NITEP 138

Cosmological Collider Signals of
Non-Gaussianity from Higgs boson in GUT

Nobuhito Marua,b and Akira Okawaa

aDepartment of Physics, Osaka Metropolitan University,
Osaka 558-8585, Japan

bNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),
Osaka Metropolitan University, Osaka 558-8585, Japan

Cosmological Collider Physics gives us the opportunity to probe high-energy
physics from observing the spacetime fluctuations generated during inflation
imprinted on the cosmic microwave background. In other words, it is a method
to investigate physics on energy scales that cannot be reached by terrestrial
accelerators by means of precise observations of the universe. In this paper,
we focus on the case where the GUT scale is close to the energy scale of
inflation, and calculate three point function of inflaton by exchanging the
Higgs boson in GUT at tree level. The results are found to be consistent with
the current observed restrictions on non-Gaussianity without a drastic fine
tuning of parameters, and it might be possible to detect the signature of the
Higgs boson in GUT by 21cm spectrum, future LSS and future CMB depending on
our model parameters.

##  1 Introduction

The Standard Model of elementary particles can explain a wide variety of
elementary particle phenomena and has acquired reliability. On the other hand,
there are certainly some phenomena that cannot be explained by the Standard
Model, for example, the mechanism of generating tiny neutrino masses, the
origin of dark matter and dark energy, and the hierarchy problem. To solve
these problems, various theories beyond the Standard Model have been
considered, such as extra dimensional scenario, supersymmetry, and Grand
Unified Theory (GUT). GUT is the theory that unifies three of the four
fundamental interactions that exist in nature: strong interaction,
electromagnetic interaction, and weak interaction. The renormalization group
method suggests that these three interactions are unified at a certain energy
scale. Since the Standard Model is described by the Weinberg-Salam model,
which unifies the electromagnetic and weak interactions, it is natural that
the strong interaction is also unified. However, the energy scale of the GUT
is expected to be about $10^{15}$ GeV, and it is difficult to verify such a
high energy theory by terrestrial accelerators. For this reason, Cosmological
Collider Physics has attracted interest [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91, 92, 93, 94]. Cosmological Collider Physics is a field that
obtains information on high energy elementary particles by observing quantum
fluctuations in space-time stretched by inflation through the cosmic microwave
background radiation. That is, precise observation of the universe can provide
information on elementary particles in high energy which cannot be reached by
terrestrial accelerators.

Non-Gaussianity is the three- or higher-point function of some quantum
fluctuation in the curvature perturbations. Three point functions in models
with only inflatons and gravitons were computed by Maldacena [95]. The
effective field theory of inflation was proposed by C. Cheung, et al. [96],
and its formalism has been used to calculate three-point functions in models
with various particles. We focus on the case where the GUT scale is close to
the energy scale of inflation and calculate three point function of inflaton
by exchanging Higgs boson in GUT. The characteristic feature of this model is
that the interaction between Higgs boson and inflaton generated by the
(non-)vanishing Higgs boson vacuum expectation value (VEV) contributes to
three point function of the inflaton at the (tree) 1-loop level as shown in
Fig.1 and Fig.2. The results are found to be consistent with the current
observational constraints on non-Gaussianity without drastic fine tuning of
parameters, and it might be possible to detect the signature of the Higgs
boson in GUT by 21cm spectrum, future LSS and future CMB depending on our
model parameters.

Figure 1: Leading graph of the inflaton three point function in the absence of
spontaneous symmetry breaking, which is inevitably becomes at one-loop level.
The rigid line represents the inflaton, and the dotted line represents the
Higgs boson in GUT. See Appendix B for notation. Figure 2: Leading graph of
the inflaton three point function in the presence of spontaneous symmetry
breaking. In the absence of spontaneous symmetry breaking, no such tree graph
exists.

This paper is organized as follows. First, we discuss the setup of our model.
The effective field theory of inflation is briefly introduced, giving the
propagators of the inflaton as NG boson $\pi$. Furthermore, we introduce the
Higgs potential in GUT and confirm that the Higgs boson interacts with the
inflaton linearly after developing the vacuum expectation value due to the
spontaneous symmetry breaking. In Section 3, we actually calculate the non-
Gaussianity. Concretely, we compute three point function of the inflatons via
the Higgs boson exchange at tree level by using the approximation in horizon
exit. Non-Gaussianity is evaluated from the obtained three-point functions and
the results are compared with the observational data by the Planck satellite.
Conclusions are given in Section 4. In addition, Appendix A presents
calculations on determinants of the metric, and Appendix B summarizes in-in
formalism [97].

##  2 A Model

We consider the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime with
fluctuations and curvature $K=0$ expressed in the ADM formalism as the
inflationary spacetime.

$ds^{2}=-N^{2}dt^{2}+\tilde{h}_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right),$
(1)

where $\tilde{h}_{ij}$ is a spatial components of the metric, $N$ is the lapse
function and $N^{i}$ is the shift function. The tilde represents a physical
quantity in the comoving gauge. The action we consider is

$S=S_{\mathrm{grav}}+S_{\text{SU(5) gauge}}+S_{\text{SU(5) Higgs}},$ (2)

where $S_{\mathrm{grav}},S_{\text{SU(5) gauge}}$ and $S_{\text{SU(5) Higgs}}$
describe the actions of the gravity, SU(5) gauge bosons and an SU(5) adjoint
Higgs boson to break GUT gauge symmetry respectively111In the following, we
consider in this paper the SU(5) GUT as an illustrating example, but it can be
easily extended to other GUT gauge group.. We will now examine each action in
detail but omit the $S_{\text{SU(5) gauge}}$ since it is unnecessary for our
computation of the graph Fig.2. From the viewpoint of the effective field
theory [96, 98], the Einstein-Hilbert action and the inflaton action after the
transformation of time coordinates $t\mapsto\tilde{t}=t-\pi(\tilde{t},\bm{x})$
can be written as

$\displaystyle S_{\mathrm{grav}}=\int d^{4}x\sqrt{-g}$
$\displaystyle\left[\frac{1}{2}M_{\mathrm{Pl}}^{2}R-M_{\mathrm{Pl}}^{2}\left(3H^{2}(\tilde{t}+\pi)+\dot{H}(\tilde{t}+\pi)\right)\right.$
(3)
$\displaystyle\left.+M_{\mathrm{PI}}^{2}\dot{H}(\tilde{t}+\pi)\left(\partial_{\mu}(\tilde{t}+\pi)\partial_{\nu}(\tilde{t}+\pi)g^{\mu\nu}\right)\right.$
$\displaystyle+\frac{M_{2}(\tilde{t}+\pi)^{4}}{2!}\left(\partial_{\mu}(\tilde{t}+\pi)\partial_{\nu}(\tilde{t}+\pi)g^{\mu\nu}+1\right)^{2}$
$\displaystyle+\left.\frac{M_{3}(\tilde{t}+\pi)^{4}}{3!}\left(\partial_{\mu}(\tilde{t}+\pi)\partial_{\nu}(\tilde{t}+\pi)g^{\mu\nu}+1\right)^{3}+\ldots\right],$

where $g$ is the determinant of the metric $g_{\mu\nu}$, $M_{\mathrm{Pl}}$ is
the Planck mass, $R$ is the Ricci curvature in four dimensions, $H$ is the
Hubble parameter, $\pi$ is a Nambu-Goldstone boson of time translation which
is identified with the inflaton. $M_{2,3}$ are the coefficients of the high-
dimensional operators.

The quadratic terms of the effective action of inflaton $\pi$ is identified as
[96, 99]

$I_{2}=M_{\text{Pl}}^{2}\int
dtd^{3}x\,a^{3}\left[-\frac{\dot{H}}{c_{s}^{2}}\left(\dot{\pi}^{2}-c^{2}_{s}\frac{\left(\partial_{i}\pi\right)^{2}}{a^{2}}\right)\right].$
(4)

The sound speed $c_{s}$ is a quantity that is not determined by the effective
field theory, but is bounded by the fundamental theory and observational data.
The mode function $w(\tau,k)$ of the inflaton $\pi$ is obtained as

$w(\tau,k)=\frac{c_{s}}{\sqrt{2\epsilon}aHM_{\text{Pl}}}\frac{1+ic_{s}k\tau}{\sqrt{2c_{s}k}c_{s}k\tau}e^{-ic_{s}k\tau},$
(5)

where $\tau$ is the conformal time and $\epsilon$ is the slow roll parameter.
Using the expression for the propagators by in-in formalism in Appendix B, the
propagators of inflaton $\pi$ is obtained as

$\displaystyle\Delta_{>}\left(\tau_{1},\tau_{2},k\right)$ $\displaystyle=$
$\displaystyle w\left(\tau_{1},k\right)w^{*}\left(\tau_{2},k\right)$ (6)
$\displaystyle=$
$\displaystyle\frac{1+ic_{s}k\left(\tau_{1}-\tau_{2}\right)+(c_{s}k)^{2}\tau_{1}\tau_{2}}{4\epsilon
M_{\text{Pl}}c_{s}k^{3}}e^{-ic_{s}k(\tau_{1}-\tau_{2})},$
$\displaystyle\Delta_{<}\left(\tau_{1},\tau_{2},k\right)$ $\displaystyle=$
$\displaystyle w\left(\tau_{2},k\right)w^{*}\left(\tau_{1},k\right)$ (7)
$\displaystyle=$
$\displaystyle\frac{1-ic_{s}k\left(\tau_{1}-\tau_{2}\right)+(c_{s}k)^{2}\tau_{1}\tau_{2}}{4\epsilon
M_{\text{Pl}}c_{s}k^{3}}e^{ic_{s}k(\tau_{1}-\tau_{2})}$

Next, we discuss the action of the Higgs boson for GUT gauge symmetry breaking
to the SM gauge symmetry. Let us denote $\Sigma^{\prime}$ for the Higgs of the
24-dimensional adjoint representation of SU(5) and its renormalizable
potential is introduced as

$V(\Sigma^{\prime})=-M^{2}\text{tr}(\Sigma^{\prime
2})+\lambda_{1}\left\\{\text{tr}(\Sigma^{\prime
2})\right\\}^{2}+\lambda_{2}\,\text{tr}(\Sigma^{\prime 4}),$ (8)

where $M,\lambda_{1}$ and $\lambda_{2}$ are constants, and $M$ is of order of
the GUT scale.

