aainstitutetext: Centro de Física Teórica de Partículas-CFTP and Departamento
de Física, Instituto Superior Técnico, Universidade de Lisboa, Av Rovisco
Pais, 1, P-1049-001 Lisboa, Portugalbbinstitutetext: Univ Coimbra, Faculdade
de Ciências e Tecnologia da Universidade de Coimbra, Rua Larga, 3004-516
Coimbra, Portugaccinstitutetext: CFisUC, Rua Larga, 3004-516 Coimbra,
Portugalddinstitutetext: Departamento de Física da Universidade de Aveiro,
Campus de Santiago, 3810-183 Aveiro, Portugaleeinstitutetext: CIDMA, Campus de
Santiago, 3810-183 Aveiro, Portugal

# Warm Inflation, Neutrinos and Dark matter:
a minimal extension of the Standard Model

Miguel Levy b,c João G. Rosa c,d,e Luís B. Ventura<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that warm inflation can be realized within a minimal extension of the
Standard Model with three right-handed neutrinos, three complex scalars and a
gauged lepton/B-L U(1) symmetry. This simple model can address all the
shortcomings of the Standard Model that are not related to fine-tuning, within
general relativity, with distinctive experimental signatures that can be
probed in the near future. The inflaton field emerges from the collective
breaking of the U(1) symmetry, and interacts with two of the right-handed
neutrinos, sustaining a high-temperature radiation bath during inflation. The
discrete interchange symmetry of the model protects the scalar potential
against large thermal corrections and leads to a stable inflaton remnant at
late times which can account for dark matter. Consistency of the model and
agreement with Cosmic Microwave Background observations naturally yield light
neutrino masses below 0.1 eV, while thermal leptogenesis occurs naturally
after a smooth exit from inflation into the radiation era.

##  1 Introduction

The Standard Model (SM) is an extremely successful quantum field theory
framework that accurately describes all processes involving known elementary
particles. Nevertheless, several crucial open problems remain unsolved within
this framework and have motivated a plethora of extensions in the literature.

On the one hand, several open problems are related to fine-tuning issues, such
as the stability of the electroweak scale (the ‘hierarchy problem’), the
smallness of the cosmological constant or the severe constraints on CP-
violation in strong interactions. These issues can, however, only be seen at
most as hints for new physics111Finely-tuned parameters are not a problem per
se within a renormalizable quantum field theory, but only a symptom of their
sensitivity to unknown physical properties at high energy scales.. On the
other hand, there are four problems that definitely point towards the
existence of new particles and/or interactions: (1) neutrino masses, (2) dark
matter, (3) the cosmological baryon asymmetry and (4) inflation. Neutrino
masses cannot be accounted for with only the field content of the SM (see e.g.
Tanabashi:2018oca  for a review); there is mounting evidence for a particle
nature of dark matter, as opposed to a modification of gravity (also reviewed
in Tanabashi:2018oca ); CP-violation within the SM is manifestly insufficient
to generate the observed matter-antimatter asymmetry (see e.g. Cline:2006ts );
and standard cosmology fails to explain the nearly scale-invariant spectrum of
Cosmic Microwave Background (CMB) anisotropies, inflation being the leading
mechanism to generate this, alongside explaining the apparent fine-tuning of
initial conditions embodied in the so-called horizon and flatness problems
(see e.g. Baumann:2009ds ).

While several ‘top-down’ frameworks can address these problems, most predict a
plethora of new physical states that have yet to be found in colliders or
require too many unknown parameters to be fully tested experimentally.
Arguably, a ‘bottom-up’ approach of building minimalistic models, with as few
novel ingredients as possible, may prove more fruitful from both the
theoretical and experimental perspectives, keeping in mind that such models
may find different embeddings in more complete theories, such as grand
unification or superstring/M-theory. In fact, if suitable minimal models can
be found to agree with experiment, this may provide further guidance into
developing UV-complete theories.

Hence, in this work, we focus on building a minimal quantum field theory model
where the four problems singled out above can be simultaneously addressed. We
elect the framework of warm inflation Berera:1995wh ; Berera:1995ie  as our
primary arena, since in this context interactions between the inflaton scalar
field and other physical states play a relevant role in both the dynamics and
observational consequences of inflation. This thus provides a natural paradigm
within which to connect the high-energy physics relevant to describe the
inflationary dynamics to low-energy observables and related open problems.

The fundamental difference between warm inflation and the more conventional
cold inflation models is the inclusion of dissipative effects that result from
energy transfer between the inflaton field and an ambient radiation bath.
These are non-equilibrium processes associated with particle production that,
if significant, may counteract the diluting effect of accelerated expansion
and keep the thermal heat bath at a slowly-varying temperature above the
Hubble scale (which sets the Hawking temperature of the de Sitter horizon)
(see Berera:2008ar  for a review). These effects also source thermal
fluctuations of the inflaton field, which in turn produce a nearly scale-
invariant spectrum of primordial curvature perturbations Taylor:2000ze ;
Hall:2003zp ; Graham:2009bf ; Ramos:2013nsa . The shape of this spectrum,
which is imprinted in CMB maps, depends on the form of the interactions
between the inflaton field and particles in the thermal bath, which is the
most interesting aspect of this alternative paradigm from a particle physics
perspective.

A finite temperature during inflation is, however, challenging to realize in
practice, due to potentially large thermal corrections to the inflaton’s mass
that could hinder the slow-roll dynamics, despite the additional dissipative
friction BGR ; YL . The typically large classical values of the inflaton field
also tend to make the particles it couples to quite heavy, greatly suppressing
dissipative effects unless a large multiplicity of fields is present, which is
a challenge in realistic extensions of the SM (see e.g. BasteroGil:2009ec ;
BasteroGil:2011mr ). These problems have been recently overcome in the ‘Warm
Little Inflaton’ (WLI) scenario Bastero-Gil:2016qru , where, as we detail
below, collective symmetry breaking of a U(1) gauge symmetry and an additional
discrete symmetry protect the flatness of the scalar potential against large
thermal corrections and keep the mass of particles coupled to the inflaton
field parametrically below the temperature of the heat bath (see also Bastero-
Gil:2017wwl ; Bastero-Gil:2018uep ).

The WLI scenario also offers new ways to connect inflation to the other SM
shortcomings outlined above. It has already been shown that the discrete
interchange symmetry also makes the inflaton stable at late times. Since it
only interacts directly with heavy particles, these interactions are only
relevant at the high-temperatures attained during inflation. This implies that
a cold, weakly interacting relic of the inflaton field remains until the
present day and can account for the dark matter abundance Rosa:2018iff 222Such
an inflaton remnant may also act as quintessence at late times Rosa:2019jci ,
although this requires particular non-renormalizable forms of the potential
that can only be justified within a more complete theory (see also
Dimopoulos:2019gpz ; Lima:2019yyv  for related scenarios)..

The simple structure of the WLI Lagrangian is also quite suggestive of the
particular implementation we seek in this work. The inflaton interacts with
heavy fermions, which in turn may decay into a light scalar and a light
fermion through standard Yukawa terms. It is thus natural to try to identify
these heavy fermions with the right-handed neutrinos missing in the SM, and
which commonly appear in unification theories based e.g. on the gauge group
SO(10). Their light decay products then correspond to the SM Higgs and lepton
doublets. We will show that consistently realizing warm inflation within this
setup requires Majorana masses for the right-handed neutrinos just below the
GUT scale, at around $10^{15}$ GeV, and neutrino Yukawa couplings $\lesssim
1$. The standard type-I seesaw mechanism then naturally yields light neutrino
masses $\lesssim$ 0.1 eV, in agreement with neutrino oscillation experiments.
We will then explore in detail how the basic WLI structure can be modified to
accommodate realistic neutrino masses and mixings by considering an additional
spontaneously broken $\mathbb{Z}_{3}$ family symmetry.

Finally, the inclusion of Majorana right-handed neutrino singlets leads to the
well-known thermal leptogenesis scenario in standard Big Bang cosmology
Fukugita:1986hr . We will show that our setup is no exception, and that this
may occur, in particular, via the CP-violating decays of the lightest right-
handed neutrino, which fall out-of-equilibrium after radiation has smoothly
taken over the inflaton as the dominant component in the Universe. As we will
see, the lightest right-handed neutrino species is not directly involved in
the dissipative processes that keep the Universe warm during inflation, but
obtaining a realistic neutrino mass spectrum at low-energies necessarily
implies that these neutrinos are thermally produced before decaying after
inflation.

This scenario constitutes the first successful implementation of warm
inflation within a concrete extension of the SM, building upon a large body of
work over the past decades in terms of understanding the intricacies of non-
equilibrium processes during inflation Berera:1996nv ; Berera:1996fm ;
Berera:1998px ; Berera:1999ws ; Berera:2001gs ; Berera:2002sp ; Berera:2004vm
; Berera:2003kg ; Moss:2006gt ; BasteroGil:2006vr ; Berera:2007qm ;
Moss:2008yb ; Graham:2008vu ; BasteroGil:2010pb ; BasteroGil:2012cm ;
BasteroGil:2012zr ; Bastero-Gil:2015nja ; Berera:2018tfc ; Das:2020lut , model
building Berera:1999wt ; BasteroGil:2004tg ; BasteroGil:2005gy ;
BuenoSanchez:2008nc ; BasteroGil:2009gh ; Sanchez:2010vj ; Cai:2010wt ;
Cerezo:2012ub ; Matsuda:2012kc ; Bastero-Gil:2013owa ; Li:2018riw ; Bastero-
Gil:2018yen  and observational predictions Hall:2004ab ; Moss:2007cv ;
Berera:2008ab ; BasteroGil:2011xd ; Bastero-Gil:2013nja ; Bartrum:2013oka ;
Bartrum:2013fia ; Bastero-Gil:2014jsa ; Bastero-Gil:2014raa ; Benetti:2016jhf
; Benetti:2019kgw ; Kamali:2019xnt . Our construction further attests to the
ability of the warm inflation framework to address several different open
problems in both particle physics and cosmology Berera:1998hv ;
BasteroGil:2011cx ; Bastero-Gil:2014oga ; Bastero-Gil:2017yzb  and, as we will
discuss, provides several experimental/observational imprints of its different
aspects that would not be possible if the inflationary Universe were cold and
empty as conventionally assumed.

This work is organized as follows. In the next section we review the warm
inflation paradigm, discussing its dynamics and observational predictions,
describing in detail the WLI scenario on which our model is based. We then
introduce our basic particle physics model in section 3, highlighting the role
of the different fields and symmetries involved. We analyze the cosmological
dynamics associated with our scenario in section 4, first during inflation and
then the post-inflationary evolution, in particular discussing how the
inflaton may account for dark matter. Section 5 is devoted to the spectrum of
light neutrino masses, obtained through the seesaw mechanism, and here we
explain how introducing a family-symmetry breaking flavon field can be
accommodated within our basic setup. We use these results to show, in section
6, that thermal leptogenesis naturally occurs after inflation in this
scenario. We summarize our main results in section 7, also discussing possible
avenues for future research.

##  2 Elements of warm inflation

Warm inflation (WI) is an inflationary paradigm where dissipative processes
transfer energy from the inflaton scalar field, $\phi$, to light degrees of
freedom in a nearly-thermal radiation bath. The cosmological dynamics of the
inflaton-radiation system is described by the dynamical equations for the
inflaton and radiation, alongside the Friedmann equation determining the
Hubble expansion rate $H$:

$\displaystyle\ddot{\phi}+3H\dot{\phi}+\Upsilon\dot{\phi}+\partial_{\phi}V(\phi)=0,$
(1) $\displaystyle\dot{\rho}_{r}+4H\rho_{r}=\Upsilon\dot{\phi}^{2},$ (2)
$\displaystyle H^{2}=\frac{\rho_{\phi}+\rho_{r}}{3M_{p}^{2}},$ (3)

where $\rho_{\phi}=\dot{\phi}^{2}/2+V(\phi)$ is the inflaton’s energy density,
$\rho_{r}\equiv(\pi^{2}/30)g_{*}T^{4}$ is the radiation energy density, with
$g_{*}$ denoting the number of relativistic degrees of freedom, and
$M_{p}\equiv\sqrt{\hbar c/(8\pi G)}\approx 2.44\times 10^{18}\text{ GeV}$ is
the reduced Planck mass. The dissipation coefficient
$\Upsilon=\Upsilon(\phi,T)$ can in general be computed using standard non-
equilibrium quantum field theory techniques from the fundamental Lagrangian of
a given particle physics scenario, at least close to thermal equilibrium
with333This condition ensures that curved-space corrections may safely be
neglected. $T\gtrsim H$.

In the slow-roll approximation, the inflaton field and radiation fluid are
slowly-varying: $\ddot{\phi}\approx 0$, $\dot{\rho}_{r}\approx 0$ while
$\rho_{\phi}\approx V(\phi)\gg\rho_{r}$, therefore

$\displaystyle\dot{\phi}\approx-\partial_{\phi}V(\phi)/(3H(1+Q)),$ (4)
$\displaystyle\rho_{r}\approx\frac{3}{4}Q\dot{\phi}^{2},$ (5) $\displaystyle
H^{2}\approx\frac{V(\phi)}{3M_{p}^{2}},$ (6)

where $Q=\Upsilon/(3H)>0$. This slow evolution occurs while the conditions:

$\epsilon_{\phi},~{}|\eta_{\phi}|<1+Q,$ (7)

are valid, which generalizes the standard slow-roll conditions on the inflaton
potential parameters
$\epsilon_{\phi}\equiv(M_{p}^{2}/2)(\partial_{\phi}V/V)^{2}$ and
$\eta_{\phi}\equiv M_{p}^{2}(\partial_{\phi\phi}V/V)$. This means that the
scalar potential needs not be as flat as in conventional models due to the
additional dissipative friction.

Since inflation is a period of exponential expansion of the universe, it is
more convenient to work with the number of e-folds of expansion, $N_{e}$,
rather than the cosmological time. As such, Eq. (4) can be written in the
form:

$\frac{\phi^{\prime}}{\phi}=-\frac{\sigma_{\phi}}{1+Q},$ (8)

where primes denote derivatives with respect to the number of e-folds and
$\sigma_{\phi}=\\!M_{p}^{2}(\partial_{\phi}V/V\phi)$.

Then, using Eqs. (4)-(6), the ratio of radiation to inflaton potential energy
density is given by;

$\frac{\rho_{r}}{V}\approx\frac{1}{2}\frac{\epsilon_{\phi}}{1+Q}\frac{Q}{1+Q}.$
(9)

The radiation energy density thus remains a subdominant component
($\rho_{r}/V<1$) while the slow-roll conditions hold, but may become
comparable to the inflaton’s energy density towards the end of inflation
($\epsilon_{\phi}\sim 1+Q$) if the system enters the strong dissipation
regime444Note that $Q$ is a dynamical quantity, so that inflation may e.g.
start in the weak dissipation regime and end in the strong dissipation regime,
or vice-versa., $Q>1$.

While the background homogeneous inflaton field evolves according to (1), the
full inhomogeneous field satisfies a Langevin-like equation with an additional
Gaussian white noise term on the right-hand side. This essentially reflects
the fluctuation-dissipation theorem, with the variance of the noise term being
determined by the dissipation coefficient $\Upsilon$, and can be rigorously
derived using non-equilibrium quantum field theory techniques. Although we
refer the interested reader to the existing literature for further details
(see e.g. Berera:2008ar ), we point out that this is a fundamental property of
any dissipative system. For instance, consider the Brownian motion of a
particle in a gas - the average effect of random collisions with the gas
molecules is a damping of the particle’s motion (analogous to the effect of
the $\Upsilon\dot{\phi}$ term in Eq. (1)), but the particle’s velocity can
never truly reach zero due to the effect of individual collisions (which
yields the random noise term).

This signals a fundamental difference between the cold and warm inflation
paradigms. Whereas in the former inflaton fluctuations start in a quantum
regime and then evolve towards a classical limit due to gravitational particle
production in de Sitter space, in warm inflation the interactions with the
heat bath source thermal inflaton fluctuations, which are born classical. The
full power spectrum of primordial curvature perturbations generated by these
thermal inflaton fluctuations has been computed in detail by several different
works in the literature (see e.g. Ramos:2013nsa  and references therein),
taking into account the dynamics of the fluctuations themselves (with the
extra friction resulting in an earlier freezeout), the occupation numbers of
inflaton particles in the thermal bath and the interplay between inflaton and
radiation perturbations:

$\Delta_{\mathcal{R}}^{2}=\frac{V(\phi_{*})}{24\pi^{2}M_{p}^{4}}\frac{(1+Q_{*})^{2}}{\epsilon_{\phi_{*}}}F(Q_{*}),$
(10)

where starred quantities are evaluated at the instant when fluctuations at the
CMB pivot scale became super-horizon during inflation, generically 50-60
e-folds before it ended. The function $F(Q_{*})$ is given by:

$F(Q_{*})=\left(1+2n_{*}+\frac{2\sqrt{3}\pi Q_{*}}{\sqrt{3+4\pi
Q_{*}}}\frac{T_{*}}{H_{*}}\right)G(Q_{*}),$ (11)

where $n_{*}$ is the phase-space distribution of the inflaton fluctuations and
$G(Q_{*})$ is the correction due to the coupled evolution of inflaton and
radiation fluctuations due to the temperature dependence of the dissipation
coefficient. In general, this has to be computed numerically and depends on
the form of $\Upsilon(T,\phi)$, as well as (mildly) on the scalar potential
$V(\phi)$. For the case $\Upsilon\propto T$ (at least near-horizon crossing)
and $V(\phi)\propto\phi^{4}$ that we will be interested in, a numerical fit
gives $G(Q_{*})\approx 1+0.0185Q_{*}^{2.315}+0.335Q_{*}^{1.364}$ Bastero-
Gil:2016qru .

