aainstitutetext: Center for Theoretical Physics of the Universe, Institute for
Basic Science (IBS),
Daejeon, 34126, Korea.bbinstitutetext: School of Physics, Sun Yat-Sen
University,
Guangzhou 510275, China.ccinstitutetext: Department of Physics, University of
California,
Berkeley, CA 94720, USA.ddinstitutetext: Kavli Institute for the Physics and
Mathematics of the Universe (WPI), University of Tokyo,
Kashiwa 277-8583, Japan.eeinstitutetext: Ernest Orlando Lawrence Berkeley
National Laboratory,
Berkeley, CA 94720, USA.ffinstitutetext: Hamamatsu Professor

# Type II Seesaw Leptogenesis

Neil D. Barrie b Chengcheng Han c,d,e,f and Hitoshi Murayama
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Type II Seesaw Mechanism provides a minimal framework to explain the
neutrino masses involving the introduction of a single triplet Higgs to the
Standard Model. However, this simple extension was believed to be unable to
successfully explain the observed baryon asymmetry of the universe through
Leptogenesis. In our previous work (Phys. Rev. Lett. 128, 141801), we
demonstrated that the triplet Higgs of the Type II Seesaw Mechanism alone can
simultaneously generate the observed baryon asymmetry of the universe and the
neutrino masses while playing a role in setting up Inflation. This is
achievable with a triplet Higgs mass as low as 1 TeV, and predicts that the
neutral component obtains a small vacuum expectation value $v_{\Delta}<10$
keV. We find that our model has very rich phenomenology and can be tested by
various terrestrial experiments as well as by astronomical observations.
Particularly, we show that the successful parameter region may be probed at a
future 100 TeV collider, upcoming lepton flavor violation experiments such as
Mu3e, and neutrinoless double beta decay experiments. Additionally, the
tensor-to-scalar ratio from the inflationary scenario will be probed by the
LiteBIRD telescope, and observable isocurvature perturbations may be produced
for some parameter choices. In this article, we present all the technical
details of our calculations and further discussion of its phenomenological
implications.

###### Keywords:

Baryogenesis, Higgs, Inflation, Neutrinos

††preprint: CTPU-PTC-22-04††arxiv: 2204.XXXXX

##  1 Introduction

One of the most significant mysteries in modern physics is the existence of
non-zero neutrino masses. In the Standard Model (SM) the neutrinos are
massless by definition, meaning that extensions to the SM must be proposed to
explain their observed mass. The leading mechanism for generating the neutrino
masses is known as the Seesaw Mechanism. The Seesaw Mechanism can be split
into three main categories called Type I, II and III. The Type I Seesaw
Mechanism is the most widely studied, consisting of an extension of the SM by
at least two right-handed neutrinos Minkowski:1977sc ; Yanagida:1979as ;
Glashow:1979nm ; GellMann:1980vs . The Type II Seesaw Mechanism involves the
inclusion of a triplet Higgs to the SM, which produces a Majorana mass term
for the neutrinos when its neutral component takes its non-zero vacuum
expectation value (VEV) Magg:1980ut ; Cheng:1980qt ; Lazarides:1980nt ;
Mohapatra:1980yp . This new $SU(2)$ triplet scalar can have rich terrestrial
phenomenology dependent on its mass scale. On the other hand, the Type III
Seesaw Mechanism requires the existence of at least two $SU(2)$ fermion
triplets Foot:1988aq ; Albright:2003xb . None of these models, or mixtures of
them, have been experimentally observed and theoretical work regarding their
phenomenological implications is ongoing. Planned future experiments will
probe the associated particles and bring us closer to understanding the origin
of the neutrino masses.

An important feature of the neutrino sector is that it may be closely related
to another unexplained observation: the baryon asymmetry of our universe.
Thanks to the sphaleron process Kuzmin:1985mm , if a lepton asymmetry can be
generated before the Electroweak Phase Transition (EWPT), it will be
transferred to the baryonic sector via the sphalerons Harvey:1990qw ; this
scenario is known as Leptogenesis Fukugita:1986hr . In general, Leptogenesis
scenarios in the Type I and Type III Seesaw Mechanisms are typically at very
high energy scales and are subsequently difficult to probe. The Type II Seesaw
Mechanism may offer opportunities to connect these high and low scale
dynamics. Unfortunately, the Type II Seesaw Mechanism is unable to
successfully lead to thermal Leptogenesis without the inclusion of additional
particles; an extra triplet Higgs or a right-handed neutrino Ma:1998dx ;
Hambye:2005tk . This is disappointing since the explanation of the neutrino
masses only requires a single triplet Higgs, destroying the minimal nature of
the model.

However, since the triplet Higgs is a scalar particle, there are alternative
approaches by which the observed baryon asymmetry may be generated. For
example, Electroweak Baryogenesis may be possible if the EWPT becomes strongly
first-order due to the new interactions introduced through the triplet Higgs,
although additional $\mathcal{CP}$ violation sources are also required.
Unfortunately, for this scenario to be successful, the triplet Higgs mass must
be below $\sim$500 GeV Zhou:2022mlz , which is already in strong tension with
current LHC searches. Another interesting alternative to consider is the
Affleck-Dine mechanism Affleck:1984fy . This mechanism requires (i) a scalar
that is charged under some mixture of the global $U(1)_{L}$ or $U(1)_{B}$
symmetries; (ii) a small term in the Lagrangian that breaks this symmetry; and
(iii) a displaced vacuum value during the early universe. Interestingly, the
triplet Higgs carries a $U(1)_{L}$ charge due to its Yukawa coupling with the
leptons, which generate the neutrino masses. Also, $U(1)_{L}$ breaking terms
naturally arise from its couplings with the SM Higgs. However, condition (iii)
is less trivially achieved.

In the Affleck-Dine mechanism, it is assumed that the scalar field exhibits a
flat direction, a trajectory where the mass term dominates the potential and
the quartic term vanishes, and thus it is easily displaced from the minimum of
its potential during the inflationary epoch. For $m_{\textrm{scalar}}\ll
H_{\textrm{inf}}$, the correlation length of the displaced value can be much
larger than the current universe and it may be assumed to take a constant
value. Thus, the first and usual implementation of the Affleck-Dine mechanism
involves the framework of Supersymmetry, since one can easily find many $D$
and $F$ flat directions for a scalar field that carries $U(1)_{L}$ or
$U(1)_{B}$ charge - the superpartners of the SM fermions Dine:1995uk ;
Dine:1995kz . Indeed, the a model of Supersymmetry that includes two triplet
superfields was considered in the past as a possible mechanism to generate the
baryon asymmetry, with two scalar triplets required to cancel anomalies
Senami:2001qn . In contrast to these scenarios, we want to consider a minimal
framework that does not include Supersymmetry. We emphasize that it is non-
trivial to consider the Affleck-Dine mechanism without Supersymmetry since a
general model is immune to the presence of flat directions. Additionally,
difficulties can arise in finding natural mechanisms to transfer the generated
asymmetry from the scalar sector to SM fermions without Supersymmetry. One may
instead wonder about the possibility of the vacuum displacement originating
from the stochastic behavior of the scalar field during the early inflationary
epoch. Since the correlation length of the vacuum value is short
$\mathcal{O}(\frac{1}{H_{\textrm{inf}}})$ for a quartic coupling of order of
$\mathcal{O}(1)$ Starobinsky:1994bd , the average of the baryon asymmetry of
our universe is approximately zero in this case due to the stochastic nature
of the initial phase.

Therefore, the remaining option is to consider the triplet scalar as a
component of the inflaton, due to the setup of the inflationary phase
requiring an initially displaced vacuum value Brout:1977ix ; Sato:1980yn ;
Guth:1980zm ; Linde:1981mu ; Albrecht:1982mp . The possibility of the inflaton
playing a role in the generation of the baryon asymmetry of the universe has
attracted significant attention in recent years Hertzberg:2013mba ;
Lozanov:2014zfa ; Yamada:2015xyr ; Bamba:2016vjs ; Bamba:2018bwl ;
Cline:2019fxx ; Barrie:2020hiu ; Lin:2020lmr ; Kawasaki:2020xyf ;
Kusenko:2014lra ; Wu:2019ohx ; Charng:2008ke ; Ferreira:2017ynu ;
Rodrigues:2020dod ; Lee:2020yaj ; Enomoto:2020lpf ; Mohapatra:2021aig ;
Mohapatra:2022ngo . The inflaton candidates that are typically considered are
difficult to connect directly to SM physics due to the large discrepancy in
their associated energy scales. The inflaton in our setup will consist of the
triplet Higgs, in combination with the SM Higgs Starobinsky:1980te ;
Whitt:1984pd ; Jakubiec:1988ef ; Maeda:1988ab ; Barrow:1988xh ;
Faulkner:2006ub ; Bezrukov:2011gp , providing a unique connection between
terrestrial experiments and the high energy dynamics of the early Universe.

In the past, there have been attempts to explain the origins of inflation,
Baryogenesis, and the neutrino masses within a simple, unified framework
Murayama:1992ua ; Murayama:1993xu ; Barrie:2020hiu . However, it is difficult
to naturally provide solutions to these unknowns simultaneously with a single
addition to the SM. We will present a model that represents a simple and well-
motivated realization of this idea.

In our initial study Barrie:2021mwi , we determined that the introduction of
the triplet Higgs of the Type II Seesaw Mechanism to the SM is able to explain
simultaneously the origins of inflation, Baryogenesis, and the neutrino
masses. In this work, we will scrutinise this scenario further, provide the
technical details of the model, and discuss its extensive phenomenological
implications. The structure of this paper is as follows: In Section 2, we
provide a summary of the Type II Seesaw Mechanism and describe the overall
framework of this scenario. Section 3 details the inflationary setup and
includes a discussion of the induced inflationary trajectory. The Affleck-Dine
mechanism for Baryogenesis and its application to the Type II Seesaw Mechanism
is explored in detail in Section 4, and the predicted baryon asymmetry is
presented. In Section 5, we examine and combine all of the phenomenological
implications of this scenario at colliders and neutrino experiments, as well
as the cosmological constraints. Finally, in Section 6, the final results and
future avenues for exploration are discussed.

##  2 The Type II Seesaw Mechanism and Model Framework

The Type II Seesaw Mechanism provides a well-motivated and natural framework
for explaining the smallness of the observed neutrino masses. This scenario
involves the minimal extension of the SM scalar sector by a $SU(2)_{L}$
triplet scalar $\Delta$ which carries a hypercharge of $1$. The triplet and SM
doublet Higgs’ are parameterized as follows,

$\displaystyle H=\left(\begin{array}[]{c}~{}~{}h^{+}\\\
h\end{array}\right),~{}~{}\Delta=\left(\begin{array}[]{cc}\Delta^{+}/\sqrt{2}&\Delta^{++}\\\
\Delta^{0}&-\Delta^{+}/\sqrt{2}\end{array}\right)~{},$ (5)

where $h$ and $\Delta^{0}$ are the neutral components of $H$ and $\Delta$
respectively. In addition to the neutral component, the charged components of
the triplet Higgs, $\Delta^{+}$ and $\Delta^{++}$, play an important role at
terrestrial collider experiments. Their phenomenological implications will be
integral to the discovery of the Type II Seesaw Mechanism and the
determination of its properties Chongdar:2021tgm ; Dev:2021axj ;
Mandal:2022zmy ; Cheng:2022jyi .

Importantly, the addition of the triplet Higgs leads to new interactions
involving the SM Higgs and the left-handed lepton doublet. In particular, the
following Yukawa interaction between the left-handed lepton doublets $L_{i}$
and the triplet Higgs $\Delta$,

${\mathcal{L}}_{\textrm{Yukawa}}=\mathcal{L}^{\rm
SM}_{\textrm{Yukawa}}-\frac{1}{2}y_{ij}\bar{L}^{c}_{i}\Delta L_{j}+h.c.$ (6)

This Yukawa interaction is integral to the generation of a non-zero neutrino
mass matrix, which is generated when the neutral component of the triplet
Higgs $\Delta^{0}$ obtains its non-zero VEV. Additionally, this interaction
allows us to assign a lepton charge of $Q_{L}=-2$ to the triplet Higgs, thus
fulfilling one of the conditions for the Affleck-Dine mechanism, and opening
the possibility for it to play a role in the origins of the baryon asymmetry
via Leptogenesis.