$M\sim\mathcal{O}(M_{\text{GUT}}).$ (9)

We expand the adjoint Higgs $\Sigma^{\prime}$ around the expectation value

$\langle\Sigma\rangle=v\ \text{diag}(2,2,2,-3,-3),\quad
v\sim\mathcal{O}(M_{\text{GUT}})$ (10)

as

$\Sigma^{\prime}=\langle\Sigma\rangle+\Sigma.$ (11)

Now, we calculate three point function of inflaton to extract non-Gaussianity
by the existence of $\Sigma$, which is given by a graph of tree level exchange
of $\Sigma$ shown in Fig. 2. In order to extract the first-order term of
$\Sigma$ in the potential, we expand $\Sigma^{\prime 2}$ around the
expectation value yields

$\displaystyle\Sigma^{\prime 2}$ $\displaystyle=$
$\displaystyle\left(\langle\Sigma\rangle+\Sigma\right)^{2}$ (12)
$\displaystyle=$
$\displaystyle\langle\Sigma\rangle^{2}+\langle\Sigma\rangle\Sigma+\Sigma\langle\Sigma\rangle+\Sigma^{2},$

and we obtain

$\displaystyle 2\text{tr}\left(\langle\Sigma\rangle\Sigma\right)$
$\displaystyle=$ $\displaystyle
2\times\left(2v\Sigma_{11}+2v\Sigma_{22}+2v\Sigma_{33}-3v\Sigma_{44}-3v\Sigma_{55}\right)$
(13) $\displaystyle=$ $\displaystyle 2\times
5v\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)$ $\displaystyle=$
$\displaystyle 10v\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)$

from the terms in $\text{tr}(\Sigma^{\prime 2})$. Note that only diagonal
components of $\Sigma$ are taken into account and the traceless condition for
$\Sigma$

$\Sigma_{11}+\Sigma_{22}+\Sigma_{33}+\Sigma_{44}+\Sigma_{55}=0.$ (14)

is used in the second equality. Then, since the first term of (12) can be
computed as

$\text{tr}\left(\langle\Sigma\rangle^{2}\right)=30v^{2},$ (15)

we have

$\text{tr}(\Sigma^{\prime
2})=30v^{2}+10v\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)+\text{tr}(\Sigma^{2}).$
(16)

Now we can obtain

$\displaystyle\lambda_{1}\left\\{\text{tr}(\Sigma^{\prime 2})\right\\}^{2}$
$\displaystyle=$
$\displaystyle\lambda_{1}\left[30v^{2}+10v\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)+\text{tr}(\Sigma^{2})\right]^{2}$
(17) $\displaystyle\supset$ $\displaystyle
600\lambda_{1}v^{3}\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)$

from the second term in potential (8). Similarly,

$\lambda_{2}\,\text{tr}(\Sigma^{\prime 4})\supset
140\lambda_{2}v^{3}\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)$ (18)

is obtained from the third term in (8).
Next, we consider third-order term for $\Sigma$ in potential (8):

$\displaystyle\lambda_{1}\left\\{\text{tr}(\Sigma^{\prime 2})\right\\}^{2}$
$\displaystyle=$
$\displaystyle\lambda_{1}\left[30v^{2}+10v\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)+\text{tr}(\Sigma^{2})\right]^{2}$
(19) $\displaystyle\supset$ $\displaystyle
20\lambda_{1}v\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)\text{tr}(\Sigma^{2})$

The last term $\text{tr}(\Sigma^{2})$ is

$\text{tr}(\Sigma^{2})=\Sigma^{2}_{11}+\Sigma^{2}_{22}+\cdots+\Sigma^{2}_{55}+2(\Sigma_{12}\Sigma_{21}+\Sigma_{13}\Sigma_{31}+\cdots+\Sigma_{45}\Sigma_{54}),$
(20)

but the only meaningful term are only
$\Sigma^{2}_{11}+\Sigma^{2}_{22}+\cdots+\Sigma^{2}_{55}$, since the
$\pi\Sigma$ term has only a diagonal component
$\Sigma_{11},\Sigma_{22},\Sigma_{33}$ when performing the graph calculation.
$\Sigma^{2}_{44}+\Sigma^{2}_{55}$ can be rewritten as

$\Sigma^{2}_{44}+\Sigma^{2}_{55}=\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)^{2}-2\Sigma_{44}\Sigma_{55}$
(21)

using (14). Hence, the meaningful part is

$\displaystyle\text{tr}(\Sigma^{2})$ $\displaystyle\supset$
$\displaystyle\Sigma^{2}_{11}+\Sigma^{2}_{22}+\Sigma^{2}_{33}+\Sigma^{2}_{44}+\Sigma^{2}_{55}$
(22) $\displaystyle\supset$
$\displaystyle\Sigma^{2}_{11}+\Sigma^{2}_{22}+\Sigma^{2}_{33}+\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)^{2}$
$\displaystyle=$ $\displaystyle
2\left(\Sigma^{2}_{11}+\Sigma^{2}_{22}+\Sigma^{2}_{33}\right)+2\left(\Sigma_{11}\Sigma_{22}+\Sigma_{22}\Sigma_{33}+\Sigma_{33}\Sigma_{11}\right)$

and

$\displaystyle\lambda_{1}\left\\{\text{tr}(\Sigma^{\prime 2})\right\\}^{2}$
$\displaystyle\supset$ $\displaystyle
40\lambda_{1}v\left(\Sigma^{3}_{11}+\Sigma_{22}^{3}+\Sigma_{33}^{3}\right.$
(23)
$\displaystyle\left.+2\Sigma_{11}^{2}\Sigma_{22}+2\Sigma_{11}^{2}\Sigma_{33}+2\Sigma_{22}^{2}\Sigma_{11}+2\Sigma_{22}^{2}\Sigma_{33}+2\Sigma_{33}^{2}\Sigma_{22}+2\Sigma_{33}^{2}\Sigma_{11}\right.$
$\displaystyle\left.+3\Sigma_{11}\Sigma_{22}\Sigma_{33}\right)$

is obtained. Although there are cross terms among diagonal components, their
contributions to non-Gaussianity are comparable, therefore we only need to
consider $40\lambda_{1}v\Sigma^{3}_{11}$ in effect. We also omitted
$\Sigma_{44}\Sigma_{55}$, since they are functions of
$\Sigma_{11},\Sigma_{22}$ and $\Sigma_{33}$ and they give the same order of
contribution as the cross-terms. Furthermore, the third-order term of $\Sigma$
is also obtained from the term $\lambda_{2}\,\text{tr}(\Sigma^{\prime 4})$ of
potential (8), since it only yields the same order of contribution, we write
them collectively as $\alpha$ later. The necessary part of action for the
adjoint Higgs boson of SU(5)

$S_{\text{SU(5) Higgs}}=\int
d^{4}x\,\sqrt{-g}\left[\left(D_{\mu}\Sigma\right)^{\dagger}D^{\mu}\Sigma-V(\Sigma)\right]$
(24)

for calculation in Fig.2 is

$\displaystyle S_{\text{SU(5) Higgs}}$ $\displaystyle\supset$
$\displaystyle\int
d^{4}x\,\sqrt{-g}\left[\left(\partial_{\mu}\Sigma\right)^{\dagger}\partial^{\mu}\Sigma+10M^{2}v\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)\right.$
$\displaystyle-\left.\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\left(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}\right)+40\lambda_{1}v\left(\Sigma^{3}_{11}+\Sigma_{22}^{3}+\Sigma_{33}^{3}\right)\right].$

To calculate the graph in Fig.2, we have to extract the terms proportional to
$\pi$ and $\pi^{2}$ from the metric $\sqrt{-g}$. From appendix A, the metric
$\sqrt{-g}$ is

$\sqrt{-g}=a^{3}(\tilde{t})+3a^{3}H\pi-a^{3}\dot{\pi}+\frac{2}{9}a^{3}H^{2}\pi^{2}-3a^{3}H\pi\dot{\pi}+a^{3}\dot{\pi}^{2}$
(26)

after the transformation

$t\mapsto\tilde{t}=t-\pi(\tilde{t},\bm{x})$ (27)

of time coordinates. Thus, the interaction with the NG boson $\pi$ with no
derivative is