The amplitude of the power spectrum $\Delta_{\mathcal{R}}^{2}\approx 2.1\times
10^{-9}$ Akrami:2018odb  leads to a constraint on the magnitude of the
inflaton potential, whereas the spectral index constrains the slow-roll
parameters:

$n_{s}-1=\frac{d\log\Delta_{\mathcal{R}}^{2}}{d\log
k}\simeq\frac{d\log\Delta_{\mathcal{R}}^{2}}{dN_{e}}\approx\frac{2\eta_{\phi_{*}}-6\epsilon_{\phi_{*}}}{1+Q_{*}}\left(1-\frac{2Q_{*}}{3+5Q_{*}}-\frac{Q_{*}(1+Q_{*})}{3+5Q_{*}}\frac{\partial_{Q_{*}}F(Q_{*})}{F(Q_{*})}\right).$
(12)

Note that this is similar to the spectral index in cold inflation,
$n_{s}-1=(2\eta_{\phi_{*}}-6\epsilon_{\phi_{*}})$, but multiplied by a
$Q_{*}$-dependent factor which is smaller than $1$ (note that $\phi_{*}$ and
$Q_{*}$ are related via Eq. (9)). Therefore, the curvature power spectrum is
typically more blue-tilted in warm inflation than in cold inflation, although
differences in the background evolution have to be taken into account.

Neither dissipative processes nor the finite temperature during inflation are,
however, expected to source additional gravitational waves, given that the
temperature is generically 3-4 orders of magnitude below the Planck scale.
Hence, the enhancement of scalar curvature perturbations due to thermal
effects typically leads to a suppression of the tensor-to-scalar ratio
relative to cold inflation:

$r\equiv\frac{\Delta_{\text{t}}^{2}}{\Delta_{\mathcal{R}}^{2}}=\frac{16\epsilon_{\phi_{*}}}{(1+Q_{*})^{2}F(Q_{*})}.$
(13)

We note again that, due to the difference in background evolution, the value
of $\phi_{*}$ in cold and warm inflation is generically different for the same
form of the inflaton potential. However, most warm scenarios are characterized
by a low tensor-to-scalar ratio, even in the weak dissipation regime at
horizon-crossing, $Q_{*}\ll 1$. This is an attractive feature of warm
inflation, which allows, in particular, for an agreement between inflationary
models with simple monomial potentials with the primordial spectrum inferred
from CMB observations Bartrum:2013fia ; Bastero-Gil:2016qru ; Bastero-
Gil:2017wwl ; Bastero-Gil:2018uep ; Bastero-Gil:2019gao . This is not the case
of the cold inflation scenario, where the simplest renormalizable monomials,
$\phi^{2}$ and $\phi^{4}$, have been ruled out by the results of the Planck
mission Akrami:2018odb .

In summary, warm inflation has several attractive features: an unobservable
late reheating period is replaced by a smooth transition into a radiation-
dominated epoch, with interactions between the inflaton field and other
particles in the heat bath leaving an imprint on the primordial curvature
spectrum; the additional friction facilitates slow-roll and the suppression of
the tensor-to-scalar ratio brings the simplest models into agreement with
observational data.

###  2.1 The Warm Little Inflaton scenario

Building successful realizations of warm inflation within quantum field theory
remained, despite its appealing features, a challenge for more than two
decades, since in general (i) the heat bath backreacts on the scalar
potential, yielding in particular potentially large thermal mass corrections
to the inflaton field that can spoil the slow-roll dynamics, despite the
dissipative friction; (ii) particles in the thermal bath gain mass from the
inflaton field, and may become too heavy to yield any significant dissipative
effects. For instance, a Yukawa coupling of the form $g\phi\bar{\psi}\psi$
yields a mass $g\phi\gg T$ for the fermions in the thermal bath, as well as a
thermal correction $\sim gT\gg H$ to the inflaton’s mass, unless the Yukawa
coupling $g\ll 1$, which would in turn also suppress the dissipation
coefficient.

These problems were overcome, for the first time, in the ‘Warm Little
Inflaton’ (WLI) model555More recently, a realization of warm inflation based
on an axion-like inflaton field with only derivative interactions with the
thermal bath has been proposed in Berghaus:2019whh ., where in essence the
inflaton field is coupled to two particle species in the thermal heat bath.
The masses of these particles are oscillatory functions of the inflaton field,
and hence bounded, and their combined effect leads to a cancellation of the
leading thermal corrections to the scalar potential Bastero-Gil:2016qru .

The original model, on which we will base our subsequent construction, extends
the SM by a U(1) gauge symmetry, under which two complex scalar fields
$\Phi_{1}$ and $\Phi_{2}$ are equally charged. Both scalar fields acquire
equal vacuum expectation values (VEVs) below a critical temperature and
collectively break the U(1) symmetry. The vacuum manifold of the theory can
then be parametrized as:

$\Phi_{1}=\frac{1}{\sqrt{2}}(M+\rho_{1})e^{i(\pi+\varphi)/(\sqrt{2}M)},\qquad\Phi_{2}=\frac{1}{\sqrt{2}}(M+\rho_{2})e^{i(\pi-\varphi)/(\sqrt{2}M)},$
(14)

where $M$ sets the scale of spontaneous symmetry breaking. The overall phase
$\pi$ can be removed, constituting the Goldstone boson that becomes the
longitudinal component of the massive gauge boson. The remaining angular
degree of freedom, $\phi$, then remains as a physical field that will act as
the inflaton. Note that this corresponds to the relative phase between the two
complex scalars, which does not change under U(1) gauge transformations. The
inflaton is therefore a gauge singlet and the U(1) symmetry does not constrain
its potential.

The complex scalars are also coupled, via standard Yukawa terms, to two
additional fermions, $\psi_{1}$ and $\psi_{2}$, in the thermal bath. While the
left-handed components of the fermions have the same charge as the complex
scalars, their right-handed counterparts are gauge singlets. To avoid the
‘$\eta$-problem’, i.e. inducing large thermal corrections to the inflaton’s
mass, one imposes a discrete interchange symmetry, $\Phi_{1}\leftrightarrow
i\Phi_{2}$ and $\psi_{1}\leftrightarrow\psi_{2}$, yielding an interaction
Lagrangian of the form666For simplicity one works with
$\phi\equiv\varphi/\sqrt{2}$, even though this is not the normalized field.:

$-\mathcal{L}_{\phi,\psi}=\frac{1}{\sqrt{2}}\left(g_{1}\Phi_{1}+g_{2}\Phi_{2}\right)\overline{\psi}_{1L}\psi_{1R}+\frac{1}{\sqrt{2}}\left(-ig_{2}\Phi_{1}+ig_{1}\Phi_{2}\right)\overline{\psi}_{2L}\psi_{2R}+h.c..$
(15)

If one chooses $g_{1}=g_{2}=g$, as in the original proposal in Bastero-
Gil:2016qru , the fermion masses after symmetry breaking are given by
$M_{1}=gM\cos{(\phi/M)}$, $M_{2}=gM\sin{(\phi/M)}$. Since these are bounded
functions, this allows the fermion fields to remain light throughout inflation
and ensures non-negligible dissipation with only two fields. Alternatively,
one can choose $g_{1,2}=g$ and $g_{2,1}=0$ Rosa:2018iff , in which case the
fermion masses are independent of the inflaton field, $M_{1}=M_{2}=gM/2$. In
both cases, the combination $M_{1}^{2}+M_{2}^{2}$ that determines the leading
finite temperature correction to the scalar potential is independent of the
inflaton field, $\phi$, thus preventing the dangerous $\mathcal{O}(T^{2})$
thermal mass corrections that can preclude slow-roll. In this work, we will
considert he general case where $g_{1}\neq g_{2}\neq 0$, which exhibits in
fact these very same features, as we show in detail in Appendix A.

Note that the interchange symmetry leads to a reflection symmetry for $\phi$:

$\Phi_{1}\leftrightarrow
i\Phi_{2}\quad\Longleftrightarrow\quad\phi\leftrightarrow\frac{\pi}{2}M-\phi,$
(16)

which in practice is a $\mathbb{Z}_{2}$ symmetry acting on $\phi/M-\pi/4$.
Therefore, only interactions involving even powers of the (shifted) inflaton
field are possible. This prevents the inflaton from decaying when close to the
minimum at the origin, thus necessarily yielding a stable remnant at late
times that can act as dark matter Rosa:2018iff , as we will explore in more
detail below in the context of the general model. Note also that the
interchange symmetry requires equal charges and VEVs for $\Phi_{1}$ and
$\Phi_{2}$, as assumed above.

With this basic setup in mind, we will now build a concrete extension of the
SM where the fermions $\psi_{1}$ and $\psi_{2}$ are identified with two of the
right-handed neutrinos.

##  3 Basic particle physics setup

Our concrete implementation of the WLI scenario is based, as mentioned above,
on an identification of the $\psi_{1,2}$ fermions with two of the missing
right-handed neutrinos in the SM, and which are singlets under the latter’s
gauge group. Naturally, the additional (gauged) U(1) symmetry can then be
identified with lepton number or $B-L$, and generically we denote it by U(1)X.
Given the three fermion families in the Standard Model, it is natural to
consider three additional right-handed neutrino Weyl fermions, $N_{1,2,3}$, as
well as three additional complex scalar fields $\Phi_{1,2,3}$. However, the
discrete interchange symmetry only acts on the fields in the first two
generations, i.e. $\Phi_{1}\leftrightarrow i\Phi_{2},N_{1}\leftrightarrow
N_{2}$. While it may a priori seem that the additional third family could be
eliminated, we will see that the third right-handed neutrino state is crucial
for obtaining realistic neutrino masses and mixings, with the VEV of
$\Phi_{3}$ setting its Majorana mass. Including three right-handed neutrinos
may also allow for a possible embedding into e.g. an $SO(10)$ GUT.

Given the additional particle content required for a realistic model, compared
to the original WLI setup, we also impose a discrete $\mathbb{Z}_{3}$ family
symmetry that ensures that both $N_{3}$ and $\Phi_{3}$ do not affect the
inflationary dynamics, in particular the cancellation of the leading thermal
corrections to the inflaton’s mass. In Table 1, we describe the particle
content of the model and corresponding charge assignments for the gauge and
flavour symmetry, with two possibilities in the latter case. Note that quarks
are not directly involved in the inflationary dynamics, such that their charge
assignments are left free.

| $\Phi_{1,2}$ | $\Phi_{3}$ | $N_{1,2}$ | $N_{3}$ | $H$ | ${L_{1,2}}$ | ${L_{3}}$ | ${\ell_{1,2}}$ | ${\ell_{3}}$
---|---|---|---|---|---|---|---|---|---
$SU(3)_{c}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$SU(2)_{L}$ | $1$ | $1$ | $1$ | $1$ | $2$ | $2$ | $2$ | $1$ | $1$
$U(1)_{Y}$ | $0$ | $0$ | $0$ | $0$ | $\frac{1}{2}$ | -$\frac{1}{2}$ | -$\frac{1}{2}$ | -$1$ | -$1$
$U(1)_{X}$ | $2$ | $2$ | -$1$ | -$1$ | $0$ | -$1$ | -$1$ | -$1$ | -$1$
$\mathbb{Z}_{3}$ (option I) | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | 1 | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$
$\mathbb{Z}_{3}$ (option II) | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | 1 | $\omega$ | $\omega$ | $\omega$ | $\omega$
Table 1: Field content of the model, for $X=B-L$. We give two optional charge
assignments for the discrete $\mathbb{Z}_{3}$ family symmetry
($\omega=e^{2i\pi/3}$), differing only for ${L_{3}}$ and $\ell_{3}$.

We assume that all three complex scalar fields collectively break the
$U(1)_{X}$ symmetry upon acquiring VEVs, with the interchange symmetry
imposing equal VEVs for $\Phi_{1}$ and $\Phi_{2}$. The vacuum manifold is then
given by:

$\displaystyle\Phi_{1}=\frac{1}{\sqrt{2}}(M+\rho_{1})e^{i\pi/M_{T}}e^{i\xi\theta/M_{T}}e^{i\phi/M},$
$\displaystyle\Phi_{2}=\frac{1}{\sqrt{2}}(M+\rho_{2})e^{i\pi/M_{T}}e^{i\xi\theta/M_{T}}e^{-i\phi/M},$
(17)
$\displaystyle\Phi_{3}=\frac{1}{\sqrt{2}}(M^{\prime}+\rho_{3})e^{i\pi/M_{T}}e^{-2i\xi^{-1}\theta/M_{T}},$

where $M_{T}^{2}=2M^{2}+M^{\prime 2}$, $\xi=M^{\prime}/M$ and in general
$M\neq M^{\prime}$. In this parametrization, $\pi$ denotes the Goldstone boson
that is ‘eaten’ by the $X$-gauge boson after spontaneous symmetry breaking,
and which can be explicitly removed by going to the unitary gauge777We note
that in the collective breaking of a U(1) symmetry by $n$ equally charged
complex scalar fields with VEVs
$\sqrt{2}\langle\Phi_{i}\rangle=v_{i}e^{i\varphi_{i}/v_{i}}$, $i=1,\ldots,n$,
the Goldstone boson is the linear combination
$\pi=\sum_{i}v_{i}\varphi_{i}/v$, where $v^{2}=\sum_{i}v_{i}^{2}$.. This
leaves two physical gauge-invariant fields, $\phi$ and $\theta$. We take
$\phi$ to play the role of the inflaton field, corresponding to the relative
phase between the fields $\Phi_{1}$ and $\Phi_{2}$, as in the original WLI
model. The interchange symmetry imposes only that the scalar potential for
$\phi$ must be an even function of $\phi/M-\pi/4$, being otherwise arbitrary.
The scalar potential for $\theta$ is arbitrary, being unconstrained by the
interchange symmetry, and we assume, to simplify our analysis, that $\theta$
has a sufficiently large mass that we may set $\theta=0$ both during and after
inflation.

We also assume, for simplicity, that the gauge coupling and scalar self-
couplings are $\mathcal{O}(1)$, such that both the gauge field and the radial
fields $\rho_{1,2}$ gain masses $\sim M$ and hence decouple from the dynamics
at temperatures below the symmetry breaking scale at which inflation occurs.
We may also neglect the radial field $\rho_{3}$, since its coupling to the
inflaton field can be taken as arbitrarily small in a technically natural way.

The relevant interactions for the inflationary dynamics are those between the
complex scalars and the right-handed neutrinos, which yield Majorana mass
terms, alongside the Yukawa couplings involving $N_{i}$ and the SM Higgs and
lepton doublets. The interchange and family symmetries then yield the general
beyond the SM Yukawa Lagrangian density:

$\mathcal{L}_{\text{Yuk}}=\sum_{i,j}\left(G_{ij}\Phi_{i}\overline{N}_{j}^{c}N_{j}-Y_{ij}\overline{N}_{i}H^{\dagger}L_{j}+\mathrm{h.c.}\right),$
(18)

with $Y_{1i}=Y_{2i}$ and

$G=\begin{pmatrix}g_{1}&g_{2}&0\\\ -ig_{2}&ig_{1}&0\\\
0&0&g_{3}\end{pmatrix}.$ (19)

This structure implies that the inflaton field only interacts with two of the
right-handed neutrinos, $N_{1,2}$, and only these are involved in the
dissipative dynamics leading to warm inflation as we describe in the next
section. The Yukawa terms allow these right-handed neutrinos to decay into SM
leptons and Higgs doublet states (the Higgs boson itself and the components
that will be ‘eaten’ by the weak gauge fields after the electroweak phase
transition). Dissipation of the inflaton’s energy then occurs via the two-
stage process $\phi\rightarrow N_{1,2}\leftrightarrow LH$ in a near-
equilibrium regime. These are therefore the light degrees of freedom that make
up the thermal bath during inflation, and later we will describe how the
remaining SM states are thermally excited close to the end of inflation.

These very same interactions are also responsible for generating light
neutrino masses at low energies through the type I seesaw mechanism
Minkowski:1977sc , such that:

$m_{\nu}=-m_{D}^{T}M_{R}^{-1}m_{D},$ (20)

where $M_{Rj}=G_{ij}\langle\Phi_{i}\rangle$ is the diagonal Majorana mass
matrix for the right-handed neutrinos and $m_{D}=vY$ is the Dirac mass matrix,
with $v\simeq 174$ GeV denoting the Higgs VEV after electroweak symmetry
breaking. For a single family, this would yield $m_{\nu}\sim
y^{2}v^{2}/M_{R}$. Taking this to match the largest mass splitting from solar
neutrino oscillations, $m_{\nu}\sim 0.05$ eV Tanabashi:2018oca , we find
$M_{R}\sim 10^{14}-10^{15}$ GeV for $y\lesssim 1$ Yukawa couplings. As we will
see in the next sections, this is precisely the parametric range required to
consistently realize warm inflation.

We note that, by imposing the $\mathbb{Z}_{3}$ family symmetry, the third
right-handed neutrino $N_{3}$ is effectively decoupled from the inflaton
field, which is required to enforce the WLI mechanism for the cancellation of
large thermal corrections to the inflaton’s mass, which as described earlier
and detailed in Appendix A involves only two fields interacting with the
inflaton. While this fixes the $\mathbb{Z}_{3}$ charge of the right-handed
neutrinos, there is still an ambiguity in the charge assignments for the SM
leptons. The first option in Table 1 assigns the same $\mathbb{Z}_{3}$ charge
for $N_{3}$ and $L_{3},\ell_{3}$, effectively discriminating all the third
family of leptons, which would allow for e.g. an embedding in an $SO(10)$ GUT.
In the second option, only $N_{3}$ has a different $\mathbb{Z}_{3}$ charge
from the other leptons, effectively decoupling it from the low-energy theory.
This leads to a two RH seesaw scenario, which is simpler and more predictive.
We note that both options yield consistent realizations of warm inflation,
although distinct low-energy phenomenology as we will explore below.

##  4 Cosmological dynamics

###  4.1 Inflation

To describe the dynamics of the inflaton field and radiation bath, we need to
specify (i) the scalar potential for $\phi$ and its thermal corrections due to
the right-handed neutrinos $N_{1}$ and $N_{2}$ and (ii) the dissipation
coefficient associated with the production of the latter in the thermal bath
by the slowly varying inflaton field.

