# Flavor Leptogenesis During Reheating Era

Arghyajit Datta<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Guwahati, Assam 781039, India  Rishav Roshan
<EMAIL_ADDRESS>Department of Physics, Kyungpook National University,
Daegu 41566, Korea  Arunansu Sil<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Guwahati, Assam 781039, India

###### Abstract

Recently, it has been shown that the presence of a non-instantaneous era of
reheating can significantly alter the charged lepton(s) equilibration
temperature(s) which plays important role in flavor leptogenesis. In this
work, we extend the analysis to a more general situation where RHNs are also
produced from the decay of the inflaton. The presence of these RHNs along with
the thermally generated ones (above its mass equivalent temperature only)
redistributes different components of the energy density of the Universe
during this reheating era, thereby affecting the charged lepton equilibration
temperature (in addition to the Hubble effect) as well as the final reheating
temperature $T_{\rm{RH}}$. Taking both the effects into account, we find that
the decay of the lightest RHN in the set-up not only provides a platform to
study flavor leptogenesis during reheating, but also an interesting framework
of $quasi$-thermal leptogenesis emerges.

##  I Introduction

The existence of non-zero neutrino masses [1, 2, 3], and the origin of baryon
asymmetry [4] of the Universe (BAU) are two of the major issues that the
Standard Model (SM) of particle physics fails to accommodate, indicating the
necessity for the physics beyond the SM. While the issue of light neutrino
mass can be elegantly handled by the introduction of three heavy SM singlet
right-handed neutrinos (RHN) having Yukawa interaction with the SM Higgs and
lepton doublets within the so called ‘seesaw’ mechanism [5, 6, 7, 8, 9, 10,
11, 12], the same offers an interesting explanation of the BAU through
leptogenesis [13, 14, 15]. Here, a lepton asymmetry is created as a result of
the CP-violating out-of-equilibrium decay of the RHNs which then be partially
converted into the baryon asymmetry through the $(B+L)$ violating sphaleron
interactions of the SM at electroweak temperature $T_{\rm EW}\gtrsim$ 100 GeV.

In most widely studied framework of ‘$thermal$’ leptogenesis, it is considered
that after the Universe enters in the radiation dominated era (the beginning
of which is marked by reheating temperature $T_{\text{RH}}$), the RHNs can be
created by thermal scattering more specifically via inverse decays and 2-2
scattering mediated by SM fields. Subsequently, considering a hierarchical RHN
masses, the lightest among the three RHNs (say $N_{1}$ with mass $M_{1}$)
starts to contribute to lepton asymmetry production via its out-of-equilibrium
decay to the SM lepton ($l_{L_{\alpha}}$) and Higgs ($H$) doublets around a
temperature $T\lesssim M_{1}$. The reheating temperature obviously satisfies
the condition $T_{\text{RH}}>M_{1}$. The abundance of the RHNs ($Y_{N_{1}}$)
and the produced lepton asymmetry in a specific flavor direction
($\Delta_{L_{\alpha}}$) in this radiation dominated epoch are connected by the
Boltzmann equations (BE) where apart from production, one needs to incorporate
all the lepton-number violating processes that can potentially erase such
asymmetry.

Provided that such decay happens at sufficiently high temperature (at or above
$5\times 10^{11}$ GeV), where all the right-handed charged leptons of
different flavors are in out of equilibrium, one can safely use the unflavored
approximation [13, 14, 15]. This is because the rate of charged lepton Yukawa
interactions remain weaker compared to that of RHN Yukawa interactions. On the
other hand, if the leptogenesis happens at a temperature ($T$) below $5\times
10^{11}$ GeV, the right-handed tau leptons equilibrate with the thermal bath
and flavor effects [16, 17, 18, 19, 20, 21, 22, 23, 24] are inevitable. Once
this tau lepton Yukawa interaction is equilibrated, it tends to destroy the
lepton asymmetry carried by the tau leptons generated from the decay of RHNs.
For the muon and electron Yukawa interactions, this happens below $10^{9}$ GeV
and $5\times 10^{4}$ GeV respectively.

There exists a lower limit on the mass of the RHNs as $M_{1}\gtrsim 10^{9}$
GeV (known as Davidson-Ibarra bound [25]) in order to satisfy the correct
baryon asymmetry of the Universe via leptogenesis which in turn indicates that
reheating temperature should be higher than this value for standard thermal
leptogenesis. Although it is feasible to have such a high $T_{\text{RH}}$,
there is no such concrete evidence in support of it too. On the contrary, it
can be as low as few MeV [26, 27, 28, 29]. In this context, it is interesting
to investigate the possibility of having reheating temperature smaller than
the mass of the RHNs in view of leptogenesis. While one such possibility is to
have non-thermal leptogenesis [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54], a different possibility
opens up where a non-instantaneous reheating period (extended from
$T_{\text{Max}}$ to $T_{\text{RH}}$) can be brought into the picture. It is
known that reheating can actually be a non-instantaneous process [55, 26, 26,
56, 57, 58, 59] where a maximum temperature $T_{\text{Max}}$ after inflation
can be realized followed by the onset of radiation dominated era indicated by
reheating temperature $T_{\text{RH}}$ with $T_{\text{Max}}>T_{\text{RH}}$. In
a recent study [60], we have shown that leptogenesis remains a viable option
even if the decaying RHN mass (lightest and hence responsible for lepton
asymmetry generation) satisfies $T_{\text{Max}}>M_{1}>T_{\text{RH}}$.

Additionally in [60], we have an important observation (for the first time)
that a prolonged reheating period modifies the equilibration temperature (ET)
of individual charged lepton Yukawa interactions and hence the study of
leptogenesis, in particular the flavor leptogenesis during this extended
reheating period becomes very rich. Here, the effective coupling of the
inflaton with the SM fermion fields provides the sole contribution to the
radiation component of the Universe defining the temperature of the thermal
bath. This, in turn, plays a non-trivial role in controlling the expansion
rate of the Universe during the course of reheating. A not-so-small choice of
the effective coupling leads to a faster expansion of the Universe in the
reheating period which carries the potential to delaying the era of
equilibration of different charged lepton Yukawa interactions. Once the RHNs
are thermally produced from the bath, and subsequently decay out of
equilibrium (beyond $T\lesssim M_{1}$) in the period of this extended
reheating, it is found that the delayed equilibration of the charged lepton
Yukawa interaction can significantly shift the flavor regimes of the
leptogenesis.

In this work, we focus on a more general picture allowing an additional
interaction between the inflaton and the RHN fields on top of the existing
effective coupling between the inflaton and SM fermion fields, designated by
$y_{\phi ff}$ coupling, as in [60]. Hence, apart from thermal scattering, RHNs
($N_{1}$ here for simplicity) may also be produced directly from the decay of
the inflaton field. This new source of RHN production can in principle alters,
depending on the relative strength of the RHN-inflaton interaction (indicated
by $y_{\phi NN}$), the individual components (such as for inflaton, RHNs and
radiation) of energy density of the Universe during the extended reheating
period. Then, a subsequent effect on the Hubble expansion not only affects the
final reheating temperature but also modifies the ET of individual charged
lepton flavor with respect to the one observed in [60].

Furthermore, for the lightest RHN mass lies in between $T_{\text{Max}}$ and
$T_{\text{RH}}$, we encounter an interesting situation for leptogenesis here.
We have already found that a modified $thermal$ (flavored) leptogenesis
results in this extended period of reheating due to the absence of radiation
domination as well as shift in the charged lepton ET. This observation can be
visualized as a limiting case of vanishing inflaton-RHN coupling of the
present proposal. In this general set-up, once this inflaton-RHN coupling
$y_{\phi NN}$ is switched on, injection of the non-thermal RHNs into the
system on top of the thermally generated ones is expected to enhance the
outcome of leptogenesis. However, with not-so-large inflaton-RHN coupling, the
result remains essentially close to thermal leptogenesis in extended reheating
scenario. However, once $y_{\phi NN}$ becomes significant enough, say
comparable to or larger than $y_{\phi ff}$, the RHNs produced from inflaton
decay along with thermally generated ones could stay out-of-equilibrium during
this reheating period itself. As a result, these RHNs may effectively decay
above the temperature $T\sim M_{1}$ and start to produce lepton asymmetry at
$T>M_{1}$. Therefore, for such moderate range of $y_{\phi NN}$ coupling, we
realize a situation which is intermediate between purely thermal and non-
thermal leptogenesis scenario, which we name as quasi-thermal leptogenesis.
Note that it does not indicate a completely new direction for leptogenesis,
rather it corresponds to a more general and detailed interplay of thermal and
non-thermal contributions of RHN production toward leptogenesis during an
extended reheating period inclusive of flavor effect (that affects differently
to thermal and non-thermal contributions) in comparison to earlier studies
[61, 62, 63]. It is found that for sufficiently large $y_{\phi NN}$, the
presence of accumulated number density of RHNs helps relaxing the lower limit
of the RHN mass $M_{1}$ to some extent for which adequate lepton asymmetry can
be produced.

The paper is organized as follows. Below in section II, we provide a brief
overview of the standard $thermal$ leptogenesis and importance of flavors
while in section III, we discuss our general set-up of quasi-thermal
leptogenesis. We devote section IV to discuss the outcome of the proposal.
Finally in section V, we conclude.

##  II Thermal Leptogenesis and effect of flavor

A mere extension of the SM by three right handed singlet neutrinos
($N_{i=1,2,3}$) as suggested by the type-I seesaw forms the basic set-up to
discuss leptogenesis, the Lagrangian of which (in the charged lepton diagonal
basis) is given by

$\displaystyle-\mathcal{L}_{T_{I}}=\overline{\ell}_{L_{\alpha}}(Y_{\nu})_{\alpha
i}\tilde{H}N_{i}+\frac{1}{2}\overline{N_{i}^{c}}(M_{R})_{ii}N_{i}+h.c.,$ (1)

where the lepton number violating Majorana mass term for RHNs, $M_{R}$, is
considered to be diagonal, $M_{R}={\rm{diag}}(M_{1},M_{2},M_{3})$ for
simplicity. The neutrino Yukawa coupling $Y_{\nu}$ matrix in general contains
CP violating phases. A Dirac mass term $m_{D}=Y_{\nu}v/\sqrt{2}$ is generated
after spontaneous breaking of electroweak symmetry with $v$ = 246 GeV. In the
see-saw limit $m_{D}\ll M_{R}$, a light neutrino mass matrix

$\displaystyle m_{\nu}=-m_{D}M_{R}^{-1}m_{D}^{T},$ (2)

results along with three heavy neutrinos. A further diagonalization of
$m_{\nu}$ by the PMNS matrix $U$ [64, 65, 66] via $U^{\dagger}m_{\nu}U^{*}$=
diag ($m_{1},m_{2},m_{3}$), leads to three light neutrino masses
$m_{i=1,2,3}$.

