††institutetext: Department of Physics, The Hong Kong University of Science
and Technology,
Clear Water Bay, Kowloon, Hong Kong, P.R. China††institutetext: The HKUST
Jockey Club Institute for Advanced Study,
The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong, P.R. China

# Cutting Rule for Cosmological Collider Signals: A Bulk Evolution Perspective

Xi Tong  Yi Wang  and Yuhang Zhu<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that the evolution of interacting massive particles in the de Sitter
bulk can be understood at leading order as a series of resonant decay and
production events. From this perspective, we classify the cosmological
collider signals into local and nonlocal categories with drastically different
physical origins. This further allows us to derive a cutting rule for
efficiently extracting these cosmological collider signals in an analytical
fashion. Our cutting rule is a practical way for extracting cosmological
collider signals in model building, and can be readily implemented as symbolic
computational packages in the future.

##  1 Introduction

As the leading paradigm of the primordial universe, inflation provides us more
than a natural genesis of a flat, causal, homogeneous and isotropic background
spacetime. Not only can the Gaussian quantum fluctuations prepared by
inflation seed the large-scale inhomogeneities and anisotropies, but also the
deviations from a purely Gaussian spectrum characterize unique physics during
the inflationary era. The study of such non-Gaussianities and its implications
for the particle spectrum during inflation has become an active field in the
recent years, under the title Cosmological Collider (CC) physics Chen:2009we ;
Chen:2009zp ; Arkani-Hamed:2015bza ; Baumann:2011nk ; Assassi:2012zq ;
Chen:2012ge ; Pi:2012gf ; Gong:2013sma ; Chen:2015lza ; Chen:2016nrs ;
Lee:2016vti ; Meerburg:2016zdz ; Chen:2016uwp ; Chen:2016hrz ; Chen:2017ryl ;
An:2017hlx ; An:2017rwo ; Kumar:2017ecc ; Chen:2018xck ; Chua:2018dqh ;
Kumar:2018jxz ; Wu:2018lmx ; Alexander:2019vtb ; Lu:2019tjj ; Hook:2019zxa ;
Hook:2019vcn ; Kumar:2019ebj ; Wang:2019gbi ; Liu:2019fag ; Wang:2019gok ;
Wang:2020uic ; Li:2020xwr ; Wang:2020ioa ; Fan:2020xgh ; Kogai:2020vzz ;
Bodas:2020yho ; Aoki:2020zbj ; Arkani-Hamed:2018kmz ; Baumann:2019oyu ;
Baumann:2020dch ; Lu:2021gso ; Lu:2021wxu ; Wang:2021qez . Powered by the fast
expansion of the universe, the CC operates at an energy scale as high as
$H\lesssim 10^{14}$GeV, where $H$ is the Hubble parameter during inflation.
The inflating spacetime stretches the wavelength of individual modes of a
quantum field, leading to the pairwise production of massive particles. If
coupled to the inflaton sector, these gravitationally produced particles decay
into curvature fluctuations, thereby seeding the non-Gaussian features on the
Cosmic Microwave Background (CMB), the Large Scale Structure (LSS), and the
21-cm tomography of neutral hydrogen atoms in the dark age. Considering the
coupling to the graviton, these particles also leave distinctive features for
the multi-spectra of Primordial Gravitational Waves (PGWs) as well as their
cross-correlators with the curvature perturbations.

More specifically, the observables in cosmological collider physics are the
$n$-point correlation functions. In the soft limits, the $n$-point correlation
functions exhibit oscillations with respect to the logarithm of momentum
ratios ($i.e.$, $\sin(\mu\ln k+\cdots)$), with the frequency $\mu$ encoding
the mass of the internal particle $m\sim\mu H$. These oscillations non-
analytic in $k$ are the essential ingredient in CC physics, and are often
referred to as CC signals or the clock signals Arkani-Hamed:2015bza ;
Chen:2015lza . Being an on-shell effect, the CC signals are the reminiscence
of the mass poles for Mandelstam variables in flat spacetime scattering
amplitudes. Furthermore, the angular dependence of the CC signal reveals the
spin of the internal particle, while its imaginary part reflects the CP
property of the internal particle.

The past decade has witnessed a rich development of cosmological collider
physics. On the theory side, in addition to the traditional in-in formalism
Weinberg:2005vy  and the Schwinger-Keldysh diagrammatics Chen:2017ryl , new
formalisms are put forward to compute and understand the processes happening
at the CC. For instance, based on dS symmetries and the singularity structure
of perturbation theory, the cosmological bootstrap program aims to
systematically reconstruct the correlation functions from a pure boundary
perspective Arkani-Hamed:2018kmz ; Baumann:2019oyu ; Baumann:2020dch . Methods
from the AdS toolkit, such as the Mellin-Barnes representation, are applied to
tree-level processes with useful results Sleight:2019mgd ; Sleight:2019hfp .
The recent cutting rules utilize locality, unitarity and analyticity, to
derive general recursion relations for the discontinuity of cosmological
wavefunctions Goodhew:2020hob ; Melville:2021lst ; Jazayeri:2021fvk ;
Goodhew:2021oqg ; Baumann:2021fxj ; Meltzer:2021zin . More formally, the
cosmological polytopes provide an intriguing geometric picture for the CC
process, while also hinting connections between cosmological correlators and
flat spacetime scattering amplitudes Arkani-Hamed:2017fdk . On the
phenomenology side, first, the detection of any Beyond Standard Model (BSM)
physics at the CC requires a clear understanding of the Standard Model (SM)
background, which has been carefully studied inChen:2016uwp ; Chen:2016hrz ;
Kumar:2017ecc ; Hook:2019vcn . BSM models such as supersymmetry Baumann:2011nk
, extra dimension Kumar:2018jxz  and monodromy Flauger:2016idt  give robust
predictions at the CC. In search of a large CC signal, CP-violating
dimension-5 operators that act as an external chemical potential can assist
gravitational production by alleviating the effective mass Chen:2018xck ;
Liu:2019fag ; Wang:2019gbi ; Wang:2020ioa ; Hook:2019zxa ; Sou:2021juh ;
Bodas:2020yho , while a warm inflationary background can provide abundant
massive particles from the furnace in the UV Berera:1995wh ; Berera:1995ie ;
Tong:2018tqf . As a channel alternative to curvature perturbations,
isocurvature colliders also encode the CC signals via modulated reheating
Lu:2019tjj ; Kumar:2019ebj ; Lu:2021gso .

However, despite the fruitful developments and immense opportunities for
cosmological collider physics, model building in cosmological collider physics
still faces a practicality problem. Namely, given a model with a set of
couplings, the process of computing CC observables itself can be laborious.
Unlike flat spacetime amplitudes, the loss of time translation invariance
implies energy non-conservation at interaction vertices, as well as non-
trivial mode functions. Due to these two complications, the well-celebrated
Lorentz-covariant perturbation theory in the 4-momentum space loses its
advantage over the time-dependent perturbation theory in the time domain.
However, the time-domain calculation has its own difficulties. A consistent
time-dependent perturbation demands specific orderings of the interaction
vertices (which, in flat spacetime amplitudes, would be easily handled by $\pm
i\epsilon$ deformations in the propagator momenta). This leads to nested
integrals too complicated to solve analytically. It is not until the recent
developments with the bootstrap method, that a single tree-level exchange
diagram is fully evaluated in a closed form. Yet the bootstrap method highly
depends on the dS symmetries, which can be spontaneously broken by a rolling
scalar background. For example, in the chemical potential scenario, the
rolling inflaton selects a preferred dS patch, enhancing dS-symmetry-breaking
particle production. So far, CC signals in this scenario have only been
evaluated numerically for particles with mass $m\lesssim 5H$, or approximated
by using the late-time expansion. However, numerics can be extremely
inefficient for large masses, while the late-time expansion does not capture
the full waveform of the CC signal, and can even be problematic in certain
cases. As a result, how to efficiently extract precise CC signals from the
complicated time integrals is a problem faced by all model builders in
cosmological collider physics.

We argue that at tree-level, there may be a practical shortcut to solve this
problem by considering the physical evolution history in the dS bulk. Although
the many nested time integrals obscure the physics, what happens in the bulk
at leading order may be described as a combination of three basic events. (i),
particles are produced from the background evolution. (ii), particles are
produced at interaction vertices. (iii), particles decay into massless
inflatons/gravitons. All three types of events happen locally in time, in the
form of resonances. Thus the proper time of the particle spent between the
resonance events is well defined and corresponds to its dynamical phase. This
is the physical origin of the oscillatory CC signals. Hence, depending on the
world line of the massive particle, $e.g.$, between (i) and (iii) or between
(ii) and (iii), the CC signals can be classified into two categories with
different origins and behaviors. On the other hand, these resonances are the
stationary points dominating the non-analytic part of the time integral. This
motivates us to concentrate on the resonant production/decay picture and
propose a practical cutting rule to analytically extract the leading order CC
signals for any IR convergent tree diagrams. For the convenience of model
builders in cosmological collider physics, we first give a condensed summary
of the procedure for this practical cutting rule below and give justifications
in the main text.

Cutting rule for CC signals (For a more detailed version, see Sect. 3.3)

  1. 1.

Given a specific tree diagram, consider only single-colored diagrams with
vertices being all-black ($+$) or all-white ($-$).

  2. 2.

Focus on one massive propagator at a time and integrate out all other massive
propagators as local EFT operators.

  3. 3.

Explicitly compute the left/right blobs. The results should be reducible to a
sum of polynomials multiplying an exponential function of conformal time.

  4. 4.

Flip the time-ordering Heaviside step function, and obtain two factorized
integrals in the form of Laplace transformations. Discard the leftover
commutator integral, which contains no CC signals.

  5. 5.

Symmetrize and traverse all the left/right injection frequency.

  6. 6.

Repeat Step 2-5 for each massive propagator, then sum them up to obtain the
total signal.

  7. 7.

At last, dress the total signal by the EFT background where all massive fields
are integrated out.

This paper is organized as follows. We first classify the oscillatory CC
signals and discuss their physical interpretations from a bulk perspective in
Sect. 2. Then, in Sect. 3, we introduce the practical cutting rule for the CC
signal and justify it using the resonant production/decay picture. This is
supplemented with several examples in Sect. 4, including the application to
the chemical potential scenario with significant dS symmetry breaking. At
last, we conclude and give future prospects in Sect. 5.

##  2 Classification and Interpretation of Cosmological Collider Signals

###  2.1 Cosmological collider physics in a nutshell

In this section, we begin with a lightning review of cosmological collider
physics, and then introduce the resonant production/decay picture for the bulk
evolution of massive particles. During inflation, the inflaton background
$\phi_{0}$ usually evolves into a slow-roll attractor phase with
$\dot{\phi}_{0}\sim$ const. The scale factor in the FRW ansatz grows
exponentially with cosmic time, $i.e.$, $a(t)\propto e^{Ht}=-/H\tau$, with
$H\sim$ const being the Hubble parameter and $-\infty<\tau<0$ being the
conformal time. Thus the background spacetime resembles that of dS space in
the Poincaré patch. Upon this quasi-dS background, perturbations in the matter
sector can be quantized and calculated. In particular, quantum fluctuations of
the inflaton field $\varphi\equiv\phi-\phi_{0}$ generate the curvature
fluctuation $\zeta$ we observed today, via the gauge transformation
$\zeta=-(H/\dot{\phi_{0}})\varphi$. These inflaton perturbations may enjoy
non-trivial couplings to various massive particles in the dS bulk, and due to
their vanishing mass, they survive the dS expansion until the dS boundary at
$\tau=0$. Thus their $n$-point correlation functions
$\langle\zeta(\tau_{f},\mathbf{x}_{1}),\dots\zeta(\tau_{f},\mathbf{x}_{n})\rangle|_{\tau_{f}\to
0}$ encode interesting information about the particle spectrum of the
primordial universe.

To calculate the correlation functions in a given particle physics model with
weak couplings, it is most straightforward to use a diagrammatic
representation to organize the perturbative expansion. This formalism is known
as the Schwinger-Keldysh path integral. The quantum fields start out in the
Bunch-Davis (BD) vacuum at $\tau_{i}\to-\infty$, then evolve according to a
local Hamiltonian density, and are finally collected by the $n$ probes placed
at $\tau_{f}\to 0$. We read out the average value of the $n$ localized probes
instead of the probability amplitudes of detecting certain on-shell states
which are non-local in spacetime. Due to this special setup, only the “in”
state is well-defined but not the “out” state. We also need two copies of
evolution histories to accomplish the averaging process, one with time
ordering ($+$) and the other with anti-time ordering ($-$). More specifically,
take the inflaton as the example, the generating functional of non-interacting
curvature fluctuations can be written as

$Z_{0}[J_{+},J_{-}]\equiv\int\mathcal{D}\zeta_{+}\mathcal{D}\zeta_{-}\text{exp}\left[i\int_{\tau_{i}}^{\tau_{f}}d\tau
d^{3}x\left(\mathcal{L}_{0}[\zeta_{+}]-\mathcal{L}_{0}[\zeta_{-}]+J_{+}\zeta_{+}-J_{-}\zeta_{-}\right)\right].$
(1)

There are totally four types of propagators which can be generated using

$\displaystyle-
iG_{ab}(\tau,\mathbf{x_{1}};\tau^{\prime},\mathbf{x_{2}})=\frac{\delta}{ia\delta
J_{a}(\tau,\mathbf{x_{1}})}\frac{\delta}{ib\delta
J_{b}(\tau^{\prime},\mathbf{x_{2}})}Z_{0}[J_{+},J_{-}]\bigg{|}_{J_{\pm}=0}~{},~{}a,b=\pm~{}.$
(2)

In momentum space, they are

$\displaystyle G_{++}(\mathbf{k},\tau,\tau^{\prime})$ $\displaystyle=$
$\displaystyle\theta(\tau-\tau^{\prime})u_{k}(\tau)u^{*}_{k}(\tau^{\prime})+\theta(\tau^{\prime}-\tau)u^{*}_{k}(\tau)u_{k}(\tau^{\prime})~{},$
(3) $\displaystyle G_{+-}(\mathbf{k},\tau,\tau^{\prime})$ $\displaystyle=$
$\displaystyle u^{*}_{k}(\tau)u_{k}(\tau^{\prime})~{},$ (4) $\displaystyle
G_{-+}(\mathbf{k},\tau,\tau^{\prime})$ $\displaystyle=$ $\displaystyle
u_{k}(\tau)u^{*}_{k}(\tau^{\prime})~{},$ (5) $\displaystyle
G_{--}(\mathbf{k},\tau,\tau^{\prime})$ $\displaystyle=$
$\displaystyle\theta(\tau-\tau^{\prime})u^{*}_{k}(\tau)u_{k}(\tau^{\prime})+\theta(\tau^{\prime}-\tau)u_{k}(\tau)u^{*}_{k}(\tau^{\prime})~{},$
(6)

where $u_{k}(\tau)$ is the inflaton mode function. The propagator $D_{ab}$ for
massive fields can be derived in a similar way, with $u_{k}(\tau)$ replaced by
their own mode functions $v_{k}(\tau)$. As a reference, under the BD initial
condition, the scalar mode function reads

$\displaystyle
u_{k}=\frac{H}{2\sqrt{\epsilon}{M_{\rm{P}}}}\frac{1}{k^{3/2}}(1+ik\tau)e^{-ik\tau}~{},$
(7) $\displaystyle
v_{k}=-ie^{-\frac{\pi}{2}\mu+i\frac{\pi}{4}}\frac{\sqrt{\pi}}{2}H(-\tau)^{3/2}H^{(1)}_{i\mu}(-k\tau)~{},~{}~{}\mu\equiv\sqrt{\frac{m^{2}}{H^{2}}-\frac{9}{4}}~{}.$
(8)

Figure 1:  Some typical diagrams one encounters in cosmological collider
physics. They are 4-point diagrams with (a) an internal massive scalar
propagator (red line), (b) an internal massive propagator (red line) and four
massive propagators of another mass (blue line) on the external legs, (c) a
massive vector propagator (red wavy line).

