compat=1.0.0

UMN-TH-4224/23

FTPI-MINN-23-16

Cosmological Collider Signatures
of Higgs-$R^{2}$ Inflation

Yohei Ema,a,b Sarunas Vernerc

a | _William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,_
---|---
| _University of Minnesota, Minneapolis, MN 55455, USA_
b | _School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA_
c | _Institute for Fundamental Theory, Physics Department,_
| _University of Florida, Gainesville, FL 32611, USA_

We study the cosmological collider signatures in the Higgs-$R^{2}$ inflation
model. We consider two distinct types of signals: one originating from the
inflaton coupling to Standard Model fermions and gauge bosons, and another
arising from the isocurvature mode interaction with the inflaton. In the
former case, we determine that the signal magnitude is likely too small for
detection by upcoming probes, primarily due to suppression by both the Planck
scale and slow-roll parameters. However, we provide a detailed computation of
the signal which could be potentially applicable to various Higgs inflation
variants. For the isocurvature mode signals, we observe that the associated
couplings remain unsuppressed when the isocurvature mode is relatively light
or comparable to the inflationary scale. In this case, we study the
Higgs-$R^{2}$ inflation parameter space that corresponds to the quasi-single-
field inflation regime and find that the signal strength could be as large as
$|f_{\rm NL}|>1$, making Higgs-$R^{2}$ inflation a viable candidate for
observation by future 21-cm surveys.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Higgs-$R^{2}$ model and Weyl transformation
    2. 2.2 Inflationary predictions
    3. 2.3 Covariant formalism and isocurvature mode
    4. 2.4 Order of magnitude estimation
  3. 3 Cosmological collider signatures of fermions and gauge bosons
    1. 3.1 Fermion contribution
    2. 3.2 Gauge boson contribution
  4. 4 Cosmological collider signatures of isocurvature mode
    1. 4.1 Mixed propagator
    2. 4.2 Bispectrum
  5. 5 Conclusions and Discussion
  6. A Conventions and notation
    1. A.1 Conventions
    2. A.2 ADM decomposition
  7. B Covariant formalism of multi-field inflation
    1. B.1 Background dynamics
    2. B.2 Perturbation: general discussion
    3. B.3 Quadratic action
    4. B.4 Cubic action
  8. C Schwinger-Keldysh propagators in de Sitter spacetime
    1. C.1 Preliminary
    2. C.2 Scalar
    3. C.3 Fermion
    4. C.4 Massive gauge boson

##  1 Introduction

Cosmic inflation plays a central role in modern cosmology [1, 2, 3,
4].\natural1\natural1$\natural$1For reviews on inflation, see [5, 6, 7, 8]. In
its simplest form, inflation is driven by the potential energy of a slowly
rolling scalar field, called the inflaton. Despite its success in explaining
the cosmic microwave background (CMB) anisotropy and the seeds of the large-
scale structure (LSS), the particle physics origin of the inflaton remains
unknown, and unraveling its nature is one of the most important goals in
modern cosmology.

Since the Higgs boson is the only elementary scalar field within the Standard
Model (SM), it naturally invites speculation about its relationship to
inflaton, another scalar field. The original Higgs model of inflation [9, 10,
11, 12], where the SM Higgs is associated with the inflaton, introduces a
large non-minimal coupling of order $\xi\sim\mathcal{O}(10^{4})$ between the
Higgs boson and the Ricci curvature scalar.\natural2\natural2$\natural$2 In
models of critical Higgs inflation, this requirement can be relaxed if the
Higgs quartic coupling $\lambda$ is small at the inflationary scale [13, 14].
As a result, the cut-off scale is lowered to $M_{P}/\xi$, where $M_{P}$ is the
reduced Planck scale [15, 16, 17, 18]. This scale is lower than the
inflationary energy scale and hence it casts doubts on the validity of the
model. In particular, while unitarity may be preserved during inflation due to
the large Higgs field value [19, 20], it is violated by the production of
longitudinal gauge bosons right after inflation during preheating [21]. This
violation occurs due to the mass term arising from the target space curvature
in the Einstein frame [22, 23].\natural3\natural3$\natural$3 This critically
depends on the fact that the SM Higgs doublet contains four scalar degrees of
freedom. While only the radial direction is important during inflation, the
Goldstone modes, or equivalently, the longitudinal gauge bosons, are
efficiently produced after inflation, leading to a violation of unitarity. For
the case involving only a single scalar degree of freedom, see [24].
Therefore, a UV-completion of Higgs inflation is necessary for understanding
the inflationary dynamics until the end of preheating and reheating.
Inflationary observables, such as the spectral index and the tensor-to-scalar
ratio, depend on the reheating temperature via the number of e-folds of
inflation, and this presents a critical issue in Higgs inflation.

Higgs-$R^{2}$ inflation introduces a squared Ricci scalar term, $R^{2}$, in
the action. With an additional scalar degree of freedom—the scalaron—arising
from the $R^{2}$ contribution [25, 26, 27, 28], the model remains perturbative
up to $M_{P}$, as long as $\xi^{2}/\alpha\lesssim 1$, where $\alpha$ is the
coefficient of $R^{2}$. This ensures that Higgs-$R^{2}$ inflation is a UV
completion of Higgs inflation [29, 30]. A large value of $\alpha$ naturally
arises from the renormalization group (RG) running [31, 32, 33, 34] when a
non-minimal coupling $\xi$ is large. This UV-completion is best understood
through the nonlinear sigma model [35, 36, 37]. In this framework, the target
space encompasses the conformal mode of the metric and remains invariant under
the Weyl transformation, with the scalaron identified with the sigma meson
that flattens the target space of Higgs inflation.

In Higgs-$R^{2}$ inflation, the inflaton is a mixture of the Higgs and the
scalaron. Consequently, the inflaton naturally couples to SM particles through
both its Higgs component and the conformal factor, $\Omega$. A distinguishing
feature of Higgs-$R^{2}$ inflation is that these couplings are all explicitly
given, allowing for unambiguous study of their effects. Specifically, the
preheating and reheating _after_ inflation, associated with these couplings,
have been thoroughly investigated in [38, 39, 40, 41, 42, 43].

In this paper, we study the cosmological signatures arising from the couplings
between the inflationary sector and the SM particles _during_ inflation. It is
well known that a particle with a mass comparable to the Hubble parameter
leaves a unique imprint in the squeezed limit of the non-gaussianity of the
curvature perturbation when it couples to the inflatonary sector. These
signals are referred to as cosmological collider signatures [44, 45, 46, 47,
48, 49, 50, 51].

In this context, SM particles are particularly interesting. In Higgs-$R^{2}$
inflation, even though the Higgs field value is typically much larger than the
Hubble parameter during inflation, there still exist particles with masses as
light as the Hubble parameter due to the large hierarchy of the Yukawa
couplings [52, 53, 54]. Furthermore, the inflaton sector now contains multiple
scalar fields, namely the Higgs and the scalaron, and the isocurvature mode
could potentially give rise to additional cosmological collider signatures.
Therefore, the main aim of this paper is to investigate these cosmological
collider signatures arising from the SM particles. While we show that these
signatures from the SM fermions and gauge bosons are too small to be
observable in the near future, the isocurvature mode may produce a substantial
effect, detectable by future 21 cm observations [55, 56].

The remainder of this paper is organized as follows: In Section 2, we review
Higgs-$R^{2}$ inflation, with a focus on the coupling between the SM particles
and the inflaton sector, as well as the isocurvature mode. We then compute the
cosmological collider signatures of the SM fermions and gauge bosons in
Section 3, and those of the isocurvature mode in Section 4. Finally, we
summarize our findings in Section 5. We aim to keep our discussion free of
technical details, focusing solely on key results, as the computations are
quite involved and may obscure the main findings. Instead, all technical
details are provided in the appendices. In Appendix A, we summarize the
conventions and notation used throughout this paper. Subsequently, we review
the covariant formalism of multi-field inflation in Appendix B. The results of
this appendix are extensively used in the computation of the cosmological
collider signatures of the isocurvature mode. Finally, we derive the
propagators for scalars, fermions, and gauge bosons in de Sitter spacetime in
Appendix C.

##  2 Preliminaries

In this section, we briefly review the inflationary dynamics of the
Higgs-$R^{2}$ model [29, 57, 58, 59, 60]. We pay particular attention to the
couplings between the inflaton field and other SM particles, as these are
essential for understanding cosmological collider signatures.

###  2.1 Higgs-$R^{2}$ model and Weyl transformation

The action of Higgs-$R^{2}$ inflation in the Jordan frame is given by

$\displaystyle S\;=\;\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R\left(1+\frac{2\xi|\Phi|^{2}}{M_{P}^{2}}\right)+\alpha
R^{2}+g^{\mu\nu}(D_{\mu}\Phi)^{\dagger}(D_{\nu}\Phi)-\lambda|\Phi|^{4}\right]+S_{\psi
A}\,,$ (2.1)

where $g_{\mu\nu}$ is the spacetime metric and $g$ is its determinant, $R$ is
the Ricci curvature scalar, $\Phi$ represents the Higgs doublet, and $\lambda$
is the Higgs quartic coupling. The covariant derivative is defined as

$\displaystyle
D_{\mu}\Phi\;=\;\left(\partial_{\mu}-i\frac{g}{2}\tau^{a}W^{a}_{\mu}+i\frac{g^{\prime}}{2}B_{\mu}\right)\Phi\,,$
(2.2)

where $W^{a}_{\mu}$ and $B_{\mu}$ are the SU(2) and U(1) gauge bosons, with
$g$ and $g^{\prime}$ their gauge couplings, respectively, and $\tau^{a}$ is
the Pauli matrix. Although the $R^{2}$ term induces the Higgs mass term and
the cosmological constant due to the RG running, its effect is suppressed
compared to the terms given above during inflation [34]. Therefore, we omit
these terms in the following discussion. The matter action is given by

$\displaystyle S_{\psi A}=\int d^{4}x\sqrt{-g}$
$\displaystyle\left[i\bar{\psi}_{i}\not{\nabla}\psi_{i}-Y_{l}^{ij}\bar{L}_{i}\Phi
l_{Rj}-Y_{d}^{ij}\bar{Q}_{i}\Phi
d_{Rj}-Y_{u}^{ij}\bar{Q}_{i}\tilde{\Phi}u_{Rj}-\frac{1}{4}W_{\mu\nu}^{a}W^{a\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}\right]+\cdots,$
(2.3)

where $\psi_{i}$ includes all the SM fermions, $Q_{i}$ is the left-handed
quark doublet, $L_{i}$ is the left-handed lepton doublet, $u_{Ri}$ and
$d_{Ri}$ are the right-handed up-type and down-type quarks, $l_{Ri}$ are the
right-handed leptons, and $\tilde{\Phi}\equiv i\tau^{2}\Phi^{*}$. The Yukawa
coupling matrices are given by $Y_{l}^{ij}$, $Y_{d}^{ij}$, and $Y_{u}^{ij}$,
with the flavor indices $i$ and $j$. We omit the hermitian conjugate of the
Yukawa interaction, as well as the gluon kinetic term and the interaction
between the fermions and gauge bosons that are irrelevant to our study. The
covariant derivative for fermions is given by

$\displaystyle\not{\nabla}\psi=e^{\mu}_{a}\gamma^{a}\left(\partial_{\mu}+\frac{1}{4}\omega_{\mu}^{bc}\gamma_{bc}\right)\psi,$
(2.4)

where $e_{\mu}^{a}$ is the tetrad defined by
$g_{\mu\nu}=\eta_{ab}e_{\mu}^{a}e_{\nu}^{b}$, $\omega_{\mu}^{ab}$ is the spin
connection, and
$\gamma^{ab}\equiv\frac{1}{2}\left[\gamma^{a},\gamma^{b}\right]$ (see App. A
for more details). We use the Latin characters for both the gauge indices and
local Lorentz indices; however, their distinction is clear from the context.

The action (2.1) is defined in the Jordan frame. We now transform to the
Einstein frame by performing a Weyl transformation, as it is more convenient
for the analysis of inflation. We first introduce an auxiliary field
$\tilde{\sigma}$ to extract the scalar degree of freedom from the $R^{2}$
term. In this case, the action becomes [27, 61]

$\displaystyle S\;=\;\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R\left(1+\frac{2\xi|\Phi|^{2}+4\tilde{\sigma}}{M_{P}^{2}}\right)-\frac{\tilde{\sigma}^{2}}{\alpha}+g^{\mu\nu}(D_{\mu}\Phi)^{\dagger}(D_{\nu}\Phi)-\lambda|\Phi|^{4}\right]+S_{\psi
A}\,.$ (2.5)

We perform the Weyl transformation by redefining the metric as

$\displaystyle
g_{\mu\nu}\to\Omega^{-2}g_{\mu\nu},\quad~{}\mathrm{where}~{}\quad\Omega^{2}\;=\;1+\frac{2\xi|\Phi|^{2}+4\tilde{\sigma}}{M_{P}^{2}}\,.$
(2.6)

The Ricci scalar transforms under the Weyl transformation according to Eq.
(A.4)\natural4\natural4$\natural$4Further details related to the Weyl
transformation are provided in App. A, and the action in the Einstein frame
becomes

$\displaystyle S\;=\;\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma+e^{-\sqrt{\frac{2}{3}}\frac{\sigma}{M_{P}}}g^{\mu\nu}(D_{\mu}\Phi)^{\dagger}(D_{\nu}\Phi)-V(\sigma,\Phi)\right]+S_{\psi
A}\,,$ (2.7)

where the scalaron field $\sigma$ is defined as

$\frac{\sigma}{M_{P}}\;\equiv\;\sqrt{6}\ln{\Omega}(\tilde{\sigma},\Phi)\,,$
(2.8)

and the scalar potential is given by

$\displaystyle
V(\sigma,\Phi)\;=\;e^{-\sqrt{\frac{8}{3}}\frac{\sigma}{M_{P}}}\left[\lambda|\Phi|^{4}+\frac{M_{P}^{4}}{16\alpha}\left(e^{\sqrt{\frac{2}{3}}\frac{\sigma}{M_{P}}}-1-\frac{2\xi|\Phi|^{2}}{M_{P}^{2}}\right)^{2}\right]\,.$
(2.9)

By redefining the fermions as $\psi_{i}\to\Omega^{3/2}\psi_{i}$, the matter
sector action (2.3) in the Einstein frame becomes

$\displaystyle S_{\psi A}\;=\;\int d^{4}x\sqrt{-g}$
$\displaystyle\left[i\bar{\psi}_{i}\not{\nabla}\psi_{i}-\frac{1}{\Omega}\left(Y_{l}^{ij}\bar{L}_{i}\Phi
l_{Rj}+Y_{d}^{ij}\bar{Q}_{i}\Phi
d_{Rj}+Y_{u}^{ij}\bar{Q}_{i}\tilde{\Phi}u_{Rj}\right)-\frac{1}{4}W_{\mu\nu}^{a}W^{a\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}\right]\,,$
(2.10)

where we only keep the terms that are relevant to our study. We work in the
unitary gauge and take the Higgs doublet as $\Phi=(0,h/\sqrt{2})^{T}$. The
electroweak (EW) gauge bosons are defined as

$\displaystyle W_{\mu}^{\pm}=\frac{1}{\sqrt{2}}(W_{\mu}^{1}\mp
iW_{\mu}^{2}),\quad
Z_{\mu}=\cos\theta_{W}W_{\mu}^{3}+\sin\theta_{W}B_{\mu},\quad\sin\theta_{W}=\frac{g^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}.$ (2.11)

The final action in the Einstein frame that we use for our analysis is given
by

$\displaystyle S\;=\;\int d^{4}x\sqrt{-g}$
$\displaystyle\left[\frac{M_{P}^{2}}{2}R+\frac{1}{2}(\partial\sigma)^{2}+\frac{1}{2\Omega^{2}}(\partial
h)^{2}-V(\sigma,h)+\bar{l}_{i}\left(i\not{\nabla}-m_{l_{i}}\right)l_{i}+\bar{u}_{i}\left(i\not{\nabla}-m_{u_{i}}\right)u_{i}\right.$
$\displaystyle\left.+\bar{d}_{i}\left(i\not{\nabla}-m_{d_{i}}\right)d_{i}-\frac{1}{2}W^{+}_{\mu\nu}W^{-\mu\nu}+{m_{W}^{2}}W^{+}_{\mu}W^{-\mu}-\frac{1}{4}Z_{\mu\nu}Z^{\mu\nu}+\frac{m_{Z}^{2}}{2}Z_{\mu}Z^{\mu}\right]\,.$
(2.12)

Here the mass terms are the functions of $\sigma$ and $h$, and can be
expressed as

$\displaystyle m_{l_{i}}\;=\;\frac{y_{l_{i}}h}{\sqrt{2}\Omega}\,,\quad
m_{u_{i}}\;=\;\frac{y_{u_{i}}h}{\sqrt{2}\Omega}\,,\quad
m_{d_{i}}\;=\;\frac{y_{d_{i}}h}{\sqrt{2}\Omega}\,,\quad
m_{W}^{2}\;=\;\frac{g^{2}h^{2}}{4\Omega^{2}}\,,\quad
m_{Z}^{2}\;=\;\frac{m_{W}^{2}}{\cos^{2}\theta_{W}}\,,$ (2.13)

and the scalar potential is given by

$\displaystyle
V(\sigma,h)\;=\;e^{-\sqrt{\frac{8}{3}}\frac{\sigma}{M_{P}}}\left[\frac{\lambda}{4}h^{4}+\frac{M_{P}^{4}}{16\alpha}\left(e^{\sqrt{\frac{2}{3}}\frac{\sigma}{M_{P}}}-1-\frac{\xi
h^{2}}{M_{P}^{2}}\right)^{2}\right]\,.$ (2.14)

Here, we rotate the fermions to the mass eigenbasis, making their mass
matrices diagonal with eigenvalues $m_{\psi_{i}}$. As before, we keep only the
terms relevant to our study. The Einstein frame action (2.1) is our starting
point of the analysis. Note that the inflaton sector couples to the SM
fermions and gauge bosons solely through the mass terms, that appear in the
combination of $h/\Omega$ terms.

Figure 1: An example of the SM mass spectrum during inflation.

In Fig. 1, we illustrate the SM mass spectrum during inflation by solving the
RG equations with SARAH [62], where the coupling values at the electroweak
(EW) scale are taken from Ref. [63]\natural5\natural5$\natural$5 Famously,
with the current central values of the Higgs mass and the top quark mass, the
Higgs quartic coupling becomes negative at an intermediate scale, $\mu\sim
10^{10}\,\mathrm{GeV}$ [64, 63]. To avoid this, we take the top quark mass to
be light, $m_{t}=170.5\,\mathrm{GeV}$, ensuring that $\lambda>0$ up to the
inflationary scale. This choice is only for illustration; any new physics
below the inflationary scale could potentially stabilize the EW vacuum for a
larger value of the top quark mass.  (see Sec. 2.2 for the discussion of the
size of the Higgs field value and the Hubble parameter during inflation). As
an example, we choose the parameter $\alpha=3\times 10^{8}$ and set the RG
scale to $\mu=H\simeq 1.4\times 10^{13}\,\rm{GeV}$ for this plot. We ignore
the scalaron’s contribution to the runnings, as it becomes important only
above the scalaron mass scale. The figure shows that even though the Higgs
field value is large compared to the Hubble parameter during inflation, with
$h/\Omega H\sim 2\times 10^{3}$ for our chosen parameters, the SM fermions can
be as light as the Hubble parameter due to the large hierarchy between the
Yukawa couplings. Consequently, these particles could produce significant
cosmological collider signatures.