$\displaystyle\sqrt{-g}\left[10M^{2}v\left(\Sigma_{11}(t)+\Sigma_{22}(t)+\Sigma_{33}(t)\right)-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\left(\Sigma_{11}(t)+\Sigma_{22}(t)+\Sigma_{33}(t)\right)\right]$
$\displaystyle=$
$\displaystyle\sqrt{-g}\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)\left(\Sigma_{11}(t)+\Sigma_{22}(t)+\Sigma_{33}(t)\right)$
$\displaystyle=$ $\displaystyle
3\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)a^{3}H\left(1+\frac{\epsilon}{3}\right)\pi\left(\Sigma_{11}(\tilde{t})+\Sigma_{22}(\tilde{t})+\Sigma_{33}(\tilde{t})\right)$
$\displaystyle+\frac{2}{9}\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)a^{3}H^{2}\left(1+\frac{1}{3}\epsilon\right)\pi^{2}\left(\Sigma_{11}(\tilde{t})+\Sigma_{22}(\tilde{t})+\Sigma_{33}(\tilde{t})\right)$
$\displaystyle+3\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)a^{3}H\left(1+\frac{\epsilon}{3}\right)\pi^{2}\left(\dot{\Sigma}_{11}(\tilde{t})+\dot{\Sigma}_{22}(\tilde{t})+\dot{\Sigma}_{33}(\tilde{t})\right),$

and with time derivative is

$\displaystyle\sqrt{-g}\left[10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right]\left(\Sigma_{11}(t)+\Sigma_{22}(t)+\Sigma_{33}(t)\right)$
(29) $\displaystyle=$
$\displaystyle-a^{3}\left\\{10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right\\}\dot{\pi}\left(\Sigma_{11}(\tilde{t})+\Sigma_{22}(\tilde{t})+\Sigma_{33}(\tilde{t})\right)$
$\displaystyle-a^{3}H\left(3+2\epsilon\right)\left\\{10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right\\}\pi\dot{\pi}\left(\Sigma_{11}(\tilde{t})+\Sigma_{22}(\tilde{t})+\Sigma_{33}(\tilde{t})\right)$
$\displaystyle-a^{3}\left\\{10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right\\}\pi\dot{\pi}\left(\dot{\Sigma}_{11}(\tilde{t})+\dot{\Sigma}_{22}(\tilde{t})+\dot{\Sigma}_{33}(\tilde{t})\right)$
$\displaystyle+a^{3}\left\\{10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right\\}\dot{\pi}^{2}\left(\Sigma_{11}(\tilde{t})+\Sigma_{22}(\tilde{t})+\Sigma_{33}(\tilde{t})\right)$

Since $\Sigma_{11},\Sigma_{22}$ and $\Sigma_{33}$ are equivalent, we consider
$\Sigma_{11}$ in the following.

Next, we consider the propagator of $\Sigma$, since the interaction has been
obtained. The second order term of potential (8) is

$\displaystyle-M^{2}\text{tr}(\Sigma^{\prime
2})\supset-M^{2}\left(\Sigma_{11}\right)^{2},$ (30)
$\displaystyle\lambda_{1}\left\\{\text{tr}(\Sigma^{\prime
2})\right\\}^{2}\supset 160\lambda_{1}v^{2}\left(\Sigma_{11}\right)^{2},$ (31)
$\displaystyle\lambda_{2}\text{tr}(\Sigma^{\prime 4})\supset
24\lambda_{2}v^{2}\left(\Sigma_{11}\right)^{2}.$ (32)

Hence, by setting

$-m^{2}:=-M^{2}+160\lambda_{1}v^{2}+24\lambda_{2}v^{2},$ (33)

$\Sigma_{11}$ is a scalar field with mass $m^{2}$ in de Sitter spacetime,
therefore it has propagator

$\displaystyle G_{>}\left(\tau_{1},\tau_{2},k\right)$ $\displaystyle=$
$\displaystyle-i\frac{\sqrt{\pi}}{2}e^{i\pi\left(\nu/2+1/4\right)}H\left(-\tau_{1}\right)^{3/2}H^{(1)}_{\nu}(-k\tau_{1})\times$
(34) $\displaystyle
i\frac{\sqrt{\pi}}{2}e^{i\pi\left(\nu^{*}/2+1/4\right)}H\left(-\tau_{2}\right)^{3/2}H^{(2)}_{\nu}(-k\tau_{2})$
$\displaystyle=$
$\displaystyle-\frac{\pi}{4}e^{-\pi\text{Im}(\nu)}H^{2}\left(\tau_{1}\tau_{2}\right)^{3/2}H^{(1)}_{\nu}(-k\tau_{1})H^{(2)}_{\nu}(-k\tau_{2}),$
$\displaystyle G_{<}\left(\tau_{1},\tau_{2},k\right)$ $\displaystyle=$
$\displaystyle-\frac{\pi}{4}e^{-\pi\text{Im}(\nu)}H^{2}\left(\tau_{1}\tau_{2}\right)^{3/2}H^{(1)}_{\nu}(-k\tau_{2})H^{(2)}_{\nu}(-k\tau_{1}).$
(35)

from appendix B. From the viewpoint of effective field theory, since $m^{2}$
should be around the Hubble scale, $\lambda_{1}$ and $\lambda_{2}$ must be an
order of $\mathcal{O}(10^{-2})$.

##  3 Calculation of non-Gaussianity

We are now ready to calculate non-Gaussianity. Since the interaction of
$\Sigma_{11}$ and the interaction of $\dot{\Sigma}_{11}$ give contributions of
the same order to non-Gaussianity, only $\Sigma_{11}$ is considered and the
latter is doubled. First, consider interactions that have no time derivative.
According to the rule of in-in formalism, there exist four types of graphs,
$(+,+)$, $(+,-)$, $(-,+)$ and $(-,-)$ as shown in Fig.3, mediated by one
$\Sigma$ between the three inflatons $\pi$.

Figure 3: Four graphs in in-in formalism. Black circles indicate $+$ and white
circles indicate $-$.

As an example, let us consider the graphs of $(+,-)$ and $(-,+)$. The sum of
the two graphs is given as follows:

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle=$
$\displaystyle\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{1})d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{2})d\tau_{2}(-i)\frac{2}{9}\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)$
(37) $\displaystyle\times a^{3}(\tau_{1})H^{2}\times
3i\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)a^{3}(\tau_{2})H$
$\displaystyle\times
G_{+-}(\tau_{1},\tau_{2},p_{3})\Delta_{+-}(0,\tau_{1},p_{1})\Delta_{+-}(0,\tau_{1},p_{2})\Delta_{++}(\tau_{2},0,p_{3})+(p_{1}\leftrightarrow
p_{3})$ $\displaystyle+(p_{2}\leftrightarrow p_{3})+\text{c.c.}$
$\displaystyle=$
$\displaystyle\frac{2}{3}\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)^{2}H^{3}$
$\displaystyle\times\int_{-(1+i\epsilon)\infty}^{0}d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}d\tau_{2}\,a^{4}(\tau_{1})a^{4}(\tau_{2})G_{+-}(\tau_{1},\tau_{2},p_{3})\Delta_{+-}(0,\tau_{1},p_{1})$
$\displaystyle\times\Delta_{+-}(0,\tau_{1},p_{2})\Delta_{++}(\tau_{2},0,p_{3})+(p_{1}\leftrightarrow
p_{3})+(p_{2}\leftrightarrow p_{3})+\text{c.c.}$

Note that we neglect the terms including $\epsilon$ which are discussed in
Appendix A in three point functions of inflaton, since they make only sub-
leading contributions to non-Gaussianity. Now, substituting the propagator
(35) of $\Sigma$ and the propagators (6) and (7) of $\pi$, we have

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle=$
$\displaystyle-\frac{\pi}{2^{7}\cdot
3}e^{-\pi\text{Im}(\nu)}\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)^{2}\frac{1}{\left(H\epsilon
M_{\text{Pl}}c_{s}p_{1}p_{2}p_{3}\right)^{3}}$ (38)
$\displaystyle\times\int_{-(1+i\epsilon)\infty}^{0}d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}d\tau_{2}\,\left(\tau_{1}\tau_{2}\right)^{-5/2}H^{(1)}_{\nu}(-p_{3}\tau_{1})H^{(2)}_{\nu}(-p_{3}\tau_{2})$
$\displaystyle\times\left(1+ic_{s}p_{1}\tau_{1}\right)\left(1+ic_{s}p_{2}\tau_{1}\right)\left(1+ic_{s}p_{3}\tau_{2}\right)e^{-ic_{s}\left(p_{1}+p_{2}\right)\tau_{1}}e^{-ic_{s}p_{3}\tau_{2}}$
$\displaystyle+(p_{1}\leftrightarrow p_{3})+(p_{2}\leftrightarrow
p_{3})+\text{c.c.}$

As discussed in [100] and [101], we use the approximation of the Hankel
function

$\displaystyle
H^{(1)}_{\nu}(-p_{3}\tau_{2})\to-i\frac{2^{\nu}}{\pi}(-p_{3}\tau_{2})^{-\nu}\Gamma(\nu),$
(39) $\displaystyle H^{(2)}_{\nu}(-p_{3}\tau_{1})\to
i\frac{2^{\nu}}{\pi}(-p_{3}\tau_{1})^{-\nu}\Gamma(\nu)$ (40)

with horizon exit $-p_{3}\tau_{1},-p_{3}\tau_{2}\to 1$ to evaluate the
integral. This is an approximation to extract the effect that the contribution
is the largest as time evolves and the fluctuations freeze. Note that we
consider the region

$0<\nu\leq\frac{3}{2},$ (41)

in other words,

$0<\frac{m^{2}}{H^{2}}\leq\frac{9}{4}$ (42)

where the suppression factor $e^{-\pi\text{Im}(\nu)}$ does not appear.
Substituting approximations (39) and (40) of the Hankel functions into (38),
three point function of inflaton $\langle\pi\pi\pi\rangle$ becomes

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle=$
$\displaystyle-\frac{\Gamma^{2}(\nu)}{2^{7-2\nu}\cdot
3\pi}\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)^{2}\frac{1}{\left(H\epsilon
M_{\text{Pl}}c_{s}p_{1}p_{2}p_{3}\right)^{3}}\frac{1}{p_{3}^{2\nu}}$
$\displaystyle\times\int_{-(1+i\epsilon)\infty}^{0}d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}d\tau_{2}\,\left(\tau_{1}\tau_{2}\right)^{-5/2-\nu}e^{-ic_{s}\left(p_{1}+p_{2}\right)\tau_{1}}e^{-ic_{s}p_{3}\tau_{2}}$
$\displaystyle\times\left(1+ic_{s}p_{1}\tau_{1}\right)\left(1+ic_{s}p_{2}\tau_{1}\right)\left(1+ic_{s}p_{3}\tau_{2}\right)+(p_{1}\leftrightarrow
p_{3})+(p_{2}\leftrightarrow p_{3})+\text{c.c.}$