The inclusion of the triplet Higgs scalar leads to new terms in the Higgs’
potential $V(H,\Delta)$, including interactions associated with $\Delta$ that
violate the global lepton number symmetry. The potential is given by,

$\displaystyle V(H,\Delta)$ $\displaystyle=$ $\displaystyle-
m_{H}^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}+m_{\Delta}^{2}{\rm
Tr}(\Delta^{\dagger}\Delta)+\lambda_{1}(H^{\dagger}H){\rm
Tr}(\Delta^{\dagger}\Delta)+\lambda_{2}({\rm Tr}(\Delta^{\dagger}\Delta))^{2}$
(7) $\displaystyle+\lambda_{3}{\rm
Tr}(\Delta^{\dagger}\Delta)^{2}+\lambda_{4}H^{\dagger}\Delta\Delta^{\dagger}H+\left[\mu(H^{T}i\sigma^{2}\Delta^{\dagger}H)+\frac{\lambda_{5}}{M_{p}}(H^{T}i\sigma^{2}\Delta^{\dagger}H)(H^{\dagger}H)\right.$
$\displaystyle\left.+\frac{\lambda^{\prime}_{5}}{M_{p}}(H^{T}i\sigma^{2}\Delta^{\dagger}H)(\Delta^{\dagger}\Delta)+h.c.\right]+...~{},$

where the terms given in the square brackets $[...]$ violate the lepton number
symmetry. The cubic $\mu$ dependent term is important for determining the VEV
obtained by $\Delta^{0}$.

In addition to the cubic term, we have included dimension five operators which
are suppressed by $M_{p}$, since these terms may dominate over the $\mu$ term
during inflation when the field values are close to the Planck scale. Even if
these interaction terms play no role in low energy physics, they could be
important during the early universe epochs of inflation and reheating,
particularly if the triplet Higgs is a component of the inflaton.

In the cosmological context, our analysis will focus on the neutral components
of $\Delta$ and $H$, which have non-trivial VEVs. Thus, the scalar potential
$V(H,\Delta)$ can be simplified as follows,

$\displaystyle V(h,\Delta^{0})$
$\displaystyle=-m_{H}^{2}|h|^{2}+m_{\Delta}^{2}|\Delta^{0}|^{2}+\lambda_{H}|h|^{4}+{\lambda_{\Delta}}|\Delta^{0}|^{4}+{\lambda_{H\Delta}}|h|^{2}|\Delta^{0}|^{2}$
(8) $\displaystyle-\left(\mu
h^{2}{\Delta^{0}}^{*}+\frac{\lambda_{5}}{M_{p}}|h|^{2}h^{2}{\Delta^{0}}^{*}+\frac{\lambda^{\prime}_{5}}{M_{p}}|\Delta^{0}|^{2}h^{2}{\Delta^{0}}^{*}+h.c.\right)+...~{},$

where $\lambda_{\Delta}=\lambda_{2}+\lambda_{3}$ and
$\lambda_{H\Delta}=\lambda_{1}+\lambda_{4}$. It is important to note that, all
of these parameters must have values such that they satisfy the vacuum
stability conditions. Also, in the early universe, we require both $h$ and
$\Delta^{0}$ to have non-vanishing vacuum values to ensure that the $U(1)_{L}$
breaking term is relevant.

From the potential given in Eq. (8), we can derive the VEV of the triplet
Higgs. In the limit where the SM Higgs VEV is much smaller than the $\Delta$
mass parameter, $m_{\Delta}\gg v_{\textrm{EW}}$, the non-vanishing
$\Delta^{0}$ VEV is approximately given by,

$v_{\Delta}\equiv\langle\Delta^{0}\rangle\simeq\frac{\mu
v_{\textrm{EW}}^{2}}{2m^{2}_{\Delta}}~{},$ (9)

where the SM Higgs VEV is $v_{\textrm{EW}}=246$ GeV. The LHC has placed a
lower limit on the mass parameter of $m_{\Delta}\gtrsim 800$ GeV from searches
for the associated doubly-charged Higgs ATLAS:2017xqs . Importantly, for the
mass range $m_{\Delta}>800$ GeV, the masses of the charged and neutral
components of the triplet Higgs are approximately equivalent,
$m_{\Delta^{++}}\simeq m_{\Delta^{+}}\simeq m_{\Delta^{0}}\simeq m_{\Delta}$.

From the Yukawa interaction term in Eq. (6), we obtain the mass matrix of the
neutrinos,

$m^{\nu}_{ij}=y_{ij}v_{\Delta}~{}.$ (10)

The neutrino masses can be derived by diagonalizing the above matrix by the
PMNS matrix. The neutrino Yukawa coupling $y_{\nu}$ should be smaller than
$\mathcal{O}(1)$ to ensure that it remains perturbative up to the Planck
scale.

The VEV of $\Delta^{0}$ has the following range of allowed values,

$\mathcal{O}(1)\textrm{~{}GeV}>|\langle\Delta^{0}\rangle|\gtrsim
0.05\textrm{~{}eV}~{},$ (11)

where the upper bound on the $\Delta^{0}$ VEV is derived from the T-parameter
constraints determined from precision measurements Kanemura:2012rs , and the
lower limit ensures the generation of the observed neutrino masses while also
requiring perturbative Yukawa couplings.

If the doublet-triplet Higgs model presented above is responsible for
inflation, it contains all of the ingredients required to generate a baryon
asymmetry during inflation through the Affleck-Dine mechanism. However, the
current observational data from Cosmic Microwave Background (CMB) measurements
exclude the simple polynomial potential found in Eq. (8) as the source of the
inflationary epoch because it is not sufficiently flat Planck:2018jri . A
solution to this problem is the addition of non-minimal couplings between the
Higgs’ and the Ricci scalar, that is a key feature of standard Higgs
inflation, which we shall discuss in detail in the next section.

##  3 The Inflationary Setup

The epoch of Inflation is an established component of standard cosmology due
to its ability to solve some of the known observational problems - such as the
flatness, horizon, and monopole problems, as well as being able to provide
measurable predictions in the form of primordial perturbations Guth:1980zm ;
Linde:1981mu ; Albrecht:1982wi ; Mukhanov:1981xt . In our scenario, the
inflationary setting will be induced by a combination of the SM and triplet
Higgs’ which have non-minimal couplings to gravity. These couplings act to
flatten the scalar potential at large field values, allowing the slow-roll
parameters to be sufficiently satisfied and the expected observational
signatures to be consistent with current CMB measurements. This form of
inflationary mechanism has been well-studied, being the setup for standard
Higgs inflation, and results in a Starobinsky-like inflationary epoch
Starobinsky:1980te ; Bezrukov:2007ep ; Bezrukov:2008ut ; GarciaBellido:2008ab
; Barbon:2009ya ; Barvinsky:2009fy ; Bezrukov:2009db ; Giudice:2010ka ;
Bezrukov:2010jz ; Burgess:2010zq ; Lebedev:2011aq ; Lee:2018esk ; Choi:2019osi
. Combining the non-minimal couplings with the scalar sector discussed in
Section 2, we find the relevant Lagrangian for our model,

$\displaystyle\frac{\mathcal{L}}{\sqrt{-g}}$
$\displaystyle=-\frac{1}{2}M_{p}^{2}R-F(H,\Delta)R-g^{\mu\nu}(D_{\mu}H)^{\dagger}(D_{\nu}H)$
(13)
$\displaystyle-g^{\mu\nu}(D_{\mu}\Delta)^{\dagger}(D_{\nu}\Delta)-V(H,\Delta)+\mathcal{L}_{\textrm{Yukawa}}~{},$

refer to Eq. (6) and Eq. (7) for the Yukawa couplings and scalar potential,
respectively. The non-minimal couplings have the form,

$F(H,\Delta)=\xi_{H}|h|^{2}+\xi_{\Delta}|\Delta^{0}|^{2}=\frac{1}{2}\xi_{H}\rho^{2}_{H}+\frac{1}{2}\xi_{\Delta}\rho^{2}_{\Delta}~{},$
(14)

where we have introduced the following polar coordinate parametrizations,

$h\equiv\frac{1}{\sqrt{2}}\rho_{H}e^{i\eta}~{}~{}\textrm{and}~{}~{}\Delta^{0}\equiv\frac{1}{\sqrt{2}}\rho_{\Delta}e^{i\theta}~{}.$
(15)

To analyse the inflationary dynamics and observables we must understand the
trajectory along which the inflaton evolves. An inflationary setting
consisting of two non-minimally coupled scalars has been found to exhibit a
unique inflationary trajectory that is dependent upon the relative size of
each scalars’ non-minimal and quartic self-couplings Lebedev:2011aq . In the
large $\xi$ and field limits, the ratio of the two Higgs scalars is fixed as,

$\frac{\rho_{H}}{\rho_{\Delta}}\equiv\tan\alpha=\sqrt{\frac{2\lambda_{\Delta}\xi_{H}-\lambda_{H\Delta}\xi_{\Delta}}{2\lambda_{H}\xi_{\Delta}-\lambda_{H\Delta}\xi_{H}}}~{}.$
(16)

To ensure this is the trajectory taken by the inflaton, we require
$2\lambda_{\Delta}\xi_{H}-\lambda_{H\Delta}\xi_{\Delta}>0$ and
$2\lambda_{H}\xi_{\Delta}-\lambda_{H\Delta}\xi_{H}>0$. Below, we provide the
details of the derivation of this trajectory, following Ref. Lebedev:2011aq ,
to demonstrate its existence and properties.

###  3.1 Derivation of the Inflationary Trajectory

Firstly, consider the following field redefinitions of $h$ and $\Delta^{0}$,

$\chi=\sqrt{\frac{3}{2}}M_{p}\log\left(1+\frac{\xi_{H}|h|^{2}}{M_{p}^{2}}+\frac{\xi_{\Delta}(|\Delta^{0}|)^{2}}{M_{p}^{2}}\
\right)\textrm{~{}~{}and~{}~{}}\kappa=\frac{\rho_{H}}{\rho_{\Delta}}~{},$ (17)

using the polar coordinate parametrization given in Eq. (15). Here we will
temporarily ignore the motion of the angular directions, $\eta$ and $\theta$,
since the potential is dominated by the dynamics of the radial directions,
$\rho_{H}$ and $\rho_{\Delta}$. Under this redefinition, the kinetic terms of
the Lagrangian become,

$\displaystyle{\cal L}_{\rm kin}=$
$\displaystyle\frac{1}{2}\biggl{(}1+\frac{1}{6}\frac{\kappa^{2}+1}{\xi_{H}\kappa^{2}+\xi_{\Delta}}\biggr{)}~{}(\partial_{\mu}\chi)^{2}+\frac{M_{p}}{\sqrt{6}}~{}\frac{(\xi_{\Delta}-\xi_{H})\kappa}{(\xi_{H}\kappa^{2}+\xi_{\Delta})^{2}}(\partial_{\mu}\chi)(\partial^{\mu}\kappa)$
$\displaystyle+\frac{M_{p}^{2}}{2}\frac{\xi_{H}^{2}\kappa^{2}+\xi_{\Delta}^{2}}{(\xi_{H}\kappa^{2}+\xi_{\Delta})^{3}}(\partial_{\mu}\kappa)^{2}~{}.$
(18)

In our analysis, we require that the non-minimal couplings are large,
$\xi_{H}$ and $\xi_{\Delta}\gg 1$, such that at leading order in $1/\xi$ the
kinetic terms are given by,

${\cal L}_{\rm
kin}=\frac{1}{2}(\partial_{\mu}\chi)^{2}+\frac{M_{p}^{2}}{2}\frac{\xi_{H}^{2}\kappa^{2}+\xi_{\Delta}^{2}}{(\xi_{H}\kappa^{2}+\xi_{\Delta})^{3}}(\partial_{\mu}\kappa)^{2}~{}.$
(19)

For large $\xi$ parameters, the mixing term
$(\partial_{\mu}\chi)(\partial^{\mu}\kappa)$ is suppressed, resulting in a
canonically normalised $\chi$ and a suppression of the kinetic term of
$\kappa$. Below we show the three main regimes for $\kappa$, that each have a
corresponding canonically normalized variable $\kappa^{\prime}$,

$\displaystyle\xi_{\Delta}\gg\xi_{H}~{}~{}{\rm or}~{}~{}\kappa\rightarrow
0~{}~{},~{}~{}~{}~{}~{}\kappa^{\prime}=\frac{\kappa}{\sqrt{\xi_{\Delta}}}~{},$
$\displaystyle\xi_{H}\gg\xi_{\Delta}~{}~{}{\rm
or}~{}~{}\kappa\rightarrow\infty~{}~{},~{}~{}~{}\kappa^{\prime}=\frac{1}{\sqrt{\xi_{H}}\kappa}~{},$
$\displaystyle\xi_{H}=\xi_{\Delta}~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\kappa^{\prime}=\frac{1}{{\sqrt{\xi_{H}}}}\arctan\kappa~{}.$
(20)

Now consider the scalar potential under this field redefinition, in terms of
$\kappa$,