###  2.2 Inflationary predictions

Here, we review the inflationary dynamics of the Higgs-$R^{2}$ model [29, 57,
58, 59, 60]. In this model, the inflaton is a combination of the Higgs field
and the scalaron, with inflation occurring in the region where $\sigma\gg
M_{P}$. We use the approximation that in the Einstein frame, the inflationary
potential takes a valley-like form, allowing us to use a single-field slow-
roll approximation, given by [57, 58]

$\displaystyle\frac{\partial V}{\partial
h}\;=\;0~{}~{}\Longrightarrow~{}~{}h_{0}^{2}\;=\;\frac{\xi
M_{P}^{2}}{\xi^{2}+4\lambda\alpha}\left(e^{\sqrt{\frac{2}{3}}\frac{\sigma_{0}}{M_{P}}}-1\right)\,,$
(2.15)

where the subscript “$0$” denotes background values. Ignoring the effects of
the isocurvature mode, the background action is given by

$\displaystyle S_{0}\;=\;\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R+\frac{1}{2}\left(1+c^{2}\right)\dot{\sigma}_{0}^{2}-\frac{\lambda
M_{P}^{4}}{4(\xi^{2}+4\lambda\alpha)}\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\sigma_{0}}{M_{P}}}\right)^{2}\right],\quad
c^{2}\;=\;\frac{\xi}{6(\xi^{2}+4\lambda\alpha)}\,,$ (2.16)

where terms suppressed by $e^{-\sqrt{\frac{2}{3}}\frac{\sigma_{0}}{M_{P}}}$ in
the kinetic term were neglected. We focus on the parameter region where
$c\lesssim 1$, necessary for a single-field slow-roll approximation as we see
later. Using the principal CMB observables from the Planck analysis [65], we
find that the CMB normalization fixes the parameters according to

$\displaystyle\left(4\alpha+\frac{\xi^{2}}{\lambda}\right)\left(1+c^{2}\right)\;\simeq\;2.4\times
10^{9}\times\left(\frac{N_{e}}{60}\right)^{2}\,,$ (2.17)

where $N_{e}$ represents the number of e-folds of inflation, given by

$\displaystyle
N_{e}\;=\;\frac{3(1+c^{2})}{4}e^{\sqrt{\frac{2}{3}}\frac{\sigma_{0}}{M_{P}}}\,,$
(2.18)

and the spectral index and tensor-to-scalar ratio are given by

$\displaystyle n_{s}\;\simeq\;1-\frac{2}{N_{e}},\qquad
r\;\simeq\;\frac{12}{N_{e}^{2}}(1+c^{2})\,.$ (2.19)

The spectral index is consistent with Planck observations [65], and the
tensor-to-scalar ratio is approximately $\mathcal{O}(10^{-3})$, and within
reach of future experiments like CMB-S4 [66] and LiteBIRD [67]. Furthermore,
the inflationary dynamics is stable against possible Planck suppressed
operators [68].

The CMB normalization (2.17) requires a large value of $\alpha$ and/or
$\xi^{2}/\lambda$. If $\alpha\gg\xi^{2}/\lambda$, the inflaton is
predominantly driven by the scalaron, whereas in the other case, it is
primarily driven by the SM Higgs field. In the remainder of this paper, we
focus on scenarios where there is no large hierarchy between $\alpha$ and
$\xi^{2}/\lambda$, allowing both the Higgs and the scalaron to contribute to
the inflationary sector. This requires a large value of $\xi\sim 10^{4}$,
provided that the coupling $\lambda$ is not too small. While this represents
the “standard” parameter region of the Higgs-$R^{2}$ model most often
discussed in the literature, we also explore cases where $\xi\gg 1$ and
$\xi\lesssim\mathcal{O}(0.1)$, with a small value of $\lambda$ at the
inflationary scale. The latter scenario is especially important when we
examine the cosmological collider signatures of the isocurvature mode.

###  2.3 Covariant formalism and isocurvature mode

Higgs-$R^{2}$ inflation has two scalar degrees of freedom, adiabatic and
isocurvature modes. To study the mode dynamics, we use the covariant formalism
of multi-field inflation [69, 70, 71, 72, 73, 74, 75]. We write down the
action in the Einstein frame as

$\displaystyle S\;=\;\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R+\frac{1}{2}h_{ab}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}-V\right]\,,$
(2.20)

where $\phi^{1}=\sigma$ and $\phi^{2}=h$. The target space metric in our case
is given by

$\displaystyle
h_{ab}\;=\;\mathrm{diag}\left(1,e^{-\sqrt{\frac{2}{3}}\frac{\sigma}{M_{P}}}\right)\,.$
(2.21)

The quadratic action of the curvature mode $\zeta$ and the isocurvature mode
$\chi$, both of which are mixtures of the Higgs field and the scalaron, is
given by

$\displaystyle S_{\mathrm{quad}}\;=\;\int
d^{4}xa^{3}\left[\frac{1}{2}\frac{\dot{\phi}_{0}^{2}}{H^{2}}\left(\dot{\zeta}^{2}-\frac{1}{a^{2}}\left(\partial_{i}\zeta\right)^{2}\right)+\frac{1}{2}\left(\dot{\chi}^{2}-\frac{1}{a^{2}}\left(\partial_{i}\chi\right)^{2}-m_{\chi}^{2}\chi^{2}\right)-\frac{2\dot{\theta}\dot{\phi}_{0}}{H}\dot{\zeta}\chi\right]\,.$
(2.22)

Here, the velocity of the background field is defined as
$\dot{\phi}_{0}=\sqrt{h_{ab}\dot{\phi}_{0}^{a}\dot{\phi}_{0}^{b}}$, and
$\dot{\theta}$ is the turning rate that parametrizes the curvature of the
inflationary trajectory, or the mixing between adiabatic and isocurvature
modes. The mass of the isocurvature mode is defined as

$\displaystyle
m_{\chi}^{2}\;=\;N^{a}N^{b}\nabla_{b}V_{a}+\frac{\dot{\phi}_{0}^{2}}{2}\mathcal{R}-\dot{\theta}^{2},\qquad\mathcal{R}\;=\;\mathcal{R}_{ab}h^{ab}\;=\;\mathcal{R}_{acbd}h^{ab}h^{cd}\,,$
(2.23)

with $\nabla_{a}$ and $\mathcal{R}_{acbd}$ constructed from the target space
metric $h_{ab}$. Further details of the derivation are provided in Appendix B.
We assume that the inflaton field value and its velocity vary along the
inflationary valley, while during inflation the Higgs field evolves according
to (2.15). In this case, the adiabatic and isocurvature directions are defined
as

$\displaystyle T^{a}\;=\;\frac{1}{\sqrt{1+c^{2}}}\begin{pmatrix}1\\\
c\,e^{\sigma_{0}/\sqrt{6}M_{P}}\end{pmatrix},\quad
N^{a}=\frac{1}{\sqrt{1+c^{2}}}\begin{pmatrix}-c\\\
e^{\sigma_{0}/\sqrt{6}M_{P}}\end{pmatrix}\,.$ (2.24)

We ignore terms suppressed by $1/N_{e}$ here and in subsequent discussion. By
substituting the explicit forms of the target space metric and potential, we
find

$\displaystyle
N^{a}N^{b}\nabla_{b}V_{a}\;\simeq\;\frac{\xi(24\lambda\alpha+\xi(1+6\xi))}{\lambda\alpha}H^{2},\quad\mathcal{R}=-\frac{1}{3M_{P}^{2}}\,,\quad\frac{\dot{\theta}}{H}\;\simeq\;3c\;=\;\sqrt{\frac{3\xi}{2(4\lambda\alpha+\xi^{2})}}\,.$
(2.25)

In the standard parameter region of the Higgs-$R^{2}$ model, $\xi\gg 1$ and
$\lambda\sim\mathcal{O}(0.1)$, the isocurvature mode is heavy and the turning
rate is small. Consequently, the isocurvature mode can be ignored for both
cosmological perturbation and cosmological collider signature studies. On the
other hand, if we take the parameters as

$\displaystyle\xi\sim\lambda\alpha\lesssim\mathcal{O}(0.1),$ (2.26)

we have $m_{\chi}^{2}/H^{2}\sim\mathcal{O}(1)$ and $c\sim\mathcal{O}(0.1)$
making the isocurvature mode light and the turning rate significant. This
corresponds to the so-called quasi-single field inflation regime [44,
45].\natural6\natural6$\natural$6 If the isocurvature mode is light enough,
with $m_{\chi}\lesssim H$, the model becomes a true multi-field type. The
inflationary dynamics for this case has been studied in [59], and we do not
consider this regime in this paper.  The observational effects of this regime
are the main focus of Sec. 4.

###  2.4 Order of magnitude estimation

Before proceeding to the actual computation, we estimate the order of
magnitude of cosmological collider signals arising from the SM particles.
Future experiments on large-scale structure [76] and 21 cm surveys [56, 77]
are expected to constrain the primordial bispectrum down to values of
$f_{NL}\sim 1$, potentially probing non-Gaussianities even below this value.
As previously demonstrated, the inflaton couples to SM fermions and gauge
bosons via a coupling of the form $h/\Omega$. By expanding this function, we
see that the coupling of the inflaton to the SM fermions and gauge bosons is
of the form

$\displaystyle{S}_{\mathrm{int}}\;=\;\int
d^{d}x\sqrt{-g}\,\mathcal{O}_{\mathrm{SM}}\left(1+c_{1}\frac{\varphi}{N_{e}M_{P}}+c_{2}\frac{\varphi^{2}}{2M_{P}^{2}}+\cdots\right)\,,$
(2.27)

where

$\mathcal{O}_{\mathrm{SM}}\;=\;\begin{cases}-m\bar{\psi}\psi\,,&\textrm{Fermions}\,,\\\
\displaystyle\frac{m^{2}}{2}g^{\mu\nu}A_{\mu}A_{\nu}\,,&\textrm{Gauge
bosons}\,,\end{cases}$ (2.28)

and $c_{1},c_{2}$ are constants of order unity. To estimate the size of non-
Gaussianity and, consequently, the signal, we first combine the linear and
quadratic couplings to form the three-point function, which leads to an
overall factor of $1/N_{e}M_{P}^{3}$. The non-Gaussianity scales as
$f_{NL}\propto P_{\zeta}^{-2}\times
H^{3}/\dot{\phi}_{0}^{3}$,\natural7\natural7$\natural$7This approximation is
valid only when the couplings between the inflaton and the SM particles do not
have any derivative couplings. where
$P_{\zeta}=H^{4}/4\pi^{2}\dot{\phi}_{0}^{2}\simeq 2.1\times 10^{-9}$ is the
amplitude of the power spectrum fixed by the CMB normalization. Lastly, since
the coupling depends on the mass, and the cosmological collider signatures are
maximized when the mass is comparable to the Hubble parameter, with $m\sim H$,
we obtain a simple estimate for non-Gaussianity using dimensional analysis

$\displaystyle|f_{\mathrm{NL}}|\;\lesssim\;\frac{\dot{\phi}_{0}H}{N_{e}M_{P}^{3}}\;\sim\;3.5\times
10^{3}\frac{H^{3}}{N_{e}M_{P}^{3}}\,.$ (2.29)

Due to the suppression by the Planck scale, this value is too small to be
observed by upcoming observations. Nevertheless, we rigorously compute this
signal in Sec. 3 to verify the accuracy of our estimation.

As discussed in Sec. 2.3, the isocurvature mode is too heavy to leave a signal
when $\xi\gg 1$. Therefore, when computing the cosmological signatures, we
focus on the case $\xi\sim\lambda\alpha\lesssim 1$, which corresponds to a
light isocurvature mode and the quasi-single field regime. The isocurvature
mode mixes with the inflaton through the turning rate $\dot{\theta}$ and also
couples to the inflaton via the potential. In particular, the derivative with
respect to the isocurvature direction $N^{a}$ is not necessarily suppressed by
the slow-roll parameters, and we find that the potential induces couplings of
the form

$\displaystyle S_{\mathrm{cubic}}\;=\;\int d\tau
d^{3}x\,a^{4}\left[-\frac{1}{6}V_{N^{3}}\chi^{3}-\frac{1}{2}V_{T^{2}N}\varphi^{2}\chi\right]\,,$
(2.30)

where $d\tau=dt/a$ is the conformal time, and

$\displaystyle\frac{V_{N^{3}}}{H}\;\sim\;\frac{V_{T^{2}N}}{H}\;\sim\;\alpha^{-1/2}\,,$
(2.31)

where we assumed $\xi\sim\lambda\alpha\sim 1$ (see Sec. 4 for the precise
forms of the couplings). Since Higgs-$R^{2}$ inflation requires $\alpha\sim
10^{9}$, the couplings are relatively suppressed compared to the Hubble
parameter. However, this suppression is compensated by the smallness of the
power spectrum. The non-Gaussianity resulting from these couplings can be
estimated as

$f_{\mathrm{NL}}\;\sim\;\begin{cases}\displaystyle
P_{\zeta}^{-1/2}\frac{V_{N^{3}}}{H}\left(\frac{\dot{\theta}}{H}\right)^{3}\,,&\textrm{for}~{}\chi^{3}\,,\vspace{1.5mm}\\\
\displaystyle
P_{\zeta}^{-1/2}\frac{V_{T^{2}N}}{H}\frac{\dot{\theta}}{H}\,,&\textrm{for}~{}\varphi^{2}\chi\,,\end{cases}$
(2.32)

where the factor $P_{\zeta}^{-1/2}$ arises from the definition of the non-
Gaussianity. Since $P_{\zeta}\times\alpha\sim\mathcal{O}(1)$ and the turning
rate $\dot{\theta}$ can be of the order of the Hubble parameter, we expect
that non-Gaussianity could potentially be as large as order unity, thus
yielding an observable signature. In Sec. 4, we compute the cosmological
collider signatures of the isocurvature mode in detail to confirm this
expectation.

##  3 Cosmological collider signatures of fermions and gauge bosons

In Higgs-$R^{2}$ inflation, the inflaton naturally couples to the SM fermions
and gauge bosons, potentially giving rise to cosmological collider signatures.
The inflaton couples to the fermions and gauge bosons through the coupling of
the form $h/\Omega$, as demonstrated in Sec. 2.1, and this coupling is
expanded as

$\displaystyle\frac{h}{\Omega}\;=\;\frac{h_{0}}{\Omega_{0}}+\varphi^{a}\nabla_{a}\left(\frac{h}{\Omega}\right)_{0}+\frac{1}{2}\varphi^{a}\varphi^{b}\nabla_{b}\nabla_{a}\left(\frac{h}{\Omega}\right)_{0}+\cdots\,,$
(3.1)

where the subscript “0” indicates that the quantities are evaluated using
background field values and $\varphi^{a}=\phi^{a}-\phi^{a}_{0}$. We provide
technical details related to this expansion in target space in App. B.
Focusing on the adiabatic direction $T^{a}$, from Eqs. (2.8) and (2.24), we
obtain

$\displaystyle\frac{h}{\Omega}\;\simeq\;\frac{h_{0}}{\Omega_{0}}\left(1+\sqrt{\frac{4\alpha\lambda+\xi^{2}}{24\alpha\lambda+\xi(1+6\xi)}}e^{-\sqrt{\frac{2}{3}}\frac{\sigma_{0}}{M_{P}}}\frac{\varphi}{2M_{P}}+\frac{\varphi^{2}}{12M_{P}^{2}}\right)\,,$
(3.2)

where we only kept the leading order terms in the slow-roll expansion. In this
section, we focus on the “standard” parameter region of the Higgs-$R^{2}$
model, with $\lambda\alpha\sim\xi^{2}\gg\xi$. In this limit, the above
expression simplifies to

$\displaystyle\frac{h}{\Omega}\;\simeq\;\frac{h_{0}}{\Omega_{0}}\left(1+\frac{\sqrt{6}}{16N_{e}}\frac{\varphi}{M_{P}}+\frac{\varphi^{2}}{12M_{P}^{2}}\right)\,,$
(3.3)

where we used the number of e-folds $(\ref{eq:numofefolds})$. We note that the
linear term is suppressed by the slow-roll parameter, or equivalently
$1/N_{e}$, while the quadratic term is not suppressed. The coupling between
the inflaton and SM fermions/gauge bosons is represented by the action (2.27),
with $(c_{1},c_{2})=(\sqrt{6}/16,1/6)$ in the case of fermion and
$(\sqrt{6}/8,1/3)$ for the gauge bosons, respectively. Here the mass $m$ is
evaluated using the background field values of $\sigma$ and $h$.

In the following analysis, we evaluate the contributions of SM fermion and
gauge bosons to the three-point functions. The computation is analogous to
Ref. [54]. We have two contributions: one with two $\mathcal{O}_{\mathrm{SM}}$
insertions and another one with three $\mathcal{O}_{\mathrm{SM}}$ insertions.
Given that the linear coupling is suppressed by an additional factor of
$1/N_{e}$, our focus is on the former case. We are primarily interested in the
squeezed limit of the bispectrum, with $k_{3}\ll k_{1},k_{2}$, and thus only
consider the diagram containing $\hat{\varphi}_{\vec{k}_{3}}$ on the linear
side and $\hat{\varphi}_{\vec{k}_{1}}\hat{\varphi}_{\vec{k}_{2}}$ on the
quadratic side of the vertices. Lastly, since the cosmological collider
signatures are free of UV divergences, there is no need for a regularization
scheme to evaluate the diagrams. Bearing these points in mind, the diagram of
interest can be expressed as follows:

$\displaystyle\langle\hat{\varphi}_{\vec{k}_{1}}\hat{\varphi}_{\vec{k}_{2}}\hat{\varphi}_{\vec{k}_{3}}\rangle$
$\displaystyle=\leavevmode\hbox to7.47pt{\vbox
to7.47pt{\pgfpicture\makeatletter\hbox{\hskip 7.34677pt\lower-3.73589pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{ \diagram*{ (f1) --
[very thick] (f5), (f2) -- (v1) -- (f3), (f4) -- (v2), (v1) -- [scalar,
quarter left] (v2) -- [scalar, quarter left] (v1), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.v1)]
\vertex(f1); \vertex[right = 0.5 of f1,label=270:${\scriptstyle k_{1}~{}}$]
(f2); \vertex[right = 0.25 of f2] (v1s); \vertex[below = 0.75 of v1s] (v1);
\vertex[right = 0.25 of v1s,label=270:${~{}~{}\scriptstyle k_{2}}$] (f3);
\vertex[right = 1.5 of f3, label=270:${~{}~{}~{}\,\scriptstyle k_{3}}$] (f4);
\vertex[right = 0.5 of f4] (f5); \vertex[below = of f4] (v2);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.075pt}{0.0pt}\pgfsys@curveto{-0.075pt}{1.95284pt}{-1.65805pt}{3.53589pt}{-3.61089pt}{3.53589pt}\pgfsys@curveto{-5.56372pt}{3.53589pt}{-7.14677pt}{1.95284pt}{-7.14677pt}{0.0pt}\pgfsys@curveto{-7.14677pt}{-1.95284pt}{-5.56372pt}{-3.53589pt}{-3.61089pt}{-3.53589pt}\pgfsys@curveto{-1.65805pt}{-3.53589pt}{-0.075pt}{-1.95284pt}{-0.075pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-3.61089pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61089pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.075pt}{0.0pt}\pgfsys@curveto{-0.075pt}{1.95284pt}{-1.65805pt}{3.53589pt}{-3.61089pt}{3.53589pt}\pgfsys@curveto{-5.56372pt}{3.53589pt}{-7.14677pt}{1.95284pt}{-7.14677pt}{0.0pt}\pgfsys@curveto{-7.14677pt}{-1.95284pt}{-5.56372pt}{-3.53589pt}{-3.61089pt}{-3.53589pt}\pgfsys@curveto{-1.65805pt}{-3.53589pt}{-0.075pt}{-1.95284pt}{-0.075pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-3.61089pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61089pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=-\frac{c_{1}c_{2}}{N_{e}M_{P}^{3}}\sum_{\lambda_{1},\lambda_{2}=\pm}\lambda_{1}\lambda_{2}\int_{-\infty}^{0}\frac{d\tau_{1}d\tau_{2}}{(H^{2}\tau_{1}\tau_{2})^{4}}\Delta_{+\lambda_{1}}(k_{1};0,\tau_{1})\Delta_{+\lambda_{1}}(k_{2};0,\tau_{1})\Delta_{+\lambda_{2}}(k_{3};0,\tau_{2})$
$\displaystyle\times\int
d^{3}x_{1}\,e^{-i(\vec{k}_{1}+\vec{k}_{2})\cdot\vec{x}_{1}}\int
d^{3}x_{2}\,e^{-i\vec{k}_{3}\cdot\vec{x}_{2}}\langle\mathcal{O}_{SM}(\tau_{1},\vec{x}_{1})\mathcal{O}_{SM}(\tau_{2},\vec{x}_{2})\rangle_{\lambda_{1}\lambda_{2}}\,,$
(3.4)

where the thick line indicates the asymptotic future time slice, the gray blob
indicates a vertex from either the time-ordered or anti-time-ordered contours
in the Schwinger-Keldysh formalism, with $\lambda_{1}$ and $\lambda_{2}$ as
their corresponding labels, and the dashed lines denote the SM fermions or
gauge bosons. The Schwinger-Keldysh formalism is reviewed in App. C. As the
correlator depends only on $\vec{x}_{12}=\vec{x}_{1}-\vec{x}_{2}$, we can
factor out the overall space integral as the delta function corresponding to
momentum conservation. We define the dimensionless non-Gaussianity function
$S_{\mathrm{NG}}$ as

$\langle\zeta_{\vec{k}_{1}}\zeta_{\vec{k}_{2}}\zeta_{\vec{k}_{3}}\rangle^{\prime}\;=\;(2\pi)^{4}S_{\mathrm{NG}}(\vec{k}_{1},\vec{k}_{2},\vec{k}_{3})\frac{P_{\zeta}^{2}}{k_{1}^{2}k_{2}^{2}k_{3}^{2}}\,,$
(3.5)

where
$\zeta_{\vec{k}}=-(H/\dot{\phi}_{0})\hat{\varphi}_{\vec{k}}$\natural8\natural8$\natural$8
This relation receives a correction at the next-to-leading order, which
contributes to the non-Gaussianity [78]. As this contribution is local and
does not generate a cosmological collider signal, we do not consider it here.
and the prime indicates that we have removed the factor
$(2\pi)^{3}\delta^{(3)}(\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3})$. The SM fermion
and gauge boson contribution to $S_{\mathrm{NG}}$ is expressed as

$\displaystyle S_{\mathrm{NG}}$
$\displaystyle=\frac{c_{1}c_{2}H\dot{\phi}_{0}}{8N_{e}M_{P}^{3}}\frac{1}{k_{1}k_{2}k_{3}}\sum_{\lambda_{1},\lambda_{2}=\pm}\lambda_{1}\lambda_{2}\int_{-\infty}^{0}\frac{d\tau_{1}d\tau_{2}}{(H^{2}\tau_{1}\tau_{2})^{4}}e^{i\lambda_{1}(k_{1}+k_{2})\tau_{1}+i\lambda_{2}k_{3}\tau_{2}}$
$\displaystyle\times\left(1-i\lambda_{1}k_{1}\tau_{1}\right)(1-i\lambda_{1}k_{2}\tau_{1})(1-i\lambda_{2}k_{3}\tau_{2})\int
d^{3}x_{12}\,e^{i\vec{k}_{3}\cdot\vec{x}_{12}}\langle\mathcal{O}_{\mathrm{SM}}(\tau_{1},\vec{x}_{1})\mathcal{O}_{\mathrm{SM}}(\tau_{2},\vec{x}_{2})\rangle_{\lambda_{1}\lambda_{2}}\,,$
(3.6)

where we used the explicit form of the massless scalar propagator $\Delta$
derived in App. C.2.

The remaining task is to evaluate the two-point function of
$\mathcal{O}_{\mathrm{SM}}$. For this, we use the late-time expansion, as
described in Ref. [54], to estimate the order of magnitude of the signal. In
the late-time expansion, we fix $\vec{x}_{12}$ and take the limit
$\tau_{1},\tau_{2}\to 0$, and different propagators become equivalent (see
App. C). Therefore, we can omit the subscripts $\lambda_{1}$ and $\lambda_{2}$
from the two-point function of $\mathcal{O}_{\mathrm{SM}}$.

###  3.1 Fermion contribution

We first discuss the fermion contribution. As we noted above, we can drop the
contour subscripts as long as we focus on the late-time behavior. Therefore,
we may evaluate the two-point function as

$\displaystyle\langle\mathcal{O}_{\mathrm{SM}}(\tau_{1},\vec{x}_{1})\mathcal{O}_{\mathrm{SM}}(\tau_{2},\vec{x}_{2})\rangle\;=\;-m^{2}\mathrm{Tr}\left[S(\vec{x}_{12};\tau_{1},\tau_{2})S(\vec{x}_{21};\tau_{2},\tau_{1})\right]\,,$
(3.7)

where $\vec{x}_{21}=\vec{x}_{2}-\vec{x}_{1}$. The fermion propagator in de
Sitter spacetime is given by

$\displaystyle
S(\vec{x}_{12};\tau_{1},\tau_{2})\;=\;\frac{H^{2}}{(4\pi)^{2}}\left[a\left(i\not{\nabla}+m\right)\right]_{1}\frac{1}{\sqrt{a_{1}a_{2}}}\left[P_{+}I_{\nu_{-}}(Z_{12})+P_{-}I_{\nu_{+}}(Z_{12})\right]\,,$
(3.8)

where $m$ represents the fermion mass, $P_{\pm}=(1\pm\gamma^{0})/2$ is the
projection operator, $a_{i}\equiv a(\tau_{i})$, $\nu_{\pm}=1/2\mp im/H$,
$Z_{12}$ is the embedding distance, and the definition of $I_{\nu}(Z_{12})$
and its derivation is given in App. C.3. To take the trace, it is convenient
to keep track of the projection operators $P_{\pm}$ and to write the
propagators as

$\displaystyle
S(\vec{x}_{12};\tau_{1},\tau_{2})=\frac{H^{2}}{(4\pi)^{2}}\frac{1}{\sqrt{a_{1}a_{2}}}$
$\displaystyle\left\\{P_{+}\left[i\left(\partial_{\tau_{1}}+\frac{\nu_{-}-3/2}{\tau_{1}}\right)I_{\nu_{-}}(Z_{12})+i\gamma^{i}\partial_{1i}I_{\nu_{+}}(Z_{12})\right]\right.$
$\displaystyle\left.+P_{-}\left[-i\left(\partial_{\tau_{1}}+\frac{\nu_{+}-3/2}{\tau_{1}}\right)I_{\nu_{+}}(Z_{12})+i\gamma^{i}\partial_{1i}I_{\nu_{-}}(Z_{12})\right]\right\\},$
(3.9)

and

$\displaystyle
S(\vec{x}_{21};\tau_{2},\tau_{1})=\frac{H^{2}}{(4\pi)^{2}}\frac{1}{\sqrt{a_{1}a_{2}}}$
$\displaystyle\left\\{\left[i\left(\partial_{\tau_{2}}+\frac{\nu_{-}-3/2}{\tau_{2}}\right)I_{\nu_{-}}(Z_{21})+i\gamma^{i}\partial_{2i}I_{\nu_{-}}(Z_{21})\right]P_{+}\right.$
$\displaystyle\left.+\left[-i\left(\partial_{\tau_{2}}+\frac{\nu_{+}-3/2}{\tau_{2}}\right)I_{\nu_{+}}(Z_{21})+i\gamma^{i}\partial_{2i}I_{\nu_{+}}(Z_{21})\right]P_{-}\right\\}.$
(3.10)

Using these definitions, we obtain

$\displaystyle\mathrm{Tr}\left[S(\vec{x}_{12};\tau_{1},\tau_{2})S(\vec{x}_{21};\tau_{2},\tau_{1})\right]$
$\displaystyle=\frac{2H^{4}}{(4\pi)^{4}}\frac{1}{a_{1}a_{2}}\left[-\left(\partial_{\tau_{1}}+\frac{\nu_{-}-3/2}{\tau_{1}}\right)I_{\nu_{-}}\left(\partial_{\tau_{2}}+\frac{\nu_{-}-3/2}{\tau_{2}}\right)I_{\nu_{-}}+(\partial_{1i}I_{\nu_{+}})(\partial_{2i}I_{\nu_{-}})+(\nu_{-}\leftrightarrow\nu_{+})\right],$
(3.11)

where we do not distinguish between the arguments $Z_{12}$ and $Z_{21}$ as it
is irrelevant for the late-time expansion. We note that the time derivative,
together with the factor $\nu_{\pm}-3/2$, eliminates the leading order term in
the late-time expansion of $I_{\nu_{\pm}}$. Using Eqs. (C.14) and (C.15), we
evaluate the late-time expansion $Z\to\infty$ as

$\displaystyle\mathrm{Tr}\left[S(\vec{x}_{12};\tau_{1},\tau_{2})S(\vec{x}_{21};\tau_{2},\tau_{1})\right]$
$\displaystyle=-\frac{H^{6}}{\pi^{4}}\frac{\frac{2\pi
m}{H}(1+\frac{m^{2}}{H^{2}})}{\sinh(2\pi
m/H)}(-2Z_{12})^{-3}-\frac{3H^{6}}{2\pi^{5}}\left[\Gamma\left(\nu_{-}\right)\Gamma\left(\nu_{-}-1\right)\Gamma^{2}\left(\frac{5}{2}-\nu_{-}\right)(-2Z_{12})^{-5+2\nu_{-}}+(\nu_{-}\to\nu_{+})\right],$
(3.12)

where we ignore the terms of $\mathcal{O}(Z_{12}^{-5})$ and keep both the
local and non-local contributions. We note that this result diverges from the
result in [54], but we verify the consistency of our findings by considering
the massless limit. If we take the limit $m\rightarrow 0$, our result
simplifies to

$\displaystyle\mathrm{Tr}\left[S(\vec{x}_{12};\tau_{1},\tau_{2})S(\vec{x}_{21};\tau_{2},\tau_{1})\right]$
$\displaystyle\simeq-\frac{1}{\pi^{4}a_{1}^{3}a_{2}^{3}}\left[\frac{1}{x_{12}^{6}}+\frac{3(\tau_{1}-\tau_{2})^{2}}{x_{12}^{8}}+\cdots\right]\,,$
(3.13)

where we expanded the $Z_{12}$ term. On the other hand, if $m=0$, the fermion
field becomes conformal, allowing the scale factor to be factored out by
redefining $\psi=a^{-3/2}\psi_{\mathrm{flat}}$. The massless fermion
propagator in flat spacetime is well-known and given in coordinate space by
(see e.g. [79])

$\displaystyle S_{\mathrm{flat}}(x)\;=\;\frac{i\not{x}}{2\pi^{2}x^{4}}\,,$
(3.14)

where $x$ here refers to a four-dimensional space-time coordinate. Therefore,
if $m=0$, we find

$\displaystyle\mathrm{Tr}\left[S(\vec{x}_{12};\tau_{1},\tau_{2})S(\vec{x}_{21};\tau_{2},\tau_{1})\right]$
$\displaystyle=\frac{1}{a_{1}^{3}a_{2}^{3}}\times\mathrm{Tr}\left[S_{\mathrm{flat}}(x_{12})S_{\mathrm{flat}}(x_{21})\right]=-\frac{1}{\pi^{4}a_{1}^{3}a_{2}^{3}}\left[\frac{1}{x_{12}^{6}}+\frac{3(\tau_{1}-\tau_{2})^{2}}{x_{12}^{8}}+\cdots\right],$
(3.15)

which coincides with our result.\natural9\natural9$\natural$9 The late-time
expansion of $A_{\pm}$, given by Eq. (4.27) in [54], is missing a factor of
$1/\sqrt{a_{1}a_{2}}$. However, we are not certain if this omission is the
source of the discrepancy.  Focusing on the non-local part and keeping only
the leading-order term, we obtain

$\displaystyle\langle\mathcal{O}_{SM}(\tau_{1},\vec{x}_{1})\mathcal{O}_{SM}(\tau_{2},\vec{x}_{2})\rangle$
$\displaystyle\simeq\frac{3H^{6}m^{2}}{2\pi^{5}}\left[\Gamma\left(\nu_{-}\right)\Gamma\left(\nu_{-}-1\right)\Gamma^{2}\left(\frac{5}{2}-\nu_{-}\right)\left(\frac{\tau_{1}\tau_{2}}{x_{12}^{2}}\right)^{5-2\nu_{-}}+(\nu_{-}\to\nu_{+})\right]\,.$
(3.16)

After performing the Fourier transformation and conformal time integrals, the
non-Gaussianity function in the squeezed limit $k_{3}\ll k_{1},k_{2}$ can be
expressed as

$\displaystyle
S_{\mathrm{NG}}\;=\;-\frac{3c_{1}c_{2}H\dot{\phi}_{0}}{8\pi^{4}N_{e}M_{P}^{3}}\frac{m^{2}}{H^{2}}C_{1/2}(\nu_{-})\left(\frac{k_{3}}{k_{1}}\right)^{4-2\nu_{-}}+(\nu_{-}\to\nu_{+})\,,$
(3.17)

where

$\displaystyle
C_{1/2}(\nu)\;=\;2^{2\nu}(3-\nu)(3-2\nu)^{2}\sin^{2}(\nu\pi)\sin(2\nu\pi)\Gamma(\nu)\Gamma(\nu-1)\Gamma(-8+4\nu_{-})\Gamma^{2}\left(\frac{5}{2}-\nu\right)\Gamma^{2}(2-2\nu)\,.$
(3.18)

The coefficient $(m^{2}/H^{2})C_{1/2}(\nu_{\pm})$ peaks at $m\sim 0.1H$ and is
generally of order unity. This expression aligns with our earlier estimation
(2.29), albeit with an additional suppression factor $\sim(2\pi)^{-4}$ arising
from the loops. Unfortunately, mainly due to Planck scale suppression, the
signal is too small to be observable in the foreseeable future.

###  3.2 Gauge boson contribution

We next compute the contribution from the gauge bosons. The two-point function
of $\mathcal{O}_{\mathrm{SM}}$ is given by

$\displaystyle\langle\mathcal{O}_{\mathrm{SM}}(\tau_{1},\vec{x}_{1})\mathcal{O}_{\mathrm{SM}}(\tau_{2},\vec{x}_{2})\rangle\;=\;\frac{m^{4}}{2a_{1}^{2}a_{2}^{2}}\eta^{\alpha\beta}\eta^{\gamma\delta}G_{\alpha\gamma}(\vec{x}_{12};\tau_{1},\tau_{2})G_{\beta\delta}(\vec{x}_{12};\tau_{1},\tau_{2})\,,$
(3.19)

where $m$ represents the gauge boson mass, and we again drop the contour
indices. The gauge boson propagator in de Sitter spacetime is given by

$\displaystyle G_{\alpha\beta}(\vec{x}_{1}-\vec{x}_{2};\tau_{1},\tau_{2})$
$\displaystyle=\frac{H^{2}}{m^{2}(4\pi)^{2}}K_{\alpha\beta}(Z_{12}),$ (3.20)

where $Z_{12}$ is the embedding distance and

$\displaystyle K_{\alpha\beta}(Z_{12})$
$\displaystyle=\left[Z_{12}I_{\nu}^{\prime\prime}+3I_{\nu}^{\prime}\right](\partial_{1\alpha}Z_{12})(\partial_{2\beta}Z_{12})+\left[(1-Z_{12}^{2})I_{\nu}^{\prime\prime}-3Z_{12}I_{\nu}^{\prime}\right]\partial_{1\alpha}\partial_{2\beta}Z_{12}\,,$
(3.21)

with $\nu=\sqrt{1/4-m^{2}/H^{2}}$ and the primes denoting the derivatives with
respect to $Z_{12}$. The full details of the derivation are given in App. C.4.
The late-time behavior of the two-point function then becomes

$\displaystyle\langle\mathcal{O}_{SM}(\tau_{1},\vec{x}_{1})\mathcal{O}_{SM}(\tau_{2},\vec{x}_{2})\rangle\;=\;\frac{3H^{8}}{128\pi^{5}}(1+2\nu)^{2}\Gamma^{2}\left(\frac{5}{2}-\nu\right)\Gamma^{2}(\nu)\left(\frac{\tau_{1}\tau_{2}}{x_{12}^{2}}\right)^{3-2\nu}+(\nu\to-\nu)\,,$
(3.22)

where we focus on the non-local
contribution.\natural10\natural10$\natural$10We note that this computation
also diverges from the results in [54]. After performing the Fourier
transformation and computing the conformal time integrals, the non-Gaussianity
function $S_{\mathrm{NG}}$ becomes

$\displaystyle S_{\mathrm{NG}}$
$\displaystyle=-\frac{3c_{1}c_{2}}{8\pi^{4}}\frac{H\dot{\phi}_{0}}{N_{e}M_{P}^{3}}\frac{m^{4}}{H^{4}}C_{1}(\nu)\left(\frac{k_{3}}{k_{1}}\right)^{2-2\nu}+(\nu\to-\nu),$
(3.23)

where

$\displaystyle
C_{1}(\nu)=2^{2\nu}(2-\nu)\sin^{2}(\nu\pi)\sin(2\nu\pi)\Gamma(-4+4\nu)\Gamma^{2}\left(\frac{5}{2}-\nu\right)\Gamma^{2}(\nu)\Gamma^{2}(-2\nu).$
(3.24)

This expression correctly reproduces our estimation (2.29), with an additional
suppression factor $\sim(2\pi)^{-4}$ arising from the loops. Similar to the
fermion case, the signal is too small for near-future observation due to
suppression by the Planck scale, $M_{P}$. Moreover, the gauge boson
contribution is likely further suppressed exponentially by $m_{W}/H$ or
$m_{Z}/H$ (see Fig. 1). Therefore, we turn our attention to the isocurvature
mode signatures.

##  4 Cosmological collider signatures of isocurvature mode

As discussed in the previous section, the inflationary sector in Higgs-$R^{2}$
inflation naturally couples to SM fermions and gauge bosons, but the resulting
cosmological collider signatures are too small for near-future detection.
Therefore, in this section, we focus on the isocurvature mode. In the
“standard” parameter region of Higgs-$R^{2}$ inflation, $\xi^{2}\gg 1$, the
isocurvature mode is too heavy to be excited during inflation, leaving no
observable signal. However, if $\lambda\ll 1$ such that
$\xi\sim\lambda\alpha\lesssim\mathcal{O}(0.1)$, the isocurvature mode can be
as light as the Hubble parameter to be excited during inflation. In this case,
the isocurvature mode can produce a substantial cosmological collider
signature, potentially detectable by future 21 cm observations [55, 56]. This
serves as a concrete example of the clock signals discussed in Refs. [44, 45,
80], originating from a UV-complete model (up to the Planck scale) motivated
by particle physics.

