with the numerical computation. Thus without integrating (3.18) directly, we
find the exact and practical solution of the primary scalar seed, which will
be extensively used in our following analysis.

##### Singularity structure

In the derivation above, the singular behaviour of $\mathcal{\hat{S}}(u)$ and
$\mathcal{\hat{H}}(u)$ at $u=0$ is manifest. Now we look at $u\rightarrow\pm
1$. For the homogeneous solution (3.25), we can see from the hypergeometric
function that there are logarithmic singularities as $u\rightarrow\pm 1$. When
$u\rightarrow 1$, this corresponds to the folded limit $k_{12}=k_{3}$, and the
homogeneous solution goes to

$\lim_{u\rightarrow
1}\mathcal{\hat{H}}(u)=\frac{i}{2}\frac{\pi}{\cosh\pi\mu}\log(1-u).$ (3.27)

For vacuum with only positive-frequency mode, we don’t expect singular
behaviour in this limit. Therefore, the homogeneous and particular parts must
cancel their logarithmic singularities against each other. For
$u\rightarrow-1$, this limit corresponds to the situation where the total
energy involved in the process vanishes, $k_{T}\equiv k_{12}+k_{3}\rightarrow
0$. The singularity of the three-point function is allowed in this limit,
which is known as the $k_{T}$-pole [80, 81]. From the homogeneous solution, we
have

$\lim_{u\rightarrow-1}\mathcal{\hat{H}}(u)=\frac{1}{2}\left(\lim_{u\rightarrow
e^{i\pi}}+\lim_{u\rightarrow
e^{-i\pi}}\right)\mathcal{\hat{H}}(u)=-\frac{i}{2}\frac{\pi}{\cosh\pi\mu}\log(1+u)\leavevmode\nobreak\
.$ (3.28)

The singularity of the series solution is not so straightforward to obtain.
The detailed derivation is in Appendix B; here we show the final result. For
the folded limit $u\rightarrow 1$

$\lim_{u\rightarrow
1}\mathcal{\hat{S}}(u)=-\frac{i}{2}\frac{\pi}{\cosh(\pi\mu)}\log(1-u)\leavevmode\nobreak\
,$ (3.29)

which precisely cancels the logarithmic singularity of the homogeneous
solution in (3.27). Meanwhile, there is a physical $k_{T}$-pole in the series
solution

$\lim_{u\rightarrow-1}\mathcal{\hat{S}}(u)=-\frac{i}{4}\log^{2}(1+u)\leavevmode\nobreak\
,$ (3.30)

which dominates over the one from the homogenous solution. Thus the three-
point function has a total-energy pole of $\log^{2}(k_{T})$, and the residue
of the pole is independent of the mass of the exchanged field. This result can
be understood by considering the early-time limit of the mixed propagator in
(3.15). Substituting it in the primary scalar seed, we find

$\lim_{k_{T}\rightarrow
0}\mathcal{\hat{I}}=\frac{i}{4}\int_{-\infty}^{0}\frac{d\eta}{\eta}\left[e^{ik_{T}\eta}\log(-2k_{3}\eta)+c.c.\right]\rightarrow-\frac{i}{4}\log^{2}k_{T}\leavevmode\nobreak\
.$ (3.31)

Thus this $k_{T}$-pole is a feature of the deformed vacuum state of the mixed
propagator. Notice that the total energy singularity here is also a partial
energy one, where the energy flowing into the cubic vertex vanishes. We will
see in the next section that these two poles become distinguishable when a
nontrivial sound speed is involved.

##### Comparison with the dS bootstrap

In the cosmological bootstrap with full de Sitter isometries, the building
block is a four-point function $\hat{F}$ of conformally coupled scalars
exchanging a massive scalar [41]. It was shown that as a consequence of
conformal symmetry, this four-point scalar seed satisfies

$\left[\Delta_{u}+\left(\mu^{2}+\frac{1}{4}\right)\right]\hat{F}(u,v)=\frac{uv}{u+v}$
(3.32)

with $u\equiv\frac{|{\bf k}_{1}+{\bf k}_{2}|}{k_{12}}$ and $v\equiv\frac{|{\bf
k}_{1}+{\bf k}_{2}|}{k_{34}}$. In the $v\to 1$ limit, the differential
equation for the four-point function is the same as (3.20). This connection
between the four-point and three-point seed functions can be made manifest in
the bulk picture. Let us look at the time integral from the right
$\varphi^{2}\sigma$ vertex in the four-point exchange diagram

$\sum_{\pm}(\pm i)\int
d\eta^{\prime}a^{4}K^{\varphi}_{\pm}(k_{3},\eta^{\prime})K^{\varphi}_{+}(k_{4},\eta^{\prime})G_{\pm+}(|{\bf
k}_{1}+{\bf k}_{2}|,\eta^{\prime},\eta)\leavevmode\nobreak\ ,$ (3.33)

where we use the $+$ propagator of the $k_{4}$ leg as an example. Next, we
take the $k_{4}\to 0$ and ${|{\bf k}_{1}+{\bf k}_{2}|\rightarrow k_{3}}$ limit
(i.e. $v\rightarrow 1$), this vertex becomes

$\sum_{\pm}(\pm i)\int\frac{d\eta^{\prime}}{\eta^{\prime
2}}e^{ik_{3}\eta^{\prime}}G_{\pm+}(k_{3},\eta^{\prime},\eta)\sim\mathcal{K}_{+}(k_{3},\eta)\leavevmode\nobreak\
,$ (3.34)

which reduces to the mixed propagator (3.12) from the $\dot{\phi}\sigma$
interaction. Therefore, the three-point function
$\langle\varphi\varphi\phi\rangle$ has the same form with the soft limit
$k_{4}\rightarrow 0$ of the $\langle\varphi\varphi\varphi\varphi\rangle$
correlator. In [41] it was shown that this is the “weight shifting” procedure
to turn the de Sitter symmetric four-point function into the slow-roll
suppressed three-point function of conformally coupled scalars and a massless
scalar. We can also check that the four-point seed solution reproduces the
$\hat{\mathcal{I}}$ solution here by taking $v\rightarrow 1$. The shape
function is identical to $\hat{F}(u,1)$, as expected.

However, it is important to comment on the distinction between these two seed
functions. In the three-point seed equation (3.20), we make no assumption of
weakly broken boosts, and thus in general, the resulting correlators such as
$\langle\varphi\varphi\phi\rangle$ are not slow-roll suppressed. This is
manifest from the schematic in Figure 2: while the dS bootstrap focuses on the
first diagram on the right-hand side, the boostless bootstrap analyzes the
diagram on the left-hand side directly. Another advantage of the three-point
scalar seed is that at the technical level it has a much simpler solution,
considering that the four-point seed solution is given by a two-variable
generalization of the hypergeometric series. As we are mainly interested in
the bispectra, it becomes more straightforward to work on the three-point
functions from the beginning. Furthermore, the primary seed function can be
simply extended to describe more complicated boost-breaking interactions, as
we will see in the following sections.

##### Unitarity and Locality

We close this section by making some remarks about how unitarity and locality
are manifested in the exchange bispectra. As two fundamental principles,
unitarity and locality have played crucial roles in the modern studies of
scattering amplitudes. For cosmological correlators, some of the consequences
of unitarity come from the cosmological optical theorem (COT) [56], while a
consequence of locality is in the manifestly local test (MLT) [52]. These
tests, which are based on the assumption of free bulk-to-boundary propagators
in both contact and exchange diagrams, previously did not take into account
the linear mixing with the external fields. Thus it is interesting to check
whether COT and MLT are still satisfied when mixed propagators are involved.
Since $\langle\varphi\varphi\phi\rangle$ in (3.17) provides the simplest
three-point function with a mixed propagator, we use this result as a
demonstration. The analysis can be easily extended to more complicated single-
exchange processes with arbitrary quadratic and cubic interactions.

For the analysis of the COT, it is convenient to use the dimensionless bulk-
to-boundary propagators $\hat{K}(k,\eta)=k^{3}K(k,\eta)$. One key step for
deriving the COT is to notice that these free propagators are Hermitian
analytic $\left[\hat{K}(-k^{*},\eta)\right]^{*}=\hat{K}(k,\eta)$, which
follows from the choice of the Bunch-Davies vacuum. For the mixed propagator,
although the expression becomes more complicated because of the quadratic
interaction, we find that this property of Hermitian analyticity is nicely
inherited by $\hat{\mathcal{K}}$, namely

$\left[\hat{\mathcal{K}}_{\pm}(-k^{*},\eta)\right]^{*}=\hat{\mathcal{K}}_{\pm}(k,\eta)\leavevmode\nobreak\
.$ (3.35)

These identities of the analytic continuation of propagators can be commuted
with the time integral. As a result, the primary scalar seed (3.18) satisfies

$\mathcal{\hat{I}}(k_{12},k_{3})+\left[\mathcal{\hat{I}}(-k_{12},-k_{3})\right]^{*}=0\leavevmode\nobreak\
,$ (3.36)

which corresponds to the COT for contact diagrams [56]. This condition
basically means that $\mathcal{\hat{I}}$ is imaginary. Interestingly, the
exchange bispectrum with a mixed propagator looks “contact”-like as far as
unitarity is concerned.

For the MLT, one may start with the observation that the free bulk-to-boundary
propagators of massless fields in de Sitter space satisfy

$\partial_{k}\hat{K}^{(n)}(k,\eta)|_{k=0}=0\leavevmode\nobreak\ ,$ (3.37)

where $n\geq 0$ corresponds to the number of time derivatives on $\hat{K}$.
This condition gives powerful boundary constraints for both contact and
exchange correlators with external free massless fields. In particular it has
been applied to bootstrap all three- and four-point functions from tree-level
boost-breaking interactions in single field inflation [52, 53, 54]. The
corresponding relation for the mixed propagator becomes more complicated than
the one in (3.37). As shown in (3.16), $\hat{\mathcal{K}}$ encodes the non-
analytic scaling of the massive field on the boundary. Intuitively, this is
because the mixed propagator describes the non-local conversion from $\sigma$
to $\phi$ in the bulk evolution. Thus the MLT in its current form is not
applicable to these correlators. It remains an interesting question about how
a generalized version could incorporate the behaviour of the mixed
propagators.

##  4 More Mixed Propagators

In the previous section, we bootstrapped the
$\langle\varphi\varphi\phi\rangle$ bispectrum with the simplest
$\dot{\phi}\sigma$ linear mixing. In this section, we consider more general
mixing vertices, incorporating all the possible quadratic interactions between
$\phi$ and $\sigma$. We will construct the general form of the mixed
propagators by first introducing their building blocks in Section 4.1. Next,
in Section 4.2 we will propose the generalized scalar seeds for the three-
point functions with higher derivative quadratic interactions. We show that
they can be obtained from the primary scalar seed with $\dot{\phi}\sigma$
mixing through recursive relations. While the exchanged field is still a
massive scalar in the analysis here, later on we will show that this
generalization of the seed functions will be crucial for bootstrapping the
three-point function of spin exchanges.

