# Probing Leptogenesis with the Cosmological Collider

Yanou Cui<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of California, Riverside, CA 92521, USA  Zhong-Zhi Xianyu
<EMAIL_ADDRESS>Department of Physics, Tsinghua University, Beijing
100084, China  Collaborative Innovation Center of Quantum Matter, Beijing,
100084, China

###### Abstract

Leptogenesis is generally challenging to directly test due to the very high
energy scales involved. In this work we propose a new probe for leptogenesis
with cosmological collider physics. With the example of a cosmological Higgs
collider, we demonstrate that during inflation leptogenesis models can produce
detectable primordial non-Gaussianity with distinctive oscillatory patterns
that encode information about the lepton-number violating couplings, the
Majorana right-hand neutrino masses, and the CP phases, which are essential to
leptogenesis.

Introduction. Leptogenesis is an attractive mechanism explaining the origin of
the baryon asymmetry in our universe, and is closely related to the seesaw
mechanism for the origin of neutrino masses Fukugita and Yanagida (1986); Luty
(1992); Buchmuller  _et al._ (2005); Davidson  _et al._ (2008); Bodeker and
Buchmuller (2021); Xing and Zhao (2021). In the simplest leptogenesis
scenario, a lepton asymmetry is generated from the decay of a heavy right-
handed (RH) neutrino $N$ which is then transferred to a baryon asymmetry via
the sphaleron process. Despite the appeals of leptogenesis, it is very
challenging to directly test it, as the very high energy scales involved are
generally beyond the reach of terrestrial experiments, in contrast to some of
the low scale baryogenesis models (E.g. Cohen  _et al._ (1991); Cirigliano
_et al._ (2006); Morrissey and Ramsey-Musolf (2012); Cui  _et al._ (2012); Cui
and Sundrum (2013); Cui and Shuve (2015); Nelson and Xiao (2019); Cui and
Shamma (2020); Beniwal  _et al._ (2017); Chala  _et al._ (2016); Cui (2015);
Cui  _et al._ (2016); Croon  _et al._ (2020); Co and Harigaya (2020); Elor and
McGehee (2021)). For instance, with hierarchical masses of $N_{i}$
($i=1,2,3$), a lower bound exists for the lightest $N$’s mass: $M_{1}\gtrsim
10^{9}$ GeV Davidson and Ibarra (2002).

In this Letter we demonstrate that, thanks to the high energies provided by
the cosmic inflation, leptogenesis may leave distinct and observable
signatures in primordial non-Gaussianity (NG), inspired by the Cosmological
Collider (CC) physics. In particular, the essential Sakharov conditions for
leptogenesis such as $L$-number violation and CP violation, as well as the
heavy RH neutrino masses, can be probed individually as they impact the CC
signals in different ways. Our finding also leads to new types of fermionic CC
signatures that are not yet considered in the literature.

CC physics has been recently developed as a method to probe heavy particles
during cosmic inflation Arkani-Hamed and Maldacena (2015); Chen  _et al._
(2016); Lee  _et al._ (2016); Chen  _et al._ (2017a, b); An  _et al._ (2018);
Kumar and Sundrum (2018); Chen  _et al._ (2017c, 2018); Wu (2019); Li  _et
al._ (2019); Lu  _et al._ (2019); Liu  _et al._ (2019); Hook  _et al._ (2019a,
b); Kumar and Sundrum (2019); Alexander  _et al._ (2019); Wang and Xianyu
(2020a); Wang (2020); Wang and Zhu (2020); Li  _et al._ (2020); Wang and
Xianyu (2020b); Fan and Xianyu (2021); Aoki and Yamaguchi (2021); Bodas  _et
al._ (2021); Maru and Okawa (2021); Lu  _et al._ (2021); Wang  _et al._
(2021); Kim  _et al._ (2021). The energy scale of inflation, characterized by
the nearly constant Hubble parameter $H$, can be up to $O(10^{13})$ GeV in
single-field slow-roll models. Different versions of CC exist. The original
version is an inflaton collider: by coupling to the inflaton, heavy particles
with masses up to $O(H)$ imprint the $n$-point ($n\geq 3$) correlators of the
curvature perturbation $\zeta$. An alternative version, called the
Cosmological Higgs Collider (CHC) Lu  _et al._ (2019), exploited the
possibility that an appreciable component of $\zeta$ is originated from the
quantum fluctuations of the Standard Model (SM) Higgs field $\mathbf{H}$
during inflation, through the modulated reheating process Dvali  _et al._
(2004); Kofman (2003); Suyama and Yamaguchi (2008); Ichikawa  _et al._ (2008).
Compared with the original CC, CHC has the advantage of yielding large and
detectable NG with less free parameters. With these considerations, along with
the fact that the SM Higgs directly participates in leptogenesis via the
Yukawa coupling, in this work we will focus on signals with the CHC.

Cosmological Higgs Collider. Quantum fluctuation during inflation generates a
space dependent VEV $v\sim H$ for the Higgs field. In the program of CHC, this
VEV controls the inflaton decay rate and by doing so imprints its spatial
dependence on the fluctuations of the local time of reheating. Consequently,
the Higgs fluctuations $\delta h$ can source at least a fraction of primordial
curvature perturbation $\zeta=\zeta_{0}+\zeta_{h}$, where $\zeta_{h}$ is the
fluctuation from Higgs modulated reheating and $\zeta_{0}$ denotes other
contributions such as from the inflaton fluctuations. Then we can write
$\zeta_{h}=(2\pi P_{\zeta}^{1/2}/H)R_{h}\delta h$ at the linear level, where
$R_{h}=\zeta_{h}/\zeta$ and $P_{\zeta}\sim 2\times 10^{-9}$ is the measured
scalar power spectrum. Thus the 3-point correlator (bispectrum) of $\zeta_{h}$
is related to the 3-point correlator of $\delta h$ as

$\langle\zeta_{h,\mathbf{k}_{1}}\zeta_{h,\mathbf{k}_{2}}\zeta_{h,\mathbf{k}_{3}}\rangle=\Big{(}\displaystyle\frac{\,{2\pi
P_{\zeta}^{1/2}R_{h}}\,}{H}\Big{)}^{3}\langle\delta h_{\mathbf{k}_{1}}\delta
h_{\mathbf{k}_{2}}\delta h_{\mathbf{k}_{3}}\rangle.$ (1)

In the original proposal Lu  _et al._ (2019) the fraction $R_{h}$ is roughly
10%, but this is model dependent. The upshot is that $\zeta_{h}$, while being
a small fraction of $\zeta$, can contribute to most of the NG. In this
_Letter_ we treat $R_{h}$ as a free parameter, $0<R_{h}<1$. Furthermore, the
working of CHC requires that the Higgs potential is stabilized and that the
Higgs is lighter than $H$ during the inflation. While this might call for new
physics beyond SM, we note that the signal considered below only depends on
the neutrino properties and is essentially independent of the new physics that
stabilized the Higgs potential.