###  4.1 Mixed propagator

We first define a mixed propagator of the adiabatic and isocurvature modes,
following Ref. [81]. The relevant part of the quadratic action is given by Eq.
(2.22), where all details of the derivation can be found in App. B.3. Since
the curvature mode $\zeta$ is related to $\varphi$ as
$\zeta=-H\varphi/\dot{\phi}_{0}$,\natural11\natural11$\natural$11 As before,
we ignore the next-to-leading order correction that generates only local-type
non-Gaussianity.  within the leading order of the slow-roll approximation, the
action is equivalently expressed as

$\displaystyle S_{\mathrm{quad}}\;\simeq\;\int d\tau
d^{3}x\left[\frac{a^{2}}{2}\left({\varphi^{\prime}}^{2}-\left(\partial_{i}\varphi\right)^{2}\right)+\frac{a^{2}}{2}\left({\chi^{\prime}}^{2}-\left(\partial_{i}\chi\right)^{2}-a^{2}m_{\chi}^{2}\chi^{2}\right)+2\dot{\theta}a^{3}\varphi^{\prime}\chi\right]\,.$
(4.1)

Ultimately, we evaluate the three-point function of the curvature perturbation
at the asymptotic future where $\tau=0$. Therefore, we define the mixed
propagator as

$\displaystyle\left\langle{\hat{\varphi}(0,\vec{x}_{1})\hat{\chi}_{\pm}(\tau,\vec{x}_{2})}\right\rangle$
$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\vec{k}\cdot(\vec{x}_{1}-\vec{x}_{2})}\mathcal{G}_{\pm}(k;\tau)\,,$
(4.2)

where the subscript $\pm$ denotes the contour on which $\chi$ resides. Using
the Feynman rules of the Schwinger-Keldysh formalism, it is computed as (see
App. C.2 for the discussion of scalar propagators)

$\displaystyle\mathcal{G}_{\pm}(k;\tau)$ $\displaystyle\equiv\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{ \diagram*{ (f1) --
[very thick] (f5), (f2) -- (v1) -- (f3), (f4) -- (v2), (v1) -- [scalar,
quarter left] (v2) -- [scalar, quarter left] (v1), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.v1)]
\vertex(f1); \vertex[right = of f1] (f2); \vertex[right = of f2] (f3);
\vertex[below = 0.75 of f2] (v1); \vertex[below = 0.75 of v1] (c1); \diagram*{
(f1) -- [very thick] (f3), (f2) -- [insertion={1.0}] (v1) -- [scalar] (c1), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to7.47pt{\vbox to7.47pt{\pgfpicture\makeatletter\hbox{\hskip
7.34677pt\lower-3.73589pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{
\diagram*{ (f1) -- [very thick] (f5), (f2) -- (v1) -- (f3), (f4) -- (v2), (v1)
-- [scalar, quarter left] (v2) -- [scalar, quarter left] (v1), }; \hss}
\diagram*{ (f1) -- [very thick] (f3), (f2) -- (v1) -- [scalar] (c1), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.v1)]
\vertex(f1); \vertex[right = of f1] (f2); \vertex[right = of f2] (f3);
\vertex[below = 0.75 of f2] (v1); \vertex[below = 0.75 of v1] (c1);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.075pt}{0.0pt}\pgfsys@curveto{-0.075pt}{1.95284pt}{-1.65805pt}{3.53589pt}{-3.61089pt}{3.53589pt}\pgfsys@curveto{-5.56372pt}{3.53589pt}{-7.14677pt}{1.95284pt}{-7.14677pt}{0.0pt}\pgfsys@curveto{-7.14677pt}{-1.95284pt}{-5.56372pt}{-3.53589pt}{-3.61089pt}{-3.53589pt}\pgfsys@curveto{-1.65805pt}{-3.53589pt}{-0.075pt}{-1.95284pt}{-0.075pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-3.61089pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61089pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to7.47pt{\vbox to7.47pt{\pgfpicture\makeatletter\hbox{\hskip
7.34677pt\lower-3.73589pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{ \diagram*{ (f1) -- [very thick] (f5), (f2) -- (v1) -- (f3), (f4) --
(v2), (v1) -- [scalar, quarter left] (v2) -- [scalar, quarter left] (v1), };
\hss} \diagram*{ (f1) -- [very thick] (f3), (f2) -- (v1) -- [scalar] (c1), };
\hss} \diagram*{ (f1) -- [very thick] (f3), (f2) -- (v1) -- [scalar] (c1), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.v1)]
\vertex(f1); \vertex[right = of f1] (f2); \vertex[right = of f2] (f3);
\vertex[below = 0.75 of f2] (v1); \vertex[below = 0.75 of v1] (c1);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.075pt}{0.0pt}\pgfsys@curveto{-0.075pt}{1.95284pt}{-1.65805pt}{3.53589pt}{-3.61089pt}{3.53589pt}\pgfsys@curveto{-5.56372pt}{3.53589pt}{-7.14677pt}{1.95284pt}{-7.14677pt}{0.0pt}\pgfsys@curveto{-7.14677pt}{-1.95284pt}{-5.56372pt}{-3.53589pt}{-3.61089pt}{-3.53589pt}\pgfsys@curveto{-1.65805pt}{-3.53589pt}{-0.075pt}{-1.95284pt}{-0.075pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-3.61089pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61089pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=2i\dot{\theta}\int_{-\infty}^{0}\frac{d\tau^{\prime}}{(-H\tau^{\prime})^{3}}\left[\partial_{\tau^{\prime}}\Delta(k;0,\tau^{\prime})\times
G_{\pm+}(k;\tau,\tau^{\prime})-\partial_{\tau^{\prime}}\Delta(k;\tau^{\prime},0)\times
G_{\pm-}(k;\tau,\tau^{\prime})\right]\,.$ (4.3)

Here, the black (white) circle represents the vertex from the time-ordered
(anti-time-ordered) contour, the solid line represents $\varphi$, and the
dashed line indicates $\chi$. Additionally, we assume that the turning rate
$\dot{\theta}$ is constant. We find that

$\displaystyle\mathcal{G}_{\pm}(k;\tau)$
$\displaystyle=\frac{\pi\dot{\theta}H}{4k^{3}}\mathcal{I}_{\pm}(-k\tau;\nu)\,,$
(4.4)

where

$\nu\;=\;\sqrt{\frac{9}{4}-\frac{m_{\chi}^{2}}{H^{2}}}\,,$ (4.5)

and

$\displaystyle\mathcal{I}_{\pm}(z;\nu)=z^{3/2}$
$\displaystyle\left[-iH_{\nu}^{(1)}(z)\int_{0}^{\infty(1-i0)}\frac{dz^{\prime}}{\sqrt{z^{\prime}}}H_{\nu}^{(2)}(z^{\prime})e^{-iz^{\prime}}+iH_{\nu}^{(2)}(z)\int_{0}^{\infty(1+i0)}\frac{dz^{\prime}}{\sqrt{z^{\prime}}}H_{\nu}^{(1)}(z^{\prime})e^{iz^{\prime}}\right.$
$\displaystyle~{}\left.+iH_{\nu}^{(1)}(z)\int_{0}^{z(1\mp
i0)}\frac{dz^{\prime}}{\sqrt{z^{\prime}}}{H_{\nu}^{(2)}}(z^{\prime})e^{\mp
iz^{\prime}}-i{H_{\nu}^{(2)}}(z)\int_{0}^{z(1\mp
i0)}\frac{dz^{\prime}}{\sqrt{z^{\prime}}}{H_{\nu}^{(1)}}(z^{\prime})e^{\mp
iz^{\prime}}\right]\,,$ (4.6)

where we explicitly show the $i\epsilon$ prescription. We note that we
correctly reproduce the result from [81] by setting their $\lambda_{2}$ equal
to our $2\dot{\theta}$. The integral can be evaluated analytically and is
expressed in terms of the generalized hypergeometric functions ${}_{2}F_{2}$.
We refer interested readers to Ref. [81] for its explicit form. We may focus
on the parameter region $m_{\chi}/H>1$ hereafter since otherwise the
isocurvature mode would not be sufficiently heavy, and the model is eventually
turning into a multi-field inflation regime.\natural12\natural12$\natural$12
The expression (4.6) is IR finite as long as $\mathrm{Re}[\nu]<3/2$ [45], or
the mass is finite, and in this sense $m/H>1$ is not a strict bound.
Nevertheless we focus on this parameter region for simplicity.  Finally, we
introduce the following asymptotic limit which is useful when considering the
squeezed limit:

$\displaystyle\mathcal{I}_{\pm}(z;\nu)\xrightarrow[z\to
0]{}-\frac{2}{\sqrt{\pi}}\left[\frac{2^{\nu}z^{3/2-\nu}\Gamma(\nu)}{\cos(\pi\nu/2)+\sin(\pi\nu/2)}+(\nu\to-\nu)\right]\,,$
(4.7)

which does not depend on the choice of $\mathcal{I}_{+}$ or $\mathcal{I}_{-}$,
and here we kept only the non-local contribution.

###  4.2 Bispectrum

Equipped with the mixed propagator, we now proceed to calculate the bispectrum
arising from the isocurvature mode. The full cubic action for both the
adiabatic and isocurvature modes is derived in App. B.4, but most terms are
suppressed by slow-roll parameters. Therefore, we focus on the dominant cubic
interactions, given by Eq. (2.30), where

$\displaystyle V_{N^{3}}$
$\displaystyle=N^{a}N^{b}N^{c}\nabla_{c}\nabla_{b}V_{a}\simeq\frac{(12\lambda\alpha+\xi(1+3\xi))\sqrt{\xi(24\lambda\alpha+\xi(1+6\xi)}}{\alpha\sqrt{2\lambda(4\alpha\lambda+\xi^{2})}}H\,,$
(4.8) $\displaystyle V_{T^{2}N}$
$\displaystyle=\frac{1}{3}\left(T^{a}T^{b}N^{c}+T^{a}N^{b}T^{c}+N^{a}T^{b}T^{c}\right)\nabla_{c}\nabla_{b}V_{a}\simeq\frac{\xi\sqrt{\xi(24\lambda\alpha+\xi(1+6\xi))}}{6\alpha\sqrt{2\lambda(4\alpha\lambda+\xi^{2})}}H\,.$
(4.9)

In this expression, we only keep the terms of the leading order in $1/N_{e}$.
Using the first coupling $V_{N^{3}}$, we obtain the expression

$\displaystyle\left.\left\langle{\hat{\varphi}_{\vec{k}_{1}}\hat{\varphi}_{\vec{k}_{2}}\hat{\varphi}_{\vec{k}_{3}}}\right\rangle^{\prime}\right|_{N^{3}}$
$\displaystyle=\leavevmode\hbox to7.47pt{\vbox
to7.47pt{\pgfpicture\makeatletter\hbox{\hskip 7.34677pt\lower-3.73589pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{ \diagram*{ (f1) -- [very thick] (f5), (f2) -- (v1) --
(f3), (f4) -- (v2), (v1) -- [scalar, quarter left] (v2) -- [scalar, quarter
left] (v1), }; \hss} \diagram*{ (f1) -- [very thick] (f3), (f2) -- (v1) --
[scalar] (c1), }; \hss} \diagram*{ (f1) -- [very thick] (f3), (f2) -- (v1) --
[scalar] (c1), }; \hss} \diagram*{ (f1) -- [very thick] (f2), (z1) --
[insertion={1.0}] (c1) -- [scalar] (v1), (z2) -- [insertion={1.0}] (c2) --
[scalar] (v1), (z3) -- [insertion={1.0}] (c3) -- [scalar] (v1), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.c2)]
\vertex(f1); \vertex[right = 0.5 of f1] (z1); \vertex[right = 0.75 of z1]
(z2); \vertex[right = 0.75 of z2] (z3); \vertex[right = 0.5 of z3] (f2);
\vertex[below = 1.5 of z2] (v1); \vertex[below = 0.75 of z2] (c2);
\vertex[left = 0.375 of c2] (c1); \vertex[right = 0.375 of c2] (c3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.075pt}{0.0pt}\pgfsys@curveto{-0.075pt}{1.95284pt}{-1.65805pt}{3.53589pt}{-3.61089pt}{3.53589pt}\pgfsys@curveto{-5.56372pt}{3.53589pt}{-7.14677pt}{1.95284pt}{-7.14677pt}{0.0pt}\pgfsys@curveto{-7.14677pt}{-1.95284pt}{-5.56372pt}{-3.53589pt}{-3.61089pt}{-3.53589pt}\pgfsys@curveto{-1.65805pt}{-3.53589pt}{-0.075pt}{-1.95284pt}{-0.075pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-3.61089pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61089pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to7.47pt{\vbox to7.47pt{\pgfpicture\makeatletter\hbox{\hskip
7.34677pt\lower-3.73589pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{ (f1) -- [very thick] (f5),
(f2) -- (v1) -- (f3), (f4) -- (v2), (v1) -- [scalar, quarter left] (v2) --
[scalar, quarter left] (v1), }; \hss} \diagram*{ (f1) -- [very thick] (f3),
(f2) -- (v1) -- [scalar] (c1), }; \hss} \diagram*{ (f1) -- [very thick] (f3),
(f2) -- (v1) -- [scalar] (c1), }; \hss} \diagram*{ (f1) -- [very thick] (f2),
(z1) -- [insertion={1.0}] (c1) -- [scalar] (v1), (z2) -- [insertion={1.0}]
(c2) -- [scalar] (v1), (z3) -- [insertion={1.0}] (c3) -- [scalar] (v1), };
\hss} \diagram*{ (f1) -- [very thick] (f2), (z1) -- [insertion={1.0}] (c1) --
[scalar] (v1), (z2) -- [insertion={1.0}] (c2) -- [scalar] (v1), (z3) --
[insertion={1.0}] (c3) -- [scalar] (v1), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.c2)]
\vertex(f1); \vertex[right = 0.5 of f1] (z1); \vertex[right = 0.75 of z1]
(z2); \vertex[right = 0.75 of z2] (z3); \vertex[right = 0.5 of z3] (f2);
\vertex[below = 1.5 of z2] (v1); \vertex[below = 0.75 of z2] (c2);
\vertex[left = 0.375 of c2] (c1); \vertex[right = 0.375 of c2] (c3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.075pt}{0.0pt}\pgfsys@curveto{-0.075pt}{1.95284pt}{-1.65805pt}{3.53589pt}{-3.61089pt}{3.53589pt}\pgfsys@curveto{-5.56372pt}{3.53589pt}{-7.14677pt}{1.95284pt}{-7.14677pt}{0.0pt}\pgfsys@curveto{-7.14677pt}{-1.95284pt}{-5.56372pt}{-3.53589pt}{-3.61089pt}{-3.53589pt}\pgfsys@curveto{-1.65805pt}{-3.53589pt}{-0.075pt}{-1.95284pt}{-0.075pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-3.61089pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61089pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=-\frac{i\pi^{3}H^{3}}{64k_{2}^{3}k_{3}^{3}}\frac{V_{N^{3}}}{H}\left(\frac{\dot{\theta}}{H}\right)^{3}\int_{0}^{\infty}\frac{dz}{z^{4}}\left[\mathcal{I}_{+}(z;\nu)\mathcal{I}_{+}\left(\frac{k_{2}}{k_{1}}z;\nu\right)\mathcal{I}_{+}\left(\frac{k_{3}}{k_{1}}z;\nu\right)-\left(\mathcal{I}_{+}\to\mathcal{I}_{-}\right)\right]\,,$
(4.10)

where the prime indicates that we have removed the factor of
$(2\pi)^{3}\delta^{(3)}(\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3})$. From the second
coupling, we find

$\displaystyle\left.\left\langle{\hat{\varphi}_{\vec{k}_{1}}\hat{\varphi}_{\vec{k}_{2}}\hat{\varphi}_{\vec{k}_{3}}}\right\rangle^{\prime}\right|_{T^{2}N}$
$\displaystyle=\leavevmode\hbox to7.47pt{\vbox
to7.47pt{\pgfpicture\makeatletter\hbox{\hskip 7.34677pt\lower-3.73589pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{ (f1) -- [very
thick] (f5), (f2) -- (v1) -- (f3), (f4) -- (v2), (v1) -- [scalar, quarter
left] (v2) -- [scalar, quarter left] (v1), }; \hss} \diagram*{ (f1) -- [very
thick] (f3), (f2) -- (v1) -- [scalar] (c1), }; \hss} \diagram*{ (f1) -- [very
thick] (f3), (f2) -- (v1) -- [scalar] (c1), }; \hss} \diagram*{ (f1) -- [very
thick] (f2), (z1) -- [insertion={1.0}] (c1) -- [scalar] (v1), (z2) --
[insertion={1.0}] (c2) -- [scalar] (v1), (z3) -- [insertion={1.0}] (c3) --
[scalar] (v1), }; \hss} \diagram*{ (f1) -- [very thick] (f2), (z1) --
[insertion={1.0}] (c1) -- [scalar] (v1), (z2) -- [insertion={1.0}] (c2) --
[scalar] (v1), (z3) -- [insertion={1.0}] (c3) -- [scalar] (v1), }; \hss}
\diagram*{ (f1) -- [very thick] (f2), (z3) -- [insertion={1.0}] (c3) --
[scalar] (v1), (z2) -- (v1), (z1) -- (v1), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.c2)]
\vertex(f1); \vertex[right = 0.5 of f1] (z1); \vertex[right = 0.75 of z1]
(z2); \vertex[right = 0.75 of z2] (z3); \vertex[right = 0.5 of z3] (f2);
\vertex[below = 1.5 of z2] (v1); \vertex[below = 0.75 of z2] (c2);
\vertex[left = 0.375 of c2] (c1); \vertex[right = 0.375 of c2] (c3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.075pt}{0.0pt}\pgfsys@curveto{-0.075pt}{1.95284pt}{-1.65805pt}{3.53589pt}{-3.61089pt}{3.53589pt}\pgfsys@curveto{-5.56372pt}{3.53589pt}{-7.14677pt}{1.95284pt}{-7.14677pt}{0.0pt}\pgfsys@curveto{-7.14677pt}{-1.95284pt}{-5.56372pt}{-3.53589pt}{-3.61089pt}{-3.53589pt}\pgfsys@curveto{-1.65805pt}{-3.53589pt}{-0.075pt}{-1.95284pt}{-0.075pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-3.61089pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61089pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to7.47pt{\vbox to7.47pt{\pgfpicture\makeatletter\hbox{\hskip
7.34677pt\lower-3.73589pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{\hbox to0.0pt{\hbox
to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{\hbox to0.0pt{ \diagram*{
(f1) -- [very thick] (f5), (f2) -- (v1) -- (f3), (f4) -- (v2), (v1) --
[scalar, quarter left] (v2) -- [scalar, quarter left] (v1), }; \hss}
\diagram*{ (f1) -- [very thick] (f3), (f2) -- (v1) -- [scalar] (c1), }; \hss}
\diagram*{ (f1) -- [very thick] (f3), (f2) -- (v1) -- [scalar] (c1), }; \hss}
\diagram*{ (f1) -- [very thick] (f2), (z1) -- [insertion={1.0}] (c1) --
[scalar] (v1), (z2) -- [insertion={1.0}] (c2) -- [scalar] (v1), (z3) --
[insertion={1.0}] (c3) -- [scalar] (v1), }; \hss} \diagram*{ (f1) -- [very
thick] (f2), (z1) -- [insertion={1.0}] (c1) -- [scalar] (v1), (z2) --
[insertion={1.0}] (c2) -- [scalar] (v1), (z3) -- [insertion={1.0}] (c3) --
[scalar] (v1), }; \hss} \diagram*{ (f1) -- [very thick] (f2), (z3) --
[insertion={1.0}] (c3) -- [scalar] (v1), (z2) -- (v1), (z1) -- (v1), }; \hss}
\diagram*{ (f1) -- [very thick] (f2), (z3) -- [insertion={1.0}] (c3) --
[scalar] (v1), (z2) -- (v1), (z1) -- (v1), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.c2)]
\vertex(f1); \vertex[right = 0.5 of f1] (z1); \vertex[right = 0.75 of z1]
(z2); \vertex[right = 0.75 of z2] (z3); \vertex[right = 0.5 of z3] (f2);
\vertex[below = 1.5 of z2] (v1); \vertex[below = 0.75 of z2] (c2);
\vertex[left = 0.375 of c2] (c1); \vertex[right = 0.375 of c2] (c3);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.075pt}{0.0pt}\pgfsys@curveto{-0.075pt}{1.95284pt}{-1.65805pt}{3.53589pt}{-3.61089pt}{3.53589pt}\pgfsys@curveto{-5.56372pt}{3.53589pt}{-7.14677pt}{1.95284pt}{-7.14677pt}{0.0pt}\pgfsys@curveto{-7.14677pt}{-1.95284pt}{-5.56372pt}{-3.53589pt}{-3.61089pt}{-3.53589pt}\pgfsys@curveto{-1.65805pt}{-3.53589pt}{-0.075pt}{-1.95284pt}{-0.075pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-3.61089pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61089pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>$\displaystyle=-\frac{i\pi
H^{3}}{16k_{2}^{3}k_{3}^{3}}\frac{V_{T^{2}N}}{H}\frac{\dot{\theta}}{H}\int_{0}^{\infty}\frac{dz}{z^{4}}\,\left[(1+iz)\left(1+i\frac{k_{2}}{k_{1}}z\right)\mathcal{I}_{+}\left(\frac{k_{3}}{k_{1}}z;\nu\right)e^{-i\left(1+\frac{k_{2}}{k_{1}}\right)z}\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-(1-iz)\left(1-i\frac{k_{2}}{k_{1}}z\right)\mathcal{I}_{-}\left(\frac{k_{3}}{k_{1}}z;\nu\right)e^{i\left(1+\frac{k_{2}}{k_{1}}\right)z}+\mathrm{perm.}\right],$
(4.11)

where “perm.” indicates the permutations $k_{3}\leftrightarrow k_{1}$ and
$k_{3}\leftrightarrow k_{2}$, which become negligible in the squeezed limit
$k_{3}\to 0$. The first expression serves as an explicit example of
contributions discussed in [44, 45], while the second corresponds to those
discussed in [80]. In the squeezed limit, $k_{3}\ll k_{1},k_{2}$, these
bispectra simplify to