###  4.1 Mixed Propagators from General Interactions

To extend our analysis of boost breaking bispectra, we must take into account
the effects of different sound speeds for the interacting fields, as well as
general quadratic interactions between them. Let us first consider effects due
to reduced sound speeds, which is typical in theories with broken boosts. We
assume that the inflaton $\phi$ and the conformally coupled scalar $\varphi$
have a sound speed $c_{s}$, while the one for the exchanged field $\sigma$ is
$c_{\sigma}$. Without loss of generality, we rescale $k\rightarrow
c_{\sigma}k$ and $c_{s}\rightarrow c_{s}/c_{\sigma}$, thus removing the
$c_{\sigma}$-dependence. After the rescaling, “$c_{s}$” can take any positive
value, being a ratio of sound speeds. Therefore we consider general (either
sub or superluminal) $c_{s}$ for the external scalars below, where the free
propagators are $K^{\phi,\varphi}_{\pm}(c_{s}k,\eta)$ and
$G^{\sigma}(k,\eta,\eta^{\prime})$. We will restore the
$c_{\sigma}$-dependence in the phenomenological analysis in Section 7.

Next, we consider the boost-breaking quadratic interactions between $\sigma$
and $\phi$. The general form of the linear mixing vertex can be written as

$\mathcal{L}^{\phi\sigma}=a^{-n_{\partial_{i}}}\partial_{i}^{n_{\partial_{i}}}\left(\partial_{t}^{n_{\partial_{t}}}\phi\right)\sigma,$
(4.1)

where $n_{\partial_{i}}$ and $n_{\partial_{t}}$ are the number of spatial and
time derivatives on $\phi$. Notice that we have moved all spatial and time
derivatives to $\phi$ via integration by parts. Naively $n_{\partial_{i}}$
should be even, but we also consider $n_{\partial_{i}}$ odd, to account for
the possible contractions between internal polarization vectors $\epsilon_{i}$
and $\partial_{i}$. This is the case for the linear mixing with spinning
fields, which we analyze in Section 6. The $\dot{\phi}\sigma$ mixing discussed
in the previous section corresponds to $n_{\partial_{i}}=0$ and
$n_{\partial_{t}}=1$.

Motivated by the analysis above, we will first propose the building blocks of
mixed propagators, and then consider how to construct the ones for all
possible mixings.

####  4.1.1 Building Blocks of the Mixed Propagators

To capture the higher-derivative interactions and effects of different sound
speeds, we introduce

$\displaystyle\hat{\mathcal{K}}_{\pm}^{(n)}(k\eta;c_{s})$
$\displaystyle\equiv$ $\displaystyle\pm
i\frac{c_{s}^{3}k^{3+n}}{H^{n+1}}\int_{-\infty}^{0}d\eta^{\prime}a(\eta^{\prime})^{3-n}\Big{[}G_{\pm\pm}^{\sigma}(k,\eta,\eta^{\prime})\partial_{\eta^{\prime}}K_{\pm}(c_{s}k,\eta^{\prime})$
(4.2) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-(-1)^{n}G_{\pm\mp}^{\sigma}(k,\eta,\eta^{\prime})\partial_{\eta^{\prime}}K_{\mp}(c_{s}k,\eta^{\prime})\Big{]}$

as the dimensionless building blocks for more general mixed propagators. We
have chosen the $k$-dependent prefactor to make $\hat{\mathcal{K}}^{(n)}$ a
function of the combination $k\eta$. For $n=0$ and $c_{s}=1$ we retrieve the
dimensionless mixed propagator for the $\dot{\phi}\sigma$ interaction. The
index $n$ counts the number of spatial derivatives in these two-point
vertices—for general $n$, this is the propagator coming from the quadratic
interaction
$a^{-n}\hat{k}_{i_{1}}...\hat{k}_{i_{n}}\partial_{i_{1}...i_{n}}\dot{\phi}\sigma$.

Following the strategy of Section 3.2, by using the inhomogeneous equation of
the $G^{\sigma}$ propagator (3.10) and then trading $\eta$ derivatives with
$k$ derivatives, we find the differential equation of
$\hat{\mathcal{K}}^{(n)}$

$\left(\mathcal{O}_{k}+{m^{2}}/{H^{2}}\right)\hat{\mathcal{K}}_{\pm}^{(n)}(k\eta;c_{s})=-\frac{1}{2}c_{s}^{2}(-k\eta)^{n+2}e^{\pm
ic_{s}k\eta}.$ (4.3)

While the left-hand side of the equation remains the same as (3.14), the
generalization to arbitrary $n$ and $c_{s}$ is manifested in the source term,
which has the form of a free bulk-to-boundary propagator $\partial_{\eta}K$.
We leave detailed discussions of $\hat{\mathcal{K}}^{(n)}$ and their
asymptotic behaviours to Appendix A. Below we show that
$\hat{\mathcal{K}}^{(n)}$ can be reduced to the simplest mixed propagators
plus a sum of free propagators.

##### Recursive relations

We focus on $\hat{\mathcal{K}}^{(n)}_{+}$, as $\hat{\mathcal{K}}^{(n)}_{-}$
can be easily obtained from complex conjugation. Consider the source term of
the generalized equation in (4.3):
$\hat{S}^{(n)}=(-k\eta)^{n+2}e^{ic_{s}k\eta}$, and notice that

$\mathcal{O}_{k}\hat{S}^{(n)}=(n+2)(n-1)\hat{S}^{(n)}-2i(n+1)c_{s}\hat{S}^{(n+1)}+(1-c_{s}^{2})\hat{S}^{(n+2)}\leavevmode\nobreak\
.$ (4.4)

This relation connects the source terms with different orders of $n$. From
this relation, there are two different results depending on the sound speed.

  * •

For $c_{s}=1$, the last term in (4.4) vanishes. By using the differential
equations of $\hat{\mathcal{K}}_{+}^{(n)}(k\eta)$ and
$\hat{\mathcal{K}}_{+}^{(n-1)}(k\eta)$, we find the recursive form

$\hat{\mathcal{K}}_{+}^{(n)}=-\frac{i}{2n}\left(\left[\mu^{2}+\left(n-\frac{1}{2}\right)^{2}\right]\hat{\mathcal{K}}_{+}^{(n-1)}+\frac{1}{2}\hat{S}^{(n-1)}\right).$
(4.5)

Thus applying this relation iteratively, the $n$-th order building block of
the mixed propagator can be expressed in terms of the simplest one with $n=0$
and a sum of source terms.

  * •

For $c_{s}\neq 1$, by using (4.4) and (4.3) with different $n$, we find

$\hat{\mathcal{K}}_{+}^{(n)}=\frac{1}{1-c_{s}^{2}}\left(2i(n-1)c_{s}\hat{\mathcal{K}}_{+}^{(n-1)}-\left[\left(n-\frac{3}{2}\right)^{2}+\mu^{2}\right]\hat{\mathcal{K}}_{+}^{(n-2)}-\frac{c_{s}^{2}}{2}\hat{S}^{(n-2)}\right).$
(4.6)

Thus in addition to $\hat{\mathcal{K}}^{(0)}$, we also need
$\hat{\mathcal{K}}^{(1)}$ to obtain $\hat{\mathcal{K}}^{(n)}$ for arbitrary
$n$. For $c_{s}\ll 1$, since $\hat{\mathcal{K}}^{(n)}\sim c_{s}^{2}$, the
relation simplifies to
$\hat{\mathcal{K}}^{(n)}\simeq-\left[\left(n-{3}/{2}\right)^{2}+\mu^{2}\right]\hat{\mathcal{K}}^{(n-2)}-{c_{s}^{2}}\hat{S}^{(n-2)}/{2}$,
and thus up to a sum of source terms, we get
$\hat{\mathcal{K}}^{(n)}\propto\hat{\mathcal{K}}^{(0)}$ for even $n$, or
$\hat{\mathcal{K}}^{(n)}\propto\hat{\mathcal{K}}^{(1)}$ for odd $n$.

These two recursive relations simplify the discussion of higher-derivative
quadratic interactions. As we shall show next, using them we can reduce the
mixed propagators from complicated interactions to simple ones with some
constant prefactors.

####  4.1.2 Building Mixed Propagators

Now we build the propagator for generic mixing, as in (4.1). We are mainly
interested in the ones with higher derivatives, and thus mixings like
$\phi\sigma$ and $\partial_{i}\phi\sigma$, as well as nonlocal interactions
with inverse Laplacians will not be included in the discussion here. In
addition, for interactions of two scalars, the number of spatial derivatives
${n}_{\partial_{i}}$ is supposed to be even, as required by the rotational
invariance. But we may have an odd number of spatial derivatives when the
massive field has spin. We leave the discussion on mixed propagators with
spinning particles to Section 6, and assume that ${n}_{\partial_{i}}$ is
positive and even in this section. Explicitly, the mixed propagator from the
general quadratic interaction in (4.1) takes the following form

$\mathcal{K}_{\pm}^{(n_{\partial_{t}},n_{\partial_{i}})}(k,\eta)=\pm i(\mp
ik)^{n_{\partial_{i}}}\int_{-\infty}^{0}d\eta^{\prime}a(\eta^{\prime})^{4-n_{\partial_{i}}-n_{\partial_{t}}}\left[G_{\pm\pm}^{\sigma}\partial_{\eta^{\prime}}^{{n}_{\partial_{t}}}K_{\pm}-G_{\pm\mp}^{\sigma}\partial_{\eta^{\prime}}^{{n}_{\partial_{t}}}K_{\mp}\right]\leavevmode\nobreak\
,$ (4.7)

where we use the upper indices $(n_{\partial_{t}},n_{\partial_{i}})$ to denote
the number of time and spatial derivatives respectively. Let us analyze
various cases separately:

  * •

For ${n}_{\partial_{t}}=1$, the quadratic interaction can be brought to the
form $a^{-n_{\partial_{i}}}\partial_{i}^{n_{\partial_{i}}}\dot{\phi}\sigma$,
and the corresponding mixed propagator is simply given by

$\mathcal{K}_{\pm}^{(1,n_{\partial_{i}})}(k,\eta)=(\mp
i)^{n_{\partial_{i}}}\frac{H^{n_{\partial_{i}}+1}}{c_{s}^{3}k^{3}}\hat{\mathcal{K}}_{\pm}^{\left(n_{\partial_{i}}\right)}\leavevmode\nobreak\
.$ (4.8)

  * •

For ${n}_{\partial_{t}}=0$, the interaction vertex is
$a^{-n_{\partial_{i}}}\partial_{i}^{n_{\partial_{i}}}\phi\sigma$. To express
its mixed propagator in terms of the building blocks, we resort to the mode
function of the massless scalar. Then we find

$\mathcal{K}_{\pm}^{(0,n_{\partial_{i}})}(k,\tau)=(\mp
i)^{n_{\partial_{i}}+1}\frac{H^{n_{\partial_{i}}}}{c_{s}^{3}k^{3}}\left[\hat{\mathcal{K}}_{\pm}^{\left(n_{\partial_{i}}-1\right)}\mp
i\hat{\mathcal{K}}_{\pm}^{\left(n_{\partial_{i}}-2\right)}\right]\leavevmode\nobreak\
.$ (4.9)

We may further rewrite the result by using the recursive relations of the
building blocks. In particular, for $c_{s}=1$ we find

$\mathcal{K}_{\pm}^{(0,n_{\partial_{i}})}(k,\eta)=\frac{-(\mp
i)^{n_{\partial_{i}}}H^{n_{\partial_{i}}}}{2(n_{\partial_{i}}-1)c_{s}^{3}k^{3}}\left[\left(\mu^{2}+(n_{\partial_{i}}-\frac{1}{2})^{2}\right)\hat{\mathcal{K}}_{\pm}^{\left(n_{\partial_{i}}-2\right)}-\frac{1}{2}(k\eta)^{n_{\partial_{i}}}e^{\pm
ik\eta}\right]\leavevmode\nobreak\ .$ (4.10)

  * •

For ${n}_{\partial_{t}}>1$, we can use the equation of motion of $\phi$ to
reduce its number of time derivatives to be ${n}_{\partial_{t}}=0,1$. Or,
equivalently we notice that from the mode function of the massless scalar the
time derivatives of the bulk-to-boundary propagator satisfy

$\partial^{{n}_{\partial_{t}}}_{\eta}K_{\pm}=(\pm
ic_{s}k)^{n_{\partial_{t}}}\left[\frac{1-{n}_{\partial_{t}}}{c_{s}^{2}k^{2}\eta}\mp\frac{i}{c_{s}k}\right]\partial_{\eta}K_{\pm}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{n}_{\partial_{t}}\geq 0\leavevmode\nobreak\ .$ (4.11)

By using this relation in the mixed propagators, the quadratic interaction
with arbitrary numbers of spatial and time derivatives leads to

$\mathcal{K}_{\pm}^{(n_{\partial_{t}},n_{\partial_{i}})}(k,\eta)=(\pm
i)^{n_{T}}\frac{H^{n_{T}}}{c_{s}^{3}k^{3}}\left[({n}_{\partial_{t}}-1)\hat{\mathcal{K}}_{\pm}^{\left(n_{T}-2\right)}\mp
i\hat{\mathcal{K}}_{\pm}^{\left(n_{T}-1\right)}\right]\leavevmode\nobreak\ ,$
(4.12)

where we have set $n_{T}={n}_{\partial_{t}}+{n}_{\partial_{i}}$ as the total
number of derivatives on $\phi$.

