††institutetext: Department of Physics and Astronomy, University of Nebraska,
Lincoln, NE 68588, USA

# Leptogenesis triggered by a first-order phase transition

Peisi Huang  Ke-Pan Xie<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose a new scenario of leptogenesis, which is triggered by a first-order
phase transition (FOPT). The right-handed neutrinos (RHNs) are massless in the
old vacuum, while they acquire a mass in the new vacuum bubbles, and the mass
gap is huge compared with the FOPT temperature. The ultra-relativistic bubble
walls sweep the RHNs into the bubbles, where the RHNs experience fast decay
and generate the lepton asymmetry, which is further converted to the baryon
asymmetry of the Universe (BAU). Since the RHNs are out of equilibrium inside
the bubble, the generated BAU does not suffer from the thermal bath washout.
We first discuss the general feature of such a FOPT leptogenesis mechanism,
and then realize it in an extended $B-L$ model. The gravitational waves from
$U(1)_{B-L}$ breaking could be detected at the future interferometers.

##  1 Introduction

Leptogenesis is a class of mechanisms for solving the matter-antimatter
asymmetry problem of the Universe Fukugita:1986hr ; Luty:1992un ;
Davidson:2008bu . In this paradigm, the heavy right-handed neutrinos (RHNs)
$\nu_{R}$ decay to the Standard Model (SM) particles via the CP violating
Dirac Yukawa interaction $\lambda_{D}\bar{\ell}_{L}\tilde{H}\nu_{R}$,
generating a lepton asymmetry which is then converted to a baryon asymmetry of
the Universe (BAU) via the electroweak (EW) sphaleron process. A particularly
attractive and elegant feature of this paradigm is that the same coupling can
also explain the origin of neutrino masses Davis:1994jw ; Super-
Kamiokande:1998kpq ; KamLAND:2002uet  via Type-I seesaw Minkowski:1977sc . In
the conventional thermal leptogenesis scenario, the generated BAU is
determined by the competition between the enhancement from the CP violating
phase in the $\lambda_{D}$ matrix and the washout effects from the thermal
bath. Typically, the washout processes are so efficient that only
$\mathcal{O}(10^{-2})$ of the originally generated BAU survives till today
Buchmuller:2005eh . If different generations of RHNs have a mass hierarchy,
then the CP violating phase has an upper bound proportional to the lightest
$\nu_{R}$ mass (denoted as $M_{1}$), known as the Davidson-Ibarra bound
Davidson:2002qv , which requires $M_{1}\gtrsim 10^{9}$ GeV to generate the
observed BAU.111If the masses of at least two generations of $\nu_{R}$’s are
nearly degenerate, the CP violating phase can be resonantly enhanced to be
$\mathcal{O}(1)$, independent of the $\nu_{R}$ mass. In that case, successful
leptogenesis can occur for $\mathcal{O}({\rm TeV})$ scale RHN Flanz:1996fb ;
Pilaftsis:1997jf ; Pilaftsis:2003gt ; Dev:2017wwc . If some of the RHNs do not
reach thermal equilibrium before the EW sphaleron process is switched off,
then leptogenesis may even apply to the sub-EW scale $\nu_{R}$ Akhmedov:1998qx
; Drewes:2017zyw .

Figure 1: The sketch of leptogenesis triggered by a FOPT. The blue and white
regions represent the new vacuum bubble (in which
$\left\langle\phi\right\rangle\neq 0$) and the old vacuum background (in which
$\left\langle\phi\right\rangle=0$), respectively. The FOPT occurs at
temperature $T_{p}$, and the bubble expands at a wall velocity $v_{w}$. Inside
a bubble, $\nu_{R}$ gains a huge mass $M_{1}\gg T_{p}$, such that the
$\nu_{R}$’s that have penetrated the bubble decay quickly, generating the BAU.
The possible washout effects (some of which are illustrated inside the yellow
rectangle) are suppressed since $M_{1}/T_{p}\gg 1$.