Interaction terms can be perturbatively expanded as usual. After separating
the Lagrangian into two parts,

$\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{\rm{int}}~{},$ (9)

the generating functional can be written as

$\displaystyle Z[J_{+},J_{-}]=\exp\left[i\int_{\tau_{i}}^{\tau_{f}}d\tau
d^{3}x\left(\mathcal{L}_{\rm{int}}\left[\frac{\delta}{i\delta
J_{+}}\right]-\mathcal{L}_{\rm{int}}\left[-\frac{\delta}{i\delta
J_{-}}\right]\right)\right]Z_{0}\left[J_{+},J_{-}\right].$ (10)

For weak coupling sizes, we are able to use time-dependent perturbation theory
and expand the above functional to any desired order. More specifically, for a
given diagram, each internal vertex may be colored to indicate their time
ordering. Black vertices ($+$) represent time ordering while white vertices
($-$) represent anti-time ordering. For a diagram with $V$ vertices, we must
sum the $2^{V}$ colored diagrams to obtain the final result. For instance,
Fig. 1 shows three typical diagrams one may encounter in cosmological collider
physics.

As stated in Sect. 1, the reason why correlation functions in dS are difficult
to compute is that the time-domain calculation requires doing nested integrals
of special functions. Take diagram (b) in Fig. 1, for example, there are
effectively $2^{6}/2=32$ colored diagrams for us to calculate111Complex
conjugation reduces the number of colored diagram by a factor of two.. Of
these 32 diagrams, one contains a $6$-layer time integral, four contain a
$5$-layer integral, and six contain a $4$-layer integral, etc. Considering
also that the integrand consists of products of ten Hankel/Whittaker
functions, it seems rather hard to extract the desired analytical form of CC
signals. With the help of effective propagators introduced in Chen:2017ryl ,
one can numerically evaluate this diagram for small masses. However, in the
presence of chemical potential, if the mass of the internal particles is large
($e.g.$, $m\gtrsim 5H$), memory consumption quickly becomes unmanageable due
to slow integral convergence. Moreover, the effective propagator is limited to
handling two-point mixing between massive field and the external inflaton
only, and is not applicable to more general couplings. Therefore, we see that
at tree level already, we are in need of a more efficient way to extract CC
signals and physics.

###  2.2 Bulk evolution as resonance events

To find an efficient way to extract the CC signals, we should understand the
evolution history of massive fields in the dS bulk. First, let us consider a
free massive field. Due to the curved background geometry, the mode function
of massive field usually takes the form of special functions ($e.g.$ Hankel or
Whitaker functions). Yet the mathematical complexity of special functions may
conceal the physics inside. Alternatively, a more physical approach is to use
the super-adiabatic approximation based on the Stokes-line method Sou:2021juh
. Consider the equation of motion of a canonically normalized massive field
with mode function $v_{k_{I}}(\tau)$,

$\displaystyle\frac{d^{2}}{d\tau^{2}}v_{k_{I}}(\tau)+w^{2}(\tau)v_{k_{I}}(\tau)=0~{},\qquad
w^{2}=k_{I}^{2}+m^{2}a^{2}-\frac{a^{\prime\prime}}{a}~{}.$ (11)

The solution in terms of the super-adiabatic approximation reads

$v_{k_{I}}(\tau)=\alpha(\tau)f(\tau)+\beta(\tau)f^{*}(\tau),$ (12)
$\text{with}~{}~{}f(\tau)\equiv\frac{1}{\sqrt{2w(\tau)}}e^{-i\int_{\tau_{i}}^{\tau}w(\tau^{\prime})d\tau^{\prime}},~{}\alpha(\tau)\approx
1~{}~{}\text{and}~{}~{}\beta(\tau)\approx-iS(\tau)e^{-F(\tau_{i})}~{}.$ (13)

Here $S(\tau)$ is the Stokes multiplier and $F(z)$ is Dingle’s singulant
variable,

$\displaystyle S(\tau)=\frac{1}{2}\left[1+\text{Erf}\left(\frac{-\imaginary
F(\tau)}{\sqrt{2|\real F(\tau)|}}\right)\right],\qquad
F(\tau)=2i\int_{\tau_{c}}^{\tau}w(\tau_{1})d\tau_{1}~{},$ (14)

and $\tau_{c}$ is the complex turning point in the lower-half complex plane,
$i.e.$, $w^{2}(\tau_{c})=0,\imaginary\tau_{c}<0$. The above division between
the positive frequency part $f(\tau)$ and negative frequency part
$f^{*}(\tau)$ minimizes the oscillations in $\alpha$ and $\beta$ by truncating
the super-adiabatic series at an optimal order Dabrowski:2016tsx . This
results in the most natural choice of a time-dependent particle number,

$\langle n_{k_{I}}(\tau)\rangle^{\prime}=|\beta(\tau)|^{2}\approx e^{-2\real
F(\tau_{i})}S^{2}(\tau)~{}.$ (15)

Therefore, it is clear that gravitational particle production happens when
$\imaginary F(\tau_{*})=0$, at

$\textbf{Production time:}\qquad\qquad|k_{I}\tau_{*}|\approx
0.66\mu~{},\qquad\qquad\qquad\qquad\quad$ (16)

with a duration

$k_{I}\Delta\tau_{*}=\frac{2\sqrt{2|\real F(\tau_{*})|}}{|\imaginary
F^{\prime}(\tau_{*})|}\approx 1.4\sqrt{\mu}~{}.$ (17)

After production, the initial BD vacuum evolves into a two-particle squeezed
state via the Bogoliubov transformation defined by (12),

$|\text{BD}\rangle=|0\rangle+\frac{\beta}{2}\int_{\mathbf{k}}|1_{\mathbf{k}}\rangle\otimes|1_{-\mathbf{k}}\rangle+\mathcal{O}(\beta^{2})~{}.$
(18)

For $|\Delta\tau_{*}/\tau_{*}|\lesssim 1$, which is typical for massive
fields, the production of the particle pair can be viewed as an event
localized in time. $\tau_{*}$ thus marks the starting tick of the clock
measured by the dynamical phase of the massive particle. Upon production, the
particle pair will propagate in opposite directions and accumulate their
dynamical phases in the process. This can be seen more clearly in the Wightman
function,

$\langle\sigma_{-}(\tau,\mathbf{k}_{I})\sigma_{+}(\tau^{\prime},\mathbf{-k}_{I})\rangle=D_{-+}(\mathbf{k}_{I},\tau,\tau^{\prime})=D_{-+}^{\text{non-
local}}(\mathbf{k}_{I},\tau,\tau^{\prime})+D_{-+}^{\text{local}}(\mathbf{k}_{I},\tau,\tau^{\prime})~{},$
(19)

with

$\displaystyle D_{-+}^{\text{non-local}}(\mathbf{k}_{I},\tau,\tau^{\prime})$
$\displaystyle=$
$\displaystyle\alpha(\tau)\beta^{*}(\tau^{\prime})f_{k_{I}}(\tau)f_{k_{I}}(\tau^{\prime})+\alpha^{*}(\tau^{\prime})\beta(\tau)f^{*}_{k_{I}}(\tau)f^{*}_{k_{I}}(\tau^{\prime})~{},$
(20) $\displaystyle D_{-+}^{\text{local}}(\mathbf{k}_{I},\tau,\tau^{\prime})$
$\displaystyle=$
$\displaystyle\alpha(\tau)\alpha^{*}(\tau^{\prime})f_{k_{I}}(\tau)f^{*}_{k_{I}}(\tau^{\prime})+\beta(\tau)\beta^{*}(\tau^{\prime})f^{*}_{k_{I}}(\tau)f_{k_{I}}(\tau^{\prime})~{}.$
(21)

The late-time limit ($\tau,\tau^{\prime}\to 0$) momentum dependence defines
the local vs non-local separation222Another way of this division is in terms
of Bessel functions Lee:2016vti , $i.e.$, $J_{i\mu}(-k\tau)$ and
$J_{-i\mu}^{*}(-k\tau)$. Nevertheless, this division between the local and
non-local part is not as sharp. Because before the production time, the
negative frequency part has not appeared yet and non-local part propagator
should still stay at zero value.,

$\displaystyle\underset{\tau,\tau^{\prime}\rightarrow
0}{\rm{lim}}D_{-+}^{\text{non-local}}(\mathbf{k}_{I},\tau,\tau^{\prime})$
$\displaystyle~{}=~{}\frac{H^{2}(\tau\tau^{\prime})^{3/2}}{4\pi}\left[\Gamma(-i\mu)^{2}\left(\frac{{k_{I}}^{2}\tau\tau^{\prime}}{4}\right)^{i\mu}+\Gamma(i\mu)^{2}\left(\frac{{k_{I}}^{2}\tau\tau^{\prime}}{4}\right)^{-i\mu}\right]~{},$
$\displaystyle\underset{\tau,\tau^{\prime}\rightarrow
0}{\rm{lim}}D_{-+}^{\text{local}}(\mathbf{k}_{I},\tau,\tau^{\prime})$
$\displaystyle~{}=~{}\frac{H^{2}(\tau\tau^{\prime})^{3/2}}{4\pi}\Gamma(-i\mu)\Gamma(i\mu)\left[e^{\pi\mu}\left(\frac{\tau}{\tau^{\prime}}\right)^{i\mu}+e^{-\pi\mu}\left(\frac{\tau}{\tau^{\prime}}\right)^{-i\mu}\right].$
(22)

The local part is a constant independent of $k_{I}$ and upon Fourier
transformation back to space coordinate, gives a contact term proportional to
$\delta^{3}(\mathbf{x}_{1}-\mathbf{x}_{2})$, hence the locality. The non-local
part is non-analytic in $k_{I}$, with the $k_{I}^{2i\mu}$ term giving rise to
long-range correlations of the form $|\mathbf{x}_{1}-\mathbf{x}_{2}|^{2i\mu}$.
The non-local part’s time dependence manifests the accumulated dynamical phase
of two propagating particles,

$(k_{I}^{2}\tau\tau^{\prime})^{\pm
i\mu}\propto\left(\frac{\tau}{\tau_{*}}\right)^{\pm
i\mu}\left(\frac{\tau^{\prime}}{\tau_{*}}\right)^{\pm i\mu}\sim e^{\mp
im(t-t_{*})}e^{\mp im(t^{\prime}-t_{*})}~{},$ (23)

where $t$ stands for the corresponding cosmic time. In contrast, the local
part’s time dependence is of the form $\left(\tau/\tau^{\prime}\right)^{\pm
i\mu}$. This can be viewed as the dynamical phase of one single particle
propagating from $\tau^{\prime}$ to $\tau$ (or $\tau$ to $\tau^{\prime}$, but
with an extra $e^{-2\pi\mu}$ suppression):

$\left(\frac{\tau}{\tau^{\prime}}\right)^{\pm i\mu}\sim e^{\mp
im(t-t^{\prime})}~{}.$ (24)

Up till now, $\tau,\tau^{\prime}$ can be any time later than the starting tick
at $\tau_{*}$. We will see later that the introduction of interactions
automatically leads to events localized in time. These interactions set the
interaction time $\tau,\tau^{\prime}$ as starting/ending ticks. Before that,
we notice a nice property about the non-local part of massive propagator, that
it is real after pair production time.

$\displaystyle D_{-+}^{\text{non-
local}}(\mathbf{k}_{I},\tau,\tau^{\prime})=D_{-+}^{\text{non-
local}}(\mathbf{k}_{I},\tau,\tau^{\prime})^{*}~{},~{}~{}\text{with}~{}~{}\tau,\tau^{\prime}>\tau_{*}~{}.$
(25)

Consequently, the commutator of field operators does not contain non-local
parts:

$\displaystyle\langle\left[\sigma_{k_{I}}(\tau),\sigma_{k_{I}}(\tau^{\prime})\right]\rangle\supset\left[v^{*}_{k_{I}}(\tau)v_{k_{I}}(\tau^{\prime})-v_{k_{I}}(\tau)v^{*}_{k_{I}}(\tau^{\prime})\right]_{\text{from
non-local }}=0~{}.$ (26)

This fact is essentially due to the requirement of microcausality. If the
commutator has non-zero non-local parts with $k_{I}^{\pm i\mu}$ dependence,
after Fourier transforming back to position space, the commutator will not
vanish out of the light-cone, in direct contradiction with microcausality. The
fact above has a direct consequence in the CC signals as we shall see later.

Now let us turn on the interactions of the massive field and the massless
inflaton $\varphi$. Any tree diagram containing a massive field propagator can
then be summarized into the form shown in Fig. 2. The left and right blob
represent the fully evaluated subdiagram on the left and right, and at time
$\tau,\tau^{\prime}$, respectively. At early times,
$\tau,\tau^{\prime}\to-\infty$, the left (right) blob has a leading time
dependence $e^{ik_{L}\tau}$ ($e^{ik_{R}\tau}$), with $k_{L}$ ($k_{R}$)
containing various momentum combinations from the left (right) subdiagram. For
a soft internal momentum $k_{I}\ll k_{L},k_{R}$, the massive propagator takes
the form (22), and the integrals over $\tau,\tau^{\prime}$ pick up the
stationary points333A more accurate calculation of the stationary points can
be obtained via the super-adiabatic mode function, which gives
$|\sqrt{k_{L}^{2}-k^{2}_{I}}\tau_{\bullet}|,|\sqrt{k_{R}^{2}-k^{2}_{I}}\tau^{\prime}_{\bullet}|\approx\mu$.
Usually, in the soft limit where $k_{I}\ll k_{L},k_{R}$, the resonance time
reduces to (28).