Therefore, as we have seen, the mixed propagators from higher derivative
quadratic interactions in general can be written in terms of linear
combinations of building blocks $\mathcal{K}^{(n)}$. By using the recursive
relations in (4.5) and (4.6), they can be further reduced to the lower-$n$
mixed propagators with some constant prefactors and a sum of source terms. At
last, let us comment on the time derivatives on $\sigma$ in the quadratic
interaction, which can be moved onto $\phi$ by repeated use of integration by
parts. This procedure also generates additional source terms in the mixed
propagators, but since they have the form of free bulk-to-boundary propagators
and lead to simple contact interactions in three-point functions, we neglect
their contribution in our analysis.

To summarize, in this section we derived the mixed propagators for arbitrary
boost-breaking quadratic interactions. For a “cosmological collider” diagram
with massive scalar exchange, the most relevant mixing is the one with lowest
derivative $\dot{\phi}\sigma$, thus we are mainly interested in
$\mathcal{K}_{\pm}(k,\eta)=H\hat{\mathcal{K}}_{\pm}^{(0)}/(c_{s}k)^{3}$.
Nonetheless, the general discussion of higher derivative mixings will be
useful for the bootstrap of spinning exchanges in Section 6.

###  4.2 Generalized Scalar Seeds

Having determined the propagator for arbitrary linear mixing, we proceed to
generalize the three-point seed function. What we have in mind is the
bispectrum between two conformally coupled scalars and one massless scalar, as
in (3.17). Replacing the linear mixing propagator $\hat{\mathcal{K}}$ with the
general building block $\hat{\mathcal{K}}^{(n)}$, defined in (4.2), we propose

$\mathcal{\hat{I}}^{(n)}(k_{12},k_{3};c_{s})=\frac{1}{k_{3}}\int_{-\infty}^{0}\frac{d\eta}{\eta^{2}}\left[e^{ik_{12}c_{s}\eta}\hat{\mathcal{K}}_{+}^{(n)}(k_{3}\eta;c_{s})-(-1)^{n}e^{-ik_{12}c_{s}\eta}\hat{\mathcal{K}}_{-}^{(n)}(k_{3}\eta;c_{s})\right]\leavevmode\nobreak\
,$ (4.13)

as the generalized scalar seeds. When $n=0$ and $c_{s}=1$, we recover the
primary scalar seed (3.18). Physically, $\mathcal{\hat{I}}^{(n)}$ are
associated with the three-point functions with higher derivative quadratic
interactions and different sound speeds. The relative sign $(-1)^{n}$ of the
second term in the integrand is introduced such that these correlators are
real. We will discuss how to derive cosmological correlators from these seeds
functions in the following sections, while here we look at the analytical form
of their shapes. By definition $\mathcal{\hat{I}}^{(n)}$ are dimensionless,
being functions of

$u\equiv\frac{k_{3}}{c_{s}k_{12}}.$ (4.14)

Using (4.3), we find the differential equation

$\left[\Delta_{u}+\left(\mu^{2}+\frac{1}{4}\right)\right]\mathcal{\hat{I}}^{(n)}(u;c_{s})=(-i)^{n-1}n!c_{s}^{2}\left(\frac{u}{1+c_{s}u}\right)^{n+1}.$
(4.15)

Comparing to the equation satisfied by the primary scalar seed, (3.20), both
the effects of higher derivative interactions and the sound speed are encoded
in the source term, while the left-hand side of the equation remaining
unchanged. Similarly this generalized equation can be solved by imposing
boundary conditions in the soft limit. We leave the explicit derivation of its
solutions and their singularity analysis to Appendix B. Below, we first show
how to obtain the answer recursively, and then qualitatively discuss the
general solution. Depending on the value of $c_{s}$, there are two different
cases to consider for the recursive relations:

  * •

For $c_{s}=1$, the inhomogeneous source in (4.15), which is a higher-order
contact term $\hat{\mathcal{C}}^{(n)}=\left[{u}/({1+u})\right]^{n+1}$,
satisfies

$\Delta_{u}\hat{\mathcal{C}}^{(n)}=n(n+1)\hat{\mathcal{C}}^{(n)}-2(n+1)^{2}\hat{\mathcal{C}}^{(n+1)}\leavevmode\nobreak\
.$ (4.16)

Using the differential equations of $\mathcal{\hat{I}}^{(n)}$ and
$\mathcal{\hat{I}}^{(n-1)}$, we find

$\mathcal{\hat{I}}^{(n)}=-\frac{i}{2n}\left[\mu^{2}+\left(n-\frac{1}{2}\right)^{2}\right]\mathcal{\hat{I}}^{(n-1)}-\frac{(-i)^{n-1}n!}{2n^{2}}\hat{\mathcal{C}}^{(n-1)}\leavevmode\nobreak\
.$ (4.17)

Applying this relation iteratively, the $n$-th order solution can be written
as a sum of $\mathcal{\hat{I}}^{(0)}$ and contact terms

$\mathcal{\hat{I}}^{(n)}=(-i)^{n}n!\left[s_{n}\mathcal{\hat{I}}^{(0)}+i\sum_{m=0}^{n-1}s_{nm}\hat{\mathcal{C}}^{(n-m-1)}\right],$
(4.18)

where the $0$-th order function $\mathcal{\hat{I}}^{(0)}$ is the primary
scalar seed we have derived in Section 3. The coefficients $s_{n}$ and
$s_{nm}$ are

$\displaystyle s_{nm}$ $\displaystyle=$
$\displaystyle-\frac{\left[\left(n-\frac{1}{2}\right)^{2}+\mu^{2}\right]\left[\left(n-\frac{3}{2}\right)^{2}+\mu^{2}\right]...\left[\left(n+\frac{1}{2}-m\right)^{2}+\mu^{2}\right]}{2^{m+1}\left[n!/(n-m-1)!\right]^{2}}\leavevmode\nobreak\
,0\leq m<n$ (4.19) $\displaystyle s_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{n}(n!)^{2}}\left[\left(n-\frac{1}{2}\right)^{2}+\mu^{2}\right]\left[\left(n-\frac{3}{2}\right)^{2}+\mu^{2}\right]...\left[\frac{1}{4}+\mu^{2}\right]\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
n>0\leavevmode\nobreak\ .$ (4.20)

The contact terms are rational polynomials of the momenta with $k_{T}$ poles.
They are degenerate with higher-derivative contact interactions of the
primordial fluctuations. For cosmological colliders, only the first term in
(4.18) is relevant, capturing the effects of massive particle production. In
this sense, for $c_{s}=1$ the generalized scalar seeds can be simply reduced
to the primary one, and increasing the number of derivatives in the quadratic
interaction will not change the shape of the seed function.

  * •

For $c_{s}\neq 1$, the $n$-th order contact term becomes
$\hat{\mathcal{C}}^{(n)}=\left[{u}/({1+c_{s}u})\right]^{n+1}$, which satisfies

$\Delta_{u}\hat{\mathcal{C}}^{(n)}=n(n+1)\hat{\mathcal{C}}^{(n)}-2c_{s}(n+1)^{2}\hat{\mathcal{C}}^{(n+1)}+(n+1)(n+2)(c_{s}^{2}-1)\hat{\mathcal{C}}^{(n+2)}\leavevmode\nobreak\
.$ (4.21)

Using this and the differential equations (4.15), we obtain

$\mathcal{\hat{I}}^{(n)}=\frac{1}{1-c_{s}^{2}}\left(2i(n-1)c_{s}\mathcal{\hat{I}}^{(n-1)}-{\left[\left(n-\frac{3}{2}\right)^{2}+\mu^{2}\right]}\mathcal{\hat{I}}^{(n-2)}-(-i)^{n-1}(n-2)!{c_{s}^{2}}\hat{\mathcal{C}}^{(n-2)}\right)\leavevmode\nobreak\
,$ (4.22)

where the $n$-th order seed is expressed as a sum of the $(n-1)$-th and
$(n-2)$-th order solutions and the $(n-2)$-th order contact term. Therefore,
to obtain $\mathcal{\hat{I}}^{(n)}$ by using recursion relations, two of the
generalized seeds are needed as input, $n=0$ and $n=1$.

Figure 4: Analytical solutions of the generalized scalar seeds for $n=0$
(left) and $n=2$ (right), with $\mu=3$ and the sound speed ratio
$c_{s}=0.2,\leavevmode\nobreak\ 1,\leavevmode\nobreak\ 5$. Since we work on
the solutions for $u\in[0,1]$, the plots are stopped at $k_{3}/k_{12}<c_{s}$
when $c_{s}<1$.

Let us comment on the situation where the sound speed ratio $c_{s}$ is not 1.
It is more informative to take a look at the explicit solutions of the
differential equation (4.15) derived in Appendix B.1. Figure 4 shows the
generalized scalar seeds with different values of $n$ and $c_{s}$. Like the
primary scalar seed, the solutions here are expressed in terms of the
homogeneous part (B.4) and the particular part (B.7). The particular solutions
are rational series expansions which are analytic around $u=0$. Thus they do
not contribute to the oscillatory features of cosmological colliders, although
the presence of $c_{s}$ leads to significant modification on the series
coefficients.666As an example, when $c_{s}\ll 1$, new non-Gaussianity shapes
are expected to arise in the series solutions. This is recently discussed in
detail by [71]. As in this work we mainly focus on the oscillatory signals of
cosmological colliders, this particular regime is not included in our
analysis. We are mainly interested in the homogeneous solutions which are non-
analytic at $u\rightarrow 0$. This non-analyticity in the soft limit
corresponds to the productions of massive particles, as we have discussed in
the primary scalar seed. For $c_{s}\neq 1$, in addition to $c_{s}$-dependent
overall prefactors, the homogeneous solutions are affected through the $c_{s}$
in the definition of $u$ in (4.14). In particular we see from Figure 4 that,
the oscillations in the squeezed limit $k_{3}/k_{12}\ll 1$, which are in terms
of $u$, are shifted away from the $c_{s}=1$ shapes. This feature leads to
interesting phenomenology in the cosmological correlator. We discuss it in
detail in Section 7.