Eq. (1) shows that the primordial bispectrum in CHC can reveal information
about Higgs interactions during inflation. In particular, particles with
masses $m\sim H$ may leave a signal in the bispectrum if they couple to the SM
Higgs. In this _Letter_ we are interested in the signals mediated by neutrinos
represented by the 1-loop diagram on the left side of Fig. 1 (we will explain
the right diagram in the next section).

$\Rightarrow$

Figure 1: 1-loop process contributing to SM and Majorana RH neutrino signals
at the CHC ($i=1,2$).

The signal shows up for squeezed bispectrum where $k_{1}\simeq k_{2}\gg k_{3}$
($k_{i}\equiv|\mathbf{k}_{i}|$, $i=1,2,3$), as a characteristic oscillatory
function of the momentum ratio $\varrho\equiv k_{3}/k_{1}$. It is useful to
express the signal in terms of a dimensionless and (nearly) scale-invariant
shape function
$\mathcal{S}(k_{1},k_{2},k_{3})\equiv(k_{1}k_{2}k_{3})^{2}/(4\pi^{2}P_{\zeta})^{2}\langle\zeta_{h,\mathbf{k}_{1}}\zeta_{h,\mathbf{k}_{2}}\zeta_{h,\mathbf{k}_{3}}\rangle^{\prime}$
(′ symbol means the momentum-conserving $\delta$-function removed), as (see
Wang  _et al._ (2021) for more details):

$\displaystyle\lim_{\varrho\rightarrow
0}\mathcal{S}(\varrho)=f_{\text{NL}}^{\text{(signal)}}\varrho^{\alpha}\sin\big{(}\omega\log\varrho+\vartheta\big{)},$
(2)

where $f_{\text{NL}}^{\text{(signal)}}$ is the amplitude of the signal. The
dimensionless quantities ($\alpha,\omega,\vartheta)$ measure the scaling
behavior, the frequency, and the phase of the oscillatory signal. They are in
principle measurable and also calculable for specific processes. It is known
that the signal with a Dirac fermion of mass $m$ in a 1-loop diagram has
$\alpha=3$ and $\omega=2m/H$ Lu  _et al._ (2019). A key result of this Letter
is that the Majorana mass of the RH neutrino and CP phases innate to the
leptogenesis mechanism can generate new patterns in the signal. In particular,
the CHC signal with a Majorana neutrino gives $\alpha=2$ instead of 3, and the
CP phases give rise to more than one value of $\omega$, and the signal would
be superpositions of different oscillation patterns.

Leptogenesis, Neutrino masses and CP phases during inflation. Leptogenesis
mechanism meets the Sakharov conditions in the following ways: 1) lepton
number violation via the Majorana masses of the RH neutrinos $N$’s; 2) CP
violation via the CP phases in the Yukawa couplings through the tree-loop
interference in $N$ decay; and 3) out-of-equilibrium via the decay process
Sakharov (1967); Fukugita and Yanagida (1986).

A key observation is that, with a Majorana mass term, the mass matrix and the
Higgs couplings cannot be simultaneously diagonalized. This results in a new
type of Yukawa coupling mixing different mass eigenstates with distinct CHC
signals. The following derivations closely follow the standard neutrino seesaw
model, with the key difference that $\mathbf{H}$’s VEV is typically around the
inflationary Hubble scale $H$, and consequently the neutrino mass/mixing
pattern is very different in the inflation and post-inflation eras.

One generation suffices to demonstrate how the mixed coupling arises. Consider
the usual Type I seesaw with one generation of $N$ and SM lepton doublet
$L=(\nu,e^{-})^{T}$. In CHC the Higgs gets a nonzero VEV $v\sim H$ during
inflation due to quantum fluctuations. Here the inflationary Hubble parameter
$H$ is required to be higher than electroweak (EW) scale for the original CHC
mechanism Lu  _et al._ (2019) to work. Parametrizing the Higgs as
$\mathbf{H}=(0,(v+h)/\sqrt{2})^{T}$ and focusing on neutrinos only, we have

$\displaystyle\Delta{\mathscr{L}}=$
$\displaystyle~{}\nu^{\dagger}\mathrm{i}\mkern
2.0mu\overline{\mkern-2.0mu\sigma\mkern-2.0mu}\mkern
2.0mu^{\mu}\partial_{\mu}\nu+N^{\dagger}\mathrm{i}\mkern
2.0mu\overline{\mkern-2.0mu\sigma\mkern-2.0mu}\mkern
2.0mu^{\mu}\partial_{\mu}N$
$\displaystyle~{}+\Big{[}m_{D}\big{(}1+\displaystyle\frac{\,{h}\,}{v}\big{)}\nu
N-\displaystyle\frac{\,{1}\,}{2}m_{N}NN+\text{c.c.}\Big{]},$ (3)

where $m_{D}\equiv yv/\sqrt{2}$. The $U(2)$ symmetry of the kinetic term
allows us to choose $m_{D}$ and $m_{N}$ to be real without loss of generality,
and also to rotate to mass eigenstates $\psi_{\pm}$ with mass eigenvalues
$m_{\pm}=\mbox{$\frac{\,{1}\,}{2}$}(m_{N}\pm\sqrt{m_{N}^{2}+4m_{D}^{2}})$. In
the post-inflationary epoch, Higgs resides in its true vacuum such that
$m_{D}\ll m_{N}$, and thus $m_{-}\ll m_{+}$, realizing the seesaw mechanism
after EWPT. But during inflation $m_{D}\sim m_{N}\sim H$ is likely, therefore
$\psi_{\pm}$ could have comparable masses.