$U=\frac{\lambda_{H}\kappa^{4}+\lambda_{h\Delta}\kappa^{2}+\lambda_{\Delta}}{4(\xi_{H}\kappa^{2}+\xi_{\Delta})^{2}}M_{p}^{4}~{},$
(21)

where we have taken the large field limit for $\chi$ and assume that the
$U(1)_{L}$ breaking terms can be ignored. This potential has the following
minima, dependent upon the relations between the different non-minimal
couplings and quartic self-couplings,

$\displaystyle(1)$
$\displaystyle~{}2\lambda_{H}\xi_{\Delta}-\lambda_{h\Delta}\xi_{H}>0~{},~{}2\lambda_{\Delta}\xi_{H}-\lambda_{h\Delta}\xi_{\Delta}>0~{},~{}~{}~{}~{}\kappa=\sqrt{\frac{2\lambda_{\Delta}\xi_{H}-\lambda_{h\Delta}\xi_{\Delta}}{2\lambda_{H}\xi_{\Delta}-\lambda_{h\Delta}\xi_{H}}}~{},$
$\displaystyle(2)$
$\displaystyle~{}2\lambda_{H}\xi_{\Delta}-\lambda_{h\Delta}\xi_{H}>0~{},~{}2\lambda_{\Delta}\xi_{H}-\lambda_{h\Delta}\xi_{\Delta}<0~{},~{}~{}~{}~{}\kappa=0~{},$
$\displaystyle(3)$
$\displaystyle~{}2\lambda_{H}\xi_{\Delta}-\lambda_{h\Delta}\xi_{H}<0~{},~{}2\lambda_{\Delta}\xi_{H}-\lambda_{h\Delta}\xi_{\Delta}>0~{},~{}~{}~{}~{}\kappa=\infty~{},$
$\displaystyle(4)$
$\displaystyle~{}2\lambda_{H}\xi_{\Delta}-\lambda_{h\Delta}\xi_{H}<0~{},~{}2\lambda_{\Delta}\xi_{H}-\lambda_{h\Delta}\xi_{\Delta}<0~{},~{}~{}~{}~{}\kappa=0,\infty~{}.$
(22)

In Cases 2 and 3, we arrive at the usual single field non-minimally coupled
inflationary potential, $\lambda_{\Delta}/(4\xi_{\Delta}^{2})M_{p}^{4}$ and
$\lambda_{H}/(4\xi_{H}^{2})M_{p}^{4}$, respectively. The inflaton in Case 1 is
characterised by a mixture of the two scalars which is the typical scenario we
require to ensure that the lepton number violating interactions are relevant
during inflation. Importantly in Case 1, the potential has the following
minimum,

$U\Bigl{|}_{\rm
min~{}(1)}=\frac{1}{16}\frac{4\lambda_{\Delta}\lambda_{H}-\lambda_{h\Delta}^{2}}{\lambda_{\Delta}\xi_{H}^{2}+\lambda_{H}\xi_{\Delta}^{2}-\lambda_{h\Delta}\xi_{\Delta}\xi_{H}}M_{p}^{4}~{}.$
(23)

The numerator of Eq. (23) must be positive to ensure that we do not obtain a
negative vacuum energy at large field values. In each of the above cases, the
canonical field $\kappa^{\prime}$ has a large mass which is of order
$M_{p}/\sqrt{\xi}$. This mass will always be greater than the Hubble parameter
during inflation, so we can safely integrate out the $\kappa$ field,
establishing the inflationary trajectory.

Having derived this inflationary trajectory, we can now proceed to analyzing
the inflaton dynamics in our Lagrangian given in Eq. (13). Since we only wish
to consider Case 1, we can simplify our analysis by defining the inflaton as
$\varphi$, through the following relations to the polar coordinate fields,

$\displaystyle\rho_{H}=\varphi\sin\alpha,~{}\rho_{\Delta}=\varphi\cos\alpha~{},$
$\displaystyle\xi\equiv\xi_{H}\sin^{2}\alpha+\xi_{\Delta}\cos^{2}\alpha~{}.$
(24)

The Lagrangian in Eq. (13) is now given by,

$\frac{\mathcal{L}}{\sqrt{-g}}=-\frac{M_{p}^{2}}{2}R-\frac{\xi}{2}\varphi^{2}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}\varphi^{2}\cos^{2}\alpha~{}g^{\mu\nu}\partial_{\mu}\theta\partial_{\nu}\theta-V(\varphi,\theta)~{},$
(25)

where

$V(\varphi,\theta)=\frac{1}{2}m^{2}\varphi^{2}+\frac{\lambda}{4}\varphi^{4}+2\varphi^{3}\left(\tilde{\mu}+\frac{\tilde{\lambda}_{5}}{M_{p}}\varphi^{2}\right)\cos\theta~{},$
(26)

and

$\displaystyle m^{2}=m^{2}_{\Delta}\cos^{2}\alpha-m_{H}^{2}\sin^{2}\alpha~{},$
$\displaystyle\lambda=\lambda_{H}\sin^{4}\alpha+\lambda_{H\Delta}\sin^{2}\alpha\cos^{2}\alpha+\lambda_{\Delta}\cos^{4}\alpha~{},$
$\displaystyle\tilde{\mu}=-\frac{1}{2\sqrt{2}}\mu\sin^{2}\alpha\cos\alpha~{},$
$\displaystyle\tilde{\lambda}_{5}=-\frac{1}{4\sqrt{2}}(\lambda_{5}\sin^{4}\alpha\cos\alpha+\lambda^{\prime}_{5}\sin^{2}\alpha\cos^{3}\alpha)~{}.$
(27)

The Affleck-Dine mechanism will be realised in our scenario through the motion
of the dynamical field $\theta$. The size of the generated lepton asymmetry
will be determined by the size of the non-trivial motion induced in $\theta$
sourced by inflation. During inflation, $m\ll\varphi$, which means that the
quartic term in the Jordan frame potential dominates the inflationary
dynamics.

###  3.2 The Single Field Inflation Approximation

We will choose input parameters that preserve the inflationary trajectory and
ensure that it is negligibly affected by the dynamics of $\theta$ and the
lepton asymmetry production. In this case, the inflationary observables in our
framework are consistent with those of the Starobinsky inflationary model, and
as such, are in excellent agreement with current observational constraints
Planck:2018jri .

Since we have derived the approximate single field inflationary behaviour
above, we can analyse the evolution of the inflationary phase and its
predictions for the inflationary parameters. To do this, we will first
translate the Lagrangian in Eq. (25) from the Jordan frame to the Einstein
frame, using the following transformations Wald:1984rg ; Faraoni:1998qx ,

$\tilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu},~{}~{}\Omega^{2}=1+\xi\frac{\varphi^{2}}{M^{2}_{p}}~{},$
(28)

and subsequently reparametrizing $\varphi$ in terms of the canonically
normalized scalar $\chi$.

In transforming from the Jordan to Einstein frame, the Jordan frame field
$\varphi$ no longer has a canonically normalised kinetic term. We must make
the following field redefinition to obtain the canonically normalized scalar,

$\frac{d\chi}{d\varphi}\;=\;\dfrac{\sqrt{(6\xi^{2}\varphi^{2}/M_{p}^{2})+\Omega^{2}}}{\Omega^{2}}\;~{},$
(29)

which gives the definition of $\chi$ in terms of $\varphi$ as,

$\chi(\varphi)=\frac{1}{\sqrt{\xi}}\left(\sqrt{1+6\xi}~{}\sinh^{-1}(\sqrt{\xi+6\xi^{2}}\varphi)-\sqrt{6\xi}~{}\sinh^{-1}(\sqrt{6\xi^{2}}\varphi/\sqrt{1+\xi\varphi^{2}})\right)~{}.$
(30)

Three distinct regimes can be observed in the evolution of the relation
between the $\chi$ and $\varphi$ fields, namely,

$\dfrac{\chi}{M_{p}}\approx\left\\{\begin{array}[]{lll}\dfrac{\varphi}{M_{p}}&\mbox{for
$\dfrac{\varphi}{M_{p}}\ll\dfrac{1}{\xi}$}&\mbox{(radiation-like)}\\\
\sqrt{\dfrac{3}{2}}\,\xi\left(\dfrac{\varphi}{M_{p}}\right)^{2}&\mbox{for
$\dfrac{1}{\xi}\ll\dfrac{\varphi}{M_{p}}\ll\dfrac{1}{\sqrt{\xi}}$\quad}&\mbox{(matter-
like)}\\\
\sqrt{\dfrac{3}{2}}\ln\Omega^{2}=\sqrt{\dfrac{3}{2}}\ln\left[1+\xi\left(\dfrac{\varphi}{M_{p}}\right)^{2}\right]&\mbox{for
$\dfrac{1}{\sqrt{\xi}}\ll\dfrac{\varphi}{M_{p}}$}&\mbox{(inflation)}\end{array}\right.$
(31)

Thus, we now obtain the final Einstein frame Lagrangian,

$\displaystyle\frac{\mathcal{L}}{\sqrt{-g}}=-\frac{M_{p}^{2}}{2}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{1}{2}f(\chi)g^{\mu\nu}\partial_{\mu}\theta\partial_{\nu}\theta-U(\chi,\theta)~{},$
(32)

where

$\displaystyle
f(\chi)\equiv\frac{\varphi(\chi)^{2}\cos^{2}\alpha}{\Omega^{2}(\chi)},~{}~{}\textrm{and}~{}~{}U(\chi,\theta)\equiv\frac{V(\varphi(\chi),\theta)}{\Omega^{4}(\chi)}~{}.$
(33)

As with the relation between the $\chi$ and $\varphi$ fields, the scalar
potential exhibits three distinct regimes in each field definition.
Importantly, it is evident that the non-minimal coupling leads to a flattening
of the potential in the large field limit. The scalar potential in each regime
is,

$U(\chi)\approx\left\\{\begin{array}[]{lll}\dfrac{1}{4}\lambda\chi^{4}&\mbox{for
$\dfrac{\chi}{M_{p}}\ll\dfrac{1}{\xi}$}&\mbox{(radiation-like)}\\\
\dfrac{1}{2}m_{S}^{2}\chi^{2}&\mbox{for
$\dfrac{1}{\xi}\ll\dfrac{\chi}{M_{p}}\ll 1$\quad}&\mbox{(matter-like)}\\\
\dfrac{3}{4}m_{S}^{2}M_{p}^{2}\left(1-e^{-\sqrt{\frac{2}{3}}(\chi/M_{p})}\right)^{2}&\mbox{for
$1\ll\dfrac{\chi}{M_{p}}$}&\mbox{(inflation)}\end{array}\right.$ (34)

Therefore, the $\chi$ potential replicates the Starobinsky form in the large
field limit,

$U_{\textrm{inf}}(\chi)=\frac{3}{4}m_{S}^{2}M_{p}^{2}\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\chi}{Mp}}\right)^{2}~{},$
(35)

where the Starobinsky mass scale is required by observational constraints to
be $m_{S}=\sqrt{\frac{\lambda M_{p}^{2}}{3\xi^{2}}}\simeq 3\cdot 10^{13}$ GeV
Faulkner:2006ub ; Planck:2018jri . This will be the inflationary framework we
will use in our scenario, and from which we will now determine the expected
observational signatures.

Before calculating the inflationary observables, we provide a plot in Figure 1
of the evolution of $\chi$ during and after inflation, where we have
numerically solved the equations of motion derived from the Lagrangian in Eq.
(32). The input parameters are fixed to $\xi=300$ and $\lambda=4.5\cdot
10^{-5}$, we take $\tilde{\mu}$ to be sufficiently small to not affect the
dynamics, and the following initial conditions are chosen,
$\dot{\theta}_{0}=0$, $\chi_{0}=6.0M_{p}$, and $\dot{\chi}_{0}=0$ with the
definition $H_{0}\equiv m_{S}/2$. The figure shows that the end of inflation
occurs near $tH_{0}=100$, and thus, horizon crossing occurs at approximately
$tH_{0}\approx 40-50$. After inflation, the universe quickly enters into a
matter-like evolution until $tH_{0}\approx 400$. The line $\chi=M_{p}/\xi$ in
Figure 1 is included to indicate the end of the matter-like epoch.

Figure 1: The evolution of $\chi$ during and after inflation is depicted, in
Planck units. The following input parameters are chosen to be $\xi=300$ and
$\lambda=4.5\cdot 10^{-5}$, with the initial conditions $\chi_{0}=6.0M_{p}$,
$\dot{\chi}_{0}=0$, and $\dot{\theta}_{0}=0$ used. We require that
$\tilde{\mu}$ is sufficiently small to not affect the inflationary dynamics,
and we define $H_{0}\equiv m_{S}/2$.