The appearance of $c_{s}$ also affects the singularity structure of the seed
functions. For general $c_{s}$, the folded configuration now corresponds to
$u\rightarrow 1/c_{s}$, while $u\rightarrow-1/c_{s}$ reflects the limit of
vanishing total energies $E_{\rm tot}\equiv c_{s}(k_{12}+k_{3})\rightarrow 0$.
Meanwhile, there is another singularity of $\mathcal{\hat{I}}^{(n)}$ at
$u\rightarrow-1$. This is the partial energy pole when the sum of energies in
the cubic vertex goes to zero $E_{L}\equiv k_{3}+c_{s}k_{12}\rightarrow
0$.777One may expect another partial energy pole at
$E_{R}=k_{3}+c_{s}k_{3}\rightarrow 0$ where the energies in the quadratic
vertex vanish. This is simply the non-analytic soft limit which we have used
as boundary conditions for solving the differential equation (4.15). When
$c_{s}=1$, the $E_{\rm tot}$\- and $E_{L}$-poles coincide with each other, as
we have seen in the primary scalar seed. But in general they are two physical
singularities in the exchange bispectra. The detailed analysis is presented in
Appendix B.2.

##### Summary

In this section, by incorporating all possible linear mixings between $\phi$
and $\sigma$, we extended the three-point scalar seed to the most general one
$\mathcal{\hat{I}}^{(n)}$ with arbitrary sound speed and high-derivative
quadratic interactions. Their analytical expressions are obtained from
recursive relations and also explicit solutions of the differential equation
(4.15). These generalized seed functions provide building blocks for
bootstrapping inflationary bispectra from both scalar and spin exchanges, as
we will show in Section 5 and Section 6 respectively.

##  5 Boost-Breaking Weight-Shifting Operators

We have derived the three-point seed functions that are associated with the
bispectra of two conformally coupled scalars and one inflaton. In this
section, we will apply these results to compute the scalar exchange bispectra
of three massless external scalars. We will introduce a set of weight-shifting
operators to map the conformally coupled scalars $\varphi$ to the massless
fields $\phi$. In particular, we generalize the $\varphi^{2}\sigma$ cubic
vertex to the ones of $\phi\phi\sigma$-type with any boost-breaking
interactions. As the inflaton fluctuations are directly related to the
primordial curvature perturbations, these correlators are most relevant for
observations.

In Section 5.1, we derive the weight-shifting operators for generic boost-
breaking cubic interactions. In Section 5.2, we apply these operators to
generate the phenomenologically interesting bispectra from massive scalar
exchanges.

###  5.1 From $\Delta=2$ to $\Delta=3$

For two cases of interest, the conformal weights $\Delta$ of the two boundary
operators in the three-point correlators are given by: $\Delta=2$,
corresponding to conformally coupled scalars; and $\Delta=3$ which are the
ones of massless fields. We perform the analysis one by one.

##### The $\langle\varphi\varphi\phi\rangle$ bispectrum

Let us first explicitly compute the three-point function with two $\Delta=2$
boundary operators. With the generalized scalar seeds, it becomes
straightforward to obtain $\langle\varphi\varphi\phi\rangle$ from the simplest
cubic coupling $\varphi^{2}\sigma$. We use the general version of the mixed
propagator in (4.12), and the three-point function in (3.17) becomes

$\displaystyle\langle\varphi_{{\bf k}_{1}}\varphi_{{\bf k}_{2}}\phi_{{\bf
k}_{3}}\rangle^{\prime}$ $\displaystyle=$
$\displaystyle\frac{i\eta_{0}^{2}}{4k_{1}k_{2}c_{s}^{2}}I_{\varphi\varphi\phi}+{\rm
perms}\leavevmode\nobreak\ ,$ (5.1)

where

$I_{\varphi\varphi\phi}=\int\frac{d\eta}{\eta^{2}}\left[e^{ic_{s}k_{12}\eta}\mathcal{K}_{+}(k_{3},\eta)-c.c.\right]=i^{n_{T}}\frac{H^{n_{T}}}{c_{s}^{3}k_{3}^{2}}\left[({n}_{\partial_{t}}-1)\mathcal{\hat{I}}^{(n_{T}-2)}-i\mathcal{\hat{I}}^{(n_{T}-1)}\right]\leavevmode\nobreak\
.$ (5.2)

For the linear mixing $\dot{\phi}\sigma$, this integral is simply given by
$I_{\varphi\varphi\phi}=\left({H}/{c_{s}^{3}k_{3}^{2}}\right)\mathcal{\hat{I}}^{(0)}$.
Thus the analysis of the generalized scalar seeds in Section 4.2 can be
directly applied for the $\langle\varphi\varphi\phi\rangle$ correlator. It is
interesting to consider an extremal case where the exchanged field is heavy,
$m^{2}\gg H^{2}$. Then $\sigma$ can be integrated out and we expect the
correlator becomes the one from the contact interaction
$\varphi^{2}\dot{\phi}$. In our formalism, this bispectrum can be obtained by
looking at the differential equation (4.15) in the $\mu\rightarrow\infty$
limit, where we may drop the differential operator $\Delta_{u}$. Thus the
scalar seed is simply given by $\mathcal{\hat{I}}^{(0)}\simeq
ic_{s}^{2}\mu^{-2}{u}/\left(1+c_{s}u\right)$, and we find

$\langle\varphi_{\bf k_{1}}\varphi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}\sim\frac{1}{k_{1}k_{2}k_{3}^{2}k_{T}}+{\rm
perms}\leavevmode\nobreak\ ,$ (5.3)

which matches the bispectrum shape from the contact interaction
$\varphi^{2}\dot{\phi}$ as expected.

For the later convenience, here let us also take a look at cubic interactions
with arbitrary time derivatives on the massive field $\sigma$

$\varphi^{2}\partial^{n_{\sigma}}_{t}\sigma=\varphi^{2}(-H\eta\partial_{\eta})^{n_{\sigma}}\sigma+...\leavevmode\nobreak\
,$ (5.4)

where we have only kept the highest derivatives term when we change to the
conformal time. These time derivatives on $\sigma$ lead to modifications on
the mixed propagators. Since the building block $\hat{\mathcal{K}}^{(n)}$ is a
function of the combination $k\eta$, we are able to trade $\eta$-derivatives
on it with $k$-derivatives and get a differential operator
$(k\partial_{k})^{n_{\sigma}}$. As a result, the mixed propagator in (4.12) is
changed to

$\mathcal{K}_{\pm}(k,\eta)=(\pm
i)^{n_{T}}\frac{H^{n_{T}+{n_{\sigma}}}}{c_{s}^{3}k^{3}}(-k\partial_{k})^{n_{\sigma}}\left[({n}_{\partial_{t}}-1)\hat{\mathcal{K}}_{\pm}^{\left(n_{T}-2\right)}\mp
i\hat{\mathcal{K}}_{\pm}^{\left(n_{T}-1\right)}\right]+...$ (5.5)

where the ellipses denote terms proportional to the free bulk-to-boundary
propagators. Accordingly, the bulk integral $I_{\varphi\varphi\phi}$ of the
$\langle\varphi\varphi\phi\rangle$ correlator now becomes

$\displaystyle
I_{\varphi\varphi\phi}=i^{n_{T}}\frac{H^{n_{T}+{n_{\sigma}}}}{c_{s}^{3}k_{3}^{3}}(-k_{3}\partial_{k_{3}})^{n_{\sigma}}k_{3}\left[({n}_{\partial_{t}}-1)\mathcal{\hat{I}}^{(n_{T}-2)}-i\mathcal{\hat{I}}^{(n_{T}-1)}\right]\leavevmode\nobreak\
.$ (5.6)

While these $k_{3}\partial_{k_{3}}$ operators are not frequently used for
scalar exchange, a similar procedure plays an important role for deriving the
spin-raising operator in Section 6.

##### The $\langle\phi\phi\phi\rangle$ bispectrum

Now we consider the three-point function where the three external fields are
massless scalars. This corresponds to the inflaton bispectra which are most
relevant for observations. To compute these correlators, one way to proceed is
to repeat what we did for the $\langle\varphi\varphi\phi\rangle$ correlator:
introduce a bulk integral $I_{\phi\phi\phi}$ based on the cubic vertex, derive
its differential equation using the mixed propagator, and solve for its
solution with proper boundary conditions. Although in principle it can be
done, this procedure may become rather complicated, since the boost-breaking
cubic vertices may take various forms, and for each of them we need to solve
an inhomogeneous equation correspondingly. Here we take a more efficient
‘bootstrap” approach by making the use of the scalar seeds. The key insight is
that the $\langle\phi\phi\phi\rangle$ bispectra can be generated by acting on
the seeds with various differential operators. This is similar in spirit to
the “weight-shifting” approach of [41, 42, 43].888Similar differential
operators that change the weights (masses) of scalar fields were first
introduced in the context of the conformal bootstrap [82]. Despite the absence
of boost symmetry, interestingly, such operators still exist, as shown in [55]
for correlators in single field inflation.

Here our goal is to systematically derive the $\langle\phi\phi\phi\rangle$
correlator from the scalar seeds. Concretely, we aim to map the
$\varphi^{2}\sigma$ vertex to a general cubic vertex of $\phi\phi\sigma$-type
with boost-breaking interactions. That is to say, the scaling dimension of two
boundary operators needs to be shifted from $\Delta=2$ to $\Delta=3$, and at
the same time we should take into account the time and spatial derivatives on
them. Let us begin by proposing the generic form of the cubic vertex as

$\partial_{i}^{n_{s}}\left(\partial_{t}^{n_{1}}\phi\partial_{t}^{n_{2}}\phi\partial_{t}^{n_{3}}\sigma\right)\leavevmode\nobreak\
,$ (5.7)

where $n_{s}$ is the total number of spatial derivatives, $n_{1}$, $n_{2}$ and
$n_{3}$ are the numbers of time derivatives for the two massless field $\phi$
and the massive scalar $\sigma$ respectively. Notice that the spatial
derivatives can act on any field in the vertex, and $n_{s}$ is an even number
for scalar interactions. In Fourier space, the above cubic vertex leads to (at
highest derivatives)

$(-{\bf k}_{a}\cdot{\bf
k}_{b})^{n_{s}/2}a^{-\tilde{n}_{T}-n_{3}}\partial^{n_{1}}_{\eta}\phi_{k_{1}}\partial^{n_{2}}_{\eta}\phi_{k_{2}}\partial^{n_{3}}_{\eta}\sigma_{k_{3}}\leavevmode\nobreak\
,$ (5.8)

where $\tilde{n}_{T}=n_{1}+n_{2}+n_{s}$ is the total number of derivatives on
the inflaton $\phi$, and ${\bf k}_{a}\cdot{\bf k}_{b}$ ($a,b=1,2,3$)
corresponds to the possible contractions of momenta. Then the three-point
function of the massless scalar is given by the following bulk integral

$\displaystyle\langle\phi_{\bf k_{1}}\phi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}=i(-{\bf k}_{a}\cdot{\bf k}_{b})^{n_{s}/2}\int d\eta
a(\eta)^{4-\tilde{n}_{T}}\left[\partial^{n_{1}}_{\eta}K_{+}(k_{1},\eta)\partial^{n_{2}}_{\eta}K_{+}(k_{2},\eta)\mathcal{K}_{+}(k_{3},\eta)-c.c.\right]+{\rm
perms}\leavevmode\nobreak\ .$ (5.9)