Although these are mostly familiar results, the important point is that the
Higgs coupling matrix cannot be simultaneously diagonalized by the above
rotations. Rather, it takes the following form after the rotation:

$\displaystyle{\mathscr{L}}\supset\displaystyle\frac{\,{m_{D}h}\,}{v\sqrt{m_{N}^{2}+4m_{D}^{2}}}\Big{[}m_{D}(\psi_{-}^{2}-\psi_{+}^{2})+m_{N}\psi_{-}\psi_{+}\Big{]},$
(4)

with a nonzero trilinear coupling that mixes $\psi_{\pm}$ when $m_{N}\neq 0$.
As we will see, this mixed Yukawa coupling in the mass eigenstates can give
rise to a unique signal in 3-point function of Higgs, and thus can be viewed
as a signature of Majorana neutrino mass.

Realistic leptogenesis model typically involves 3 generations of RH $N$’s.
With nonzero Majorana mass terms, the above result of Yukawa coupling to mixed
mass eigenstates still apply, but with the new feature that these couplings
generally contain irremovable CP phases. As we will see below, these phases
introduce yet another new pattern in the CHC signal.

Cosmological (Higgs) Collider Signals of Leptogenesis. A key result from the
above discussion is that the Yukawa couplings mix different mass eigenstates
and carry nonzero complex phases. Now we demonstrate their distinctive
physical consequences on the CHC observables.

The central object is the 3-point correlator of the Higgs fluctuation, namely
the left diagram in Fig. 1. The CHC signal appears when the blue lines carry
momenta much smaller than the black lines, i.e., in the squeezed limit. In
this configuration it proves advantageous to take an EFT limit for the bottom
black line in the triangle, which effectively shrinks the line into a point
vertex Tong  _et al._ (2021), as shown in the right diagram of Fig. 1. Then it
is straightforward to use the diagrammatic rule Chen  _et al._ (2017c) to
write down an expression for this process:

$\displaystyle\langle\delta h_{\mathbf{k}_{1}}\delta h_{\mathbf{k}_{2}}\delta
h_{\mathbf{k}_{3}}\rangle^{\prime}=\sum_{\mathsf{a,b}=\pm}\mathsf{ab}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}a^{4}(\tau_{1})a^{4}(\tau_{2})$
$\displaystyle~{}~{}\times
G_{\mathsf{a}}(k_{1},\tau_{1})G_{\mathsf{a}}(k_{2},\tau_{1})G_{\mathsf{b}}(k_{3},\tau_{2})\mathcal{I}(k_{3};\tau_{1},\tau_{2}),$
(5)

where $\tau$ is conformal time, $a=-1/(H\tau)$ is the scale factor,
$\mathsf{a},\mathsf{b}=\pm$ are Schwinger-Keldysh indices, $G_{\mathsf{a}}$ is
the boundary-to-bulk $\delta h$ propagator and $\mathcal{I}$ is the fermion
1-loop integral including couplings:

$\mathcal{I}(k;\tau_{1},\tau_{2})=\int{\mathrm{d}}^{3}X\,e^{-\mathrm{i}\mathbf{k}\cdot\mathbf{X}}\langle\mathcal{O}_{1}(\tau_{1},\mathbf{X})\mathcal{O}_{2}(\tau_{2},\mathbf{0})\rangle.$
(6)

Now we explain the effective operators $\mathcal{O}_{1,2}$. We can write the
most general Yukawa coupling mixing a pair of mass eigenstate as

$\Delta{\mathscr{L}}=\delta h\mathcal{O}_{1},~{}~{}\mathcal{O}_{1}\equiv
y_{12}(e^{\mathrm{i}\varphi_{12}}\psi_{1}\psi_{2}+\text{c.c.}),$ (7)

where $\psi_{1,2}$ are Weyl spinors and mass eigenstates with masses
$m_{1,2}$. Generally, $m_{1}\neq m_{2}$ ($m_{1}=m_{2}$) with (without) the
presence of Majorana mass in the original Lagrangian. We set $y_{12}$ real and
positive and isolate the CP phase $\varphi_{12}$. $\mathcal{O}_{1}$ accounts
for the upper vertex in the right diagram in Fig. 1. The diagonal case
$\mathcal{O}_{1}=y_{ii}\psi_{i}\psi_{i}+$ c.c. $(i=1,2)$ can also contribute
to the signal, but this is not mass mixing and the signal would be identical
to the case of “Dirac” to be discussed below. When the diagonal case
contribution is present and sizable, the new signal in this work can be
extracted by dedicated template fitting.

The lower vertex in the right diagram in Fig. 1 is an effective vertex with
the “local” neutrino integrated out as explained above. This effective
coupling takes the form

$\Delta{\mathscr{L}}=\displaystyle\frac{\,{1}\,}{2}(\delta
h)^{2}\mathcal{O}_{2},~{}~{}\mathcal{O}_{2}\equiv\displaystyle\frac{\,{1}\,}{\Lambda}(e^{\mathrm{i}\varphi_{5}}\psi_{1}\psi_{2}+\text{c.c.}),$
(8)

where we denote the absolute value of effective coupling
$y_{12}(y_{11}/m_{1}+y_{22}/m_{2})$ by an effective cutoff $1/\Lambda$ and its
phase by $\varphi_{5}$.

We take Wick contraction and the spinor trace to get:

$\displaystyle\langle\mathcal{O}_{1}(x)\mathcal{O}_{2}(y)\rangle=-\displaystyle\frac{\,{4y_{12}}\,}{\Lambda}\Big{[}\cos(\varphi_{12}+\varphi_{5})g_{m_{1}}(x,y)g_{m_{2}}(x,y)$
$\displaystyle~{}+\cos(\varphi_{12}-\varphi_{5})f_{m_{1}}(x,y)f_{m_{2}}(x,y)\Big{]}.$
(9)

Here $g_{m}$ and $f_{m}$ are functions of mass $m$ and positions
$x=(\tau_{1},\mathbf{X})$ and $y=(\tau_{2},\mathbf{0})$, given explicitly in
Chen  _et al._ (2018). The minus sign comes from the anticommutativity of
spinors.