The same seesaw Lagrangian also provides a natural explanation of the matter
anti-matter asymmetry of the Universe via leptogenesis [13, 14, 15] where the
CP-violating decays of heavy RHNs into SM lepton and Higgs doublets,
$N_{i}\to\ell_{L_{\alpha}}+H$, are instrumental. In the early Universe,
provided the temperature (after inflation) was high enough, these heavy RHNs
can be produced from thermal bath via inverse decay (mediated by the same
neutrino Yukawa interaction) and attain thermal equilibrium. Thereafter, as
the temperature drops below the individual mass of a RHN ($i.e.~{}T<M_{i}$),
the decay of the respective heavy field $N_{i}$ becomes relevant for
generating a CP-asymmetry along a particular lepton flavor ($\alpha$)
direction parametrized by,

$\displaystyle\varepsilon^{(i)}_{\ell_{\alpha}}=\frac{\Gamma(N_{i}\to\ell_{L_{\alpha}}+H)-\Gamma(N_{i}\to\overline{\ell}_{L_{\alpha}}+\overline{H})}{\sum_{\alpha}\Gamma(N_{i}\to\ell_{L_{\alpha}}+H)+\Gamma(N_{i}\to\overline{\ell}_{L_{\alpha}}+\overline{H})},$
(3)

where the denominator corresponds to the total decay rate of $N_{i}$ at tree
level, given by:

$\displaystyle\Gamma_{N_{i}}$
$\displaystyle=\sum_{\alpha}\Gamma(N_{i}\to\ell_{L_{\alpha}}+H)+\Gamma(N_{i}\to\overline{\ell}_{L_{\alpha}}+\overline{H})$
$\displaystyle=\frac{(Y_{\nu}^{\dagger}Y_{\nu})_{ii}}{8\pi}M_{i}.$ (4)

The out-of-equilibrium condition necessary for lepton asymmetry production is
satisfied when the decay rate of $N_{i}$ remains smaller than the expansion
rate of the Universe.

###  II.1 Unflavored estimate

A non-zero $\varepsilon^{(i)}_{\ell_{\alpha}}$ would follow due to the
interference between the tree level and loop-level decay amplitudes. With a
hierarchical RHN masses as $M_{1}\ll M_{2}\ll M_{3}$, the CP-asymmetries
generated by $N_{2}$ and $N_{3}$ are however expected to be washed out by the
lepton number violating interactions of $N_{1}$, leaving
$\varepsilon^{(1)}_{\ell_{\alpha}}$ ($\equiv\varepsilon_{\ell_{\alpha}}$ where
we omit the generation index (1) henceforth) as the only relevant one for
leptogenesis, given by

$\displaystyle\varepsilon_{\ell_{\alpha}}=\frac{1}{8\pi(Y_{\nu}^{\dagger}Y_{\nu})_{11}}\sum_{j\neq
1}\Bigg{\\{}\rm{Im}\left[(Y_{\nu}^{*})_{\alpha 1}(Y_{\nu})_{\alpha
j}(Y_{\nu}^{\dagger}Y_{\nu})_{1j}\right]\mathbf{F}\left(\frac{M_{j}^{2}}{M_{1}^{2}}\right)+\rm{Im}\left[(Y_{\nu}^{*})_{\alpha
1}(Y_{\nu})_{\alpha
j}(Y_{\nu}^{\dagger}Y_{\nu})_{j1}\right]\mathbf{G}\left(\frac{M_{j}^{2}}{M_{1}^{2}}\right)\Bigg{\\}}.$
(5)

Here
$\mathbb{F}(x)=\sqrt{x}\left[1+\frac{1}{1-x}+(1+x)\ln\left(\frac{x}{1+x}\right)\right]$
and $\mathbb{G}(x)=1/(1-x)$ are the loop functions originated from both vertex
and self-energy corrections to the decay of $N_{1}$. Conventionally the total
CP asymmetry is calculated after having the flavor sum as
$\varepsilon_{\ell}=\sum_{\alpha}\varepsilon_{\ell_{\alpha}}$ which results a
vanishing contribution to the second term of the r.h.s of Eq. (5). However,
both the terms remain important for flavored leptogenesis.

It is shown in [25] that a maximum CP-asymmetry, generated from the $N_{1}$
decay, can be extracted from Eq. (5) as,

$\displaystyle|\varepsilon_{\ell}|\lesssim\frac{3}{8\pi}\frac{M_{1}}{v^{2}}(m_{3}-m_{1})=\varepsilon_{\ell}^{\text{Max}}.$
(6)

Such an asymmetry however be washed out partially due to the lepton number
violating interactions so as to write down the final $B-L$ asymmetry including
the efficiency factor $\kappa_{f}$ by

$\displaystyle Y_{B-L}\equiv
n_{B-L}/s=-\frac{1}{7.04}\frac{3}{4}\varepsilon_{\ell}\kappa_{f}.$ (7)

where, $Y_{x}=n_{x}/s$ represents the number density to entropy density ratio
for $x$-species. Hence, in case the $N_{1}$ responsible for the asymmetry
generation was in thermal equilibrium at the early Universe, the limit on CP-
asymmetry as in Eq. (6) leads to an estimate of maximal baryon asymmetry
$Y_{B}^{\text{Max}}$. Requirement of $Y_{B}^{\text{Max}}\geq
Y_{B}^{\rm{exp}}=8.718\times 10^{-11}$ [67, 4] eventually provides a lower
limit on lightest RHN mass [25, 61] as:

$\displaystyle M_{1}$ $\displaystyle\gtrsim\frac{7.04}{0.96\times
10^{-2}}\frac{8\pi v^{2}}{3m_{3}}\frac{Y_{B}^{\rm{exp}}}{\kappa_{f}}$
$\displaystyle\approx\frac{6\times
10^{8}~{}\rm{GeV}}{\kappa_{f}}~{}\left(\frac{Y_{B}^{\rm{exp}}}{8.718\times
10^{-11}}\right)~{}\left(\frac{0.05~{}\rm{eV}}{m_{3}}\right),$ (8)

where light neutrino masses are considered to be hierarchical and consistent
with neutrino oscillation data [65].

An accurate estimate for the final asymmetry or in other words the efficiency
factor $\kappa_{f}$ would however follow if one solves coupled BEs that
correlate the abundance of the lightest RHN with the lepton number asymmetry
produced. Note that in this case, the reheating temperature (considering
instantaneous reheating) $T_{\text{RH}}$ should be more than $M_{1}$
indicating $T_{\text{RH}}\gtrsim 10^{10}$ GeV or so.

###  II.2 Flavored Regime and Charged Lepton Equilibration

In evaluating the final lepton asymmetry above, a flavor sum is performed.
However, it has been found that the situation can actually be more complicated
as soon as charged lepton Yukawa interactions
($Y_{\alpha}\overline{\ell}_{L_{\alpha}}He_{R_{\alpha}}$ with $e_{R\alpha}$ as
representative of right handed electron/muon/tau) become faster compared to
$N_{1}-\ell_{L}H$ interaction [21]. In that case, during the out-of-
equilibrium decay process of the $N_{1}$, the charged lepton Yukawa
interaction for one or more flavor(s) may enter equilibrium leading to the
breaking of quantum coherence of the lepton doublet state along different
flavor directions produced from the $N_{1}$ decay [16, 18, 19, 22, 23]. As a
result, lepton asymmetry along individual flavors may start to become
distinguishable. In this case, one needs to look for the evolution of the
individual flavor lepton asymmetries instead of total lepton number asymmetry
by constructing BEs for lepton asymmetries along individual flavors.

####  II.2.1 Evaluation of Equilibration Temperature

In order to check whether the charged lepton Yukawa interaction of a
particular flavor $\alpha$ is fast enough at a given temperature to be in
thermal equilibrium, the associated interaction rate ($\Gamma_{\alpha}$) has
to be more than the expansion rate of the Universe [18]. Since, this charged
lepton equilibrium temperature (ET) plays a decisive role in determining the
flavor effect, we elaborate on it in case of thermal leptogenesis here. In the
standard scenario, assuming all these phenomena are occurring in a radiation
dominated Universe, the thermally averaged interaction rates of the SM Higgs
doublet decaying to left-handed lepton doublets and right-handed charged
lepton singlets (more specifically
$H\leftrightarrow\ell_{L_{\alpha}}e_{R_{\alpha}}$[68]) can be estimated as
[68, 69, 70]:

$\displaystyle\langle\Gamma_{\alpha}\rangle=\int\frac{d^{3}p}{(2\pi)^{3}2E_{P}}\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}2E_{k^{\prime}}}(2\pi)^{4}\delta^{(4)}(p-k-k^{\prime})|\mathcal{M}|^{2}\frac{f_{p}}{n_{p}},$
(9)

where $p$ is the $4$-momentum of the Higgs while $k$ and $k^{\prime}$ are the
$4$-momentum of lepton doublet and singlet right handed charged lepton
respectively. The thermal distribution of Higgs $f_{p}$ and number density
$n_{p}$ are taken as:

$\displaystyle
f_{p}=\frac{1}{e^{E_{p}/T}-1}\,,~{}n_{p}=\frac{\zeta(3)T^{3}}{\pi^{2}}\,.$
(10)

The matrix amplitude squared $|\mathcal{M}|^{2}$ for such decay would be
(assuming final state particles have negligible mass):

$\displaystyle|\mathcal{M}|^{2}=2Y_{\alpha}^{2}k.k^{\prime}=Y_{\alpha}^{2}M_{H}^{2}\,,\quad\alpha=e,\mu,\tau.$
(11)

Evaluation of the integrals in Eq. (9) for $T\gg M_{H}$ yields [68]:

$\displaystyle\langle\Gamma_{\alpha}\rangle=\frac{Y_{\alpha}^{2}\pi}{192\zeta(3)T}M_{H}^{2}.$
(12)

Considering the thermal mass of the Higgs to be[71, 72, 73]:

$\displaystyle
M_{H}=M_{H}(T)\simeq\frac{T}{4}\sqrt{3g^{2}+g^{\prime^{2}}+4y_{t}^{2}+8\lambda},$
(13)

where $g,g^{\prime}$ are SM gauge coupling constants and $y_{t}$, $\lambda$
are the top Yukawa and the Higgs quartic couplings respectively, the thermally
averaged interaction rate for the decay processes will become [20]
$\mathcal{O}(5\times 10^{-3})Y_{\alpha}^{2}T$.

However, thermal corrections for the final state particles can also be
important as they are [71]:
$m_{\ell_{L}}(T)=\frac{1}{4}\sqrt{\left(3g^{2}+g^{\prime 2}\right)}T$ and
$m_{e_{R}}(T)=\frac{1}{2}g^{\prime}T$ for $\ell_{L}$ and $e_{R}$ respectively.
Though the hierarchy $M_{H}(T)>m_{\ell_{L}}(T)>m_{e_{R}}(T)$ is always
maintained for $T>T_{EW}$, a situation can be achieved at some high
temperature where this Higgs decay channel may actually be closed with
$M_{H}(T)$ being smaller than $m_{\ell_{L}}(T)+m_{e_{R}}(T)$. This happens due
to the decrease of top Yukawa coupling $y_{t}$ with the increase in
temperature [74].

Figure 1: Variation of $\langle\Gamma_{\alpha}\rangle/H$ with respect to
($w.r.t.$) $T$ in radiation dominated phase. Horizontal dashed line denotes
$\langle\Gamma_{\alpha}\rangle/H=1$, while the vertical dashed lines indicate
the ETs of three charged Yukawa interactions.