As mentioned above, the inflaton is a gauge singlet and its potential function
is arbitrary up to the reflection symmetry imposed by the underlying
interchange symmetry. We note that this is very different from e.g. an axion
field, which is a (pseudo-)Goldstone boson and therefore inherits a shift
symmetry from the underlying U(1) Peccei-Quinn symmetry. In our model, the
Goldstone mode corresponds to the $\pi$ phase and not the inflaton field,
which is invariant under U(1) transformations. Within an effective field
theory approach, which underlies our ‘bottom-up’ philosophy, we consider the
most general renormalizable tree-level potential for $\phi$ including a
quadratic and a quartic term:

$V(\phi)=\frac{1}{2}m_{\phi}^{2}\left(\phi-{\pi\over
4}M\right)^{2}+\lambda\left(\phi-{\pi\over 4}M\right)^{4}.$ (21)

During inflation, we will see that the field will take large background values
$\phi\gg M$ for which the quartic term dominates and is well approximated by
$\lambda\phi^{4}$, while at late times the field behaves as non-relativistic
matter while oscillating close to the minimum, where the quadratic mass term
dominates.

This potential is corrected at finite temperature due to the $\phi$-dependence
of the $N_{1}$ and $N_{2}$ masses, which are given by888We use $N_{1,2}$ for
the flavor and mass states interchangeably and distinguish between the two
when necessary.:

$\displaystyle M_{1}^{2}$ $\displaystyle=$ $\displaystyle{M^{2}\over
4}\left[g_{+}^{2}\cos^{2}(\phi/M)+g_{-}^{2}\sin^{2}(\phi/M)\right],$ (22)
$\displaystyle M_{2}^{2}$ $\displaystyle=$ $\displaystyle{M^{2}\over
4}\left[g_{+}^{2}\sin^{2}(\phi/M)+g_{-}^{2}\cos^{2}(\phi/M)\right],$

where $g_{\pm}=g_{1}\pm g_{2}$. Inflation will occur mostly in the regime
where $T\gtrsim M_{1,2}$, where the high-temperature expansion of the
effective potential is a good approximation and given by (per fermionic degree
of freedom) Dolan:1973qd ; Cline:1996mga :

$\displaystyle V_{\text{high-T}}(T)=$ $\displaystyle
T^{4}\left[-\frac{7\pi^{2}}{720}+\frac{1}{48}\frac{m^{2}}{T^{2}}+\frac{1}{64\pi^{2}}\frac{m^{4}}{T^{4}}\left(\ln\left(\frac{m^{2}}{T^{2}}\right)-c_{f}\right)\right.$
(23)
$\displaystyle-\left.\frac{1}{2}\frac{m^{2}}{T^{2}}\sum_{l=2}^{n}\left(\frac{-m^{2}}{4\pi^{2}T^{2}}\right)^{l}\frac{(2l-3)!!\zeta(2l-1)}{(2l)!!(l+1)}\left(2^{2l-1}-1\right)\right],$

where $c_{f}\simeq 2.635$. Since
$M_{1}^{2}+M_{2}^{2}=M^{2}(g_{+}^{2}+g_{-}^{2})/4$ is independent of the
inflaton field, the leading thermal corrections to the inflaton’s mass cancel
out when adding the contributions of $N_{1}$ and $N_{2}$, which is the essence
of the WLI mechanism that is therefore preserved in our implementation. An
alternative way of showing this is to compute the thermally corrected inflaton
self-energy in the high temperature regime, a computation we describe in
detail in Appendix A. We note that quadratic divergences in the quantum loop-
corrections to the inflaton’s mass also cancel for the same reason, which is
reminiscent of the ‘Little Higgs’ mechanism to address the electroweak
hierarchy problem.

There remain sub-leading thermal corrections of the Coleman-Weinberg form,
which add up with the corresponding zero-temperature corrections. Remarkably,
the two contributions add up in such a way that the argument of the logarithm
in Eq. (LABEL:HighTPot) above becomes simply $\mu^{2}/T^{2}$, where $\mu$ is
the renormalization scale. It is therefore natural to take this close to the
temperature of the thermal bath, for instance at horizon-crossing. Since the
temperature varies slowly during inflation, the magnitude of this contribution
is suppressed and, more importantly, it only induces oscillatory contributions
to the slow-roll parameters, which have a negligible effect as described in
the original WLI proposal Bastero-Gil:2016qru .

Since we will consider the post-inflationary evolution, when the temperature
falls below the mass scale of the right-handed neutrinos, we must consider
also the low-temperature approximation of the effective potential (per
fermionic degree of freedom)Dolan:1973qd ; Cline:1996mga :

$V_{\text{low-T}}(T)=-T^{4}e^{-m/T}\left(\frac{m/T}{2\pi}\right)^{3/2}\sum_{l=0}^{t}\frac{1}{2^{l}l!}\frac{\Gamma(l+5/2)}{\Gamma(l-5/2)}\left(\frac{T}{m}\right)^{l},$
(24)

where the Boltzmann-suppression of thermal effects is manifest. In our
numerical simulations we use both forms of the effective potential matched in
such a way (similar to Sec. 2.2. of Cline:1996mga ) that a precise description
of the full effective potential is obtained. We find that using $n=3$
(high-$T$) and $t=3$ (low-$T$), with the latter turned on at $T/M_{1}\sim 0.5$
provides a smooth enough transition, yielding an error around $1\%$ (see also
e.g. Borges:2016nne ).

The thermal dissipation coefficient $\Upsilon$ can be computed using non-
equilibrium quantum field theory techniques at finite temperature, provided
that (i) the temperature exceeds the Hawking temperature of de Sitter space
$H/2\pi$, so that the computation can be done in flat space; (ii) the thermal
bath remains close to thermal equilibrium, which requires equilibration
processes to occur faster than expansion; and (iii) the inflaton’s dynamics is
‘adiabatic’ compared to the thermal processes, which is justified within the
slow-roll approximation provided that the latter condition is satisfied.
Physically, the dissipative friction will correspond to a non-equilibrium
production of right-handed neutrinos $N_{1}$ and $N_{2}$, which are kept close
to equilibrium via their decays into light leptons and Higgs particles
(including the would-be $W$ and $Z$ longitudinal modes). The above conditions
may then be satisfied in the regime $T\gtrsim\Gamma_{N_{1,2}}\gtrsim H$, which
will constrain the parametric regimes where warm inflaton can successfully be
realized.

We detail the computation of the dissipation coefficient in Appendices B and
C, where in the former we describe the computation of the thermal mass and
decay width of the right-handed neutrinos, which are needed to compute
$\Upsilon$. For clarity, here we give only the approximate expressions for the
contribution of each right-handed neutrino to the dissipation coefficient in
the high and low-temperature regimes999This is the leading result in the
narrow width approximation for the right-handed neutrinos, which vanishes if
only one of the couplings is non-zero (i.e. $g_{-}=\pm g_{+}$). In the latter
case one needs to go beyond this leading approximation, as shown in
Rosa:2018iff .:

$\displaystyle\Upsilon_{i}=$
$\displaystyle\frac{\left(g_{1}^{2}+g_{2}^{2}\right)\cos^{2}\delta}{4y^{2}}\frac{\pi}{9-6\ln{(M_{i}/T)}}T,\quad
T\gtrsim M_{i}$ (25) $\displaystyle\Upsilon_{i}=$
$\displaystyle\frac{\left(g_{1}^{2}+g_{2}^{2}\right)\cos^{2}\delta}{2y^{2}}\left(\\!{2M_{i}T\over\pi}\right)^{\\!1/2}\\!\\!e^{-{M_{i}\over
T}},\quad T\ll M_{i}$ (26)

for $i=1,2$, where we have defined the effective neutrino Yukawa coupling
$y^{2}=\sum_{j=1}^{3}Y_{1j}^{2}=\sum_{j=1}^{3}Y_{2j}^{2}$ and the inflaton-
neutrino ‘mixing angle’:

$-\delta=\tan^{-1}\left({g_{-}\over
g_{+}}\tan(\phi/M)\right)+\tan^{-1}\left({g_{-}\over
g_{+}}\cot(\phi/M)\right)~{}.$ (27)

We note that in the expressions above it is implicit that the right-handed
neutrino masses include thermal corrections, although for simplicity we used
the same notation as for the zero-temperature inflaton-dependent masses.
Furthermore, we use, in our numerical simulations, the more complete
expressions given in Appendix C, that are suitable to explore the regime
$T\sim M_{i}$.

In the high-temperature regime relevant for inflation, the dissipation
coefficient is essentially proportional to the temperature, as in the original
WLI model101010There is an oscillatory dependence on the inflaton field
through the mixing angle $\delta$, but for large field excursions
$\Delta\phi\gg M$ we may take an averaged value of $\cos^{2}\delta$.. In the
opposite limit dissipative effects become Boltzmann-suppressed111111There are
still residual contributions from virtual right-handed neutrino modes, but
these are too suppressed to have any significant impact on the cosmological
dynamics., and the right-handed neutrinos effectively decay away. As we will
see below and originally shown in Rosa:2018iff , this leaves a stable relic
inflaton field that interacts very weakly with SM particles, and is therefore
a natural dark matter candidate.

We now have everything in place to study the dynamics of the inflaton-
radiation system, and it is convenient to rewrite Eqs. (1)-(3) in terms of the
number of e-folds of expansion, $dt=HdN_{e}$,

$\displaystyle\phi^{\prime\prime}+3\left(1+\frac{H^{\prime}}{H}+\frac{\Upsilon}{H}\right)\phi^{\prime}+\frac{\partial_{\phi}V(\phi)}{H^{2}}=0,$
(28) $\displaystyle\rho_{r}^{\prime}+4\rho_{r}=\Upsilon H\phi^{\prime 2},$
(29) $\displaystyle H^{2}=\frac{\rho_{\phi}+\rho_{r}}{3M_{p}^{2}}.$ (30)

Note that the radiation energy density can be computed from the contribution
of all light particles to the thermal effective potential given above via
$\rho_{r}=V_{T}+Ts$ with $s=-dV_{T}/dT$. In the high-temperature regime, this
yields at leading order the well-known expression
$\rho_{r}\simeq(\pi^{2}/30)g_{*}T^{4}$, but this allows us to accurately track
the contribution of $N_{1,2}$ to $g_{*}(T)$ as the temperature falls below
their mass threshold.

The above system of equations can be solved analytically in the high-
temperature, slow-roll regime, where $\Upsilon\propto T$ and
$V(\phi)\propto\phi^{4}$. However, since the temperature of the thermal bath
is close to the masses of the right-handed neutrinos in the interesting
parametric regimes, it is more accurate to numerically solve the above
dynamical system using the full forms of the dissipation coefficient and
effective potential with minimal approximations. In figure 1 we give a
representative example of the dynamics of warm inflation in our model.

Figure 1: Inflationary dynamics for $V(\phi)=\lambda\phi^{4}$ as a function of
the number of e-folds. The total number of e-folds of inflation is
$N_{\text{eq}}\approx 58$ with $(n_{s},r)=(0.967,~{}2.6\times 10^{-4})$
(nearly-thermal inflaton fluctuations). The number of relativistic degrees of
freedom during inflation is approximately constant, $g_{*}\approx 19$. The
‘reheating’ temperature is $T_{R}\approx 7.5\times 10^{13}$ GeV. Parameters:
$Q_{*}=0.8,~{}\phi_{*}\simeq 12M_{p},~{}\lambda\approx 5\times 10^{-16}$
($y=0.74,~{}g_{1}=0.21,g_{2}=0.17,~{}M\approx 3.3\times 10^{15}$ GeV).

As one can see, the system remains in a warm ($T>H$), near-equilibrium
($\Gamma_{N}>H$), slow-roll ($\epsilon_{\phi}<1+Q$) regime for nearly 60
e-folds of accelerated expansion. At horizon-crossing, dissipative effects are
relatively weak $Q_{*}\lesssim 1$, but become sufficiently strong towards the
end of inflation for radiation to become a sizeable component and soon come to
dominate over the inflaton field, thus smoothly ending inflation (see also
figure 3). We find that the temperature at which radiation takes over, which
constitutes an effective ‘reheating temperature’ although the Universe never
cooled down during inflation, is typically $T_{R}\sim 10^{13}-10^{14}$ GeV.

We note that accelerated expansion only truly ends when
$\rho_{r}\approx\rho_{\phi}$, since $\ddot{a}\propto 2(\rho_{\phi}-\rho_{r})$,
which typically takes place 2-3 e-folds after the slow-roll conditions fail.
Hence, the example in figure 1 has $\simeq 60$ e-folds of accelerated
expansion.

Although the temperature (bottom red curve in figure 1) remains close to the
right-handed neutrino masses throughout inflation, it falls significantly
below it at the end of the slow-roll phase, which in turn shuts down
dissipative effects due to Boltzmann suppression as discussed above. Once
dissipation can no longer counteract the diluting effect of expansion at this
stage, the temperature of the thermal bath simply redshifts as $T\propto
a^{-1}$. One must also take into account that other SM degrees of freedom also
start being thermally excited towards the end of inflation as the ratio
$T/H\sim 10^{4}$ and typical gauge processes start competing with the
expansion rate. Consequently, the temperature falls by an additional factor
$(g_{*,\text{inf}}/g_{*,\text{SM}})^{1/3}\sim 2$, which also speeds up the
shut down of dissipation at the end of inflation.

Inflaton particles are also produced in the thermal bath via several
processes, including e.g. right-handed neutrino annihilation and Landau
damping processes, as well as lepton-Higgs annihilation via virtual right-
handed neutrinos. The rate of each individual process is typically suppressed
compared to the right-handed neutrino decay rate, since $g_{1,2}<y$ is
required to have sufficiently light right-handed neutrinos. Nevertheless, the
number of different processes producing inflaton particles in the thermal bath
should partially compensate for this, such that inflaton particle production
may compete with Hubble expansion already at horizon-crossing. This has
implications for observational predictions, since inflaton occupation numbers
at horizon crossing influence the shape of the primordial spectrum of
perturbations, namely through the factor $2n_{*}$ in Eq. (11), with $n_{*}\sim
T_{*}/H_{*}$ for a nearly-thermal distribution. Since a detailed analysis of
inflaton particle production rates is beyond the scope of this work, we
consider the two limiting regimes $n_{*}\ll 1$ and $n_{*}\simeq T_{*}/H_{*}$
in computing the spectrum of primordial perturbations.

In figure 2 we show the observational predictions for the curvature spectral
index $n_{s}$ and tensor-to-scalar ratio $r$ in our model as a function of the
dissipative ratio at horizon-crossing, $Q_{*}$ (with other dynamical variables
fixed by the number of e-folds of inflation and the amplitude of the
primordial curvature perturbation spectrum), in these two limiting regimes.

These results are analogous to the original WLI scenario, which also
considered a quartic potential for the inflaton field.

Figure 2: Inflationary spectrum ($n_{s},~{}r$) for $V(\phi)=\lambda\phi^{4}$
as a function of $Q_{*}$, with the $r$ axis in $\log$-scale. The red (blue)
shaded region corresponds to the case where the inflaton fluctuations have a
nearly-thermal (negligible) spectrum at horizon crossing. The left (right)
contour of these regions correspond to a $N_{e}=50$ ($60$) of inflation. As
$Q_{*}\rightarrow 0$, both regions converge to a straight line encompassing
the predictions of cold inflation, $(n_{s},~{}r)\in(0.94-0.95,0.32-0.27)$ for
$N_{e}=50-60$. The black star represents the case of figure 1. The black
(gray) contour signals the $95\%$ ($68\%$) confidence limit of Akrami:2018odb
for the TT,TE,EE+lowE+lensing +BK14+BAO dataset.

As one can see, the spectrum is more blue-tilted than the corresponding cold
inflation case, which is essentially a consequence of the thermal nature of
inflaton fluctuations and of the associated growth of the dissipative ratio
$Q$ and of the ratio $T/H$ during inflation, with modes leaving the horizon
later during inflation having a (slightly) larger amplitude. Alongside the
suppression of the tensor-to-scalar ratio typical of warm models, this makes
the quartic inflaton potential agree with observational data from Planck for
$Q_{*}\sim 10^{-2}-1$, which is quite remarkable since in the absence of
dissipative effects this model has been completely ruled out, as first pointed
out in Bastero-Gil:2016qru . For $Q_{*}\gtrsim 1$ the interplay between
inflaton and radiation fluctuations leads to the appearance of a growing mode
in the spectrum, parametrised by $G(Q_{*})$ in (11), which renders the
spectrum too blue-tilted. One should bear in mind that such a growing mode may
potentially be damped by additional viscous effects in the thermal bath
BasteroGil:2011xd , such that $Q_{*}\gtrsim 1$ scenarios may also be in
agreement with observations. Such a possibility is, however, still the object
of an ongoing discussion so we will not pursue it any further, and restrict
our analysis below to the regime where $Q_{*}\lesssim 1$.

Another distinctive observational signature of warm inflation that is also
present in our scenario is non-Gaussianity, with a particular bispectral
shape. For $\Upsilon\propto T$, $|f_{NL}^{\mathrm{warm}}|\lesssim 10$ for the
range of $Q_{*}$ in agreement with observations mentioned above Moss:2007cv ;
Bastero-Gil:2014raa . Dedicated searches made by the Planck collaboration for
the ‘warm shape’ of the bispectrum are, however, still far from reaching the
required sensitivity, allowing for $Q_{*}<(3.2-4)\times 10^{3}$ (95% C.L.)
Ade:2015ava ; Akrami:2019izv , which we hope may be improved with future
experiments.

Finally, we note that one of the appealing features of warm inflation is the
elimination of the uncertainty in the number of e-folds of inflation after
horizon-crossing of the relevant CMB scales, due to the fast and smooth
transition into a radiation-dominated universe, as recently observed in
Das:2020xmh . In fact, one may argue that warm and cold inflation involve a
comparable number of free parameters, since interactions between the inflaton
and other fields are necessarily present in both inflationary paradigms. In
warm inflation these determine the full dynamics of accelerated expansion,
whereas in cold inflation they only affect the reheating period and therefore
result in an uncertainty in the total number of e-folds. In figure 2 we
nevertheless included a $50-60$ e-folds uncertainty to accommodate possible
non-standard cosmic histories after inflation. For instance, one or more short
periods of thermal inflation Lyth:1995hj ; Lyth:1995ka 121212Note that thermal
and warm inflation are distinct scenarios, since in the former there is no
dissipative dynamics sustaining the temperature of the thermal bath and the
inflaton is held by thermal effects at a metastable minimum. In fact, the
latter correspond to the very same thermal mass corrections that need to be
suppressed to realize warm inflation in the slow-roll regime. may be necessary
to dilute any unobserved relics potentially produced after warm inflation,
given the large temperatures attained.