In this article, we propose a new scenario of leptogenesis, which is triggered
by a first-order phase transition (FOPT). The idea is quite simple: in many
models such as the $B-L$ or Majoron model, RHNs obtain masses through the
vacuum expectation value (VEV) of a scalar field $\phi$. If in the early
Universe $\phi$ experiences a FOPT from $\left\langle\phi\right\rangle=0$ to
$\left\langle\phi\right\rangle\neq 0$, then during the phase transition
$\nu_{R}$ would be massless in the old vacuum, while massive in the new vacuum
bubbles. If the mass gap is much larger than the FOPT temperature $T_{p}$,
then the $\nu_{R}$’s that have penetrated into the new vacuum will be out of
equilibrium and decay rapidly, generating the lepton asymmetry and hence the
BAU. Since $M_{1}\gg T_{p}$, the washout effects are Boltzmann suppressed, and
hence almost all the generated BAU can survive till today. The mechanism is
sketched in Fig. 1. The idea of generating BAU via the fast decay of heavy
particles crossing or being produced at the bubble wall is first proposed in
Ref. Baldes:2021vyz , where the general features are discussed, and a
benchmark model on a color triplet heavy scalar is given. Our work provides
the first detailed study of applying this idea to leptogenesis. This scenario
is distinct from the mechanisms involving the generation and diffusion of
chiral and/or lepton asymmetry in the vicinity of bubble Chung:2009cb ;
Chiang:2016vgf ; Guo:2016ixx ; DeVries:2018aul ; Xie:2020wzn ; Cline:2021dkf
(see also Pascoli:2016gkf ; Long:2017rdo ).222See Refs. Pilaftsis:2008qt ;
Shuve:2017jgj  for leptogenesis during a second-order phase transition.

One might concern that in case of $M_{1}/T_{p}\gg 1$ the RHNs do not have
sufficient energy to cross the wall, as the average kinetic energy of
$\nu_{R}$ is $\mathcal{O}(T_{p})$. Were that true, most of the RHNs will be
trapped in the old vacuum Hong:2020est ; Baker:2021nyl ; Kawana:2021tde ;
Baker:2021sno ; Arakawa:2021wgz , and only a tiny fraction of them can be
“filtered” to the new vacuum Baker:2019ndr ; Chway:2019kft ; Chao:2020adk ;
Baker:2021zsf , resulting in a much suppressed $\nu_{R}$ number density in the
$\left\langle\phi\right\rangle\neq 0$ phase, and the resultant BAU is also
negligible, as pointed out in Shuve:2017jgj ; Ahmadvand:2021vxs . This issue,
however, can be solved, provided that the bubbles are expanding in an ultra-
relativistic velocity, i.e. $\gamma_{w}\equiv(1-v_{w}^{2})^{-1/2}\gg 1$. In
that case, in the wall rest frame the RHNs have an average kinetic energy
$\mathcal{O}(\gamma_{w}T_{p})$, and almost all the $\nu_{R}$’s can penetrate
into the bubble, yielding an unsuppressed number density $\sim T_{p}^{3}$
inside the new vacuum. We will show that $\gamma_{w}\gg 1$ can be easily
achieved in a supercooling FOPT. See Refs. Katz:2016adq ; Azatov:2020ufh ;
Azatov:2021ifm ; Azatov:2021irb  for other cosmological implications of the
ultra-relativistic walls.

Compared with the conventional thermal leptogenesis scenario, our FOPT
leptogenesis scenario has an enhanced RHN number density ($T_{p}^{3}$ instead
of $(M_{1}T_{p})^{3/2}e^{-M_{1}/T_{p}}$) and does not suffer from thermal
washout effects (which in general suppress the BAU by a factor of
$\mathcal{O}(10^{-2})$). Therefore, naively we expect the CP violating phase
needed by the FOPT scenario is much smaller than that in the conventional
scenario, and hence the FOPT scenario is able to explain the BAU at a lower
$M_{1}\lesssim 10^{9}$ GeV assuming the Davidson-Ibarra bound. However, the
FOPT scenario suffers from the washout and dilution effects after the FOPT.
This is because the ultra-relativistic wall requires a strong FOPT, which
releases a large amount of latent heat and then reheats the Universe to a high
temperature $T_{\rm rh}>T_{p}$. It is difficult to satisfy $M_{1}/T_{\rm
rh}\gg 1$, which is the condition to suppress the washout effect after
reheating. In addition, the generated BAU will be diluted by a factor of
$(T_{p}/T_{\rm rh})^{3}$.

In this article, we will provide a realization of the FOPT leptogenesis
scenario in the classically conformal $B-L$ model with $M_{1}\gtrsim 10^{11}$
GeV, taking account of the reheating washout and dilution effects. While in
the same parameter space, the conventional thermal leptogenesis generates a
BAU much smaller than the observed value. Therefore, our research extends the
parameter space for leptogenesis. This article is organized as follows. Before
moving to the concrete model building, we will first study the dynamics of the
FOPT leptogenesis in Section 2, keeping the discussions as general as we can.
Then Section 3 introduces a concrete extended $B-L$ model and demonstrates the
parameter space realizing a FOPT leptogenesis scenario. The possible
gravitational wave (GW) signals are also studied. Finally, we conclude in
Section 4.