Note that apart from the 1$\rightarrow$ 2 decay, the 2$\leftrightarrow$2
scatterings involving the specific Yukawa interaction $Y_{\alpha}$ (such as
$XH^{\dagger}\rightarrow\bar{\ell}_{L}e_{R},\ell_{L}H^{\dagger}\rightarrow
Xe_{R}$ $etc.$ where $X=B,W$ gauge bosons) are also important at high
temperature as their contributions to the interaction rate falls in the range:
$(5.19-4.83)\times 10^{-3}~{}Y^{2}_{\alpha}T$ [75, 76] which is found to be in
the same ballpark of the naive decay estimate above. Hence, for the purpose of
our study, we consider the interaction rate $\langle\Gamma_{\alpha}\rangle$
associated to the charged lepton Yukawa $Y_{\alpha}$ to be

$\displaystyle\langle\Gamma_{\alpha}\rangle\simeq 5\times
10^{-3}Y_{\alpha}^{2}T.$ (14)

Comparison between the obtained interaction rates
$\langle\Gamma_{\alpha}\rangle$ with Hubble constant
($\mathcal{H}=1.66g_{*}^{1/2}T^{2}/M_{pl}$ in radiation dominated Universe,
where $M_{pl}$ is the Planck mass.) will lead to the ET of right-handed
charged lepton singlets. Figure 1 shows the variation of
$\langle\Gamma_{\alpha}\rangle/{\mathcal{H}}$ with respect to temperature $T$
for different lepton flavors: $\alpha=e,\mu,\tau$. Note that $\tau$ Yukawa
interaction becomes fast enough around $T=T^{*}_{0(\tau)}\simeq 5\times
10^{11}$ GeV (evident from the intersection of
$\langle\Gamma_{\tau}\rangle/{\mathcal{H}}$ line in blue with 1) while muon
Yukawa interaction comes to equilibrium at $T^{*}_{0(\mu)}\simeq 10^{9}$ GeV
as seen from $\langle\Gamma_{\mu}\rangle/{\mathcal{H}}=1$ point of Fig. 1.

####  II.2.2 Effects of Flavor and Boltzmann Equations

As a result, if a $N_{1}$ decays while staying in out-of-equilibrium in a
temperature range $10^{9}\lesssim T\lesssim 5\times 10^{11}$ GeV, lepton
asymmetry becomes distinguishable along two orthogonal directions denoted by
$\alpha=a$ (specifying a coherent superposition of $e$ and $\mu$ lepton
flavors) and $\tau$. Hence contrary to the unflavored case, here we need to
study the evolution of $B/3-L_{\alpha}\equiv\Delta_{\alpha}$ charges with
$\alpha=a$ and $\tau$. With a further reduction of the temperature below
$T\lesssim 10^{9}$ GeV, the lepton doublets completely loose their quantum
coherence. Hence, at this stage, lepton asymmetry becomes distinguishable
along all three flavor directions $e$, $\mu$ and $\tau$. As a result,
evolution of lepton asymmetry along all three directions becomes relevant to
produce final baryon asymmetry.

Incorporating the flavor effects into account, the evolution of the abundance
of decaying $N_{1}$s ($Y_{N_{1}}$) and lepton asymmetries along individual
flavor direction ($Y_{\Delta_{\alpha=e,\mu,\tau}}$) can be represented by the
following sets of flavored classical BEs (neglecting $\Delta L=1,2$ scattering
processes and subtracting the on-shell contribution from $\Delta L=2$ process)
[17, 18]:

$\displaystyle sHz\frac{dY_{N_{1}}}{dz}$
$\displaystyle=-\left(\frac{Y_{N_{1}}}{Y_{N_{1}}^{\rm
eq}}-1\right)\gamma_{D}\,,$ (15) $\displaystyle
sHz\frac{dY_{\Delta_{\alpha}}}{dz}$
$\displaystyle=-\Bigg{\\{}\left(\frac{Y_{N_{1}}}{Y_{N_{1}}^{\rm
eq}}-1\right)\varepsilon_{\ell\alpha}\gamma_{D}+K^{0}_{\alpha}\sum_{\beta}\Bigg{[}\frac{1}{2}(C^{\ell}_{\alpha\beta}+C^{H}_{\beta})\gamma_{D}\Bigg{]}\frac{Y_{\Delta_{\beta}}}{Y_{\ell}^{\rm
eq}}\Bigg{\\}},$ (16)

where $K^{0}_{\alpha}=\frac{(Y_{\nu}^{*})_{\alpha 1}(Y_{\nu})_{\alpha
1}}{(Y_{\nu}^{\dagger}Y_{\nu})_{11}}$ is known as flavor projector [18, 22]
and $C^{\ell},C^{H}$ matrices connect the asymmetries in lepton and Higgs
sectors to asymmetries in $\Delta_{\alpha}$ expressed in terms of
$Y_{\Delta{\alpha=e,\mu,\tau}}$ or $Y_{\Delta{\alpha=a,\tau}}$ depending on
the leptogenesis scale [18]. Eq. (16) can be a set of two (three) equations if
$10^{9}<M_{1}<5\times 10^{11}$ GeV ($M_{1}<10^{9}$ GeV). Here $\gamma_{D}$
represents the total decay rate density of $N_{1}$:

$\displaystyle\gamma_{D}=n_{N_{1}}^{\rm{eq}}\langle\Gamma_{N_{1}}\rangle,\qquad\langle\Gamma_{N_{1}}\rangle=\frac{K_{1}(z)}{K_{2}(z)}\frac{(Y_{\nu}^{\dagger}Y_{\nu})_{11}}{8\pi}M_{1}\,,$
(17)

where $K_{1}(z)$ and $K_{2}(z)$ are the modified Bessels functions.
$z=M_{1}/T$ is a dimensionless quantity, with respect to which we will look
for the evolution of $Y_{x}$. The equilibrium number density of the $N_{1}$
can be expressed as:

$\displaystyle
n_{N_{1}}^{\rm{eq}}=\frac{gM_{1}^{3}}{2\pi^{2}z}K_{2}\left(z\right),$ (18)

where $g$ is the number of degrees of freedom of $N_{1}$.

To study the evolutions numerically, we need to provide inputs for the
neutrino Yukawa coupling $Y_{\nu}$ matrix the structure of which can be
extracted using Casas-Ibarra (CI) parametrization [77] as:

$\displaystyle
Y_{\nu}=-i\frac{\sqrt{2}}{v}UD_{\sqrt{m}}\mathbf{R}D_{\sqrt{M}}\,,$ (19)

where $U$ is the Pontecorvo-Maki-Nakagawa-Sakat (PMNS) matrix [66] which
connects the flavor basis with mass basis for light neutrinos.
$D_{\sqrt{m}}=\rm{diag}(\sqrt{m_{1}},\sqrt{m_{2}},\sqrt{m_{3}})$ is the
diagonal matrix containing the square root of light neutrino mass and
similarly $D_{\sqrt{M}}=\rm{diag}(\sqrt{M_{1}},\sqrt{M_{2}},\sqrt{M_{3}})$
represents the diagonal matrix for RHN masses. $\mathbf{R}$ is an orthogonal
matrix satisfying $\mathbf{R}^{\rm{T}}\mathbf{R}=1$.

Figure 2:  Evolution of various comoving number densities with respect to $z$
for $M_{1}=5\times 10^{9}$ GeV. Horizontal black dashed line indicates the
observerd baryon asymmetry $Y_{B}^{\text{exp}}$.

Fig. 2 depicts a scenario where evolution of $\tau$ and $a$ lepton flavor
become essential as the mass of the lightest RHN is chosen to be
$M_{1}=5\times 10^{9}$ GeV. While constructing the $Y_{\nu}$ for this case, we
consider a typical hierarchy among the RHNs as
$M_{3}=100M_{2},~{}M_{2}=100M_{1}$ along with we use the best fit values of
solar and atmospheric mass-squared differences (considering $m_{1}=0$) and
mixing angles, and CP-violating phase $\delta$ to define $U$ via Eq. (19). For
such a set of hierarchical RHNs, $\mathbf{R}$ can be considered to have the
following structure:

$\displaystyle\mathbf{R}=\begin{pmatrix}0&\cos\theta_{R}&\sin\theta_{R}\\\
0&-\sin\theta_{R}&\cos\theta_{R}\\\ 1&0&0\end{pmatrix},$ (20)

Where $\theta_{R}$ can be, in general, a complex angle, chosen to be
$\theta_{R}=2.83+0.24i$ for this case so as to realize the final asymmetry to
be consistent with the correct baryon asymmetry of the Universe. As can be
seen from the Fig. 2, starting from a symmetric Universe, lepton asymmetries
along $a$ and $\tau$ direction (blue dot and red dashed lines) start to grow
due to out-of-equilibrium decay of lightest RHN (the corresponding abundance
of $N_{1}$ is indicated by solid blue line) and saturates around $z\sim 2$.
Similar behavior can be seen for the number density of baryon asymmetry
(magenta line) as well. Because of the sphaleron processes, produced lepton
asymmetries get converted to baryon asymmetry via the relation:
$Y_{B}=28/79\sum_{\alpha}Y_{\Delta_{\alpha}}$. Eventually with large value of
$z$, this baryon asymmetry saturates to experimentally observed baryon
asymmetry: $Y_{B}^{\rm{exp}}=(8.718\pm 0.012)\times 10^{-11}$ [67, 4]
(dictated by black dashed lines of Fig. 2). The discussion in this section
embarks on the importance of flavor aspects in thermal leptogenesis which is
now being explored in the context of non-instantaneous reheating period in
this work. Before that, we discuss the non-thermal leptogenesis in brief.

##  III RHNs Produced from Inflaton Decay

We may now turn our attention to a situation where the RHNs are being produced
from inflaton ($\phi$) decay. In case the inflaton decays solely to $N_{1}$s
having decay width $\Gamma_{\phi}$, the radiation component of the Universe
arises as a result of further decay of those RHNs (with decay width
$\Gamma_{N_{1}}$) into SM lepton and Higgs doublets which thermalize rapidly.
With $\Gamma_{\phi}<<\Gamma_{N_{1}}$, the reheating temperature assuming
instantaneous reheating111Here contribution from preheating [78, 79, 80] is
not taken into account. is governed by the condition:
$\Gamma_{\phi}=\mathcal{H}$ given by:

$\displaystyle
T_{\text{RH}}=\left(\frac{45}{4\pi^{3}g_{*}}\right)^{1/4}\sqrt{\Gamma_{\phi}M_{pl}}\,.$
(21)

where $g_{*}$ represents the effective number of relativistic degrees of
freedom in the SM. In this case, as $T_{\text{RH}}<M_{1}$, the $N_{1}$s would
decay immediately after being produced from inflaton.

The decay of such non-thermally produced $N_{1}$ also generates a lepton
asymmetry which can be converted to baryon asymmetry (provided
$T_{\text{RH}}>100$ GeV), expressed as [30, 34, 43]

$\displaystyle
Y_{B}=-\frac{28}{79}\frac{n_{B-L}}{s}=-\frac{28}{79}\varepsilon_{\ell}\frac{n_{N_{1}}}{s}=-\frac{28}{79}\frac{3}{4}\varepsilon_{\ell}\frac{T_{\text{RH}}}{m_{\phi}}\,.$
(22)

Here we assume $\phi$ decays to $N_{1}$ only. Also, washout by the lepton
number violating process turns out to be insignificant. The additional
suppression factor, proportional to $T_{\text{RH}}/m_{\phi}$ is related to the
ratio of the number density of produced $N_{1}$ and entropy $s$ and follows
from the equality $\rho_{\phi}=\rho_{R}$ at $T=T_{\text{RH}}$. Note that, due
to the condition $M_{1}>T_{\text{RH}}$, the inverse decay process
$\ell+H\rightarrow N_{1}$ can not take place and consequently, the loss of
flavor coherence is not expected to occur. However, the situation alters once
we consider an extended period of reheating as we plan to discuss next.

##  IV flavor effect during reheating period

As previously discussed, a thermal leptogenesis scenario with $N_{1}$ mass
$M_{1}\lesssim 5\times 10^{11}$ GeV experiences flavor effects since the
charged lepton Yukawa interactions start to enter equilibrium below this
temperature. An estimate of such ETs for different flavors is provided in Fig.
1. Note that the ETs of these Yukawa interactions (associated to different
flavors of right handed charged leptons) are calculated assuming that the
Universe is already in a radiation domination era followed by an instantaneous
reheating after inflation. Obviously, such consideration leads to
$T_{\text{RH}}>M_{1}$ indicating the presence of high reheating temperature.
Now, it is also known that the reheating might not be an instantaneous process
[55, 26, 58, 59]. On top of that, the reheating temperature can be low enough
(though larger than few MeV from BBN limit [26, 27, 28, 29]).