The oscillatory signal in CHC bispectrum can be found by expanding $f_{m}$ and
$g_{m}$ in the late time limit $\tau_{1,2}\rightarrow 0$:

$\displaystyle\mathrm{i}f_{m}(x,y)=2\text{Re}\bigg{\\{}\displaystyle\frac{\,{\Gamma(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma(\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)}\,}{4\pi^{5/2}}\Big{(}\displaystyle\frac{\,{\tau_{1}\tau_{2}}\,}{X^{2}}\Big{)}^{3/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\bigg{[}1+\displaystyle\frac{\,{\big{(}3-4\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu(2\mathrm{i}+\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\big{)}(\tau_{1}^{2}+\tau_{2}^{2})-6\tau_{1}\tau_{2}}\,}{2(1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)X^{2}}\bigg{]}\bigg{\\}},$ (10) $\displaystyle
g_{m}(x,y)=2\text{Re}\bigg{\\{}\displaystyle\frac{\,{\Gamma(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma(\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)}\,}{4\pi^{5/2}}\Big{(}\displaystyle\frac{\,{\tau_{1}\tau_{2}}\,}{X^{2}}\Big{)}^{3/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle~{}\times\bigg{[}1+\displaystyle\frac{\,{\big{(}3-4\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu(2\mathrm{i}+\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\big{)}(\tau_{1}^{2}+\tau_{2}^{2})+6\tau_{1}\tau_{2}}\,}{2(1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)X^{2}}\bigg{]}\bigg{\\}}.$ (11)

Here and later, any mass with a tilde denotes its dimensionless value in unit
of $H$, namely $\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu=m/H$, and $X\equiv|\mathbf{X}|$. We see that $\mathrm{i}f_{m}$ and
$g_{m}$ are identical at the leading order and differ at subleading orders.
This has important consequences:

1\. For pure Dirac mass, the Yukawa coupling is diagonalizable with real
eigenvalues. Therefore, in (Probing Leptogenesis with the Cosmological
Collider), $\langle\mathcal{O}_{1}\mathcal{O}_{2}\rangle$ is proportional to
$f_{m_{i}}^{2}+g_{m_{i}}^{2}$ $(i=1,2)$. This combination is zero at the
leading order in $\tau\rightarrow 0$ limit, and shows up only at subleading
orders in $\tau$. In the in-in correlator, the power-law dependence on $\tau$
gets translated to a scaling in the $k$-ratio. Consequently, the signal with
only Dirac mass decays faster than naively expected in the squeezed limit.
This result is known Chen  _et al._ (2018); Lu  _et al._ (2019) and is
compatible with the scenario of Dirac neutrino/Dirac leptogenesis.

2\. If there is a Majorana-mass-induced mixed Yukawa coupling but without CP
phases, the correlator (Probing Leptogenesis with the Cosmological Collider)
would be proportional to $f_{m_{1}}f_{m_{2}}+g_{m_{1}}g_{m_{2}}$ with
$m_{1}\neq m_{2}$. Consequently, a piece proportional to
$(\tau_{1}\tau_{2})^{\pm\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}-\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})}$ is not canceled
out at the leading order. We then expect to see an oscillating signal at the
leading order of squeezeness with a _single_ frequency given by $\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}-\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2}$. The other
combination $\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1}+\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2}$
gets cancelled for the same reason as the Dirac case.

3\. If, in addition, the mixed Yukawa coupling contains irremovable CP phases
(as in the realistic leptogenesis models), the signal would be proportional to
$\cos(\varphi_{12}+\varphi_{5})f_{m_{1}}f_{m_{2}}+\cos(\varphi_{12}-\varphi_{5})g_{m_{1}}g_{m_{2}}$.
Then, both $(\tau_{1}\tau_{2})^{\pm\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}-\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})}$ and
$(\tau_{1}\tau_{2})^{\pm\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}+\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})}$ show up at the
leading order for generic $\varphi_{12}$ and $\varphi_{5}$. Therefore, this
new feature would reveal itself as two distinct sets of oscillation modes in
the CHC signature at the leading order of the squeezeness.

Figure 2: The oscillatory shape function of the primordial bispectrum (Probing
Leptogenesis with the Cosmological Collider) as a function of $k_{1}/k_{3}$
for the three cases we consider in the text. We take $m_{D}=0.5H$ for “Dirac”,
and take $m_{1}=0.5H$ and $m_{2}=1.5H$ for both “Majorana” and
“Majorana$+$CP.” We take $\varphi_{12}=\pi/4$ and $\varphi_{5}=-\pi/3$ for
“Majorana$+$CP.” For all cases we take $R_{h}^{3}y_{12}H/\Lambda=1$.

It is straightforward to compute the 3-point correlator of $\delta h$ and the
shape function in (2). The result for non-Dirac case is: (See Chen  _et al._
(2018); Lu  _et al._ (2019) for results of pure Dirac mass.)

$\displaystyle\lim_{\varrho\rightarrow
0}\mathcal{S}(\varrho)=2R_{h}^{3}P_{\zeta}^{-1/2}\displaystyle\frac{\,{y_{12}}\,}{\Lambda}$
$\displaystyle\times\big{\\{}\cos\varphi_{5}\cos\varphi_{12}\text{Re}\big{[}\mathcal{C}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\varrho^{2+\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}-\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})}\big{]}$
$\displaystyle+\sin\varphi_{5}\sin\varphi_{12}\text{Re}\big{[}\mathcal{C}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1},-\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\varrho^{2+\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}+\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})}\big{]}\big{\\}},$
(12)

where

$\displaystyle\mathcal{C}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})=\displaystyle\frac{\,{\mathrm{i}}\,}{2^{1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}}\pi^{5}}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}^{2}}(1+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}}\,}{4})(1-\cosh\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}\pi)$
$\displaystyle\times\sinh(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}\pi)\Gamma(\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1})$
$\displaystyle\times\Gamma(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma^{2}(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12})\Gamma(-4-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}),$ (13)

with $\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}\equiv\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1}-\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2}$.
The phase Arg $\mathcal{C}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})$ gives rise to the
phase of the oscillatory signal, namely $\vartheta$ in (2), which can be used
to distinguish our signal from other oscillatory loop signals Qin and Xianyu
(2022).

Fig. 2 illustrates the signal shape function for the three cases outlines
above, with “Dirac” known in the literature, and “Majorana” and “Majorana+CP
phases” our new results featuring different oscillation patterns. “Majorana+CP
phases” is most relevant to a realistic leptogenesis model. The “Majorana”
case is also worth noting as it represents a class of models with mass-mixing
couplings with new oscillation pattern. We only present result for one pair of
mixed mass eigenstates. With three generations of neutrinos, more than one
pair of mass eigenstates could contribute to the signal. However, according to
(Probing Leptogenesis with the Cosmological Collider), the signal strength
depends sensitively on the masses. Thus, in general, we expect only one pair
of states contribute predominantly.