Note that the dependence on the external momentum $p_{3}$ is found to be non-
local. This implies that the contribution for the three point function
actually comes from the $\Sigma$ field. Non-Gaussianity is defined by

$\langle\zeta\zeta\zeta\rangle=(2\pi)^{7}\delta^{(3)}(\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3})\tilde{p}^{2}_{\zeta}\left(\frac{9}{10}f_{\text{NL}}\right)\frac{1}{(p_{1}p_{2}p_{3})^{2}}$
(44)

in the case where the configuration of the external momentum are equilateral,

$p:=p_{1}=p_{2}=p_{3}$ (45)

in the following. Note that $\zeta$ is the curvature fluctuation, which is
related to inflaton $\pi$ by $\zeta=-H\pi$. Since the integral only has an
effect up to the horizon exit $-p_{3}\tau_{1},-p_{3}\tau_{2}\to 1$, the upper
limit of the integral should be [101]

$\tau_{1*}=\tau_{2*}=-p^{-1}.$ (46)

In this case, the higher order term of the factor
$e^{-ic_{s}\left(p_{1}+p_{2}\right)\tau_{1}}e^{-ic_{s}p_{3}\tau_{2}}$ is a
small quantity, then the integral of can be calculated as follows:

$\displaystyle\int_{-(1+i\epsilon)\infty}^{0}d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}d\tau_{2}\,\left(\tau_{1}\tau_{2}\right)^{-5/2-\nu}\left(1+ic_{s}p_{1}\tau_{1}\right)\left(1+ic_{s}p_{2}\tau_{1}\right)\left(1+ic_{s}p_{3}\tau_{2}\right)$
(47) $\displaystyle\times
e^{-ic_{s}\left(p_{1}+p_{2}\right)\tau_{1}}e^{-ic_{s}p_{3}\tau_{2}}$
$\displaystyle=$
$\displaystyle\left(\frac{-2}{3+2\nu}-\frac{4ic_{s}}{1+2\nu}\right)\left(\frac{-2}{3+2\nu}-\frac{2ic_{s}}{1+2\nu}\right)p^{3+2\nu}.$

Therefore, the three point function of inflaton $\pi$ can be computed as

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle\simeq$
$\displaystyle-\frac{\Gamma^{2}(\nu)}{2^{7-2\nu}\cdot
3\pi}\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)^{2}\frac{1}{\left(H\epsilon
M_{\text{Pl}}c_{s}p_{1}p_{2}p_{3}\right)^{3}}\frac{1}{p_{3}^{2\nu}}$ (48)
$\displaystyle\times\left(\frac{-2}{3+2\nu}-\frac{4ic_{s}}{1+2\nu}\right)\left(\frac{-2}{3+2\nu}-\frac{2ic_{s}}{1+2\nu}\right)p^{3+2\nu}+(p_{1}\leftrightarrow
p_{3})+(p_{2}\leftrightarrow p_{3})+\text{c.c.}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma^{2}(\nu)}{2^{7-2\nu}\cdot
3\pi}\left(10M^{2}v+\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)^{2}$
$\displaystyle\times\frac{1}{\left(H\epsilon
M_{\text{Pl}}c_{s}\right)^{3}}p^{-6}\left(\frac{8}{(3+2\nu)^{2}}-\frac{16c^{2}_{s}}{(1+2\nu)^{2}}\right)\times
3$

Since the relation between the curvature fluctuation $\zeta$ and inflaton
$\pi$ is $\zeta=-H\pi$, the non-Gaussianity can be estimated as

$\displaystyle f_{\text{NL}}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\frac{1}{(2\pi)^{7}\tilde{p}^{2}_{\zeta}}p^{6}M^{-3}_{\text{Pl}}(-H)^{3}$
(49) $\displaystyle\times\frac{-\Gamma^{2}(\nu)}{2^{7-2\nu}\cdot
3\pi}\left(10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}\right)^{2}$
$\displaystyle\times\frac{1}{\left(H\epsilon
M_{\text{Pl}}c_{s}\right)^{3}}p^{-6}\left(\frac{8}{(3+2\nu)^{2}}-\frac{16c^{2}_{s}}{(1+2\nu)^{2}}\right)\times
3$

by reviving $M_{\text{Pl}}$ for $\zeta$. Here, the constant $c_{s}$ has the
relation $r=16\epsilon c_{s}$ with the tensor-scalar ratio $r$ and the upper
bound $r<0.036$ [102] gives

$\epsilon c_{s}\sim\mathcal{O}(10^{-3\sim-4}).$ (50)

Since the constant in (49) has the order of

$M_{\text{Pl}}\sim\mathcal{O}(10^{19})\,\text{GeV},\quad v\sim
M\sim\mathcal{O}(10^{15})\,\text{GeV},\quad\tilde{p}_{\zeta}\sim 6.1\times
10^{-9}$ (51)

respectively, the non-Gaussianity is found as

$f_{\text{NL}}\sim\mathcal{O}(10^{-4\sim-1})\times\alpha,$ (52)

where $\alpha$ represents the similar contributions from other components such
as $\Sigma_{22},\Sigma_{33}$ and $\dot{\Sigma}_{11}$ and the additional group
theoretical numerical factor that appears in extending to larger GUT gauge
groups. The order is $\alpha\sim\mathcal{O}(10^{0\sim 1})$.Since the current
observation limit (Fig.4) is

$|f_{\text{NL}}|\lesssim 1,$ (53)

this result is consistent with the observation and it might be possible to
detect the signature of the Higgs boson in GUT by 21cm spectrum, future LSS
and future CMB depending on our model parameters.

Figure 4: Schematic illustration of current and future constraints on the non-
Gaussianity (Figure taken from [103]).

Next, we consider interactions involving time derivatives. The sum of the
graphs of $(+,-)$ and $(-,+)$ is given as follows:

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle=$
$\displaystyle\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{1})d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{2})d\tau_{2}(-i)a^{3}(\tau_{1})Aia^{3}(\tau_{2})A$
(54) $\displaystyle\times
G_{+-}(\tau_{1},\tau_{2},p_{3})\frac{1}{a(\tau_{1})}\dot{\Delta}_{+-}(0,\tau_{1},p_{1})\frac{1}{a(\tau_{1})}\dot{\Delta}_{+-}(0,\tau_{1},p_{2})\frac{1}{a(\tau_{2})}\dot{\Delta}_{++}(\tau_{2},0,p_{3})$
$\displaystyle+(p_{1}\leftrightarrow p_{3})+(p_{2}\leftrightarrow
p_{3})+\text{c.c.}$

where $A$ is defined as

$A:=10M^{2}v-\left(600\lambda_{1}+140\lambda_{2}\right)v^{3}.$ (55)

Differentiating the propagation function
$\Delta_{++}\left(\tau_{2},0,p\right)$ with respect to the first time yields

$\displaystyle\dot{\Delta}_{++}\left(\tau_{2},0,p\right)$ $\displaystyle=$
$\displaystyle\dot{\Delta}_{>}\left(\tau_{2},0,p\right)\theta\left(\tau_{2}-0\right)+\Delta_{>}\left(\tau_{2},0,p\right)\delta\left(\tau_{2}-0\right)$
(56)
$\displaystyle+\dot{\Delta}_{<}\left(\tau_{2},0,p\right)\theta\left(0-\tau_{2}\right)-\Delta_{<}\left(\tau_{2},0,p\right)\delta\left(0-\tau_{2}\right)$
$\displaystyle=$ $\displaystyle\frac{1+ic_{s}p\tau_{2}}{4\epsilon
M_{\text{Pl}}c_{s}p^{3}}e^{-ic_{s}p\tau_{2}}\delta(\tau_{2})$
$\displaystyle+\left(\frac{-ic_{s}p}{4\epsilon
M_{\text{Pl}}c_{s}p^{3}}e^{ic_{s}p\tau_{2}}+ic_{s}p\frac{1-ic_{s}p\tau_{2}}{4\epsilon
M_{\text{Pl}}c_{s}p^{3}}e^{ic_{s}p\tau_{2}}\right)\theta(-\tau_{2})-\frac{1-ic_{s}p\tau_{2}}{4\epsilon
M_{\text{Pl}}c_{s}p^{3}}e^{ic_{s}p\tau_{2}}\delta(-\tau_{2})$ $\displaystyle=$
$\displaystyle\frac{1+ic_{s}p\tau_{2}}{4\epsilon
M_{\text{Pl}}c_{s}p^{3}}e^{-ic_{s}p\tau_{2}}\delta(\tau_{2})-\frac{1-ic_{s}p\tau_{2}}{4\epsilon
M_{\text{Pl}}c_{s}p^{3}}e^{ic_{s}p\tau_{2}}\delta(-\tau_{2})+\frac{c_{s}\tau_{2}}{4\epsilon
M_{\text{Pl}}p}e^{ic_{s}p\tau_{2}}\theta(-\tau_{2}),$

which leads to a simple form

$\dot{\Delta}_{++}\left(\tau_{2},0,p\right)=\frac{c_{s}\tau_{2}}{4\epsilon
M_{\text{Pl}}p}e^{ic_{s}p\tau_{2}}$ (57)

because of the $\tau_{2}$-integral. Similarly, differentiating the propagation
function $\Delta_{+-}(0,\tau_{1},p)$ with respect to the second time yields

$\displaystyle\dot{\Delta}_{+-}(0,\tau_{1},p)=\dot{\Delta}_{<}(0,\tau_{1},p)$
$\displaystyle=$ $\displaystyle\frac{ic_{s}p}{4\epsilon
M_{\text{Pl}}c_{s}p^{3}}e^{-ic_{s}p\tau_{1}}+(-ic_{s}p)\frac{1+ic_{s}p\tau_{1}}{4\epsilon
M_{\text{Pl}}c_{s}p^{3}}e^{-ic_{s}p\tau_{1}}$ (58) $\displaystyle=$
$\displaystyle\frac{c_{s}\tau_{1}}{4\epsilon
M_{\text{Pl}}p}e^{-ic_{s}p\tau_{1}}.$