$\displaystyle\left.\left\langle{\hat{\varphi}_{\vec{k}_{1}}\hat{\varphi}_{\vec{k}_{2}}\hat{\varphi}_{\vec{k}_{3}}}\right\rangle^{\prime}\right|_{N^{3}}$
$\displaystyle=\frac{2\pi
H^{3}}{k_{1}^{3}k_{3}^{3}}\frac{V_{N^{3}}}{H}\left(\frac{\dot{\theta}}{H}\right)^{3}C_{N^{3}}(\nu)\left(\frac{k_{3}}{k_{1}}\right)^{3/2-\nu}+(\nu\to-\nu)\,,$
(4.12)

where

$\displaystyle
C_{N^{3}}(\nu)\;=\;\frac{i\pi^{3/2}2^{\nu-6}\Gamma(\nu)}{\cos(\pi\nu/2)+\sin(\pi\nu/2)}\int_{0}^{\infty}dz\,z^{-(5/2+\nu)}\left[\mathcal{I}_{+}^{2}(z;\nu)-\mathcal{I}_{-}^{2}(z;\nu)\right]\,,$
(4.13)

for the former coupling, and

$\displaystyle\left.\left\langle{\hat{\varphi}_{\vec{k}_{1}}\hat{\varphi}_{\vec{k}_{2}}\hat{\varphi}_{\vec{k}_{3}}}\right\rangle^{\prime}\right|_{T^{2}N}$
$\displaystyle=\frac{2\pi
H^{3}}{k_{1}^{3}k_{3}^{3}}\frac{V_{T^{2}N}}{H}\frac{\dot{\theta}}{H}C_{T^{2}N}(\nu)\left(\frac{k_{3}}{k_{1}}\right)^{3/2-\nu}+(\nu\to-\nu)\,,$
(4.14)

where

$\displaystyle C_{T^{2}N}(\nu)$
$\displaystyle=\frac{2^{\nu-4}(5-2\nu)}{\sqrt{\pi}(3+2\nu)}\Gamma(\nu)\Gamma\left(\frac{1}{2}-\nu\right)\tan\left(\frac{\pi(1-2\nu)}{4}\right)\,,$
(4.15)

for the latter coupling. In particular, the conformal time integral can be
analytically computed in the latter case. The corresponding non-Gaussianity
function is given by

$\displaystyle S_{\mathrm{NG}}$
$\displaystyle=P_{\zeta}^{-1/2}\left[\frac{V_{N^{3}}}{H}\left(\frac{\dot{\theta}}{H}\right)^{3}C_{N^{3}}(\nu)+\frac{V_{T^{2}N}}{H}\frac{\dot{\theta}}{H}C_{T^{2}N}(\nu)\right]\left(\frac{k_{3}}{k_{1}}\right)^{1/2-\nu}+(\nu\to-\nu)\,,$
(4.16)

where we assumed $\dot{\phi}_{0}<0$ so that
$P_{\zeta}^{-1/2}=-2\pi\dot{\phi}_{0}/H^{2}$. This accurately reproduces our
estimate, given by Eq. (2.32), up to the numerical coefficients $C_{N^{3}}$
and $C_{T^{2}N}$. We represent the coefficient of the non-Gaussianity function
as

$\displaystyle
f_{\mathrm{NL}}\;=\;\frac{10}{9}P_{\zeta}^{-1/2}\left[\frac{V_{N^{3}}}{H}\left(\frac{\dot{\theta}}{H}\right)^{3}C_{N^{3}}(\nu)+\frac{V_{T^{2}N}}{H}\frac{\dot{\theta}}{H}C_{T^{2}N}(\nu)\right]\,,$
(4.17)

which characterizes the size of the non-
Gaussianity.\natural13\natural13$\natural$13 This coefficient is motivated by
the normalization given by Ref. [8]. However, this is just a convention since
we evaluate non-Gaussianity only in the squeezed limit and not in the
equilateral configuration.  However, when $m/H>3/2$, the coefficient $\nu$
becomes imaginary, and this quantity cannot be evaluated at the equilateral
configuration.

Figure 2: _Left panel:_ $C_{N^{3}}(\nu)$ and $C_{T^{2}N}(\nu)$ as functions of
$m_{\chi}/H$. Note that $\nu$ is real for $m_{\chi}/H<3/2$ and it is purely
imaginary for $m_{\chi}/H>3/2$. We show only the cases where $\nu>0$ for
$m_{\chi}/H<3/2$ because this contribution is less suppressed by $k_{3}/k_{1}$
in the squeezed limit. On the other hand, for $m_{\chi}/H>3/2$, the
coefficients $C_{N^{3}}(-\nu)$ and $C_{T^{2}N}(-\nu)$ can be related to the
complex conjugated functions $C_{N^{3}}^{*}(\nu)$ and $C_{T^{2}N}^{*}(\nu)$.
_Right panel:_ The coefficients of the non-Gaussianity function for different
values of $\xi$ and $\alpha$, with $\lambda$ fixed using the CMB normalization
(2.17) with $N_{e}=60$ e-folds (ignoring the $c^{2}$ contribution). The small
region, where $|f_{\mathrm{NL}}|>1$ with $\xi\sim 0.08$, corresponds to
$m_{\chi}/H\simeq 3/2$, and for larger $\xi$, $m_{\chi}/H>3/2$, and the signal
exhibits oscillatory behavior with respect to $k_{3}/k_{1}$. We focus on the
region $m_{\chi}/H>1$ and $|\dot{\theta}/H|<1$ so that the model is in the
quasi-single field regime. Notably, these two conditions imply that
$|\dot{\theta}/m_{\chi}|<1$.

In the left panel of Fig. 2, we display the coefficients $C_{N^{3}}(\nu)$ and
$C_{T^{2}N}(\nu)$ as functions of $m_{\chi}/H$. We note that the parameter
$\nu$ is real for $m_{\chi}/H<3/2$ and purely imaginary for $m_{\chi}/H>3/2$.
These function are exponentially suppressed for large values of $m_{\chi}/H$,
aligning with the expectation that cosmological collider signatures arise from
the non-local propagation of particles. In the large mass limit $m_{\chi}\gg
H$, correlation functions usually include two contributions: one power-
suppressed and the other exponentially suppressed by the mass term. The former
corresponds to higher-dimensional operators and gives rise only to local
effects (see, for example, Refs. [82, 83]), while the latter is associated
with particle production and, consequently, cosmological collider signatures.

In the right panel of Fig. 2, we show the non-Gaussianity coefficient
$|f_{\mathrm{NL}}|$ as a function of $\xi$ and $\alpha$. In this figure, we
set $\lambda$ according to the CMB normalization (2.17) (with $c^{2}$
ignored). As is evident from the figure, the coefficient can attain values as
large as order unity while satisfying the conditions $m_{\chi}/H>1$ and
$|\dot{\theta}/H|<1$,\natural14\natural14$\natural$14 For a larger value of
the turning rate $\dot{\theta}$, the isocurvature mode can be classically
excited, and the inflationary trajectory can be oscillatory, giving rise to
unique features in the curvature perturbation [84, 85, 86]. This is distinct
from the cosmological collider signatures arising from quantum particle
production, and we do not consider it here.  so that the model is in the
quasi-single field inflation regime. Therefore, this signal is potentially
detectable by future 21 cm observations [55, 56] within this specific
parameter space region.

In Fig. 3, we plot the full non-Gaussianity function in the squeezed limit
(4.16) for different values of $m_{\chi}/H$. Here we consider a choice of
Higgs-$R^{2}$ inflation parameters in the range $0.04\lesssim\xi\lesssim 0.4$,
$4\times 10^{8}\lesssim\alpha\lesssim 6\times 10^{8}$, and using the CMB
normalization (2.17) (where we ignore the $c^{2}$ contribution) with
$N_{e}=60$ e-folds, we determine the value of $\lambda$ and the mass of the
isocurvature mode (2.23). The left panel of Fig. 2 illustrates that the signal
peaks when $|C_{N^{3}}|\sim\mathcal{O}(1)$, with $m_{\chi}/H\sim 1$. However,
we show in Fig. 3, that when $m_{\chi}/H>3/2$, the parameter $\nu$ becomes
imaginary and the squeezed limit displays an oscillatory feature, leading to a
distinct clock signal. However, this signal has a reduced non-Gaussianity
amplitude compared to the $m_{\chi}\simeq H$ case. We highlight these
important oscillatory features in non-Gaussianities across three panels, where
$\nu$ is imaginary for the values $m_{\chi}/H=2,3,$ and $4$. For a more
detailed discussion of clock signals, see Ref. [80].

Figure 3: The non-Gaussianity function in the squeezed limit (4.16) as a
function of $k_{1}/k_{3}$ for the mass range $m_{\chi}/H=1,2,3,$ and $4$ . To
enhance the oscillatory components of the clock signals, we scale $S_{\rm NG}$
by a factor of $k_{1}/k_{3}$. We consider a nominal choice of the
Higgs-$R^{2}$ inflation parameters in the range $0.04\lesssim\xi\lesssim 0.4$,
$4\times 10^{8}\lesssim\alpha\lesssim 6\times 10^{8}$, where $\lambda$ is
fixed by the CMB normalization (2.17) (where we ignore the $c^{2}$
contribution) with $N_{e}=60$ e-folds, and the mass of the isocurvature mode
is determined from Eq. (2.23). Although the signal strength peaks when
$m_{\chi}\sim H$ and $\nu$ is real, which maximizes the value of the coupling
$|C_{N^{3}}(\nu)|$, the oscillatory behavior is only evident for the imaginary
values of $\nu$, with $m_{\chi}/H=2,3,$ and $4$. The panels also illustrate
that the $N^{3}$ coupling is always more significant than the $T^{2}N$
coupling.

Finally, we comment on the trispectrum. In Higgs-$R^{2}$ inflation, we find
that the quartic couplings of the adiabatic and isocurvature modes are given
by

$\displaystyle S_{\mathrm{quartic}}\;=\;\int d\tau
d^{3}x\,a^{4}\left[\frac{1}{24}V_{N^{4}}\chi^{4}+\frac{1}{4}V_{T^{2}N^{2}}\chi^{2}\varphi^{2}\right]\,,$
(4.18)

where

$\displaystyle V_{N^{4}}$
$\displaystyle\simeq\frac{7+3\xi(14+9\xi)}{18\alpha}+\lambda\left(6-\frac{14}{9(\xi^{2}+4\lambda\alpha)}\right),\quad
V_{T^{2}N^{2}}\simeq\frac{\xi(132\lambda\alpha+\xi(7+33\xi)}{108\alpha(\xi^{2}+4\lambda\alpha))},$
(4.19)

with the other combinations suppressed by the slow-roll parameters. These
couplings both scale as $V_{N^{4}}\sim V_{T^{2}N^{2}}\sim 1/\alpha$ for
$\xi\sim\lambda\alpha\sim\mathcal{O}(1)$. Therefore, an estimation similar to
Sec. 2.4 provides us with

$\displaystyle g_{\mathrm{NL}},~{}\tau_{\mathrm{NL}}\sim
P_{\zeta}^{-1}\frac{V_{N^{4}}}{H}\left(\frac{\dot{\theta}}{H}\right)^{4},~{}~{}P_{\zeta}^{-1}\frac{V_{T^{2}N^{2}}}{H}\left(\frac{\dot{\theta}}{H}\right)^{2}.$
(4.20)

We expect a similar size of the signal from the combination of the cubic
interactions. Since $P_{\zeta}\times\alpha\sim 1$, this can be of order unity
for $\xi\sim\lambda\alpha\sim 1$, and hence we expect a sizable trispectrum in
the parameter region of our interest in this section. We leave a detailed
study on the trispectrum, including its precise size and spectral feature in
the squeezed limit, as a possible future work.

##  5 Conclusions and Discussion

In this work, we examined and computed the cosmological collider signals for
Higgs-$R^{2}$ inflation. We considered two distinct types of potential
signals. The first originates from the inflaton coupling to the SM fermions
and gauge bosons through the coupling $h/\Omega$. The second is an
isocurvature mode coupling that couples to the inflaton through the turning
rate $\dot{\theta}$.

We found that the cosmological collider signatures from the SM fermions and
gauge bosons are relatively weak due to their Planck suppression and further
suppression by slow-roll parameters, or the number of e-folds $N_{e}$.
Consequently, they are unlikely to be detected even by forthcoming 21 cm
probes. However, a considerably stronger signal might emerge from the
isocurvature mode and inflaton couplings. In the parameter space, where
$\xi\sim\lambda\alpha\sim\mathcal{O}(0.1)$, the isocurvature mode remains
light, and the turning rate $\dot{\theta}$, and subsequently coupling to the
inflaton, is large. This parameter region aligns with the quasi-single
inflation regime of Higgs-$R^{2}$ inflation, with parameters spanning from
$0.5\lesssim\xi\lesssim 2$ and $4\times 10^{8}\lesssim\alpha\lesssim 6\times
10^{8}$. Importantly, in this scenario, the non-Gaussianity could be
significant, with $|f_{\rm NL}|\sim\mathcal{O}(1)$, peaking when
$m_{\chi}/H\simeq 3/2$. Although the isocurvature mode contribution to $f_{\rm
NL}$ is large in this parameter space, it might be difficult for future 21 cm
observations to isolate it from the background. However, when
$m_{\chi}/H\gtrsim 3/2$, the cubic interaction of the isocurvature mode, and
consequently $f_{\rm NL}$, still remain nearly of order unity. This
oscillatory clock signal might be distinguishable from the 21 cm background,
offering hope for detection by future experiments.

We note that detecting a signal originating from the isocurvature mode
couplings could provide strong evidence for multi-field models of inflation.
The Higgs-$R^{2}$ model remains a highly appealing scenario due to several of
its features, including a UV-completion. It would be interesting to explore
the full features and the parameter space of a multi-field Higgs-$R^{2}$ model
and its associated cosmological collider signatures. However, such a study is
quite involved, and we hope to investigate it in future work.

In this paper, we only focused on the SM particles. However, once we introduce
new particles, the inflaton typically couples to them through the conformal
factor, $\Omega$, unless these new particles are conformal. For example, the
inflaton can couple to right-handed neutrinos via the Yukawa interaction and
Majorana mass terms. Our estimates from Sec. 2.4 likely remain valid in this
scenario, with $c_{1},c_{2}\sim\mathcal{O}(1)$. Therefore, we do not expect
that these new particles would significantly enhance the cosmological collider
signatures in Higgs-$R^{2}$ inflation. The situation might change if a
stronger coupling between the inflaton and the new particles is introduced,
although this might spoil the UV-completeness of the model by introducing an
additional scale.

Another intriguing question is if there are variants of Higgs inflation where
the SM fermions and gauge bosons leave observable cosmological collider
signatures. One such variant of Higgs inflation scenario, known as Palatini
Higgs inflation [87, 88, 89], has recently attracted attention. This model has
the same action as the original Higgs inflation model but relies on the
Palatini formalism of gravity, where the spin connection and vierbein are
treated independently. This model avoids the unitarity issues both during [88]
and after inflation [90, 23, 91], in contrast to the metric formalism. In
general, using the same arguments as in Sec. 2.4, we expect that the non-
Gaussianity arising from the SM particles takes the form

$\displaystyle f_{\mathrm{NL}}\;\sim\;\frac{\dot{\phi}_{0}H}{\Lambda^{3}}\sim
3.5\times 10^{3}\frac{H^{3}}{\Lambda^{3}}\,,$ (5.1)

where $\Lambda$ is the scale of the inflaton coupling to the SM particles. In
Palatini Higgs inflation, the cut-off scale is given by $\Lambda\sim
M_{P}/\sqrt{\xi}$, which is significantly smaller than the Planck scale since
from the CMB normalization, we find $\xi\sim 5\times 10^{10}\lambda$. However,
in this model the Hubble parameter is also relatively small, with
$H\sim\sqrt{\lambda}M_{P}/\xi$. Therefore, the size of non-Gaussianity can be
estimated as

$\displaystyle f_{\mathrm{NL}}\;\sim\;3.5\times
10^{3}\times\left(\frac{\lambda}{\xi}\right)^{3/2}\sim 10^{-13}\,,$ (5.2)

which is too small to be observed in the near future. Nonetheless, it is still
interesting to explore if there are other variants that predict a larger
cosmological collider signal.

##### Acknowledgements —

The work of Y.E. and S.V. was performed in part at the Aspen Center for
Physics, which is supported by National Science Foundation grant PHY-2210452.
We would also like to thank Soubhik Kumar and Yiming Zhong for useful
discussions. Y.E. is supported in part by DOE grant DE-SC0011842. The work of
S.V. was supported in part by DOE grant DE-SC0022148. The Feynman diagrams in
this paper are drawn with TikZ-Feynman [92].