$\int d\tau\tau^{\alpha_{L}}e^{ik_{L}\tau}\tau^{\pm
i\mu}\sim\frac{\sqrt{2\pi\mu}}{k_{L}}e^{\pm
i\frac{\pi}{4}}\tau_{\bullet}^{\alpha_{L}}e^{ik_{L}\tau_{\bullet}}\tau_{\bullet}^{\pm
i\mu}$ (27)

at

Resonance time:
$\displaystyle|k_{L}\tau_{\bullet}|\approx\mu~{}~{}~{},~{}~{}~{}|k_{R}\tau^{\prime}_{\bullet}|\approx\mu~{}.\qquad\qquad\qquad$
(28)

These resonances also happen as events localized in time. Physically they can
be understood as the decay of the massive particle into massless modes, or its
resonant production at the vertex from massless modes. These events
effectively mark the ending ticks at $\tau_{\bullet},\tau^{\prime}_{\bullet}$.

Figure 2: Illustrations of the physical interpretation of two different types
of CC signals, (I) shows the non-local type and (II) is the local type. The
planes indicate events happening at different locations in spacetime. In both
cases, the dynamical phases accumulated between the events are encoded as
oscillations in momentum ratios.

As a result, we have three types of events happening in the dS bulk. Namely,
non-local gravitational production, local vertex resonant production, and
local vertex resonant decay. Therefore the massive particle’s dynamical phases
are accumulated either (I) from the gravitational production event at
$\tau_{*}$ to the resonant decay event at
$\tau_{\bullet},\tau^{\prime}_{\bullet}$, or (II) from the resonance
production event at $\tau^{\prime}_{\bullet}$ to the resonant decay event at
$\tau_{\bullet}$ (or vice versa). From this classification criterion, we can
deduce two possible types of CC signals,

I. Non-local Type:
$\displaystyle\quad\left(\frac{k_{L}k_{R}}{k_{I}^{2}}\right)^{\mp
i\mu}\sim\left(\frac{\tau_{\bullet}}{\tau_{*}}\right)^{\pm
i\mu}\left(\frac{\tau_{\bullet}^{\prime}}{\tau_{*}}\right)^{\pm i\mu}$
$\displaystyle\subset$ $\displaystyle~{}D_{-+}^{\text{non-
local}}(\mathbf{k}_{I},\tau_{\bullet},\tau^{\prime}_{\bullet})~{},$ II. Local
Type: $\displaystyle\quad\left(\frac{k_{L}}{k_{R}}\right)^{\mp
i\mu}\sim\left(\frac{\tau_{\bullet}}{\tau_{\bullet}^{\prime}}\right)^{\pm
i\mu}$ $\displaystyle\subset$
$\displaystyle~{}D_{-+}^{\text{local}}(\mathbf{k}_{I},\tau_{\bullet},\tau^{\prime}_{\bullet})~{}.$

This physical picture of bulk evolution is illustrated in Fig. 2. According to
their different origins (dependence about soft momentum), we will name the
signals as the non-local type or the local type.

Before concluding this section, we make a few remarks on our classification of
CC signals.

  1. 1.

Mathematically speaking, these two categories of CC signals are indeed
independent features. Because scale invariant quantities only depend on a
subspace in the 3-dimensional parameter space $\mathbb{R}^{3}$ spanned by
$k_{L},k_{R},k_{I}$, namely the projective plane $\mathbb{R}P^{2}$. One can
freely choose $k_{L}k_{R}/k_{I}^{2}$ and $k_{L}/k_{R}$ to parametrize two
orthogonal directions in this scale-invariant subspace $\mathbb{R}P^{2}$.

  2. 2.

Physically speaking, these two categories of CC signals have drastically
different physical origins. As mentioned above, the non-local signal comes
from the gravitational production and vertex decay of two massive particles,
whereas the local signal comes from the vertex production and vertex decay of
one massive particle.

  3. 3.

Following the above argument, the two CC signals can have different strengths,
because of their intrinsically distinct production mechanisms ($i.e.$, one
from background at a linear level, one from perturbation at a non-linear
level). The non-local part $D_{-+}^{\text{non-local}}$ is suppressed by
$|\alpha\beta^{*}|\sim\sqrt{\langle n_{\mathbf{k}_{I}}\rangle^{\prime}}$,
while the local part $D_{-+}^{\text{local}}$ is, to leading order, independent
of the particle number $\langle n_{\mathbf{k}_{I}}\rangle^{\prime}$. This is
most clearly demonstrated in Sect. 4.3 in the presence of a chemical
potential, which can alter the particle number. We will see that the non-local
CC signal strength can be greatly influenced by the chemical potential,
whereas the local CC signal strength is, to leading order, fixed by dS
geometry alone.

  4. 4.

The resonance picture also has the advantage that it always picks up the
leading order contribution, since they represent the stationary points in the
time integral. If the stationary point lies in the unphysical region
($\tau_{\bullet}>0$), there is an overall suppression
$(-1)^{i\mu}=e^{-\pi\mu}$, which is the penalty for violating energy
conservation in dS.

  5. 5.

Notice that in the left and right blobs, the particles need not to be
massless. At early times (or equivalently, with hard momenta), all mode
functions approach to that of a massless particle with a dynamical phase
$e^{-ik\tau}$. Therefore, the intermediate massive particle can resonate with
all possible frequencies in the left/right blob, giving rise to various
patterns of CC signals444In cases with two-point mixing, the early-time limit
of the left/right blobs may be in conflict with the late-time limit of the
massive propagator. However, one can still obtain the $e^{ik\tau}$ dynamical
phase by integrating out the massive fields in the left/right blobs as EFT
operators of massless fields. In fact, this will be our general strategy to be
mentioned in Sect. 3.2.. Then the physical interpretation is that the massive
particle in consideration can be produced by or decay into all possible
particle combinations in the left/right blobs, which are still in the UV
“massless” phase.

  6. 6.

Most of the past studies on CC signals focus on the non-local type, with a few
exceptions that explicitly mentioned the local type signal Wang:2020ioa ;
McAneny:2019epy . We point out that local signals themselves are not newly
revealed result, yet their distinction and interpretation is something we wish
to emphasize and exploit in this work.

##  3 A Cutting Rule for CC Signals

Based on the above observation on bulk evolution, in this section, we propose
a practical cutting rule for the efficient extraction of both types of CC
signals.

###  3.1 The general method

Figure 3:  A general tree level diagram. The red line is the massive
propagator and other black lines are massless. The left and right blobs may
have their own substructure and we collect it into $\mathcal{F}_{L}$ and
$\mathcal{F}_{R}$ .

Consider a general $n$-point tree-level diagram (see Fig. 3) with one massive
propagator in the Schwinger-Keldysh formalism. Focusing on the internal
massive propagator, the diagram can be expressed as the following form,

$\displaystyle\langle\zeta^{n}\rangle\supset\sum_{a,b=\pm}\int_{-\infty}^{0}d\tau
d\tau^{\prime}\mathcal{F}^{a}_{L}(\tau)\mathcal{F}^{b}_{R}(\tau^{\prime})D_{ab}(\mathbf{k}_{I},\tau,\tau^{\prime}),$
(29)

where we have absorbed terms related to the left blob and right blob into
$\mathcal{F}_{L}$ and $\mathcal{F}_{R}$. In general, they consist of products
of the scale factor, coupling constants, as well as the detailed expression of
the substructures. We also assume that the time integrals inside the
substructures have been performed, so that the only time dependences come from
the two endpoints ($\tau,\tau^{\prime}$), to which the massive propagator is
attached. In the early-time limit $\tau\to-\infty$, or in the soft limit
$k_{I}\to 0$ (namely, when the massive mode momentum is much smaller than any
momentum combinations in the left/right blob) the time dependence of the
left/right blob becomes simple:

$\mathcal{F}_{L,R}(\tau)\sim e^{\pm ik_{L,R}\tau}~{},$ (30)

where $k_{L,R}$ represents various combinations of momentum magnitude sums
flowing into the internal massive propagator.

Depending on the vertex coloring, the integrand can be separated into the
time-ordered part (TO, $ab=++$), the anti-time-ordered part (ATO, $ab=--$) and
the non-time-ordered part (NTO, $ab=+-$ or $+-$). The NTO part is easier to
deal with, because these two time integrals are factorized and can be
performed independently. In other words, diagrams with opposite coloring on
any adjacent pair of vertices are already cut and become factorized. What
actually complicates the calculation is the nested time integral from the TO
part555The ATO part is obtained from the TO part via complex conjugation and
momentum reversal., which has bounded integrals such as $\int_{\tau}^{0}$ or
$\int_{-\infty}^{\tau}$. However, we find that if one focus on the oscillatory
CC signals, which is the most salient feature of cosmological collider
physics, then after re-organizing the integrand, one can cut through the time-
ordering, and factorize the whole diagram. Consequently, the TO part becomes
as simple as the NTO part.

To demonstrate the idea, we focus on the TO part,

$\displaystyle\langle\zeta^{n}\rangle_{\rm{TO}}\sim$
$\displaystyle\int_{-\infty}^{0}d\tau
d\tau^{\prime}\mathcal{F}_{L}(\tau)\mathcal{F}_{R}(\tau^{\prime})$
$\displaystyle\times\left[\theta(\tau-\tau^{\prime})v_{k_{I}}(\tau)v^{*}_{k_{I}}(\tau^{\prime})+\theta(\tau^{\prime}-\tau)v^{*}_{k_{I}}(\tau)v_{k_{I}}(\tau^{\prime})\right]~{}.$
(31)

Without loss of generality, let us assume $k_{L}>k_{R}$. We can re-organize
the integral by flipping one of the Heaviside step function,

$\theta(\tau-\tau^{\prime})=1-\theta(\tau^{\prime}-\tau)~{},$ (32)

after which the integration is split into two terms,

$\displaystyle\langle\zeta^{n}\rangle_{\rm{TO}}\sim$
$\displaystyle\int_{-\infty}^{0}d\tau\mathcal{F}_{L}(\tau)v_{k_{I}}(\tau)\int_{-\infty}^{0}d\tau^{\prime}\mathcal{F}_{R}(\tau^{\prime})v_{k_{I}}^{*}(\tau^{\prime})$
$\displaystyle+\int_{-\infty}^{0}d\tau\int_{\tau}^{0}d\tau^{\prime}\mathcal{F}_{L}(\tau)\mathcal{F}_{R}(\tau^{\prime})\left[v^{*}_{k_{I}}(\tau)v_{k_{I}}(\tau^{\prime})-v_{k_{I}}(\tau)v^{*}_{k_{I}}(\tau^{\prime})\right]~{}.$
(33)

The first term factorizes into a product of two integrals over $(-\infty,0)$,
resembling the easier NTO part. Then we only need to examine the second term
with bounded integral, which is essentially the field commutator mentioned in
Sect. 2. We will argue that this second term is irrelevant for both types of
CC signal if $k_{L}>k_{R}$.

First of all, it is easy to check that this commutator nevertheless cannot
contribute to non-local type CC signals. By using the identity (26), the
commutator from non-local part is exactly canceled for
$\tau^{\prime}>\tau>\tau_{*}$ and for $\tau_{*}>\tau^{\prime}>\tau$. The only
non-zero contribution is with the hierarchy

$\tau^{\prime}>\tau_{*}>\tau~{}.$ (34)

Recall that the non-local signal requires a pair production event at
$\tau_{*}$ and two resonant decay events at
$\tau_{\bullet},\tau^{\prime}_{\bullet}$. Thus we must require

$\tau=\tau_{\bullet}>\tau_{*}~{}.$ (35)

However, (34) and (35) cannot be satisfied simultaneously, suggesting the
absence of the non-local type CC signal.

Then the remaining question is whether this commutator contributes to the
local type signal. According to our previous physical picture of bulk
evolution, the local CC signal can be thought of as the dynamical phase
between two resonance events happening at different vertices. The resonance
condition at the vertices gives

$|k_{L}\tau_{\bullet}|=\mu\qquad\text{and}\qquad|k_{R}\tau^{\prime}_{\bullet}|=\mu~{}.$
(36)

On the other hand, the integration region constrains

$\displaystyle
k_{L}>k_{R}~{}\qquad\text{and}\qquad|\tau_{\bullet}|>|\tau^{\prime}_{\bullet}|~{},$
(37)

However, (36) and (37) cannot be satisfied simultaneously. Therefore, the
local type CC signals are also absent in the commutator integral.

Figure 4: A schematic diagram showing how to cut the nested integral and
extract the CC signals. The first row is the TO diagram before the cut. The
second row shows a factorized diagram (the cut result) and a commutator
integral, which does not contain the CC signals and can be discarded.

As summarized in Fig. 4, we have obtained a method to cut the TO integral into
factorized integrals, with a small leftover piece free of any desired signals.
Note that our initial assumption of $k_{L}>k_{R}$ is crucial for us to argue
against the presence of local signals in the leftover commutator integral.
However, this is not a limitation to our method. The case with $k_{L}<k_{R}$
is completely analogous. Namely, we only need to flip the other Heaviside step
function666Alternatively, one can obtain the cut result in the parameter
region $k_{L}>k_{R}$ and then perform a symmetrization $k_{L}\leftrightarrow
k_{R}$. in (32), and the rest of the argument carries over. The complete cut
result is then necessarily a piecewise function of $k_{L}$ and $k_{R}$.

Before going forward, we can already have a rough estimation of the signal
strength using the cut result. After the cut, the signal $S^{>}_{I}$ in the TO
part can be estimated as

$\displaystyle S^{>}_{I}$ $\displaystyle\sim$
$\displaystyle\int_{-\infty}^{0}d\tau\mathcal{F}_{L}(\tau)\left[\alpha
f_{k_{I}}(\tau)+\beta
f^{*}_{k_{I}}(\tau)\right]\int_{-\infty}^{0}d\tau^{\prime}\mathcal{F}_{R}(\tau^{\prime})\left[\alpha^{*}f^{*}_{k_{I}}(\tau^{\prime})+\beta^{*}f_{k_{I}}(\tau^{\prime})\right]$
(38) $\displaystyle\sim$
$\displaystyle\mathcal{O}(e^{-\pi\mu})|\alpha|^{2}\left(\frac{k_{R}}{k_{L}}\right)^{i\mu}+\mathcal{O}(e^{-\pi\mu})|\beta|^{2}\left(\frac{k_{R}}{k_{L}}\right)^{-i\mu}$
$\displaystyle+$
$\displaystyle\mathcal{O}(1)\alpha\beta^{*}\left(\frac{k_{L}k_{R}}{k_{I}^{2}}\right)^{-i\mu}+\mathcal{O}(e^{-2\pi\mu})\alpha^{*}\beta\left(\frac{k_{L}k_{R}}{k_{I}^{2}}\right)^{i\mu}~{}.$

Clearly, the two types of signals are suppressed by different factors. The
non-local signal is due to gravitational particle production at a linear
level, hence it is suppressed by $\beta\sim\sqrt{\langle n\rangle^{\prime}}$.
In contrast, the local signal comes from vertex particle production at a non-
linear level, thus it is suppressed by the dS intrinsic Boltzmann factor
$e^{-\pi\mu}\sim\sqrt{e^{-m/T_{dS}}}$. This is again the penalty of violating
energy conservation in dS as mentioned in Point 4 of the summary above. If the
gravitational particle production respects dS symmetries, $\beta=e^{-\pi\mu}$
and the two signal strengths are degenerate. However, with dS-symmetry
breaking chemical potential, the particle number can deviate from that of a
simple Boltzmann form, leading to the lifting of the signal strength
degeneracy. This is explicitly demonstrated with a vector field later in Sect.
4.3.