Substituting these results, $\langle\pi\pi\pi\rangle$ becomes

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle=$
$\displaystyle\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{1})d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{2})d\tau_{2}(-i)a^{3}(\tau_{1})Aia^{3}(\tau_{2})AG_{+-}(\tau_{1},\tau_{2},p_{3})$
(59)
$\displaystyle\frac{1}{a(\tau_{1})}\dot{\Delta}_{+-}(0,\tau_{1},p_{1})\frac{1}{a(\tau_{1})}\dot{\Delta}_{+-}(0,\tau_{1},p_{2})\frac{1}{a(\tau_{2})}\dot{\Delta}_{++}(\tau_{2},0,p_{3})+(p_{1}\leftrightarrow
p_{3})+(p_{2}\leftrightarrow p_{3})+\text{c.c.}$ $\displaystyle=$
$\displaystyle\int d\tau_{1}\,\int
d\tau_{2}\,\left(-\frac{1}{H\tau_{1}}\right)^{2}\left(-\frac{1}{H\tau_{2}}\right)^{3}A^{2}\left(-\frac{\pi}{4}\right)e^{-\pi\text{Im}(\nu)}H^{2}\left(\tau_{1}\tau_{2}\right)^{3/2}H^{(1)}_{\nu}(-p\tau_{2})H^{(2)}_{\nu}(-p\tau_{1})$
$\displaystyle\times\frac{c_{s}\tau_{1}}{4\epsilon
M_{\text{Pl}}p_{1}}e^{-ic_{s}p_{1}\tau_{1}}\frac{c_{s}\tau_{1}}{4\epsilon
M_{\text{Pl}}p_{2}}e^{-ic_{s}p_{2}\tau_{1}}\frac{c_{s}\tau_{2}}{4\epsilon
M_{\text{Pl}}p_{3}}e^{ic_{s}p_{3}\tau_{2}}$
$\displaystyle+(p_{1}\leftrightarrow p_{3})+(p_{2}\leftrightarrow
p_{3})+\text{c.c.}$

Furthermore, using horizon exit approximations, we obtain

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle\sim$
$\displaystyle\int^{\tau_{*}}_{-(1+i\epsilon)\infty}d\tau_{1}\,\int^{\tau_{*}}_{-(1+i\epsilon)\infty}d\tau_{2}\,\frac{-1}{H^{5}\tau^{2}_{1}\tau^{3}_{2}}A^{2}$
(60)
$\displaystyle\times\left(-\frac{\pi}{4}\right)H^{2}\left(\tau_{1}\tau_{2}\right)^{3/2}(-i)\frac{2^{\nu}}{\pi}(-p_{3}\tau_{2})^{-\nu}\Gamma(\nu)i\frac{2^{\nu}}{\pi}(-p_{3}\tau_{1})^{-\nu}\Gamma(\nu)\left(\frac{c_{s}}{4\epsilon
M_{\text{Pl}}}\right)^{3}\frac{1}{p_{1}p_{2}p_{3}}\tau^{2}_{1}\tau_{2}$
$\displaystyle+(p_{1}\leftrightarrow p_{3})+(p_{2}\leftrightarrow
p_{3})+\text{c.c.}$ $\displaystyle=$
$\displaystyle\frac{2^{2\nu}}{4\pi}\Gamma^{2}(\nu)\frac{1}{H^{3}}\left(\frac{c_{s}}{4\epsilon
M_{\text{Pl}}}\right)^{3}A^{2}\frac{1}{p_{1}p_{2}p_{3}}p^{-2\nu}_{3}$
$\displaystyle\times\int^{\tau_{*}}_{-(1+i\epsilon)\infty}d\tau_{1}\,\int^{\tau_{*}}_{-(1+i\epsilon)\infty}d\tau_{2}\,\tau^{3/2}_{1}\tau^{-1/2}_{2}(\tau_{1}\tau_{2})^{-\nu}.$

Here, if we introduce the UV cutoff

$\tau_{\Lambda}=-\Lambda^{-1},$ (61)

we can calculate as

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle\sim$
$\displaystyle\frac{2^{2\nu}}{4\pi}\Gamma^{2}(\nu)\frac{1}{H^{3}}\left(\frac{c_{s}}{4\epsilon
M_{\text{Pl}}}\right)^{3}A^{2}\frac{1}{p_{1}p_{2}p_{3}}p^{-2\nu}_{3}\left.\frac{1}{5/2-\nu}\tau_{1}^{5/2-\nu}\right|^{\tau_{*}}_{\tau_{\Lambda}}\left.\frac{1}{1/2-\nu}\tau_{2}^{1/2-\nu}\right|^{\tau_{*}}_{\tau_{\Lambda}}$

Since the cutoff scale has a dimension of momentum, we can write

$\Lambda=\gamma p,$ (63)

using the dimensionless parameter $\gamma$. Therefore we can obtain

$\left.\frac{1}{5/2-\nu}\tau_{1}^{5/2-\nu}\right|^{\tau_{*}}_{\tau_{\Lambda}}\left.\frac{1}{1/2-\nu}\tau_{2}^{1/2-\nu}\right|^{\tau_{*}}_{\tau_{\Lambda}}=\frac{4}{\left(5-2\nu\right)\left(1-2\nu\right)}(-p)^{2\nu-3}\left(1-\gamma^{\nu-1/2}-\gamma^{\nu-5/2}+\gamma^{2\nu-3}\right).$
(64)

From the definition, non-Gaussianity is computed as

$\displaystyle f_{\text{NL}}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\frac{1}{(2\pi)^{7}\tilde{p}^{2}_{\zeta}}p^{6}M^{-3}_{\text{Pl}}(-H)^{3}\langle\pi\pi\pi\rangle$
(65) $\displaystyle=$
$\displaystyle\frac{10}{9}\frac{1}{(2\pi)^{7}\tilde{p}^{2}_{\zeta}}p^{6}M^{-3}_{\text{Pl}}(-H)^{3}$
$\displaystyle\times\frac{2^{2\nu}}{4\pi}\Gamma^{2}(\nu)\frac{1}{H^{3}}\left(\frac{c_{s}}{4\epsilon
M_{\text{Pl}}}\right)^{3}A^{2}\frac{1}{p^{3}}p^{-2\nu}$
$\displaystyle\times\frac{4}{\left(5-2\nu\right)\left(1-2\nu\right)}(-p)^{2\nu-3}\left(1-\gamma^{\nu-1/2}-\gamma^{\nu-5/2}+\gamma^{2\nu-3}\right)$
$\displaystyle\sim$ $\displaystyle
10^{-12}\left(\frac{c_{s}}{\epsilon}\right)^{3}\left(1-\gamma^{\nu-1/2}-\gamma^{\nu-5/2}+\gamma^{2\nu-3}\right).$

Now, recalling that

$\epsilon c_{s}\sim 10^{-3\sim-4},$ (66)

we have

$f_{\text{NL}}\sim 10^{-3\sim
0}c^{6}_{s}\left(1-\gamma^{\nu-1/2}-\gamma^{\nu-5/2}+\gamma^{2\nu-3}\right).$
(67)

From the viewpoint of effective field theory, the cutoff is the inflationary
scale, and $\gamma$ is

$\gamma\sim 1.$ (68)

In this case, non-Gaussianity has the width of

$f_{\text{NL}}\sim 10^{-3\sim 0}c^{6}_{s}.$ (69)

$c_{s}$ is a quantity such that it is 1 in the simplest model, and is not
expected to vary significantly in order estimation. This result is comparable
to that of case (40) where the time derivative is not included. The reasons
are shown in the following table.

| factor | propagator
---|---|---
$\langle\pi\pi\pi\rangle$ | $\frac{2}{9}AH^{2}\left(1+\frac{\epsilon}{3}\right)\times 3AH\left(1+\frac{\epsilon}{3}\right)$ | $G_{+-}\Delta_{+-}\Delta_{+-}\Delta_{++}$
$\langle\pi\pi\dot{\pi}\rangle$ | $\frac{2}{9}AH^{2}\left(1+\frac{\epsilon}{3}\right)\times A$ | $G_{+-}\Delta_{+-}\Delta_{+-}\dot{\Delta}_{++}$
| $AH\left(3+2\epsilon\right)\times 3AH\left(1+\frac{\epsilon}{3}\right)$ | $G_{+-}\Delta_{+-}\dot{\Delta}_{+-}\Delta_{++}$
$\langle\pi\dot{\pi}\dot{\pi}\rangle$ | $A\times 3AH\left(1+\frac{\epsilon}{3}\right)$ | $G_{+-}\dot{\Delta}_{+-}\dot{\Delta}_{+-}\Delta_{++}$
| $AH\left(3+2\epsilon\right)\times A$ | $G_{+-}\dot{\Delta}_{+-}\Delta_{+-}\dot{\Delta}_{++}$
$\langle\dot{\pi}\dot{\pi}\dot{\pi}\rangle$ | $A\times A$ | $G_{+-}\dot{\Delta}_{+-}\dot{\Delta}_{+-}\dot{\Delta}_{++}$
Table 1: The factors and propagator for three point function of inflaton with
various number of time derivatives.

Table 1 shows the coefficients and propagator for three point function of
inflaton with various number of time derivatives. In the table, the case where
the three point function of inflaton without a time derivative is denoted by
$\langle\pi\pi\pi\rangle$, with one, two and three time derivatives by
$\langle\pi\pi\dot{\pi}\rangle$ , $\langle\pi\dot{\pi}\dot{\pi}\rangle$, and
$\langle\dot{\pi}\dot{\pi}\dot{\pi}\rangle$. For example,
$\langle\pi\pi\pi\rangle$ represents equation (37) and
$\langle\dot{\pi}\dot{\pi}\dot{\pi}\rangle$ represents equation (54). Taking
into account that $a^{-1}=H\tau$ is added when performing the time derivative
of the propagator, the coefficients are all of the form $A^{2}H^{3}$. The
coefficients except for $A^{2}H^{3}$ of the interaction are all
$\mathcal{O}(1)$, and the propagator only differs by about $\mathcal{O}(1)$ as
long as the cutoff is set to $\Lambda=\gamma p$. Thus, they all give the same
contribution to non-Gaussianity.