Next we need to take care of the time derivatives inside the integral. For the
mixed propagator, we have absorbed its time derivatives of
$a^{-n_{3}}\partial^{n_{3}}_{\eta}\sigma$ into its expression, and trade them
with $k$-derivatives. This leads to a differential operator
$k^{n_{3}}\partial_{k}^{n_{3}}$ as we discussed around (5.5). For the free
propagator $K(k,\eta)$ of massless field $\phi$, in general its $n$-th order
time derivative takes a simple form

$\partial^{n}_{\eta}K_{\pm}(k,\eta)=\frac{H^{2}}{2c_{s}^{3}k^{3}}(\pm
ic_{s}k)^{n}\left(1-n-k\partial_{k}\right)e^{\pm
ic_{s}k\eta}\leavevmode\nobreak\ .$ (5.10)

This is a key observation that helps us convert the $\varphi$ propagators into
the ones of the massless field $\phi$ through differential operations.
Substituting it into (5.9), and taking the $k$-derivatives outside of the bulk
integral, we find

$\displaystyle\langle\phi_{\bf k_{1}}\phi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}=(-1)^{\tilde{n}_{T}}\frac{iH^{\tilde{n}_{T}}}{4c_{s}^{6}k_{1}^{2}k_{2}^{2}}\mathcal{W}_{12}I_{\varphi\varphi\phi}(k_{12},k_{3})+{\rm
perms}\leavevmode\nobreak\ ,$ (5.11)

where $I_{\varphi\varphi\phi}(k_{12},k_{3})$ is the bulk integral associated
with the $\langle\varphi\varphi\phi\rangle$ correlator in (5.6). Meanwhile, we
have introduced a dimensionless differential operator

$\mathcal{W}_{12}\equiv-c_{s}^{2-n_{s}}({\bf k}_{a}\cdot{\bf
k}_{b})^{n_{s}/2}k_{1}^{n_{1}-1}k_{2}^{n_{2}-1}\left(1-n_{1}-k_{1}\partial_{k_{1}}\right)\left(1-n_{2}-k_{2}\partial_{k_{2}}\right)\partial_{k_{12}}^{\tilde{n}_{T}-2}\leavevmode\nobreak\
,$ (5.12)

which is the boost-breaking weight-shifting operator999We wish to thank Enrico
Pajer for helping with its derivation.. From the intuition of bulk
calculation, this operator exactly maps $\varphi\varphi$ to
$\partial_{i}^{n_{s}}\left(\partial_{t}^{n_{1}}\phi\partial_{t}^{n_{2}}\phi\right)$
in time integral of the cubic vertex. As we will show shortly in the examples,
the form of $\mathcal{W}_{12}$ returns to the one in de Sitter bootstrap when
we consider the dS-invariant cubic interaction. In general, this operator is
capable of generating all the boost-breaking $\phi\phi\sigma$-type vertices
from the $\varphi^{2}\sigma$ one.

The result in (5.11) directly works on the boundary correlators. This general
expression provides all the possible inflaton bispectra from the exchange of
one massive scalar field. Starting with the generalized scalar seeds, we can
map them to arbitrary boost-breaking cubic interactions by performing
differential operators $k_{3}\partial_{k_{3}}$ (for higher derivatives on
$\sigma$) and $\mathcal{W}_{12}$ (for higher derivatives on $\phi$). In the
following we shall consider simple but observationally relevant interactions,
and then apply the operator (5.12) to bootstrap examples of the inflaton
bispectra from massive scalar exchange.

###  5.2 Scalar Exchange Bispectra

For scalar exchange, we focus on the simplest quadratic interaction
$\dot{\phi}\sigma$, which gives the mixed propagator
$\mathcal{K}_{\pm}(k,\eta;c_{s})=H/(c_{s}^{3}k^{3})\hat{\mathcal{K}}^{(0)}_{\pm}(k\eta;c_{s})$.
The most relevant cubic interactions are the ones with lowest derivatives

$\dot{\phi}^{2}\sigma\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
(\partial_{i}\phi)^{2}\sigma\leavevmode\nobreak\ .$ (5.13)

When the de Sitter boosts are not broken, they are restricted to take the
particular combination $(\partial_{\mu}\phi)^{2}\sigma$. This cubic coupling
automatically induces a small linear mixing, which is slow-roll suppressed. In
boost-breaking theories, these two cubic vertices can appear independently
with large interactions, as we have shown in the EFT analysis around (2.6).
For these low-derivative interactions, the exchange three-point function can
be expressed as

$\displaystyle\langle\phi_{\bf k_{1}}\phi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}=\frac{iH^{3}}{4c_{s}^{9}k_{1}^{2}k_{2}^{2}k_{3}^{2}}\mathcal{W}_{12}\mathcal{\hat{I}}^{(0)}+{\rm
perms}\leavevmode\nobreak\ .$ (5.14)

Then the bispectra shapes are fully determined once we specify the weight-
shifting operators based on cubic vertices. Explicitly, they are given by:

  * •

The $\dot{\phi}^{2}\sigma$ vertex. The weight-shifting operator is given by

$\mathcal{W}^{\dot{\phi}^{2}\sigma}_{12}=-c_{s}^{2}k_{1}k_{2}\partial_{k_{12}}^{2}\leavevmode\nobreak\
.$ (5.15)

In the squeezed limit $k_{3}\rightarrow 0$ and $k_{1}=k_{2}$, the bispectrum
(5.14) becomes

$\lim_{k_{3}\rightarrow 0}\langle\phi_{\bf k_{1}}\phi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}=\frac{iA_{0}H^{3}}{4c_{s}^{4}k_{1}^{3}k_{3}^{3}}\left[\frac{\Gamma\left(\frac{5}{2}+i\mu\right)}{\Gamma\left(1+i\mu\right)}(1+i\sinh\pi\mu)\left(\frac{k_{3}}{4c_{s}k_{1}}\right)^{\frac{3}{2}+i\mu}-c.c\right]\leavevmode\nobreak\
,$ (5.16)

where $A_{0}={\sqrt{\pi}\Xi_{0}(\mu,c_{s})}/{\sinh\pi\mu}$ with the $\Xi_{0}$
function defined in (A.6).

  * •

The $(\partial_{i}\phi)^{2}\sigma$ vertex. It leads to a different operator

$\mathcal{W}^{(\partial_{i}\phi)^{2}\sigma}_{12}=-\frac{1}{2k_{1}k_{2}}(k_{3}^{2}-k_{1}^{2}-k_{2}^{2}){\left(1-k_{1}\partial_{k_{1}}\right)\left(1-k_{2}\partial_{k_{2}}\right)}\leavevmode\nobreak\
,$ (5.17)

where we have used the momentum conservation to rewrite ${\bf k}_{1}\cdot{\bf
k}_{2}=(k_{3}^{2}-k_{1}^{2}-k_{2}^{2})/2$. For this bispectrum, the squeezed
limit is

$\displaystyle\lim_{k_{3}\rightarrow 0}\langle\phi_{\bf k_{1}}\phi_{\bf
k_{2}}\phi_{\bf k_{3}}\rangle^{\prime}$ $\displaystyle=$
$\displaystyle\frac{-iA_{0}H^{3}}{4c_{s}^{6}k_{1}^{3}k_{3}^{3}}\left[\frac{\Gamma\left(\frac{1}{2}+i\mu\right)}{\Gamma\left(1+i\mu\right)}\left(\frac{3}{2}+i\mu\right)\left(\frac{9}{2}+i\mu\right)\right.$
(5.18) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left.\times(1+i\sinh\pi\mu)\left(\frac{k_{3}}{4c_{s}k_{1}}\right)^{\frac{3}{2}+i\mu}-c.c\right]\leavevmode\nobreak\
.$

  * •

As a nontrivial check, we also reconstruct the bispectrum from de Sitter
invariant interaction $(\partial_{\mu}\phi)^{2}\sigma$ from the two results
above. Setting $c_{s}=1$, we get the dS-invariant weight-shifting operator

$\mathcal{W}_{12}^{\rm
dS}=-\mathcal{W}^{\dot{\phi}^{2}\sigma}_{12}+\mathcal{W}^{(\partial_{i}\phi)^{2}\sigma}_{12}=\frac{1}{2}\left(k_{12}^{2}-k_{3}^{2}\right)\partial^{2}_{k_{12}}-\frac{1}{2k_{1}k_{2}}\left(k_{3}^{2}-k_{1}^{2}-k_{2}^{2}\right)\left(1-k_{12}\partial_{k_{12}}\right)\leavevmode\nobreak\
,$ (5.19)

which reproduces the one in the de Sitter bootstrap [41]. We also check the
soft limit of this bispectrum

$\lim_{k_{3}\rightarrow 0}\langle\phi_{\bf k_{1}}\phi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}=\frac{iA_{0}H^{3}}{4k_{1}^{3}k_{3}^{3}}\left[\frac{\Gamma\left(\frac{7}{2}+i\mu\right)}{\Gamma\left(1+i\mu\right)}\frac{(1+i\sinh\pi\mu)}{\frac{1}{2}+i\mu}\left(\frac{k_{3}}{4k_{1}}\right)^{\frac{3}{2}+i\mu}-c.c\right]\leavevmode\nobreak\
,$ (5.20)

and find that it is in agreement with (6.130) in [3].

Through these simple examples, we find the boost-breaking interactions
generate new bispectrum shapes for the cosmological collider physics, which
differ from the one with all the de Sitter symmetries. Furthermore, their
sizes are not supposed to be slow-roll suppressed. We leave the detailed
discussion on the phenomenological implications in Section 7.