##  Appendix A Conventions and notation

###  A.1 Conventions

Here we summarize the conventions used in this paper.

#### Metric sector

We take almost the plus convention,

$\displaystyle\eta_{\mu\nu}\;=\;\mathrm{diag}(+1,-1,-1,-1)\,.$ (A.1)

The geometrical quantities are defined as

$\displaystyle\Gamma^{\mu}_{\nu\rho}$
$\displaystyle=\frac{1}{2}g^{\mu\alpha}\left(\partial_{\nu}g_{\rho\alpha}+\partial_{\rho}g_{\nu\alpha}-\partial_{\alpha}g_{\nu\rho}\right),\quad{R_{\mu\nu\rho}}^{\sigma}=\partial_{\mu}\Gamma^{\sigma}_{\nu\rho}-\partial_{\nu}\Gamma^{\sigma}_{\mu\rho}+\Gamma^{\alpha}_{\nu\rho}\Gamma^{\sigma}_{\alpha\mu}-\Gamma^{\alpha}_{\mu\rho}\Gamma^{\sigma}_{\alpha\nu},$
(A.2) $\displaystyle R_{\mu\nu}$
$\displaystyle={R_{\mu\alpha\nu}}^{\alpha}=\partial_{\mu}\Gamma^{\alpha}_{\alpha\nu}-\partial_{\alpha}\Gamma^{\alpha}_{\mu\nu}+\Gamma^{\alpha}_{\beta\nu}\Gamma^{\beta}_{\alpha\mu}-\Gamma^{\alpha}_{\mu\nu}\Gamma^{\beta}_{\alpha\beta},\quad
R=g^{\mu\nu}R_{\mu\nu}\,.$ (A.3)

With this definition, the sign in front of the Einstein-Hilbert action is
positive, and the conformal coupling corresponds to $\xi=-1/6$. The Ricci
scalar transforms under the Weyl transformation as

$\displaystyle{g}_{\mu\nu}\to\Omega^{-2}g_{\mu\nu},\quad{R}\to\Omega^{2}\left[R+\frac{3}{2}(\partial_{\mu}\ln\Omega^{2})(\partial^{\mu}\ln\Omega^{2})-3\Box\ln\Omega^{2}\right]\,.$
(A.4)

Finally the commutator acting on a vector field $V^{\alpha}$ satisfies

$\displaystyle[\nabla_{\mu},\nabla_{\nu}]V^{\alpha}={R_{\mu\nu\beta}}^{\alpha}V^{\beta}.$
(A.5)

#### Target space sector

Since the target space metric is positive definite, we may change the sign
convention from the metric sector. The Christoffel symbol is defined as

$\displaystyle\Gamma^{a}_{bc}\;=\;\frac{1}{2}h^{ad}\left(h_{db,c}+h_{dc,b}-h_{bc,d}\right)\,,$
(A.6)

and the Riemann tensor is defined as

$\displaystyle{\mathcal{R}^{a}}_{bcd}\;=\;\partial_{c}\Gamma^{a}_{db}-\partial_{d}\Gamma^{a}_{cb}+\Gamma^{a}_{ce}\Gamma^{e}_{db}-\Gamma^{a}_{de}\Gamma^{e}_{cb}\,,$
(A.7)

which has the opposite sign convention from the metric sector. The Ricci
tensor and Ricci scalar are defined as

$\displaystyle\mathcal{R}_{ab}\;=\;h^{cd}\mathcal{R}_{acbd}\,,\quad\mathcal{R}\;=\;h^{ab}\mathcal{R}_{ab}\,.$
(A.8)

#### Fermion sector

The covariant derivative in the fermion sector is given by

$\displaystyle\nabla_{\mu}\psi$
$\displaystyle=\left(\partial_{\mu}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\right)\psi.$
(A.9)

We define the gamma matrices as

$\displaystyle\left\\{\gamma^{a},\gamma^{b}\right\\}$
$\displaystyle=2\eta^{ab},\quad\gamma_{ab}=\frac{1}{2}\left(\gamma_{a}\gamma_{b}-\gamma_{b}\gamma_{a}\right),\quad\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-\frac{i}{4!}\epsilon^{abcd}\gamma_{a}\gamma_{b}\gamma_{c}\gamma_{d},$
(A.10)

with $\epsilon^{0123}=+1$. We use $a,b,\cdots$ for the local Lorentz indices
and $\mu,\nu,\cdots$ for the spacetime indices. The spin connection and the
vierbein are defined as

$\displaystyle\omega_{\mu}^{ab}$
$\displaystyle=e^{a}_{\nu}\left[\partial_{\mu}e^{\nu
b}+{\Gamma^{\nu}}_{\sigma\mu}e^{\sigma b}\right],\quad
g_{\mu\nu}=e_{\mu}^{a}\eta_{ab}e_{\nu}^{b}.$ (A.11)

Under the Weyl transformation, the fermion kinetic term transforms as

$\displaystyle{\not{\nabla}}{\psi}\to\Omega^{5/2}\not{\nabla}\left(\Omega^{-3/2}\psi\right)\,,$
(A.12)

in four spacetime dimensions.

###  A.2 ADM decomposition

Here we may explain the $3+1$ decomposition of the Ricci scalar that we use in
the computation of the perturbation. We may use the ADM decomposition [93]

$\displaystyle
ds^{2}\;=\;N^{2}dt^{2}-\gamma^{ij}(dx^{i}+\beta^{i}dt)(dx^{j}+\beta^{j}dt)\,,$
(A.13)

or equivalently

$\displaystyle
g_{\mu\nu}\;=\;\begin{pmatrix}N^{2}-\beta^{i}\beta_{i}&-\beta_{i}\\\
-\beta_{j}&-\gamma_{ij}\end{pmatrix}\,,\quad
g^{\mu\nu}\;=\;\begin{pmatrix}N^{-2}&-N^{-2}\beta^{i}\\\
-N^{-2}\beta^{j}&-\gamma^{ij}+N^{-2}\beta^{i}\beta^{j}\end{pmatrix}\,,$ (A.14)

where the spatial indices $i,j$ are raised and lowered by $\gamma_{ij}$. We
may define the temporal part of the vierbein $n_{\mu}$ and the extrinsic
curvature $K_{\mu\nu}$ as

$\displaystyle n_{\mu}\;\equiv\;(N,\vec{0}),\quad
K_{\mu\nu}\;=\;\nabla_{\nu}n_{\mu}\,.$ (A.15)

Our goal is to express the Ricci scalar in terms of $K_{\mu\nu}$ and the
three-dimensional curvature tensors. For this purpose, we first take the Gauss
normal coordinate, $N=1$ and $\beta^{i}=0$, and compute everything in these
coordinates. We then express the result by the covariant quantities so that we
can go back to the general coordinate. In the Gauss normal coordinates, we can
easily show that

$\displaystyle\Gamma^{0}_{00}\;=\;\Gamma^{0}_{0i}=\Gamma^{i}_{00}=0\,,\quad\Gamma^{0}_{ij}\;=\;\frac{1}{2}\dot{\gamma}_{ij}=-K_{ij}\,,\quad\Gamma^{i}_{0j}\;=\;\frac{1}{2}\gamma^{ik}\dot{\gamma}_{jk}\;=\;-{K^{i}}_{j}\,.$
(A.16)

The Ricci scalar is decomposed as

$\displaystyle R\;=\;2{R_{0\alpha 0}}^{\alpha}-\gamma^{ij}{R_{ikj}}^{k}\,.$
(A.17)

The first term is equally expressed as

$\displaystyle{R_{0\alpha
0}}^{\alpha}\;=\;n^{\mu}n^{\nu}{R_{\mu\alpha\nu}}^{\alpha}=n^{\mu}[\nabla_{\mu},\nabla_{\nu}]n^{\nu}\,,$
(A.18)

which is now in covariant form. The second term is also easily computed as

$\displaystyle-\gamma^{ij}{R_{ikj}}^{k}=R^{(3)}-K_{ij}K^{ij}+K^{2},$ (A.19)

where $K=\gamma^{ij}K_{ij}$. We further note that

$\displaystyle n^{\mu}K_{\mu\rho}\;=\;n^{\mu}\nabla_{\rho}n_{\mu}=0\,.$ (A.20)

Therefore we obtain after integration by parts

$\displaystyle
R\;=\;R^{(3)}+K_{ij}K^{ij}-K^{2}+(\mathrm{total}~{}\mathrm{deriv})\,,$ (A.21)

which is the desired $3+1$ decomposition of the Ricci scalar. Following [78],
we define

$\displaystyle K_{ij}\;=\;-\frac{1}{N}E_{ij}\,.$ (A.22)

In the general ADM coordinate, it is given by

$\displaystyle
K_{ij}\;=\;-\frac{1}{2N}\left(\dot{\gamma}_{ij}-\nabla_{j}^{(3)}\beta_{i}-\nabla_{i}^{(3)}\beta_{j}\right)\,.$
(A.23)

##  Appendix B Covariant formalism of multi-field inflation

In this appendix, we review the covariant formalism of multi-field inflation
[69, 70, 71, 72, 73, 74, 75].

###  B.1 Background dynamics

We first consider the background dynamics. We take

$\displaystyle
ds^{2}\;=\;N^{2}(t)dt^{2}-a^{2}(t)dx^{i}dx^{i}\,,\quad\phi^{a}\;=\;\phi^{a}_{0}(t)\,.$
(B.1)

The Ricci scalar is given by

$\displaystyle
R\;=\;-\frac{6H^{2}}{N^{2}}+(\mathrm{total}~{}\mathrm{deriv}),\quad
H=\frac{\dot{a}}{a}\,.$ (B.2)

By taking the derivative of the action with respect to $\phi^{a}$, $N$, and
$a$, and then setting $N=1$, we obtain

$\displaystyle\mathcal{D}_{t}\dot{\phi}^{a}_{0}+3H\dot{\phi}_{0}^{a}+\partial^{a}V\;=\;0\,,\quad
H^{2}\;=\;\frac{1}{3M_{P}^{2}}\left(\frac{1}{2}h_{ab}\dot{\phi}^{a}_{0}\dot{\phi}^{b}_{0}+V\right)\,,\quad\dot{H}\;=\;-\frac{1}{2M_{P}^{2}}h_{ab}\dot{\phi}_{0}^{a}\dot{\phi}_{0}^{b}\,,$
(B.3)

where the covariant derivative is defined as

$\displaystyle\mathcal{D}_{t}X^{a}\;=\;\dot{X}^{a}+\Gamma^{a}_{bc}\dot{\phi}_{0}^{b}X^{c}\,,$
(B.4)

with $\Gamma^{a}_{bc}$ being the Christoffel symbol constructed from $h_{ab}$.
We define

$\displaystyle
T^{a}\;\equiv\;\frac{\dot{\phi}_{0}^{a}}{\dot{\phi}_{0}}\,,\quad\mathcal{D}_{t}T^{a}\;\equiv\;-\dot{\theta}N^{a}\,,\quad\dot{\phi}_{0}\;=\;\sqrt{h_{ab}\dot{\phi}_{0}^{a}\dot{\phi}_{0}^{b}}\,,$
(B.5)

where $T^{a}$ indicates the direction of the inflationary trajectory while
$N^{a}$ the orthogonal direction with the normalization $N^{a}N_{a}=1$. The
turning rate $\dot{\theta}$ parametrizes the curvature of the inflationary
trajectory, or the mixing between the adiabatic and isocurvature modes as we
will see. By using the equation of motion of $\phi_{0}^{a}$, we see that

$\displaystyle\mathcal{D}_{t}T^{a}\;=\;-\frac{T^{a}}{\dot{\phi}_{0}}\left(\ddot{\phi}_{0}+3H\dot{\phi}_{0}+V_{\phi}\right)-\frac{1}{\dot{\phi}_{0}}\left(V^{a}-T^{a}V_{\phi}\right)\,,\quad
V_{\phi}\;=\;T^{a}V_{a}\,,$ (B.6)

where $V^{a}=\partial^{a}V$ and so on. Since $T^{a}$ is normalized, we have
$T_{a}(DT^{a}/dt)=0$, from which we obtain

$\displaystyle\ddot{\phi}_{0}+3H\dot{\phi}_{0}+V_{\phi}$
$\displaystyle=0,\quad\dot{\theta}N^{a}=\frac{1}{\dot{\phi}_{0}}\left(V^{a}-T^{a}V_{\phi}\right)\,.$
(B.7)

The latter tells us that

$\displaystyle V_{a}\;=\;N_{a}V_{N}+T_{a}V_{\phi}\,,\quad
V_{N}\;=\;N^{a}V_{a}\,,\quad\dot{\theta}\;=\;\frac{V_{N}}{\dot{\phi}_{0}}\,,\quad
N^{a}\;=\;\frac{V^{a}-T^{a}V_{\phi}}{\sqrt{h_{ab}(V^{a}-T^{a}V_{\phi})(V^{b}-T^{b}V_{\phi})}}\,.$
(B.8)

Note that $\dot{\theta}$ depends on the derivative of the potential in the
$N$-direction, not in the $T$-direction, and thus this can be large during
slow-roll inflation. We define the slow-roll parameters as

$\displaystyle\epsilon\;=\;-\frac{\dot{H}}{H^{2}}\;=\;\frac{\dot{\phi}_{0}^{2}}{2M_{P}^{2}H^{2}}\,,\quad\eta^{a}\;=\;-\frac{1}{H\dot{\phi}_{0}}\mathcal{D}_{t}\dot{\phi}_{0}^{a}\,.$
(B.9)

By using the equation of motion of $\phi_{0}^{a}$, we can show that

$\displaystyle\eta^{a}\;=\;\eta_{\parallel}T^{a}+\eta_{\perp}N^{a},\quad\eta_{\parallel}\;=\;-\frac{\ddot{\phi}_{0}}{H\dot{\phi}_{0}}\,,\quad\eta_{\perp}\;=\;\frac{\dot{\theta}}{H}\;=\;\frac{V_{N}}{H\dot{\phi}_{0}}\,.$
(B.10)

The slow-roll condition requires $\epsilon,\eta_{\parallel}\ll 1$, but not
that $\eta_{\perp}$ is small. Finally, we derive the time evolution of
$N^{a}$. We note that

$\displaystyle\frac{d}{dt}\left(T_{a}N^{a}\right)\;=\;0\;=\;-\dot{\theta}+T_{a}\mathcal{D}_{t}N^{a},\quad
N_{a}\mathcal{D}_{t}N^{a}\;=\;0\,,$ (B.11)

and thus we can write down

$\displaystyle\mathcal{D}_{t}N^{a}\;=\;\dot{\theta}T^{a}+\xi^{a}\,,$ (B.12)

where $\xi^{a}$ satisfies

$\displaystyle P^{ab}\xi_{b}\;=\;\xi^{a}\,,\quad
P_{ab}\;=\;h_{ab}-T_{a}T_{b}-N_{a}N_{b}\,.$ (B.13)

It is then straightforward to show that

$\displaystyle
P_{ab}\mathcal{D}_{t}^{2}T^{b}\;=\;-P_{ab}\nabla_{\phi}V^{b}\,,\quad\nabla_{\phi}\;=\;T^{c}\nabla_{c}\,,$
(B.14)

and thus we obtain

$\displaystyle\xi^{a}\;=\;\frac{1}{H\eta_{\perp}}P^{ab}\nabla_{\phi}V_{b}\,.$
(B.15)

In summary, the background equations of motion are governed by

$\displaystyle\ddot{\phi}_{0}+3H\dot{\phi}_{0}+V_{\phi}\;=\;0\,,\quad
H^{2}\;=\;\frac{1}{3M_{P}^{2}}\left(\frac{1}{2}\dot{\phi}_{0}^{2}+V\right)\,,\quad\dot{H}\;=\;-\frac{\dot{\phi}_{0}^{2}}{2M_{P}^{2}}\,,$
(B.16) $\displaystyle
T^{a}\equiv\frac{\dot{\phi}_{0}^{a}}{\dot{\phi}_{0}}\,,\quad\mathcal{D}_{t}T^{a}\;=\;-\dot{\theta}N^{a}\;=\;-H\eta_{\perp}N^{a}\,,\quad\mathcal{D}_{t}N^{a}\;=\;H\eta_{\perp}T^{a}+\frac{1}{H\eta_{\perp}}P^{ab}\nabla_{\phi}V_{b}\,,$
(B.17)
$\displaystyle\epsilon\;=\;-\frac{\dot{H}}{H^{2}}\,,\quad\eta^{a}\;=\;-\frac{1}{H\dot{\phi}_{0}}\mathcal{D}_{t}\dot{\phi}_{0}^{a}\;=\;\eta_{\parallel}T^{a}+\eta_{\perp}N^{a}\,,\quad\eta_{\parallel}\;=\;-\frac{\ddot{\phi}_{0}}{H\dot{\phi}_{0}}\,,\quad\eta_{\perp}\;=\;\frac{V_{N}}{H\dot{\phi}_{0}}\,.$
(B.18)

The equations are not closed if we have more than two-fields, as the equation
of motion of $N^{a}$ includes the vector not spanned by $T^{a}$ nor $N^{a}$.
However, Higgs-$R^{2}$ inflation has only two fields, the scalaron and the
radial mode of the Higgs (Goldstone modes, or equivalently, the longitudinal
gauge bosons, are treated separately). In this case, $P_{ab}=0$, and the above
equations are closed.

###  B.2 Perturbation: general discussion

Next we discuss the perturbation around the above background in the covariant
formalism. We may use the 3+1 decomposition in the ADM formalism discussed in
App. A.2. By using the decomposition of the Ricci scalar, the action is given
by

$\displaystyle S\;=\;\int d^{4}x\sqrt{\gamma}$
$\displaystyle\left[\frac{M_{P}^{2}}{2}NR^{(3)}+\frac{M_{P}^{2}}{2N}\left(E_{ij}E^{ij}-E^{2}\right)+\frac{N}{2}h_{ab}g^{\mu\nu}\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}-NV\right]\,,$
(B.19)

where the extrinsic curvature is given by

$\displaystyle
E_{ij}\;=\;\frac{1}{2}\left(\dot{\gamma}_{ij}-\nabla_{i}^{(3)}\beta_{j}-\nabla_{j}^{(3)}\beta_{i}\right)\,,$
(B.20)

and the quantities with the superscript “$(3)$” are constructed from the
spatial metric $\gamma_{ij}$. We take the flat gauge and expand the metric as

$\displaystyle
N\;=\;1+\alpha\,,\quad\beta_{i}\;=\;a^{2}\partial_{i}\beta\,,\quad\gamma_{ij}\;=\;a^{2}\delta_{ij}\,,$
(B.21)

where we ignore the vector and tensor parts.

Since we expand the fields up to the third order, extra care is required when
expanding the scalar fields. Our goal is to express everything in terms of
geometrical quantities up to the third order, as discussed in [74, 94]. We may
think of $\phi^{a}$ as connected from $\phi^{a}_{0}$ by the geodesics with an
appropriate initial velocity $\varphi^{a}$. Thus, we expand the fields as

$\displaystyle\phi^{a}\;=\;\left[\phi^{a}_{0}+\frac{d\phi^{a}}{d\lambda}+\frac{1}{2}\frac{d^{2}\phi^{a}}{d\lambda^{2}}+\frac{1}{6}\frac{d^{3}\phi^{a}}{d\lambda^{3}}+\cdots\right]_{\lambda=0}\,,$
(B.22)

where $\lambda$ is the affine parameter and the geodesic satisfies

$\displaystyle\mathcal{D}_{\lambda}\frac{d\phi^{a}}{d\lambda}\;\equiv\;\frac{d^{2}\phi^{a}}{d\lambda^{2}}+\Gamma^{a}_{bc}\frac{d\phi^{b}}{d\lambda}\frac{d\phi^{c}}{d\lambda}=0\,.$
(B.23)

We take $\varphi^{a}$ as the initial velocity, which means

$\displaystyle\left.\frac{d\phi^{a}}{d\lambda}\right|_{\lambda=0}\;=\;\varphi^{a}\,.$
(B.24)

Then, by using the geodesic equation repeatedly, we
obtain\natural15\natural15$\natural$15 There seems to be a typo in Eq. (2.4)
of [94]. This expression agrees with [74].