Next, we consider the graph created by the three-point interaction of $\Sigma$
(Fig.5).

Figure 5: The graph generated by the three-point interaction of $\Sigma$.
There are $2^{4}$ in total, and this is an example, representing the graph of
$(-,-,-,+)$.

Let $\tau_{4}$ be the time for the part of 3-point interaction and
$\tau_{1},\tau_{2},\tau_{3}$ be the time for the rest of $\pi\Sigma$
interaction. According to the rule of in-in formalism, there exist $2^{4}$
types of graphs. Consider the $(-,-,-,+)$ graphs and its complex conjugation
$(+,+,+,-)$ in the order $(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$. The case
where the interaction does not have time derivative is written down as

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle=$
$\displaystyle\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{1})d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{2})d\tau_{2}\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{3})d\tau_{3}\int_{-(1+i\epsilon)\infty}^{0}a(\tau_{4})d\tau_{4}$
$\displaystyle\times(-i)a(\tau_{1})3a^{3}(\tau_{1})AH\left(1+\frac{\epsilon}{3}\right)\Delta_{+-}(0,\tau_{1},p_{1})G_{-+}(\tau_{1},\tau_{4},p_{1})$
$\displaystyle\times(-i)a(\tau_{2})3a^{3}(\tau_{2})AH\left(1+\frac{\epsilon}{3}\right)\Delta_{+-}(0,\tau_{2},p_{2})G_{-+}(\tau_{2},\tau_{4},p_{2})$
$\displaystyle\times
ia(\tau_{4})(-1)a^{3}(\tau_{4})40\lambda_{1}vG_{+-}(\tau_{4},\tau_{3},p_{3})$
$\displaystyle\times(-i)a(\tau_{3})3a^{3}(\tau_{3})AH\left(1+\frac{\epsilon}{3}\right)\Delta_{-+}(\tau_{3},0,p_{3})+(p_{1}\leftrightarrow
p_{3})+(p_{2}\leftrightarrow p_{3})+\text{c.c.}$

Substituting the propagator, we obtain

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle=$ $\displaystyle
40\times
3^{3}\lambda_{1}v\left(1+\frac{\epsilon}{3}\right)^{3}A^{3}H^{-7}\left(\frac{1}{4\epsilon
M_{\text{Pl}}c_{s}}\right)^{3}\left(-\frac{\pi}{4}\right)^{3}e^{-3\pi\text{Im}(\nu)}\left(\frac{1}{p_{1}p_{2}p_{3}}\right)^{3}$
(71)
$\displaystyle\times\int_{-(1+i\epsilon)\infty}^{0}d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{0}d\tau_{2}\int_{-(1+i\epsilon)\infty}^{0}d\tau_{3}\int_{-(1+i\epsilon)\infty}^{0}d\tau_{4}\left(1+ic_{s}p_{1}\tau_{1}\right)\left(1+ic_{s}p_{2}\tau_{2}\right)\left(1+ic_{s}p_{3}\tau_{3}\right)$
$\displaystyle\times
e^{-ic_{s}p_{1}\tau_{1}}e^{-ic_{s}p_{2}\tau_{2}}e^{-ic_{s}p_{3}\tau_{3}}\left(\tau_{1}\tau_{2}\tau_{3}\right)^{-5/2}\tau_{4}^{1/2}$
$\displaystyle\times
H^{(1)}_{\nu}(-p_{1}\tau_{1})H^{(1)}_{\nu}(-p_{2}\tau_{2})H^{(1)}_{\nu}(-p_{3}\tau_{3})H^{(2)}_{\nu}(-p_{1}\tau_{4})H^{(2)}_{\nu}(-p_{2}\tau_{4})H^{(2)}_{\nu}(-p_{3}\tau_{4})$
$\displaystyle+(p_{1}\leftrightarrow p_{3})+(p_{2}\leftrightarrow
p_{3})+\text{c.c.}$

Using horizon exit approximations, we can compute

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle\simeq$
$\displaystyle-6\times 40\times
3^{3}\lambda_{1}v\left(1+\frac{\epsilon}{3}\right)^{3}A^{3}H^{-7}\left(\frac{1}{4\epsilon
M_{\text{Pl}}c_{s}}\right)^{3}\left(\frac{\pi}{4}\right)^{3}\left(\frac{2^{\nu}}{\pi}\right)^{6}\Gamma^{6}(\nu)\left(\frac{1}{p_{1}p_{2}p_{3}}\right)^{3}$
$\displaystyle\times\int_{-(1+i\epsilon)\infty}^{\tau_{1*}}d\tau_{1}\,\int_{-(1+i\epsilon)\infty}^{\tau_{2*}}d\tau_{2}\int_{-(1+i\epsilon)\infty}^{\tau_{3*}}d\tau_{3}\int_{-(1+i\epsilon)\infty}^{\tau_{4*}}d\tau_{4}$
$\displaystyle\times\left(\tau_{1}\tau_{2}\tau_{3}\right)^{-5/2}\tau_{4}^{1/2}\left(-p_{1}\tau_{1}\right)^{-\nu}\left(-p_{2}\tau_{2}\right)^{-\nu}\left(-p_{3}\tau_{3}\right)^{-\nu}\left(-p_{1}\tau_{4}\right)^{-\nu}\left(-p_{2}\tau_{4}\right)^{-\nu}\left(-p_{3}\tau_{4}\right)^{-\nu}$

The first factor 6 comes from momentum exchange and contributions from complex
conjugation. Now, if we insert the cutoff $\Lambda=\gamma p$ for $\tau_{4}$,
we obtain

$\displaystyle\langle\pi\pi\pi\rangle$ $\displaystyle\simeq$ $\displaystyle
6\times 40\times
3^{3}\lambda_{1}v\left(1+\frac{\epsilon}{3}\right)^{3}A^{3}H^{-7}\left(\frac{1}{4\epsilon
M_{\text{Pl}}c_{s}}\right)^{3}\left(\frac{\pi}{4}\right)^{3}\left(\frac{2^{\nu}}{\pi}\right)^{6}\Gamma^{6}(\nu)\left(\frac{1}{p_{1}p_{2}p_{3}}\right)^{3}$
(73)
$\displaystyle\times\left(\frac{2}{3+2\nu}\right)^{3}\frac{2}{3-6\nu}(p_{1}p_{2})^{3/2-\nu}(-1)^{3\nu-3/2}p_{3}^{2\nu}\left(1-\gamma^{3\nu-3/2}\right),$

thus non-Gaussianity is given by

$\displaystyle f_{\text{NL}}$ $\displaystyle=$
$\displaystyle\frac{10}{9}\frac{1}{(2\pi)^{7}\tilde{p}^{2}_{\zeta}}p^{6}M^{-3}_{\text{Pl}}(-H)^{3}\langle\pi\pi\pi\rangle$
(74) $\displaystyle=$
$\displaystyle\frac{10}{9}\frac{1}{(2\pi)^{7}\tilde{p}^{2}_{\zeta}}p^{6}M^{-3}_{\text{Pl}}(-H)^{3}$
$\displaystyle\times 6\times 40\times
3^{3}\lambda_{1}vA^{3}H^{-7}\left(\frac{1}{4\epsilon
M_{\text{Pl}}c_{s}}\right)^{3}\left(\frac{\pi}{4}\right)^{3}\left(\frac{2^{\nu}}{\pi}\right)^{6}\Gamma^{6}(\nu)\left(\frac{1}{p_{1}p_{2}p_{3}}\right)^{3}$
$\displaystyle\times\left(\frac{2}{3+2\nu}\right)^{3}\frac{2}{3-6\nu}(p_{1}p_{2})^{3/2-\nu}(-1)^{3\nu-3/2}p_{3}^{2\nu}\left(1-\gamma^{3\nu-3/2}\right).$

Substituting numerical values as before gives the result

$|f_{\text{NL}}|=\mathcal{O}(10^{-3\sim 0}).$ (75)

The calculations for the case where the interaction involves time derivatives
can be performed similarly to the discussion in Table 1, and they all yield
the same contribution to non-Gaussianity.

From the above, non-Gaussianity has the form

$|f_{\text{NL}}|=\mathcal{O}(10^{-3\sim 0})\times\alpha$ (76)

in summary. $\alpha$ is the contribution from components other than
$\Sigma_{11}$, or the additional group theoretical numerical factor that
appears in extending to larger GUT gauge groups, and so on, that is the
quantity summarizes the factors that bring about the same level of
contribution, and has an order of magnitude of $\mathcal{O}(10^{0\sim 1})$.

##  4 Conclusion

The Standard Model of elementary particles, which has successfully explained
many physical phenomena, is probably one of the most successful physical
theories. However, the nature is full of rich phenomena that cannot be
explained by the Standard Model alone. GUT is one of the attempts to describe
these interesting phenomena. GUT is a fascinating theory that unifies the
three interactions that exist in nature: strong interaction, electromagnetic
interaction, and weak interaction. Although many researchers have tried to
verify it, it has not yet been confirmed. For example, we have been looking
for proton decay in Super Kamiokande, but have not obtained that reaction. In
addition, it is difficult to investigate by using accelerators because the GUT
scale is very high energy ($10^{15}$GeV). For this reason, Cosmological
Collider Physics has been the focus of much attention in recent years.
Cosmological Collider Physics is a method to obtain information on elementary
particles by using the effective field theory of inflation. Quantum
fluctuations generated in the short time after the birth of the universe are
stretched by inflation. It appears in the form of non-Gaussianity by observing
the cosmic microwave background radiation. This means that Cosmological
Collider Physics is a very interesting way to obtain information on high
energy elementary particles that cannot be reached by terrestrial accelerators
by means of precise observation of the universe.