As a side remark, it is easy to check the signals in the NTO part,

$\displaystyle\mathcal{\langle}\zeta^{4}\rangle^{\prime}_{NTO}$
$\displaystyle\sim$
$\displaystyle\int_{-\infty}^{0}d\tau\mathcal{F}_{L}(\tau)\left[\alpha^{*}f_{k_{I}}^{*}(\tau)+\beta^{*}f_{k_{I}}(\tau)\right]\int_{-\infty}^{0}d\tau^{\prime}\mathcal{F}^{*}_{R}(\tau^{\prime})\left[\alpha
f_{k_{I}}(\tau^{\prime})+\beta f^{*}_{k_{I}}(\tau^{\prime})\right]$ (39)
$\displaystyle\sim$
$\displaystyle\mathcal{O}(e^{-2\pi\mu})|\alpha|^{2}\left(\frac{k_{R}}{k_{L}}\right)^{-i\mu}+|\beta|^{2}\left(\frac{k_{R}}{k_{L}}\right)^{i\mu}$
$\displaystyle+$
$\displaystyle\mathcal{O}(e^{-\pi\mu})\alpha\beta^{*}\left(\frac{k_{L}k_{R}}{k_{I}^{2}}\right)^{-i\mu}+\mathcal{O}(e^{-\pi\mu})\alpha^{*}\beta\left(\frac{k_{L}k_{R}}{k_{I}^{2}}\right)^{i\mu}~{}.$

Obviously, the non-time-order part is heavily suppressed by
$|\beta|^{2},e^{-\pi\mu}|\beta|$ or $e^{-2\pi\mu}$, thus we will henceforth
neglect all contributions from NTO parts.

###  3.2 The left/right blob substructures

The above discussion is based on the early-time oscillation behavior of
left/right blobs. Namely, the dynamical phase is linearly growing with the
conformal time $\tau$. This will indeed be the case if the substructure of
left/right blobs contains solely massless fields. Then the massive field will
resonate with all possible frequencies contained in $F_{L,R}$. However, one
can imagine that the substructure may also involve massive fields, which allow
for dynamical phases growing logarithmically with conformal time. The
resonance condition then becomes much more complicated than (36), and new
types of CC signals would appear. Physically, this corresponds to bulk
processes of higher complexity, such as the cascade decay of massive
particles. Nevertheless, these processes are heavily suppressed, and there are
two ways to treat these extra massive propagators if we focus on the leading
CC signal.

First, if the massive propagator has nothing to do with the CC signal under
consideration, there will be no resonance at the corresponding vertices. Thus
it can be integrated out using a partial EFT. In other words, we mimic its
effect by some local operators, whose contribution is perturbative in
$\mu^{-1}$ and analytic in the momenta.

Second, if the massive propagator does resonate at the corresponding vertices,
the whole diagram will gain more suppression factors (each extra resonant
massive propagator contributes an $e^{-\pi\mu}$ suppression). Thus, we can
ignore this possibility at leading order safely.

To summarize, our strategy is to work with one resonant massive propagator at
a time, while contracting other massive propagators to local EFT operators and
absorbing them into the left/right blobs. We then traverse every massive
propagator in the tree diagram and sum their CC signals together. This should
take care of all leading-order signals non-perturbative in $\mu^{-1}$. In the
end, we can integrate out all massive fields and put back the total EFT tower
as a background.

Now that we have reduced all massive modes in the left/right blobs, we can
work out the possible forms of $k_{L}$ and $k_{R}$, and hence that of the CC
signals. Although their detailed expressions will depend on the substructure
of the left/right blobs shown in the Fig.(3), their general forms can be found
via mathematical induction. We can prove that in the absence of IR
divergences, the time dependence of the left/right blobs is always in the
following form:

$\mathcal{F}_{L}(\tau)=\sum_{k_{L}}\mathcal{P}_{L}(\tau)e^{ik_{L}\tau}\quad,\quad\mathcal{F}_{R}(\tau^{\prime})=\sum_{k_{R}}\mathcal{P}_{R}(\tau^{\prime})e^{ik_{R}\tau^{\prime}}~{},$
(40)

where $\mathcal{P}_{L,R}$ are polynomials. The summation on $k_{L,R}$ ranges
over all possible frequencies contained in the left/right blob. The proof is
quite straightforward. First, notice the integral

$\int x^{n}e^{ix}dx=(-i)^{3n+1}\Gamma(n+1,-ix)~{}.$ (41)

For a positive integer $n\in\mathbb{N}$ (which is always the case for IR
convergent massless fields), the incomplete gamma function can be simplified
recursively to a polynomial multiplied by an exponential function,

$\displaystyle\Gamma(n+1,-ix)$ $\displaystyle=n\Gamma(n,-ix)+(-ix)^{n}e^{ix}$
$\displaystyle=\cdots$ $\displaystyle=n!\Gamma(1,-ix)+\cdots+(-ix)^{n}e^{ix}$
$\displaystyle=\left[n!+\cdots+(-ix)^{n}\right]e^{ix}~{}.$ (42)

Now, we focus on the substructure of the left blob.

Figure 5: A larger left blob $L_{1}\cup L_{2}$ can be constructed by gluing
the roots of two smaller blob trees with a massless propagator of momentum
$p$.

We start with the assumption that, a left blob with $n_{L_{1}}$ external lines
takes the form

$\mathcal{F}_{L_{1}}(\tau_{1})=\sum_{k_{L_{1}}}\mathcal{P}_{L_{1}}(\tau_{1})e^{ik_{L_{1}}\tau_{1}}~{},$
(43)

with $\mathcal{P}_{L_{1}}$ being a polynomial and $k_{L_{1}}$ traversing all
frequencies in $L_{1}$. This is obviously true for the simplest blob
substructure, namely an $n_{L_{1}}$-point contact vertex. More complex left
blobs can be constructed by gluing two smaller left blobs. For instance,
consider another left blob $L_{2}$, with

$\mathcal{F}_{L_{2}}(\tau_{2})=\sum_{k_{L_{2}}}\mathcal{P}_{L_{2}}(\tau_{2})e^{ik_{L_{2}}\tau_{2}}~{}.$
(44)

Gluing the root of the two trees together by a massless propagator of internal
momentum $\mathbf{p}=\sum_{i=1}^{n_{L_{1}}}\mathbf{k}_{i}$, we have

$\displaystyle\mathcal{F}_{L_{1}\cup L_{2}}(\tau_{2})$
$\displaystyle=\int_{-\infty}^{0}d\tau_{1}\mathcal{F}_{L_{1}}(\tau_{1})\hat{V}_{1}(\tau_{1})G_{++}(\mathbf{p},\tau_{1},\tau_{2})\hat{V}_{2}(\tau_{2})\mathcal{F}_{L_{2}}(\tau_{2})~{},$
(45)

where $\hat{V}_{1,2}$ are interaction vertices that may contain integer
multiple of derivative operators and scale factors. The spatial derivatives
factor out from the integral, while time derivatives only change the detailed
form of the polynomial, which is unimportant here. Hence, we can reduce the
blob $L_{1}\cup L_{2}$ as follows,

$\displaystyle\mathcal{F}_{L_{1}\cup L_{2}}(\tau_{2})$
$\displaystyle=\sum_{k_{L_{1}},k_{L_{2}}}\Bigg{[}\int_{-\infty}^{\tau_{2}}d\tau_{1}\hat{\mathcal{P}}_{L_{1}}(\tau_{1})e^{i(k_{L_{1}}+p)\tau_{1}}\hat{\mathcal{P}}_{L_{2}}(\tau_{2})e^{i(k_{L_{2}}-p)\tau_{2}}$
$\displaystyle\quad\quad\quad\quad\quad+\int_{\tau_{2}}^{0}d\tau_{1}\hat{\mathcal{P}}_{L_{1}}(\tau_{1})e^{i(k_{L_{1}}-p)\tau_{1}}\hat{\mathcal{P}}_{L_{2}}(\tau_{2})e^{i(k_{L_{2}}+p)\tau_{2}}\Bigg{]}$
$\displaystyle=\sum_{k_{L_{1}},k_{L_{2}}}\Bigg{[}\mathcal{Q}(\tau_{2})e^{i(k_{L_{1}}+p)\tau_{2}}\hat{\mathcal{P}}_{L_{2}}(\tau_{2})e^{i(k_{L_{2}}-p)\tau_{2}}$
$\displaystyle\quad\quad\quad\quad\quad+\mathcal{R}(\tau_{2})e^{i(k_{L_{1}}-p)\tau_{2}}\hat{\mathcal{P}}_{L_{2}}(\tau_{2})e^{i(k_{L_{2}}+p)\tau_{2}}+\mathcal{S}(\tau_{2})\hat{\mathcal{P}}_{L_{2}}(\tau_{2})e^{i(k_{L_{2}}+p)\tau_{2}}\Bigg{]}$
$\displaystyle=\sum_{k_{L_{1}},k_{L_{2}}}\Bigg{[}\left(\mathcal{Q}(\tau_{2})+\mathcal{R}(\tau_{2})\right)\hat{\mathcal{P}}_{L_{2}}(\tau_{2})e^{i(k_{L_{1}}+k_{L_{2}})\tau_{2}}+\mathcal{S}(\tau_{2})\hat{\mathcal{P}}_{L_{2}}(\tau_{2})e^{i(p+k_{L_{2}})\tau_{2}}\Bigg{]}~{}.$
(46)

Here
$\hat{\mathcal{P}}_{L_{1}},\hat{\mathcal{P}}_{L_{2}},\mathcal{Q},\mathcal{R},\mathcal{S}$
are polynomials whose detailed forms are of no importance. Notice that in the
second step of (46), we have formally performed the integral over $\tau_{1}$,
using the fact that such integrals can be reduced as in (42), to polynomials
multiplied by an exponential function. Interestingly, after integrating out
$\tau_{1}$, two different exponential factors appear. The first one depends on
the injecting frequency ($k_{L_{1}}+k_{L_{2}}$), the other one depends on the
internal momentum ($p+k_{L_{2}}$). In this way, we have shown that gluing two
smaller left blobs gives rise to a larger left blob that shares the same form
of time dependence,

$\mathcal{F}_{L_{1}\cup L_{2}}(\tau_{2})=\sum_{k_{L_{1}\cup
L_{2}}}\mathcal{P}_{L_{1}\cup L_{2}}(\tau_{2})e^{ik_{L_{1}\cup
L_{2}}\tau_{2}}~{},$ (47)

where the summation ranges over $k_{L_{1}\cup
L_{2}}=k_{L_{1}}+k_{L_{2}},p+k_{L_{2}}$ for all possible values of
$k_{L_{1}},k_{L_{2}}$.

By iterating this gluing procedure many times, one can find all possible
oscillation frequencies of the left blob $L=L_{1}\cup L_{2}\cup\cdots$. The
right blob is treated the same way. Hence all CC signal patterns can be found
out as

$\displaystyle\text{Non-Local
Type}\quad\left[\frac{\mathcal{T}(\\{k_{L}\\}\mathcal{T}(\\{k_{R}\\}))}{\mathcal{T}(\\{k_{I}\\})}\right]^{\mp
i\mu}~{},$ (48) $\displaystyle\text{Local
Type}\quad\left[\frac{\mathcal{T}(\\{k_{L}\\})}{\mathcal{T}(\\{k_{R}\\})}\right]^{\mp
i\mu}~{},$ (49)

where $\mathcal{T}$ denotes all possible combinations of frequency sets.

###  3.3 Summary of the cutting algorithm

In this subsection, we summarize the procedures of applying our cutting rule
for CC signals. For a $n$-point correlation function, denote the set of all
massive propagators as $\Sigma$.

Cutting algorithm for CC signals

  1. 1.

Given a specific tree diagram, consider only single-colored diagrams with
vertices being all-black ($+$) or all-white ($-$). Any mixed-colored diagrams
are negligible by at least a factor $\mathcal{O}(e^{-2\pi\mu})$.

  2. 2.

Focus on one massive propagator $I\in\Sigma$ and integrate out all other
massive propagators as local EFT operators. This reduces the tree topology to
that of Fig. 3.

  3. 3.

Explicitly compute the left/right blobs. The results should be reducible to
$\mathcal{F}_{L}(\tau)=\sum_{k_{L}}\mathcal{P}_{L}(\tau)e^{ik_{L}\tau}$ and
$\mathcal{F}_{R}(\tau^{\prime})=\sum_{k_{R}}\mathcal{P}_{R}(\tau^{\prime})e^{ik_{R}\tau^{\prime}}$
in the absence of IR divergences, where $\mathcal{P}_{L,R}$ are polynomials,
and the sums over $k_{L,R}$ take into account of all possible frequencies.

  4. 4.

For $k_{L}>k_{R}$, flip
$\theta(\tau-\tau^{\prime})=1-\theta(\tau^{\prime}-\tau)$ and perform two
factorized integrals,

$S_{I}^{>}(k_{L},k_{R})=\int_{-\infty}^{0}d\tau\mathcal{F}_{L}(\tau)v_{k_{I}}(\tau)\times\int_{-\infty}^{0}d\tau^{\prime}\mathcal{F}_{R}(\tau^{\prime})v_{k_{I}}^{*}(\tau^{\prime})~{}.$
(50)

The integral takes the form of a Laplace transformation of special functions,
which often yields analytical expressions,

$S_{I}^{>}(k_{L},k_{R})=\mathcal{L}\\{\mathcal{P}_{L}v_{k_{I}}\\}(ik_{I})\times\mathcal{L}\\{\mathcal{P}_{R}v_{k_{I}}^{*}\\}(ik_{I})~{}.$
(51)

The leftover commutator integral has no CC signals in it, and can be
discarded.

  5. 5.

Account for the case $k_{L}<k_{R}$ by symmetrization, and traverse all
possible frequencies,

$S_{I}=\sum_{k_{L},k_{R}}\left[\theta(k_{L}-k_{R})S_{I}^{>}(k_{L},k_{R})+\theta(k_{R}-k_{L})S_{I}^{>}(k_{R},k_{L})\right]~{}.$
(52)

  6. 6.

Repeat Step 2-5 for each massive propagator $I\in\Sigma$, then sum them up to
obtain the total signal,

$S=\sum_{I\in\Sigma}S_{I}~{}.$ (53)

  7. 7.

At last, dress the total signal $S$ by the EFT background $B$ where all
massive fields are integrated out.

$\langle\zeta^{n}\rangle\simeq(S+B)\sim\Big{[}\mathcal{O}(|\beta|)\times(\text{non-
local})+\mathcal{O}(e^{-\pi\mu})\times(\text{local})\Big{]}+\mathcal{O}\left(\frac{\text{EFT}}{\mu^{\\#}}\right)~{}.$
(54)

Before going to explicit examples, we make a few comments on this cutting
algorithm.