Finally, let us look at the bispectrum shapes that arise from integrating out
a heavy scalar with $m\gg H$. The boost-breaking weight-shifting operators can
also help us derive these contact three-point correlators with any number of
derivatives. Since in this situation the solutions of the generalized scalar
seeds are well approximated by the contact term $\mathcal{\hat{I}}^{(n)}\sim
u^{n+1}/(1+c_{s}u)^{n+1}$, schematically the inflaton bispectrum can be
written as

$\langle\phi_{\bf k_{1}}\phi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}\supset\frac{1}{k_{1}^{2}k_{2}^{2}k_{3}^{3}}\mathcal{W}_{12}\left(k_{3}\partial_{k_{3}}\right)^{m}\left[k_{3}\left(\frac{k_{3}}{k_{T}}\right)^{n+1}\right]+{\rm
perms}\leavevmode\nobreak\ .$ (5.21)

These shape functions are rational polynomials of the absolute values of the
momenta. They can be interpreted as coming from higher-derivative inflaton
self-interactions. A complete set of boostless contact bispectra from single-
field inflation have been derived from symmetries and locality constraints in
[51, 52, 53], while our approach (5.21) provides a consistency check for the
computation of these shapes. For illustration, we take the lowest-derivative
vertices again as examples. Consider the $\dot{\phi}\sigma$ mixing, and the
scalar seed is given by $\mathcal{\hat{I}}^{(0)}\sim u/(1+c_{s}u)$. Then the
cubic vertex $\dot{\phi}^{2}\sigma$ leads to the following shape of the three-
point function

$\langle\phi_{\bf k_{1}}\phi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}\sim\frac{1}{k_{1}k_{2}k_{3}^{2}}\partial_{k_{12}}^{2}\left(\frac{k_{3}}{k_{T}}\right)+{\rm
perms}\sim\frac{1}{k_{1}k_{2}k_{3}k_{T}^{3}}\leavevmode\nobreak\ ,$ (5.22)

which is the one from the contact interaction $\dot{\phi}^{3}$. Similarly
integrating out the $\sigma$ field in the $(\partial_{i}\phi)^{2}\sigma$
vertex leads to the bispectrum from $(\partial_{i}\phi)^{2}\dot{\phi}$

$\displaystyle\langle\phi_{\bf k_{1}}\phi_{\bf k_{2}}\phi_{\bf
k_{3}}\rangle^{\prime}$ $\displaystyle\sim$
$\displaystyle\frac{k_{3}^{2}-k_{1}^{2}-k_{2}^{2}}{k_{1}^{3}k_{2}^{3}k_{3}^{2}}\left(1-k_{1}\partial_{k_{1}}\right)\left(1-k_{2}\partial_{k_{2}}\right)\left(\frac{k_{3}}{k_{T}}\right)+{\rm
perms}$ (5.23) $\displaystyle=$
$\displaystyle\left(k_{3}^{2}-k_{1}^{2}-k_{2}^{2}\right)\frac{2k_{1}k_{2}+k_{12}k_{T}+k_{T}^{2}}{k_{T}^{3}k_{1}^{3}k_{2}^{3}k_{3}}+{\rm
perms}\leavevmode\nobreak\ .$

Both of the above bispectra have the equilateral type scaling in the soft
limit, unlike the non-analytical behaviour of massive field exchange. If we
consider higher derivative vertices, they correspond to more derivatives in
the weight-shifting operators and/or higher order seed functions, both of
which lead to higher powers of $k_{T}$ in the denominator of the shape
function. This is in agreement with the analysis in [51, 52, 53].

##  6 Exchange of Spinning Particles

Massive spinning particles leave unique imprints in primordial non-
Gaussianity. In particular, they modulate the bispectrum with a spin-dependent
envelope. In this section, we extend to compute the boostless bispectra from
the single exchange of spinning particles during inflation. We make the use of
the generalized scalar seeds in Section 4.2, and map them to spin-exchange
correlators by “spin-raising” operators.

In Section 6.1 we discuss the exchange of spin-1 fields in detail, using it as
a case study for the general strategy. In Section 6.2, we introduce the basics
of free and mixed propagators of fields with arbitrary spins during inflation,
while the free theory of spinning fields in de Sitter space is summarized in
Appendix D. In Section 6.3 we derive the three-point functions from spinning
exchanges with generic boost-breaking interactions.

###  6.1 Spin-1 Exchange

In this section, we describe in detail the derivation of the bispectrum from
spin-1 exchange. We introduce the necessary “spin-raising” operator to obtain
it from scalar exchange seeds. It is helpful to study this simplest case to
gain insight about how to bootstrap the generic spinning exchange bispectra.
However, from a phenomenological perspective, this case is not the most
interesting one, as the signals in the squeezed limit from odd spin exchange
is more suppressed than the one with even spins.

####  6.1.1 Spin-1 Propagators

First, let us derive the mixed propagators with spin-1 fields. We will focus
on the longitudinal mode of the massive spin-1 particle (the only component
that contributes to the exchange diagram) and establish its connection with
the massive scalars.

##### Free Theory in de Sitter

The notion of spin is less unambiguous in the EFT of inflation. With the full
dS isometries, we can have a dS-invariant description for the spinning fields,
which leads to the EFT in [13]. While the dS boosts are broken, it becomes
possible to construct another type of EFT [23], where more general theories of
spinning fields are allowed but it remains unclear about how to embed them in
a UV-complete theory. Meanwhile, for the interest of this work, we notice that
only the helicity-0 longitudinal mode will contribute to the scalar bispectra
of cosmological colliders. As long as this single component of the spinning
field is concerned, there is no much difference in these two EFT approaches.
Thus in this paper we shall take the first approach with dS isometries, where
the UV completion is much better understood.

For a massive spin-1 field $\sigma_{\mu}$ in de Sitter space, its quadratic
action is given by

$S_{2}=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}\nabla_{\mu}\sigma_{\nu}\nabla^{\mu}\sigma^{\nu}+\frac{1}{2}(\nabla^{\mu}\sigma_{\mu})^{2}-\frac{1}{2}(m^{2}+3H^{2})\sigma^{\mu}\sigma_{\mu}\right]\leavevmode\nobreak\
,$ (6.1)

which is equivalent to the Proca action up to integration by parts. Here we
have chosen the mass definition such that $m$ becomes the mass of
$\sigma_{\mu}$ in the flat space limit, and we can check that the action
becomes gauge invariant when $m\rightarrow 0$. From this action, the equation
of motion of the $\sigma_{\mu}$ field is

$\left[\nabla^{\nu}\nabla_{\nu}-\left(m^{2}+3H^{2}\right)\right]\sigma_{\mu}=0\leavevmode\nobreak\
,$ (6.2)

and we also find the transverse condition

$\nabla^{\mu}\sigma_{\mu}=0\leavevmode\nobreak\ .$ (6.3)

The spinning field $\sigma_{\mu}$ may also have a nontrivial sound speed
$c_{\sigma}$, even though it appears in dS-invariant forms in the above
expressions. Like in the massive scalar case, in the free theory we can always
rescale this sound speed into the spatial coordinates
$c_{\sigma}\partial_{i}\rightarrow\partial_{i}$ (or, equivalently
$c_{\sigma}k\rightarrow k$ in the Fourier space), such that it disappears in
the final expression. As long as all the components of $\sigma_{\mu}$ have the
same sound speed, this rescaling can lead us back to (6.1) – (6.3). The
spinning fields with different sound speeds for each component were discussed
in [23], but since only one component makes contribution to the final scalar
three-point function, the theories with multiple sound speeds do not lead to
additional bispectrum shapes. Therefore for convenience, our strategy is to
focus on the dS-invariant theories but allow one uniform sound speed for the
spinning field.

To discuss the mode functions, it is more convenient to expand the spinning
fields into their helicity basis. For spin-1 fields, this decomposition
becomes

$\sigma_{\mu}=\sum_{\lambda=-1}^{1}\sigma_{\mu}^{(\lambda)}\leavevmode\nobreak\
,$ (6.4)

where $\lambda=0$ gives the longitudinal ($l$) mode and the transverse ($t$)
ones correspond to $\lambda=\pm 1$. The transverse modes have only the spatial
components $\sigma^{(\pm 1)}_{i}=\sigma_{t}^{\pm 1}\epsilon^{(\pm 1)}_{i}$,
with the polarization vectors satisfying $k_{i}\epsilon^{(\pm 1)}_{i}({\bf
k})=0$. The temporal ($T$) and spatial ($S$) components of the longitudinal
mode with $\lambda=0$ can be further expressed as

$\sigma_{\eta}^{(0)}=\sigma^{T}_{l}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\sigma_{i}^{(0)}=\sigma^{S}_{l}\epsilon_{i}^{(0)}\leavevmode\nobreak\ ,$ (6.5)

with the longitudinal polarization vector $\epsilon_{i}^{(0)}({\bf
k})=\hat{k}_{i}$. As we will show very soon, only the longitudinal mode
contributes to the spin-1 mixed propagator and thus leads to nonzero exchange
bispectra. In the following we will focus on the $\lambda=0$ modes, and drop
the upper index (λ) in the mode functions and polarization vector.

##### Longitudinal modes

For future convenience, we introduce the new notation for $\lambda=0$
longitudinal modes

$\sigma^{(0)}_{1}=\sigma^{T}_{l}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \sigma^{(1)}_{1}=\sigma^{S}_{l}\leavevmode\nobreak\ .$
(6.6)

For a general expression $\sigma^{(n)}_{s}$, the lower index represents the
spin101010Note that for the spinning mode functions we do not use the lower
index to represent the momentum $k$, which differs from the notation of scalar
fields. and the upper index $n\leq s$ is used for the number of spatial
polarization directions. For the Fourier mode $\sigma_{1}^{(0)}$, the equation
(6.2) becomes

$\left(\mathcal{O}_{\eta}+\mu_{1}^{2}+\frac{9}{4}\right)\sigma^{(0)}_{0}=0\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm with}\leavevmode\nobreak\ \leavevmode\nobreak\
\mu_{1}=\sqrt{\frac{m^{2}}{H^{2}}-\frac{1}{4}}$ (6.7)

which is the same as the one of massive scalars in (3.3). Imposing the Bunch-
Davies initial condition and the correct normalization [13], we obtain

$\displaystyle\sigma_{1}^{(0)}(k,\eta)=\frac{H}{2m}\sqrt{\frac{\pi}{k}}e^{i\pi/4}e^{-\pi\mu_{1}/2}(-k\eta)^{3/2}H^{(1)}_{i\mu_{1}}(-k\eta)\leavevmode\nobreak\
,$ (6.8)

which is related to the scalar mode function $\sigma_{k}(\eta)$ in (3.4) by

$\sigma_{1}^{(0)}(k,\eta)=kN_{1}\sigma_{k}(\eta)\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
with}\leavevmode\nobreak\ \leavevmode\nobreak\
N_{1}=\frac{i}{m}\leavevmode\nobreak\ .$ (6.9)

The equation and mode function solution of the $\sigma_{1}^{(1)}$ mode are
more complicated, but we notice that the two longitudinal modes are related
through the constraint (6.3)

$\sigma^{(1)}_{1}=-iU^{(1)}_{\eta}\sigma^{(0)}_{1}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
with}\leavevmode\nobreak\ \leavevmode\nobreak\
U^{(1)}_{\eta}\equiv\frac{1}{k}\left(\partial_{\eta}-\frac{2}{\eta}\right).$
(6.10)

Therefore we are able to connect the $\lambda=0$ longitudinal mode of
$\sigma_{\mu}$ with the solution of the massive scalar field discussed in
Section 3.1. As we will show below, this is the key observation that allows us
to map the results of mixed propagators and seed functions with massive
scalars to the spinning case. Specifically, the bulk-to-bulk propagators of
the $\sigma_{\eta}=\sigma^{(0)}_{1}$ mode are simply given by

$G^{(0)}_{\pm\pm}(k,\eta,\eta^{\prime})=\frac{k^{2}}{m^{2}}G^{\sigma}_{\pm\pm}(k,\eta,\eta^{\prime})\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
G^{(0)}_{\pm\mp}(k,\eta,\eta^{\prime})=\frac{k^{2}}{m^{2}}G^{\sigma}_{\pm\mp}(k,\eta,\eta^{\prime})\leavevmode\nobreak\
.$ (6.11)

For the $\sigma_{i}=\sigma^{(1)}_{1}\epsilon_{i}$ mode, its propagators are a
bit more complicated, but can also be expressed in terms of $G^{\sigma}$ as

$\displaystyle G_{ij,\pm\mp}^{(1)}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle|N_{1}|^{2}k^{2}U^{(1)}_{\eta}U^{(1)}_{\eta^{\prime}}G_{\pm\mp}^{\sigma}(k,\eta,\eta^{\prime})\epsilon_{i}\epsilon_{j}\leavevmode\nobreak\
,$ $\displaystyle G_{ij,\pm\pm}^{(1)}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle|N_{1}|^{2}k^{2}U^{(1)}_{\eta}U^{(1)}_{\eta^{\prime}}G_{\pm\pm}^{\sigma}(k,\eta,\eta^{\prime})\epsilon_{i}\epsilon_{j}\mp
i\eta^{2}\delta(\eta-\eta^{\prime})\frac{H^{2}}{m^{2}}\epsilon_{i}\epsilon_{j}\leavevmode\nobreak\
.$ (6.12)

The $\delta$-function term in $G_{ij,\pm\pm}^{(1)}$ is to cancel the ones
generated when the time derivatives hit the $\Theta$-functions in the bulk-to-
bulk propagator.