Conditions for Baryon Asymmetry and Results. We numerically scan parameter
space to identify the regions with observable CHC signals and successful
leptogenesis. As we can see from (Probing Leptogenesis with the Cosmological
Collider), detectable CHC signals favor large Yukawa couplings. But large
Yukawa couplings poses a potential tension with leptogenesis as they generally
lead to strong washout, which we briefly review now.

The baryon asymmetry $Y_{B}$ as predicted from leptogenesis can be generally
expressed as

$\displaystyle
Y_{B}=\displaystyle\frac{\,{c_{s}}\,}{c_{s}-1}\kappa\displaystyle\frac{\,{\epsilon_{1}}\,}{g_{*}},$
(14)

where $c_{s}=(8N_{f}+4)/(22N_{f}+13)$ is the sphaleron conversion efficiency
factor, and $c_{s}=28/79\simeq 0.35$ for $N_{f}=3$. $g_{*}$ is the number of
relativistic degrees during leptogenesis epoch, which is $106.75$ for SM.
$\epsilon_{1}$ is the asymmetry from the the decay of heavy RH neutrinos
(assuming the lightest $N_{1}$ dominates the contribution), given by Chen
(2007)

$\epsilon_{1}\simeq-\displaystyle\frac{\,{3}\,}{8\pi}\displaystyle\frac{\,{1}\,}{(y_{\nu}y_{\nu}^{\dagger})_{11}}\sum_{i=2,3}\text{Im}\Big{[}(y_{\nu}y_{\nu}^{\dagger})_{1i}^{2}\Big{]}\displaystyle\frac{\,{m_{1}}\,}{m_{i}}.$
(15)

The above baryon asymmetry is subject to potential reduction by “washout”
processes, including the inverse decay and $2\rightarrow 2$ $\cancel{L}$
scattering, characterized by the washout factor $\kappa$, which is the ratio
of the decay rate of the RH neutrino, $\Gamma_{1}$, to the the Hubble scale
$H$ when the temperature equals $m_{1}$, $r=\Gamma_{1}/H(T=m_{1})$, i.e.,

$r=\displaystyle\frac{\,{M_{\text{Pl}}}\,}{32\pi\times
1.7\sqrt{g_{*}}}\displaystyle\frac{\,{(y_{\nu}y_{\nu}^{\dagger})_{11}}\,}{m_{1}}.$
(16)

$r\ll 1$ is the weak washout regime, while $r\gg 1$ leads to strong washout
that significantly suppresses the yield of $Y_{B}$. The relation between $r$
and $\kappa$ can be obtained by solving Boltzmann equations relevant for
$Y_{B}$ evolution Kolb and Wolfram (1980); Kolb and Turner (1990). For
parameters in Fig. 3 we are always in the moderate washout scenario
($10<r<10^{6}$) where the approximation $\kappa\simeq 0.3/(r\log r)^{0.6}$
works well Kolb and Turner (1990).

In Fig. 3 we show the signal size in (Probing Leptogenesis with the
Cosmological Collider) with $m_{2}=2m_{1}$, and scan over a range of mass
$m_{1}$ and Yukawa coupling $y_{12}$ within the perturbative regime. More
concretely, we take $R_{h}^{3}=1/2$, $1/\Lambda=y_{12}^{2}/m_{1}$,
$\varphi_{12}=\pi/4$, $\varphi_{5}=-\pi/3$. We also show the predicted baryon
asymmetry $Y_{B}$ with its observed value today
$Y_{B0}\equiv\frac{n_{B}-n_{\bar{B}}}{s}\simeq 8.7\times 10^{-11}$ from CMB
and BBN related measurements Aghanim  _et al._ (2020); Tanabashi  _et al._
(2018). In Fig. 3, leptogenesis parameters realizing the observed value is
shown with the solid magenta line, while contours with larger or smaller
$Y_{B}$ are also shown to account for possible late-time dilution or the
presence of other sources for baryogenesis.

In Fig. 3 we also include the reach of current CMB and forecast for future
LSS/21cm observations Wang and Xianyu (2020a). As we can see, a good range of
the parameter space for viable leptogenesis lead to signals within reach of
future CMB/LSS/21 cm line experiments Meerburg  _et al._ (2017); Moradinezhad
Dizgah  _et al._ (2018); Meerburg  _et al._ (2019); Kogai  _et al._ (2021).
Furthermore, with a CHC calibrated with known SM processes Lu  _et al._
(2019), it is possible to identify the neutrino signals out of SM
“backgrounds.” The prospect for CHC signals is most distinct and promising
when $m_{1}\sim H$: for $m_{1}\gg H$, the signal would be strongly suppressed
as shown in Fig. 2, while for $m_{1}\ll H$ the Majorana mass becomes
subdominant to the Dirac mass term–in this case a CHC signal can be observable
but restores known patterns in the literature. Furthermore, since the signal
strength depends on $H$ mainly through the ratio $m_{1}/H$, for a fixed
$m_{1}$, increasing (decreasing) $H$ in Fig. 3 amounts to shifting all
shadings horizontally towards the right (left) side, while the lines remain
the same.

Figure 3: Contour plots for the amplitude of neutrino signal at CHC
$f_{\text{NL}}^{\text{(signal)}}$ and baryon density $Y_{B}$ predicted by the
leptogenesis model on the 2D plane of Yukawa coupling $y_{12}$ and Majorana RH
neutrino mass $m_{1}$. The solid red line indicates the observed value of
baryon asymmetry, $Y_{B0}$. The solid blue lines represent experimental
sensitivities to $f_{\mathrm{N}L}$.