$\displaystyle\phi^{a}=\phi^{a}_{0}+\varphi^{a}-\frac{1}{2}\Gamma^{a}_{bc}\varphi^{b}\varphi^{c}+\frac{1}{6}\left(2\Gamma^{a}_{de}\Gamma^{e}_{bc}-\partial_{d}\Gamma^{a}_{bc}\right)\varphi^{b}\varphi^{c}\varphi^{d}+\cdots\,.$
(B.25)

With this definition, $\varphi^{a}$ lives in the tangent space, allowing
everything to be expressed in terms of geometrical quantities. The potential
is expanded as

$\displaystyle
V(\phi^{a})=\left[V+\frac{dV}{d\lambda}+\frac{1}{2}\frac{d^{2}V}{d\lambda^{2}}+\frac{1}{6}\frac{d^{3}V}{d\lambda^{3}}+\cdots\right]_{\lambda=0}.$
(B.26)

For instance, the second derivative is computed as

$\displaystyle\frac{d^{2}V}{d\lambda^{2}}=\frac{d}{d\lambda}\left(\frac{d\phi^{a}}{d\lambda}V_{a}\right)=\left(\frac{d^{2}\phi^{a}}{d\lambda^{2}}V_{a}+\frac{d\phi^{a}}{d\lambda}\frac{d\phi^{b}}{d\lambda}\partial_{b}V_{a}\right)=\frac{d\phi^{a}}{d\lambda}\frac{d\phi^{b}}{d\lambda}\left(\partial_{b}V_{a}-\Gamma^{c}_{ab}V_{c}\right)=\frac{d\phi^{a}}{d\lambda}\frac{d\phi^{b}}{d\lambda}\nabla_{b}V_{a},$
(B.27)

where we used the geodesic equation in the intermediate step. By repeating a
similar computation for the third order term, we can expand the potential as

$\displaystyle
V(\phi)=V+\varphi^{a}V_{a}+\frac{1}{2}\varphi^{a}\varphi^{b}\nabla_{b}V_{a}+\frac{1}{6}\varphi^{a}\varphi^{b}\varphi^{c}\nabla_{c}\nabla_{b}V_{a}+\cdots,$
(B.28)

where the quantities without the arguments are evaluated by $\phi_{0}^{a}$.
The expansion of the kinetic term is more non-trivial. The first order term is
given by

$\displaystyle\frac{d}{d\lambda}\left[\frac{1}{2}h_{ab}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}\right]\;=\;h_{ab}\partial_{\mu}\left(\frac{d\phi^{a}}{d\lambda}\right)\partial^{\mu}\phi^{b}+\frac{1}{2}h_{ab,c}\frac{d\phi^{c}}{d\lambda}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}\,.$
(B.29)

We may use that

$\displaystyle h_{ab,c}\;=\;h_{ad}\Gamma^{d}_{bc}+h_{bd}\Gamma^{d}_{ac}\,.$
(B.30)

The above is then simplified as

$\displaystyle\frac{d}{d\lambda}\left[\frac{1}{2}h_{ab}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}\right]\;=\;h_{ab}\left[\partial_{\mu}\left(\frac{d\phi^{a}}{d\lambda}\right)+\Gamma^{a}_{cd}(\partial_{\mu}\phi^{c})\frac{d\phi^{d}}{d\lambda}\right]\partial^{\mu}\phi^{b}\;\equiv\;h_{ab}\mathcal{D}_{\mu}\left(\frac{d\phi^{a}}{d\lambda}\right)\partial^{\mu}\phi^{b}\,.$
(B.31)

To compute the second and third order terms, it is convenient to note that

$\displaystyle\left[\mathcal{D}_{\lambda},\mathcal{D}_{\mu}\right]V^{a}$
$\displaystyle={\mathcal{R}^{a}}_{bcd}V^{b}\frac{d\phi^{c}}{d\lambda}\partial_{\mu}\phi^{d}\,,$
(B.32)

for an arbitrary vector $V^{a}$. Notice that here $\lambda$ in
$\mathcal{D}_{\lambda}$ is the affine parameter and not the spacetime
coordinate index. The second order term is then easily obtained as

$\displaystyle\frac{d^{2}}{d\lambda^{2}}\left[\frac{1}{2}h_{ab}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}\right]$
$\displaystyle=h_{ab}\left[\mathcal{D}^{\mu}\left(\frac{d\phi^{a}}{d\lambda}\right)\mathcal{D}_{\mu}\left(\frac{d\phi^{b}}{d\lambda}\right)+\partial^{\mu}\phi^{b}\mathcal{D}_{\lambda}\mathcal{D}_{\mu}\left(\frac{d\phi^{a}}{d\lambda}\right)\right]$
$\displaystyle=h_{ab}\mathcal{D}^{\mu}\left(\frac{d\phi^{a}}{d\lambda}\right)\mathcal{D}_{\mu}\left(\frac{d\phi^{b}}{d\lambda}\right)+\mathcal{R}_{abcd}\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{d}\frac{d\phi^{b}}{d\lambda}\frac{d\phi^{c}}{d\lambda}\,,$
(B.33)

where we used the geodesic equation
$\mathcal{D}_{\lambda}(d\phi^{a}/d\lambda)=0$ and the commutator in the last
line. The third order term is also easy to obtain:

$\displaystyle\frac{d^{3}}{d\lambda^{3}}\left[\frac{1}{2}h_{ab}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}\right]$
$\displaystyle=(\nabla_{e}\mathcal{R}_{abcd})\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{d}\frac{d\phi^{b}}{d\lambda}\frac{d\phi^{c}}{d\lambda}\frac{d\phi^{e}}{d\lambda}+4\mathcal{R}_{abcd}D^{\mu}\left(\frac{d\phi^{a}}{d\lambda}\right)\partial_{\mu}\phi^{d}\frac{d\phi^{b}}{d\lambda}\frac{d\phi^{c}}{d\lambda}\,.$
(B.34)

In summary, the kinetic term and the potential are expanded as

$\displaystyle\frac{1}{2}h_{ab}(\phi)\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}=\frac{1}{2}g^{00}\dot{\phi}_{0}^{2}+\dot{\phi}_{0}^{a}g^{0\mu}\mathcal{D}_{\mu}\varphi_{a}+\frac{1}{2}\left[h_{ab}g^{\mu\nu}\mathcal{D}_{\mu}\varphi^{a}\mathcal{D}_{\nu}\varphi^{b}+g^{00}\dot{\phi}_{0}^{c}\dot{\phi}_{0}^{d}\mathcal{R}_{acdb}\varphi^{a}\varphi^{b}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{6}\left[(\nabla_{c}\mathcal{R}_{adeb})g^{00}\dot{\phi}_{0}^{d}\dot{\phi}_{0}^{e}\varphi^{a}\varphi^{b}\varphi^{c}+4\mathcal{R}_{abcd}\dot{\phi}_{0}^{d}g^{0\mu}(\mathcal{D}_{\mu}\varphi^{a})\varphi^{b}\varphi^{c}\right]+\cdots,$
(B.35) $\displaystyle
V(\phi)=V+V_{a}\varphi^{a}+\frac{1}{2}(\nabla_{b}V_{a})\varphi^{a}\varphi^{b}+\frac{1}{6}(\nabla_{c}\nabla_{b}V_{a})\varphi^{a}\varphi^{b}\varphi^{c}+\cdots,$
(B.36)

where we take $\partial_{\mu}\phi_{0}^{a}=\delta_{\mu}^{0}\dot{\phi}_{0}^{a}$.
Note that we have not expanded the metric in the kinetic term yet.

Next, we discuss the constraint equations. The lapse function $N$ and the
shift vector $\beta^{i}$ do not have the time derivatives, and thus they are
constrained quantities. To compute the action up to third order, we need to
solve the constraints only up to first order [78]. With this in mind, the
quadratic action is given by

$\displaystyle S_{\mathrm{quad}}\;=\;\int d^{4}xa^{3}$
$\displaystyle\left[\frac{1}{2}h_{ab}\mathcal{D}_{t}\varphi^{a}\mathcal{D}_{t}\varphi^{b}-\frac{1}{2a^{2}}h_{ab}\partial_{i}\varphi^{a}\partial_{i}\varphi^{b}-\frac{1}{2}\left(\nabla_{b}V_{a}-\dot{\phi}_{0}^{c}\dot{\phi}_{0}^{d}\mathcal{R}_{acdb}\right)\varphi^{a}\varphi^{b}\right.$
$\displaystyle\left.-2M_{P}^{2}H\left(\alpha-\frac{\dot{\phi}_{0}}{2M_{P}^{2}H}\varphi\right)\partial_{i}^{2}\beta-\left(3M_{P}^{2}H^{2}-\frac{\dot{\phi}_{0}^{2}}{2}\right)\alpha^{2}-\left(\dot{\phi}_{0}^{a}\mathcal{D}_{t}\varphi_{a}+V_{a}\varphi^{a}\right)\alpha\right]\,,$
(B.37)

where $\varphi=T^{a}\varphi_{a}$, and we used

$\displaystyle
NR^{(3)}+\frac{1}{N}(E_{ij}E^{ij}-E^{2})=\frac{1}{N}\left(-6H^{2}+4H\partial_{i}^{2}\beta+(\partial_{i}\partial_{j}\beta)^{2}-(\partial_{i}^{2}\beta)^{2}\right)\,.$
(B.38)

The constraint equation is solved as

$\displaystyle\alpha\;=\;\frac{H}{\dot{\phi}_{0}}\epsilon\varphi,\quad\partial_{i}^{2}\beta\;=\;-\frac{1}{2HM_{P}^{2}}\left[\left(6M_{P}^{2}H^{2}-\dot{\phi}_{0}^{2}\right)\alpha+\dot{\phi}_{0}T_{a}\mathcal{D}_{t}\varphi^{a}+V_{a}\varphi^{a}\right]\,.$
(B.39)

The latter is equivalent to

$\displaystyle\partial_{i}^{2}\beta\;=\;-\frac{1}{2HM_{P}^{2}}\left[\left(6M_{P}^{2}H^{2}-\dot{\phi}_{0}^{2}\right)\alpha+\dot{\phi}_{0}\dot{\varphi}+V_{\phi}\varphi+2\dot{\phi}_{0}H\eta_{\perp}\chi\right]\,.$
(B.40)

###  B.3 Quadratic action

After substituting the solution of the constraint equations to the action and
performing integration by parts, we arrive at the quadratic action

$\displaystyle S_{\mathrm{quad}}$ $\displaystyle=\int
dtd^{3}xa^{3}\left[\frac{1}{2}h_{ab}\mathcal{D}_{t}\varphi^{a}\mathcal{D}_{t}\varphi^{b}-\frac{1}{2a^{2}}h_{ab}\partial_{i}\varphi^{a}\partial_{i}\varphi^{b}-\frac{1}{2}M_{ab}^{2}\varphi^{a}\varphi^{b}\right]\,,$
(B.41)

where the mass term is given by

$\displaystyle M_{ab}^{2}$
$\displaystyle=\nabla_{b}V_{a}-\dot{\phi}_{0}^{2}T^{c}T^{d}\mathcal{R}_{acdb}+\frac{2H}{\dot{\phi}_{0}}\epsilon(V_{a}T_{b}+V_{b}T_{a})+2(3-\epsilon)\epsilon
H^{2}T_{a}T_{b}\,.$ (B.42)

Note that we have not used the slow-roll approximation to derive it. In the
two-field case, we parametrize the fields as

$\displaystyle\varphi^{a}\;=\;T^{a}\varphi+N^{a}\chi=-T^{a}\frac{\dot{\phi}_{0}}{H}\zeta+N^{a}\chi\,,$
(B.43)

where $\varphi=-\dot{\phi}_{0}\zeta/H$ is the adiabatic mode and $\chi$ is the
isocurvature mode. We note that

$\displaystyle\dot{\varphi}\;=\;-\frac{\dot{\phi}_{0}}{H}\left[\dot{\zeta}-\left((3-\epsilon)H+\frac{V_{\phi}}{\dot{\phi}_{0}}\right)\zeta\right]\,,$
(B.44)
$\displaystyle\frac{1}{\dot{\phi}_{0}}\frac{dV_{\phi}}{dt}\;=\;-\dot{\theta}^{2}+T^{a}T^{b}\nabla_{b}V_{a},\quad\frac{1}{\dot{\phi}_{0}}\frac{dV_{N}}{dt}\;=\;\dot{\theta}\frac{V_{\phi}}{\dot{\phi}_{0}}+T^{a}N^{b}\nabla_{b}V_{a}\,.$
(B.45)

After some computation, the action is simplified as

$\displaystyle S_{\mathrm{quad}}\;=\;\int
d^{4}xa^{3}\left[\frac{1}{2}\frac{\dot{\phi}_{0}^{2}}{H^{2}}\left(\dot{\zeta}^{2}-\frac{1}{a^{2}}\left(\partial_{i}\zeta\right)^{2}\right)+\frac{1}{2}\left(\dot{\chi}^{2}-\frac{1}{a^{2}}\left(\partial_{i}\chi\right)^{2}-m_{\chi}^{2}\chi^{2}\right)-\frac{2\dot{\theta}\dot{\phi}_{0}}{H}\dot{\zeta}\chi\right],$
(B.46)

where

$\displaystyle
m_{\chi}^{2}\;=\;N^{a}N^{b}\nabla_{b}V_{a}+\frac{\dot{\phi}_{0}^{2}}{2}\mathcal{R}-\dot{\theta}^{2},\quad\mathcal{R}=\mathcal{R}_{ab}h^{ab}\;=\;\mathcal{R}_{acbd}h^{ab}h^{cd}\,.$
(B.47)

It is now clear that the turning rate $\dot{\theta}$ controls the mixing
between the adiabatic and isocurvature modes.

###  B.4 Cubic action

Using the results from App. B.2, the cubic action is straightforward to
derive. The pure gravity part is given by

$\displaystyle S_{\mathrm{cubic}}^{(\mathrm{grav})}$ $\displaystyle=\int
d^{4}xa^{3}\left[\left(3M_{P}^{2}H^{2}-\frac{\dot{\phi}_{0}^{2}}{2}\right)\alpha^{3}+2M_{P}^{2}H\alpha^{2}\partial_{i}^{2}\beta-\frac{M_{P}^{2}}{2}\alpha\left((\partial_{i}\partial_{j}\beta)^{2}-(\partial_{i}^{2}\beta)^{2}\right)\right]\,.$
(B.48)

The gravity-matter mixing part is given by

$\displaystyle S_{\mathrm{cubic}}^{(\mathrm{mix})}=\int d^{4}xa^{3}$
$\displaystyle\left[(\partial_{i}\beta)\left(\dot{\phi}_{0}\alpha\partial_{i}\varphi-\partial_{i}\varphi_{a}\mathcal{D}_{t}\varphi^{a}\right)+\alpha^{2}\dot{\phi}_{0}^{a}\mathcal{D}_{t}\varphi_{a}\right.$
$\displaystyle\left.-\frac{\alpha}{2}\left(\mathcal{D}_{t}\varphi^{a}\mathcal{D}_{t}\varphi_{a}+\frac{1}{a^{2}}h_{ab}\partial_{i}\varphi^{a}\partial_{i}\varphi^{b}+\left(\nabla_{b}V_{a}+\dot{\phi}_{0}^{2}T^{c}T^{d}\mathcal{R}_{acdb}\right)\varphi^{a}\varphi^{b}\right)\right]\,.$
(B.49)

Finally, the pure matter sector is given by

$\displaystyle S_{\mathrm{cubic}}^{(\mathrm{matter})}=\int
d^{4}xa^{3}\left[\frac{1}{6}(\nabla_{c}\mathcal{R}_{adeb})\dot{\phi}_{0}^{d}\dot{\phi}_{0}^{e}\varphi^{a}\varphi^{b}\varphi^{c}+\frac{2}{3}\mathcal{R}_{abcd}\dot{\phi}_{0}^{d}(\mathcal{D}_{t}\varphi^{a})\varphi^{b}\varphi^{c}-\frac{1}{6}(\nabla_{c}\nabla_{b}V_{a})\varphi^{a}\varphi^{b}\varphi^{c}\right]\,.$
(B.50)

They correctly reproduce the results from Refs. [74, 94].

##  Appendix C Schwinger-Keldysh propagators in de Sitter spacetime

In this appendix, we review the Schwinger-Keldysh propagators of the scalar,
fermion, and massive gauge boson during inflation. We ignore the slow-roll
parameters and take the background spacetime as the pure de Sitter one. The
spacetime dimension is taken to be $d$ in this appendix. See e.g. [81] and
references therein for more details on the Schwinger-Keldysh formalism in the
context of cosmological collider physics.

###  C.1 Preliminary

First, we summarize several equations that are repeatedly used in the
derivation of the propagators. We will encounter the mode equation of the form

$\displaystyle
0\;=\;\left[\frac{d^{2}}{d\tau^{2}}+k^{2}-\frac{\nu^{2}-1/4}{\tau^{2}}\right]v_{k}\,,$
(C.1)

where $\tau$ is the conformal time and $v_{k}$ is the mode function with $k$
the size of its momentum. By defining $z=-k\tau$ and
$\tilde{v}_{k}=v_{k}/\sqrt{z}$, this is recast as Bessel’s differential
equation

$\displaystyle
z^{2}\frac{d^{2}\tilde{v}_{k}}{dz^{2}}+z\frac{d\tilde{v}_{k}}{dz}+\left(z^{2}-\nu^{2}\right)\tilde{v}_{k}\;=\;0.$
(C.2)

A general solution is given by a linear combination of the Hankel functions as

$\displaystyle
v_{k}\;=\;z^{1/2}\left(c_{1}H_{\nu}^{(1)}(z)+c_{2}H_{\nu}^{(2)}(z)\right)\,.$
(C.3)

The Hankel function of the first (second) kind corresponds to the positive
(negative) frequency mode in the asymptotic past $z\to\infty$. Consequently,
the Bunch-Davies vacuum condition eliminates the latter mode.