It is important to note that, the reason we choose $\xi=300$ and
$\lambda=4.5\cdot 10^{-5}$ is that for larger $\xi$ values, or equivalently
small $\lambda$, issues can arise regarding unitarity during the oscillatory
stage that follows inflation in Higgs inflation models Ema:2016dny ;
DeCross:2015uza ; DeCross:2016cbs ; DeCross:2016fdz . It has been found that
this problem can be avoided if these models are UV completed by the addition
of an $R^{2}$ term Giudice:2010ka ; Gorbunov:2018llf ; Ema:2017rqn ;
Ema:2019fdd ; Ema:2020zvg .

Another subtlety is that the $\lambda$ coupling will obtain radiative
corrections from the gauge couplings and Yukawa couplings of the SM and
triplet Higgs’. A naive estimation of the size of these radiative corrections
is given approximately by - gauge couplings: $\frac{g^{4}}{16\pi^{2}}$
($\mathcal{O}(10^{-(3-4)})$); neutrino Yukawa couplings:
$-\frac{y^{4}}{16\pi^{2}}$; or the top Yukawa coupling:
$-\frac{y_{t}^{4}}{16\pi^{2}}$, for a general mixing angle $\alpha$. To obtain
$\lambda=4.5\cdot 10^{-5}$, a small tuning of $y$ or $y_{t}$ might be
necessary to cancel the contributions from the Higgs’ gauge couplings.
However, we stress that such a small $\lambda$ is not a necessary condition
for our mechanism to work, and we will give a comment on our final result for
larger couplings. In any case, for couplings of $\lambda\sim 10^{-2}$, the
dynamics will be stable against these radiative corrections. Nevertheless, we
will use this benchmark coupling for our study to simplify the numerical
analysis during the oscillation stage.

###  3.3 Inflationary Observables

Considering the Einstein frame inflationary potential in Eq. (35), we derive
the following characteristic slow-roll parameters,

$\epsilon_{s}\simeq\frac{3}{4N_{*}^{2}}~{},~{}~{}\textrm{and}~{}~{}\eta_{s}\simeq\frac{1}{N_{*}}~{},$
(36)

where we have given them in terms of the number of e-foldings of expansion
from horizon crossing to the end of the inflation, $N_{*}$. Subsequently, we
obtain the predicted spectral index and tensor-to-scalar ratio in our
inflationary scenario that should be observed in CMB measurements,

$n_{s}\simeq
1-\frac{2}{N_{*}}~{},~{}~{}\textrm{and}~{}~{}r\simeq\frac{12}{N_{*}^{2}}~{},$
(37)

respectively. The required number of inflationary e-folds to solve the
flatness and horizon problems is between $50<N_{*}<60$. Thus, our model gives
the following predicted ranges for the spectral index and tensor-to-scalar
ratio,

$\displaystyle 0.96\lesssim n_{s}\lesssim 0.9667~{},$ (38) $\displaystyle
0.0033\lesssim r\lesssim 0.0048~{}.$ (39)

We can now compare these to the current best constraints on the inflationary
predictions from the recent Planck results Planck:2018jri ,

$\displaystyle n_{s}=0.9649\pm 0.042~{}~{}{\rm(68\%C.L.)}~{},$ (40)
$\displaystyle r_{0.002}<0.056~{}~{}{\rm(95\%C.L.)}~{},$ (41)

for the $\Lambda-$CDM$+r$ model. Comparing these with those in Eq. (38) and
(39), we see that our scenario is in extremely good agreement with current
observational results for the required number of inflationary e-folds. The
prediction for the tensor-to-scalar ratio is well below the current
constraints, but it will be observable by the upcoming LiteBIRD telescope
Hazumi:2019lys .

In addition to these results, the scalar perturbations observed in the CMB
provide a constraint on the Starobinsky mass scale $m_{S}$. It is necessary to
satisfy the following relation for consistency with this observation,

$\frac{\lambda}{\xi^{2}}\simeq 5\cdot 10^{-10}~{},$ (42)

with the corresponding Starobinsky mass scale being $m_{S}\simeq 3\cdot
10^{13}$ GeV. This relation shall inform our parameter choices in the upcoming
sections, and was used to make the parameter choices in Figure 1.

##  4 The Affleck-Dine Mechanism and Generated Baryon Asymmetry

The origin of the baryon asymmetry of the universe is a fundamental mystery of
modern physics. The size of the observed baryon asymmetry is parametrized by
the asymmetry parameter $\eta_{B}$ Aghanim:2018eyx ,

$\eta_{B}^{\textrm{obs}}=\frac{n_{B}}{s}\simeq 8.5\cdot 10^{-11}~{},$ (43)

where $n_{B}$ and $s$ are the baryon number and entropy densities of the
universe, respectively. Many models have been proposed to explain this
asymmetry, with one of the most well-studied scenarios being the Affleck-Dine
mechanism Affleck:1984fy .

The core idea behind the Affleck-Dine mechanism is the generation of non-zero
angular motion in the phase of a complex scalar field $\phi$ that is charged
under a global $U(1)$ symmetry. Assuming the complex scalar acquires a large
initial field value in the early universe, $\phi$ will start to oscillate once
the Hubble parameter decreases below the $\phi$ mass $m$. If the scalar
potential $V(\phi)$ contains an explicit $U(1)$ breaking term, a net $U(1)$
charge asymmetry will be generated by this motion. Consequently, if the $U(1)$
symmetry is composed of the global $U(1)_{B}$ or $U(1)_{L}$ symmetries, a
baryon asymmetry can be generated prior to the EWPT. In the following, we will
show that the role of the $\phi$ field can be taken by our inflaton, the mixed
state of the SM and triplet Higgs’ discussed in the previous section, which
carries a complex phase associated with the global $U(1)_{L}$ symmetry.

If we identify this scalar with our inflaton and the complex phase of the
triplet Higgs, the asymmetry number density associated with the $U(1)_{L}$
charge will be given by,

$n_{L}=Q_{L}\varphi^{2}(\chi)\dot{\theta}\cos^{2}\alpha~{}.$ (44)

where $\alpha$ is the mixing angle between the SM and triplet Higgs’ during
inflation defined in the previous section. Therefore, in order to obtain a
non-zero lepton number asymmetry density $n_{L}$, we require a non-zero vacuum
value for $\chi$ and non-trivial motion in the complex phase $\theta$. This
net lepton asymmetry shall be transferred to the baryonic sector through
equilibrium sphaleron processes prior to the EWPT.

###  4.1 Asymmetry Generation from the Triplet Higgs

To begin our analysis, consider the Lagrangian of our model in the Einstein
frame, as given in Eq. (32),

$\frac{\mathcal{L}}{\sqrt{-g}}=-\frac{M_{p}^{2}}{2}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{1}{2}f(\chi)g^{\mu\nu}\partial_{\mu}\theta\partial_{\nu}\theta-U(\chi,\theta)~{},$
(45)

where

$f(\chi)\equiv\frac{\varphi(\chi)^{2}\cos^{2}\alpha}{\Omega^{2}(\chi)}~{},$
(46)

and

$U(\chi,\theta)\equiv\frac{V(\varphi(\chi),\theta)}{\Omega^{4}(\chi)}~{}=\frac{m^{2}\varphi^{2}(\chi)+\lambda\varphi^{4}(\chi)}{\Omega^{4}(\chi)}+\frac{2\tilde{\mu}\varphi^{3}(\chi)+2\frac{\tilde{\lambda}_{5}}{M_{p}}\varphi^{5}(\chi)}{\Omega^{4}(\chi)}\cos\theta~{}.$
(47)

The equations of motion for $\chi$ and $\theta$ can be simply derived as
follows,

$\displaystyle\ddot{\chi}-\frac{1}{2}f_{,\chi}\dot{\theta}^{2}+3H\dot{\chi}+U_{,\chi}=0~{},$
$\displaystyle\ddot{\theta}+\frac{f_{,\chi}}{f(\chi)}\dot{\theta}\dot{\chi}+3H\dot{\theta}+\frac{1}{f(\chi)}U_{,\theta}=0~{}.$
(48)

Evaluating these equations of motion during the inflationary epoch, it is
found that both $\chi$ and $\theta$ enter approximate slow-roll regimes
described by,

$\dot{\chi}\simeq-\frac{M_{p}U_{,\chi}}{\sqrt{3U}},~{}~{}\textrm{and}~{}~{}\dot{\theta}\simeq-\frac{M_{p}U_{,\theta}}{f(\chi)\sqrt{3U}}~{}.$
(49)

To simplify our analysis, we make use of the following parameterization
$\tau=tH_{0}$, where $H_{0}=m_{S}/2$ and define prime as the derivative with
respect to $\tau$. We also define the reduced Hubble parameter $\tilde{H}$ by,

$\tilde{H}^{2}=\frac{1}{3}\left(\frac{1}{2}{\chi^{\prime}}^{2}+\frac{1}{2}f(\chi){\theta^{\prime}}^{2}+U/H_{0}^{2}\right).$
(50)

Under these redefinitions, the equations of motion from Eq. (48) become,

$\displaystyle\chi^{\prime\prime}-\frac{1}{2}f_{,\chi}{\theta^{\prime}}^{2}+3\tilde{H}\chi^{\prime}+\frac{U_{,\chi}}{H_{0}^{2}}=0~{},$
$\displaystyle\theta^{\prime\prime}+\frac{f_{,\chi}}{f(\chi)}\theta^{\prime}\chi^{\prime}+3\tilde{H}\theta^{\prime}+\frac{1}{f(\chi)H_{0}^{2}}U_{,\theta}=0~{},$
(51)

and the associated lepton number density is,

$\dot{\theta}=\theta^{\prime}H_{0},~{}{\rm
and}~{}n_{L}=Q_{L}\varphi^{2}(\chi)\theta^{\prime}\cos^{2}\alpha H_{0}~{}.$
(52)

Figure 2:  The dynamics of (left) $\dot{\theta}$ and (right) $\theta$ during
inflation are depicted, in Planck units. The left figure provides a comparison
of the exact numerical result with the analytical result found in Eq. (48) for
$\tilde{\lambda}_{5}=10^{-9}$. The other input parameters selected are
$\xi=300$ and $\lambda=4.5\cdot 10^{-5}$, with the following initial
conditions $\chi_{0}=6.0M_{p}$, $\dot{\chi}_{0}=0$, and $\dot{\theta}_{0}=0$
used. We choose $\tilde{\mu}$ to be sufficiently small to not affect the
dynamics, and where we define $H_{0}=m_{S}/2$.

In Figure 2, we depict the evolution of $\theta$ and $\dot{\theta}$ during
inflation, providing a comparison with the estimation of $\dot{\theta}$
presented in Eq. (49). We find that $\dot{\theta}$ is well described by the
derived slow-roll approximation. Due to the non-zero $\dot{\theta}$ induced,
there is an increase in $\theta$ during inflation. However, as shown in the
right panel of Figure 2, for fixed $\theta_{0}$, the rate of increase is
proportional to $\tilde{\lambda}_{5}$, which is indicated by the parameter
dependencies found in Eq. (49).

During inflation, the $\chi$ field approaches $M_{p}$ and so we can ignore the
subdominant $m$ and $\tilde{\mu}$ terms that are much less than $M_{p}$. Using
these approximations, we can determine the lepton number asymmetry generated
by the motion of the triplet Higgs phase $\theta$ at the end of inflation,

$\displaystyle{n_{L}}_{\textrm{end}}$ $\displaystyle=$ $\displaystyle
Q_{L}\varphi^{2}_{\textrm{end}}\dot{\theta}_{\textrm{end}}\cos^{2}\alpha$ (53)
$\displaystyle\simeq$
$\displaystyle-\mathcal{O}(1)Q_{L}\varphi^{2}_{\textrm{end}}\frac{M_{p}U_{,\theta}}{f(\chi_{\rm
end})\sqrt{3U_{\rm end}}}\cos^{2}\alpha$ $\displaystyle\simeq$
$\displaystyle-\mathcal{O}(1)Q_{L}\tilde{\lambda}_{5}\varphi^{3}_{\textrm{end}}\sin\theta_{\rm
end}/\sqrt{3\lambda}~{},$

where the $\mathcal{O}(1)$ factor that appears in the second step is typically
$\sim 3$ from numerical calculations, and accounts for the breakdown of the
slow-roll approximation at the end of inflation, which we have defined as when
the slow-roll parameter satisfies $\epsilon=1$. In the last step, we assume
the quartic term dominates the inflationary potential and that the
$\tilde{\lambda}_{5}$ coupling dominates the lepton number violating
interactions. After inflation, the lepton number density is solely red-shifted
by $a^{3}$ alongside another $\mathcal{O}(1)$ factor given by $\Omega$.