###  4.2 Inflaton dark matter

As we have seen above, once inflation ends dissipative effects quickly shut
down and the inflaton field becomes underdamped and starts oscillating about
the minimum of the potential. Since the potential is still dominated by the
quartic term following inflation, the oscillating inflaton field first behaves
as a dark radiation component, with an amplitude $\phi\propto a^{-1}$. This
phase lasts until the amplitude of the inflaton field drops below
$\phi_{\text{DM}}=m_{\phi}/\sqrt{2\lambda}$, after which the quadratic term of
the potential becomes dominant and the inflaton remnant behaves as non-
relativistic (pressureless) matter, with $\phi\propto a^{-3/2}$.

Figure 3: Cosmological dynamics. The parameters are identical to those of
figure 1. The dashed lines represent the two inflaton-radiation equalities.
Big bang nucleosynthesis (BBN) ($T\sim 1$ MeV) and the last scattering surface
(CMB) ($T\sim 0.3$ eV) are also highlighted. The inset shows a close-up of the
energy densities during BBN, where $f\equiv\rho_{\phi}/\rho_{r}\sim 10^{-5}$.

Due to the underlying interchange symmetry, which as discussed earlier
corresponds to a $\mathbb{Z}_{2}$ reflection symmetry, the inflaton field
becomes stable and, moreover, since the temperature falls below the right-
handed neutrino mass threshold, the inflaton can only interact with SM
particles via the exchange of virtual right-handed neutrinos. The inflaton
field thus becomes a cold, weakly interacting relic after inflation, making it
a natural dark matter candidate.

Scattering processes mediated by virtual right-handed neutrinos may
nevertheless lead to the thermalization of the oscillating inflaton
condensate, in which case the inflaton would become a WIMP-like candidate that
would eventually decouple from the thermal bath. The analysis made in
Rosa:2018iff  for a generic WLI model shows, however, that the large right-
handed neutrino masses $M_{1}=M_{2}=\sqrt{g_{1}^{2}+g_{2}^{2}}M/2$ greatly
suppress such processes, which therefore do not significantly affect the
field’s dynamics131313The results of Rosa:2018iff  can be immediately
translated to our model with the appropriate identification of the generic
scalar and fermion fields considered there with the right-handed neutrinos,
leptons and Higgs fields.. The oscillating inflaton field may nevertheless
produce $\phi$ particles through the $\lambda\phi^{4}$ self-interaction while
in the quartic regime. From the point of view of the cosmological dynamics,
such inflaton particles are indistinguishable from an oscillating field since
they do not thermalize and behave first as dark radiation and then as cold
dark matter. As argued in Rosa:2018iff , the transition between these two
regimes occurs essentially at the same time as for the oscillating field,
since they are produced with a momentum $p\sim\sqrt{\lambda}\phi$ in the dark
radiation phase, which redshifts exactly like the amplitude of the oscillating
field in the quartic potential. The amount of expansion required to reach the
non-relativistic regime $p\lesssim m_{\phi}$ thus corresponds to the amount
needed for $\phi\lesssim\phi_{\text{DM}}$.

After inflation, there are three distinct contributions to the energy density
of the Universe, including (i) the dominant radiation composed by relativistic
SM particles, (ii) the remnant background inflaton field that behaves
initially as dark radiation while the field oscillates about the quartic
potential, and (iii) inflaton particles in the thermal bath, which thermalized
with the remainder relativistic degrees of freedom during inflation and
decoupled from the latter once the right-handed neutrinos $N_{1,2}$ became
non-relativistic. Note that thermalization of inflaton particles should occur
before the end of inflation even if not already at horizon-crossing as
discussed above, given that the $g_{1,2}$ couplings are not too suppressed in
parametric regimes where the strong dissipation regime is reached before
inflation ends.

The fractional contribution of the decoupled inflaton particles is given by:

$\frac{\rho_{\delta\phi}}{\rho_{r}}(T)=\frac{1}{g_{*,\text{SM}}}\left[\frac{g_{*}(T)}{g_{*,\text{SM}}}\right]^{1/3},$
(31)

where we have used entropy conservation, while the background inflaton
remnant, in its dark radiation regime, contributes as:

$\frac{\rho_{\phi}}{\rho_{r}}(T)=f\left[\frac{g_{*}(T)}{g_{*,\text{SM}}}\right]^{1/3},$
(32)

where $f$ is the ratio between the background inflaton and the radiation
energy densities at the onset of field oscillations, which remains constant
apart from temperature jumps due to varying $g_{*}$. Numerically, we find
$f\sim 10^{-(4-5)}$ in the parametric range of interest for a successful
realization of warm inflation.

Both these inflaton components will contribute to the present dark matter
abundance, and to assess which is the dominant one, we note, on the one hand,
that the background remnant starts behaving as dark matter (i.e. the quadratic
term in the potential becomes dominant) for field amplitudes
$\lesssim\phi_{\text{DM}}=m_{\phi}/\sqrt{2\lambda}$. This corresponds to a
temperature:

$T_{\mathrm{DM}}=m_{\phi}\left(\frac{30}{2\pi^{2}\lambda
f}\right)^{1/4}{g_{*,\text{SM}}^{1/12}\over g_{*}(T_{1})^{1/3}}~{}.$ (33)

On the other hand, the decoupled inflaton particles behave as dark radiation
for longer (and hence are more diluted) than the background remnant, as they
only become non-relativistic at $T=m_{\phi}\lesssim T_{\mathrm{DM}}$, yielding
for $T<m_{\phi}$:

$\frac{\rho_{\delta\phi}}{\rho_{\phi}}\simeq 6\times
10^{-4}\left(\frac{10^{-4}}{f}\right)^{3/4}\left(\frac{\lambda}{10^{-16}}\right)^{1/4}g_{*}(T_{1})^{1/3},$
(34)

so that the background remnant is generically the dominant dark matter
component. Then, using the present value of the dark matter abundance, we find
for the inflaton mass and temperature at which the remnant starts behaving as
cold dark matter:

$\displaystyle m_{\phi}$
$\displaystyle\approx\left(\frac{\Omega_{c}}{0.25}\right)\left(\frac{g_{*}}{106.75}\right)\left(\frac{10^{-15}}{\lambda}\right)^{\frac{1}{2}}\left(\frac{10^{3}}{\phi/T}\right)^{3}\text{
eV},$ (35) $\displaystyle T_{\text{DM}}$ $\displaystyle\approx
0.11\left(\frac{\Omega_{c}}{0.25}\right)\left(\frac{g_{*}}{106.75}\right)^{\frac{4}{3}}\left(\frac{10^{-15}}{\lambda}\right)\left(\frac{10^{3}}{\phi/T}\right)^{4}\text{MeV},$

where $\Omega_{c}$ is the relative dark matter density present today. For the
example depicted in figures 1 and 3, $m_{\phi}\approx 0.4\ \text{eV}$ and
$T_{\text{DM}}\approx 40\text{ keV}$. We note that one should take into
account the radiative corrections from the two right-handed neutrinos to the
inflaton’s mass to assess whether such small values of the inflaton mass
require fine-tuning. In particular, one would expect Coleman-Weinberg
contributions to the inflaton potential $\sim{M_{i}^{4}/64\pi^{2}}$ from both
$N_{1}$ and $N_{2}$. However, it is well known that this standard form of the
Coleman-Weinberg potential is obtained via mass-independent renormalization
schemes, such as $\overline{MS}$, which do not respect the Appelquist-
Carazonne decoupling theorem Appelquist:1974tg . A computation of the 1-loop
effective potential (at zero temperature) using a mass-dependent
renormalization scheme gives a modified form of the Coleman-Weinberg term,
generically given by BasteroGil:2010vq :

$\Delta V^{(1)}=\sum_{i}(-1)^{F}{m_{i}^{4}\over
64\pi^{2}}\left[\log\left({m_{i}^{2}\over\mu^{2}}\right)-I\left({m_{i}^{2}\over\mu^{2}}\right)\right]~{},$
(36)

where $\mu$ is the renormalization scale, the sum is over all bosons and
fermions with masses $m_{i}$, and

$I(x)=\log(x)-2-\sqrt{1+4x}\log\left({\sqrt{1+4x}-1\over\sqrt{1+4x}+1}\right)~{}.$
(37)

While for $\mu\gtrsim m_{i}$ this yields the standard Coleman-Weinberg
expression, this is no longer the case at low energies $\mu\ll m_{i}$. Thus,
for $\mu\ll M_{1,2}$ the leading contribution of the right-handed neutrinos to
the effective potential is $\Delta
V^{(1)}_{N_{1,2}}=\mu^{2}(M_{1}^{2}+M_{2}^{2})/192\pi^{2}+\mathcal{O}(\mu^{4})$.
As we have seen above, this is independent of the inflaton field, regardless
of the specific choice of $\mu$ in this limit. Thus, the right-handed
neutrinos do not generate large radiative corrections to the inflaton’s mass
at low energies/temperatures, through exactly the same mechanism that cancels
large thermal corrections in the high-temperature limit. Hence, no fine-tuning
of the inflaton mass is needed to obtain the observed dark matter abundance
within the proposed setup.

As originally discussed in Rosa:2018iff , the fact that the inflaton field
plays a role in the post-inflationary Universe, and in particular naturally
accounts for all dark matter, yields specific observational signatures in
addition to the warm inflation observables discussed in the previous section.

First, inflaton fluctuations on super-horizon scales generate not only
curvature perturbations but also perturbations in the dark matter density
itself, since $\rho_{\phi}\propto\phi^{2}$ 141414Here we have shifted the
inflaton field value $\phi$ by $M\pi/4$ without loss of generality.. Hence,
the model predicts cold dark matter isocurvature modes that are correlated
with the leading adiabatic curvature fluctuations. These can be parametrized
via Lyth:2002 :

$S_{c}=-3H\left(\frac{\delta\rho_{c}}{\dot{\rho}_{c}}-\frac{\delta\rho_{r}}{\dot{\rho}_{r}}\right)=\frac{\delta\rho_{c}}{\rho_{c}}-\frac{3}{4}\frac{\delta\rho_{r}}{\rho_{r}}=2\frac{\delta\phi}{\phi}-3\frac{\delta
T}{T}~{}.$ (38)

Since, during warm inflation with $\Upsilon\propto T$ as in our scenario, we
have at horizon-crossing:

$\frac{\delta
T_{*}}{T_{*}}=\frac{4Q_{*}}{3+5Q_{*}}\frac{\delta\phi_{*}}{\phi_{*}}~{},$ (39)

we find for the ratio between isocurvature and adiabatic modes:

$B_{c}=\frac{S_{c}}{\mathcal{R}}=-\left(2-\frac{12Q_{*}}{3+5Q_{*}}\right)\frac{\sigma_{\phi_{*}}}{1+Q_{*}}~{},$
(40)

where we have used that
$\mathcal{R}=H(\delta\phi/\dot{\phi})=\delta\phi/\phi^{\prime}$ and Eq. (8).
Since the primordial spectrum of curvature perturbations in our setup requires
$Q_{*}<3$, we conclude that cold dark matter isocurvature perturbations are
anti-correlated with the main adiabatic component.

The Planck collaboration has placed stringent bounds on the magnitude of such
modes, typically parametrized in terms of
$\beta_{\text{Iso}}=B_{c}^{2}/(B_{c}^{2}+1)$. Using the values of $Q$ and
$\phi$ at horizon crossing of the example in figures 1 and 3, we obtain
$\beta_{\text{Iso}}=9.6\times 10^{-5}$, approximately one order of magnitude
below Planck’s constraint on fully anti-correlated isocurvature perturbations,
$\beta_{\text{Iso}}^{\text{Planck}}<8\times 10^{-4}$ Akrami:2018odb .

The isocurvature spectral index is generically different from the adiabatic
one, and in our case we obtain:

$n_{I}=n_{s}+\frac{36(6\epsilon_{\phi_{*}}-2\eta_{\phi_{*}})Q_{*}}{(-3+Q_{*})(3+5Q_{*})^{2}}+\frac{12\sigma_{\phi_{*}}+4(-3\epsilon_{\phi_{*}}+\eta_{\phi_{*}}+\sigma_{\phi_{*}})Q_{*}}{(1+Q_{*})(3+5Q_{*})}.$
(41)

Nevertheless, for our illustrative example, this yields $n_{I}\approx 0.963$,
which differs from $n_{s}$ by only $\simeq 0.4$%.

In addition to isocurvature modes, the inflaton also contributes to the
effective number of relativistic degrees of freedom while in the dark
radiation phase. In this case, the contribution from the background condensate
is $\Delta N_{\text{eff}}^{\phi}=4.4(\rho_{\phi}/\rho_{\gamma})$ which is
typically quite suppressed, e.g. $\Delta N_{\text{eff}}^{\phi}\simeq 9\times
10^{-5}$ for the parameters employed in figures 1 and 3. More significant is
the contribution of decoupled inflaton particles, $\Delta
N_{\text{eff}}^{\delta\phi}~{}=~{}(4/7)\left(43/4g_{*}\right)^{\frac{4}{3}}~{}\approx~{}0.027$,
which may be probed with the next generation CMB experiments and large-scale
structure surveys Baumann:2018 . An interesting aspect to emphasize is the
fact that, while both these contributions are present during nucleosynthesis,
the background field already behaves as dark matter at recombination, while
the decoupled inflaton particles typically become non-relativistic around this
time or slightly afterwards. Hence, our setup generically predicts a distinct
effective number of relativistic species as inferred from light nuclear
element abundances and the CMB spectrum, which is a quite distinctive
signature.

We note that in the WLI scenario analyzed in Rosa:2018iff , where the
possibility of inflaton dark matter in warm inflation was first identified,
corresponding to the case where only one of the couplings $g_{1,2}$ was non-
zero, the inflaton-to-radiation ratio in the post-inflationary dark radiation
phase was larger than in the more general case analyzed in this work, where
$g_{1}\neq g_{2}\neq 0$, yielding larger values for $\Delta
N_{\text{eff}}^{\phi}$. This is essentially due to the smaller values of the
dissipation coefficient in the case where only one of the couplings in non-
zero, which trigger earlier inflaton oscillations in the quartic potential and
thus with a larger amplitude.

##  5 Neutrino masses

As we have showed in the previous section, the Yukawa terms of (18) lead to
dissipative dynamics during an inflationary period with $T_{\text{inf}}\sim
10^{15}$ GeV. These terms also lead to the generation of neutrinos masses at
low energies through the seesaw mechanism (see section 3): in this model, the
high-energy and low-energy physics are related because both dissipation and
the seesaw mechanism are realized through the same operators.

Recalling Eq. (20) and Table 1, and taking, without loss of generality,
$M_{R}$ and the charged lepton mass matrices to be diagonal, only the Dirac
mass matrix $m_{D}$ remains to be determined. This matrix will be shaped by
the $\mathbb{Z}_{3}$ charge assignments of the left-handed leptons, and the
interchange symmetry (the latter imposing that the first and second rows must
be identical). Recalling the two possible $\mathbb{Z}_{3}$ configurations in
Table 1, we have the following possible structures for the Dirac mass matrix:

  1. 1.

Option I:

$m^{I}_{D}=v\begin{pmatrix}a&b&0\\\ a&b&0\\\ 0&0&c\end{pmatrix},$ (42)

resulting in the following light neutrino masses:

$m_{\nu}^{(I)}=\begin{pmatrix}0,&v\frac{\lvert
c\rvert^{2}}{M_{3}},v\frac{(\lvert a\rvert^{2}+\lvert
b\rvert^{2})(M_{1}+M_{2})}{M_{1}M_{2}}\end{pmatrix},$ (43)

where $\lvert a\rvert^{2}+\lvert b\rvert^{2}=y^{2}$.

  2. 2.

Option II:

$m_{D}^{(II)}=v\begin{pmatrix}a&b&c\\\ a&b&c\end{pmatrix}~{},$ (44)

which features two zero eigenvalues:

$m^{II}_{\nu}=\begin{pmatrix}0,&0,&v\frac{M_{1}+M_{2}}{M_{1}M_{2}}(\lvert
a\rvert^{2}+\lvert b^{2}\rvert+\lvert c\rvert^{2})\end{pmatrix}~{},$ (45)

where, in this case, $\lvert a\rvert^{2}+\lvert b^{2}\rvert+\lvert
c\rvert^{2}=y^{2}$. Although a spectrum with two massless neutrinos is
experimentally ruled out, the separation between the solar and atmospheric
neutrino mass-squared differences indicates that two of the neutrinos have
relatively similar masses. One may then envisage scenarios where small
deviations from the above $\mathbb{Z}_{3}$-symmetric structure are responsible
for the necessary mass-splitting.

Regardless of the charge assignment, given the common prediction of a massless
neutrino, it is ultimately possible to identify the effective Yukawa coupling
used in the dynamics of warm inflation as being proportional to the mass scale
$\sqrt{\Delta m^{2}_{\text{sol}}}\sim 0.05$ eV. There is, therefore, a very
close relation between inflationary dynamics and low-energy phenomenology, one
of the attractive features of this model.

In order to obtain a more realistic mass spectrum, one has to account for the
dynamical nature of the Yukawa couplings caused by the large separation
between the right-handed neutrino mass scale ($\sim 10^{15}$ GeV) and the
electroweak scale ($\sim 10^{2}$ GeV), through the use of the renormalization
group equations (RGEs). In particular, we are interested in the evolution of
effective Yukawa coupling $y$, as this simultaneously determines the amount of
dissipation during inflation and the scale of the light neutrino masses.

For this, we use the REAP Mathematica package Antusch:2005gp , which solves
the RGEs of the leptonic sector for the Standard Model plus right-handed
neutrinos (SM+RHN) model151515This is well justified because the energy scales
of interest here are below the SSB scale of $U(1)_{X}$ gauge symmetry group:
the theory is an effective SM+RHN.. One finds that $y^{2}(\mu\sim
10^{15}\text{ GeV})/y^{2}(\mu\sim 10^{2}\text{ GeV})\sim 1.4-1.5$, as
explicitly shown in figure 4 for the “$\mathbb{Z}_{3}$-II” charge assignment.
To obtain the correct neutrino masses at the low-energy scale, one has to
account for this effect.