##  2 Dynamics of the FOPT leptogenesis

###  2.1 Basic setup

In this section, we do not specify a concrete model. The discussions apply to
any model that contains the following two features. First, the RHNs
$\nu_{R}^{i}$ have the Majorana Yukawa interaction

$\mathcal{L}\supset-\sum_{i,j}\frac{1}{2}\left(\lambda_{R}^{ij}\bar{\nu}_{R}^{i,c}\nu_{R}^{j}\frac{\phi}{\sqrt{2}}+\text{h.c.}\right),$
(1)

where $i$, $j$ are family indices, $\phi$ is a real scalar field that
experiences a FOPT from $\left\langle\phi\right\rangle=0$ to
$\left\langle\phi\right\rangle=v_{p}$ at temperature $T_{p}$. Therefore, the
RHNs are massless in the old vacuum but obtain masses
$M_{R}^{ij}=\lambda_{R}^{ij}v_{p}/\sqrt{2}$ inside the new vacuum bubble. For
simplicity we set $M_{R}^{ij}={\rm diag}\\{M_{1},M_{2},M_{3}\\}$ and let
$\nu_{R}^{1}$ be the lightest RHN. The second feature is that the RHNs should
couple to the SM leptons and bosons via the Dirac Yukawa interaction

$\mathcal{L}\supset-\sum_{i,j}\left(\lambda_{D}^{ij}\bar{\ell}_{L}^{i}\tilde{H}\nu_{R}^{j}+\text{h.c.}\right),$
(2)

where $\ell_{L}^{i}=(\nu_{L}^{i},e_{L}^{i})^{T}$ is the lepton doublet, and
$\tilde{H}=i\tau^{2}H^{*}$ is the charge conjugation of the Higgs doublet. Eq.
(2) allows the RHNs to decay via
$\nu_{R}^{i}\to\ell_{L}^{j}H/\bar{\ell}_{L}^{j}H^{*}$, and hence to generate
the lepton asymmetry.

The magnitude of the $\lambda_{D}^{ij}$ matrix can be estimated by the seesaw
relation $m_{\nu}\approx|\lambda_{D}|^{2}v_{\rm EW}^{2}/(2M_{R})$ as

$|\lambda_{D}|\approx 10^{-2}\times\left(\frac{M_{R}}{10^{11}~{}{\rm
GeV}}\right)^{1/2}\left(\frac{m_{\nu}}{0.05~{}{\rm eV}}\right)^{1/2},$ (3)

where $v_{\rm EW}=246$ GeV is the Higgs VEV. The CP violating effect is
characterized by the RHN decay width asymmetry

$\epsilon_{i}=\frac{\sum_{j}\Gamma(\nu_{R}^{i}\to\ell_{L}^{j}H)-\Gamma(\nu_{R}^{i}\to\bar{\ell}_{L}^{j}H^{*})}{\sum_{j}\Gamma(\nu_{R}^{i}\to\ell_{L}^{j}H)+\Gamma(\nu_{R}^{i}\to\bar{\ell}_{L}^{j}H^{*})},$
(4)

which is related to the imaginary part of
$(\lambda_{D}\lambda_{D}^{\dagger})^{2}$. A nonzero $\epsilon_{1}$ is needed
for the generation of BAU. According to the Davidson-Ibarra bound
Davidson:2002qv ,

$|\epsilon_{1}|\leqslant\frac{3}{8\pi}\frac{M_{1}(m_{3}-m_{1})}{v_{\rm
EW}^{2}}\approx 10^{-5}\times\left(\frac{M_{1}}{10^{11}~{}{\rm
GeV}}\right)\left(\frac{m_{\nu}}{0.05~{}{\rm eV}}\right).$ (5)

We can see that $\epsilon_{1}$ is quite small even for a rather heavy
$\nu_{R}^{1}$.

Above is the basic setup of the FOPT leptogenesis mechanism. When applying
this mechanism, we allow a concrete model to have more ingredients, such as a
$Z^{\prime}$ boson from the gauged $U(1)_{B-L}$ group or other additional
scalars and fermions. To realize our mechanism, three things must be checked.
First, right after penetration, the RHNs should decay instead of annihilating
with each other, or scattering with the particles in the thermal bath. Second,
as the penetrated RHNs are typically boosted, so are the decay products, and
it is necessary to check that they do not cause additional washout effects for
the BAU. Third, after the FOPT, the Universe is reheated to $T_{\rm rh}$ and
we have to confirm the thermal bath washout effects are still Boltzmann
suppressed even at this temperature. Also, the dilution factor $(T_{p}/T_{\rm
rh})^{3}$ should be included. All those issues will be addressed one by one in
the subsequent subsections.