###  4.2 The Lepton Number Density Evolution after Inflation

When inflation ends, at $\chi_{\textrm{end}}\simeq 0.67M_{p}$, the oscillatory
epoch begins for which the universe is characterized by an approximately
matter-like evolution. The inflationary potential at this time is given in Eq.
(34). At the end of the inflationary stage, we define the lepton asymmetry
number density as,

${n_{L}}_{\textrm{end}}=Q_{L}\varphi^{2}_{\textrm{end}}\dot{\theta}_{\textrm{end}}\cos^{2}\alpha~{}.$
(54)

Now we wish to analyse the evolution of the lepton number density after
inflation. To determine this, first consider the following relation derived
from the equation of motion in Eq. (48),

$\frac{d[a^{3}f(\chi)\dot{\theta}]}{dt}=\frac{d[a^{3}n_{L}/(Q_{L}\Omega^{2})]}{dt}=a^{3}U_{,\theta}~{}.$
(55)

During the matter-like phase, if the term $U_{,\theta}$ red-shifts faster than
matter-like, $a^{-3}$, then it can be ignored and the quantity
$a^{3}n_{L}/(Q_{L}\Omega^{2})$ will be conserved. This is guaranteed when the
dominant $U(1)_{L}$ breaking term in $V(\varphi,\theta)$ is a polynomial
function of $\varphi$ larger than degree four because the quartic term of
$\varphi$ in $V(\varphi,\theta)$ is equivalent to the mass term of the $\chi$
field, which red-shifts as $a^{3}$ after inflation. Thus, we consider the
quintic $\tilde{\lambda}_{5}$ term to be the dominant $U(1)_{L}$ breaking term
at large field values.

If instead, the cubic term was to dominate, the $U(1)_{L}$ breaking term would
become increasingly relevant as the expansion of the universe proceeds, and
subsequently destroy the lepton number generated during inflation. In the
following subsections, we will show that the cubic term is already required to
be small to avoid the washout of the lepton asymmetry after reheating.

As shown in Eq. (55), after inflation the lepton asymmetry number density is
red-shifted by the usual matter-like scale factor dependence, with the
addition of a factor of $1/\Omega^{2}$ which is of order of $\mathcal{O}(1)$.
It is then possible to estimate the lepton asymmetry number density at any
time after inflation using,

$n_{L}(t)={n_{L}}_{\textrm{end}}\frac{\Omega^{2}(\chi)}{\Omega^{2}(\chi_{\textrm{end}})}\left(\frac{a}{a_{\textrm{end}}}\right)^{-3}~{}.$
(56)

In Figure 3, we depict a comparison of the numerical simulations to the
analytical result given in Eq. (56) to demonstrate the accuracy of this
relation. The input parameters have been chosen to fit the current CMB
constraints.

Figure 3: Comparison of the lepton number density $n_{L}$ derived from the
full numerical calculations (Coloured lines) and the analytical estimation
(Black dotted lines) in Eq. (56), through the inflationary and oscillatory
epochs. The results are presented for varying $\theta_{0}$ and
$\tilde{\lambda}_{5}$, with all parameters are in Planck units and we define
$H_{0}=m_{S}/2$. The fixed input parameters are $\xi=300$ and
$\lambda=4.5\cdot 10^{-5}$. We take $\tilde{\mu}$ to be sufficiently small to
not affect the dynamics, and the following initial conditions are chosen,
$\chi_{0}=6.0M_{p}$, $\dot{\chi}_{0}=0$, and $\dot{\theta}_{0}=0$.

In the preceding analysis, it has been assumed that the mixing angle $\alpha$
is fixed throughout the inflation and oscillation epochs. It is observed that
the $\Omega^{2}$ factor quickly approaches 1 after inflation, and thus, the
scalar potential can be approximated by $V(h,\Delta^{0})$ in Eq. (8). This
potential also exhibits a characteristic minimum direction which is defined
by,

$\frac{\rho_{H}}{\rho_{\Delta}}\equiv\tan\beta=\sqrt{\frac{2\lambda_{\Delta}-\lambda_{H\Delta}}{2\lambda_{H}-\lambda_{H\Delta}}}~{},$
(57)

where we require $2\lambda_{\Delta}-\lambda_{H\Delta}>0$ and
$2\lambda_{H}-\lambda_{H\Delta}>0$ for stability. By comparing this relation
to Eq. (16), we see that if $\xi_{H}=\xi_{\Delta}$, then the two mixing angles
$\alpha$ and $\beta$ converge, and we can use the same mixing angle to
describe the inflationary and oscillation epochs. If instead, we have
$\xi_{H}\neq\xi_{\Delta}$, the $\alpha$ and $\beta$ are generally different
and a detailed understanding of the evolution during the oscillation stage
after inflation requires a dedicated analysis. However, we believe that the
lepton asymmetry would only be negligibly affected if this is the case because
almost all of the lepton asymmetry is generated during inflation, with its
evolution just determined by red-shifting due to the expansion of the universe
in the oscillatory phase. It should be noted that the detail of the
oscillation epoch, and the transition between unequal $\alpha$ and $\beta$,
might affect the expected preheating dynamics and the reheating temperature.
The analysis of these dependencies is beyond the scope of this work, and so,
we adopt $\xi_{H}=\xi_{\Delta}$ for simplicity.

###  4.3 Behaviour of the $\mu$ Term

We now provide a discussion of the dynamics of the cubic coupling
$\tilde{\mu}$. Unlike the quartic and quintic terms, the cubic term becomes
increasingly relevant as $\varphi$ decreases. If the cubic term is dominant
during the oscillatory phase, the $\theta$ field will not exhibit directed
motion in phase space, and will oscillate instead. This means that the lepton
asymmetry starts to oscillate and the predictability of our model is lost.
This is depicted in the top-left panel of Figure 4, where the $U(1)_{L}$
breaking term is taken to consist only of the $\tilde{\mu}$ term.

Figure 4:  The lepton number density during inflation and and the oscillatory
epoch for varying values of the cubic coupling term $\tilde{\mu}$, where we
define $H_{0}=m_{S}/2$. The blue and green curves denote positive and negative
lepton number densities respectively, and $n_{L}$ and $\tilde{\mu}$ are given
in Planck units. The input parameters are fixed to $\xi=300$ and
$\lambda=4.5\cdot 10^{-5}$, with initial conditions $\chi_{0}=6.0M_{p}$,
$\dot{\chi}_{0}=0$, and $\dot{\theta}_{0}=0$ chosen.

We must determine the condition that preserves the lepton asymmetry and does
not allow this oscillatory behaviour to become significant. During the matter-
like epoch, the lepton number density generated from the cubic term during one
oscillation time $1/m_{S}$ can be given approximately by,

$\Delta n_{L}\approx\frac{2Q_{L}\tilde{\mu}\varphi^{3}}{m_{S}}~{}.$ (58)

To preserve the lepton number density generated during inflation, it is
necessary to ensure that $\Delta n_{L}\lesssim n_{L}$ throughout the matter-
like epoch, prior to the completion of reheating. That is,

${\tilde{\mu}\lesssim\frac{m_{S}{n_{L}}_{\rm reh}}{4\varphi_{\rm
reh}^{3}}~{}.}$ (59)

Numerically, we find the following constraint $\tilde{\mu}\lesssim
10^{-13}M_{p}$ when requiring $\Delta n_{L}\lesssim n_{L}$, where we have made
the parameter choices $\theta_{0}=0.1$, $\tilde{\lambda}_{5}=10^{-9},$ and
subsequently ${n_{L}}_{\textrm{end}}=1.1\cdot 10^{-11}M_{p}^{3}$. In Figure 4,
we depict the evolution of the lepton asymmetry number density for different
values of $\tilde{\mu}$. It is found that when $\tilde{\mu}<10^{-13}M_{p}$ the
lepton asymmetry does indeed become stable, showing consistency with the
analytical constraint. For parameter values that lead to the observed baryon
asymmetry today, we obtain the following upper bound $\tilde{\mu}\lesssim
10^{-18}M_{p}$.

###  4.4 The Predicted Baryon Asymmetry

Once reheating is completed, the generated non-zero lepton number density
$n_{L}$ will be present in the form of neutrinos. The lepton number carried by
these neutrinos will be redistributed into baryons through the equilibrium
electroweak sphalerons, with the ratio defined by
$n_{B}\simeq-\frac{28}{79}n_{L}$ Klinkhamer:1984di ; Kuzmin:1985mm ;
Trodden:1998ym ; Sugamoto:1982cn . To calculate the baryon asymmetry parameter
generated in our scenario, we must determine the reheating temperature.

The process of reheating in standard Higgs inflation was first analysed in
Ref. Garcia-Bellido:2008ycs ; Bezrukov:2008ut , where it was found that the
parametric resonance production of $W/Z$ bosons plays a significant role in
the preheating process. It has since been found that the preheating process is
more violent than was previously expected Ema:2016dny ; DeCross:2015uza ;
DeCross:2016cbs ; DeCross:2016fdz , with unitarity being violated for SM Higgs
non-minimal couplings of $\xi>350$. However, recently it has been found that
for such large $\xi$ values, it is possible for the model to be UV complete in
Higgs-$R^{2}$ scenarios of inflation Giudice:2010ka ; Gorbunov:2018llf ;
Ema:2017rqn ; Ema:2019fdd ; Ema:2020zvg  for which the preheating process must
be recalculated He:2018mgb ; He:2020ivk ; He:2020qcb . In our scenario, we
choose $\xi=300$ to avoid any possible issues with unitarity. A recent
analysis of the preheating process in SM Higgs inflation Sfakianakis:2018lzf
has shown that for models with $\xi>100$, the reheating happens at
approximately $\sim 3$ e-folds after the end of inflation, so we adopt $\Delta
N=N_{\textrm{reh}}-N_{\textrm{end}}=3$ for simplicity. The details of
reheating may be different in our scenario due to the mixing of the SM and
triplet Higgs’, and thus, it is necessary to undertake a comprehensive
analysis, which will be left to future work.

For the typical parameters chosen, the reheating process occurs at
$t_{\textrm{reh}}=223/H_{0}$, which has the corresponding Hubble parameter
$H_{\textrm{reh}}=0.0047H_{0}$. From the relation
$H_{\textrm{reh}}^{2}\simeq\frac{\pi^{2}}{90}g_{*}\frac{T_{\textrm{reh}}^{4}}{M_{p}^{2}}$,
we then obtain the reheating temperature $T_{\textrm{reh}}\approx 2.2\cdot
10^{14}$ GeV. Considering the entropy density just after reheating defined by
$s=\frac{2\pi^{2}}{45}g_{*}T_{\textrm{reh}}^{3}$ Husdal:2016haj , we then
determine that the baryon asymmetry parameter generated in our model is given
by,

$\eta_{B}=\frac{{n_{B}}}{s}\bigg{|}_{\rm
reh}=\eta_{B}^{\textrm{obs}}\left(\frac{{|n_{L}}_{\textrm{end}}|/M_{p}^{3}}{1.3\cdot
10^{-16}}\right)\left(\frac{g_{*}}{112.75}\right)^{-\frac{1}{4}}~{},$ (60)

where $\eta_{B}^{\textrm{obs}}\simeq 8.5\cdot 10^{-11}$ is the observed baryon
asymmetry parameter Aghanim:2018eyx . This relation shows that at the end of
inflation, it is necessary to have generated a lepton number asymmetry density
of $1.3\cdot 10^{-16}M_{p}^{3}$ to produce the observed baryon asymmetry,
which corresponds to the example parameter sets: $\tilde{\lambda}_{5}=7\cdot
10^{-15}$ for $\theta_{0}=0.1$ and $\tilde{\lambda}_{5}=10^{-10}$ for
$\theta_{0}=6.5\cdot 10^{-6}$ from numerical calculations. These example
parameters illustrate the wide range of parameters over which successful
Leptogenesis can occur. Note that in both of these example cases, the typical
parameters escape the isocurvature constraints Byrnes:2006fr ; Gordon:2000hv ;
Kaiser:2012ak  placed by CMB observations Planck:2018jri ; see the next
section for the detailed calculations of the expected isocurvature
perturbations in our framework.

As detailed in the previous subsection, if the $\tilde{\mu}$ coupling is too
large, the lepton asymmetry will begin to rapidly oscillate during the matter-
like epoch and the predictability of our model breaks down. On the other hand,
a smaller $\tilde{\mu}$ term helps to avoid the washout of the lepton
asymmetry after reheating. In the following, we require $|\tilde{\mu}|\lesssim
10^{-18}M_{p}$ for the initial $\theta_{0}=0.1$ to ensure the generated baryon
asymmetry is preserved and consistent with observation.