In this paper, we focus on the case where the energy scale of the inflation is
close to the GUT scale, and discuss if the GUT can be verified by calculating
the non-Gaussianity due to the Higgs boson in GUT. Concretely, in addition to
the effective action of inflation, we considered the action of the adjoint
Higgs scalar field in SU(5) GUT. A characteristic feature of this model is
that the Higgs boson has a vacuum expectation value due to spontaneous
symmetry breaking, which leads to linear interactions of Higgs boson with the
inflation. Using these interactions, the three point function of the inflatons
is generated by the tree level exchange of the adjoint Higgs boson. The graphs
contributing to the inflaton three point function can be computed by
performing horizon exit approximation, and non-Gaussianity is evaluated from
the obtained values. As a result, we have shown

$|f_{\text{NL}}|\lesssim 1$ (77)

for non-Gaussianity without a drastic fine-tuning of parameters. This result
is consistent with the current observed limit and suggests the existence of
the adjoint Higgs boson in GUT and it might be possible to detect the
signature of the Higgs boson in GUT by 21cm spectrum, future LSS and future
CMB depending on our model parameters.

##  Appendix A Calculation of $\sqrt{-g}$

For the determinant of the metric, starting from the curvature fluctuation in
unitary gauge $\zeta(t,\bm{x})$, we derive inflaton We can write determinant
of the metric in unitary gauge as

$\sqrt{-g}=Na^{3}(t)=\left(1+\frac{1}{H(t)}\frac{d}{dt}\zeta(t)\right)a^{3}(t)$
(78)

using ADM formalism. Curvature fluctuations $\zeta(t,\bm{x})$ and NG boson
$\pi(\tilde{t},\bm{x})$ are related by the relation

$\zeta(t,\bm{x})=-H(\tilde{t})\pi(\tilde{t},\bm{x})$ (79)

by the transformation of time

$t\mapsto\tilde{t}=t-\pi(\tilde{t},\bm{x}).$ (80)

By rewriting the derivative of $\zeta(t,\bm{x})$ with respect to $t$ into the
derivative with respect to $\tilde{t}$, we obtain

$\displaystyle\frac{d}{dt}\zeta(t,\bm{x})$ $\displaystyle=$
$\displaystyle\frac{d\tilde{t}}{dt}\frac{d}{d\tilde{t}}\left(-H(\tilde{t})\pi(\tilde{t},\bm{x})\right)$
(81) $\displaystyle=$
$\displaystyle-\left[1-\frac{d\tilde{t}}{dt}\dot{\pi}(\tilde{t})\right]\left(\dot{H}\pi+H\dot{\pi}\right)$
$\displaystyle=$
$\displaystyle-\left(1-\dot{\pi}+\dot{\pi}^{2}\right)\left(\dot{H}\pi+H\dot{\pi}\right)$
$\displaystyle=$
$\displaystyle\dot{H}(\tilde{t})\left(-\pi+\pi\dot{\pi}\right)+H(\tilde{t})\left(-\dot{\pi}+\dot{\pi}^{2}\right)$

up to the second order of $\pi$, where the dot represents the derivative with
respect to $\tilde{t}$. After applying the time transformation to $H^{-1}(t)$,
we obtain

$H^{-1}(t)=H^{-1}(\tilde{t}+\pi)=\left(H(\tilde{t})+\dot{H}(\tilde{t})\pi+\frac{1}{2}\ddot{H}(\tilde{t})\pi^{2}\right)^{-1}$
(82)

which can be written as

$\displaystyle H^{-1}(t)$ $\displaystyle\simeq$
$\displaystyle\left(H(\tilde{t})+\dot{H}(\tilde{t})\pi\right)^{-1}$ (83)
$\displaystyle=$
$\displaystyle\frac{1}{H(\tilde{t})}\left(1+\frac{\dot{H}(\tilde{t})}{H(\tilde{t})}\pi\right)^{-1}$
$\displaystyle=$
$\displaystyle\frac{1}{H(\tilde{t})}\left(1-H(\tilde{t})\epsilon(\tilde{t})\pi\right)^{-1}$
$\displaystyle=$
$\displaystyle\frac{1}{H(\tilde{t})}+\epsilon(\tilde{t})\pi+H\epsilon^{2}\pi^{2}$

Note that $\ddot{H}$ is in the second order of $\epsilon=-\dot{H}/H^{2}$.
Similarly, applying the time transformation to the scale factor $a(t)$, we
obtain

$\displaystyle a^{3}(t)=a^{3}(\tilde{t}+\pi)$ $\displaystyle=$
$\displaystyle\left[a(\tilde{t})+\dot{a}(\tilde{t})\pi+\frac{1}{2}\ddot{a}(\tilde{t})\pi^{2}\right]^{3}$
(84) $\displaystyle=$ $\displaystyle
a^{3}(\tilde{t})+3a^{2}(\tilde{t})\dot{a}(\tilde{t})\pi(\tilde{t})+\frac{3}{2}a^{2}\ddot{a}\pi^{2}+3a\dot{a}^{2}\pi^{2}.$

Putting them together, the determinant of the metric (78) expands to

$\displaystyle\sqrt{-g}$ $\displaystyle=$
$\displaystyle\left(1+\frac{1}{H(t)}\frac{d}{dt}\zeta(t,\bm{x})\right)a^{3}(t)$
(85) $\displaystyle=$
$\displaystyle\left[1+\frac{\dot{H}(\tilde{t})}{H(\tilde{t})}\left\\{-\pi+\pi\dot{\pi}\right\\}-\dot{H}\epsilon\pi^{2}-\dot{\pi}+\dot{\pi}^{2}-H\epsilon\pi\dot{\pi}\right]$
$\displaystyle\times\left[a^{3}(\tilde{t})+3a^{2}(\tilde{t})\dot{a}(\tilde{t})\pi(\tilde{t})+\frac{3}{2}a^{2}\ddot{a}\pi^{2}+3a\dot{a}^{2}\pi^{2}\right]$
$\displaystyle=$ $\displaystyle
a^{3}(\tilde{t})+3a^{2}\dot{a}\pi+\frac{3}{2}a^{2}\ddot{a}\pi^{2}+3a\dot{a}^{2}\pi^{2}-\frac{\dot{H}}{H}\left(a^{3}\pi+3a^{2}\dot{a}\pi^{2}\right)+\frac{\dot{H}}{H}a^{3}\pi\dot{\pi}$
$\displaystyle-\dot{H}\epsilon
a^{3}\pi^{2}-a^{3}\dot{\pi}-3a^{2}\dot{a}\pi\dot{\pi}+a^{3}\dot{\pi}^{2}-H\epsilon
a^{3}\pi\dot{\pi}.$

Using the relation

$\dot{a}=aH,\quad\ddot{a}=aH^{2}+a\dot{H},$ (86)

and expressing the derivative of the scale factor $a$ in terms of the Hubble
$H$, we can obtain

$\displaystyle\sqrt{-g}$ $\displaystyle=$ $\displaystyle
a^{3}(\tilde{t})+3a^{3}H\pi+\frac{3}{2}a^{3}\left(H^{2}+\dot{H}\right)\pi^{2}+3a^{3}H^{2}\pi^{2}$
$\displaystyle-\frac{\dot{H}}{H}a^{3}\pi-3a^{3}\dot{H}\pi^{2}+\frac{\dot{H}}{H}a^{3}\pi\dot{\pi}-\dot{H}\epsilon
a^{3}\pi^{2}-a^{3}\dot{\pi}-3a^{3}H\pi\dot{\pi}+a^{3}\dot{\pi}^{2}-H\epsilon
a^{3}\pi\dot{\pi}$ $\displaystyle=$ $\displaystyle
a^{3}(\tilde{t})+3a^{3}H\left(1+\frac{\epsilon}{3}\right)\pi-a^{3}\dot{\pi}+a^{3}H^{2}\left(\frac{9}{2}+\frac{3}{2}\epsilon\right)\pi^{2}-a^{3}H\left(3+2\epsilon\right)\pi\dot{\pi}+a^{3}\dot{\pi}^{2}.$

##  Appendix B In-in formalism

In this section, we briefly describe the propagators and Feynman diagram in
the in-in formalism (Schwinger-Keldysh formalism), which is a formalism
describing non-equilibrium systems appearing in the inflationary universe. For
details, please refer to [104].

For simplicity, we now take a real scalar field $\varphi$ as an example, but
the generalization to other fields is straightforward. To consider the
generating function for a real scalar field $\varphi$ in in-in formalism, we
need two fields $\varphi_{\pm}(\eta,\bm{x})$ and their corresponding sources
$J_{\pm}(\eta,\bm{x})$.

$Z\left[J_{+},J_{-}\right]=\int\mathcal{D}\varphi_{+}\mathcal{D}\varphi_{-}\exp\left[i\int_{\eta_{0}}^{\eta_{f}}\mathrm{d}\eta
d^{3}x\left(\mathscr{L}_{\mathrm{cl}}\left[\varphi_{+}\right]-\mathscr{L}_{\mathrm{cl}}\left[\varphi_{-}\right]+J_{+}\varphi_{+}-J_{-}\varphi_{-}\right)\right].$
(88)

This is the form of the generating functional in flat spacetime with the
addition of the complex conjugate contribution. The reason for adding such
contributions is that the expectation values of physical quantities in flat
spacetime field theories are calculated assuming that the initial vacuum and
the final vacuum are in the same state, whereas in inflationary spacetime the
initial vacuum and the final vacuum are in different states. The physical
meaning of this operation, which changes $i$ to $-i$, is that the time
evolution from the initial state to the final state and backward in time from
there, which implies that the correct time evolution contribution of the
vacuum is taken into account. Once written in a form that incorporates this
contribution, we can proceed in the same way as in flat spacetime field
theories. Dividing the Lagrangian into the free field part $\mathscr{L}_{0}$
and the interaction part $\mathscr{L}_{\mathrm{int}}$, we express the
generating functional as follows.