##### Spin-1 Mixed Propagator

Next, we consider the linear mixing between the massive spin-1 field and the
inflaton. At leading order in derivatives, there are two quadratic
interactions: $\partial_{\eta}\phi\sigma_{\eta}$ and
$\partial_{i}\phi\sigma_{i}$. Since they are related from the constraint
equation in (6.3), we can mainly focus our analysis on the
$\partial_{i}\phi\sigma_{i}$ mixing [13]. From this quadratic vertex, we find
the contraction from the transverse mode $k_{i}\epsilon_{i}^{(\pm 1)}$
vanishes, and thus only the spatial component of the longitudinal mode
$\sigma_{i}=\sigma^{(1)}_{1}\epsilon_{i}$ gives the nonzero contribution to
the spin-1 mixed propagator. Explicitly, we introduce the bulk-to-boundary
propagator from $\sigma_{i}$ to $\phi$ as

$\displaystyle\mathcal{K}^{(1)}_{i,\leavevmode\nobreak\ \pm}(k,\eta)$
$\displaystyle=$ $\displaystyle\pm
i\int_{-\infty}^{0}d\eta^{\prime}a(\eta^{\prime})^{2}\left[G_{ij,\leavevmode\nobreak\
\pm\pm}^{(1)}(k,\eta,\eta^{\prime})(\mp
ik_{j})K_{\pm}(c_{s}k,\eta^{\prime})\right.$ (6.13)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left.-G_{ij,\leavevmode\nobreak\ \pm\mp}^{(1)}(k,\eta,\eta^{\prime})(\pm
ik_{j})K_{\mp}(c_{s}k,\eta^{\prime})\right]$

Using (6.1.1), $\mathcal{K}^{(1)}_{i,\leavevmode\nobreak\ \pm}$ can be
expressed as

$\displaystyle\mathcal{K}^{(1)}_{i,\leavevmode\nobreak\ \pm}$ $\displaystyle=$
$\displaystyle-\epsilon_{i}|N_{1}|^{2}k^{2}U^{(1)}_{\eta}\int_{-\infty}^{0}d\eta^{\prime}a(\eta^{\prime})^{2}\left[G_{\pm\pm}^{\sigma}\partial_{\eta^{\prime}}K_{\pm}+G_{\pm\mp}^{\sigma}\partial_{\eta^{\prime}}K_{\mp}\right]\mp\frac{ik}{m^{2}}\epsilon_{i}K_{\pm}(k,\eta)\leavevmode\nobreak\
,$ (6.14)

where we have applied integration by parts in $\eta^{\prime}$ to reduce the
$U^{(1)}_{\eta^{\prime}}$ operator in the integrand. This expression can be
written in terms of the scalar mixed propagators

$\displaystyle\mathcal{K}^{(1)}_{i,\leavevmode\nobreak\ \pm}(k,\eta)=\pm
i\epsilon_{i}\frac{H^{2}|N_{1}|^{2}}{c_{s}^{3}k^{3}\eta}\mathcal{U}^{(1)}_{k}\hat{\mathcal{K}}^{(1)}_{\pm}(k\eta;c_{s})\mp\frac{ik}{m^{2}}\epsilon_{i}K_{\pm}(c_{s}k,\eta)$
(6.15)

where $\hat{\mathcal{K}}^{(1)}_{\pm}$ is the dimensionless building block of
the mixed propagator in (4.2) with $n=1$. For the operator
$\mathcal{U}^{(1)}_{k}$, we have used the fact that $\hat{\mathcal{K}}^{(1)}$
is a function of the combination $k\eta$, such that we are able to trade
$\eta$-derivatives with $k$-derivatives. This gives us

$U^{(1)}_{\eta}=\frac{1}{k}\left(\partial_{\eta}-\frac{2}{\eta}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{1}{k\eta}\mathcal{U}^{(1)}_{k}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
with}\leavevmode\nobreak\ \mathcal{U}^{(1)}_{k}\equiv
k\left(\partial_{k}-\frac{2}{k}\right)\leavevmode\nobreak\ ,$ (6.16)

which is the spin-raising operator for $s=1$. Using this differential
operator, we can derive the spin-1 mixed propagator from the scalar one.
Notice that the piece which comes from the $\delta$-function term in (6.1.1)
is the free propagator $K_{\pm}$ of a massless scalar in (6.15). This term
leads to a standard contact interaction in the exchange diagram, whose
contribution to the bispectrum is the same with the ones from single field
inflation. In the following, we shall drop this contribution, and focus on the
effect of the first term of the spin-1 mixed propagator in (6.15).

It is straightforward to extend the analysis to the quadratic interactions
with higher derivatives. We can move all the derivatives to the scalar field
$\phi$ via integration by parts, then we get interactions like
$\partial_{t}^{n}\partial_{i}\phi\sigma_{i}$. These additional derivatives on
$\phi$ change the form of the integrand in (6.14). As we have seen in Section
4, the integral can always be written as a linear combination of the building
blocks $\hat{\mathcal{K}}^{(n)}$, while the spin-raising operator remains
unaffected.

####  6.1.2 Spin-1 Exchange Bispectra

With the spin-1 mixed propagator, we now bootstrap the three-point functions
from the single exchange diagram. We start with the bispectrum of two
conformally coupled scalars $\varphi$ with a massless scalar $\phi$. As one
leg of the cubic vertex should be attached to the mixed propagator with the
$\sigma_{i}$ longitudinal mode, the lowest derivative interaction is given by
$\varphi\partial_{i}\varphi\sigma_{i}$. As a result, the bispectrum is given
by

$\langle\varphi_{{\bf k}_{1}}\varphi_{{\bf k}_{2}}\phi_{{\bf
k}_{3}}\rangle^{\prime}=i\int_{-\infty}^{0}d\eta
a^{2}\left[K_{+}^{\varphi}(k_{1},\eta)(-ik_{2}^{i})K_{+}^{\varphi}(k_{2},\eta)\mathcal{K}^{(1)}_{i,\leavevmode\nobreak\
+}(k_{3},\eta)-c.c.\right]+{\rm perm.}$ (6.17)

By using the mixed propagator (6.15) from the $\partial_{i}\phi\sigma_{i}$
mixing, we can rewrite the bulk integral above in the form

$\displaystyle\langle\varphi_{{\bf k}_{1}}\varphi_{{\bf k}_{2}}\phi_{{\bf
k}_{3}}\rangle^{\prime}$ $\displaystyle=$ $\displaystyle
i\frac{A^{\varphi}_{1}c_{s}}{k_{1}k_{3}^{3}}(\hat{\bf k}_{2}\cdot\hat{\bf
k}_{3})\mathcal{U}^{(1)}_{k_{3}}\int_{-\infty}^{0}\frac{d\eta}{\eta}\left[e^{ic_{s}k_{12}\eta}\hat{\mathcal{K}}^{(1)}_{+}(k_{3}\eta)-c.c.\right]+{\rm
perm.}$ (6.18) $\displaystyle=$
$\displaystyle\frac{A^{\varphi}_{1}}{k_{1}k_{3}^{3}}(\hat{\bf
k}_{2}\cdot\hat{\bf k}_{3})\leavevmode\nobreak\
\mathcal{U}^{(1)}_{k_{3}}\leavevmode\nobreak\
\left[k_{3}\partial_{k_{12}}\mathcal{\hat{I}}^{(1)}\left(u;c_{s}\right)\right]+{\rm
perm.}\leavevmode\nobreak\ ,$

where $A^{\varphi}_{1}=H^{4}|N_{1}|^{2}\eta_{0}^{2}/4c_{s}^{6}$ and
$\mathcal{\hat{I}}^{(1)}$ is the generalized scalar seeds in (4.13) with
$n=1$. Therefore, we derive the spin-1 exchange bispectrum from the seed
functions by using the spin-raising operator $\mathcal{U}^{(1)}_{k_{3}}$. This
example demonstrates the generic structure of the three-point correlator from
spinning exchange in our approach. In addition to the spin-raising operator
and the scalar seed, the $\hat{\bf k}_{2}\cdot\hat{\bf k}_{3}=\cos\theta$
factor can be written as the Legendre polynomial $P_{1}(\cos\theta)$, which
encodes the angular dependent signal of the spinning particle; the
$k_{3}\partial_{k_{12}}$ operator is associated with the form of the cubic
interaction.

Next, we consider the inflaton bispectrum with three massless scalars. From
the EFT of inflation with spinning fields [13], the lowest derivative cubic
interaction with two inflatons and one massive spin-1 field is given by
$\dot{\phi}\partial_{i}\phi\sigma_{i}$, as shown in (2.10). This vertex only
arises in boost-breaking theories.111111In theories with full de Sitter
isometries, the cubic interaction appears as
$(\nabla_{\mu}\phi\nabla^{\nu}\nabla^{\mu}\phi-\nabla^{\nu}\nabla_{\mu}\phi\nabla^{\mu}\phi)\sigma_{\nu}$,
which leads to vanishing bispectra after permutation [3, 41]. Using the mixed
propagator $\mathcal{K}^{(1)}_{i,\leavevmode\nobreak\ \pm}(k,\eta)$, we find

$\displaystyle\langle\phi_{{\bf k}_{1}}\phi_{{\bf k}_{2}}\phi_{{\bf
k}_{3}}\rangle^{\prime}$ $\displaystyle=$ $\displaystyle
i\int_{-\infty}^{0}d\eta
a(\eta)\left[\partial_{\eta}K_{+}(k_{1},\eta)(-ik_{2}^{i})K_{+}(k_{2},\eta)\mathcal{K}^{(1)}_{i,\leavevmode\nobreak\
+}(k_{3},\eta)-c.c.\right]+{\rm perm.}$ (6.19) $\displaystyle=$
$\displaystyle\frac{A_{1}}{k_{1}^{2}k_{2}^{2}k_{3}^{3}}(\hat{\bf
k}_{2}\cdot\hat{\bf k}_{3})\leavevmode\nobreak\
\mathcal{U}^{(1)}_{k_{3}}\leavevmode\nobreak\
\mathcal{W}^{(1)}\left[k_{3}\mathcal{\hat{I}}^{(1)}(u)\right]+{\rm
perm.}\leavevmode\nobreak\ ,$

where the coefficient $A_{1}=-{H^{5}|N_{1}|^{2}}/{4c_{s}^{8}}$, and we have
used the following weight-shifting operator

$\mathcal{W}_{12}\equiv(1-k_{2}\partial_{k_{2}})k_{1}\partial_{k_{1}}\leavevmode\nobreak\
.$ (6.20)

This result provides the analytical shape of the inflationary bispectrum from
the boostless spin-1 exchange. We will discuss its phenomenological
implications in Section 7. Here let us simply look at its behaviour in the
squeezed limit121212There is a minor difference with the results in [13]. In
Eq. (C.23) of [13] and the discussions below, for odd spins, the squeezed
limit has $i\cosh\pi\mu_{1}$ instead of $(1+i\sinh\pi\mu_{1})$. We expect this
is because the relative signs of the sum in Eq.(C.2) of [13] should become
different for odd spin exchanges.