Discussion and Conclusion. In this Letter we propose a new cosmological probe
of the well-motivated leptogenesis with Majorana neutrinos, which to date is
considered challenging to test directly due to the high energies involved.
Specifically, we consider the scenario with SM Higgs contributing to the
primordial fluctuation during inflation. Based on Cosmological (Higgs)
Collider Physics, we demonstrated that this scenario can lead to observable
and distinct imprints of the Higgs’ Yukawa couplings to heavy RH Majorana
neutrinos and SM neutrinos in the primordial bispectrum. Essential information
about leptogenesis, such as the $L$-violation, heavy RH neutrino masses (in
the unit of the inflationary Hubble scale), and the CP phases, can be
extracted from delicate measurements of the primordial non-Gaussianity. With a
distinct oscillatory feature, our signal is easily distinguishable from the
local-shaped NG produced by the Higgs self-coupling De Simone  _et al._
(2013). Also, by using the phase information Qin and Xianyu (2022), we can
further select our signal out of other possible oscillatory signals
contributed from addition heavy states, and thus establish the uniqueness of
our signal. Our work presents an intriguing case of how CC physics may shed
light on the profound puzzle of matter-antimatter asymmetry in our Universe,
in light of the rich incoming data from CMB and LSS observations targeting
primordial non-Gaussianity.

###### Acknowledgements.

Acknowledgements. We thank KITP (supported by National Science Foundation
under Grant No. NSF PHY-1748958) for hospitality. YC is supported in part by
the US Department of Energy under award number DE-SC0008541. ZX is supported
in part by the National Key R&D Program of China (2021YFC2203100), an Open
Research Fund of the Key Laboratory of Particle Astrophysics and Cosmology,
Ministry of Education of China, and a Tsinghua University Initiative
Scientific Research Program.

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Probing Leptogenesis with the Cosmological Collider
Supplementary Material
Yanou Cui and Zhong-Zhi Xianyu

In this Supplemental Material we explain details about the Higgs Yukawa
couplings with three generations of Majorana neutrinos and present the full
calculation for the 3-point correlator of the Higgs fluctuations for the
leptogenesis model we consider.

##  I Higgs-Yukawa with 3 generations

The discussion in the main text about the Higgs-Yukawa coupling with Majorana
mass term can be directly generalized to 3 generations. The Lagrangian can be
written as

$\displaystyle\Delta{\mathscr{L}}=$
$\displaystyle~{}\nu_{i}^{\dagger}\mathrm{i}\mkern
2.0mu\overline{\mkern-2.0mu\sigma\mkern-2.0mu}\mkern
2.0mu^{\mu}\partial_{\mu}\nu_{i}+N_{i}^{\dagger}\mathrm{i}\mkern
2.0mu\overline{\mkern-2.0mu\sigma\mkern-2.0mu}\mkern
2.0mu^{\mu}\partial_{\mu}N_{i}+\Big{[}m_{Dij}\big{(}1+\displaystyle\frac{\,{h}\,}{v}\big{)}\nu_{i}N_{j}-\displaystyle\frac{\,{1}\,}{2}m_{Nij}N_{i}N_{j}+\text{c.c.}\Big{]}.$
(S1)

Here both $m_{Dij}$ and $m_{Nij}$ are complex 3$\times$3 matrices. We can use
the $U(6)$ symmetry of the kinetic terms to diagonalize the full $6\times 6$
mass matrix similar as we did in the main text for 1 generation. Again we
expect that the Higgs coupling matrix cannot be simultaneously diagonalized in
the presence of Majorana mass, thus we have Yukawa couplings that mix
different mass eigenstates and lead to unique cosmological collider signals.
More explicitly, we can parameterize the $U(6)$ rotation $O$ that diagonalize
the mass matrix as

$O=\begin{pmatrix}A&B\\\ C&D\end{pmatrix},$ (S2)

where $A,B,C,D$ are $3\times 3$ complex matrices, and $O$ satisfies the
unitary condition $OO^{\dagger}=I$. By construction,

$O^{T}\begin{pmatrix}0&-m_{D}\\\
-m_{D}&m_{N}\end{pmatrix}O=\begin{pmatrix}M_{1}&0\\\ 0&M_{2}\end{pmatrix},$
(S3)

where $M_{1}=\text{diag}(m_{1},m_{2},m_{3})$ and
$M_{2}=\text{diag}(m_{4},m_{5},m_{6})$ represent the two diagonal blocks of
the diagonalized mass matrix $M$. With this we can find the Higgs coupling
matrix in the mass eigenbasis, as,

${\mathscr{L}}\supset\displaystyle\frac{\,{h}\,}{2v}\begin{pmatrix}\psi_{1},\cdots,\psi_{6}\end{pmatrix}\begin{pmatrix}M_{1}-C^{T}m_{N}C&-C^{T}m_{N}D\\\
-D^{T}m_{N}C&M_{2}-D^{T}m_{N}D\end{pmatrix}\begin{pmatrix}\psi_{1}\\\
\vdots\\\ \psi_{6}\end{pmatrix},$ (S4)

where $\psi_{i}$ ($i=1,\cdots,6$) denotes the mass eigenstate.

It is clear that in the case of $m_{N}=0$, the Higgs coupling matrix is simply
proportional to the mass matrix and thus the two can be simultaneously
diagonalized, with non-negative real diagonal elements. No mixing could occur
in the Yukawa couplings in this case. But when there is nonzero Majorana mass,
the Higgs Yukawa matrix is no longer diagonalized in the mass eigenbasis. In
addition, we have no further freedom to rotate $\psi_{i}$ as long as $m_{i}$
is not zero. So we would in general expect that the Yukawa matrix contains
irremovable complex phases. These phases can generate the desired CP violation
when the heavy neutrino decays.

##  II Diagrammatic calculations

Here we present the full calculation using the diagrammatic rule à la
Schwinger-Keldysh. The 3-point correlator of the Higgs fluctuation mediated by
a fermion loop can be expressed as

$\displaystyle\langle\delta h(\mathbf{k}_{1})\delta h(\mathbf{k}_{2})\delta
h(\mathbf{k}_{3})\rangle^{\prime}=\sum_{\mathsf{a,b}=\pm}\mathsf{ab}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{|H\tau_{1}|^{4}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{|H\tau_{2}|^{4}}G_{\mathsf{a}}(k_{1},\tau_{1})G_{\mathsf{a}}(k_{2}\tau_{1})G_{\mathsf{b}}(k_{3},\tau_{2})\mathcal{I}_{\mathsf{ab}}(k_{3},\tau_{1},\tau_{2})$
(S5)

The function $\mathcal{I}$ is the fermionic 1-loop integral including the
Yukawa couplings. According to our discussion above, there are several
possibilities with distinct forms of $\mathcal{I}$. Here we show the details.
First, let us review the case of a Dirac fermion, already studied in Lu  _et
al._ (2019) which can be thought of as two Weyl fermions with degenerate mass.
Therefore,