$\displaystyle Z\left[J_{+},J_{-}\right]$ $\displaystyle=$
$\displaystyle\exp\left[i\int_{\eta_{0}}^{\eta_{f}}d\eta
d^{3}x\left(\mathscr{L}_{\mathrm{int}}\left[\frac{\delta}{i\delta
J_{+}}\right]-\mathscr{L}_{\mathrm{int}}\left[-\frac{\delta}{i\delta
J_{-}}\right]\right)\right]Z_{0}\left[J_{+},J_{-}\right],$ (89) $\displaystyle
Z_{0}\left[J_{+},J_{-}\right]$ $\displaystyle\equiv$
$\displaystyle\int\mathcal{D}\varphi_{+}\mathcal{D}\varphi_{-}\exp\left[i\int_{\eta_{0}}^{\eta_{f}}d\eta
d^{3}x\left(\mathscr{L}_{0}\left[\varphi_{+}\right]-\mathscr{L}_{0}\left[\varphi_{-}\right]+J_{+}\varphi_{+}-J_{-}\varphi_{-}\right)\right].$

The difference from the flat spacetime field theory is that there are four
kinds of propagators, according to the subscripts of $+$ and $-$.

$-iG_{ab}\left(\eta_{1},\bm{x}_{1};\eta_{2},\bm{x}_{2}\right)\equiv\left.\frac{\delta}{ia\delta
J_{a}\left(\eta_{1},\bm{x}_{1}\right)}\frac{\delta}{ib\delta
J_{b}\left(\eta_{2},\bm{x}_{2}\right)}Z_{0}\left[J_{+},J_{-}\right]\right|_{J_{\pm}=0},$
(91)

where $a$ and $b$ denote $a,b=\pm$. To give an example, a concrete propagator
of type $(+,+)$ is

$\displaystyle-iG_{++}\left(\eta_{1},\bm{x}_{1};\eta_{2},\bm{x}_{2}\right)$
$\displaystyle=$ $\displaystyle\left.\frac{\delta}{i\delta
J_{+}\left(\eta_{1},\bm{x}_{1}\right)}\frac{\delta}{i\delta
J_{+}\left(\eta_{2},\bm{x}_{2}\right)}Z_{0}\left[J_{+},J_{-}\right]\right|_{J_{\pm}=0}$
(92) $\displaystyle=$
$\displaystyle\int\mathcal{D}\varphi_{+}\mathcal{D}\varphi_{-}\varphi_{+}\left(\eta_{1},\bm{x}_{1}\right)\varphi_{+}\left(\eta_{2},\bm{x}_{2}\right)e^{i\int
d\eta
d^{3}x\left(\mathscr{L}_{0}\left[\varphi_{+}\right]-\mathscr{L}_{0}\left[\varphi_{-}\right]\right)}$
$\displaystyle=$
$\displaystyle\left\langle\Omega\left|\mathrm{T}\left\\{\varphi\left(\eta_{1},\bm{x}_{1}\right)\varphi\left(\eta_{2},\bm{x}_{2}\right)\right\\}\right|\Omega\right\rangle,$

where T is the time-ordered product operator. Using the translational and
rotational symmetries at each time in the background spacetime, we can perform
the Fourier transformation and move to the three-dimensional momentum space.
We can obtain the propagators in momentum space

$\Delta_{ab}\left(\eta_{1},\eta_{2},k\right)=-i\int
d^{3}xe^{-i\bm{k}\cdot\bm{x}}G_{ab}\left(\eta_{1},\bm{x};\eta_{2},\bm{0}\right),$
(93)

where $-i$ is added to simplify the Feynman rule in momentum space. Here
$k=|\bm{k}|$ is the magnitude of the three-dimensional momentum, and the
propagator in momentum space $\Delta_{ab}\left(\eta_{1},\eta_{2},k\right)$ is
a function of magnitude due to the rotational symmetry, independent of the
direction of the three-dimensional momentum. The field $\varphi$ can be
represented by a mode function $v(\eta,\bm{k})$ and creation and annihilation
operators for a given three-dimensional momentum $\bm{k}$. The four types of
propagators are concretely written as

$\displaystyle\Delta_{++}\left(\eta_{1},\eta_{2},k\right)$ $\displaystyle=$
$\displaystyle\Delta_{>}\left(\eta_{1},\eta_{2},k\right)\theta\left(\eta_{1}-\eta_{2}\right)+\Delta_{<}\left(\eta_{1},\eta_{2},k\right)\theta\left(\eta_{2}-\eta_{1}\right),$
(94) $\displaystyle\Delta_{+-}\left(\eta_{1},\eta_{2},k\right)$
$\displaystyle=$ $\displaystyle\Delta_{<}\left(\eta_{1},\eta_{2},k\right),$
(95) $\displaystyle\Delta_{-+}\left(\eta_{1},\eta_{2},k\right)$
$\displaystyle=$ $\displaystyle\Delta_{>}\left(\eta_{1},\eta_{2},k\right),$
(96) $\displaystyle\Delta_{--}\left(\eta_{1},\eta_{2},k\right)$
$\displaystyle=$
$\displaystyle\Delta_{<}\left(\eta_{1},\eta_{2},k\right)\theta\left(\eta_{1}-\eta_{2}\right)+\Delta_{>}\left(\eta_{1},\eta_{2},k\right)\theta\left(\eta_{2}-\eta_{1}\right),$
(97)

where $\Delta_{>}\left(\eta_{1},\eta_{2},k\right)$ and
$\Delta_{<}\left(\eta_{1},\eta_{2},k\right)$ are defined by

$\displaystyle\Delta_{>}\left(\eta_{1},\eta_{2},k\right)\equiv
v\left(\eta_{1},k\right)v^{*}\left(\eta_{2},k\right),$ (98)
$\displaystyle\Delta_{<}\left(\eta_{1},\eta_{2},k\right)\equiv
v^{*}\left(\eta_{1},k\right)v\left(\eta_{2},k\right)$ (99)

and $\theta(\eta)$ is a step function of $\eta$. To represent these four
propagators in terms of a diagram, we can use black and white circles for $+$
and $-$, respectively (Fig.6).

Figure 6: Graphical representation for an internal line in in-in formalism.

These are graphical rules for internal lines. The external line connects the
time slice with end time $\eta=\eta_{f}$ (boundary point) to the other time
slice. Since the boundary points do not have the distinction between $+$ and
$-$, there are only two types of propagators for the external lines, which are
called as bulk-to-boundary propagators. To represent this boundary point, we
use a square in a diagrams (Fig.7).

Figure 7: Graphical representation for an external line in in-in formalism.

As an example, we consider the propagator of a complex scalar field $\phi$
with mass $m$ in the de Sitter spacetime. By using conformal time $\eta$ and
the Minkowski metric $\eta^{\mu\rho}$ to write down the action, we have

$S=\int
d^{4}x\,\left[\eta^{\mu\rho}\phi^{*}\partial_{\mu}\partial_{\rho}\phi-a^{2}m^{2}|\phi|^{2}\right].$
(100)

From this action, we obtain the equation of motion

$\left(\eta^{\mu\rho}\partial_{\mu}\partial_{\rho}-a^{2}m^{2}\right)\phi=0,$
(101)

which is the Klein-Gordon equation with mass $a^{2}m^{2}$. Due to the
rotational symmetry, we can transform the scalar field $\phi$ into three
dimensional momentum space and write its mode function as $u(\eta,k)$, the
Klein-Gordon equation (101) is

$\frac{\partial^{2}u}{\partial\eta^{2}}(\eta,k)+\left(k^{2}+\frac{m^{2}}{H^{2}\eta^{2}}\right)u(\eta,k)=0.$
(102)

The solution to this Klein-Gordon equation is

$u(\eta,k)=-i\frac{\sqrt{\pi}}{2}e^{i\pi\left(\nu/2+1/4\right)}\left(-\eta\right)^{\frac{1}{2}}H_{\nu}(-k\eta),$
(103)

where $H_{\nu}(-k\eta)$ is a Hankel function of the first kind and we define
the index

$\nu\equiv\sqrt{\frac{1}{4}-\left(\frac{m}{H}\right)^{2}}.$ (104)

Therefore, a scalar field with mass $m^{2}$ in de Sitter spacetime has
propagators

$\displaystyle G_{>}\left(\eta_{1},\eta_{2},k\right)$ $\displaystyle=$
$\displaystyle-i\frac{\sqrt{\pi}}{2}e^{i\pi\left(\nu/2+1/4\right)}H\left(-\eta_{1}\right)^{3/2}H^{(1)}_{\nu}(-k\eta_{1})\times$
(105) $\displaystyle
i\frac{\sqrt{\pi}}{2}e^{i\pi\left(\nu^{*}/2+1/4\right)}H\left(-\eta_{2}\right)^{3/2}H^{(2)}_{\nu}(-k\eta_{2})$
$\displaystyle=$
$\displaystyle-\frac{\pi}{4}e^{-\pi\text{Im}(\nu)}H^{2}\left(\eta_{1}\eta_{2}\right)^{3/2}H^{(1)}_{\nu}(-k\eta_{1})H^{(2)}_{\nu}(-k\eta_{2}),$
$\displaystyle G_{<}\left(\eta_{1},\eta_{2},k\right)$ $\displaystyle=$
$\displaystyle-\frac{\pi}{4}e^{-\pi\text{Im}(\nu)}H^{2}\left(\eta_{1}\eta_{2}\right)^{3/2}H^{(1)}_{\nu}(-k\eta_{2})H^{(2)}_{\nu}(-k\eta_{1})$
(106)

from eq.(98) and eq.(99).

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