Figure 4: Running of the squared effective Yukawa coupling $y^{2}$ from the
$Z$ mass scale to the U(1) symmetry breaking scale $M$. The mass threshold of
the lightest RHN ($N_{3}$) is shown in light gray, while the mass threshold of
the heaviest degenerate RHN ($N_{1,2}$) is shown in dark gray.

Let us take the results obtained in section 4, and focus on the example of
figures 1 and 2, with $y=0.74,~{}g_{1}=0.21,g_{2}=0.17,~{}M\approx 3.3\times
10^{15}$ GeV. Using Eq. (45), it becomes clear that the required Yukawa
couplings obtained in section 4 already result in masses compatible with the
low energy physics:

$m^{(\mu=M)}_{3}=2\frac{y^{2}v^{2}}{M_{1}}\sim 0.07\text{
eV}\xrightarrow{\text{running}}m_{3}^{(\mu=m_{Z})}\sim 0.05\text{ eV}.$

A more detailed account of RGEs and the running of parameters is outside the
scope of this paper, and the interested reader is pointed to, e.g.
Antusch:2003kp ; Antusch:2005gp .

We can relate inflationary observables and neutrino masses within our setup in
the following way. For each value of the dissipative ratio at horizon-
crossing, $Q_{*}$, the number of e-folds of inflation and the amplitude of the
primordial curvature spectrum can be used to infer $\phi_{*}$ and
$T_{*}/H_{*}$. In turn, these values can be used to determine the combination
$(g_{1}^{2}+g_{2}^{2})\cos^{2}\delta/y^{2}$ that yields the magnitude of the
dissipation coefficient. We set the effective Yukawa coupling $y$ at the
minimum value for which the equilibrium condition $\Gamma_{N}>H$ is satisfied
at horizon-crossing, and the mass scale $M$ can be set by demanding that
$T\gtrsim M_{1,2}$ throughout inflation161616More specifically, we find
numerically that requiring $T_{*}>1.7M_{1}$ is enough to ensure that the
right-handed neutrinos remain sufficiently light for $\sim$60 e-folds of
inflation after horizon-crossing of CMB scales.. In this analysis we set
$g_{2}=0.8g_{1}$ for concreteness, noting that similar results are obtained if
there is no large hierarchy between these couplings. In figure 5 we show the
results of this analysis, simultaneously showing the predictions for the
inflationary observables and the mass $m_{3}$ of the heaviest left-handed
neutrino in the “$\mathbb{Z}_{3}$-II” charge assignment (which is independent
of the mass of the third right-handed neutrino, $M_{3}$).

Figure 5: Predictions for inflationary observables ($n_{s},~{}r$) and the
largest left-handed neutrino mass eigenstate, for $V(\phi)=\lambda\phi^{4}$,
considering the two limiting cases where inflaton particles are nearly
thermalized or have negligible occupation numbers at horizon-crossing of the
relevant CMB scales. The blue contours correspond to the Planck
TT,TE,EE+lowE+lensing +BK14+BAO 68% and 95% C.L. regions Akrami:2018odb . The
ellipses highlight the parametric regions where the neutrino mass is in
agreement with the experimental value.

As one can see in this figure, agreement with CMB observations corresponds to
a heaviest neutrino mass $0.04\ \mathrm{eV}<m_{3}<0.1\ \mathrm{eV}$, which is
quite remarkable. The most realistic scenarios, with $m_{3}\simeq 0.05$ eV,
correspond to dissipative ratios at horizon-crossing $0.1\lesssim
Q_{*}\lesssim 1$.

Neither of the above charge assignments can, however, account for a realistic
neutrino mixing pattern compatible with neutrino oscillation experiments,
requiring additional degrees of freedom involved in the seesaw mechanism. By
including these, we should be careful in not spoiling the already successful
realization of warm inflation with a realistic mass scale for the heaviest
left-handed neutrino.

One possible option is to consider an $SU(2)_{L}$ scalar triplet $\Delta_{L}$.
The vacuum expectation value of this field generates a renormalizable Majorana
mass term for the light neutrinos through the type II seesaw mechanism
Schechter:1980gr . This would then result in a mixed type I+II seesaw, with
$m_{\nu}\rightarrow m_{LL}-m^{T}_{D}M^{-1}_{R}m_{D}$. These sort of scalars
are common in UV-completions of the SM, such as LRSM Mohapatra:1974gc  and
$SO(10)$ GUTs Lazarides:1980nt ; Hambye:2003ka . The downside of this approach
is the introduction of a large number of new parameters in the theory.

A more economical path, on which we will focus our discussion, is the
introduction of a flavour-symmetry breaking “flavon” field, $\chi$, singlet
under the gauge group but with a non-trivial $\mathbb{Z}_{3}$-charge. For
instance, choosing a charge $\omega^{2}$ for the flavon field allows for a
dimension-5 term $\chi\overline{N}_{3}HL_{L}/\Lambda$, where $\Lambda\gtrsim
M$ is a heavy mass scale depending on the UV-completion of the model. Upon
acquiring an expectation value $\langle\chi\rangle\equiv v_{\chi}\ll\Lambda$,
the flavon field therefore allows for small Yukawa couplings involving the
third right-handed neutrino, which were thus far forbidden by the
$\mathbb{Z}_{3}$ symmetry.

Since the $\mathbb{Z}_{3}$-II charge assignment yields a simpler neutrino mass
spectrum, depending only on a single effective Yukawa coupling in the
$\mathbb{Z}_{3}$-symmetric limit, we will henceforth consider only this
option. The non-zero flavon VEV then allows for a Dirac mass matrix of the
form:

$m_{D}=\begin{pmatrix}a&b&c\\\ a&b&c\\\
0&0&0\end{pmatrix}\xrightarrow{\chi\rightarrow
v_{\chi}}\begin{pmatrix}a&b&c\\\ a&b&c\\\ d&e&f\end{pmatrix}.$ (46)

The interchange symmetry is unaffected by these terms, and consequently the
setup still predicts one massless neutrino. The new entries $(d,e,f)$ in (46)
in the Dirac mass matrix are $\mathcal{O}(v_{\chi}/\Lambda)$ and can therefore
be treated as small deviations, yielding a light neutrino mass matrix of the
form:

$m_{\nu}=-v^{2}\begin{pmatrix}\frac{2a^{2}}{M_{1}}+\frac{d^{2}}{M_{3}}&\frac{2ab}{M_{1}}+\frac{de}{M_{3}}&\frac{2ac}{M_{1}}+\frac{df}{M_{3}}\\\
.&\frac{2b^{2}}{M_{1}}+\frac{y^{2}}{M_{3}}&\frac{2bc}{M_{1}}+\frac{ef}{M_{3}}\\\
.&.&\frac{2c^{2}}{M_{1}}+\frac{f^{2}}{M_{3}}\end{pmatrix}.$ (47)

Note that $M_{3}$ is, so far, a free parameter, therefore a suppression of
$(d,e,f)\propto v_{\chi}/\Lambda$ affecting the flavon terms can be countered
by a lower $M_{3}$. Therefore, a hierarchy between $M_{1}$ and $M_{3}$,
$M_{1}\gg M_{3}$, may lead to sizable flavonic contributions, e.g.
$(2a^{2}/M_{1})\gtrsim(d^{2}/M_{3})$, to the effective neutrino mass matrix.

As a simple example, let us consider the case $d,e=0$, which leads to:

$\displaystyle m_{2}\simeq
f^{2}\frac{v^{2}}{M_{3}}\frac{a^{2}+b^{2}}{y^{2}},\quad
m_{3}\simeq\frac{2y^{2}v^{2}}{M_{1}}+f^{2}\frac{v^{2}}{M_{3}}\frac{c^{2}}{y^{2}},$
(48)

where we expanded the non-trivial eigenvalues of $m_{\nu}m_{\nu}^{\dagger}$ to
second-order in $f$ and, for simplicity, took all couplings to be real. This
shows that a single flavonic term in the Dirac Yukawa matrix is is sufficient
to lift the unrealistic degeneracy of the neutrino mass spectrum. Inspecting
Eq. (48), it is clear that the choice $c=0$ protects $m_{3}$ against unwanted
contributions. In this limit, a realistic value for the second non-vanishing
neutrino mass $m_{2}$ can be obtained for:

$f^{2}\approx 4\times 10^{-6}\frac{M_{3}}{10^{10}\text{ GeV}}~{}.$ (49)

where we have included the Yukawa running. This illustrates that the third
right-handed neutrino, which we recall does not interact directly with the
inflaton field, can be much lighter than the other two, noting that the
flavon-induced Yukawa couplings can be suppressed for $v_{\chi}\ll\Lambda$.

We must ensure that the introduction of an additional flavon field and its
induced $N_{3}$ Yukawa couplings do not spoil the successful realization of
warm inflation obtained earlier with only two right-handed neutrinos. If, on
the one hand, the flavon field acquires its VEV during inflation, this may
lead to a direct coupling between the inflaton and the third right-handed
neutrino through dimension-5 operators of the form
$g_{3}\overline{N_{3}^{c}}\Phi_{1,2}N_{3}$, where
$g_{3}=\mathcal{O}(v_{\chi}/\Lambda)$. These reintroduce dangerous thermal
corrections to the inflaton mass that could spoil inflation if $N_{3}$ is
thermalized, unless $g_{3}\ll 1$. Given that $N_{3}$ can be thermalized via
decays and inverse decays, with $\Gamma_{N_{3}}/H\sim|y_{3}|^{2}(T/H)$ during
inflation, with $|y_{3}|^{2}\equiv|d|^{2}+|e|^{2}+|f|^{2}$, this leads to an
upper bound on this effective coupling $|y_{3}|^{2}\lesssim 10^{-5}$.

On the other hand, it is possible that the flavon field only acquires a VEV
after inflation ends, in which case $N_{3}$ is not thermally excited during
inflation and therefore there are no additional thermal corrections to the
inflaton’s mass. This would require $m_{\chi}\gtrsim H\gtrsim v_{\chi}$ during
inflation. This could be due to e.g.- Planck-suppressed non-renormalizable
contributions to the flavon potential or a non-minimal coupling to gravity,
$\xi\chi^{2}R$, which easily induce $\mathcal{O}(H)$ masses to scalar fields
unprotected by any symmetry. In the case of a non-minimal coupling to gravity,
the induced flavon mass becomes highly suppressed once radiation becomes
dominant ($R\simeq 0$), so that the spontaneous breaking of the
$\mathbb{Z}_{3}$ symmetry by the flavon field would naturally be triggered
once inflation ends. One may worry that this could lead to the formation of
domain-walls that would eventually overclose the Universe, but we note that
the spontaneous breaking of the U(1) symmetry via the $\Phi_{i}$ VEVs already
leads to soft $\mathbb{Z}_{3}$-breaking terms such as
$\chi^{2}\Phi_{1}\Phi_{3}^{\dagger}\xrightarrow{SSB}MM^{\prime}\chi^{2}$ that
ensure that the flavon potential has a single global minimum. In this scenario
$v_{\chi}\lesssim 10^{12}-10^{13}$ GeV, such that if the scale of the non-
renormalizable flavon operators $\Lambda\gtrsim M\sim 10^{15}$ GeV, the
$N_{3}$ Yukawa couplings should also be suppressed in this case,
$y_{3}\lesssim 10^{-3}-10^{-2}$.

We thus conclude that, in both scenarios, the third right-handed neutrino must
be relatively light to accommodate a realistic neutrino mass spectrum,
$M_{3}\lesssim 10^{10}-10^{11}$ GeV.

##  6 Leptogenesis

It is well known that the addition of right-handed neutrinos provides an
elegant mechanism for generating the cosmological baryon asymmetry known as
leptogenesis Fukugita:2003en , where lepton number violation is a consequence
of the Majorana mass terms and C/CP-violation arises through the neutrino
Yukawa couplings. In its simplest realization, a lepton asymmetry is then
generated via the out-of-equilibrium decays of an initially thermal population
of right-handed neutrinos. For hierarchical Majorana masses, the lightest
right-handed neutrino is responsible for producing the largest contribution to
the lepton asymmetry, which is later converted into a baryon asymmetry by
electroweak sphaleron processes, which violate $B+L$ while preserving the
combination $B-L$ Hirsch:2001dg .

It is then clear that leptogenesis can naturally be incorporated within our
setup, where the underlying U(1) gauge symmetry can be identified with lepton
number or $B-L$, two of the right-handed neutrinos are degenerate (at the
minimum of the inflaton potential) and the third right-handed neutrino is
typically much lighter in order to obtain a realistic light neutrino mass
spectrum.

As we have seen in the previous section, consistency of warm inflation
requires $N_{3}$ to only be thermally produced after inflation, either because
its Yukawa couplings are suppressed by the ratio $v_{\chi}/\Lambda\ll 1$ or
because the flavon field only breaks the $\mathbb{Z}_{3}$ family symmetry
after inflation ends. In either case, a thermal population of $N_{3}$ is
produced in the radiation bath after inflation if the effective $N_{3}$ Yukawa
coupling $y_{3}$ is not too suppressed.

The CP asymmetry due to $N_{3}$ decays is given by Davidson:2008bu

$\displaystyle\varepsilon$ $\displaystyle=\frac{\Gamma(N_{3}\rightarrow
L_{j}+H)-\Gamma(N^{\dagger}_{3}\rightarrow
L^{\dagger}_{j}+H^{\dagger})}{\Gamma(N_{3}\rightarrow
L_{j}+H)+\Gamma(N^{\dagger}_{3}\rightarrow L^{\dagger}_{j}+H^{\dagger})}$ (50)
$\displaystyle=\frac{1}{8\pi(Y^{\dagger}_{\nu}Y_{\nu})_{33}}\sum_{i\neq
3}\text{Im}\left(\left[(Y_{\nu}^{\dagger}Y_{\nu})_{3i}\right]^{2}\right)\left(f\left(\frac{M_{i}^{2}}{M_{3}^{2}}\right)+g\left(\frac{M_{i}^{2}}{M_{3}^{2}}\right)\right),$

where

$f(x)=\sqrt{x}\left[1-(1+x)\ln\left(\frac{1+x}{x}\right)\right],\quad
g(x)=\frac{\sqrt{x}}{1-x},$ (51)

and $f(x)$ results from the interference between the tree-level decay
amplitude and the 1-loop vertex correction, whereas $g(x)$ stems from the
absorptive part of the neutrino self-energy171717Note that the expression (50)
differs from the one presented in Hirsch:2001dg  due to a different convention
in the definition of $Y_{\nu}$, which are related through a $\mathrm{h.c.}$
transformation..

This CP asymmetry of the $N_{3}$ decays leads to a lepton asymmetry, diluted
by wash-out processes, such as scattering and inverse decays. The final
lepton-to-entropy ratio is then given by:

$Y_{L}=d\frac{\epsilon}{g_{*}},$ (52)

where $g_{*}$ is the effective number of degrees of freedom, and $d$ is the
dilution factor, which is well approximated by Kolb:1983ni ; Nielsen:2001fy :

$d=\frac{1}{2\sqrt{K^{2}+9}},\qquad K=\left.{\Gamma_{N_{3}}\over
2H}\right|_{T=m_{3}}\simeq\frac{M_{P}}{1.7\times
8\pi\sqrt{g_{*}}}\frac{(Y_{\nu}^{\dagger}Y_{\nu})_{33}}{M_{3}},$ (53)

for $K\lesssim 10$, which is the region of interest in the present case.

Finally, the leptonic asymmetry is converted into a baryonic asymmetry through
sphaleron processes, yielding a baryon-to-entropy ratio:

$Y_{B}=\frac{C}{C-1}Y_{L},\qquad C=\frac{8N_{F}+4N_{H}}{22N_{F}+13N_{H}},$
(54)

where $N_{F}$ is the number of families and $N_{H}$ the number of Higgs
doublets. In this model, which after inflation becomes simply the SM with
right-handed neutrinos, $N_{F}=3$ and $N_{H}=1$, giving $C\simeq 1/3$. The
baryon asymmetry can be experimentally measured through the BBN and CMB, with
$Y_{B}\simeq(8-10)\times 10^{-11}$ Cyburt .

We have performed a numerical parameter scan of our model, matching the full
neutrino mass matrix to the measured values of the neutrino mass differences
and mixing parameters within three standard deviations deSalas:2017kay ,
fixing the effective Yukawa coupling $y$ and the heaviest neutrino masses
$M_{1}=M_{2}$ to values yielding a technically and observationally consistent
warm inflation scenario. Requiring that the baryon symmetry resulting from
out-of-equilibrium $N_{3}$ decays matches the observational value, it is then
possible to relate the mass of the lightest right-handed neutrino, $M_{3}$
(unconstrained by inflation), with its effective Yukawa coupling $y_{3}$
defined above. The results of this analysis are shown in figure 6.

Figure 6: Mass of the lightest right-handed neutrino, $N_{3}$, as a function
of its effective Yukawa coupling, $|y_{3}|$. All neutrino observables are
required to be within the 3$\sigma$ experimental range and we have fixed
$y=0.74$, $M_{1}=4.6\times 10^{14}$ GeV, and $Y_{B}=7.5\times 10^{-11}$. The
light-gray shaded region shows the forbidden region if the $N_{3}$ neutrino
thermalizes during inflation, and the gray shaded region estimates the
breakdown of the approximation $M_{3}\ll M_{1}$.

Taking into account our previous discussion on the size of the $N_{3}$ Yukawa
couplings, we see that a successful model consistently yielding inflation, a
realistic neutrino mass spectrum and the observed cosmological baryon
asymmetry requires $M_{3}\sim 10^{9}-10^{11}$ GeV, consistently with the
Davidson-Ibarra bound Davidson:2002qv , with suppressed Yukawa coupling
$|y_{3}|\sim 10^{-3}-10^{-2}$ as expected from their flavon-induced origin.

##  7 Summary and future prospects

The present work addresses the main shortcomings of the SM except those
involving fine-tuning issues (which arguably are not true inconsistencies of
the theory), within a unified model: 1) inflation is driven by a slowly
rolling scalar field, 2) neutrino masses are generated via the seesaw
mechanism, with right-handed neutrinos being the only particles interacting
directly with the inflaton field; 3) the remnant inflaton field naturally
accounting for dark matter at late times; and 4) a cosmological baryon
asymmetry is generated via thermal leptogenesis. This is implemented within
the first realization of warm inflation within a concrete extension of the SM,
based on the generic Warm Little Inflaton scenario. This is a minimal
extension with the addition of three right-handed neutrinos and three complex
scalar fields that are singlets under the SM gauge group but charged under a
gauged U(1) symmetry identified with lepton number or $B-L$. Moreover, all
these problems are addressed without modifying the gravitational sector,
making it less sensitive to the UV-completion of the model.