The Hankel function satisfies several relations useful for our purpose. First,
the Hankel functions of the first and second kind are related to each other by
the complex conjugates as

$\displaystyle\left(H_{\nu}^{(1)}(z)\right)^{*}\;=\;H_{\nu^{*}}^{(2)}(z)\,,$
(C.4)

where we assume that the argument $z$ is real. They also satisfy

$\displaystyle\left(\frac{d}{dz}+\frac{\nu}{z}\right)H_{\nu}^{(i)}(z)\;=\;H_{\nu-1}^{(i)}(z)\,,\quad
H_{-\nu}^{(1)}\;=\;e^{i\pi\nu}H_{\nu}^{(1)}\,,\quad
H_{-\nu}^{(2)}\;=\;e^{-i\pi\nu}H_{\nu}^{(2)}\,,$ (C.5)

and the latter two equations indicate that

$\displaystyle\left(e^{i\pi\nu/2}H_{\nu}^{(1)}(z)\right)^{*}\;=\;e^{-i\pi\nu/2}H_{\nu}^{(2)}(z)\,,$
(C.6)

if $\nu$ is either real or pure imaginary. The Wronskian of the Hankel
functions is given by

$\displaystyle
H_{\nu+1}^{(1)}(z)H_{\nu}^{(2)}(z)-H_{\nu}^{(1)}(z)H_{\nu+1}^{(2)}(z)\;=\;-\frac{4i}{\pi
z}.$ (C.7)

When we compute the propagators in the coordinate space, we need to integrate
the product of the Hankel functions. The result is the hypergeometric function

$\displaystyle\frac{\pi}{4}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}e^{i\vec{k}\cdot(\vec{x}_{1}-\vec{x}_{2})}H_{\nu}^{(1)}(z_{1})H_{\nu}^{(2)}(z_{2})\;=\;\frac{I_{\nu}(Z_{12})}{(4\pi)^{d/2}(\tau_{1}\tau_{2})^{\frac{d-1}{2}}}\,,$
(C.8)

where $z_{1}=-k\tau_{1}$, $z_{2}=-k\tau_{2}$,

$\displaystyle{I}_{\nu}(Z)$
$\displaystyle\equiv\frac{\Gamma\left(\frac{d-1}{2}+\nu\right)\Gamma\left(\frac{d-1}{2}-\nu\right)}{\Gamma(d/2)}{}_{2}F_{1}\left(\frac{d-1}{2}+\nu,\frac{d-1}{2}-\nu;\frac{d}{2};\frac{1+Z}{2}\right)\,,$
(C.9)

and

$\displaystyle
Z_{ij}\;=\;1-\frac{x_{ij}^{2}-(\tau_{i}-\tau_{j}-i0)^{2}}{2\tau_{i}\tau_{j}}\,,$
(C.10)

with $x_{ij}=|\vec{x}_{i}-\vec{x}_{j}|$. Here we keep the $i\epsilon$
prescription required to make the integral convergent explicit. We may note
that $I_{\nu}$ satisfies the differential equation

$\displaystyle 0$
$\displaystyle=(1-Z^{2})I_{\nu}^{\prime\prime}(Z)-dZI_{\nu}^{\prime}(Z)-\left(d-2+\frac{m^{2}}{H^{2}}\right)I_{\nu}(Z)\,.$
(C.11)

We also define

$\displaystyle Z_{++}$
$\displaystyle=1-\frac{x_{12}^{2}-(|\tau_{1}-\tau_{2}|-i0)^{2}}{2\tau_{1}\tau_{2}}\,,\quad
Z_{--}=1-\frac{x_{12}^{2}-(|\tau_{1}-\tau_{2}|+i0)^{2}}{2\tau_{1}\tau_{2}}\,,$
(C.12) $\displaystyle Z_{+-}$
$\displaystyle=1-\frac{x_{12}^{2}-(\tau_{1}-\tau_{2}+i0)^{2}}{2\tau_{1}\tau_{2}}\,,\quad~{}~{}Z_{-+}=1-\frac{x_{12}^{2}-(\tau_{1}-\tau_{2}-i0)^{2}}{2\tau_{1}\tau_{2}}\,,$
(C.13)

where the subscripts indicate different contours in the Schwinger-Keldysh
formalism. Finally, the following relations turn out to be useful to derive
the gauge boson propagator:

$\displaystyle\partial_{\tau_{1}}Z_{12}=\frac{1}{\tau_{2}}-\frac{Z_{12}}{\tau_{1}},\quad\partial_{\tau_{2}}Z_{12}=\frac{1}{\tau_{1}}-\frac{Z_{12}}{\tau_{2}},\quad\partial_{\tau_{1}}\partial_{\tau_{2}}Z_{12}=-\left(\frac{1}{\tau_{1}^{2}}+\frac{1}{\tau_{2}^{2}}\right)+\frac{Z_{12}}{\tau_{1}\tau_{2}},$
(C.14)
$\displaystyle\partial_{1i}Z_{12}=-\partial_{2i}Z_{12}=-\frac{(x_{1}-x_{2})_{i}}{\tau_{1}\tau_{2}},\quad\partial_{1i}\partial_{2j}Z_{12}=\frac{\delta_{ij}}{\tau_{1}\tau_{2}},\quad\frac{x_{12}^{2}}{\tau_{1}^{2}\tau_{2}^{2}}=\frac{1}{\tau_{1}^{2}}+\frac{1}{\tau_{2}^{2}}-\frac{2Z_{12}}{\tau_{1}\tau_{2}}.$
(C.15)

It follows that

$\displaystyle(\partial_{1\alpha}Z_{12})(\partial_{1}^{\alpha}Z_{12})=\frac{Z_{12}^{2}-1}{\tau_{1}^{2}},\quad(\partial_{2\alpha}Z_{12})(\partial_{2}^{\alpha}Z_{12})=\frac{Z_{12}^{2}-1}{\tau_{2}^{2}},$
(C.16)
$\displaystyle(\partial_{1\alpha}\partial_{2\beta}Z_{12})(\partial_{1}^{\alpha}\partial_{2}^{\beta}Z_{12})=\frac{Z_{12}^{2}+3}{\tau_{1}^{2}\tau_{2}^{2}},\quad(\partial_{1\alpha}Z_{12})(\partial_{2\beta}Z_{12})\partial_{1}^{\alpha}\partial_{2}^{\beta}Z_{12}=\frac{Z_{12}(Z_{12}^{2}-1)}{\tau_{1}^{2}\tau_{2}^{2}}\,,$
(C.17)

where the contractions are taken with respect to $\eta_{\alpha\beta}$.

###  C.2 Scalar

We now review the derivation of the scalar field propagators in de Sitter
spacetime. We consider a massive real scalar field $\chi$ with a non-minimal
coupling $\xi$ (just for completeness) as

$\displaystyle S_{\chi}\;=\;\int
d^{d}x\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{m^{2}}{2}\chi^{2}+\frac{\xi}{2}R\chi^{2}\right]\,.$
(C.18)

Here $\chi$ can be either adiabatic or isocurvature modes.

#### Mode equation

We may quantize the field as

$\displaystyle\hat{\chi}(\tau,\vec{x})\;=\;a^{-\frac{d-2}{2}}\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\vec{k}\cdot\vec{x}}\left[\chi(\vec{k},\tau)\hat{a}_{\vec{k}}+\chi^{*}(-\vec{k},\tau)\hat{a}_{-\vec{k}}^{\dagger}\right]\,.$
(C.19)

The mode function satisfies

$\displaystyle\left[\frac{d^{2}}{d\tau^{2}}+k^{2}+\left(\frac{m^{2}}{H^{2}}-\frac{d(d-2)}{4}-d(d-1)\xi\right)\frac{1}{\tau^{2}}\right]\chi\;=\;0\,,$
(C.20)

and the creation-annihilation operator satisfies

$\displaystyle\left[\hat{a}_{\vec{k}},\hat{a}^{\dagger}_{\vec{k}^{\prime}}\right]$
$\displaystyle=(2\pi)^{d-1}\delta^{(d-1)}(\vec{k}-\vec{k}^{\prime})\,.$ (C.21)

This is Bessel differential equation (C.1), and hence the solution is given by

$\displaystyle\chi\;=\;\sqrt{\frac{-\pi\tau}{4}}e^{i\pi\nu/2}H_{\nu}^{(1)}(-k\tau)\,,\quad\nu\;=\;\sqrt{\frac{(d-1)^{2}}{4}+d(d-1)\xi-\frac{m^{2}}{H^{2}}}\,.$
(C.22)

where we assume the Bunch-Davies vacuum condition.

#### Propagators

For our purpose, it is convenient to define the scalar propagators in the
momentum space. We define the scalar two-point function as

$\displaystyle\left\langle{\hat{\chi}(\tau_{1},\vec{x}_{1})\hat{\chi}(\tau_{2},\vec{x}_{2})}\right\rangle$
$\displaystyle=\int\frac{d^{d-1}k}{(2\pi)^{d-1}}e^{i\vec{k}\cdot(\vec{x}_{1}-\vec{x}_{2})}G(k;\tau_{1},\tau_{2}),\quad
G(k;\tau_{1},\tau_{2})=\frac{\pi\sqrt{\tau_{1}\tau_{2}}}{4(a_{1}a_{2})^{\frac{d-2}{2}}}H_{\nu}^{(1)}(-k\tau_{1})H_{\nu}^{(2)}(-k\tau_{2}),$
(C.23)

where we used Eq. (C.6). The corresponding expression in the coordinate space
is given by

$\displaystyle\left\langle{\hat{\chi}(\tau_{1},\vec{x}_{1})\hat{\chi}(\tau_{2},\vec{x}_{2})}\right\rangle$
$\displaystyle=\frac{H^{d-2}}{(4\pi)^{d/2}}I_{\nu}(Z_{12})\,,$ (C.24)

where we used Eq. (C.8). In the Schwinger-Keldysh formalism, there are four
distinct propagators, each corresponding to the selection of fields on
different contours. These are given by

$\displaystyle G_{++}(k;\tau_{1},\tau_{2})$
$\displaystyle=G(k;\tau_{1},\tau_{2})\theta(\tau_{1}-\tau_{2})+G(k;\tau_{2},\tau_{1})\theta(\tau_{2}-\tau_{1})\,,$
(C.25) $\displaystyle G_{--}(k;\tau_{1},\tau_{2})$
$\displaystyle=G(k;\tau_{2},\tau_{1})\theta(\tau_{1}-\tau_{2})+G(k;\tau_{1},\tau_{2})\theta(\tau_{2}-\tau_{1})=G_{++}^{*}(k;\tau_{1},\tau_{2})\,,$
(C.26) $\displaystyle G_{-+}(k;\tau_{1},\tau_{2})$
$\displaystyle=G(k;\tau_{1},\tau_{2})\,,\quad
G_{+-}(k;\tau_{1},\tau_{2})=G(k;\tau_{2},\tau_{1})=G_{-+}^{*}(k;\tau_{1},\tau_{2})\,,$
(C.27)

where the subscript “$\pm$” indicates the time-ordered and anti-time-ordered
contours. We may use $G$ for a general massive scalar field (and hence the
isocurvature mode), and $\Delta$ specific to a massless scalar field with
$\xi=0$ (including the adiabatic mode), corresponding to $\nu=(d-1)/2$. In
particular, in $d=4$, we obtain

$\displaystyle\Delta(k;\tau_{1},\tau_{2})$
$\displaystyle=\frac{H^{2}}{2k^{3}}(1+ik\tau_{1})(1-ik\tau_{2})e^{-ik(\tau_{1}-\tau_{2})}\,.$
(C.28)

It is useful to note that

$\displaystyle\partial_{\tau}\Delta(k;\tau,0)\;=\;\frac{H^{2}\tau}{2k}e^{-ik\tau}\,,\quad\partial_{\tau}\Delta(k;0,\tau)\;=\;\frac{H^{2}\tau}{2k}e^{ik\tau}\,.$
(C.29)

###  C.3 Fermion

Next, we review the derivation of the massive Dirac field propagator in de
Sitter spacetime [95, 96, 97, 98]. The action is given by

$\displaystyle S_{\psi}\;=\;\int
d^{d}x\sqrt{-g}\,\bar{\psi}\left(i\not{\nabla}-m\right)\psi\,.$ (C.30)

In de Sitter spacetime, this reduces to

$\displaystyle S_{\psi}\;=\;\int d\tau
d^{d-1}x\,\bar{\tilde{\psi}}\left[i\gamma^{b}\partial_{b}-am\right]\tilde{\psi}\,,$
(C.31)

where we used

$\displaystyle
i\not{\nabla}\;=\;a^{-\frac{d+1}{2}}i\gamma^{b}\partial_{b}\,a^{\frac{d-1}{2}}\,,$
(C.32)

and redefined the fermion as $\tilde{\psi}=a^{(d-1)/2}\psi$, where $b$ runs
the local Lorentz indices. This indicates that the massless fermion is
conformal in an arbitrary spacetime dimension. The Dirac equation follows as

$\displaystyle\left[i\gamma^{b}\partial_{b}-am\right]\tilde{\psi}\;=\;0\,.$
(C.33)

#### Mode equation

By going to the Fourier space

$\displaystyle\psi\;=\;a^{-\frac{d-1}{2}}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}e^{i\vec{k}\cdot\vec{x}}\tilde{\psi}(\vec{k},\tau)\,,$
(C.34)

we obtain the mode equation

$\displaystyle\left[i\gamma^{0}\frac{d}{d\tau}-\vec{k}\cdot\vec{\gamma}-am\right]\tilde{\psi}\;=\;0\,.$
(C.35)

To solve this, we act the operator
“$i\gamma^{0}d/d\tau-\vec{k}\cdot\vec{\gamma}+am$”, and obtain

$\displaystyle\left[\frac{d^{2}}{d\tau^{2}}+k^{2}+im\frac{da}{d\tau}\gamma^{0}+a^{2}m^{2}\right]\tilde{\psi}\;=\;0\,.$
(C.36)

We decompose the modes as

$\displaystyle\tilde{\psi}\;=\;\sum_{\lambda,h=\pm}\tilde{\psi}_{\lambda
h}\,\xi_{\lambda}\otimes\chi_{h}\,,$ (C.37)

where we define the spinors in the subspaces as

$\displaystyle\gamma^{0}\xi_{\lambda}\;=\;\lambda\xi_{\lambda},\quad\vec{\sigma}\cdot\vec{k}\,\chi_{h}=hk\chi_{h}\,,\quad\sum_{\lambda}\xi_{\lambda}\xi^{\dagger}_{\lambda}\;=\;\sum_{h}\chi_{h}\chi^{\dagger}_{h}=\mathbbm{1}_{2\times
2}\,,$ (C.38)

with $h$ denoting the helicity and $\lambda$ denoting the eigenvalue of
$\gamma^{0}$. The mode equation is then given by

$\displaystyle\left[\partial_{\tau}^{2}+k^{2}+\frac{1}{\tau^{2}}\left(\frac{m^{2}}{H^{2}}+\lambda\frac{im}{H}\right)\right]\tilde{\psi}_{\lambda
h}\;=\;0\,,$ (C.39)

which is of the form (C.1), and thus we obtain

$\displaystyle\tilde{\psi}_{\lambda h}$ $\displaystyle=\sqrt{\frac{\pi
z}{4}}\left[e^{i\pi\nu_{\lambda}/2}H_{\nu_{\lambda}}^{(1)}(z)\times b_{\lambda
h}+e^{-i\pi\nu_{\lambda}/2}H_{\nu_{\lambda}}^{(2)}(z)\times
d_{\lambda-h}^{*}\right]\,,$ (C.40)

where $z=-k\tau$. Here we keep both the positive and negative energy
solutions, corresponding to the particle and anti-particle, and define

$\displaystyle\nu_{\lambda}\;=\;\frac{1}{2}-\lambda\frac{im}{H}\,.$ (C.41)

We now plug this expression back into Eq. (C.35) and
obtain\natural16\natural16$\natural$16 Here we implicitly fix the convention
for the relative sign between $\xi_{+}$ and $\xi_{-}$, such that it can be
expressed as $\xi_{\pm}^{\dagger}=(1,\pm 1)/\sqrt{2}$ in the Weyl
representation.

$\displaystyle\left(i\frac{d}{d\tau}+\frac{m}{H\tau}\right)\tilde{\psi}_{+h}+hk\tilde{\psi}_{-h}=0,\quad\left(i\frac{d}{d\tau}-\frac{m}{H\tau}\right)\tilde{\psi}_{-h}+hk\tilde{\psi}_{+h}=0\,.$
(C.42)

With the relations (C.5), this is solved as

$\displaystyle b_{-h}\;=\;-hb_{+h}\,,\quad d_{-h}\;=\;-hd_{+h}\,.$ (C.43)

After quantization, we obtain

$\displaystyle\hat{\psi}(\vec{x},\tau)$
$\displaystyle=a^{-(d-1)/2}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\sqrt{\frac{-\pi
k\tau}{4}}e^{i\vec{k}\cdot\vec{x}}\sum_{h=\pm}\chi_{h}$
$\displaystyle\otimes\left[\left(e^{i\pi\nu_{+}/2}H_{\nu_{+}}^{(1)}\xi_{+}-he^{i\pi\nu_{-}/2}H_{\nu_{-}}^{(1)}\xi_{-}\right)\hat{b}_{\vec{k},h}+\left(e^{-i\pi\nu_{+}/2}H_{\nu_{+}}^{(2)}\xi_{+}+he^{-i\pi\nu_{-}/2}H_{\nu_{-}}^{(2)}\xi_{-}\right)\hat{d}^{\dagger}_{-\vec{k},-h}\right]\,.$
(C.44)

The quantization condition is

$\displaystyle
a^{d-1}\left\\{\hat{\psi}(\vec{x},\tau),\hat{\psi}^{\dagger}(\vec{y},\tau)\right\\}=\delta^{(d-1)}(\vec{x}-\vec{y})\times\mathbbm{1}\,,$
(C.45)

which results in

$\displaystyle\left\\{\hat{b}_{\vec{k},h},\hat{b}_{\vec{k}^{\prime},h^{\prime}}^{\dagger}\right\\}$
$\displaystyle=\left\\{\hat{d}_{\vec{k},h},\hat{d}_{\vec{k}^{\prime},h^{\prime}}^{\dagger}\right\\}=(2\pi)^{d-1}\delta^{(d-1)}(\vec{k}-\vec{k}^{\prime})\delta_{hh^{\prime}}\,.$
(C.46)

Here we used

$\displaystyle
e^{i\pi(\nu_{+}-\nu_{-})/2}H_{\nu_{+}}^{(1)}H_{\nu_{-}}^{(2)}+e^{-i\pi(\nu_{+}-\nu_{-})/2}H_{\nu_{-}}^{(1)}H_{\nu_{+}}^{(2)}=\frac{4}{\pi
z}\,,$ (C.47)

which follows from Eqs. (C.5) and (C.7).

#### Propagators

Next, we compute the propagators. We define them as

$\displaystyle\left[S_{1}(\vec{x}_{1}-\vec{x}_{2};\tau_{1},\tau_{2})\right]_{ab}$
$\displaystyle=\langle{\psi}_{a}(\tau_{1},\vec{x}_{1})\bar{\psi}_{b}(\tau_{2},\vec{x}_{2})\rangle\,,\quad\left[S_{2}(\vec{x}_{1}-\vec{x}_{2};\tau_{1},\tau_{2})\right]_{ab}=-\langle\bar{\psi}_{b}(\tau_{2},\vec{x}_{2}){\psi}_{a}(\tau_{1},\vec{x}_{1})\rangle\,.$
(C.48)

To evaluate these expressions, we note that\natural17\natural17$\natural$17 We
assume the convention for the relative sign between $\xi_{\pm}$ that is
consistent with Eq. (C.42). With this and the previous choices, the
propagators do not depend on the sign convention.

$\displaystyle\sum_{h}\chi_{h}\chi_{h}^{\dagger}\otimes\xi_{\pm}\xi^{\dagger}_{\pm}=\frac{1\pm\gamma^{0}}{2},\quad\sum_{h}h\chi_{h}\chi_{h}^{\dagger}\otimes\xi_{\mp}\xi^{\dagger}_{\pm}=\pm\frac{\vec{k}\cdot\vec{\gamma}}{k}\frac{1\pm\gamma^{0}}{2},$
(C.49)