$\displaystyle\mathcal{I}_{\text{D}}(k_{3},\tau_{1},\tau_{2})=\displaystyle\frac{\,{2y}\,}{\Lambda}\int{\mathrm{d}}^{3}X\,e^{-\mathrm{i}\mathbf{k}_{s}\cdot\mathbf{X}}\big{\langle}\big{[}\psi\psi+\psi^{\dagger}\psi^{\dagger}\big{]}(x_{1})\big{[}\psi\psi+\psi^{\dagger}\psi^{\dagger}\big{]}(x_{2})\big{\rangle},$
(S6)

where we simply write $y_{12}=y$ and $x_{1}=(\tau_{1},\mathbf{X})$ and
$x_{2}=(\tau_{2},\mathbf{0})$ in conformal coordinates. The correlator can be
worked out as

$\displaystyle\big{\langle}\big{[}\psi\psi+\psi^{\dagger}\psi^{\dagger}\big{]}(x_{1})\big{[}\psi\psi+\psi^{\dagger}\psi^{\dagger}\big{]}(x_{2})\big{\rangle}=-8\Big{[}g_{m}^{2}(x_{1},x_{2})+f_{m}^{2}(x_{1},x_{2})\Big{]}.$
(S7)

From Chen  _et al._ (2018),

$\displaystyle
f_{m}(x,y)=-\displaystyle\frac{\,{\mathrm{i}H^{3}\Gamma(2-\mathrm{i}m/H)\Gamma(2+\mathrm{i}m/H)}\,}{16\sqrt{2}\pi^{2}}\sqrt{1-Z}\;_{2}F_{1}\Big{(}2-\displaystyle\frac{\,{\mathrm{i}m}\,}{H},2+\displaystyle\frac{\,{\mathrm{i}m}\,}{H};2;\displaystyle\frac{\,{1+Z}\,}{2}\Big{)},$
(S8) $\displaystyle
g_{m}(x,y)=\displaystyle\frac{\,{H^{3}\Gamma(2-\mathrm{i}m/H)\Gamma(2+\mathrm{i}m/H)}\,}{32\sqrt{2}\pi^{2}}\displaystyle\frac{\,{m}\,}{H}\sqrt{1+Z}\;_{2}F_{1}\Big{(}2-\displaystyle\frac{\,{\mathrm{i}m}\,}{H},2+\displaystyle\frac{\,{\mathrm{i}m}\,}{H};3;\displaystyle\frac{\,{1+Z}\,}{2}\Big{)},$
(S9)

where $Z=Z(x,y)$ is the imbedding distance between $x$ and $y$. Let
$x=(\tau_{1},\mathbf{x})$ and $y=(\tau_{2},\mathbf{y})$ in the conformal
coordinates, and let $X\equiv|\mathbf{x}-\mathbf{y}|$, then
$Z=1-[X^{2}-(\tau_{1}-\tau_{2})^{2}]/(2\tau_{1}\tau_{2})$.

To get the CHC signal at the squeezed limit, we expand this correlator at
large distance limit. The result is

$\displaystyle-8(g_{m}^{2}+f_{m}^{2})=-\displaystyle\frac{\,{3H^{6}}\,}{2\pi^{5}}(1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma^{2}(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma^{2}(-\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Big{(}\displaystyle\frac{\,{\tau\tau^{\prime}}\,}{x^{2}}\Big{)}^{4-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.}$ (S10)

Complete the Fourier transform, we then get the nonlocal part of $\mathcal{I}$
to be

$\displaystyle\mathcal{I}_{\text{D}}(k,\tau_{1},\tau_{2})=\displaystyle\frac{\,{12\mathrm{i}yH^{6}}\,}{\pi^{4}\Lambda
k^{3}}(1-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma^{2}(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma^{2}(-\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma(-6+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu)\sinh(2\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu\pi)(k^{2}\tau_{1}\tau_{2})^{4-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.}$ (S11)

Next, consider the case of Majorana neutrino where the Higgs coupling mixes
different mass eigenstates. So

$\displaystyle\mathcal{I}_{\text{M}}(k_{s},\tau_{1},\tau_{2})=\displaystyle\frac{\,{y_{12}}\,}{\Lambda}\int{\mathrm{d}}^{3}X\,e^{-\mathrm{i}\mathbf{k}_{s}\cdot\mathbf{X}}\big{\langle}\big{[}\psi_{1}\psi_{2}+\psi_{1}^{\dagger}\psi_{2}^{\dagger}\big{]}(x_{1})\big{[}\psi_{1}\psi_{2}+\psi_{1}^{\dagger}\psi_{2}^{\dagger}\big{]}(x_{2})\big{\rangle}.$
(S12)

This time the correlator is

$\displaystyle\big{\langle}\big{[}\psi_{1}\psi_{2}+\psi_{1}^{\dagger}\psi_{2}^{\dagger}\big{]}(x_{1})\big{[}\psi_{1}\psi_{2}+\psi_{1}^{\dagger}\psi_{2}^{\dagger}\big{]}(x_{2})\big{\rangle}=-4\Big{[}g_{m_{1}}(x_{1},x_{2})g_{m_{2}}(x_{1},x_{2})+f_{m_{1}}(x_{1},x_{2})f_{m_{2}}(x_{1},x_{2})\Big{]}.$
(S13)

At late-time / large distance limit, we get quite different a scaling
behavior,

$\displaystyle-4(g_{m_{1}}g_{m_{2}}+f_{m_{1}}f_{m_{2}})=\displaystyle\frac{\,{H^{6}}\,}{2\pi^{5}}\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Big{(}\displaystyle\frac{\,{\tau_{1}\tau_{2}}\,}{x^{2}}\Big{)}^{3+\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}-\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})}+\text{c.c.}$
(S14)

That is, the correlator decreases slower than the previous case in the large
$x$ limit. Fourier transform this result, we get

$\displaystyle\mathcal{I}_{\text{M}}(k,\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{-2\mathrm{i}y_{12}H^{6}}\,}{\pi^{4}\Lambda
k^{3}}\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})$
$\displaystyle~{}\times\Gamma(-4-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12})\sinh(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}\pi)(k^{2}\tau_{1}\tau_{2})^{3+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}}+\text{c.c.},$
(S15)

where $\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}=\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1}-\mkern 2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2}$
and the tilde on any mass parameter $m$ means the dimensionless ratio $\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu=m/H$.