$\displaystyle\lim_{k_{3}\rightarrow 0}\langle\phi_{{\bf k}_{1}}\phi_{{\bf
k}_{2}}\phi_{{\bf k}_{3}}\rangle^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{H^{3}\sqrt{\pi}}{32c_{s}^{5}k_{1}^{3}k_{3}^{3}}\frac{\Xi_{1}(\mu_{1},c_{s})}{\cosh\pi\mu_{1}}(\hat{\bf
k}_{1}\cdot\hat{\bf k}_{3}+\hat{\bf k}_{2}\cdot\hat{\bf k}_{3})$ (6.21)
$\displaystyle\times\left[(7+2i\mu_{1})(1+i\sinh\pi\mu_{1})\frac{\Gamma(-i\mu_{1})}{\Gamma(\frac{1}{2}-i\mu_{1})}\left(\frac{k_{3}}{4c_{s}k_{1}}\right)^{\frac{3}{2}+i\mu_{1}}+c.c.\right]\leavevmode\nobreak\
,$

where the function $\Xi_{1}$ is defined in (A.6). Since from momentum
conservation we have

$\lim_{k_{3}\rightarrow 0}(\hat{\bf k}_{1}\cdot\hat{\bf k}_{3}+\hat{\bf
k}_{2}\cdot\hat{\bf k}_{3})=-\frac{k_{3}}{k_{1}}\left[1-(\hat{\bf
k}_{1}\cdot\hat{\bf
k}_{3})^{2}\right]+\mathcal{O}(k_{3}^{2}/k_{1}^{2})\leavevmode\nobreak\ ,$
(6.22)

the squeezed limit scales as $(k_{3}/k_{1})^{5/2}$, which is more suppressed
than the equilateral shape of the bispectrum. This behaviour is generic for
odd spin exchange, but not for even spins, as we show below.

To summarize, we have shown from this simplest example of spinning exchange
that the spinning propagators are related to the scalar ones by applying
differential operators. This is because only the helicity-$0$ longitudinal
mode of the massive field contributes to the linear mixing. Therefore, spin-
exchange bispectra can be mapped from scalar-exchange correlators in a simple
way. In the following we will follow the same strategy to bootstrap the
bispectra of higher spin exchange.

###  6.2 Spinning Propagators

Now we consider correlators involving bosonic particles
$\sigma_{\mu_{1}...\mu_{s}}$ of higher spin $s$ and arbitrary mass $m$. We
present a detailed review of its free theory in de Sitter space in Appendix D
(see also Appendix A of [13]). Again, like in the analysis of the massive
scalar and spin-1 fields , we have absorbed the sound speed by rescaling
$c_{\sigma}k\rightarrow k$.

The first step is to derive the spinning mixed propagators. For this purpose,
we discuss only the longitudinal mode $\lambda=0$ and consider its projection
on the spatial slicing $\sigma_{i_{1}...i_{n}\eta...\eta}$. In the
polarization basis, it is given by

$\sigma_{i_{1}...i_{n}\eta...\eta}=\sigma^{(n)}_{s}\epsilon_{i_{1}...i_{n}}\leavevmode\nobreak\
.$ (6.23)

The polarization tensor $\epsilon_{i_{1}...i_{n}}$ satisfies

$\hat{k}_{i_{1}}...\hat{k}_{i_{n}}\epsilon_{i_{1}...i_{n}}({\bf
k})=1\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\hat{q}_{i_{1}}...\hat{q}_{i_{n}}\epsilon_{i_{1}...i_{n}}({\bf
k})=P_{n}(\hat{\bf q}\cdot\hat{\bf k})\leavevmode\nobreak\ ,$ (6.24)

where $P_{n}(\hat{\bf q}\cdot\hat{\bf k})$ is the Legendre polynomial of order
$n$. We focus on the mode functions $\sigma^{(n)}_{s}$. Like in the spin-1
case, we use the upper index $n\leq s$ to denote the number of polarization
directions, and the lower index for the spin. In particular, the $n=0$ mode
$\sigma_{s}^{(0)}$ can be expressed in terms of the scalar mode function
(3.4), with $\mu=\mu_{s}$

$\displaystyle\sigma_{s}^{(0)}(k,\eta)=N_{s}k^{s}\sigma_{k}(\eta)\leavevmode\nobreak\
,$ (6.25)

where $N_{s}$ is a normalization constant, defined in (D.12), and

$\mu_{s}=\sqrt{\frac{m^{2}}{H^{2}}-\left(s-\frac{1}{2}\right)^{2}}\leavevmode\nobreak\
.$ (6.26)

For cosmological collider bispectra, we are interested in the $n=s$ mode,
$\sigma_{s}^{(s)}$, whose equation of motion becomes rather complicated.
However, its solution can be obtained iteratively from the transverse
condition $\nabla^{\mu_{1}}\sigma_{\mu_{1}...\mu_{s}}=0$, which in Fourier
space becomes

$\sigma_{s}^{(s)}=\left(-i\right)^{s}U^{(s)}_{\eta}\sigma_{s}^{(0)}.$ (6.27)

Here we have introduced the differential operator

$U^{(s)}_{\eta}\equiv\sum_{m=0}^{s}\frac{a^{(s)}_{m}}{k^{m}}\left(\partial_{\eta}-\frac{2}{\eta}\right)^{m},$
(6.28)

with $a^{(s)}_{s}=1$, $a^{(s)}_{s-1}=a^{(s)}_{s-3}=...=0$ and $a^{(s)}_{s-2n}$
are real constants fixed by the transverse condition. 131313See (D.15) for the
explicit formulae. For example, the $s=1$ operator is given in (6.10), while
the one for spin-2 fields is

$U^{(2)}_{\eta}=\frac{1}{k^{2}}\left(\partial_{\eta}-\frac{2}{\eta}\right)^{2}+\frac{1}{3}$
(6.29)

We leave more examples and further details of higher spins to Appendix D.
Using (6.25) and (6.27), we can establish the connection between the
$\sigma_{s}^{(s)}$ mode and the massive scalar mode function in general. In
particular, we are interested in the bulk-to-bulk propagators of the
$\sigma_{i_{1}i_{2}...i_{s}}$ mode. They are expressed in terms of the
$G^{\sigma}$-propagators as

$\displaystyle
G_{i_{1}...i_{s}j_{1}...j_{s},\pm\mp}^{(s)}(k,\eta,\eta^{\prime})$
$\displaystyle=$
$\displaystyle|N_{s}|^{2}k^{2s}U^{(s)}_{\eta}{U}^{(s)}_{\eta^{\prime}}G_{\pm\mp}^{\sigma}(k,\eta,\eta^{\prime})\epsilon_{i_{1}...i_{s}}\epsilon_{j_{1}...j_{s}}\leavevmode\nobreak\
,$ (6.30) $\displaystyle
G_{i_{1}...i_{s}j_{1}...j_{s},\pm\pm}^{(s)}(k,\eta,\eta^{\prime})$
$\displaystyle=$
$\displaystyle|N_{s}|^{2}k^{2s}U^{(s)}_{\eta}{U}^{(s)}_{\eta^{\prime}}G_{\pm\pm}^{\sigma}(k,\eta,\eta^{\prime})\epsilon_{i_{1}...i_{s}}\epsilon_{j_{1}...j_{s}}+\cdots$
(6.31)

where the $\cdots$ represent terms with $\delta(\eta-\eta^{\prime})$. As we
have seen in the spin-1 case, these extra pieces lead to contact terms in the
final bispectra. Since they are degenerate with the non-Gaussian shapes from
single field inflation, we drop them in the following.

Now we derive the mixed propagator with a spin-$s$ massive field. From the EFT
analysis in (2.13), the lowest derivative quadratic interaction between the
inflaton and a spin-$s$ field is

$\partial_{i_{1}...i_{s}}\phi\sigma_{i_{1}...i_{s}}\leavevmode\nobreak\ ,$
(6.32)

which is allowed to have large couplings in boost-breaking theories. From this
two-point vertex, only the helicity-$0$ longitudinal mode with maximal number
of polarization directions ($n=s$) gives a nonzero contribution, and we find
the spin-$s$ mixed propagator 141414For simplicity, we focus on
$\mathcal{K}^{(s)}_{i_{1}...i_{s},+}$, while
$\mathcal{K}^{(s)}_{i_{1}...i_{s},-}$ is given by complex conjugation.

$\displaystyle\mathcal{K}^{(s)}_{i_{1}...i_{s},+}(k,\eta;c_{s})$
$\displaystyle=$ $\displaystyle
i(-i)^{s}k_{j_{1}}...k_{j_{s}}\int_{-\infty}^{0}d\eta^{\prime}a(\eta^{\prime})^{4-2s}\left[G_{i_{1}...i_{s}j_{1}...j_{s},++}^{(s)}(k,\eta,\eta^{\prime})K_{+}(k,\eta^{\prime};c_{s})\right.$
(6.33) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\left.-(-1)^{s}G_{i_{1}...i_{s}j_{1}...j_{s},+-}^{(s)}(k,\eta,\eta^{\prime})K_{-}(k,\eta^{\prime};c_{s})\right]$
$\displaystyle=$ $\displaystyle
i(-i)^{s}|N_{s}|^{2}k^{3s}\epsilon_{i_{1}...i_{s}}{U}^{(s)}_{\eta}\int_{-\infty}^{0}d\eta^{\prime}a(\eta^{\prime})^{4-2s}\left[{U}^{(s)}_{\eta^{\prime}}G_{++}^{\sigma}(k,\eta,\eta^{\prime})K_{+}(k,\eta^{\prime};c_{s})\right.$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left.-(-1)^{s}{U}^{(s)}_{\eta^{\prime}}G_{+-}^{\sigma}(k,\eta,\eta^{\prime})K_{-}(k,\eta^{\prime};c_{s})\right]+\cdots$

where in the second equality we used (6.30) and (6.31). The ellipses represent
terms proportional to the conventional bulk-to-boundary propagator
$K_{\pm}(k,\eta;c_{s})$.

Notice that the constant factors and the $U^{(s)}_{\eta}$ operator in (6.30)
and (6.31) have been moved outside of the integral, while we still need to
address the $U^{(s)}_{\eta^{\prime}}$ operator inside in order to rewrite the
spinning mixed propagators in terms of $\hat{\mathcal{K}}^{(n)}$. For the
spin-1 case, $U^{(1)}_{\eta^{\prime}}$ has only one derivative and can be
simplified via integration by parts. For particles with arbitrary spin, it
becomes more complicated. In general, the higher order time derivatives on the
massive field (i.e. the $U^{(s)}_{\eta^{\prime}}$ operator on the $G^{\sigma}$
propagators here) can be reduced to terms with at most one derivative by using
the equation of motion of the $\sigma$ field. Let us take the spin-2 fields as
an explicit example whose $U^{(s)}_{\eta^{\prime}}$ operator is given in
(6.29). Applying the scalar mode equation of motion, the $\sigma_{2}^{(2)}$
mode is

$\sigma_{2}^{(2)}(k,\eta^{\prime})=-N_{2}k^{2}U^{(2)}_{\eta^{\prime}}\sigma_{k}(\eta^{\prime})=-N_{2}k^{2}\left(\frac{2}{k^{2}\eta^{\prime}}\partial_{\eta^{\prime}}+\frac{\mu_{2}^{2}-{15}/{4}}{k^{2}\eta^{\prime