The era of reheating begins when after inflation, the inflaton field $\phi$
starts to decay. Neglecting the possibility via preheating [78, 79, 80], as
this field $\phi$ starts to decay to the lighter SM degrees of freedoms, the
thermalization of these light decay products helps the Universe to attain a
maximum temperature $T_{\text{Max}}$. Subsequently, the temperature of the
Universe falls at a rate much slower than the standard scaling $T\sim a^{-1}$
($a$ is the Friedmann–Robertson–Walker scale factor). This continues till a
point (defining $T_{\text{RH}}$) where the radiation component becomes the
dominating one in the Universe. This nontrivial behavior of the temperature as
function of scale factor $a$ results due to the faster expansion rate compared
to the standard scenario (quantified by Hubble $\mathcal{H}$) during the
lifetime of the inflaton.

As a result of this modified $\mathcal{H}$, a change in the behaviour of
$\langle\Gamma_{\alpha}\rangle/{\mathcal{H}}$ is also expected in the period
in between $T_{\text{Max}}$ and $T_{\text{RH}}$, compared to the standard
scenario represented in section II as estimated in [60] by the present
authors. This prolonged reheating was dictated by the size of the effective
coupling of the inflaton with SM fields. This would be particularly prominent
provided $T_{\text{RH}}$ falls below $T^{*}_{0(\tau)}$ and $T_{\text{Max}}$
maintains a relation $T_{\text{Max}}>T^{*}_{0(e)}$. In fact, as a result of
such a delayed entry (due to the modified $\mathcal{H}$ for
$T_{\text{Max}}>T>T_{\text{RH}}$) of the charged lepton Yukawa interactions
into the equilibrium, a shift of the flavor regime(s) of leptogenesis compared
to the standard scenario is expected in this case of extended reheating
period.

If the lepton asymmetry is generated from the out-of-equilibrium decay of the
lightest RHN, then depending on the scale of leptogenesis, three possibilities
can be realized in presence of this extended reheating: (A)
$M_{1}<T_{\text{RH}}$, (B) $T_{\text{Max}}>M_{1}>T_{\text{RH}}$ and (C)
$M_{1}>T_{\text{Max}}$. Case (A) corresponds to the standard thermal
leptogenesis as by the time decay of $M_{1}$ becomes effective for
leptogenesis, Universe is already in radiation dominated era. So, effect of
this extended period of reheating does not carry any additional impact here.
On the other hand, with case (C), $N_{1}$ can never be produced thermally.
They can only be produced non-thermally from the decay of some heavy particle
(provided that coupling exists), like inflaton resembling the case of purely
non-thermal leptogenesis as discussed in section III where the effect of
flavor does not play any vital role.

The case (B) is however the most interesting one. In [60], it was shown that
the lightest RHN can be produced from the thermal bath during
$T_{\text{Max}}>T\gtrsim M_{1}$ while it decays thereafter ($T<M_{1}$) into SM
lepton and Higgs doublets, hence generating the lepton asymmetry. Now, during
this extended period of reheating, the shift in the ET of right handed charged
leptons affects the lepton asymmetry production. For example, we have shown
[60] that with an effective coupling of inflaton to SM fermions ($y_{\phi
ff}$) of order $10^{-4}$, the ET of $\tau_{R}$ reduces by almost an order of
magnitude compared to the standard case. Such inflaton-SM fermion coupling
also sets222Note that there exists another parameter $\lambda$, involved in
the inflation potential, which also takes part in determining these
$T_{\text{Max}}$ and $T_{\text{RH}}$ values. However $\lambda$ is fixed so as
to have correct prediction for the normalization of CMB anisotropies. The
present analysis is also based on the same inflationary potential considered
in [60] as we elaborate upon soon. $T_{\text{Max}}\sim 7\times 10^{11}$ GeV
while $T_{\text{RH}}$ becomes $4\times 10^{10}$ GeV. Consequently, $N_{1}$ of
mass $M_{1}\simeq 10^{11}$ GeV can be produced thermally during reheating (as
$T_{\text{Max}}\sim 7\times 10^{11}$ GeV) while it decays around $T\lesssim
M_{1}$. The shift in the ET renders the corresponding leptogenesis as an
unflavored case which otherwise falls in the ballpark of flavored leptogenesis
(two flavor regime).

Motivated by the above result, in this work, we further consider an intriguing
extension (in the context of case (B) itself) where in addition to the
inflaton-SM fermion effective coupling ($y_{\phi ff}$) there exists a direct
coupling ($y_{\phi NN}$) between the inflaton and $N_{1}$s. Introduction of
such a coupling not only induces $N_{1}$s in the system from the decay of the
inflaton (in addition to the thermally generated $N_{1}$), but also opens up
the possibility of modifying the Hubble further, hence affecting
$T_{\text{RH}}$ as well as the ET depending on its relative coupling strength
compared to inflaton-SM fermion coupling.

Hence, with a nonzero branching of the inflaton to $N_{1}$ in addition to the
inflaton-SM fermion one, we expect to have $N_{1}$ production from inflaton
decay as well as thermal production of it via inverse decay for a temperature
range $T_{\text{Max}}>T\gtrsim M_{1}$. These $N_{1}$ however find themselves
in out of equilibrium as the Hubble $\mathcal{H}$ at this temperature regime
remains large enough ($\phi$ dominates). Therefore, $N_{1}$ would decay and
may contribute to lepton asymmetry generation even at temperature above
$M_{1}$ unless it has been washed out by inverse decay. In case $\rho_{N_{1}}$
dominates over $\rho_{R}$, the washout by inverse decay turns out to be weak
so as not to erase the asymmetry. This particular era of leptogenesis turns
out to be somewhat different from a purely thermal or non-thermal one since
the lepton asymmetry here is generated from the decay of both the thermally
produced $N_{1}$s and those produced from inflaton decays in this regime. We
call it as ‘quasi’-thermal leptogenesis. Additionally for temperature below
$T\sim M_{1}$, leptogenesis proceeds in the usual way. However, with a
significantly dominant coupling of $y_{\phi NN}$ over $y_{\phi ff}$, $N_{1}$s
can even be produced beyond $T=M_{1}$ point. In this case ($T<M_{1}$), such
non-thermally produced $N_{1}$ would instantaneously decay and contribute to
lepton asymmetry production similar to the usual non-thermal leptogenesis
scenario.

Following the above discussion, we now construct the relevant Lagrangian
(apart from the SM one) as given by,

$\displaystyle-\mathcal{L}=y_{\phi ff}\phi\overline{f}f+y_{\phi
NN}\phi\overline{N^{c}_{1}}N_{1}+V(\phi),$ (23)

in addition to the Type-I seesaw Lagrangian of Eq. (1). Here, $y_{\phi ff}$ is
only an effective coupling and $f(\overline{f})$ are the SM fermions. For
simplicity, we only keep the coupling of inflaton with the lightest $N_{1}$.
Here, $V(\phi)$ corresponds to the scalar potential of inflaton $\phi$
describing the post-inflationary epoch during which the inflaton field
oscillates about the minimum (origin), relevant for realizing reheating. We
have taken a power-law form for $V(\phi)$ about the minima [58] as:

$\displaystyle V(\phi)=\lambda\frac{\left|\phi\right|^{n}}{M_{P}^{n-4}}\,,$
(24)

where $M_{P}$ is the reduced Planck mass. The magnitude of the coupling
$\lambda$ can be estimated from the CMB observables such as spectral index and
tensor-to-scalar ratio and depends on the order of the polynomial $n$. Origin
of such choice of potential near minimum ($\phi=0$) can be traced back to
T-attractor models of inflation in no-scale supergravity [81, 82],

$\displaystyle V_{\rm{Inf}}(\phi)=\lambda
M_{P}^{4}\left[\sqrt{6}\tanh\left(\frac{\phi}{\sqrt{6}M_{P}}\right)\right]^{n},$
(25)

such that the $V(\phi)$ of Eq. (24) follows after expanding the above
$V_{\rm{Inf}}(\phi)$ about the origin ($\phi\ll M_{P}$) to the leading order.

After the end of the inflation, the inflaton starts to perform damped
oscillations about the minima of the potential and eventually decays to the SM
fermion-antifermion pair as well as to $N_{1}$ following Eq. (23). Here we
ignore any potential contribution that may come from preheating [80, 33].
Consequently, the energy density of the inflaton field $\rho_{\phi}$ satisfies
the equation:

$\displaystyle\frac{d\rho_{\phi}}{dt}+3\left(\frac{2n}{n+2}\right)\mathcal{H}\rho_{\phi}$
$\displaystyle=-(\Gamma_{\phi ff}+\Gamma_{\phi NN})\rho_{\phi}.$ (26)

Here the term proportional to $\mathcal{H}$ indicates the dilution of energy
density due to the expansion of the Universe while the term on the right hand
side of the BE represents the dilution (hence comes with a negative sign) of
the energy density of the $\phi$ as a result of its decay to $N_{1}$ and SM
fermion/anti-fermions. In the right hand side, $\Gamma_{\phi ff}$ and
$\Gamma_{\phi NN}$ represent the decay widths of inflaton to SM fermions and
$N_{1}$ respectively and are expressed as

$\displaystyle\Gamma_{\phi ff}$ $\displaystyle=\frac{y_{\phi
ff}^{2}}{8\pi}m_{\phi}\,,\quad\Gamma_{\phi NN}=\frac{y_{\phi
NN}^{2}}{8\pi}m_{\phi}\,.$ (27)

At this place, it is pertinent to discuss on the inflaton mass $m_{\phi}$. In
such setups, the effective inflaton mass $m_{\phi}$ (=
$[\partial^{2}_{\phi}V(\phi)]^{1/2}$ at minimum) of the oscillating inflaton
field $\phi(t)$ becomes a function of the inflaton field itself, since

$\displaystyle\partial^{2}_{\phi}V(\phi)=\;\lambda~{}n(n-1)\phi^{n-2}M_{P}^{4-n}.$
(28)

For $n=2$, a definite mass is associated with the inflaton and a perturbative
decay of the inflaton happens naturally via the $y_{\phi ff}$ and $y_{\phi
NN}$ interactions. We restrict ourselves with $n=2$ in this work. However, for
$n>2$, though the inflaton is apparently massless at origin, a condensate of
the oscillating inflaton seems plausible [58, 83] for which a time-dependent
effective mass of the oscillating inflaton can be associated as,

$\displaystyle
m_{\phi}^{2}=n(n-1)M_{P}^{\frac{2(4-n)}{n}}\lambda^{\frac{2}{n}}\rho_{\phi}^{\frac{n-2}{n}},$
(29)

where $\rho_{\phi}$ is the energy density of the inflaton field $\phi$. This
is obtained after taking average of the equation of motion of the $\phi(t)$
field over one complete oscillation under envelope approximation333 The
envelope approximation can be stated as $\phi(t)=\phi_{0}(t)\mathcal{A}(t)$.
Here $\phi_{0}(t)$ describes the envelope-function representative of red-shift
and decay in a longer time scale while $\mathcal{A}$ stands for oscillatory
behaviour (within the envelope) of it at short time scale such that
$\rho_{\phi}=V(\phi_{0})$ (for more details, see [84, 58, 83])., as suggested
in [84, 58].