In our analysis, we choose the parameter inputs $\lambda_{H}=0.1$, and
$\xi_{H}=\xi_{\Delta}=300$ based on the arguments laid out above. For the
other parameters, we set $\lambda_{\Delta}=4.5\cdot 10^{-5}$ to accommodate
the current CMB measurements, while there exists two options for
$\lambda_{H\Delta}$ that ensure the required inflationary trajectory is
preserved. In the case of $\lambda_{H\Delta}>0$, we require
$2\lambda_{\Delta}\xi_{H}-\lambda_{H\Delta}\xi_{\Delta}>0$ and
$2\lambda_{H}\xi_{\Delta}-\lambda_{H\Delta}\xi_{H}>0$ so that we have a mixed
state of $h$ and $\Delta^{0}$ as the inflaton. The typical value we can
consider is $\lambda_{H\Delta}=10^{-5}$, which gives the mixing angle as
$\alpha\simeq 0.02$. In the case of $\lambda_{H\Delta}<0$, we require that
$|\lambda_{H\Delta}|<2\sqrt{\lambda_{H}\lambda_{\Delta}}$ to avoid the
potential becoming unbounded from below. We can then choose
$\lambda_{\Delta}=4.5\cdot 10^{-5}$, $\lambda_{H\Delta}=-0.001$, which gives
$\alpha\simeq 0.07$.

We stress here that the above parameter choices are just a simplified version
of our benchmark model considered in previous subsections. For the parameters
in a real model, we need to also consider the contributions of radiative
corrections, as well as the satisfaction of the vacuum stability conditions.
As we have mentioned before, the selection of
$\lambda_{H\Delta}\sim\lambda_{\Delta}\sim\mathcal{O}(0.01-0.1)$ may be a more
realistic coupling choice to avoid any issues that could arise from radiative
corrections.

Numerically, an example set of parameters that successfully generates the
observed baryon asymmetry is $\lambda_{5}\sim\lambda_{5}^{\prime}=8.8\cdot
10^{-11}~{}(7.9\cdot 10^{-12})$ with $\theta_{0}\sim 0.1$ in combination with
the typical parameter choices given above, for the $\lambda_{H\Delta}>0$
($\lambda_{H\Delta}<0$) cases respectively. In addition, we obtain an upper
limit on the cubic term coupling of $|\mu|\lesssim 15~{}(1.4)$ TeV. Note that,
all these parameters are defined at a renormalization scale near $M_{p}$.
Given that the $U(1)_{L}$ breaking terms exhibit small couplings, it is
natural to consider that they originate from a spurion field that carries a
$U(1)_{L}$ charge of $+2$. The $U(1)_{L}$ breaking terms are subsequently
generated by requiring that the spurion field obtains a VEV of the order of
$\mathcal{O}(10^{4})$ TeV.

###  4.5 Lepton Number Washout Effects

Since we expect a large reheating temperature in this scenario, the triplet
Higgs will rapidly thermalize at the beginning of the radiation epoch, and so
we must consider possible washout processes. Firstly, we require that the
process $LL\leftrightarrow HH$ is never in thermal equilibrium,

$\Gamma|_{T=m_{\Delta}}=n\langle\sigma v\rangle\approx
y^{2}\mu^{2}/m_{\Delta}<H|_{T=m_{\Delta}}~{},$ (61)

where
$H|_{T=m_{\Delta}}=\sqrt{\frac{\pi^{2}g_{*}}{90}}\frac{m_{\Delta}^{2}}{M_{p}}$.

Remember that the triplet Higgs generates the neutrino masses through the
following definition $m_{\nu}\simeq y\frac{|\mu|v^{2}}{2m_{\Delta}^{2}}$, and
that there should be at least one neutrino mass at least of the order of the
$m_{\nu}\sim 0.05$ eV. Combining this with the above relation, we obtain a
limit on the triplet Higgs mass of $m_{\Delta}<10^{12}$ GeV for $m_{\nu}=0.05$
eV.

The other potentially dangerous processes that we must consider are
$LL\leftrightarrow\Delta$ and $HH\leftrightarrow\Delta$. They must not co-
exist, otherwise, the lepton number asymmetry will be rapidly washed out.
Following Ref. Harvey:1990qw , one can easily find that the only solution for
the corresponding chemical potentials, if both of these processes are in
equilibrium, is $\mu_{L}=\mu_{u_{L}}=\mu_{H}=0$. However, to maintain the
lepton asymmetry generated during inflation, the process
$LL\leftrightarrow\Delta$ must be efficient while $HH\leftrightarrow\Delta$ is
out of equilibrium. This condition can be easily realised for a moderate
$m_{\Delta}$ and a sizable $y$, for example, $m_{\Delta}<10^{8}$ GeV and
$y>10^{-5}$. The requirement to ensure that these two processes do not coexist
is given by,

$\Gamma_{ID}(HH\leftrightarrow\Delta)|_{T=m_{\Delta}}<H|_{T=m_{\Delta}}~{}.$
(62)

where

$\Gamma_{ID}(HH\leftrightarrow\Delta)|_{T=m_{\Delta}}\approx\Gamma_{D}(\Delta\rightarrow
HH)\simeq\frac{\mu^{2}}{32\pi m_{\Delta}}~{}.$ (63)

Using that $v_{\Delta}\simeq-\frac{\mu v_{\textrm{EW}}^{2}}{2m^{2}_{\Delta}}$
with Eq. (62), the necessary condition is found to be,

$v_{\Delta}\lesssim 10^{-5}~{}{\rm GeV}\left(\frac{m_{\Delta}}{1~{}{\rm
TeV}}\right)^{-1/2}~{},$ (64)

hence, for $m_{\Delta}\gtrsim 1$ TeV, we generally require that
$v_{\Delta}\lesssim 10$ keV to prevent the washout of the lepton asymmetry.
This is an important feature of our model with significant phenomenological
implications, as will be discussed in the following section.

##  5 Phenomenological and Cosmological Implications

The Type II Seesaw Mechanism alone has rich phenomenological implications both
terrestrially and cosmologically. The inclusion of the associated triplet
Higgs in the inflationary setup and its responsibility for the production of
the matter-antimatter asymmetry places unique constraints on the allowed
parameter space of the model, and allows for additional avenues for
experimental verification. Here we will discuss some of these possibilities,
and provide a summary of the parameter space for successful Baryogenesis that
is consistent with each of these constraints. We shall also consider the
detection prospects from future experimental results.

It should be noted, that although not discussed below, there are potential
Gravitational Wave signals that may be associated with our model, in addition
to the predicted tensor-to-scalar ratio. Namely, from the dynamics of the
preheating process Figueroa:2017vfa ; Caprini:2018mtu , and from the
inducement of a first-order phase transition through the new interactions
introduced into the scalar potential Zhou:2022mlz .

###  5.1 Current and Future Collider Constraints

Searches for the charged components of the triplet Higgs at terrestrial
colliders provide important constraints on its allowed mass. Depending upon
the VEV of the triplet Higgs and its Yukawa couplings, they can decay
dominantly into gauge bosons or leptons Melfo:2011nx . Generally, for
$v_{\Delta}\lesssim 0.1$ MeV, the triplet Higgs will dominantly decay into
leptons, while for $v_{\Delta}\gtrsim 1$ MeV, it will dominantly decay into
gauge bosons. Cascade decays are also possible if the mass splitting between
the triplet Higgs components is sizable, namely $\gtrsim$ 10 GeV. However,
such large mass splittings require a large $\lambda_{4}$ coupling. To keep
perturbativity and vacuum stability up to $M_{p}$ scale, the $\lambda_{4}$
coupling cannot be much larger than $\mathcal{O}(1)$, and subsequently it has
been shown that the mass splitting between the different states of the triplet
Higgs is generally less than 10 GeV for $m_{\Delta}>1$ TeV Bonilla:2015eha .

In our scenario, $v_{\Delta}$ can only be accommodated within the range 0.05
eV $\lesssim v_{\Delta}\lesssim$ 10 keV, meaning that the triplet Higgs will
dominantly decay into leptons. If we observed the triplet Higgs in such a
channel, it would provide a smoking gun for our model. In fact, the LHC has
already performed searches for the triplet Higgs via decays to the leptonic
channel from which a lower limit of $\sim 800$ GeV has been determined for the
doubly-charged Higgs ATLAS:2017xqs . The future 100 TeV collider could explore
larger mass regions, and preliminary studies Du:2018eaw  show that a doubly-
charged Higgs of mass of up to 4 TeV will be probed at the 5$\sigma$ level.

###  5.2 Lepton Flavor Violation Experimental Tests

The doubly-charged component of the triplet Higgs is responsible for lepton
flavor violating processes such as $\mu\rightarrow e\gamma$ and
$\mu\rightarrow eee$. In particular, the channel that can be probed to the
highest precision is provided by the decay $\mu^{+}\to e^{+}e^{-}e^{+}$ which
has the following branching ratio Dinh:2012bp ,

${\cal B}(\mu^{+}\to e^{+}e^{-}e^{+})=\frac{|y_{\mu
e}y^{\dagger}_{ee}|^{2}}{16G_{F}^{2}m_{\Delta^{++}}^{4}}\,~{},$ (65)

which can then be compared with the current best upper bound SINDRUM:1987nra ,

${\cal B}(\mu^{+}\to e^{+}e^{-}e^{+})\leqslant 1.0\cdot 10^{-12}\,~{}.$ (66)

The lepton flavor violating decay process $\mu\to e\gamma$ is generated at
one-loop level with the help of the doubly and singly-charged scalar
components of the triplet Higgs. As a result, the partial width for this
process is given by Dinh:2012bp ,

${\cal B}(\mu\to
e\gamma)\simeq\frac{\alpha}{3072\pi}\frac{\left|(y^{\dagger}y)_{e\mu}\right|^{2}}{G_{F}^{2}}\left(\frac{1}{m_{\Delta^{+}}^{2}}+\frac{8}{m_{\Delta^{++}}^{2}}\right)^{2}\,~{},$
(67)

Nowadays, the most stringent constraint on $\mu\to e\gamma$ is given by the
MEG collaboration which has determined the following upper bound on this
process MEG:2016leq ,

${\cal B}(\mu\to e\gamma)<4.2\cdot 10^{-13}\,~{}.$ (68)

To translate the full implications of these constraints to the parameter space
of our model, it is necessary to perform a thorough parameter scan which
includes the current neutrino data. This is beyond the scope of this paper and
here we just assume all the couplings are of the same order for simplicity.
The current experimental limits on these lepton flavor violating decay
processes already provide constraints on the triplet Higgs properties, see
Figure 7, with future experiments such as Mu3e Perrevoort:2018ttp  to improve
upon the $\mu\rightarrow eee$ limits by two orders of magnitude. Thus, the
allowed parameter space of our model will be tested through these processes in
the near future.

###  5.3 Neutrino Experiment Tests

The Type II Seesaw Mechanism can be probed through precision tests of rare
processes in the neutrino sector. These are associated with the new couplings
introduced between the triplet Higgs and the neutrinos, as well as through the
generation of Majorana mass terms for the neutrinos.

Neutrinoless double beta decay is a predicted experimental feature of models
containing Majorana mass terms for the neutrinos, which will be probed in the
near future. Theories that provide unique predictions may be differentiated or
ruled out by increased experimental precision in the measurement of this
process. In this model, the observed neutrino masses are of the Majorana type.
This is in contrast to models which include right-handed neutrinos, where the
observed neutrinos can have both Dirac and Majorana type mass terms. Thus, our
model can be probed in near future neutrinoless double beta decay experiments.

Additionally, the baryon asymmetry generated in this model is independent of
the leptonic $\mathcal{CP}$ phase, with $\mathcal{CP}$ spontaneously broken in
the leptonic sector at early times in the universe. There are currently
conflicting measurements of the leptonic $\mathcal{CP}$ phase coming from the
T2K and NOvA experiments, with T2K disfavouring a $\mathcal{CP}$ conserving
angle T2K:2019bcf , which is inconsistent with the NOvA result NOvA:2021nfi .
In this context, our model provides an interesting theoretical possibility for
Leptogenesis.

###  5.4 Vacuum Stability

The current experimentally determined values of the SM parameters strongly
suggest that the Higgs vacuum is only metastable EliasMiro:2011aa ;
Degrassi:2012ry ; Lebedev:2012sy ; Salvio:2013rja ; Branchina:2014usa ;
Bezrukov:2014ipa . This can be seen in the running of the renormalization
group equations (RGE) of the Higgs quartic coupling $\lambda_{h}$. It should
be noted that the running of $\lambda_{h}$ is sensitive to the top pole mass,
which has yet to be determined with high precision. There have been different
models proposed to avoid the metastability problem. One interesting
possibility is the introduction of a quintessence coupling to the SM Higgs,
which ensures that in the early universe the Higgs self-coupling is kept
positive Han:2018yrk .