  1. 1.

The cut result $S+B$ is only an approximation to the full $n$-point function.
The error comes from two sources. First, to obtain closed expressions, one
must truncate the large-mass EFT in both Step 2 and Step 7. This results in
errors suppressed by powers of $\mu^{-2}$, which can be reduced if we include
more EFT operators. Second, we have thrown away all the commutator integrals
since they do not contain CC signals. However, they are important for
restoring differentiability at $k_{L}=k_{R}$ and canceling the folded limit
pole at $k_{L,R}\to k_{I}$. This results in an $\mathcal{O}(e^{-\pi\mu})$
error in these special limits, one that cannot be reduced within our method.
Yet in practice, as long as we stay away from these special limits (which only
take up a small fraction of the total phase space), this source of error is
generically small, as will be demonstrated in Sect. 4.

  2. 2.

The Laplace transformation of familiar mode functions (Hankel, Whittaker) can
be evaluated to hypergeometric functions. This would not have been possible
without cutting the nested time integral. In addition, EFT contributions are
also analytically calculable. Thus our cutting rule typically results in
completely analytical results777Yet, necessarily non-analytic.. Even if the
Laplace transform of unfamiliar mode functions may not be evaluated in a
closed form, the original two-dimensional integral is now factorized into two
one-dimensional integrals. This greatly reduces the computational cost of
numerical integration.

  3. 3.

Each step in the above algorithm is specific but tedious to perform by hand.
Therefore, the best way to systematically carry out the cutting procedure is
to promote this into a computer program, a goal that we hope to accomplish in
the future. This would be beneficial for model builders in cosmological
collider physics.

Finally, a comparison must be made between our method and the recently
proposed cosmological cutting rules Goodhew:2020hob ; Melville:2021lst ;
Goodhew:2021oqg ; Baumann:2021fxj ; Meltzer:2021zin . Based on general
principles such as locality, unitarity and analyticity, the cosmological
cutting rules give precise relations among the discontinuities of wavefunction
exponents $\text{Disc}_{k_{I}}\psi_{n}$. From these relations, one may
recursively construct the whole wavefunction. In contrast, our cutting rule
directly applies to correlation functions $\langle\zeta^{n}\rangle$. It is
based on the physical picture of bulk evolution with the specific aim of
extracting CC signals of massive fields. Our cut result is also approximate,
with reducible EFT truncation errors and irreducible errors from neglecting
commutator integrals. However, as stated above, the algorithm itself is well-
defined and can be readily implemented as computer programs yielding
analytical expressions. We have traded mathematical rigor and formality for
practicality and efficiency.

Of course, there are also similarities. The fact that the non-local CC signal
is associated with a branch cut discontinuity in $k_{I}$ and that it actually
comes from a factorized integral seem to hint a connection to the cosmological
cutting rules. It is certainly interesting to further explore the similarities
and dissimilarities of these two approaches in the future.

##  4 Application to Typical Diagrams

In this section, we will explicitly demonstrate how to apply our cutting rule
to extract CC signals for the three typical diagrams shown in Fig. 1. For a
better illustration, all diagrams calculated in this section correspond to the
curvature trispectrum (4-point function). The oft studied bispectrum (3-point
function) can be easily obtained by taking the soft limit of one external
momentum, and we give more related discussions at the end of Sect. 4.1.

###  4.1 Example 1: Direct scalar exchange

The first example (Fig. 6) is that of a simple exchange diagram with an
intermediate massive scalar $\sigma$ of mass
$\frac{m}{H}=\sqrt{\mu^{2}+\frac{9}{4}}$. The interaction vertex is chosen to
be

$\mathcal{L}_{3}=c_{3}a^{2}\varphi^{\prime 2}\delta\sigma~{},$ (55)

where $\varphi$ is the massless inflaton.

Figure 6:  Example 1: A direct massive scalar exchange.

Setp 1, according to the cutting algorithm, we only focus on the TO diagram
with the vertices being black. The ATO diagram is obtained via complex
conjugation and momenta reversal. More explicitly,

$\displaystyle{\langle\zeta_{\mathbf{k_{1}}}\zeta_{\mathbf{k_{2}}}\zeta_{\mathbf{k_{3}}}\zeta_{\mathbf{k_{4}}}\rangle}^{\prime}_{s,\text{TO}}=$
$\displaystyle-\tilde{c_{3}}^{2}\int_{-\infty}^{0}\int_{-\infty}^{0}d\tau
d\tau^{\prime}a^{2}(\tau)a^{2}(\tau^{\prime})\partial_{\tau}G_{++}(\mathbf{k_{1}},\tau,0)\partial_{\tau}G_{++}(\mathbf{k_{2}},\tau,0)$
$\displaystyle\times
D_{++}(\mathbf{k}_{I},\tau,\tau^{\prime})\partial_{\tau^{\prime}}G_{++}(\mathbf{k_{3}},\tau^{\prime},0)\partial_{\tau^{\prime}}G_{++}(\mathbf{k_{4}},\tau^{\prime},0)~{}.$
(56)

Here $\mathbf{k}_{I}=\mathbf{k}_{1}+\mathbf{k}_{2}$,
$\tilde{c_{3}}\equiv\frac{c_{3}\dot{\phi_{0}}^{2}}{H^{2}}$ and $G,D$ are the
propagators for $\varphi,\sigma$, respectively. Without loss of generality, we
take the $s$-channel as an example. The $t,u$-channels can be included after
permutation.

Step 2 and 3 are trivial since no other massive propagators exist in this
simple diagram. The left/right blobs are

$\displaystyle\mathcal{F}_{L}(\tau)$ $\displaystyle=$ $\displaystyle
2i\tilde{c_{3}}u_{k_{1}}u_{k_{2}}(0)a^{2}u^{\prime*}_{k_{1}}u^{\prime*}_{k_{2}}(\tau)~{},$
(57) $\displaystyle\mathcal{F}_{R}(\tau^{\prime})$ $\displaystyle=$
$\displaystyle
2i\tilde{c_{3}}u_{k_{3}}u_{k_{4}}(0)a^{2}u^{\prime*}_{k_{3}}u^{\prime*}_{k_{4}}(\tau^{\prime})~{}.$
(58)

Then at Step 4, denoting $k_{i_{1}\cdots i_{m}}\equiv
k_{i_{1}}+\cdots+k_{i_{m}}$, we take $k_{12}>k_{34}$ and flip one of the
Heaviside step functions to split the TO diagram into

$\displaystyle{\langle\zeta^{4}\rangle}_{s,TO}$ $\displaystyle=$
$\displaystyle S^{>}_{I}+\mathcal{I}_{I,com}~{},$ (59)

with the cut result

$\displaystyle S^{>}_{I}(k_{12},k_{34})$
$\displaystyle=\int_{-\infty}^{0}d\tau\mathcal{F}_{L}v_{k_{I}}(\tau)\int_{-\infty}^{0}d\tau^{\prime}\mathcal{F}_{R}v^{*}_{k_{I}}(\tau^{\prime})$
$\displaystyle=-4\tilde{c_{3}}^{2}u_{k_{1}}u_{k_{2}}u_{k_{3}}u_{k_{4}}(0)\left[\int_{-\infty}^{0}d\tau
a^{2}u^{\prime*}_{k_{1}}u^{\prime*}_{k_{2}}v_{k_{I}}\int_{-\infty}^{0}d\tau^{\prime}a^{2}u^{\prime*}_{k_{3}}u^{\prime*}_{k_{4}}v^{*}_{k_{I}}\right]~{},$
(60)

and the commutator integral

$\displaystyle\mathcal{I}_{I,com}=$
$\displaystyle\int_{-\infty}^{0}d\tau\int_{\tau}^{0}d\tau^{\prime}\mathcal{F}_{L}(\tau)\mathcal{F}_{R}(\tau^{\prime})\left(v^{*}_{k_{I}}(\tau)v_{k_{I}}(\tau^{\prime})-v_{k_{I}}(\tau)v^{*}_{k_{I}}(\tau^{\prime})\right)$
$\displaystyle=$
$\displaystyle-\tilde{4c_{3}}^{2}u_{k_{1}}u_{k_{2}}u_{k_{3}}u_{k_{4}}(0)$
$\displaystyle\times\int_{-\infty}^{0}d\tau
a^{2}u^{\prime*}_{k_{1}}u^{\prime*}_{k_{2}}(\tau)\int_{\tau}^{0}d\tau^{\prime}a^{2}u^{\prime*}_{k_{3}}u^{\prime*}_{k_{4}}(\tau^{\prime})\left(v^{*}_{k_{I}}(\tau)v_{k_{I}}(\tau^{\prime})-v_{k_{I}}(\tau)v^{*}_{k_{I}}(\tau^{\prime})\right)~{}.$
(61)

The CC signals can be evaluated via Laplace transformation as888The negligible
NTO diagram also have a similar form, more details can be found in Appendix B.

$\displaystyle S^{>}_{I}(k_{12},k_{34})$
$\displaystyle=-\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{\pi^{2}{c_{3}}^{2}H^{6}(16\mu^{4}+40\mu^{2}+9)^{2}\sech^{2}(\pi\mu)}{2^{17}k_{1}k_{2}k_{3}k_{4}k_{I}^{5}}$
$\displaystyle\times{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\end{array}\Bigg{|}\,\frac{k_{I}+k_{12}}{2k_{I}}\Bigg{]}\times{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\end{array}\Bigg{|}\,\frac{k_{I}-k_{34}}{2k_{I}}\Bigg{]}~{}.$ (66)

Here $k_{12}$ is assumed to have a small negative imaginary part in order to
select the correct branch of the hypergeometric function. As argued above, the
commutator integral contains neither local nor non-local signals. We
numerically verify this fact in Appendix A.

Step 5, the total cut CC signals is obtained via symmetrization,

$S_{I}=\theta(k_{12}-k_{34})S_{I}^{>}(k_{12},k_{34})+\theta(k_{34}-k_{12})S_{I}^{>}(k_{34},k_{12})+\text{c.c.}~{}.$
(67)

Step 6 is trivial and Step 7 is to supplement the CC signals with an EFT
background. This background admits a perturbative expansion in powers of
$\mu^{-1}$. These are analytic functions of momenta that can be mimicked by
local operators in the single-field EFT. The EFT background is usually smaller
than CC signals under soft limits. Nevertheless, it plays an important role in
“equilateral” regions where all momenta (both external and internal) are
comparable in magnitude. In this example, the leading order EFT contribution
is generated from the operator

$\displaystyle\Delta\mathcal{L}^{EFT}_{I}=\frac{c_{3}^{2}}{2\mu^{2}H^{2}}\varphi^{\prime
4}~{}.$ (68)

In the “$s$”-channel, its impact on the 4-point function is

$\displaystyle
B_{I}=\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{12H^{6}c_{3}^{2}}{\mu^{2}k_{1}k_{2}k_{3}k_{4}k_{1234}^{5}}~{}.$
(69)

Finally, we get the analytical approximation for the 4-point function in the
$s$-channel:

$\displaystyle\langle\zeta_{\mathbf{k_{1}}}\zeta_{\mathbf{k_{2}}}\zeta_{\mathbf{k_{3}}}\zeta_{\mathbf{k_{4}}}\rangle_{s}^{\prime}\approx
S_{I}+B_{I}~{},$ (70)

with $S_{I}$ and $B_{I}$ given by (67) and (69).

Figure 7: The comparison between the 4-point functions obtained from the
cutting rule (blue lines) and numerical integration (red dashed lines). Left
panel: Non-local type CC signals, where we have fixed $k_{12}/k_{34}=1$, and
multiplied a factor $\left(k_{12}k_{34}/k_{I}^{2}\right)^{9/2}$ for better
visualization. Right panel: Local type CC signals, where we have fixed
$k_{12}k_{34}/k_{I}^{2}=4\times 10^{4}$. All results in the upper panels are
normalized by their maxima. In the lower panels, we compare the CC signals
filtered from the numerical result and those directly obtained via the cut. We
have adopted a high-pass filter with Hamming window.

In Fig. 7, we compare our cut result to that of brute-force numerical
integration. The left column shows the non-local type CC signal where we keep
$k_{12}\approx k_{34}$ and vary $\frac{k_{12}k_{34}}{k_{I}^{2}}$. The right
column shows the local type signal where we keep
$\frac{k_{12}k_{34}}{k_{I}^{2}}=4\times 10^{4}$ and vary
$\frac{k_{12}}{k_{34}}$. The upper panels show the full 4-point function while
the lower panels show the CC signals alone. In order to extract oscillatory
signals from the total numerical result, we have evoked high-pass filters to
eliminate the low-frequency EFT background Wang:2021qez . As we can see from
the plots, the numerical results (red dashed lines) match our analytical cut
results (blue line) very well, especially for the signal part. The slight
mismatch comes around in two places. First, when $k_{12}\approx k_{34}$, the
overall signal size differ relatively by
$\mathcal{O}(\mu^{-2})\sim\frac{1}{9}$, which is the next-order contribution
in the EFT tower. We expect that including more EFT operators induced by the
exchange diagram can systematically cure this mismatch. The second type of
mismatch manifest itself in the folded limit ($k_{12},k_{34}\to k_{I}$) pole
as well as in the derivative discontinuity at $k_{12}=k_{34}$. As mentioned
before, these are caused by the disposal of the commutator integral, which,
despite being small, plays an important role in restoring BD vacuum in the
early-time limit. This intrinsic mismatch shows a defect of our method.
Nevertheless, as long as we stay away from the special configurations, the CC
signals we care about will not be influenced much.

Finally, we can gain some insight by studying the large-mass asymptotic
behavior of $S_{I}$ in the soft limit,

$S_{I}^{>}\xrightarrow[k_{I}\to 0]{\mu\gg
1}~{}\propto\mu^{3}e^{-\pi\mu}\left[\cos\left(\mu\ln\frac{{4k_{12}k_{34}}}{k_{I}^{2}}\right)-\sin\left(\mu\ln\frac{k_{12}}{k_{34}}\right)\right]~{},$
(71)

where we have only kept the leading contribution. Clearly, there are two
independent oscillation patterns with degenerate amplitudes. In the
literature, this large-mass result is often obtained via direct late-time (IR)
expansion (22). Here, we see that the late-time expansion is qualitatively
justified for large masses in this 4-point function case. However, there are
two disadvantages for the traditional late-time expansion. First, while the
late-time expansion gives the correct exponential suppression factor
$e^{-\pi\mu}$, it may fail to capture the correct mass power dependence
($\mu^{n}$), which is also important in the estimation of signal strength.
Such is the case, for example, in the 3-point diagram frequently encountered
in cosmological collider physics. If one performs a late-time expansion before
time integration Chua:2018dqh , the CC signal amplitude is proportional to
$\mu^{2}e^{-\pi\mu}$. However, as we can see from tending $k_{4}\to 0$ in
(71), the correct amplitude should be $\mu^{3}e^{-\pi\mu}$. This can reproduce
the results of Chen:2015lza . The problem arises because performing a late-
time expansion does not commute with completing the time integral. This fact
was previously pointed out in Wang:2020ioa . Second, the signal waveform
obtained from the late-time expansion is only valid in the soft limit
$k_{I}\to 0$. Significant dephasing of the waveform will appear when the
internal momentum is less soft. In other words, its validity is limited to a
small part of the phase space. In contrast to the late-time expansion, our
method can get the correct amplitude as well as the correct waveform of the CC
signals, greatly enlarging the phase space we can take advantage of.