The model is endowed with two additional discrete symmetries: an interchange
symmetry under which $N_{1}\leftrightarrow N_{2}$ and $\Phi_{1}\leftrightarrow
i\Phi_{2}$, leaving $N_{3}$ and $\Phi_{3}$ unchanged, and a $\mathbb{Z}_{3}$
family symmetry. The inflaton field is identified with the relative phase of
the complex fields $\Phi_{1}$ and $\Phi_{2}$, and interacts directly only with
the right-handed neutrinos $N_{1}$ and $N_{2}$. These interactions, alongside
the decay of the right-handed neutrinos into leptons and Higgs degrees of
freedom, are responsible for dissipative effects that sustain a warm thermal
bath during inflation, with the interchange symmetry protecting the scalar
potential against large thermal corrections that could prevent a slow-roll
evolution. We have shown that such a scenario consistently allows for a
sufficiently long inflationary period with a quartic scalar potential, with a
scalar spectral index and tensor-to-scalar ratio within the range allowed by
Planck data, as in the original WLI model. In addition, the strong dissipation
regime is attained towards the end of inflation, causing the radiation (i.e.
right-handed neutrinos, leptons and Higgs doublets) to dominate smoothly at
the end of the slow-roll regime. At this stage, the ratio $T/H$ also becomes
sufficiently large to allow for the thermal production of the remaining SM
degrees of freedom.

One of the most interesting features of this setup is that the scale of the
right-handed neutrino masses, $M_{1,2}\sim 10^{14}-10^{15}$ GeV, and effective
Yukawa couplings, $y\lesssim 1$, is fixed by the constraints of realizing warm
inflation near thermal equilibrium at temperatures above the right-handed
neutrino mass threshold, since otherwise dissipative effects would be
exponentially suppressed. This automatically yields the scale of the heaviest
left-handed neutrino mass, via the seesaw mechanism, in agreement with
neutrino oscillation experiments, $m_{3}\simeq 0.05$ eV (taking into account
the small running of the Yukawa couplings). The interchange symmetry also
leads to a specific prediction of at least one massless neutrino.

A full agreement with experimental results for neutrino masses and mixings
requires an extension of this basic setup with e.g. a flavon field that breaks
the $\mathbb{Z}_{3}$ family symmetry, since otherwise the third-right handed
neutrino is not involved in the seesaw mechanism. We have shown that this can
be done without spoiling the successful realization of warm inflation, since
$N_{3}$ need not be thermally excited during inflation.

Our setup also offers natural solutions to two other important cosmological
problems. First, the interchange symmetry makes the inflaton stable at late
times and protects its mass from large radiative corrections. Since the right-
handed neutrino masses are bounded, $N_{1,2}$ are only relativistic and
thermally produced during inflation, decoupling from the dynamics once the
temperature drops below their mass threshold at the end of inflation. The
resulting inflaton remnant thus corresponds to a cold and weakly interacting
fluid that may naturally account for dark matter for inflaton masses
$\lesssim$ 1 eV. Second, a thermal population of the lightest right-handed
neutrino is necessarily generated after inflation, and its out-of-equilibrium
decays can produce a lepton asymmetry that is later converted into the
observed cosmological baryon asymmetry by electroweak sphaleron processes. In
our setup, the three right-handed neutrinos play separate roles, with $N_{1}$
and $N_{2}$ being responsible for warm inflation and $N_{3}$ for thermal
leptogenesis after warm inflation, but all within the same framework yielding
a realistic pattern for neutrino masses and mixings.

In addition, the model can be probed via a plethora of different observables,
with both low-energy neutrino experiments and astrophysical observations, as
best exemplified by figure 5. Warm inflation leads to a specific consistency
relation between the tensor spectral index and the tensor-to-scalar ratio
Cai:2010wt ; Bartrum:2013fia , and within our particular setup we predict the
latter to lie in the range $r\sim 10^{-4}-10^{-3}$ given the measured scale of
neutrino masses. Other observables include non-Gaussian features with a
particular “warm” shape that may be accessible in the near future and cold
dark matter isocurvature modes roughly an order of magnitude below the current
Planck sensitivity, and anti-correlated with the main adiabatic curvature
perturbations. The fact that the inflaton remnant behaves as dark radiation
during BBN and cold dark matter at recombination also yields a very
distinctive probe of our scenario. In conjunction, all these observables make
our setup quite distinguishable from other interesting attempts to
simultaneously address the same problems (e.g. Ballesteros:2016euj ;
Ballesteros:2016xej ).

It would be interesting to try to extend the present warm inflation scenario
into the full strong dissipation regime, i.e. $Q_{*}\gtrsim 100$, where the
inflationary dynamics becomes less reliable on the UV completion Berera:1999ws
; Berera:2004vm ; Motaharfar:2018zyb ; Das:2018rpg . Within the current
understanding of fluctuations in warm inflation, our setup would predict a too
blue-tilted power spectrum in this regime, due to the interplay between
inflaton and radiation fluctuations. This may require modifying the particle
content and the inflaton potential, as recently proposed in the modified WLI-
construction of Bastero-Gil:2019gao , or a better understanding of the non-
equilibrium properties of the thermal bath, which may have a significant
impact on observational predictions (e.g. BasteroGil:2011xd ).

A natural possibility to investigate in the future is also embedding the
minimalistic setup presented in this work within grande unified theories,
particularly those based on $SO(10)$ or larger gauge groups containing the
latter as a sub-group, since right-handed neutrinos are automatically included
in the fundamental representation of $SO(10)$ alongside the known quarks and
leptons. As we mentioned in section 3, the first option for the
$\mathbb{Z}_{3}$ charge assignments could accommodate such an embedding, and
potentially both the latter and the discrete interchange symmetry can have
measurable effects on the low-energy particle spectrum that could further help
testing this model.

###### Acknowledgements.

We thank Mar Bastero-Gil, Arjun Berera and Rudnei Ramos for useful discussions
on this topic. This work was supported by the FCT Grant No. IF/01597/2015. The
work of M. L. is funded by Fundação para a Ciência e Tecnologia-FCT Grant
No.PD/BD/150488/2019, in the framework of the Doctoral Programme IDPASC-PT. ML
was initially supported by Grant No. BI/UI97/8364/2019, in the context of the
FCT Grant No. IF/01597/2015. J. G. R. and L. B. V. are supported by the CFisUC
project No. UID/FIS/04564/2019. J. G.R. is also supported by the projects
PTDC/FIS-OUT/28407/2017 and the ENGAGE SKA (POCI-01-0145-FEDER-022217). L.B.V
is also supported by FCT Grant PD/BD/140917/2019 and partially by the CIDMA
project No. UID/MAT/04106/2020.

##  Appendix A Inflaton self-energy

One of the most important features of the WLI paradigm is the removal of the
leading contributions to $V(\phi)$ that preclude inflation, as discussed in
the main text. An alternative way to see this is by computing the inflaton
self-energy generated from the couplings to the $N_{1,2}$ fermions. To do so,
one can start with the relevant Lagrangian (18) after SSB of the $U(1)_{X}$,
working in the RH neutrino mass basis181818Hence the appearance of the phases
$\delta_{i}$ resulting from the appropriate rotations.:

$-\mathcal{L}_{\phi,N}=\overline{N^{c}}_{i}\left(\frac{1}{2}M_{i}+\frac{1}{2}V_{i}e^{i\delta_{i}}\delta\phi+\frac{1}{4}f_{i}\delta\phi\delta\phi\right)N_{i}+h.c.,$
(55)

where summation over the two RH neutrinos is implied and the background value
of the inflaton field was separated from its fluctuations, by replacing
$\phi\rightarrow\phi+\delta\phi$ and expanding the exponential in powers of
$\delta\phi/M$ up to quadratic order

$e^{i\phi/M}=e^{i\phi/M}e^{i\delta\phi/M}=e^{i\phi/M}\sum_{n=0}^{\infty}\frac{(i\delta\phi/M)^{n}}{n!}=e^{\phi/M}\left(1+i\delta\phi/M+\delta\phi\delta\phi/M^{2}+...\right).$
(56)

Here, $M_{i}$, $V_{i}$ and $f_{i}$ take real values, and are related to each
other (see below). One can now define Majorana fermions as

$\mathbf{N}_{i}=(N_{i}+N_{i}^{c}),\qquad\overline{\mathbf{N}_{i}}=\overline{(N_{i}+N_{i}^{c})},\qquad
N_{i}=P_{R}\mathbf{N}_{i},\qquad N_{i}^{c}=P_{L}\mathbf{N}_{i},$ (57)

and write (55) as

$-\mathcal{L}_{\phi,N}=\bigg{(}\frac{1}{2}M_{i}\overline{\mathbf{N}_{i}}\mathbf{N}_{i}+\frac{1}{2}V_{i}\overline{\mathbf{N}_{i}}e^{i\gamma_{5}\delta}\delta\phi\mathbf{N}_{i}+\frac{1}{4}f_{i}\overline{\mathbf{N}_{i}}\delta\phi\delta\phi\mathbf{N}_{i}\bigg{)},$
(58)

where $f_{i}=-M_{i}/M^{2}$ and $V_{i}^{2}=M_{j}^{2}/M^{2}$, $i=1,2$ and $j\neq
i=2,1$, with

$\displaystyle
M_{1}^{2}=\frac{M^{2}}{4}\left[(g_{1}+g_{2})^{2}\cos^{2}(\phi/M)+(g_{1}-g_{2})^{2}\sin^{2}(\phi/M)\right],$
(59) $\displaystyle
M_{2}^{2}=\frac{M^{2}}{4}\left[(g_{1}-g_{2})^{2}\cos^{2}(\phi/M)+(g_{1}+g_{2})^{2}\sin^{2}(\phi/M)\right],$
$\displaystyle-\delta=\tan^{-1}\left(\frac{g_{-}}{g_{+}}\cot(\phi/M)\right)+\tan^{-1}\left(\frac{g_{-}}{g_{+}}\tan(\phi/M)\right),$

where $g_{\pm}$ was defined in (22).

Figure 7: Feynman diagrams contributing to the inflaton self-energy, (64) and
(60), respectively. The dashed line represents $\delta\phi$ while the solid
one represents $\mathbf{N}_{i}$.

There are two different contributions to the inflaton self-energy i) from the
4-point interaction $\delta\phi\delta\phi NN$ terms, ii) from the 3-point
Yukawa interaction $\delta\phi NN$ terms, represented in figure 7. The former
is readily computed:

$\displaystyle-i\Pi_{\text{4P}}(0)$
$\displaystyle=\frac{1}{2}(-1)\int\frac{d^{4}p}{(2\pi)^{4}}\text{Tr}\left[if_{1}\frac{i(\not{p}-M_{1})}{p^{2}-M_{1}^{2}}+if_{2}\frac{i(\not{p}-M_{2})}{p^{2}-M_{2}^{2}}\right]$
(60)
$\displaystyle=\frac{1}{2}\frac{1}{M^{2}}\int\frac{d^{4}p}{(2\pi)^{4}}\left(\frac{4M_{1}^{2}}{p^{2}-M_{1}^{2}}+\frac{4M_{2}^{2}}{p^{2}-M_{2}^{2}}\right).$

where the symmetry factor of $1/2$ is due to the Majorana nature of $N$
Gluza:1991wj  and we used the relation between $f_{i}$ and $M_{i}$.

The cubic interactions are given by

$-i\Pi_{\text{3P}}^{i}(0)=-\frac{1}{2}V_{i}^{2}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{\text{Tr}\left[\left((\not{p}+M_{i})(e^{i\delta}P_{R}+e^{-i\delta}P_{L})\right)^{2}\right]}{(p^{2}-M_{i}^{2})^{2}},$
(61)

where

$\text{Tr}\left[\left((\not{p}+M_{i})(e^{i\delta}P_{R}+e^{-i\delta}P_{L})\right)^{2}\right]=4p^{2}+4(\cos^{2}\delta-\sin^{2}\delta)M_{i}^{2}=4p^{2}+4\cos(2\delta)M_{i}^{2}.$
(62)

so that the inflaton self-energy correction given by the Yukawa interaction is

$-i\Pi_{\text{3P}}^{i}(0)=-\frac{1}{2}V_{i}^{2}\int\frac{d^{4}p}{(2\pi)^{4}}\left(\frac{4p^{2}+4\cos(2\delta)M_{i}^{2}}{(p^{2}-M_{i}^{2})^{2}}\right),$
(63)

Summing over the two fermionic contributions, from $N_{1}$ and $N_{2}$,

$-i\Pi_{\text{3P}}(0)=-\frac{1}{2}\frac{1}{M^{2}}\int\frac{d^{4}p}{(2\pi)^{4}}\bigg{(}\frac{4M_{2}^{2}}{(p^{2}-M_{1}^{2})}+\frac{4M_{1}^{2}}{(p^{2}-M_{2}^{2})}+\frac{8M_{1}^{2}M_{2}^{2}\cos^{2}\delta}{(p^{2}-M_{1}^{2})^{2}}+\frac{8M_{1}^{2}M_{2}^{2}\cos^{2}\delta}{(p^{2}-M_{2}^{2})^{2}}\bigg{)}.$
(64)

By combining (60) with (64), the leading order contributions (in $p$) to the
inflaton self-energy cancel. As a result, there are no quadratically divergent
terms: only log-divergent terms and finite terms. This holds true in the high
temperature limit (HTL), where the masses in the propagators can be neglected
when compared to the magnitude of $M_{i}~{}\ll~{}p~{}\sim~{}T$.

##  Appendix B Fermion decay width and thermal masses

### Right-handed (RH) neutrino Decay Width

Knowledge of the RH neutrino decay width is required for finding the
dissipation coefficient, essential for the WI inflationary dynamics. The
relevant Lagrangian after the $SU(2)$ contractions is:

$\mathcal{L}_{\text{Yuk},\text{lep}}=-Y(\overline{\mathbf{N}}P_{L}h_{0}\nu_{L}-\overline{\mathbf{N}}P_{L}h^{+}l^{\pm}_{L})+h.c.,$
(65)

following the notation of (57). Above the EWSB scale, both charged and neutral
processes contribute to the RH neutrino decay width in a similar fashion, as
they are mediated by the same Yukawa couplings. Furthermore, the $h.c.$ term
above can be recast as
$\overline{\nu_{L}}h_{0}^{*}P_{R}\mathbf{N}\rightarrow\overline{\mathbf{N}}P_{R}\nu_{L}^{c}h_{0}^{*}$,
by making use of $-\mathbf{N}^{T}C^{-1}=\overline{\mathbf{N}}$ Gluza:1991wj ,
and $\gamma^{0}C\psi^{*}=\psi^{c}$Pal:2010ih , resulting in four different
processes, all of which identical due to the nature of the Cutkosky rules
Cutkosky:1960sp .

Following the finite-temperature generalization of the Cutkosky rules
Kobes:1986wb ,

$\text{Im}\Sigma=(1+e^{-\beta
p_{0}})\hat{\Gamma}_{d},\qquad\hat{\Gamma}_{d}=\frac{1}{2}\sum_{z}F(y_{1},y_{2},z),$
(66)

where $\Sigma$ and $\hat{\Gamma}_{d}$ is the self-energy and decay width of
the RH neutrinos, respectively. The sum in $z$ is over different possibilities
of circling internal vertices, and $y_{1}$ ($y_{2}$) is the vertex associated
with the incoming (outgoing) momenta, and is uncircled (circled)191919In the
Cutkosky rules and their finite-temperature generalization, circled vertices
are the $h.c.$ of the uncircled vertices.. At the 1-loop level, there are no
internal vertices

$\text{Im}\Sigma_{i}(p)=\frac{1}{2}(1+e^{-\beta
p_{0}}){(YY^{\dagger})}_{ii}\int\frac{d^{4}k}{(2\pi)^{4}}P_{R}\Delta_{s}^{+}(k)P_{R}\Delta_{f}^{+}(p-k),$
(67)

in which no summation over repeated indexes is implied, and Kobes:1986wb

$\displaystyle\Delta_{s}^{+}(p)$ $\displaystyle=$ $\displaystyle
2\pi[\theta(p_{0})+n_{B}(p_{0})]\delta(p^{2}-m^{2}),$ (68)
$\displaystyle\Delta_{f}^{+}(p)$ $\displaystyle=$ $\displaystyle
2\pi[\theta(p_{0})-n_{F}(p_{0})](\not{p}-m)\delta(p^{2}-m^{2}).$ (69)

In the massless limit, the above product is equal to

$\displaystyle\Delta_{s}^{+}(k)\Delta_{f}^{+}(p-k)=$
$\displaystyle(2\pi)^{2}(\not{p}-\not{k})\delta((p-k)^{2})\delta(k^{2})\theta(p_{0}-k_{0})\theta(k_{0})$
(70a)
$\displaystyle+(2\pi)^{2}(\not{p}-\not{k})\delta((p-k)^{2})\delta(k^{2})\theta(p_{0}-k_{0})n_{B}(k_{0})$
(70b)
$\displaystyle-(2\pi)^{2}(\not{p}-\not{k})\delta((p-k)^{2})\delta(k^{2})\theta(k_{0})n_{F}(p_{0}-k_{0})$
(70c)
$\displaystyle-(2\pi)^{2}(\not{p}-\not{k})\delta((p-k)^{2})\delta(k^{2})n_{B}(k_{0})n_{F}(p_{0}-k_{0}).$
(70d)

For (70a), using

$\int d^{4}k\theta(k_{0})\delta(k^{2})=\int
d^{3}k\frac{1}{2k_{0}},\quad\text{and}\quad\int d^{4}p\delta((p-k)^{2})=\int
d^{4}pd^{4}k^{\prime}\delta(k^{\prime 2})\delta((p-k)-k^{\prime}),$ (71)

one obtains

$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}(2\pi)^{2}(P_{R})^{2}(\not{p}-\not{k})\theta(p_{0}-k_{0})\theta(k_{0})\delta(k^{2})\delta((p-k)^{2})$
(72)
$\displaystyle=(2\pi)\int\frac{d^{3}k}{(2\pi)^{3}}P_{R}\frac{\not{p}-\not{k}}{4\mathbf{|k|}\mathbf{|p-k|}}\delta(p_{0}-\mathbf{|k|}-\mathbf{|p-k|}).$