In the context of our model, it is important to consider the change in the
renormalization group running for the Higgs self-coupling induced by the
introduction of the triplet Higgs, and whether the vacuum can be stabilized
through its couplings. Investigations into the implications of the Type II
Seesaw Mechanism for vacuum stability have been explored extensively
Arhrib:2011uy ; Chun:2012jw ; BhupalDev:2013xol ; Bonilla:2015eha ;
Haba:2016zbu ; Mandal:2022zmy ; Moultaka:2020dmb . This has resulted in the
determination of the following conditions, which ensure vacuum stability up to
the Planck scale,

$\displaystyle\mathcal{B}_{0}=\biggl{\\{}\lambda_{H}>0;~{}~{}~{}~{}\lambda_{2}+\lambda_{3}>0;~{}~{}~{}~{}\lambda_{2}+\frac{\lambda_{3}}{2}>0\biggr{\\}}~{},$
(69)
$\displaystyle\mathcal{B}_{1}=\biggl{\\{}\lambda_{1}+2\sqrt{\lambda_{H}(\lambda_{2}+\lambda_{3})}>0;~{}~{}~{}~{}\lambda_{1}+\lambda_{4}+2\sqrt{\lambda_{H}(\lambda_{2}+\lambda_{3})}>0;$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}2\sqrt{\lambda_{H}}\lambda_{3}\leq\sqrt{(\lambda_{2}+\lambda_{3})\lambda_{4}^{2}}\biggr{\\}}~{},$
(70)
$\displaystyle\mathcal{B}_{2}=\biggl{\\{}2\sqrt{\lambda_{H}}\lambda_{3}\geq\sqrt{(\lambda_{2}+\lambda_{3})\lambda_{4}^{2}};$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\lambda_{1}+\frac{\lambda_{4}}{2}+2\sqrt{\lambda_{H}(\lambda_{2}+\frac{\lambda_{3}}{2})(1-\frac{\lambda^{2}_{4}}{8\lambda_{H}\lambda_{3}})}>0\biggr{\\}}~{},$
(71)

where it is required that
$\mathcal{B}_{0}\land\\{\mathcal{B}_{1}\lor\mathcal{B}_{2}\\}$ be satisfied
when imposed at any renormalization scale up to $M_{p}$ to avoid vacuum
instability below the Planck scale Bonilla:2015eha ; Moultaka:2020dmb . It is
important to note that, the widely used older conditions found in Ref.
Arhrib:2011uy ; Chun:2012jw ; BhupalDev:2013xol ; Haba:2016zbu ;
Mandal:2022zmy  are sufficient but not necessary. Note that, the threshold
corrections from the triplet Higgs can be ignored due to the smallness of the
$\mu$ term. In summary, it has been found that there are vast regions of
parameter space where the stability condition can be satisfied Bonilla:2015eha
; Moultaka:2020dmb .

###  5.5 Isocurvature Perturbations

In our scenario, we have assumed that the inflationary period is induced by
two scalar fields. Generally, such inflationary setups can generate non-
trivial non-Gaussian features Kaiser:2012ak . Additionally, we must consider
whether the isocurvature perturbations produced by the inflationary dynamics
of $\theta$ are within the current observational limits. In the calculation
presented below, we follow the works of Ref. Cline:2019fxx  and
Kawasaki:2020xyf , and use the formalism employed in Ref. Gordon:2000hv ;
Kaiser:2012ak .

Besides the $\chi$ field, in our model, the only other relevant dynamics
during inflation are from the $\theta$ field which is responsible for the
generation of the lepton number asymmetry. We calculate the isocurvature
perturbations by considering the dynamics of our two-field inflation framework
$(\chi,\theta)$. A general Einstein frame action can be written as follows,

$S_{\rm E}=\int
d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R-\frac{1}{2}h_{IJ}g^{\mu\nu}\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{J}-V(\phi)\right]~{},$
(72)

where $h_{IJ}$ is the field space metric. We make the following definition
$\phi^{I}(x^{\mu})=\varphi^{I}(t)+\delta\phi^{I}(x^{\mu})$, which gives the
following equations of motion for the background fields $\varphi^{I}(t)$,

$\mathcal{D}_{t}\dot{\varphi}^{I}+3H\dot{\varphi}^{I}+h^{IJ}V_{,K}=0~{},$ (73)

where $\mathcal{D}_{t}$ is the covariant directional derivative defined by
$\mathcal{D}_{t}A^{I}\equiv\dot{\varphi}^{I}\mathcal{D}_{J}A^{I}=\dot{A}^{I}+\Gamma^{I}_{JK}A^{J}\dot{\varphi}^{K}$.
The gauge invariant scalar perturbations, known as the Mukhanov-Sasaki
variables, are defined as,

$Q^{I}=\delta\phi^{I}+\frac{\dot{\varphi}^{I}}{H}\psi~{}.$ (74)

Thus, we obtain,

$\mathcal{D}_{t}^{2}Q^{I}+3H\mathcal{D}_{t}Q^{I}+\left[\frac{k^{2}}{a^{2}}\delta^{I}_{J}+\mathcal{M}^{I}_{J}-\frac{1}{M_{p}^{2}a^{3}}\mathcal{D}_{t}\left(\frac{a^{3}}{H}\dot{\varphi}^{I}\dot{\varphi}_{J}\right)\right]Q^{J}=0~{},$
(75)

where

$\mathcal{M}^{I}_{J}\equiv
h^{IK}(\mathcal{D}_{J}\mathcal{D}_{K}V)-\mathcal{R}^{I}_{LMJ}\dot{\varphi}^{L}\dot{\varphi}^{M}~{}.$
(76)

The adiabatic field $\sigma$ and its direction $\hat{\sigma}^{I}$ can be
calculated using,

$\dot{\sigma}^{2}=h_{IJ}\dot{\varphi}^{I}\dot{\varphi}^{J}~{},~{}~{}~{}\textrm{and}~{}~{}~{}\hat{\sigma}^{I}\equiv\frac{\dot{\varphi}^{I}}{\dot{\sigma}}~{}.$
(77)

Then we have,

$\displaystyle\ddot{\sigma}+3H\dot{\sigma}+V_{,\sigma}=0~{},$ (78)
$\displaystyle
H^{2}=\frac{1}{3M_{p}^{2}}\left[\frac{1}{2}\dot{\sigma}^{2}+V\right]~{},$ (79)
$\displaystyle\dot{H}=-\frac{1}{2M_{p}^{2}}(\dot{\sigma})^{2}~{},$ (80)

where $V_{,\sigma}\equiv\hat{\sigma}^{I}V_{,I}$. We then define the entropy
direction $\hat{s}^{I}$ as,

$\hat{s}^{I}\equiv\frac{\omega^{I}}{\omega}~{},~{}~{}~{}\textrm{and}~{}~{}~{}\omega^{I}\equiv\mathcal{D}_{t}\hat{\sigma}^{I}~{},$
(81)

where $\omega=\sqrt{h_{IJ}\omega^{I}\omega^{J}}$.

The slow-roll parameters are then given by,

$\displaystyle\epsilon$ $\displaystyle\equiv$
$\displaystyle-\frac{\dot{H}}{H^{2}}=\frac{3\dot{\sigma}^{2}}{\dot{\sigma}^{2}+2V}~{},$
(82) $\displaystyle\eta_{ss}$ $\displaystyle\equiv$ $\displaystyle
M_{p}^{2}\frac{\hat{s}_{I}\hat{s}^{J}\mathcal{M}^{I}_{J}}{V}~{},$ (83)
$\displaystyle\eta_{\sigma\sigma}$ $\displaystyle\equiv$ $\displaystyle
M_{p}^{2}\frac{\hat{\sigma}_{I}\hat{\sigma}^{J}\mathcal{M}^{I}_{J}}{V}~{}.$
(84)

The gauge invariant adiabatic and isocurvature perturbations are parameterized
by,

$\mathcal{R}_{c}=(H/\dot{\sigma})\hat{\sigma}_{I}Q^{I}~{},~{}~{}~{}\textrm{and}~{}~{}~{}\mathcal{S}=(H/\dot{\sigma})\hat{s}_{I}Q^{I}~{}.$
(85)

The evolution of $\mathcal{R}_{c}$ and $\mathcal{S}$ after horizon crossing is
given by,

$\dot{\mathcal{R}}_{c}=\alpha
H\mathcal{S}+\mathcal{O}(\frac{k^{2}}{a^{2}H^{2}})~{},~{}~{}~{}\textrm{and}~{}~{}~{}\dot{\mathcal{S}}=\beta
H\mathcal{S}+\mathcal{O}(\frac{k^{2}}{a^{2}H^{2}})~{},$ (86)

where $\alpha=\frac{2\omega(t)}{H(t)}$ and
$\beta(t)=-2\epsilon-\eta_{ss}+\eta_{\sigma\sigma}-\frac{4}{3}\frac{\omega^{2}}{H^{2}}$.

The transfer functions can be defined as,

$\displaystyle\left(\begin{array}[]{c}\mathcal{R}_{c}\\\
\mathcal{S}\end{array}\right)=\left(\begin{array}[]{cc}1&T_{\mathcal{R}S}\\\
0&T_{\mathcal{S}S}\end{array}\right)\left(\begin{array}[]{c}\mathcal{R}_{c}\\\
\mathcal{S}\end{array}\right)_{*}~{},$ (93)

with

$\displaystyle T_{\mathcal{RS}}(t_{*},t)$ $\displaystyle=$
$\displaystyle\int^{t}_{t_{*}}dt^{\prime}2\omega(t^{\prime})T_{\mathcal{SS}}(t_{*},t)~{},$
(94) $\displaystyle T_{\mathcal{SS}}(t_{*},t)$ $\displaystyle=$
$\displaystyle\exp\left[\int^{t}_{t_{*}}dt^{\prime}\beta(t^{\prime})H(t^{\prime})\right]~{}.$
(95)

The correlation of the curvature and isocurvature modes is typically defined
by,

$\cos\Delta\equiv{T_{\mathcal{RS}}}/(1+T^{2}_{\mathcal{RS}})^{1/2}~{}.$ (96)

A limit can be placed on this parameter from the current Planck data, namely
$\cos\Delta\lesssim 0.1$ Planck:2018jri , which is the parameter we wish to
derive in our scenario.

Now we have all the ingredients necessary to calculate the isocurvature
perturbations in our model. Firstly, the scalars in our scenario are,

$\phi^{1}=\chi,~{}~{}\phi^{2}=\theta,~{}~{}h_{IJ}=h_{IJ}(\chi)=\left(\begin{array}[]{cc}1&0\\\
0&f(\chi)\end{array}\right)~{},$ (97)

where $f(\chi)\equiv\frac{\varphi(\chi)^{2}\cos^{2}\alpha}{\Omega^{2}(\chi)}$
.

These give the following,

$\Gamma^{1}_{ij}=\left(\begin{array}[]{ccc}0&0\\\
0&-\frac{1}{2}f^{\prime}\end{array}\right),~{}~{}~{}\Gamma^{2}_{ij}=\left(\begin{array}[]{ccc}0&\frac{f^{\prime}}{2f}\\\
\frac{f^{\prime}}{2f}&0\end{array}\right)~{}.$ (98)

Figure 5: The evolution of $T_{\mathcal{R}S}$ during inflation for different
parameter values for $\lambda_{5}$ and $\theta_{0}$. The other input
parameters are fixed to $\xi=300$ and $\lambda=4.5\cdot 10^{-5}$, with initial
conditions $\chi_{0}=6.0M_{p}$, $\dot{\chi}_{0}=0$, and $\dot{\theta}_{0}=0$
chosen. We choose $\tilde{\mu}$ to be sufficiently small to not affect the
dynamics.

Thus, we find that the only non-vanishing components of the Riemann tensor
$R^{l}_{kji}$ are,

$R^{1}_{212}=-R^{1}_{221}=-R^{2}_{112}=R^{2}_{121}=\frac{{f^{\prime}}^{2}}{4f}-\frac{f^{\prime\prime}}{2}~{},$
(99)

which lead to the Ricci curvature tensor $R_{ji}$ and Ricci scalar $R$ as,

$R_{12}=R_{21}=\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f^{2}}~{},~{}~{}~{}\textrm{and}~{}~{}~{}R=\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{2f^{2}}~{}.$
(100)

In Figure 5, we depict the numerically calculated evolution of
$T_{\mathcal{R}S}$ throughout inflation, which must be smaller than $0.1$ to
avoid the current observational constraints from the Planck telescope. As
exhibited in the figure, the scenario with $\tilde{\lambda}_{5}=10^{-9}$ and
$\theta_{0}=0.1$ produces isocurvature perturbations close to the current CMB
observational sensitivity. However, this parameter set massively overproduces
the observed baryon asymmetry. It is observed that a reduction of
$\tilde{\lambda}_{5}$ or $\theta_{0}$ leads to a smaller $T_{\mathcal{R}S}$,
and thus, successful Leptogenesis in our scenario does not produce sizeable
isocurvature perturbations. This is consistent with the expectation that if
the parameters are $\tilde{\lambda}_{5}=0$ or $\theta_{0}=0$, there will be no
lepton asymmetry generated and the isocurvature mode would disappear.