###  4.2 Example 2: Scalar exchange with two-point mixing and different masses

The second diagram (Fig. 8) involving two-point mixing is considerably more
complicated. As described in Sect. 2.1, the TO diagram alone consists of a
6-layer integral. However, with the cutting rule, we can extract the CC
signals without too much effort, as we will demonstrate below.

Figure 8:  Example 2: Massive scalar exchange with two-point mixing.

Let the mass of exchanged massive field $\sigma$ (red line) be $\mu$ and the
mass of the massive field $s$ on the external lines (cyan) be $\nu$. The
interactions are set by the following couplings,

$\displaystyle\Delta\mathcal{L}_{2}=\rho
a^{3}\varphi^{\prime}s~{}~{},~{}~{}\Delta\mathcal{L}_{3}=\lambda
a^{4}s^{2}\sigma~{}.$ (72)

Without loss of generality, we assume that all propagators are made
distinguishable. It is not hard to include permutations according to the Bose
statistics.

First, focusing on the TO diagram with all vertices being black, the
correlation function reads

$\displaystyle{\langle\zeta_{\mathbf{k_{1}}}\zeta_{\mathbf{k_{2}}}\zeta_{\mathbf{k_{3}}}\zeta_{\mathbf{k_{4}}}\rangle}^{\prime}_{s,\text{TO}}$
$\displaystyle=$
$\displaystyle-\lambda^{2}\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\int_{-\infty}^{0}d\tau\tau^{\prime}a^{4}(\tau)a^{4}(\tau^{\prime})\mathcal{G}_{+}(k_{1},\tau)\mathcal{G}_{+}(k_{2},\tau)\mathcal{G}_{+}(k_{3},\tau^{\prime})\mathcal{G}_{+}(k_{4},\tau^{\prime})D^{\sigma}_{++}(k_{I},\tau,\tau^{\prime})~{}.$
(73)

The effective propagator $\mathcal{G}$ is defined as Chen:2018sce

$\displaystyle\mathcal{G}_{\pm}(k,\tau_{1})=i\rho\int_{-\infty}^{0}d\tau_{2}a^{3}(\tau_{2})\left[D^{s}_{++}(k,\tau_{1},\tau_{2})\partial_{\tau_{2}}G_{++}(k,\tau_{2},0)\right]~{},$
(74)

where $D^{s}$ and $D^{\sigma}$ are the massive propagators of the field $s$
and the field $\sigma$.

Figure 9: The strategy of the cutting algorithm for the diagram in Fig. 8. We
traverse all massive propagators and cut one at a time ($(12,34)$ on the first
row and $(1,234)$ on the second row), while integrating out others as EFTs. At
last, we integrate all of them to obtain a total EFT background $B$.

Step 2, there are five massive propagators in this diagram. Naming them
according to the external lines separated by each one of them, we have the set
of all massive propagators,

$\Sigma=\Big{\\{}(12,34),~{}(1,234),~{}(2,134),~{}(3,124),~{}(4,123)\Big{\\}}~{}.$
(75)

We focus on one massive propagator $I\in\Sigma$ at a time, integrating out the
others as EFT operators. This whole procedure is illustrated in Fig. 9. For
example, doing so for $I=(12,34)$ yields a contact EFT vertex at leading
order,

$\displaystyle\Delta\mathcal{L}^{EFT}_{(12,34)}=\frac{\lambda\rho^{2}}{(H\nu)^{4}}a^{2}\varphi^{\prime}\varphi^{\prime}\sigma~{}.$
(76)

Step 3, the left/right blobs are now easily computable:

$\displaystyle\mathcal{F}_{L}(\tau)$ $\displaystyle=$
$\displaystyle\frac{i\lambda\rho^{2}}{(H\nu)^{4}}u_{k_{1}}u_{k_{2}}(0)a^{2}u^{\prime*}_{k_{1}}u^{\prime*}_{k_{2}}(\tau)~{},$
(77) $\displaystyle\mathcal{F}_{R}(\tau^{\prime})$ $\displaystyle=$
$\displaystyle\frac{i\lambda\rho^{2}}{(H\nu)^{4}}u_{k_{3}}u_{k_{4}}(0)a^{2}u^{\prime*}_{k_{3}}u^{\prime*}_{k_{4}}(\tau^{\prime})~{}.$
(78)

Step 4, for $k_{12}>k_{34}$, we cut the massive propagator $I=(12,34)$ by

$\displaystyle{\langle\zeta^{4}\rangle}_{(12,34),\rm{TO}}$ $\displaystyle=$
$\displaystyle S^{>}_{(12,34)}+\mathcal{I}_{(12,34),com}~{},$ (79)

with the cut result

$\displaystyle S^{>}_{(12,34)}(k_{12},k_{34})$
$\displaystyle=\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\int_{-\infty}^{0}d\tau\mathcal{F}_{L}v_{k_{I}}(\tau)\int_{-\infty}^{0}d\tau^{\prime}\mathcal{F}_{R}v^{*}_{k_{I}}(\tau^{\prime})$
$\displaystyle=-\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{\lambda^{2}\rho^{4}}{(H\nu)^{8}}u_{k_{1}}u_{k_{2}}u_{k_{3}}u_{k_{4}}(0)\left[\int_{-\infty}^{0}d\tau
a^{2}u^{\prime*}_{k_{1}}u^{\prime*}_{k_{2}}v_{k_{I}}\int_{-\infty}^{0}d\tau^{\prime}a^{2}u^{\prime*}_{k_{3}}u^{\prime*}_{k_{4}}v^{*}_{k_{I}}\right]~{}.$
(80)

Step 5, symmetrization gives the complete CC signal in the channel
$I=(12,34)$,

$S_{(12,34)}=\theta(k_{12}-k_{34})S_{(12,34)}^{>}(k_{12},k_{34})+\theta(k_{34}-k_{12})S_{(12,34)}^{>}(k_{34},k_{12})+\text{c.c.}~{}.$
(81)

Step 6, we need to loop back to Step 2 and go through the same procedure for
the other massive propagators $I\in\Sigma$. For instance, in the channel
$I=(1,234)$, large-mass EFT expansion of other propagators gives the following
local operator at leading order:

$\displaystyle\Delta\mathcal{L}^{EFT}_{(1,234)}=\frac{\lambda^{2}\rho^{3}}{H^{8}\mu^{2}\nu^{6}}a\varphi^{\prime}\varphi^{\prime}\varphi^{\prime}s~{}.$
(82)

Thus the left/right blobs are

$\displaystyle\mathcal{F}_{L}(\tau)$ $\displaystyle=$ $\displaystyle i\rho
u_{k_{1}}(0)a^{3}u^{\prime*}_{k_{1}}(\tau)~{},$ (83)
$\displaystyle\mathcal{F}_{R}(\tau^{\prime})$ $\displaystyle=$
$\displaystyle\frac{i\lambda^{2}\rho^{3}}{H^{8}\mu^{2}\nu^{6}}u_{k_{2}}u_{k_{3}}u_{k_{4}}(0)au^{\prime*}_{k_{2}}u^{\prime*}_{k_{3}}u^{\prime*}_{k_{4}}(\tau^{\prime})~{}.$
(84)

After cut and symmetrization (which is trivial since $k_{1}<k_{234}$ by the
triangle inequality), the corresponding CC signal in the channel $I=(12,34)$
is then

$S_{(1,234)}=S_{(1,234)}^{>}(k_{234},k_{1})+\text{c.c.}~{},$ (85)

with

$\displaystyle S^{>}_{(1,234)}(k_{234},k_{1})$
$\displaystyle=\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\int_{-\infty}^{0}d\tau\mathcal{F}_{L}v_{k_{I}}^{*}(\tau)\int_{-\infty}^{0}d\tau^{\prime}\mathcal{F}_{R}v_{k_{I}}(\tau^{\prime})$
$\displaystyle=-\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{\lambda^{2}\rho^{4}}{H^{8}\mu^{2}\nu^{6}}u_{k_{1}}u_{k_{2}}u_{k_{3}}u_{k_{4}}(0)\left[\int_{-\infty}^{0}d\tau
a^{3}u^{\prime*}_{k_{1}}v^{*}_{k_{I}}\int_{-\infty}^{0}d\tau^{\prime}au^{\prime*}_{k_{2}}u^{\prime*}_{k_{3}}u^{\prime*}_{k_{4}}v_{k_{I}}\right]~{}.$
(86)

Signals in the other channels differ from $(1,234)$ only by a rename of the
external momenta. We need to sum all their contributions,

$S=S_{(12,34)}+S_{(1,234)}+S_{(2,134)}+S_{(3,124)}+S_{(4,123)}~{}.$ (87)

Finally, in Step 7, we dress the signals with an overall EFT background. At
leading order, it is sourced by the operator

$\Delta\mathcal{L}^{EFT}_{\Sigma}=\frac{\lambda^{2}\rho^{4}}{H^{10}\mu^{2}\nu^{8}}\varphi^{\prime}\varphi^{\prime}\varphi^{\prime}\varphi^{\prime}~{},$
(88)

which gives

$B=\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{3\lambda^{2}\rho^{4}}{\mu^{2}\nu^{8}H^{2}k_{1}k_{2}k_{3}k_{4}k_{1234}^{5}}~{}.$
(89)

As a result, the cutting algorithm gives an analytical approximation to
$\langle\zeta^{4}\rangle^{\prime}$ as $\langle\zeta^{4}\rangle^{\prime}=S+B$,
with the detailed expressions of each $S_{I}$ shown in Appendix C.

Figure 10: The 4-point functions obtained from the cut rule (blue lines) and
numerical integration (red dashed lines). Left column: Non-local type CC
signals in the $(12,34)$-channel, where we have fixed $k_{12}/k_{34}=1$, and
multiplied a factor $\left(k_{12}k_{34}/k_{I}^{2}\right)^{9/2}$ for better
visualization. Middle column: Local type CC signals in the $(12,34)$-channel,
where we have fixed $k_{12}k_{34}/k_{I}^{2}=4\times 10^{4}$. Right column:
Mixed type (non-local+local) CC signals in the $(1,234)$-channel, where we
have fixed $k_{2}=k_{3}=k_{4}$. The first row corresponds to the mass choice
$\mu=3,\nu=10$, while the second row corresponds to the mass choice
$\mu=10,\nu=4$. The 4-point functions are normalized by their maxima. In the
lower panels, we compare the CC signals filtered from the numerical result and
that directly obtained via the cut. In cases where the CC signal strength is
too weak, we have plotted the analytical cut result only, because the
numerical accuracy is inadequate for signal filtering.

We compare our cut result with numerics in Fig. 10. We have plotted three
types of CC signals for two sets of mass parameters. The upper panels
correspond to the full 4-point function, while the lower panels show CC
signals obtained via the cutting algorithm compared to that filtered from the
numeric result. One can see that the oscillatory CC signals match well to that
filtered from numerical integration, except at aforementioned special limits
where our approximation breaks down. Yet the overall background has an
$\mathcal{O}(1)$ mismatch. This is because in Step 2 and 7, we have only
considered lowest order EFT operators induced by the massive fields999In fact,
the EFT error receives amplification from the number of massive propagators.
One can estimate the EFT error by $R\sim
2\times\left(\frac{1}{\mu^{2}}+\frac{4}{\nu^{2}}\right)$, which gives $R\sim
30\%$ and $R\sim 52\%$ for the two parameter choices in Fig. 10. This is why
it seems to be larger than that in Fig. 7.. This EFT error is also responsible
for the small phase mismatch in the $\frac{k_{234}}{3k_{1}}$ signal. However,
we expect such error can be systematically reduced by including higher-order
EFT operators. Despite being tedious, the process of including more EFT
operators is well-understood and can be performed in an analytical and
mechanical fashion. We leave the methodical analysis of the EFT background to
future studies.

###  4.3 Example 3: Vector exchange with chemical potential

The third example (Fig. 11) entails a spin-1 massive vector boson with
chemical potential. The chemical potential of a vector boson $A_{\mu}$ can be
introduced via coupling the gradient of a rolling scalar field $\theta(\tau)$
to the Chern-Simons current,

$\Delta\mathcal{L}_{\text{chem}}=\frac{1}{2}\sqrt{-g}\kappa_{\mu}J_{CS}^{\mu}=-\frac{1}{4}\theta(\tau)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}+\text{total
derivatives}~{},$ (90)

where
$J_{CS}^{\mu}=\frac{1}{\sqrt{-g}}\epsilon^{\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}$
and $\kappa_{\mu}=\partial_{\mu}\theta(\tau)\equiv
a(\tau)\kappa\delta_{\mu}^{0}$. The free-theory dynamics of the vector boson
is then described by the Lagrangian

$\displaystyle\mathcal{L}_{V}=\sqrt{-g}\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}A_{\mu}A^{\mu}\right)+\Delta\mathcal{L}_{\text{chem}}~{}.$
(91)

The mode functions with helicity $\lambda=\pm,0$ are expressed in terms of the
Whittaker function,

$\displaystyle
v_{k_{I}}^{\pm}=\frac{1}{\sqrt{2k_{I}}}e^{\mp\pi\tilde{\kappa}/2}W_{\pm
i\tilde{\kappa},i\mu}(2ik_{I}\tau)~{},$ $\displaystyle
v_{k_{I}}^{0}=\frac{1}{\sqrt{2k_{I}}}\frac{1}{\sqrt{1+4\mu^{2}}}\left[2k\tau
W_{0,1+i\mu}(2ik\tau)-(2\mu+i)W_{0,i\mu}(2ik\tau)\right]~{},$ (92)

where
$\mu=\sqrt{\frac{m^{2}}{H^{2}}-\frac{1}{4}},\tilde{\kappa}\equiv\frac{\kappa}{H}$.
The particle production amount can be computed using the Stokes-line method,

$\langle n^{\lambda}_{\mathbf{k}_{I}}\rangle^{\prime}|_{\tau\to
0}=|\beta^{\lambda}_{\mathbf{k}_{I}}(0)|^{2}\simeq
e^{-\pi(\mu+\lambda\tilde{\kappa})}~{}.$ (93)

with the production time and duration given by Sou:2021juh

$\qquad\qquad|k_{I}\tau_{*}|\approx
0.66\mu-0.34\lambda\tilde{\kappa}~{}~{},~{}~{}k_{I}\Delta\tau_{*}\approx
1.4\sqrt{\mu}\left(1-\lambda\frac{0.32\tilde{\kappa}}{\mu}\right)~{}.$ (94)