Dropping the $\delta(k_{0}+\mathbf{|k|})$ contributions (associated with
scatterings and other processes Laine:2016hma ) and keeping in mind that

$n_{B}(k_{0})n_{F}(p_{0}-k_{0})=(1+e^{\beta
p_{0}})^{-1}(1+n_{B}(k_{0})-n_{F}(p_{0}-k_{0})),$ (73)

we find the results

$\displaystyle\eqref{second-term}=\eqref{first-term-
result}*n_{B}(\mathbf{|k|}),$ $\displaystyle\eqref{third-term}=-\eqref{first-
term-result}*n_{F}(\mathbf{|p-k|}),$ (74) $\displaystyle\eqref{fourth-
term}=-(1+e^{\beta p_{0}})^{-1}\big{(}\eqref{first-term}+\eqref{second-
term}+\eqref{third-term}\big{)}.$

Therefore, the full contribution to the loop is

$\left(1-(1+e^{\beta
p_{0}})^{-1}\right)\,2\pi\,P_{R}\,\int\frac{d^{3}k}{(2\pi)^{3}}(1+n_{B}(\mathbf{|k|})-n_{F}(\mathbf{|p-k|}))\delta(p_{0}-\mathbf{|k|}-\mathbf{|p-k|}).$
(75)

Recalling (67) and using $(1+e^{-\beta p_{0}})(1-(1+e^{\beta p_{0}})^{-1})=1$,

$\displaystyle\text{Im}\Sigma_{i}(p)=$ (76)
$\displaystyle\frac{1}{2}{(YY^{\dagger})}_{ii}(2\pi)\int\frac{d^{3}k}{(2\pi)^{3}}(P_{R})\frac{\not{p}-\not{k}}{4\mathbf{|k|}\mathbf{|p-k|}}(1+n_{B}(\mathbf{|k|})-n_{F}(\mathbf{|p-k|}))\delta(p_{0}-\mathbf{|k|}-\mathbf{|p-k|}).$

Taking the limit where $p$ and $k$ are aligned,

$\not{p}-\not{k}=\not{p}\left(\frac{p_{0}-|\mathbf{k}|}{p_{0}}\right),$ (77)

it is possible to compute $\Gamma$ through Kiessig:2010pr ,

$\displaystyle\Gamma_{i}$
$\displaystyle=-\frac{1}{2p_{0}}\text{Tr}[(\not{p}+m_{i})\text{Im}\Sigma]=\frac{-(YY^{\dagger})_{ii}}{4p_{0}}(2\pi)\times$
(78)
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\text{Tr}\left[(\not{p}+m_{i})(P_{R})\frac{\not{p}-\not{k}}{4\mathbf{|k|}\mathbf{|p-k|}}\right](1+n_{B}(\mathbf{|k|})-n_{F}(\mathbf{|p-k|}))\delta(p_{0}-\mathbf{|k|}-\mathbf{|p-k|})$
$\displaystyle=\frac{-(YY^{\dagger})_{ii}p^{2}}{16\pi
p_{0}^{2}\mathbf{|p|}}\int^{k_{+}}_{k_{-}}dk(1+n_{B}(\mathbf{|k|})-n_{F}(\mathbf{|p-k|}))(p_{0}-|\mathbf{k}|),$

where the angular integration was performed through the Dirac delta, which
binds the moduli integration by imposing the appropriate bounds on the cosine
function:

$\delta(p_{0}-|\mathbf{k}|-|\mathbf{p-k}|)=\frac{|\mathbf{p-k}|}{|\mathbf{p}||\mathbf{k}|}\delta(\cos\theta-\cos\theta_{0}),\\\
$ (79)
$\cos\theta_{0}\equiv\frac{p_{0}}{\mathbf{|p|}}+\frac{\mathbf{|p|}^{2}-p_{0}^{2}}{2\mathbf{|p|}\mathbf{|k|}},\quad
k_{\pm}=\frac{1}{2}\left(p_{0}\pm|\mathbf{p}|\right).$ (80)

Taking into account all contributions,

$\Gamma_{N_{i}}=4\times\frac{1}{2}\frac{m_{i}^{2}}{\omega_{p_{i}}}\frac{{T^{2}(YY^{\dagger})}_{ii}}{8\pi|p|\omega_{p_{i}}}F(\mathbf{|p|}/T,m_{i}/T),$
(81)

where $F(\mathbf{|p|}/T,m_{i}/T)$ is given by Bastero-Gil:2016qru

$\displaystyle F(x,w)=f(x_{+},\omega_{p}/T)-f(x_{+},\omega_{p}/T),$ (82)

with $x_{\pm}=k_{\pm}/T$, $\omega_{p}=\sqrt{\mathbf{p}^{2}+m^{2}(T)}$ and

$\displaystyle f(x,w)=$ $\displaystyle\int
dx(w-x)\left[1+\frac{1}{e^{x}-1}-\frac{1}{e^{(w-x)}+1}\right]$ (83)
$\displaystyle=$
$\displaystyle-\frac{\pi^{2}}{3}-w^{2}+xw-\frac{x^{2}}{2}+(w-x)\ln\left(\frac{1-e^{-x}}{1+e^{-w+x}}\right)+\text{Li}_{2}\left(e^{-x}\right)+\text{Li}_{2}\left(-e^{-w+x}\right).$

where $w=\omega_{p}/T$. As a consistency check, it can be seen that taking the
zero-temperature limit of (81) yields the result of delAguila:2005pin .

### Thermal Masses

The thermal contribution to the masses of the right-handed neutrinos result
from the real part of the fermion self-energy diagram

$\Sigma_{i}=i\int\frac{d^{4}k}{(2\pi)^{4}}V_{1}G^{(11)}(k)V_{2}S^{(11)}(p-k),$
(84)

composed by a left-handed neutrino/Higgs loop. $V_{1,2}$ are the interaction
vertices, $V_{1}=V_{2}=P_{L}y_{ij}$, with a sum over the SM lepton families
implied, where $\sum^{3}_{j=1}y^{2}_{1j}=\sum^{3}_{j=1}y^{2}_{2j}=y^{2}$.
Using the finite-temperature propagators

$\Delta(p)=\frac{i}{p^{2}-m^{2}+i\varepsilon}=iP\left(\frac{1}{p^{2}-m^{2}}\right)+\pi\delta(p^{2}-m^{2}),$
(85)

where $P(x)$ is the Cauchy principal value. The real and imaginary parts of
the propagators are Bellac:2011kqa :

$\displaystyle G^{(11)}(k)$
$\displaystyle=\left[iP\left(\frac{1}{k^{2}}\right)+2\pi\delta(k^{2})(1/2+n_{B}(k))\right],$
(86) $\displaystyle S^{(11)}(k)$
$\displaystyle=\left[iP\left(\frac{1}{k^{2}}\right)+2\pi\delta(k^{2})(1/2-n_{F}(k))\right]\not{k},$

for massless particles. Accounting for the factor $i$ in (84), the real part
of the self-energy comes from the imaginary component of the propagator
product:

$\displaystyle\text{Re}\Sigma_{i}(p)=$ (87) $\displaystyle
4P_{L}y^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left[\frac{(\not{p}-\not{k})2\pi\delta(k^{2})(1/2+n_{B}(k))}{(p-k)^{2}}+\frac{(\not{p}-\not{k})2\pi\delta((p-k)^{2})(1/2-n_{F}(p-k))}{k^{2}}\right].$

Redefining $k\rightarrow p-k$ in the second term, and separating the zero-
temperature and finite-temperature contributions,
$\Sigma=\Sigma^{0}+\Sigma^{T}$:

$\\!\\!\\!\\!\frac{\text{Re}\Sigma_{i}^{T}(p)}{4y^{2}}=P_{L}\int\frac{d^{4}k}{(2\pi)^{4}}\left[(\not{p}-\not{k})n_{B}(k)-\not{k}n_{F}(k)\right]\frac{2\pi\delta(k^{2})}{(p-k)^{2}}.$
(88)

Unfolding
$\delta(k^{2})=\delta(k_{0}^{2}-\mathbf{k}^{2})=(\delta(k_{0}\pm\mathbf{k}))/(2\mathbf{k})$,
taking the nearly-massless limit ($p$ and $k$ are aligned), and performing the
integration in $k_{0}$, one obtains

$\displaystyle\text{Re}\Sigma_{i}^{T}(p)=$ (89)
$\displaystyle\frac{2P_{L}y^{2}}{4\pi^{3}}\int\frac{d^{3}k}{2k}\left[\frac{(p_{0}-k)n_{B}(k)-kn_{F}(k)}{2k(p-p_{0})+(p_{0}^{2}-p^{2})}+\frac{(p_{0}-(-k)n_{B}(-k)-(-k)n_{F}(-k)}{2k(p+p_{0})+(p_{0}^{2}-p^{2})}\right]\frac{\not{p}}{p_{0}}.$

In the limit $p^{2}\approx p_{0}^{2}$ and making use of

$n_{B}(-x)=-1-n_{B}(x),\qquad n_{F}(-x)=1-n_{F}(x),$ (90)

one obtains

$\displaystyle\text{Re}\Sigma_{i}^{T}(p)$
$\displaystyle=\frac{y^{2}P_{L}}{8\pi^{2}p_{0}p^{2}}\not{p}\int
dk\bigg{[}(p+p_{0})\big{(}(p_{0}-k)n_{B}(k)-kn_{F}(k)\big{)}+$ (91)
$\displaystyle+(p-p_{0})\big{(}-p_{0}(1+n_{B}(k))-k(n_{B}(k)+n_{F}(k))\big{)}\bigg{]}$

Following Weldon:1982bn , the leading $T^{2}$ contributions are generated by
would-be quadratically divergent terms if no thermal distribution were present
to act as a $k\sim\mathcal{O}(T)$ cut-off in the integration.

$\displaystyle\text{Re}\Sigma_{i}^{T^{\prime}}(p)$
$\displaystyle=\frac{P_{L}y^{2}}{8\pi^{2}p_{0}p^{2}}\not{p}\int
dk\bigg{(}(n_{B}(k)+n_{F}(k))(p_{0}+p)-(n_{B}(k)+n_{F}(k))(p_{0}-p)\bigg{)}k$
(92) $\displaystyle=\frac{P_{L}y^{2}}{4\pi^{2}}\frac{\not{p}}{p^{2}}\int
dk\left(n_{B}(k)+n_{F}(k)\right)k,$

using $p=p_{0}$. Since

$\int^{\infty}_{0}dkkn_{B}(k)=\frac{\pi^{2}T^{2}}{6},\qquad\int^{\infty}_{0}dkkn_{F}(k)=\frac{\pi^{2}T^{2}}{12},$
(93)

one obtains

$\text{Re}\Sigma_{i}^{T^{\prime}}(p)=\frac{P_{L}y^{2}}{p^{2}}\frac{T^{2}}{16}\not{p}.$
(94)

Taking into account all contributions,
$\text{Re}\Sigma_{f}^{T^{\prime}}(p)=4\text{Re}\Sigma_{i}^{T^{\prime}}(p)$,
and the thermal contribution to the fermion masses is Weldon:1982bn

$M_{T}^{2}=\frac{\text{Tr}}{4}\left[\not{p}\text{Re}\Sigma_{f}^{T^{\prime}}(p)\right]=\frac{4y^{2}T^{2}}{4\times
16}\text{Tr}\left[\frac{1-\gamma_{5}}{2}\not{p}\not{p}\right]=\frac{y^{2}T^{2}}{8}.$
(95)

##  Appendix C Dissipation coefficient

The relevant Lagrangian for the computation of the dissipation coefficient has
two vertices for each Majorana fermion:

$-\mathcal{L}^{\phi}_{\text{Yuk}}=\frac{M_{i}}{2}\overline{\mathbf{N}}_{i}\mathbf{N}_{i}+\frac{V_{i}}{2}\overline{\mathbf{N}}_{i}\delta\phi\left(e^{i\delta}P_{R}+e^{-i\delta}P_{L}\right)\mathbf{N}_{i}.$
(96)

To obtain the correct result, one needs to take into account that the
canonically normalized inflaton field is $\varphi=\phi/\sqrt{2}$. Doing so,
the dissipation coefficient for each neutrino is given by BasteroGil:2010pb

$\displaystyle\Upsilon$
$\displaystyle=\frac{1}{4T}\int\frac{d^{4}p}{(2\pi)^{4}}\text{Tr}[\rho_{\psi_{i}}\mathcal{V}\rho_{\psi_{i}}\mathcal{V}]n_{F}(p_{0})(1-n_{F}(p_{0})).$
(97)

where $\mathcal{V}$ is the term inside parenthesis in (96) times $V_{i}$,
defined in (LABEL:Vertices_and_Phase). Note that one still has to sum over
both fermionic contributions ($i=1,2$). The extra factor of $1/2$ compared to
BasteroGil:2010pb  is due to the symmetry factors of the Feynman diagrams
(related to having identical, Majorana, particles running in the loops).
Expanding the fermionic trace above

$\displaystyle
V_{i}^{2}e^{2i\delta}\text{Tr}\left[(\not{p}+M_{i})P_{R}(\not{p}+M_{i})P_{R}\right]$
$\displaystyle=2M_{i}^{2}V_{i}^{2}e^{2i\delta},$ (98) $\displaystyle
V_{i}^{2}e^{-2i\delta}\text{Tr}\left[(\not{p}+M_{i})P_{L}(\not{p}+M_{i})P_{L}\right]$
$\displaystyle=2M_{i}^{2}V_{i}^{2}e^{-2i\delta},$ $\displaystyle
2V_{i}^{2}\text{Tr}\left[(\not{p}+M_{i})P_{R}(\not{p}+M_{i})P_{L}\right]$
$\displaystyle=4M_{i}^{2}V_{i}^{2}.$

where we used that $p^{2}=M_{i}^{2}$. Using

$\rho_{B}^{2}=\frac{\pi}{2\omega_{p_{i}}^{2}\Gamma_{N_{i}}}\left(\delta(p_{0}-\omega_{p_{i}})+\delta(p_{0}+\omega_{p_{i}})\right),$
(99)

the dissipation coefficient can be written as

$\Upsilon_{i}=\frac{V_{i}^{2}}{T}\cos^{2}\delta\int\frac{d^{3}p}{(2\pi)^{3}}\frac{M_{i}^{2}}{\omega_{p_{i}}^{2}\Gamma_{N_{i}}}n_{F}(p_{0})(1-n_{F}(p_{0})).$
(100)

Summing over the two $\mathbf{N}$ contributions,

$\Upsilon=\Upsilon_{1}+\Upsilon_{2}=\frac{V_{1}^{2}+V_{2}^{2}}{T}\cos^{2}\delta\text{Int.},$
(101)

where we used $M_{1}\simeq M_{2}$, a good approximation when the thermal
contribution dominates the right-handed neutrino masses. The integral is then
defined as

$\text{Int.}=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{M_{i}^{2}}{\omega^{2}_{p}\Gamma_{N}}n_{F}(p_{0})(1-n_{F}(p_{0})).$
(102)

Accounting for (81), the dissipation coefficient is:

$\Upsilon_{i}=\frac{\pi
V_{i}^{2}}{y^{2}T^{3}}\cos^{2}\delta\int\frac{d^{3}p}{(2\pi)^{3}}|\mathbf{p}|\frac{n_{F}(p_{0})(1-n_{F}(p_{0}))}{F(|\mathbf{p}|/T,M_{i}/T)},$
(103)

where $y^{2}$ is the effective Yukawa coupling defined in (26), with
$F(|\mathbf{p}|/T,M_{i}/T)$ is defined in (82). Performing the integration
over the solid angle and changing the integration variable to
$x\equiv|\mathbf{p}|/T$,

$\Upsilon_{i}=\frac{\pi}{4\pi^{3}}\frac{V_{i}^{2}}{y^{2}}\cos^{2}(\delta)T\int_{0}^{\infty}\frac{d^{3}p}{T^{3}}\frac{|\mathbf{p}|}{T}\frac{n_{F}(1-n_{F})}{F(|\mathbf{p}|/T,M_{i}/T)}=\frac{2V_{i}^{2}}{\pi
y^{2}}T\cos^{2}(\delta)\int_{0}^{\infty}dxx^{3}\frac{n_{F}(1-n_{F})}{F(x,w)}.$
(104)

where $w\equiv M_{i}/T$. In order to find a simple expression for the
dissipation coefficient (104), we analyzed two asymptotic regimes: i) the
high-temperature limit ($w\ll 1$) and ii) the low-temperature limit ($w\gg
1$). The final result is an interpolation between the two.

Assuming $x\sim 1$, which is the most relevant region of the integral, the
high-temperature limit becomes

$F(x,w)\sim-2x\log(w)\Rightarrow\int_{0}^{\infty}dx\frac{x^{2}}{\beta-2\log(w)}n_{F}(1-n_{F})=\frac{\pi^{2}}{6(\beta-2\log(w))},$
(105)

where $\beta$ is a constant to be determined. Numerically, we find that
$\beta=3$ results in a low error in the estimate of the dissipation
coefficient in the high-temperature limit. For the low-temperature case,

$\displaystyle F(x,w)$
$\displaystyle\sim\frac{xw}{2}\coth{\left(\frac{w}{2}\right)}\Rightarrow\int_{0}^{\infty}dx\frac{x^{2}}{w\coth{\left(\frac{w}{2}\right)}}e^{-w-\frac{x^{2}}{2w}}\left(1-e^{-w-\frac{x^{2}}{2w}}\right)$
(106)
$\displaystyle=\frac{1}{2}\sqrt{\pi}e^{-2w}\left(2\sqrt{2}e^{w}-1\right)\sqrt{w}\tanh{\left(\frac{w}{2}\right)}$

Finally, we may introduce Boltzmann factors to combine these two asymptotic
limits in a smooth way, yielding:

$\Upsilon_{i}=\frac{V_{i}^{2}}{y^{2}}T\cos^{2}\delta\left(\frac{\pi
e^{-2(M_{i}/T)}}{(9-6e^{-(M_{i}/T)}\log{(M_{i}/T)})}+\frac{e^{-(M_{i}/T)}(2\sqrt{2}-e^{-(M_{i}/T)})\tanh{(M_{i}/T)}}{\sqrt{\pi}}\right),$
(107)

which features a relative error never greater than $4\%$ (see Figure 8).