Thus, a sizable isocurvature signature could be produced only if considerable
washout effects are allowed at late times. This possibility may be probed by
future observations.

###  5.6 Q-ball Formation

In mechanisms for Baryogenesis that involve charged scalar fields, such as the
Affleck-Dine mechanism, the formation of Q-balls can occur. If they are
produced, it is imperative to investigate their stability and regime of
formation to ascertain their phenomenological implications for observations
and Baryogenesis. If the Q-balls are absolutely stable, they can make up part
of the dark matter relic density, and in doing so, jeopardise successful
Baryogenesis by sequestering the generated asymmetry from the thermal plasma.

In our model, any Q-balls that are produced will have decay pathways into
fermions through the neutrino Yukawa coupling with the triplet Higgs. Thus, as
long as the decay time of these processes is such that the Q-balls decay
before the EWPT, there should be no phenomenological implications of Q-ball
formation during the inflation and oscillation epochs Coleman:1985ki ;
Lee:1988ag ; Enqvist:2002si ; Enqvist:2003gh ; Kusenko:1997ad ;
Rinaldi:2013lsa . It should be noted that if Q-Ball formation does occur it
can lead to the production of Gravitational Waves which may be probed by
future experiments White:2021hwi .

We need to first determine if Q-ball formation is possible in our scenario. To
do this we must consider the properties of the effective potential
$U_{\textrm{eff}}(\chi,\omega)=U(\chi)-\frac{1}{2}\omega^{2}\chi^{2}$, and
determine the range of $\omega$ frequencies for which bounce solutions exist.
The upper and lower bounds on the frequency, $\omega+$ and $\omega_{-}$, are
given by,

$\omega_{+}^{2}=U^{\prime\prime}(0)~{},\textrm{~{}~{}~{}and~{}~{}~{}}\omega_{-}^{2}=\left.\frac{2U(\chi)}{\chi^{2}}\right|_{\textrm{min}}~{},$
(101)

where $U(\chi)$ is given in Eq. (34). These relations lead to the following
requirement for the formation of Q-balls,

$U^{\prime\prime}(0)>\left.\frac{2U(\chi)}{\chi^{2}}\right|_{\textrm{min}}~{},$
(102)

where we have taken $m_{\Delta}\simeq\sqrt{U^{\prime\prime}(0)}$.

Figure 6: An example of the effective potential for bounce solutions in our
framework, with all of the parameters given in Planck units.

A non-trivial $\omega_{-}$ is obtained when the potential just obtains two
degenerate minima, as depicted in Figure 6, with one at the origin and the
other at a large $\chi$ field value. Given that the potential becomes flat for
$\chi\gg M_{p}$, all the values of $\omega_{-}>0$ will allow for Q-ball
formation at sufficiently large $\chi$. However, it is expected that higher-
dimensional operators of $\phi$ should exist in the Jordan frame potential.
Translating these terms into the canonically normalised Einstein frame field
$\chi$, they are exponentially enhanced by factors of
$e^{\frac{n}{\sqrt{6}}\frac{\chi}{M_{p}}}$ depending on the dimensionality of
the term, where $n+4$ is the dimension of the operator. A dimension 6 operator
of the form ${c_{6}}{\frac{\phi^{6}}{M_{p}^{2}}}$, increases faster with
$\chi$ than the $\omega_{-}$ dependent term, with the lower bound on $\omega$
then dependent upon the choice of the coupling ${c_{6}}$. It should be noted
that $c_{6}$ must also be sufficiently small to not destroy the inflationary
dynamics of our framework.

An example parameter choice is depicted in Figure 6, for which
$\omega_{-}\simeq 1.27\cdot 10^{-6}M_{p}$ is the lower bound for a dimension 6
coupling, $c_{6}=10^{-8}$. This value of $\omega_{-}$ is much larger than the
range of masses $m_{\Delta}$ that we consider in our scenario. An approximate
requirement for Q-Ball formation can then be derived that relates the $c_{6}$
coupling and the frequency $\omega_{-}$,

$c_{6}\simeq 430\frac{\omega_{-}}{M_{p}}e^{-\frac{m_{S}}{\omega_{-}}}~{}.$
(103)

Thus, we can see that significant exponential suppression is required to allow
Q-ball formation for small frequencies. If we take $\omega_{-}\sim
m_{\Delta}\sim 1$ TeV, then it is necessary for $c_{6}$ to be incredibly tiny,
being much less than $\sim 2\cdot 10^{-13}e^{-3\cdot 10^{10}}$. Therefore, we
conclude that the formation of stable Q-balls is unlikely to occur in our
scenario.

###  5.7 Summary of Parameter Space and Experimental Reach

In Figure 7, we show the region of allowed parameter space for which the
lepton number density generated during inflation leads to successful
Baryogenesis. Requiring perturbative neutrino Yukawa $y$ couplings up to the
Planck scale $M_{p}$ ($y\lesssim 1$) results in the black exclusion region.
However, to avoid fine-tuning at the high energy scale, a small Yukawa
coupling $y<0.1$ is preferred for the size of quartic coupling we typically
consider, namely $\lambda_{\Delta}\simeq 4.5\cdot 10^{-5}$. To ensure the
$\mu$-term does not destroy the generated lepton number asymmetry, as
discussed in the previous section, we exclude the grey region. Additionally,
significant lepton asymmetry washout effects exist after reheating for
parameters within the blue exclusion region.

Figure 7: The allowed region of parameter space is depicted (White), avoiding
non-perturbative neutrino Yukawa couplings (Black), cubic term domination of
the $U(1)_{L}$ breaking interactions (Grey), and lepton asymmetry washout
processes (Blue). The red region indicates the current limits from lepton
violating processes SINDRUM:1987nra , with green denoting the future Mu3e
experimental sensitivity Perrevoort:2018ttp . The future 100 TeV collider
constraints are shown in the light red region Du:2018eaw . We provide dotted
red lines to illustrate the corresponding values of the neutrino Yukawa
couplings and triplet Higgs VEV within the white allowed region.

We note that the input parameters used in Figure 7 are not the only allowed
parameter set. For example, we can take
$\lambda_{H\Delta}\lesssim\lambda_{\Delta}\sim 10^{-2}$, and subsequently to
obtain the required lepton asymmetry for Eq. (53), one can easily find that
the coupling $\lambda_{5}^{\prime}$ must be one order of magnitude larger.
Simultaneously, in this example, the upper limit on the $\mu$ term, given by
the gray region, will be weakened by an order of magnitude.

A limit is obtained from precision measurements of the triplet Higgs VEV
$\langle\Delta^{0}\rangle$, namely, it must be less than a few GeV. However,
this leads to no additional constraints on our model because we require that
the VEV must be less than 10 keV to avoid lepton asymmetry washout effects
after reheating. The LHC searches provide lower limits on the masses of the
triplet Higgs components, with the current bound on the mass of the doubly
charged Higgs being $\gtrsim 800$ GeV ATLAS:2017xqs , due to which we depict
triplet Higgs masses $\geq 1$ TeV in Figure 7. These bounds will be improved
at the upgraded high luminosity LHC and at future collider experiments
Du:2018eaw .

If we assume that each of the neutrino Yukawa couplings are of approximately
the same order of magnitude, the lepton flavor violating processes induced by
the doubly-charged component of the triplet Higgs already constrain the
allowed parameter space SINDRUM:1987nra ; Han:2021nod . The current bounds
will be improved by two orders of magnitude by the upcoming Mu3e experiment
Perrevoort:2018ttp , as indicated in Figure 7.

##  6 Conclusions and Future Prospects

The Type II Seesaw Mechanism provides a minimal framework in which to explain
the neutrino masses through the introduction of a single triplet Higgs to the
SM. It has been believed that this simple extension cannot provide a
satisfactory solution to the unknown origin of the observed baryon asymmetry
of the universe, in contrast to the Type I or Type III Seesaw Mechanisms. In
this paper, we have demonstrated that the introduction of the triplet Higgs of
the Type II Seesaw Mechanism can provide a natural way to generate the baryon
asymmetry if it also plays a role in setting up the inflationary dynamics. In
doing so, a successful and economic Affleck-Dine Baryogenesis scenario is
realised without the need for Supersymmetry. Beyond these successes, the model
exhibits an intriguing and unique combination of phenomenological predictions.
We now summarize these possibilities and their future experimental prospects:

  * •

The dominant decay processes exhibited by the triplet Higgs is dependent upon
the VEV of the triplet and its Yukawa couplings. These decays can be mainly
into gauge bosons or leptons. In our framework, the $v_{\Delta}$ is
constrained to be within the range of 0.05 eV - 10 keV, with this upper bound
required to ensure that the lepton number asymmetry is unaffected by lepton
washout effects after reheating. Importantly, within this range of
$v_{\Delta}$ values, the triplet Higgs dominantly decays into leptons. If we
were to observe the triplet Higgs in the leptonic channel, it would provide a
smoking gun for our model.

  * •

The doubly-charged component of the triplet Higgs is responsible for lepton
flavor violating processes such as $\mu\rightarrow e\gamma$ and
$\mu\rightarrow eee$. The current experimental limits on these processes are
already able to place interesting constraints on the allowed properties of the
triplet Higgs in our model, depicted in Figure 7. Planned future experiments,
such as Mu3e, will improve upon the $\mu\rightarrow eee$ bounds by
approximately two orders of magnitude. Thus, one of the most interesting
regions of the allowed parameter space of our model will be able to be tested
in the near future.

  * •

The neutrino masses generated in the Type II Seesaw Mechanism are of the
Majorana type. This is in contrast to models that include right-handed
neutrinos, in which the observed neutrinos exhibit both Dirac and Majorana
type mass terms. The presence of the Majorana mass term in our model leads to
rare processes such as neutrinoless double beta decay. Thus, the upcoming
experiments that will seek to observe neutrinoless double beta decay will be
essential tests of our model.

  * •

The Leptogenesis scenario we present in this work is independent of the
leptonic $\mathcal{CP}$ phase, with the leptonic $\mathcal{CP}$ symmetry
instead being spontaneously broken during the early universe. The most up-to-
date experimental measurements of the leptonic $\mathcal{CP}$ phase, from the
T2K and NOvA experiments, are in conflict. The T2K result disfavours a
$\mathcal{CP}$ conserving angle T2K:2019bcf , which is inconsistent with the
NOvA measurement NOvA:2021nfi . In this context, our model provides an
interesting theoretical possibility for successful Leptogenesis.

  * •

Gravitational Waves may play an important role in verifying this Leptogenesis
mechanism. We predict that the tensor-to-scalar ratio signal from inflation is
$0.0033<r<0.0048$, which may be probed by the near future telescope LiteBIRD
Hazumi:2019lys . Also, the preheating process may also provide additional
sources of Gravitational Waves with characteristic signatures Figueroa:2017vfa
; Caprini:2018mtu .

  * •

The inflationary epoch in our model is set up by the mixture of two scalar
fields, the SM and triplet Higgs’. Generally, such inflationary scenarios are
able to generate non-trivial non-Gaussian features that can be observed in
measurements of the CMB Kaiser:2012ak . In addition, it is possible that
sizable isocurvature signatures are produced by our Leptogenesis mechanism if
considerable lepton asymmetry washout effects are allowed to occur after
reheating. These possibilities will be probed by future observational efforts.

###### Acknowledgements.

We would like to thank Tsutomu T. Yanagida, Misao Sasaki, Shi Pi and Jiajie
Ling for their helpful discussions. C. H. is supported by the Guangzhou Basic
and Applied Basic Research Foundation under Grant No. 202102020885, and the
Sun Yat-Sen University Science Foundation. NDB was supported by IBS under the
project code, IBS-R018-D1. The work of HM was supported by the Director,
Office of Science, Office of High Energy Physics of the U.S. Department of
Energy under the Contract No. DE-AC02-05CH11231, by the NSF grant PHY-1915314,
by the JSPS Grant-in-Aid for Scientific Research JP20K03942, MEXT Grant-in-Aid
for Transformative Research Areas (A) JP20H05850, JP20A203, by WPI, MEXT,
Japan, and Hamamatsu Photonics, K.K.

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