Finally we consider neutrinos with nonzero CP phases.

$\displaystyle\mathcal{I}_{\text{CP}}(k_{s},\tau_{1},\tau_{2})=|y_{5}y_{12}|\int{\mathrm{d}}^{3}x\,e^{\mathrm{i}\mathbf{k}_{s}\cdot\mathbf{x}}\big{\langle}\big{[}e^{\mathrm{i}\varphi_{5}}\psi_{1}\psi_{2}+e^{-\mathrm{i}\varphi_{5}}\psi_{1}^{\dagger}\psi_{2}^{\dagger}\big{]}(x_{1})\big{[}e^{\mathrm{i}\varphi_{12}}\psi_{1}\psi_{2}+e^{-\mathrm{i}\varphi_{12}}\psi_{1}^{\dagger}\psi_{2}^{\dagger}\big{]}(x_{2})\big{\rangle}.$
(S16)

The correlator can be worked out similarly,

$\displaystyle\big{\langle}\big{[}e^{\mathrm{i}\varphi_{5}}\psi_{1}\psi_{2}+e^{-\mathrm{i}\varphi_{5}}\psi_{1}^{\dagger}\psi_{2}^{\dagger}\big{]}(x_{1})\big{[}e^{\mathrm{i}\varphi_{12}}\psi_{1}\psi_{2}+e^{-\mathrm{i}\varphi_{12}}\psi_{1}^{\dagger}\psi_{2}^{\dagger}\big{]}(x_{2})\big{\rangle}$
$\displaystyle=$
$\displaystyle-4\Big{[}\cos(\varphi_{5}+\varphi_{12})g_{m_{1}}(x_{1},x_{2})g_{m_{2}}(x_{1},x_{2})+\cos(\varphi_{5}-\varphi_{12})f_{m_{1}}(x_{1},x_{2})f_{m_{2}}(x_{1},x_{2})\Big{]}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{H^{6}}\,}{2\pi^{5}}\bigg{[}\cos\varphi_{5}\cos\varphi_{12}\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Big{(}\displaystyle\frac{\,{\tau_{1}\tau_{2}}\,}{x^{2}}\Big{)}^{3+\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}-\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})}$
$\displaystyle~{}+\sin\varphi_{5}\sin\varphi_{12}\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Big{(}\displaystyle\frac{\,{\tau_{1}\tau_{2}}\,}{x^{2}}\Big{)}^{3+\mathrm{i}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1}+\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})}\bigg{]}+\text{c.c.}.$ (S17)

The result for the loop-integral in this case is:

$\displaystyle\mathcal{I}_{\text{CP}}(k,\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{-2\mathrm{i}|y_{12}y_{5}|H^{6}}\,}{\pi^{4}k^{3}}\bigg{[}\cos\varphi_{5}\cos\varphi_{12}\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})$
$\displaystyle~{}\times\Gamma(-4-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12})\sinh(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}\pi)(k^{2}\tau_{1}\tau_{2})^{3+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}}$
$\displaystyle~{}+\sin\varphi_{5}\sin\varphi_{12}\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{2})$
$\displaystyle~{}\times\Gamma(-4-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu)\sinh(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu\pi)(k^{2}\tau_{1}\tau_{2})^{3+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu}\bigg{]}+\text{c.c.}.$
(S18)

Now we are ready to calculate the 3-point correlator (S5) for the three cases:
1) Dirac neutrino, 2) Majorana neutrino without CP phases, 3) Majorana
neutrino with CP phases. The loop integrals $\mathcal{I}$ in (S5) correspond
to these three cases are given in (S11), (II), and (II), respectively.

$\displaystyle\mathcal{S}_{\text{D}}=2\,\text{Re}\,\bigg{[}R_{h}^{3}P_{\zeta}^{-1/2}\displaystyle\frac{\,{y}\,}{\Lambda}\mathcal{C}_{\text{D}}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Big{(}\displaystyle\frac{\,{k_{3}}\,}{2k_{1}}\Big{)}^{3-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu}\bigg{]},$ (S19)
$\displaystyle\mathcal{C}_{\text{D}}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)=\displaystyle\frac{\,{6\mathrm{i}}\,}{\pi^{5}}(5-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu)(1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)^{4}(1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu)\cosh^{2}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu\pi)\sinh(2\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu\pi)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Gamma^{2}(1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma^{2}(-\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}m)\Gamma^{2}(1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu)\Gamma(-6+4\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu).$ (S20)
$\displaystyle\mathcal{S}_{\text{M}}=2\,\text{Re}\,\bigg{[}R_{h}^{3}P_{\zeta}^{-1/2}\displaystyle\frac{\,{y_{12}}\,}{\Lambda}\mathcal{C}_{\text{M}}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Big{(}\displaystyle\frac{\,{k_{3}}\,}{2k_{1}}\Big{)}^{2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}}\bigg{]},$ (S21)
$\displaystyle\mathcal{C}_{\text{M}}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})=\displaystyle\frac{\,{\mathrm{i}}\,}{2\pi^{5}}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}^{2}}(1+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12}}\,}{4})(1-\cosh\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}\pi)\sinh(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}\pi)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Gamma(\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{1})\Gamma(2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Gamma^{2}(2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{12})\Gamma(-4-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}).$ (S22)

$\displaystyle\mathcal{S}_{\text{CP}}=2\,\text{Re}\,\bigg{\\{}R_{h}^{3}P_{\zeta}^{-1/2}\displaystyle\frac{\,{y_{12}}\,}{\Lambda}\bigg{[}\cos\varphi_{5}\cos\varphi_{12}\mathcal{C}_{\text{M}}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1},\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Big{(}\displaystyle\frac{\,{k_{3}}\,}{2k_{1}}\Big{)}^{2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{12}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sin\varphi_{5}\sin\varphi_{12}\mathcal{C}_{\text{M}}(\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu_{1},-\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern
2.0mu_{2})\Big{(}\displaystyle\frac{\,{k_{3}}\,}{2k_{1}}\Big{)}^{2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mum\mkern-2.0mu}\mkern 2.0mu}\bigg{]}\bigg{\\}},$
(S23)

where $\mathcal{C}_{\text{M}}$ is given in (S22).