We see that the particle production of transverse modes is suppressed
(enhanced) for positive (negative) helicity, whereas the longitudinal mode is
not affected by the chemical potential. The vector boson propagators are

$\displaystyle
D^{+-}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})=\sum_{\lambda}\left[\epsilon^{\lambda}_{i_{1}}(-\hat{\mathbf{k}}_{I})v^{\lambda}_{k_{I}}(\tau)\right]^{*}\epsilon^{\lambda}_{i_{2}}(-\hat{\mathbf{k}}_{I})v^{\lambda}_{k_{I}}(\tau^{\prime})~{},$
(95) $\displaystyle
D^{-+}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})=\sum_{\lambda}\epsilon^{\lambda}_{i_{1}}(\hat{\mathbf{k}}_{I})v^{\lambda}_{k_{I}}(\tau)\left[\epsilon^{\lambda}_{i_{2}}(\hat{\mathbf{k}}_{I})v^{\lambda}_{k_{I}}(\tau^{\prime})\right]^{*}~{},$
(96) $\displaystyle
D^{++}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})={\theta}(\tau-\tau^{\prime})D^{-+}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})+\theta(\tau^{\prime}-\tau)D^{+-}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})~{},$
(97) $\displaystyle
D^{--}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})={\theta}(\tau-\tau^{\prime})D^{+-}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})+\theta(\tau^{\prime}-\tau)D^{-+}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})~{}.$
(98)

The polarization vector reads

$\displaystyle\mathbf{\epsilon}^{\pm}(\hat{\mathbf{k}}_{I})=\frac{1}{\sqrt{2(1-(\hat{\mathbf{n}}\cdot{\hat{\mathbf{k}}_{I}})^{2})}}\left[(\hat{\mathbf{n}}-(\hat{\mathbf{n}}\cdot\hat{\mathbf{k}}_{I})\hat{\mathbf{k}}_{I})\pm
i(\hat{\mathbf{k}}_{I}\times\hat{\mathbf{n}})\right]~{},$
$\displaystyle\mathbb{\epsilon}^{0}(\hat{\mathbf{k}}_{I})=\hat{\mathbf{k}}_{I}~{}.$
(99)

with $\hat{\mathbf{n}}$ being an arbitrary unit vector with
$\hat{\mathbf{k}}_{I}\times{\hat{\mathbf{n}}}\neq 0$.

Figure 11:  Example 3: Massive vector exchange with chemical potential.

After introducing the basic ingredients above, we now compute the 4-point
function mediated by the vector boson. We assume the coupling

$\displaystyle\Delta\mathcal{L}_{int}=\lambda_{3}a\varphi^{\prime}\partial_{i}\varphi
A_{i}~{}.$ (100)

For simplicity, we imagine that the external lines are distinguishable. One
can always permute the external momentum after obtaining the result in which
they are distinguishable. Following the cutting rule, as a first step, we
focus on the TO part of the 4-point function,

$\displaystyle{\langle\zeta_{\mathbf{k_{1}}}\zeta_{\mathbf{k_{2}}}\zeta_{\mathbf{k_{3}}}\zeta_{\mathbf{k_{4}}}\rangle}^{\prime}_{s,\rm{TO}}=$
$\displaystyle\lambda_{3}^{2}\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\int_{-\infty}^{0}\int_{-\infty}^{0}d\tau
d\tau^{\prime}a(\tau)a(\tau^{\prime})G_{++}(\mathbf{k_{1}},\tau,0)\partial_{\tau}G_{++}(\mathbf{k_{2}},\tau,0)$
$\displaystyle\times
k_{1}^{i_{1}}k_{3}^{i_{2}}D^{++}_{i_{1}i_{2}}(\mathbf{k}_{I},\tau,\tau^{\prime})G_{++}(\mathbf{k_{3}},\tau^{\prime},0)\partial_{\tau^{\prime}}G_{++}(\mathbf{k_{4}},\tau^{\prime},0)~{}.$
(101)

The kinematic factor reads

$\displaystyle\Pi^{\lambda}(\mathbf{k}_{1},\mathbf{k}_{3},\mathbf{k}_{I})$
$\displaystyle\equiv\left(\mathbf{k}_{1}\cdot\mathbf{\epsilon}^{\lambda}(\hat{\mathbf{k}}_{I})\right)\left(\mathbf{k}_{3}\cdot\mathbf{\epsilon}^{\lambda}(\hat{\mathbf{k}}_{I})\right)^{*}$
$\displaystyle=\left\\{\begin{aligned}
&\frac{1}{2}\left[\mathbf{k}_{1}\cdot\mathbf{k}_{3}-(\mathbf{k}_{1}\cdot\hat{\mathbf{k}}_{I})(\mathbf{k}_{3}\cdot\hat{\mathbf{k}}_{I})-i\lambda\hat{\mathbf{k}}_{I}\cdot(\mathbf{k}_{1}\crossproduct\mathbf{k}_{3})\right]&\qquad\lambda=\pm~{},\\\
&(\mathbf{k}_{1}\cdot\hat{\mathbf{k}}_{I})(\mathbf{k}_{3}\cdot\hat{\mathbf{k}}_{I})&\qquad\lambda=0~{}.\end{aligned}\right.$
(102)

Step 2 being trivial, Step 3 gives the left/right blobs as

$\displaystyle\mathcal{F}_{L}(\tau)$ $\displaystyle=$
$\displaystyle\lambda_{3}u_{k_{1}}u_{k_{2}}(0)au^{*}_{k_{1}}u^{\prime*}_{k_{2}}(\tau)~{},$
(103) $\displaystyle\mathcal{F}_{R}(\tau^{\prime})$ $\displaystyle=$
$\displaystyle\lambda_{3}u_{k_{3}}u_{k_{4}}(0)au^{*}_{k_{3}}u^{\prime*}_{k_{4}}(\tau)~{}.$
(104)

Step 4, cut the nested integral and discard the leftover commutator integral,

$\displaystyle{\langle\zeta^{4}\rangle}^{\prime}_{s,TO}$ $\displaystyle=$
$\displaystyle S^{>}_{I}+\mathcal{I}_{I,com},$ (105)

where

$\displaystyle S^{>}_{I}(k_{12},k_{34})=$
$\displaystyle\sum_{\lambda}\Pi^{\lambda}(\mathbf{k}_{1},\mathbf{k}_{3},\mathbf{k}_{I})\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\int_{-\infty}^{0}d\tau\mathcal{F}_{L}v_{k_{I}}^{\lambda}(\tau)\int_{-\infty}^{0}d\tau^{\prime}\mathcal{F}_{R}v^{\lambda*}_{k_{I}}(\tau^{\prime})~{},$
(106)

and

$\mathcal{I}_{I,com}=\sum_{\lambda}\Pi^{\lambda}(\mathbf{k}_{1},\mathbf{k}_{3},\mathbf{k}_{I})\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\int_{-\infty}^{0}d\tau\int_{\tau}^{0}d\tau^{\prime}\mathcal{F}_{L}(\tau)\mathcal{F}_{R}(\tau^{\prime})\left(v^{\lambda*}_{k_{I}}(\tau)v^{\lambda}_{k_{I}}(\tau^{\prime})-v^{\lambda}_{k_{I}}(\tau)v^{\lambda*}_{k_{I}}(\tau^{\prime})\right)~{}.$
(107)

Again, $S^{>}_{I}$ can be analytically computed, we provide its expression in
Appendix D . Then in Step 5, we symmetrize the signal and obtain

$S_{I}=\theta(k_{12}-k_{34})S_{I}^{>}(k_{12},k_{34})+\theta(k_{34}-k_{12})S_{I}^{>}(k_{34},k_{12})+(\text{c.c.}|_{\mathbf{k}_{i}\to-\mathbf{k}_{i}})~{}.$
(108)

Step 6 and 7, the EFT background is sourced at leading order by the local
operator

$\displaystyle\Delta\mathcal{L}^{EFT}_{I}=\frac{\lambda_{3}^{2}}{\mu^{2}H^{2}}\varphi^{\prime
2}(\partial_{i}\varphi)^{2}~{}.$ (109)

In the “$s$”-channel, its contribution is

$B_{I}=\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{\lambda_{3}^{2}H^{6}\mathbf{k_{1}}\cdot\mathbf{k_{3}}}{\mu^{2}}\left[\frac{4k_{1}^{2}+(k_{1234}+4k_{1})\left(k_{234}+3k_{3}\right)}{4k_{1}^{3}k_{2}k_{3}^{3}k_{4}k_{1234}^{5}}\right]~{}.$
(110)

Finally, we arrive at the result

$\displaystyle\langle\zeta_{\mathbf{k_{1}}}\zeta_{\mathbf{k_{2}}}\zeta_{\mathbf{k_{3}}}\zeta_{\mathbf{k_{4}}}\rangle_{s}^{\prime}\approx
S_{I}+B_{I}~{},$ (111)

with $S_{I}$ and $B_{I}$ given by (108) and (110). We compare our cut result
to numerics in Fig. 12. Clearly, both the CC signals and the EFT background
match the numerical result very well.

Figure 12: The 4-point functions obtained from the cut rule (blue lines) and
numerical integration (red dashed lines). Left panel: Non-local type CC
signals, where we have fixed $k_{12}/k_{34}=1$, and multiplied a factor
$\left(k_{12}k_{34}/k_{I}^{2}\right)^{9/2}$ for better visualization. Right
panel: Local type CC signals, where we have fixed
$k_{12}k_{34}/k_{I}^{2}=4\times 10^{4}$. All results in the upper panels are
normalized by their maxima. In the lower panels, we compare the CC signals
filtered from the numerical result and that directly obtained via the cut.

To have a closer look at the signal strengths, we take the large-mass limit
and the soft limit, then $S_{I}$ becomes

$\displaystyle S_{I}\xrightarrow[k_{I}\to 0]{\mu\to\infty}~{}\propto\mu^{3}$
$\displaystyle\sum_{\lambda=-1}^{1}\Bigg{\\{}e^{-\pi(\mu+\lambda\tilde{\kappa})}\cos\left[\mu\ln\frac{4k_{12}k_{34}}{k_{I}^{2}}-\frac{\lambda^{2}\tilde{\kappa}^{2}}{\mu}\right]\times
e^{i\lambda\phi_{(12,34)}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-e^{-\pi\mu}\sin\left[\mu\ln\frac{k_{12}}{k_{34}}\right]\times\cos[\lambda\phi_{(12,34)}]\Bigg{\\}}~{}.$
(112)

We observe that both the amplitude and the phase of the non-local type CC
signal is strongly dependent on the chemical potential $\tilde{\kappa}$. For
instance, the non-local signal of the negative helicity mode $\lambda=-1$ is
enhanced for $\tilde{\kappa}>0$, and suppressed for $\tilde{\kappa}<0$. In
contrast, its local signal is chemical-potential independent, since it
reflects the vertex production/decay process, whose amplitude only depends on
the spacetime geometry. This is demonstrated in Fig. 13 by plotting the
negative-helicity signals on the projective plane spanned by
$\frac{k_{12}k_{34}}{k_{I}^{2}}$ and $\frac{k_{12}}{k_{34}}$. To better
visualize the oscillations, we have added a factor
$(k_{12}k_{34}/k_{I}^{2})^{9/2}$ and normalized each plots by their maximum.
As we can see from the figure, there are two different oscillations along the
horizontal and vertical directions, respectively. In the absence of chemical
potential, two CC signals share the same magnitude. However, as soon as we
turn on a non-zero chemical potential ($|\tilde{\kappa}|=0.5$), the non-local
type becomes dominant (subdominant) if the chemical potential is positive
(negative). The positive helicity mode behaves exactly in the opposite
fashion, whereas the longitudinal mode is not affected at all.
Observationally, the impact of chemical potential on CC signals of different
helicities can be distinguished from their dihedral angle dependence. In fact,
the case without chemical potential resembles linearly polarized light while
that with chemical potential resembles elliptically polarized light. In this
way, we can confirm that the two types of signals stem from drastically
different phenomena. Notice also that, from the large-mass approximation
(112), the parity-violating imaginary part of the 4-point function is only
present in the non-local signal, not the local signal nor the EFT background.
This agrees with the no-go theorem on parity-violating trispectra in single-
field EFTs Liu:2019fag .

Figure 13: Signals of the negative helicity component,
$(k_{12}k_{34}/k_{I}^{2})^{9/2}S^{>}_{I}(k_{12},k_{34})|_{\lambda=-1}$, for
different chemical potential choices ($\tilde{\kappa}=0.5,0,-0.5$), with the
mass fixed to be $\mu=3$. Each plot is normalized by their maximum. We see
that oscillations in different directions dominates different parameter
ranges. Without chemical potential, two CC signals share the same magnitude.
As the chemical potential is turned on, the non-local type becomes dominant
(subdominant) if the chemical potential is positive (negative). Figure 14: The
computational cost of a single data point in the configuration space using
conventional numerics. The blue line and the red line show the time and memory
consumption of computing Fig. 11 with $k_{12}=k_{34}=20k_{I}$, where the
relative numerical error is targeted at $\mathcal{O}(10^{-3})$. The chemical
potential $\tilde{\kappa}$ is chosen as $\mu-2$ . As a comparison, the
computational cost using the cut result is much lower, being around
$\mathcal{O}(10^{-3})$s and $\mathcal{O}(10^{-2})$MB.

Before concluding this section, we emphasize that with our cutting algorithm,
the numerical efficiency is greatly enhanced. As mentioned in Sect. 1, the
conventional numerical integration approach to CC signals is extremely
inefficient, and one has to balance between the integration precision and the
computational cost. Take the diagram (Fig. 11) considered in this section as
an example, we wish to monitor the computational cost of one data point in the
configuration space, namely $(k_{12}=k_{34}=20k_{I})$ for the non-local CC
signal of the $\lambda=-1$ mode. In Fig. 14, we plot the time (blue line) and
memory (red line) consumption for different masses $\mu$, with the chemical
potential chosen to be $\tilde{\kappa}=\mu-2$. Here we have targeted the
relative numerical error at $\mathcal{O}(10^{-3})$. We see clearly that as
mass grows larger, both the time and memory cost increase rapidly. For
example, when $\mu=10,\tilde{\kappa}=8$, we need around one hour and
approximately $3$GB of memory to compute even one data point. In contrast,
with the help of the cutting rule, evaluating an analytical cut result only
costs around $\mathcal{O}(1)$ms and $\mathcal{O}(10)$KB. This is an
improvement of 5-6 orders of magnitude, which can be extremely helpful when
sampling lots of data points in the trispectrum configuration space.