###  2.2 RHNs right after penetration

In the vicinity of the bubble wall, we can model the bubble expansion as a
one-dimension problem: the wall is a plane perpendicular to the $z$-axis and
moving in a velocity $-v_{w}$ with $v_{w}>0$. The $z\to-\infty$ region is the
old phase with $\left\langle\phi\right\rangle=0$, where the RHNs are assumed
to be in thermal equilibrium. Therefore, the lightest RHN $\nu_{R}^{1}$
follows a boosted massless Fermi-Dirac distribution in the wall frame

$f_{\mathfrak{s}}^{\rm
wa}(p_{x},p_{y},p_{z})=\frac{1}{e^{\gamma_{w}(E_{0}-v_{w}p_{z})/T_{p}}+1},$
(6)

where $E_{0}=\sqrt{p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}$. The corresponding particle
number density is

$n_{\mathfrak{s}}^{\rm
wa}=g_{\nu}\int\frac{d^{3}p}{(2\pi)^{3}}f_{\mathfrak{s}}^{\rm
wa}(p_{x},p_{y},p_{z})=\gamma_{w}\times
g_{\nu}\frac{3\zeta_{3}}{4\pi^{2}}T_{p}^{3}\equiv\gamma_{w}n_{\mathfrak{s}}^{\rm
pl},$ (7)

where $\zeta_{3}\approx 1.202$, and $g_{\nu}=2$ is the spin degeneracy factor.
$n_{\mathfrak{s}}^{\rm wa}$ is enhanced by a factor of $\gamma_{w}$ compared
with $n_{\mathfrak{s}}^{\rm pl}$ in the plasma frame, which can be understood
as the Lorentz contraction of the volume element. Note that we use a
superscript “wa” (“pl”) to label the wall frame (plasma frame), and a
subscript “$\mathfrak{s}$” (“$\mathfrak{h}$”) to label the old vacuum with
$\left\langle\phi\right\rangle=0$ (new vacuum with
$\left\langle\phi\right\rangle=v_{p}$), respectively.

In the wall frame the average $z$-component momentum is $\left\langle
p_{z}\right\rangle^{\rm
wa}_{\mathfrak{s}}=7\pi^{4}v_{w}\gamma_{w}T_{p}/(135\zeta_{3})$, and hence the
RHNs are boosted along the $+z$ direction. If $\gamma_{w}$ is large enough,
$\left\langle p_{z}\right\rangle^{\rm wa}_{\mathfrak{s}}$ is enhanced that
most $\nu_{R}^{1}$’s have sufficient energy to overcome the mass gap $M_{1}$
between the new and old vacua. Hereafter we only consider the
$\gamma_{w}\gtrsim M_{1}/T_{p}\gg 1$ limit, then the $\nu_{R}^{1}$’s can
generally cross the wall and enter the new vacuum. Due to energy conservation
in the wall frame, after crossing the wall, in the new vacuum the average
energy and momentum of $\nu_{R}^{1}$ should be

$\left\langle E\right\rangle^{\rm
wa}_{\mathfrak{h}}\sim\gamma_{w}T_{p},\quad\left\langle
p_{z}\right\rangle^{\rm
wa}_{\mathfrak{h}}\sim\sqrt{\gamma_{w}^{2}T_{p}^{2}-M_{1}^{2}}\sim\gamma_{w}T_{p}-\frac{M_{1}^{2}}{2\gamma_{w}T_{p}}.$
(8)

Transforming back to the plasma frame, one obtains the typical $\nu_{R}^{1}$
energy and momentum after penetrating into the bubble

$\left\langle E\right\rangle^{\rm pl}_{\mathfrak{h}}\sim
M_{1}\frac{M_{1}}{T_{p}},\quad\left\langle p_{z}\right\rangle^{\rm
pl}_{\mathfrak{h}}\sim-M_{1}\frac{M_{1}}{T_{p}},$ (9)

which means in the plasma frame the $\nu_{R}^{1}$’s that have entered the new
vacuum are boosted in the $-z$ direction by a Lorentz factor of
$\gamma_{1}\equiv M_{1}/T_{p}\gg 1$. In other words, in the plasma frame, part
of the wall kinetic energy is converted into the rest mass and kinetic energy
of the RHNs that enter the bubble. This causes the energy loss of the wall and
serves as a source of the friction force acting on the wall, as we will
discuss in Eq. (38).