The produced fermion-antifermion pairs would interact quickly among themselves
to produce other SM particles and rapidly thermalizes producing the radiation
energy density component $\rho_{R}$. At this stage, we can define the
temperature of the Universe via

$\displaystyle T=\left[\frac{30\rho_{R}}{\pi^{2}g_{*}}\right]^{1/4}.$ (30)

On the other hand, the $N_{1}$s produced from the inflaton decay further
decays to the SM particles which will eventually contribute to $\rho_{R}$ too.
Additionally, as per our consideration in case (B), the thermal bath can also
produce back $N_{1}$ particularly for the temperature of the Universe
$T_{\text{Max}}>T\gtrsim M_{1}$. The BEs for $\rho_{N_{1}}$ and $\rho_{R}$ can
therefore be written as,

$\displaystyle\frac{d\rho_{N_{1}}}{dt}+3\mathcal{H}\rho_{N_{1}}$
$\displaystyle=-(\rho_{N_{1}}-\rho_{N_{1}}^{eq})\langle\Gamma_{N_{1}}\rangle+\Gamma_{\phi
NN}\rho_{\phi}\,,$ (31)
$\displaystyle\frac{d\rho_{R}}{dt}+4\mathcal{H}\rho_{R}$
$\displaystyle=(\rho_{N_{1}}-\rho_{N_{1}}^{eq})\langle\Gamma_{N_{1}}\rangle+\Gamma_{\phi
ff}\rho_{\phi}.$ (32)

In all the BEs above, $\mathcal{H}$ represents the Hubble expansion rate to be
written as

$\displaystyle\mathcal{H}^{2}=\frac{\rho_{\phi}+\rho_{N_{1}}+\rho_{R}}{3M_{P}^{2}},$
(33)

since in this epoch $T_{\text{Max}}>T>T_{\text{RH}}$, the energy density of
the Universe comprises of the components $\rho_{\phi},\rho_{N_{1}}$ and
$\rho_{R}$. The $\rho_{N_{1}}^{\rm{eq}}$ is the equilibrium energy density of
$N_{1}$ as given by

$\displaystyle\rho_{N_{1}}^{\rm{eq}}=\frac{M_{1}^{4}\left[\frac{3}{z^{2}}K_{2}(z)+\frac{1}{z}K_{1}(z)\right]}{\pi^{2}},$
(34)

where $K_{1},K_{2},$ and $z$ have already been defined while explaining Eq.
(17). The presence of this term in the above BEs is related to the existence
of the inverse decay from radiation bath to produce $N_{1}$. Eq.(26),
(31)-(32) therefore together represent the most general set of equations to
study the scenario under consideration.

After discussing the $N_{1}$ production and the related BEs to study the
respective components of the energy densities of the Universe, we now turn our
attention to construct the BEs relevant for leptogenesis. As discussed
earlier, being Majorana particle, the decay of the lightest RHN $N_{1}$ is a
lepton number violating one and can produce CP asymmetry, which will
eventually generate lepton asymmetry of the Universe. In order to take care
effects of charged lepton Yukawa equilibration of different flavors, the
following classical flavored BE can be constructed:

$\displaystyle\frac{dn_{\Delta_{\alpha}}}{dt}+3\mathcal{H}n_{\Delta_{\alpha}}\;=\;-\langle\Gamma_{N_{1}}$
$\displaystyle\rangle\Big{[}\frac{\varepsilon_{\ell_{\alpha}}}{M_{1}}(\rho_{N_{1}}-\rho_{N_{1}}^{\rm{eq}})+\frac{1}{2}K^{0}_{\alpha}$
$\displaystyle\times\sum_{\beta}(C^{\ell}_{\alpha\beta}+C^{H}_{\beta})\frac{n_{N_{1}}^{\rm{eq}}}{n_{\ell}^{\rm{eq}}}n_{\Delta_{\beta}}\Big{]}.$
(35)

The equation remains identical to the one of Eq. (16) except the fact that
$\mathcal{H}$ is now comprised of all the contributions from inflaton,
$N_{1}$, and radiation energy densities in line with Eq. (33). This will
certainly influence the lepton asymmetry production differently compared to
the scenario discussed in Section II. The first term (within the first
parenthesis) on the right hand side of Eq. (35) represents the production of
lepton asymmetry from the decay of lightest RHN $N_{1}$ while the remaining
terms denote the washout of the produced asymmetry along individual lepton
directions due to the inverse decay of the $N_{1}$. Apart from the flavored
setup, a situation may arise where flavor effects are not that important in
discussing leptogenesis. In that case, an unflavored scenario exhibits where
the evolution of the $B-L$ asymmetry is governed by the single BE given by:

$\displaystyle\frac{dn_{B-L}}{dt}+3\mathcal{H}n_{B-L}=-\langle\Gamma_{N_{1}}\rangle\left[\frac{\varepsilon_{\ell}}{M_{1}}(\rho_{N_{1}}-\rho_{N_{1}}^{\rm{eq}})+\frac{n_{N_{1}}^{\rm{eq}}}{2n_{\ell}^{\rm{eq}}}n_{B-L}\right].$
(36)

Solving Eq. (26), (31), (32) and (35) or (36) simultaneously, will lead to the
evolution of energy density of relevant elements of the Universe and produced
lepton asymmetry from the time of the end of inflaton till today. However,
while solving these BEs, it is convenient to use new variables [26, 43] for
which we use the following transformations:

$\displaystyle E_{\phi}$ $\displaystyle=\rho_{\phi}a^{3}\,,\quad
E_{N_{1}}=\rho_{N_{1}}a^{3}\,,\quad R=\rho_{R}a^{4}\,,$ $\displaystyle
N_{B-L}=n_{B-L}a^{3}\,,\quad N_{\Delta_{\alpha}}=n_{\Delta_{\alpha}}a^{3}$
(37)

Moreover, it is convenient to write the BEs as functions of the scale factor
$(a)$ rather than time ($t$). More precisely, we use the ratio of the scale
factor to its value at the end of inflation,

$\displaystyle A=\frac{a}{a_{\rm{end}}}.$ (38)

Using the newly introduced dimensionless variables, BEs in Eq. (26), (31),
(32), (35) and (36) will look like:

$\displaystyle\frac{dE_{\phi}}{dA}=3\left(\frac{2-n}{n+2}\right)\frac{E_{\phi}}{A}-\frac{(\Gamma_{\phi
ff}+\Gamma_{\phi NN})E_{\phi}}{A\mathcal{H}}\,,$ (39)
$\displaystyle\frac{dR}{dA}=\frac{\langle\Gamma_{N_{1}}\rangle
a_{\rm{end}}}{\mathcal{H}}(E_{N_{1}}-E_{N_{1}}^{\rm{eq}})+\frac{\Gamma_{\phi
ff}E_{\phi}}{\mathcal{H}}\,,$ (40)
$\displaystyle\frac{dE_{N_{1}}}{dA}=\frac{\Gamma_{\phi
NN}E_{\phi}}{A\mathcal{H}}-\frac{\langle\Gamma_{N_{1}}\rangle}{A\mathcal{H}}(E_{N_{1}}-E_{N_{1}}^{\rm{eq}})\,,$
(41)
$\displaystyle\frac{dN_{\Delta_{\alpha}}}{dA}=-\frac{\langle\Gamma_{N_{1}}\rangle}{A\mathcal{H}}\Bigg{[}\frac{\varepsilon_{\ell_{\alpha}}}{M_{1}}(E_{N_{1}}-E_{N_{1}}^{\rm{eq}})+\frac{1}{2}K^{0}_{\alpha}$
$\displaystyle\hskip
85.35826pt\times\sum_{\beta}(C^{\ell}_{\alpha\beta}+C^{H}_{\beta})\frac{Y_{N_{1}}^{\rm{eq}}}{Y_{\ell}^{\rm{eq}}}N_{\Delta_{\beta}}\Bigg{]}\,,$
(42)
$\displaystyle\frac{dN_{B-L}}{dA}=-\frac{\langle\Gamma_{N_{1}}\rangle}{A\mathcal{H}}\left[\frac{\varepsilon_{\ell}}{M_{1}}(E_{N_{1}}-E_{N_{1}}^{\rm{eq}})+\frac{Y_{N_{1}}^{\rm{eq}}}{2Y_{\ell}^{\rm{eq}}}N_{B-L}\right].$
(43)

Finally, the produced lepton asymmetry can be converted to baryon asymmetry
using the relation:

$\displaystyle
Y_{B}=\frac{28}{79}\frac{1}{sA^{3}}N_{B-L}=\frac{28}{79}\frac{1}{sA^{3}}\sum_{\alpha}N_{\Delta_{\alpha}}.$
(44)

##  V Results

Case | $y_{\phi NN}$ |  $T_{\text{RH}}$ (GeV) |  $T^{*}_{\tau}$ (GeV) | $Y_{B}$
---|---|---|---|---
I | $0$ | $1.67\times 10^{10}$ | $4.7\times 10^{10}$ | $8.72\times 10^{-11}$
II | $10^{-7}$ | $1.67\times 10^{10}$ | $4.7\times 10^{10}$ | $8.72\times 10^{-11}$
III | $10^{-5}$ | $1.72\times 10^{10}$ | $4.8\times 10^{10}$ | $3.52\times 10^{-10}$
IV | $4\times 10^{-5}$ | $2.37\times 10^{10}$ | $5\times 10^{10}$ | $4.29\times 10^{-10}$
Table 1: Values of $T_{\text{RH}},~{}T^{*}_{\tau}$ and final BAU $Y_{B}$ for
different choices of $y_{\Phi NN}$. Here we have fixed $y_{\phi ff}=4\times
10^{-5}$ and $M_{1}=6\times 10^{10}$ GeV. $T_{\text{Max}}=4.45\times 10^{12}$
GeV is found to be constant for all four cases.

We now employ the set of BEs Eq. (39)-(41) simultaneously in order to estimate
the individual components of energy densities such as $\rho_{\phi},\rho_{R}$
and $\rho_{N_{1}}$ which are connected to $E_{\phi},E_{R}$ and $E_{N_{1}}$
respectively via Eq. (37). By knowing $\rho_{R}$ as a function of the scale
factor $a$ or the rescaled one $A$, the temperature can be defined by Eq.
(30). Then, using Eq. (33), we estimate the shift of the ET if any from their
standard estimate (see Fig. 1) by comparing the interaction rate of charged
lepton Yukawa interaction of individual flavor $\langle\Gamma_{\alpha}\rangle$
with $\mathcal{H}$. Afterward, depending on the shift of ET of individual
flavor, we proceed for evaluating the flavored (unflavored) $B-L$ asymmetries
by solving Eq. (42) (Eq. (43)) where we also feed the solutions of other Eqs.
(39)-(41).

In order to evaluate the BAU today following the above strategy, we notice
that the mechanism is controlled by the following independent parameters ($i$)
$y_{\phi ff}$ [inflaton-SM fermion effective coupling], ($ii$) $y_{\phi NN}$
[inflaton-RHN coupling], ($iii$) $M_{1}$ [the lightest RHN mass] and ($iv$)
$\\{Re[\theta_{R}],~{}Im[\theta_{R}]\\}$ [constituents of $R$ matrix to
estimate $Y_{\nu}$]. We will maintain a typical hierarchy of RHN masses as:
$M_{3}=10^{2}M_{2}=10^{4}M_{1}$. Furthermore, the dynamics also depends on the
input parameters from the inflaton potential: $\\{n,\lambda\\}$. As stated
earlier, we consider $n=2$ in $V(\phi)$ of Eq. (24) and correspondingly the
value of $\lambda=2\times 10^{-11}$ is determined such that the inflationary
observables like spectral index and tensor-to-scalar ratio are found to be
within $95\%$ C.L. of the Planck+BICEP2/$Keck$ constraints. For a more
detailed discussion, see the Appendix and ref. [58, 28, 29, 85]. Like the
previous study [60], here also we confine ourselves by choosing $y_{\phi
ff}\lesssim\mathcal{O}(10^{-5}-10^{-4})$ [86, 58] as above this value, non-
perturbative production of fermions from inflaton decay may start to dominate.
As a result, perturbative prescription for particle production would be
invalid.