##  5 Summary and outlook

Complicated by nested integrals of special functions, computing inflationary
correlators and extracting oscillatory signals are difficult tasks in
cosmological collider physics. In most cases, one can only rely on either
numerics or work with approximate results of considerable systematic error. In
this work, we leverage on the physical picture of bulk evolution, and to make
a first step toward a systematic and efficient extraction method of CC
signals. We started by a brief review of cosmological collider physics and the
Schwinger-Keldysh formalism. After pointing out the technical difficulties for
extracting CC signals, we moved on to a physical understanding of the bulk
evolution of massive particles. We argued that at leading order, the bulk
evolution history can be understood as a collection of resonance events. The
oscillatory CC signals can be interpreted as the dynamical phases accumulated
between these events, and are classified into two categories with distinct
origins. Then, based on this resonance picture, we proposed a cutting rule for
extracting analytical CC signals from a general tree diagram. The cutting
algorithm is supplemented with three application examples, in which we find
good agreement between the cut CC signals and those filtered from numerics.
The computational efficiency is also found to be greatly improved compared to
traditional numerical methods.

As mentioned above, though this work attempts to understand the dynamics of
massive fields during inflation and to develop efficient shortcuts to
observables, there are certainly many improvements to make and future
directions to explore. We conclude by listing some of them below.

  1. $\bullet$

First, our cutting rule is subjected to two sources of errors. The EFT
truncation error can sometimes be significant, especially when the particle
mass is small or when the number of massive propagators is large. Therefore,
we need to take into account the higher order terms in the large-mass EFT in a
systematical way. Despite being tedious, this refinement is doable in
principle. In contrast, the non-perturbative error at special momentum
configurations may require new insights. We hope to reduce both errors in
future works.

  2. $\bullet$

Second, the specificity of the cutting algorithm makes it convenient to be
implemented as a computer program. We will try to achieve this aim in the
future. This may benefit model builders in cosmological collider who wish to
skip the technicalities to direct signals.

  3. $\bullet$

Another limitation of our cutting rule is that, so far, it is only applicable
to tree diagrams. At loop level, in addition to background pair production and
vertex resonant production/decay of massive particles, more complex phenomena
may happen, and the cutting rule may need further modifications.

  4. $\bullet$

It is also interesting to study the relation between our phenomenological
cutting rule to other more formal cutting rules. As mentioned in Sect. 3.3,
they have crucial differences, but their shared features (most importantly,
the factorization of non-local CC signals on the branch cut of the internal
momentum) seem to hint a hidden connection between them. It is also helpful to
understand our physical-picture-based cutting rule from a more fundamental
perspective, which may aid to its generalization to other scenarios.

###### Acknowledgements.

We would like to thank Zhong-Zhi Xianyu and Xingang Chen for helpful
discussions. This work is supported in part by the National Key R&D Program of
China (2021YFC2203100), GRF Grants 16304418 and 16303819 from the Research
Grants Council of Hong Kong, and the NSFC Excellent Young Scientist (EYS)
Scheme (Hong Kong and Macau) Grant No. 12022516.

##  Appendix A The commutator integral

In this appendix, we numerically verify that the commutator integrals in
Example 1 and Example 3 contain neither non-local nor local signals. The
result is shown in Fig. 15. Clearly, the commutator integrals do not oscillate
at all in the parameter region $k_{12}>k_{34}$. Thus it does not contain any
CC signals, and can be safely deserted.

Figure 15: The numerical result of the commutator integral
$\mathcal{I}_{I,com}$. First row: The scalar exchange case in Example 1.
Second row: The vector exchange case in Example 3 with chemical potential. The
left column shows the absence of non-local signals, where we have fixed
$k_{12}/k_{34}=1$. $\mathcal{I}_{I,com}$ is multiplied by a factor
$(k_{12}k_{34}/k_{I}^{2})^{9/2}$ for better visualization, and is normalized
by its value at $k_{12}k_{34}/k_{I}^{2}\gg 1$. The right column shows the
absence of local signals in the region $k_{12}/k_{34}>1$. The region
$k_{12}/k_{34}<1$ with oscillations is not our concern and is painted gray.
Here we have fixed $k_{12}k_{34}/k_{I}^{2}=4\times 10^{4}$ and normalized
$\mathcal{I}_{I,com}$ by its maximum.

##  Appendix B NTO part in Example 1

In this appendix, we examine the NTO part of the diagram Fig. 6. With the
coupling (55), the $s$-channel NTO part is

$\displaystyle{\langle\zeta^{4}\rangle}_{s,\rm{NTO}}$
$\displaystyle=\tilde{c_{3}}^{2}u_{k_{1}}u_{k_{2}}u^{*}_{k_{3}}u^{*}_{k_{4}}(0)\int_{-\infty}^{0}d\tau
a^{2}u^{\prime*}_{k_{1}}u^{\prime*}_{k_{2}}v^{*}_{k_{I}}\int_{-\infty}^{0}d\tau^{\prime}a^{2}u^{\prime}_{k_{3}}u^{\prime}_{k_{4}}v_{k_{I}}+\text{c.c.}$
$\displaystyle=\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{c^{2}_{3}H^{6}e^{-\pi\mu}\pi}{16k_{1}k_{2}k_{3}k_{4}k_{I}^{5}}$
$\displaystyle\quad\times\left[\int_{0}^{\infty}dx_{1}{x_{1}}^{\frac{3}{2}}e^{-i\frac{k_{12}}{k_{I}}x_{1}}{H}^{(2)}_{-i\mu}(x_{1})\int_{0}^{\infty}dx_{2}{x_{2}}^{\frac{3}{2}}e^{i\frac{k_{34}}{k_{I}}x_{2}}{H}^{(1)}_{i\mu}(x_{2})+\text{c.c.}\right].$
(113)

Using the Laplace transformation

$\displaystyle\int_{0}^{\infty}dxx^{n}H^{(1)}_{i\mu}(x)e^{isx}$
$\displaystyle=\mathcal{L}\\{x^{n}H^{(1)}_{i\mu}(x)\\}(-is)$
$\displaystyle=\frac{i^{n}e^{\frac{\pi\mu}{2}}\big{|}\Gamma(1+n-i\mu)\big{|}^{2}}{\sqrt{\pi}2^{n}\Gamma\left(\frac{3}{2}+n\right)}{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}1+n-i\mu,1+n+i\mu\\\\[2.0pt]
\frac{3}{2}+n\end{array}\Bigg{|}\,\frac{1-s}{2}\Bigg{]}~{},$ (116)

we can reduce the NTO CC signal to

$\displaystyle{\langle\zeta^{4}\rangle}_{s,NTO}=$
$\displaystyle\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{c^{2}_{3}H^{6}\pi^{2}(16\mu^{4}+40\mu^{2}+9)^{2}\sech^{2}(\pi\mu)}{2^{17}k_{1}k_{2}k_{3}k_{4}k_{I}^{5}}$
$\displaystyle\times{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\end{array}\Bigg{|}\,\frac{k_{I}-k_{12}}{2k_{I}}\Bigg{]}\times{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\end{array}\Bigg{|}\,\frac{k_{I}-k_{34}}{2k_{I}}\Bigg{]}+\text{c.c.}~{}.$
(121)

To see the Boltzmann suppression factor more explicitly, we take the soft
limit of internal momentum as well as the large-mass expansion,

$\displaystyle{\langle\zeta^{4}\rangle}_{s,NTO}\propto
e^{-2\pi\mu}\mu^{3}\left\\{\sin\left[\mu\log\left(\frac{4k_{12}k_{34}}{k_{I}^{2}}\right)\right]+\cos\left[\mu\log\left(\frac{k_{34}}{k_{12}}\right)\right]\right\\}~{}.$
(122)

As a result of the smallness of the Boltzmann factor $e^{-2\pi\mu}$, the NTO
part is always subdominant compared to the TO part, thus we can safely neglect
it at leading order.

##  Appendix C Expressions of CC signals in Example 2

Using the Laplace transformation for Hankel functions (116), we can obtain the
analytical form of CC signals as

$\displaystyle S^{>}_{(12,34)}(k_{12},k_{34})=$
$\displaystyle-\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{\lambda^{2}\rho^{4}\pi^{2}\left(16\mu^{4}+40\mu^{2}+9\right)^{2}\sech(\pi\mu)^{2}}{2^{19}k_{1}k_{2}k_{3}k_{4}k_{I}^{5}\nu^{8}H^{2}}$
$\displaystyle\times{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\end{array}\Bigg{|}\,\frac{k_{I}+k_{12}}{2k_{I}}\Bigg{]}\times{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\end{array}\Bigg{|}\,\frac{k_{I}-k_{34}}{2k_{I}}\Bigg{]}$ (127)

in the $(12,34)$ channel and

$\displaystyle S^{>}_{(1,234)}(k_{1},k_{234})$
$\displaystyle=-\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{\lambda^{2}\rho^{4}\pi\sech(\pi\nu)\bigg{|}\Gamma\left(\frac{9}{2}-i\nu\right)\bigg{|}^{2}}{12288\mu^{2}\nu^{6}H^{2}k_{1}^{6}k_{2}k_{3}k_{4}}{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{9}{2}-i\nu,\frac{9}{2}+i\nu\\\\[2.0pt]
5\end{array}\Bigg{|}\,\frac{k_{1234}}{2k_{1}}\Bigg{]}$ (130)

in the $(1,234)$ channel. Signals in the other three channels ($(2,134)$,
etc.) are obtained via simple replacements.

##  Appendix D Expressions of CC signals in Example 3

The analytical expression of CC signals in Example 3 can be computed by
noticing the Laplace transformation of the Whittaker function,

$\displaystyle\int_{0}^{\infty}x^{n}W_{i\lambda\tilde{\kappa},i\mu}(cx)e^{-isx}$
$\displaystyle=\mathcal{L}\\{x^{n}W_{i\lambda\tilde{\kappa},i\mu}(cx)\\}(is)$
$\displaystyle=c^{-1-n}\bigg{|}\Gamma\left(\frac{3}{2}+n-i\mu\right)\bigg{|}^{2}{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{3}{2}+n-i\mu,\frac{3}{2}+n+i\mu\\\\[2.0pt]
2+n-i\kappa\lambda\end{array}\Bigg{|}\,\frac{1}{2}-\frac{is}{c}\Bigg{]}~{}.$
(133)

Here ${}_{2}\mathbf{F}_{1}$ is the regularized hypergeometric function

$\displaystyle{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}a,b\\\\[2.0pt]
c\end{array}\Bigg{|}\,z\Bigg{]}\equiv{}_{2}{F}_{1}\Bigg{[}\begin{array}[]{c}a,b\\\\[2.0pt]
c\end{array}\Bigg{|}\,z\Bigg{]}/\Gamma(c)~{}.$ (138)

Denoting the factorized time integrals for transverse modes as

$\displaystyle\mathcal{I}^{\lambda}_{L}$
$\displaystyle\equiv\int_{0}^{\infty}\left(1+i\frac{k_{1}}{k_{I}}x\right)W_{i\lambda\tilde{\kappa},i\mu}e^{-i\frac{k_{12}}{k_{I}}x}(-2ix)dx$
$\displaystyle={i\pi(4\mu^{2}+1)\sech(\pi\mu)}$
$\displaystyle\times\frac{1}{8}\left\\{{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{3}{2}-i\mu,\frac{3}{2}+i\mu\\\\[2.0pt]
2-i\lambda\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{I}+k_{12}}{2k_{I}}\Bigg{]}-\frac{\left(\mu^{2}+\frac{9}{4}\right)k_{1}}{2k_{I}}{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3-i\lambda\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{I}+k_{12}}{2k_{I}}\Bigg{]}\right\\}~{},$
(143) $\displaystyle\mathcal{I}^{\lambda}_{R}$
$\displaystyle\equiv\int_{0}^{\infty}\left(1+i\frac{k_{3}}{k_{I}}x\right)W_{-i\lambda\tilde{\kappa},-i\mu}e^{-i\frac{k_{34}}{k_{I}}x}(2ix)dx$
$\displaystyle=\mathcal{I}^{\lambda}_{L}\bigg{|}_{k_{1,2}\rightarrow-
k_{3,4}}^{*}~{},$ (144)

while for the longitudinal mode

$\displaystyle\mathcal{I}^{0}_{L}$
$\displaystyle\equiv\int_{0}^{\infty}\left(1+i\frac{k_{1}}{k_{I}}x\right)\left(\frac{-2xW_{0,1+i\mu}(-2ix)-(2\mu+i)W_{0,i\mu}(-2ix)}{\sqrt{1+4\mu^{2}}}\right)e^{-i\frac{k_{12}}{k_{I}}x}dx$
$\displaystyle=\frac{\pi\sech(\pi\mu)}{16(k_{12}-k_{I})^{2}(k_{12}+k_{I})(1+4\mu^{2})^{\frac{1}{2}}}$
$\displaystyle\times\left\\{c^{(1)}{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{3}{2}-i\mu,\frac{3}{2}+i\mu\\\\[2.0pt]
2\end{array}\Bigg{|}\,\frac{k_{I}+k_{12}}{2k_{I}}\Bigg{]}+c^{(2)}{}_{2}F_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{3}{2}+i\mu\\\\[2.0pt]
2\end{array}\Bigg{|}\,\frac{k_{I}+k_{12}}{2k_{I}}\Bigg{]}\right\\}~{},$ (149)
$\displaystyle\mathcal{I}^{0}_{R}$
$\displaystyle=\mathcal{I}^{0}_{L}\bigg{|}_{k_{1,2}\rightarrow-
k_{3,4}}^{*}~{},$ (150)

with

$\displaystyle c^{(1)}$
$\displaystyle=(1+\mu^{2})\left\\{2k_{I}^{2}(-4k_{I}+k_{2}(9-6i\mu))+k_{12}k_{I}\left(4k_{I}(2i\mu-5)-k_{2}(4\mu^{2}+9)\right)\right.$
$\displaystyle\left.+k_{12}^{2}\left(-3k_{2}+17k_{I}-4ik_{2}\mu+4(k_{I}-k_{2})\mu^{2}\right)+k_{12}^{3}(4\mu^{2}+8i\mu+5)\right\\}~{},$
(151) $\displaystyle c^{(2)}$
$\displaystyle=2\left(3-2i\mu(1+6i\mu+4\mu^{2})\right)\left(k_{12}^{3}+k_{12}^{2}k_{1}+(2k_{1}-k_{2})k_{I}^{2}\right)~{},$
(152)

we obtain the analytical result of CC signals in the diagram Fig. 11,

$\displaystyle S^{>}_{I}(k_{12},k_{34})=$
$\displaystyle\sum_{\lambda}\Pi^{\lambda}(\mathbf{k}_{1},\mathbf{k}_{3},\mathbf{k}_{I})\left(\frac{H}{\dot{\phi_{0}}}\right)^{4}\frac{\lambda^{2}_{3}H^{6}e^{-\pi\lambda\tilde{\kappa}}}{2^{5}k_{1}^{3}k_{2}k_{3}^{3}k_{4}k_{I}^{3}}\mathcal{I}^{\lambda}_{L}\mathcal{I}^{\lambda}_{R}~{}.$
(153)

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