Figure 8: Numerical evaluation of the integral in (104) and its analytical
approximation, $\Upsilon_{i}/B$ with $B=T\cos^{2}(\delta)V_{i}^{2}/y^{2}$.

Taking the thermal corrections to be the dominant contribution to the heavy
neutrino masses during inflation, i.e., $M_{t}\gg M_{0}$, then $M_{i}\propto
M_{t}T$ and the contributions from $N_{1}$ and $N_{2}$ are easily combined:

$\Upsilon=\frac{\left(g_{1}^{2}+g_{2}^{2}\right)}{2y^{2}}T\cos^{2}(\delta)\left(\frac{\pi
e^{-M_{t}}}{9e^{M_{t}}-6\ln{M_{t}}}+\frac{e^{-2M_{t}}\left(2\sqrt{2}e^{M_{t}}-1\right)\sqrt{M_{t}}\tanh{(\frac{M_{t}}{2})}}{\sqrt{\pi}}\right).$
(108)

Replacing $M_{T}$ with the result obtained from the thermal mass computation,
one arrives at

$\Upsilon=\frac{(g_{1}^{2}+g_{2}^{2})}{2y^{2}}\cos^{2}\delta\left(\frac{\pi}{9-6\ln(y/2\sqrt{2})}\right)T\equiv
C_{T}T,$ (109)

which is a function of the three coupling constants relevant to inflationary
dynamics: $g_{1},g_{2}$, and $y$.

## References

  * (1) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98 (2018) no.3, 030001
  * (2) J. M. Cline, [arXiv:hep-ph/0609145 [hep-ph]].
  * (3) D. Baumann, arXiv:0907.5424 [hep-th].
  * (4) A. Berera and L. Z. Fang, Phys. Rev. Lett. 74 (1995), 1912-1915 [arXiv:astro-ph/9501024 [astro-ph]].
  * (5) A. Berera, Phys. Rev. Lett. 75 (1995), 3218-3221 [arXiv:astro-ph/9509049 [astro-ph]].
  * (6) A. Berera, I. G. Moss and R. O. Ramos, Rept. Prog. Phys. 72 (2009), 026901 [arXiv:0808.1855 [hep-ph]].
  * (7) A. N. Taylor and A. Berera, Phys. Rev. D 62 (2000), 083517 [arXiv:astro-ph/0006077 [astro-ph]].
  * (8) L. M. Hall, I. G. Moss and A. Berera, Phys. Rev. D 69, 083525 (2004) [astro-ph/0305015].
  * (9) C. Graham and I. G. Moss, JCAP 0907, 013 (2009) [arXiv:0905.3500 [astro-ph.CO]].
  * (10) R. O. Ramos and L. A. da Silva, JCAP 1303, 032 (2013) [arXiv:1302.3544 [astro-ph.CO]].
  * (11) A. Berera, M. Gleiser and R. O. Ramos, Phys. Rev. D 58, 123508 (1998) [hep-ph/9803394].
  * (12) J. Yokoyama and A. D. Linde, Phys. Rev. D 60, 083509 (1999) [hep-ph/9809409].
  * (13) M. Bastero-Gil and A. Berera, Int. J. Mod. Phys. A 24, 2207 (2009) [arXiv:0902.0521 [hep-ph]].
  * (14) M. Bastero-Gil, A. Berera and J. G. Rosa, Phys. Rev. D 84 (2011), 103503 [arXiv:1103.5623 [hep-th]].
  * (15) M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, Phys. Rev. Lett. 117, no. 15, 151301 (2016) [arXiv:1604.08838 [hep-ph]].
  * (16) M. Bastero-Gil, S. Bhattacharya, K. Dutta and M. R. Gangopadhyay, JCAP 02 (2018), 054 [arXiv:1710.10008 [astro-ph.CO]].
  * (17) M. Bastero-Gil, A. Berera, R. Hernandez Jimenez and J. G. Rosa, Phys. Rev. D 98 (2018) no.8, 083502 [arXiv:1805.07186 [astro-ph.CO]].
  * (18) J. G. Rosa and L. B. Ventura, Phys. Rev. Lett. 122, no. 16, 161301 (2019) [arXiv:1811.05493 [hep-ph]].
  * (19) J. G. Rosa and L. B. Ventura, Phys. Lett. B 798 (2019), 134984 [arXiv:1906.11835 [hep-ph]].
  * (20) K. Dimopoulos and L. Donaldson-Wood, Phys. Lett. B 796 (2019), 26-31 [arXiv:1906.09648 [gr-qc]].
  * (21) G. B. F. Lima and R. O. Ramos, Phys. Rev. D 100 (2019) no.12, 123529 [arXiv:1910.05185 [astro-ph.CO]].
  * (22) M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986), 45-47
  * (23) A. Berera, Phys. Rev. D 54 (1996), 2519-2534 [arXiv:hep-th/9601134 [hep-th]].
  * (24) A. Berera, Phys. Rev. D 55 (1997), 3346-3357 [arXiv:hep-ph/9612239 [hep-ph]].
  * (25) A. Berera, M. Gleiser and R. O. Ramos, Phys. Rev. Lett. 83 (1999), 264-267 [arXiv:hep-ph/9809583 [hep-ph]].
  * (26) A. Berera, Nucl. Phys. B 585 (2000), 666-714 [arXiv:hep-ph/9904409 [hep-ph]].
  * (27) A. Berera and R. O. Ramos, Phys. Rev. D 63 (2001), 103509 [arXiv:hep-ph/0101049 [hep-ph]].
  * (28) A. Berera and R. O. Ramos, Phys. Lett. B 567 (2003), 294-304 [arXiv:hep-ph/0210301 [hep-ph]].
  * (29) A. Berera, PoS AHEP2003 (2003), 069 [arXiv:hep-ph/0401139 [hep-ph]].
  * (30) A. Berera and R. O. Ramos, Phys. Lett. B 607 (2005), 1-7 [arXiv:hep-ph/0308211 [hep-ph]].
  * (31) I. G. Moss and C. Xiong, [arXiv:hep-ph/0603266 [hep-ph]].
  * (32) M. Bastero-Gil and A. Berera, Phys. Rev. D 76 (2007), 043515 [arXiv:hep-ph/0610343 [hep-ph]].
  * (33) A. Berera, I. G. Moss and R. O. Ramos, Phys. Rev. D 76 (2007), 083520 [arXiv:0706.2793 [hep-ph]].
  * (34) I. G. Moss and C. Xiong, JCAP 11 (2008), 023 [arXiv:0808.0261 [astro-ph]].
  * (35) I. G. Moss and C. M. Graham, Phys. Rev. D 78 (2008), 123526 [arXiv:0810.2039 [hep-ph]].
  * (36) M Bastero-Gil, A. Berera and R. O. Ramos, JCAP 09 (2011), 033 [arXiv:1008.1929 [hep-ph]].
  * (37) M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, JCAP 01 (2013), 016 [arXiv:1207.0445 [hep-ph]].
  * (38) M. Bastero-Gil, A. Berera, R. Cerezo, R. O. Ramos and G. S. Vicente, JCAP 11 (2012), 042 [arXiv:1209.0712 [astro-ph.CO]].
  * (39) M. Bastero-Gil, A. Berera and N. Kronberg, JCAP 12 (2015), 046 [arXiv:1509.07604 [hep-ph]].
  * (40) A. Berera, J. Mabillard, M. Pieroni and R. O. Ramos, JCAP 07 (2018), 021 [arXiv:1803.04982 [astro-ph.CO]].
  * (41) S. Das and R. O. Ramos, [arXiv:2005.01122 [gr-qc]].
  * (42) A. Berera and T. W. Kephart, Phys. Rev. Lett. 83 (1999), 1084-1087 [arXiv:hep-ph/9904410 [hep-ph]].
  * (43) M. Bastero-Gil and A. Berera, Phys. Rev. D 71 (2005), 063515 [arXiv:hep-ph/0411144 [hep-ph]].
  * (44) M. Bastero-Gil and A. Berera, Phys. Rev. D 72 (2005), 103526 [arXiv:hep-ph/0507124 [hep-ph]].
  * (45) J. C. Bueno Sanchez, M. Bastero-Gil, A. Berera and K. Dimopoulos, Phys. Rev. D 77 (2008), 123527 [arXiv:0802.4354 [hep-ph]].
  * (46) M. Bastero-Gil, A. Berera, J. B. Dent and T. W. Kephart, [arXiv:0904.2195 [astro-ph.CO]].
  * (47) J. C. Bueno Sanchez, M. Bastero-Gil, A. Berera, K. Dimopoulos and K. Kohri, JCAP 03 (2011), 020 [arXiv:1011.2398 [hep-ph]].
  * (48) Y. F. Cai, J. B. Dent and D. A. Easson, Phys. Rev. D 83 (2011), 101301 [arXiv:1011.4074 [hep-th]].
  * (49) R. Cerezo and J. G. Rosa, JHEP 01 (2013), 024 [arXiv:1210.7975 [hep-ph]].
  * (50) T. Matsuda, Phys. Rev. D 87 (2013) no.2, 026001 [arXiv:1212.3030 [hep-th]].
  * (51) M. Bastero-Gil, A. Berera, T. P. Metcalf and J. G. Rosa, JCAP 03 (2014), 023 [arXiv:1312.2961 [hep-ph]].
  * (52) K. Li, X. M. Zhang, H. Y. Ma and J. Y. Zhu, Phys. Rev. D 98 (2018) no.12, 123528 [arXiv:1804.00276 [gr-qc]].
  * (53) M. Bastero-Gil, A. Berera, R. Hernandez-Jimenez and J. G. Rosa, Phys. Rev. D 99 (2019) no.10, 103520 [arXiv:1812.07296 [hep-ph]].
  * (54) L. M. H. Hall, I. G. Moss and A. Berera, Phys. Lett. B 589 (2004), 1-6 [arXiv:astro-ph/0402299 [astro-ph]].
  * (55) I. G. Moss and C. Xiong, JCAP 04 (2007), 007 [arXiv:astro-ph/0701302 [astro-ph]].
  * (56) A. Berera, L. M. H. Hall, I. G. Moss and H. V. Peiris, [arXiv:0805.2112 [astro-ph]].
  * (57) M. Bastero-Gil, A. Berera and R. O. Ramos, JCAP 07 (2011), 030 [arXiv:1106.0701 [astro-ph.CO]].
  * (58) M. Bastero-Gil, A. Berera, N. Mahajan and R. Rangarajan, Phys. Rev. D 87 (2013) no.8, 087302 [arXiv:1302.2995 [astro-ph.CO]].
  * (59) S. Bartrum, A. Berera and J. G. Rosa, JCAP 06 (2013), 025 [arXiv:1303.3508 [astro-ph.CO]].
  * (60) S. Bartrum, M. Bastero-Gil, A. Berera, R. Cerezo, R. O. Ramos and J. G. Rosa, Phys. Lett. B 732 (2014), 116-121 [arXiv:1307.5868 [hep-ph]].
  * (61) M. Bastero-Gil, A. Berera, I. G. Moss and R. O. Ramos, JCAP 05 (2014), 004 [arXiv:1401.1149 [astro-ph.CO]].
  * (62) M. Bastero-Gil, A. Berera, I. G. Moss and R. O. Ramos, JCAP 12 (2014), 008 [arXiv:1408.4391 [astro-ph.CO]].
  * (63) M. Benetti and R. O. Ramos, Phys. Rev. D 95 (2017) no.2, 023517 [arXiv:1610.08758 [astro-ph.CO]].
  * (64) M. Benetti, L. Graef and R. O. Ramos, JCAP 10 (2019), 066 [arXiv:1907.03633 [astro-ph.CO]].
  * (65) V. Kamali, M. Motaharfar and R. O. Ramos, Phys. Rev. D 101 (2020) no.2, 023535 [arXiv:1910.06796 [gr-qc]].
  * (66) A. Berera, T. W. Kephart and S. D. Wick, Phys. Rev. D 59 (1999), 043510 [arXiv:hep-ph/9809404 [hep-ph]].
  * (67) M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, Phys. Lett. B 712 (2012), 425-429 [arXiv:1110.3971 [hep-ph]].
  * (68) M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, JCAP 10 (2014), 053 [arXiv:1404.4976 [astro-ph.CO]].
  * (69) M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, JHEP 02 (2018), 063 [arXiv:1711.09023 [hep-ph]].
  * (70) Y. Akrami et al. [Planck Collaboration], arXiv:1807.06211 [astro-ph.CO].
  * (71) M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, arXiv:1907.13410 [hep-ph].
  * (72) K. V. Berghaus, P. W. Graham and D. E. Kaplan, arXiv:1910.07525 [hep-ph].
  * (73) P. Minkowski, Phys. Lett. 67B, 421 (1977).
  * (74) J. M. Cline and P. A. Lemieux, Phys. Rev. D 55, 3873 (1997) [hep-ph/9609240].
  * (75) L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974).
  * (76) J. S. Borges and R. O. Ramos, Eur. Phys. J. C 76, no.6, 344 (2016) [arXiv:1602.08165 [hep-ph]].
  * (77) P. A. R. Ade et al. [Planck], Astron. Astrophys. 594, A17 (2016) [arXiv:1502.01592 [astro-ph.CO]].
  * (78) Y. Akrami et al. [Planck], Astron. Astrophys. 641, A9 (2020) [arXiv:1905.05697 [astro-ph.CO]].
  * (79) S. Das and R. O. Ramos, [arXiv:2007.15268 [hep-th]].
  * (80) D. H. Lyth and E. D. Stewart, Phys. Rev. Lett. 75, 201-204 (1995) [arXiv:hep-ph/9502417 [hep-ph]].
  * (81) D. H. Lyth and E. D. Stewart, Phys. Rev. D 53, 1784-1798 (1996) [arXiv:hep-ph/9510204 [hep-ph]].
  * (82) T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 (1975)
  * (83) M. Bastero-Gil, A. Berera and B. M. Jackson, JCAP 07, 010 (2011) [arXiv:1003.5636 [hep-ph]].
  * (84) D. Wands et al., Phys. Rev. D 67, 023503 (2003) [astro-ph/0208055v3].
  * (85) D. Baumann, D. Green, B. Wallisch, JCAP 08, 029 (2018) [arXiv:1712.08067 [astro-ph.CO]].
  * (86) S. Antusch, J. Kersten, M. Lindner, M. Ratz and M. A. Schmidt, JHEP 0503, 024 (2005) [hep-ph/0501272].
  * (87) S. Antusch, J. Kersten, M. Lindner, and M. Ratz, Nucl. Phys. B674 (2003), 401 [hep-ph/0305273].
  * (88) J. Schechter and J. W. F. Valle, Phys. Rev. D 22, 2227 (1980).
  * (89) R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 2558 (1975).
  * (90) G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181, 287 (1981).
  * (91) T. Hambye and G. Senjanovic, Phys. Lett. B 582, 73 (2004) [hep-ph/0307237].
  * (92) M. Fukugita and T. Yanagida, Berlin, Germany: Springer (2003) 593 p
  * (93) M. Hirsch and S. F. King, Phys. Rev. D 64, 113005 (2001) [hep-ph/0107014].
  * (94) S. Davidson, E. Nardi and Y. Nir, Phys. Rept. 466, 105 (2008) [arXiv:0802.2962 [hep-ph]].
  * (95) E. W. Kolb and M. S. Turner, Ann. Rev. Nucl. Part. Sci. 33, 645-696 (1983)
  * (96) H. B. Nielsen and Y. Takanishi, Phys. Lett. B 507, 241 (2001) [hep-ph/0101307].
  * (97) R. H. Cyburt, B. D. Fields, K. A. Olive, Tsung-Han Yeh, Rev. Mod. Phys. 88, 015004 (2016) [arXiv:1505.01076v1 [astro-ph.CO]].
  * (98) P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola and J. W. F. Valle, Phys. Lett. B 782, 633 (2018) [arXiv:1708.01186 [hep-ph]].
  * (99) S. Davidson and A. Ibarra, Phys. Lett. B 535, 25-32 (2002) [arXiv:hep-ph/0202239 [hep-ph]].
  * (100) G. Ballesteros, J. Redondo, A. Ringwald and C. Tamarit, Phys. Rev. Lett. 118, no.7, 071802 (2017) [arXiv:1608.05414 [hep-ph]].
  * (101) G. Ballesteros, J. Redondo, A. Ringwald and C. Tamarit, JCAP 08, 001 (2017) [arXiv:1610.01639 [hep-ph]].
  * (102) M. Motaharfar, V. Kamali and R. O. Ramos, Phys. Rev. D 99, no.6, 063513 (2019) [arXiv:1810.02816 [astro-ph.CO]].
  * (103) S. Das, Phys. Rev. D 99, no.6, 063514 (2019) [arXiv:1810.05038 [hep-th]].
  * (104) J. Gluza and M. Zralek, Phys. Rev. D 45, 1693 (1992).
  * (105) P. B. Pal, Am. J. Phys. 79, 485 (2011) [arXiv:1006.1718 [hep-ph]].
  * (106) R. E. Cutkosky, J. Math. Phys. 1, 429 (1960).
  * (107) R. L. Kobes, Can. J. Phys. 64, 602 (1986).
  * (108) M. Laine and A. Vuorinen, Lect. Notes Phys. 925, pp.1 (2016) [arXiv:1701.01554 [hep-ph]].
  * (109) C. P. Kiessig, M. Plumacher and M. H. Thoma, Phys. Rev. D 82, 036007 (2010) [arXiv:1003.3016 [hep-ph]].
  * (110) F. del Aguila and J. A. Aguilar-Saavedra, JHEP 0505, 026 (2005) [hep-ph/0503026].
  * (111) M. L. Bellac, “Thermal Field Theory,”, Cambridge University Press, 1996, Cambridge (UK)
  * (112) H. A. Weldon, Phys. Rev. D 26, 2789 (1982).
  * (113) S. R. Coleman and E. J. Weinberg, Phys. Rev. D 7, 1888 (1973).