[In absence of inflaton-RHN coupling:] Note that the present scenario differs
from the previous one due to the inclusion of $y_{\phi NN}$ coupling in this
work. Hence, a choice of the parameter $y_{\phi NN}=0$ should reproduce the
outcome of our previous work. We therefore start studying the phenomenology by
choosing $y_{\phi NN}=0$ (case I) first and then turning on $y_{\phi NN}$
gradually to a value comparable to $y_{\phi ff}$. We choose $y_{\phi
ff}=4\times 10^{-5}$ so as to be consistent with the perturbative limit on it.
The Left plot of Fig. 3 shows the variation of temperature (along $y$ axis)
with the parametrized scale factor $A$ (along $x$ axis) where we use the
solution for the radiation energy density of Eq. (40) (ignoring the first term
in the right hand side as $y_{\phi NN}=0$) coupled with Eqs. (39) and put it
back in Eq. (30). After inflation, the temperature of the Universe then
attains a maximum value $\sim T_{\text{Max}}=4.45\times 10^{12}$ GeV and
thereafter falls (having a different slope compared to the radiation dominated
epoch) to a point where $\rho_{\phi}=\rho_{R}$ is reached which defined the
end of reheating as $T_{\text{RH}}=1.67\times 10^{10}$ GeV.

Figure 3:  Evolution of temperature $T$ (left panel) and various energy
densities (right panel) $w.r.t.$ rescaled scale factor for $M_{1}=6\times
10^{10}$ GeV and $y_{\phi NN}=0$. In the middle plot we show the dependence of
$\langle\Gamma_{\alpha}\rangle/\mathcal{H}$ on $T$ for the same choice of
$M_{1}$ and $y_{\phi NN}$.

Due to such faster expansion and nontrivial scale factor dependence of
temperature during this extended reheating period
$T_{\text{Max}}>T>T_{\text{RH}}$, the charged lepton Yukawa interaction
(particularly for $\tau_{R}$ in this case) will come to thermal equilibrium at
a smaller temperature than the standard radiation-dominated case. To provide a
concrete evaluation of the same, we include middle plot of Fig. 3 where
$\langle\Gamma_{\alpha}\rangle/\mathcal{H}$ evolution ($\alpha=e,\mu,\tau$
with blue, green and red lines respectively) are plotted against temperature
$T$ variation. This shift in $\tau_{R}$ ET is depicted clearly by the
intersection point of the blue line and the horizontal dashed line indicating
$\langle\Gamma_{\tau}\rangle/\mathcal{H}=1$ in middle plot of Fig. 3. The
relevant ET in this extended period of reheating turn out to be
$T^{*}_{\tau}=4.7\times 10^{10}$ GeV and is included in Table 1 along with
values of other parameters. The reheating temperature being bounded by
$\sim\mathcal{O}(10^{10})$ GeV, no change in $\mu_{R}$ or $\tau_{R}$ ET has
been found as expected.

With such a shift in the ET of $\tau_{R}$ in this particular case (first row
of Table 1), flavor leptogenesis would get affected. In order to have an
impact of it on flavor leptogenesis, we choose a value of $N_{1}$ mass
$M_{1}=6\times 10^{10}$ GeV which falls intermediate between the associated
$T_{\text{Max}}$ and $T_{\text{RH}}$. First we estimate the evolution of
various energy densities $\rho_{\phi},~{}\rho_{R},~{}\rho_{N_{1}}$ in this
scenario against $A$ by solving Eqs. (39)-(41) simultaneously as shown in the
rightmost plot of Fig. 3 indicated by red, blue and green solid lines
respectively. In evaluating $Y_{\nu}$, we consider
$Re[\theta_{R}]=6.03,~{}Im[\theta_{R}]=0.22$ (the reason behind such a choice
is to have correct BAU finally). Note that in absence of $y_{\phi NN}$
coupling, $N_{1}$s are thermally generated during $T_{\text{Max}}>T>M_{1}$
from the thermal bath consisting of SM fields, thanks to radiation production
from inflaton decay via $y_{\phi ff}$. As the Universe expands, the radiation
energy density is being diluted non trivially till an equality with inflaton
energy density defining $T_{\text{RH}}$ (intersection of blue and red lines).
Standard radiation domination follows only beyond this point.

These $N_{1}$s would effectively decay around $T\sim M_{1}$ and produce the
lepton asymmetry via leptogenesis. The $N_{1}$ being thermally produced, this
scenario is similar to the standard flavored thermal leptogenesis scenario,
though impacted by the shift in $T^{*}_{\tau}$ as seen above. We find
$M_{1}>T^{*}_{\tau}$, all the charged lepton Yukawa interactions remain out of
equilibrium in this phase. As a consequence, an unflavored leptogenesis
prevails here in case of extended period of reheating. This is the main
difference we experience while comparing it with standard flavored thermal
leptogenesis scenario in which with leptogenesis scale $T\sim M_{1}$, $\tau$
lepton Yukawa interaction occurs rapidly (as it is already in equilibrium, see
section II) following which a two-flavor setup must be incorporated (compared
to the unflavored one in present case). The corresponding evolution of lepton
asymmetry is shown by black line in bottom portion of Fig. 4 as a function of
the modified scale factor $A$ which saturates to a lepton asymmetry value that
eventually converts to the observed BAU value. To make the correspondence
between $\rho_{N_{1}}$ and the production of $Y_{B-L}$, we also incorporate
the $\rho_{N_{1}}$ evolution in the top portion of the figure.

Figure 4:  Evolution of energy density of $N_{1}$ (upper panel) and produced
baryon asymmetry (lower panel) $w.r.t.$ rescaled scale factor for different
values of $y_{\phi NN}$ for $M_{1}=6\times 10^{10}$ GeV and $y_{\phi
ff}=4\times 10^{-5}$.

Figure 5:  Evolution of different energy densities $w.r.t.$ rescaled scale
factor for $y_{\phi NN}=10^{-7}$(left panel) and $y_{\phi NN}=10^{-5}$ (right
panel). Here we set $M_{1}=6\times 10^{10}$ GeV and $y_{\phi ff}=4\times
10^{-5}$.

[In presence of inflaton-RHN coupling:] We now turn on inflaton-RHN coupling
and observe its impact on the charged lepton Yukawa equilibration and
consequently on the produced baryon asymmetry during reheating era. Let us
begin with a sufficiently small $y_{\phi NN}=10^{-7}$ (tabulated in Table 1 as
case II) as compared to $y_{\phi ff}$ chosen. As shown in the Fig. 5,
switching on $y_{\phi NN}$ causes the inflaton to produce a large number of
$N_{1}$s (indicated by green line) initially. This can be understood if we
compare the $\rho_{N_{1}}$ evolution (green line in Fig. 5) above temperature
$T=M_{1}$ (indicated by the vertical black small dashed line) in this case
versus the case with solely thermally produced $N_{1}$s (see Fig. 3). However,
as the temperature drops, the production of $N_{1}$ from inflaton decay does
not keep up with the Universe’s expansion rate due to its feeble coupling
chosen. As a result, the energy density of these $N_{1}$s (as decay products
of inflaton) gets diluted and at some stage the production of $N_{1}$s from
the inverse decay dominates over it. This is evident in the left plot of Fig.
5 by the sudden change of slope of $\rho_{N_{1}}$ just before $T=M_{1}$ which
coincides with the energy density of thermally produced $N_{1}$s of Fig. 3.
This continues till a point beyond which $N_{1}$ decay starts to contribute to
lepton asymmetry production.

We also notice that due to the dominant decay of $\phi$ into SM fermions, the
$\rho_{\phi}$ and $\rho_{R}$ (mainly contributed from $y_{\phi ff}$ coupling)
do not alter by a noticeable amount and hence $\mathcal{H}$ remains
essentially unchanged compared to the previous case with $y_{\phi NN}$ =0. As
a result, together with $T_{\text{Max}}$ and $T_{\text{RH}}$, the $\tau_{R}$
ET $T^{*}_{\tau}$ remains identical with the thermal case (see second row of
Table 1). Hence the present situation falls in the category of unflavored
leptogenesis. The evolution of the $B-L$ asymmetry for $y_{\phi NN}=10^{-7}$
scenario is presented by the red dash-dotted line in the bottom plot of Fig. 4
which overlaps with the $y_{\phi NN}$ = 0 case (solid black line), thereby
satisfying the observed baryon asymmetry of the Universe.

As we further increase the strength of the inflaton-$N_{1}$ coupling, $i.e.$
$y_{\phi NN}=10^{-5}$ (as case III of Table 1 while keeping other parameters
fixed at their previous values), a change in $\rho_{N_{1}}$ becomes visible.
The right plot of Fig. 5 shows the behavior of the energy densities of
different components of the Universe in this case. Note that contrary to cases
I and II, dominant contribution to $\rho_{N_{1}}$ here follows from the
$N_{1}$s being decay products of inflaton as it supersedes the thermally
produced ones from inverse decay. The radiation component (blue line) however
still remains dominant compared to $\rho_{N_{1}}$ (green line). Note that
$T_{\text{Max}}$ remains unaffected as it is mainly controlled by $y_{\phi
ff}$ coupling (fixed for cases I-IV) responsible for initial radiation
production. However a small shift in the $T_{\text{RH}}$ as compared to cases
I and II, is observed and indicated in Table 1. This happens as a result of
higher $y_{\phi NN}$ coupling which causes the inflaton to decay earlier than
cases I and II so that $\rho_{\phi}=\rho_{R}$ defining the $T_{\text{RH}}$ is
realized at a slightly higher temperature. Due to the dominance of $\rho_{R}$
(almost unchanged compared case I and II) over $\rho_{N_{1}}$, the temperature
evolution above $M_{1}$ remains close to the two earlier cases. For the same
reason, $\mathcal{H}$ is also almost unaffected and this is reflected in the
evaluation of $T^{*}_{\tau}$ (only a slight change) as included in Table 1. As
previously discussed, here also, even though we have a slightly higher value
of $T^{*}_{\tau}$ than the previous cases, the leptogenesis scale however
remains larger with respect to $T^{*}_{\tau}$. Hence, an unflavored
prescription is still adequate for estimating the lepton asymmetry.

The evolution of produced lepton asymmetry in this case III is shown by the
blue dash-dotted line in the bottom panel of Fig. 4. As seen in this plot, the
lepton asymmetry starts to being produced at a stage above temperature $T\sim
M_{1}$. This is related to the fact that $N_{1}$s produced during this era of
reheating find themselves in out of equilibrium ($\mathcal{H}$ is larger than
decay width of $N_{1}$) and would decay. Additionally, $\rho_{N_{1}}$ being
comparable to $\rho_{R}$, the inverse decay process (related with neutrino
Yukawa coupling) remains subdominant compared to the $N_{1}$ decay. As a
result, lepton asymmetry (still unflavored though) produced from such decay of
$N_{1}$ would not be washed out completely and a non-zero asymmetry survives.
The amount of asymmetry production increases till a point where $\rho_{N_{1}}$
ceases to exist. Beyond it, $Y_{B-L}$ falls to some extent, before attaining
its asymptotic value which is larger than the value of the lepton asymmetry
necessary for the production of observed BAU, as the produced asymmetry gets
diluted due to the increase of entropy ($N_{1}$ decay produces a sizeable
$\rho_{R}$ and hence, entropy) in the Universe. Note that this phase of
leptogenesis is different from thermal leptogenesis scenario as $N_{1}$s are
never in thermal equilibrium. On the other hand, this is not purely the case
of non-thermal leptogenesis which happens with $N_{1}$, as the decay product
of inflaton, finding itself in an environment with $T\ll M_{1}$ (so, thermal
generation is ruled out). So here with case (B), we find a nonstandard
generation of lepton asymmetry as a consequence of extended reheating where
the inflaton has a sizeable coupling with the lightest RHN. As discussed in
the beginning of section, we call it a ‘quasi’- thermal leptogenesis as
neither it is the case of a purely thermal nor that of non-thermal
leptogenesis. The value of $Y_{B}$ can be brought down to the correct BAU by
decreasing $Im[\theta_{R}]=0.031$, while all other parameters/outcomes are
unaffected.

Finally, the above discussed effect becomes prominent if we choose to increase
the $y_{\phi NN}$ coupling further, say $y_{\phi NN}=y_{\phi ff}$ as included
in case IV of Table 1. In this case, we obtain $\rho_{N_{1}}=\rho_{R}$. As
radiation and $N_{1}$s contribute equally to the energy density of the
Universe during the reheating period, the expansion rate of the Universe gets
modified in this scenario, affecting the $T_{\text{RH}}$ as well as
$T^{*}_{\tau}$. In this case, we get a larger $T_{\text{RH}}$ as inflaton
decays earlier than the previous case owing to the larger $y_{\phi NN}$
coupling. The related numerical estimates for this case IV are listed in the
fourth row of Table 1. In this case also, a larger baryon asymmetry of the
Universe is created which can be settled to the observed $Y_{B}$ value without
altering any other parameter/predicted values once we reduce the value to
$Im[\theta_{R}]=0.02$.

Figure 6: Variation of final baryon asymmetry $Y_{B}$ $w.r.t.$ $y_{\phi NN}$ for $M_{1}=6\times 10^{10}$ GeV and $y_{\phi ff}=4\times 10^{-5}$. The points indicated by “star” represent the case-II-IV from Table 1. Point | $y_{\phi NN}$ | $y_{\phi ff}$ |  $T_{\text{Max}}$ (GeV) |  $M_{1}$ (GeV) |  $T_{\text{RH}}$ (GeV) |  $T^{*}_{\tau}$ (GeV) |  $T^{*}_{\mu}$ (GeV) | $Y_{B}$
---|---|---|---|---|---|---|---|---
BP1 | $10^{-6}$ | $10^{-7}$ | $2.23\times 10^{11}$ | $5\times 10^{9}$ | $4.2\times 10^{8}$ | $2\times 10^{9}$ | $7\times 10^{8}$ | $5.67\times 10^{-9}$
BP2 | $5\times 10^{-7}$ | $5\times 10^{-8}$ | $1.6\times 10^{11}$ | $5\times 10^{9}$ | $2.1\times 10^{8}$ | $1.5\times 10^{9}$ | $4\times 10^{8}$ | $3.60\times 10^{-9}$
BP3 | $10^{-7}$ | $10^{-8}$ | $7.04\times 10^{10}$ | $5\times 10^{9}$ | $4.2\times 10^{7}$ | $6\times 10^{8}$ | $1.3\times 10^{8}$ | $7.28\times 10^{-10}$
Table 2:  We list three benchmark points (BP) where the leptogenesis scale
falls in between $T_{\text{Max}}$ and $T_{\text{RH}}$. While BP1 represents
the case where $M_{1}$ is closer to $T_{\text{RH}}$, BP2 represents an
intermediate scenario and BP3 indicates a scenario where $M_{1}$ lies closer
to $T_{\text{Max}}$.

In Table 1, we have listed a few specific values of $y_{\phi NN}$ coupling to
describe the impact of reheating on leptogenesis. In Fig. 6, we provide the
estimate of final baryon asymmetry (via unflavored leptogenesis) once the
Yukawa coupling $y_{\phi NN}$ is varied ($y_{\phi NN}\leq y_{\phi ff}$). As
already found in cases I-III, with tiny $y_{\phi NN}$ coupling ($y_{\phi
NN}\lesssim 10^{-6}$), the final baryon asymmetry $Y_{B}$ almost remains
independent of $y_{\phi NN}$. Thereafter, a rise in $Y_{B}$ can be seen due to
the fact that the production of RHN $N_{1}$ from the inflaton decay also
becomes significant. This additional production channel causes a significant
rise in the $N_{1}$’s abundance $\rho_{N_{1}}$ which further leads to a larger
production of lepton asymmetry (also the baryon asymmetry). This behavior is
also clear from Fig. 4. A peak in $Y_{B}$ is observed when $y_{\phi NN}\simeq
2\times 10^{-5}$ after which $Y_{B}$ is reduced once the $y_{\phi NN}$ is
further increased. This fall can be understood by looking at the third term of
Eq. (43) where one notes that a larger production of asymmetry also results in
a larger washout of the asymmetry.

[With dominant inflaton-RHN coupling:] So far the discussion we have, we find
that the gradual increase of $y_{\phi NN}$ coupling not only affects the
temperature behavior and expansion rate of the Universe during reheating
period but also impacts the lepton asymmetry production in this quasi-thermal
regime. However we have restricted ourselves with values of couplings
associated to the inflaton below $\mathcal{O}(10^{-5}-10^{-4})$ so as to keep
the analysis consistent with perturbative reheating era [86, 58]. Alongside,
we take $y_{\phi ff}$ at a borderline value $4\times 10^{-5}$ and hence we are
unable to make $y_{\phi NN}$ larger than $y_{\phi ff}$ by order(s) of
magnitude and discuss the impact of such consideration. Also, with such choice
of $y_{\phi ff}$, the reheating temperature turns out to be high enough so as
to keep $M_{1}$ accordingly large (to realize scenario [B]).

With an aim to observe the consequence of $y_{\phi NN}>y_{\phi ff}$ while
keeping things more flexible such as lowering the scale of leptogenesis
impacted by the extended period of reheating, we now consider three specific
situations: (i) $M_{1}$ is close to $T_{\text{RH}}$ [BP1], (ii) $M_{1}$ is
intermediate between $T_{\text{Max}}$ and $T_{\text{RH}}$ [BP2], and (iii)
$M_{1}$ is close to $T_{\text{Max}}$ [BP3] where the mass of the lightest RHN
is fixed at $M_{1}=5\times 10^{9}$ GeV while $y_{\phi ff}$ and $y_{\phi NN}$
are floated to realize such considerations. We choose the values of
$Re[\theta_{R}]=2.83$ and $Im[\theta_{R}]=0.24$ used in thermal flavored
leptogenesis scenario (in two flavor regime) of section II (see Fig. 2) the
result of which is consistent with correct BAU. The purpose of such a choice
is to compare the outcome of the extended period of reheating on final baryon
asymmetry generation with $y_{\phi NN}>y_{\phi ff}$.

Figure 7:  Evolution of different energy densities (upper panel) and
temperature $T$ (lower panel) $w.r.t.$ rescaled scale factor for BP1 (left
panel), BP2 (middle panel), and BP3 (right panel). Here we fix $M_{1}=5\times
10^{9}$ GeV and $\theta_{R}=2.83+0.24i$. Figure 8:  Variation of
$\langle\Gamma_{\alpha}\rangle/\mathcal{H}$ $w.r.t.$ $T$ for $M_{1}=5\times
10^{9}$ GeV and $\theta_{R}=2.83+0.24i$. Here solid lines indicate the BP1,
dashed dotted line represents BP2, and BP3 is denoted by dashed lines.  Figure
9:  Evolution of produced lepton asymmetry $w.r.t.$ rescaled scale factor for
all the three BPs for $M_{1}=5\times 10^{9}$ GeV and $\theta_{R}=2.83+0.24i$.

Note that as inflaton-SM fermion coupling essentially define the
$T_{\text{Max}}$ while $y_{\phi NN}$ has some role to play in determining
$T_{\text{RH}}$, we first make some appropriate choices of these two
parameters in defining the three benchmark cases BP-1,2,3. They are listed in
Table 2. For all these sets, $y_{\phi NN}$ remains one order of magnitude
larger than $y_{\phi ff}$ coupling. In evaluating the temperature evolution
during the reheating, we solve Eqs. (39)-(41) as a function of the rescaled
scale factor $A$ simultaneously and using Eq. (30), temperature is evaluated.
Bottom panels of all three plots of Fig. 7 represent the temperature variation
with $A$ and upper panels depict the same for energy densities of different
components.

As seen from the plots, immediately after the end of inflation, the
temperature reaches a maximum value $T_{\text{Max}}$. Then it starts to
decrease in accordance with our previous discussion (in section IV) due to
faster expansion of the Universe during this period of extended reheating.
However, an interesting departure of $T$ from this fall is observed around
$T\sim M_{1}$. This is related to the emergence of a new production channel,
producing radiation from $N_{1}$ decay asa result of $y_{\phi NN}$ dominance.
An interplay between such additional injection of radiation in the bath (which
tries to increase $\rho_{R}$) and the depletion of $\rho_{R}$ due to Hubble
expansion, a plateau like region is formed in $T$ evolution plot. This period
however does not last long as eventually Hubble expansion rate overtakes this
radiation production rate from $N_{1}$ decay ($\rho_{N_{1}}$ decreases sharply
beyond a point). Eventually, radiation dominates over matter beyond
$T_{\text{RH}}$ and temperature of the Universe drops as $A^{-1}$. In the
upper panels of Fig. 7, we note that due to choice(s) of smaller coupling(s)
$y_{\phi ff}$ ($y_{\phi NN}$) in going from BP1 to BP3, inflaton takes larger
time to decay for BP3 (BP2)compared to BP1. As a result, matter radiation
equality shifts at a later epoch resulting in lower reheating temperature for
BP3 (BP2) relative to BP1.

This nontrivial behavior of temperature along with the larger expansion rate
of the Universe during reheating period affects the ETs of charged lepton
Yukawa interactions as can be seen from Fig. 8. For RHN mass $M_{1}=5\times
10^{9}$ GeV, though there is no shift in ET for right-handed electron, ample
amount of change can be seen for ET for $\mu_{R}$ and $\tau_{R}$ compared to
the standard radiation dominated scenario for all three BPs. This change makes
the charged Yukawa interactions come to equilibrium at a much lower
temperature which are included in Table 2.

As a consequence of considerably lower values of $T^{*}_{\mu}$ and
$T^{*}_{\tau}$, we expect the quantum decoherence of the SM lepton doublet
states to take place here at much lower temperatures (same as charged lepton
ET) compared to the standard radiation dominated scenario. Hence, for all
three BPs, lepton asymmetry generation process turns out to be not affected by
individual charged lepton doublets at leptogenesis scale $M_{1}=5\times
10^{9}$ GeV and an unflavored leptogenesis prevails here. This is a new result
as compared with earlier standard analysis presented in section II, where at
this leptogenesis scale, $\tau_{R}$ was already in equilibrium affecting the
lepton asymmetry generation along $\tau$ direction distinctively. Accordingly,
a two flavor leptogenesis was incorporated for correct generation of lepton
asymmetry. On the other hand, in this present case incorporating the effect
extended reheating (with inflaton-RHN dominance), unflavor approach to
evaluate the baryon asymmetry of the Universe would be enough.

For all three BPs, a different rate of washout during the reheating period
accounts for the main difference in the produced final baryon asymmetry.
Dashed purple line of Fig. 9 shows that as $M_{1}$ is closer to
$T_{\text{Max}}$ for BP3 as a result of which the produced asymmetry suffers a
larger amount of washout (due to Hubble expansion) contrary to BP1 and BP2.
Finally, even with a relatively low $M_{1}$ in this quasi-thermal regime, an
overproduction of baryon asymmetry by one to two order(s) of magnitude is
observed for these three BPs relaxing the parameter space even further with
respect to the modified thermal leptogenesis scenario studied in section V.
For completeness purpose, we notice that such final $Y_{B}$ values can be
brought down to correct level of BAU by changing $\theta_{R}$ without altering
other parameter values or the outcomes such as $T_{\text{RH}}$ and
$T^{*}_{\alpha}$. For example, for BP1 (BP2), one needs to set
$Im[\theta_{R}]=0.002~{}(0.005)$, while for BP3, $Im[\theta_{R}]$ can be fixed
at $0.02$ so that observed BAU can be generated.

##  VI Conclusion

In this work, we have shown that an extended period of reheating resulting
from inflaton-decay into radiation together with the lightest RHN can
significantly alter the equilibration temperature of the charged lepton Yukawa
interactions. Consequently, flavored leptogenesis mechanism gets affected. We
start with a discussion on how the equilibration temperature(s) of charged
lepton Yukawa interaction(s) can be estimated in a radiation dominated
Universe and its impact on lepton asymmetry generation known as flavored
leptogenesis. In such a setup, the reheating process is generally assumed to
be instantaneous and happens to be higher than the mass of the decaying RHN
whose decay contributes to lepton asymmetry production. However, depending on
the inflaton coupling to SM particles, the reheating process may survive a
longer period creating a prolonged era of reheating, from $T_{\text{Max}}$ to
$T_{\text{RH}}$. Motivated by our recent finding on the impact of this
extended era of reheating on charged lepton equilibration temperature and
flavored leptogenesis, here we extend the setup by including additional
inflaton-RHN coupling.

We find that with relatively large value of inflaton-RHN coupling compared to
the inflaton-SM fermion effective coupling, the reheating period gets further
modified. While the inflaton-SM fermion coupling mainly controls the maximum
temperature of the Universe immediately after inflation, the inflaton-RHN
coupling has the potential to impact the reheating temperature. The production
of RHN and SM bath from the inflaton decay during this period of prolonged
reheating helps the Universe to expand at a much faster rate (depending on the
inflaton-RHN coupling though) in comparison to the scenario where inflaton
decays directly to radiation solely. As a result of such faster expansion,
along with the modified temperature behavior, the charged Yukawa interactions
enter into equilibrium in a delayed fashion. We also observe that such a
delayed equilibration of charged lepton Yukawa interactions can significantly
modify the lepton asymmetry generation compared to what is observed in thermal
leptogenesis. For example, a flavored leptogenesis scenario found to be in two
flavor regime in standard thermal leptogenesis may emerge as an unflavored one
here.

Another interesting outcome of the present scenario is revealed with a
dominant inflaton-RHN coupling with respect to inflaton-SM fermion effective
coupling. Here we encounter an unusual situation where the lepton asymmetry
starts to be produced at a temperature above the mass of the lightest RHN
without being completely washed out. In fact, the reheating era produces an
environment where the lightest RHN find itself in out-of-equilibrium in this
regime and its decay therefore contributes to lepton asymmetry production. In
a way, this helps to reduce the scale of leptogenesis since the inclusion of
inflaton-RHN coupling may inject a large amount of RHN into the system on top
of thermally produced ones (whose decay also contribute to produce lepton
asymmetry) during reheating thereby resulting an enhanced lepton asymmetry.
The framework however can be extended beyond our present consideration. In
some of the low scale leptogenesis scenarios, the framework might alter the
prediction as well as allowing the leptogenesis scale to drop even further
opening the possibility to explore leptogenesis in collider experiments. The
related study is beyond the scope of this paper and we plan for some more work
in these directions in future.

##  VII Acknowledgements

A.D. acknowledges the financial support received from the grant
CRG/2021/005080. A.S. acknowledges the support from grants CRG/2021/005080 and
MTR/2021/000774 from SERB, Govt. of India. R.R. acknowledges the National
Research Foundation of Korea (NRF) grant funded by Korea government
(NRF-2020R1C1C1012452).

##  Appendix A Predictions of inflatonary observables

The inflation potential in our work follows from supergravity framework [81]
and is of the form

$\displaystyle V(\phi)=\lambda
M^{4}_{P}\left[\sqrt{6}\tanh\left(\frac{\phi}{\sqrt{6}M_{P}}\right)\right]^{n}.$
(45)

It contains free parameters $\lambda$ and $n$. In order to estimate the
inflationary predictions followed from such a potential, the slow-roll
parameters need to be considered, which are given by:

$\displaystyle\epsilon=\frac{1}{2}M_{P}^{2}\left(\frac{V^{\prime}}{V}\right)^{2},\quad\eta=M_{P}^{2}\left(\frac{V^{\prime\prime}}{V}\right).$
(46)

On the other hand, the number of $e$-folds can be estimated as:

$\displaystyle N$ $\displaystyle=\frac{1}{M_{P}^{2}}\int_{\phi_{\rm
end}}^{\phi_{*}}\frac{V}{V^{\prime}}d\phi\simeq\int_{\phi_{\rm
end}}^{\phi_{*}}\frac{1}{\sqrt{2\epsilon}M_{P}}d\phi,$
$\displaystyle\simeq\frac{3}{2n}\cosh\left(\sqrt{\frac{2}{3}}\frac{\phi_{*}}{M_{P}}\right),$
(47)

where $\phi_{*}$ corresponds to the crossing horizon value of the inflaton and
$\phi_{\rm end}$ represents the field value at the end of inflation.

Consequently, within the slow-roll approximation, the inflationary observables
such as spectral index ($n_{s}$) and the tensor-to-scalar ratio ($r$) can be
expressed as:

$\displaystyle n_{s}$ $\displaystyle\simeq 1-6\epsilon+2\eta,$ (48)
$\displaystyle r$ $\displaystyle\simeq 16\epsilon=\frac{12}{N^{2}}.$ (49)

Finally, the remaining inflationary observable, the amplitude of the curvature
power spectrum $P_{s}$, is given by:

$\displaystyle P_{s}=\frac{V^{3}}{12\pi^{2}M_{P}^{6}V^{\prime 2}}.$ (50)

All these three inflationary observables are to be evaluated at
$\phi=\phi_{*}$.

Note that contrary to two other parameters, $P_{s}$ is a function of
normalization constant $\lambda$ (involved in the potential of Eq. (45)).
Therefore, using the relation between $\phi_{*}$ and $N$ from Eq. (47),
$P_{s}$ at $\phi_{*}$ ($\equiv P_{s*}$) can be simplified as:

$\displaystyle P_{s*}\simeq\frac{6^{n/2}\lambda N^{2}}{18\pi^{2}},$ (51)

which helps to determine the value of $\lambda$ for a specific $n$ since at
the Planck pivot scale, $k_{*}=0.05$ Mpc-1, $\ln(10^{10}P_{s*})=3.044$ holds
[4]. A precise estimate of the number of $e$-foldings $N$ would be helpful in
determining these observables. In general, $N$ can be influenced by the
duration of the reheating process which turns out to be important to consider
here as our analysis is intricately connected to the reheating temperature. A
better estimate of $N$ (considering no additional entropy production between
the end of reheating $T_{RH}$ and the re-entry of the pivot scale $k_{*}$ to
the horizon at a later epoch) follows as [28, 87]

$\displaystyle N=\frac{1}{4}\ln\left[\frac{V(\phi_{*})^{2}}{M_{P}^{4}\rho_{\rm
end}}\right]+\frac{1-3\langle\omega\rangle}{12(1+\langle\omega\rangle)}\ln\left[\frac{\rho_{\rm
RH}}{\rho_{\rm
end}}\right]+\ln\left[\frac{1}{\sqrt{3}}\left(\frac{\pi^{2}}{30}\right)^{1/4}\left(\frac{43}{11}\right)^{1/3}\frac{T_{0}}{H_{0}}\right]-\ln\left[\frac{k_{*}}{a_{0}H_{0}}\right]-\frac{1}{12}\ln
g_{\rm RH},$ (52)

where $\langle\omega\rangle$ is the $e$-fold average of the equation-of-state
parameter $\omega$ during reheating and $g_{\rm RH}$ denotes the effective
relativistic degrees-of-freedom at reheating, taken as the SM value = $106$.
The present day temperature and Hubble parameter are $T_{0}=2.7255$ K and
$H_{0}=67.36$ km s-1 Mpc-1 respectively following the CMB observation [4]. The
present-day scale factor $a_{0}$ can be set to unity without loss of any
generality. Using Eqs. (50)-(51) and an approximated $N$ expression:
$N\simeq\sqrt{3/2}V(\phi_{*})/[M_{P}V^{\prime}(\phi_{*})]$, the value of
$V(\phi_{*})$ can be estimated as $V(\phi_{*})\simeq 6^{n/2}\lambda
M_{P}^{4}$. The energy density at the end of inflation, $\rho_{\rm end}$, can
be evaluated using $\phi_{\rm end}$ obtained by setting the slow roll
parameters to be one. On the other hand, $\rho_{\rm RH}$ corresponds to energy
density of radiation at $T_{\rm{RH}}$. Finally, $\langle\omega\rangle$ is
determined by identifying it for the $\phi$ field only
($\langle\omega_{\phi}\rangle$) and that too is estimated over a complete
cycle of inflaton oscillation about origin in post-inflationary reheating
phase, which is found to be [58]
$\langle\omega_{\phi}\rangle=\frac{n-2}{n+2}$.

Figure 10: $68\%$ (light patch) and $95\%$ (dark patch) contours in the $r$
vs. $n_{s}$ plane with $Planck$ 2018[4] \+ BICEP$/Keck$ (BK) 2015 data[88]
[$Planck$ 2018 + BICEP$/Keck$ 2018[89] data]: drawn in purple [green] patches.
Red and brown stars are the ($n_{s},r$) values generated from Table 1 while
BPs from Table 2 produce the blue blocks. For a better view, the inset plot
provides a zoomed view of the calculated $(n_{s},r)$ from Table 1.

We restrict ourselves with $n=2$ in the study. However the analysis can easily
be extended for a more general set-up with the discussion above. Now, with
$n=2$, we use different values of $\rho_{\rm RH}$ from Table 1 and 2 of main
text and proceed to solve Eq. (52) numerically to finally determine the set of
$n_{s}$ and $r$ parameters (from each values of $T_{\rm{RH}}$ values) using
Eqs. (48) and (49) and an estimate of $\lambda$ follows as $2\times 10^{-11}$.
We find $\phi_{\rm end}\simeq M_{P}$. The evaluated values of $n_{s}$ and $r$
are plotted in Fig. 10, where red, brown and black stars indicate the set of
$(n_{s},r)$ values generated from cases I, III and IV of Table 1 respectively.
On the other hand, BP1, BP2 and BP3 from Table 2 produce the $(n_{s},r)$
values represented by dark blue, blue and light blue blocks respectively. The
light and dark purple (green) patches of the figure represent the allowed
range of $(n_{s},r)$ obtained from the combined Planck 2018[4] and
BICEP/$Keck$ 2015 analysis [88] (combined Planck 2018 and BICEP/$Keck$ 2018
analysis [89]) at $68\%$ confidence level (C.L.) and $95\%$ C.L. respectively.
As can be seen from the figure, cases discussed in Table 1 produce $(n_{s},r)$
values which are within the $95\%$ C.L. (or close to $95\%$ C.L. with updated
analysis). On the other hand, the $(n_{s},r)$ values produced from the BP1,
BP2 and BP3 of Table 2 fall within the $68\%$ C.L.

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