After entering the new vacuum, a $\nu_{R}^{1}$ may decay, or annihilate with
another $\nu_{R}^{1}$, or scatter with the particles in the plasma. When
calculating these interaction rates, it is convenient to work in the
“$\nu_{R}^{1}$ gas frame”, which is boosted along the $-z$ direction with a
Lorentz factor $\gamma_{1}$. In that frame, the $\nu_{R}$’s are on average at
rest, and with a relative velocity $v^{\rm ga}_{\rm rel}\sim T_{p}/M_{1}$ to
each other, and the number density is $n^{\rm
ga}_{\mathfrak{h}}\approx\gamma_{1}n^{\rm pl}_{\mathfrak{s}}$ Baldes:2021vyz .
In the gas frame, the $\nu_{R}^{1}$ decay rate is

$\Gamma_{D}=\frac{|\lambda_{D}|^{2}}{8\pi}M_{1}\approx\frac{m_{\nu}}{4\pi}\left(\frac{M_{1}}{v_{\rm
EW}}\right)^{2},$ (10)

where we have assumed one-flavor SM final state for simplicity, and the second
approximate equality is from the seesaw relation.

Depending on the concrete model, the $\nu_{R}^{1}$’s can annihilate with each
other via various channels. For example, the Majorana interaction Eq. (1)
induces the annihilation to a pair of scalars, i.e.
$\nu_{R}^{1}\nu_{R}^{1}\to\phi\phi$, if kinematically allowed, while the Dirac
interaction Eq. (2) induces
$\nu_{R}^{1}\nu_{R}^{1}\to\ell_{L}\bar{\ell}_{L}/HH^{*}$. If the model is
embedded into a gauged $U(1)_{B-L}$ group, then the $\nu_{R}^{1}\nu_{R}^{1}\to
Z^{\prime*}\to f\bar{f}$ and $\nu_{R}^{1}\nu_{R}^{1}\to Z^{\prime}Z^{\prime}$
channels may be important, where $Z^{\prime}$ and $f$ denote the $U(1)_{B-L}$
gauge boson and SM fermions, respectively. Then the annihilation rate can be
expressed as

$\Gamma_{\rm ann}=\sum_{X}n_{\mathfrak{h}}^{\rm
ga}\left\langle\sigma_{\nu_{R}^{1}\nu_{R}^{1}\to X}v_{\rm
rel}\right\rangle_{\rm ga},$ (11)

summing over all possible annihilation final states. The subscript “ga” of
$\left\langle\sigma v\right\rangle$ is to remind us that this is an average
performed in the gas frame. $\Gamma_{\rm ann}$ scales as
$T_{p}^{3}/M_{1}^{2}$.

Another possible fate of the penetrated $\nu_{R}^{1}$’s is to scatter with the
particles in the plasma. The Dirac Yukawa interaction can mediate scattering
channels such as $\nu_{R}^{1}\ell_{L}\to q_{L}\bar{t}_{R}$ or
$\nu_{R}^{1}t_{R}\to q_{L}\bar{\ell}_{L}$ and their charge conjugations and
crossings. The corresponding interaction rates are

$\Gamma_{\rm sca}=\sum_{a,X}\gamma_{1}n_{a}^{\rm
pl}\left\langle\sigma_{\nu_{R}^{1}a\to X}\right\rangle_{\rm ga},$ (12)

summing over all possible initial states $a$ and final states $X$. In the gas
frame, the plasma species $a$ is boosted by a Lorentz factor of $\gamma_{1}$,
therefore the number density is enhanced by $\gamma_{1}$ compared with
$n_{a}^{\rm pl}\sim T_{p}^{3}$ in the plasma frame, and we have taken the
relative velocity between $\nu_{R}^{1}$ and $a$ to be approximately 1. The
scattering cross section $\langle\sigma_{\nu_{R}^{1}a\to X}\rangle_{\rm ga}$
scales as $1/M_{1}^{2}$, thus $\Gamma_{\rm sca}\sim T_{p}^{2}/M_{1}$.

For the sake of leptogenesis, we want the $\nu_{R}^{1}$’s to decay rather than
annihilate with each other or scatter with the particles in the plasma, i.e.

$\Gamma_{D}>\Gamma_{\rm ann},\quad\Gamma_{D}>\Gamma_{\rm sca}.$ (13)

Under this condition, the $\nu_{R}^{1}$’s swept by the bubble wall decay
immediately and generate a BAU of

$Y_{B}^{p}=-c_{s}\epsilon_{1}\frac{n_{\mathfrak{s}}^{\rm
pl}}{s}=-c_{s}\epsilon_{1}\frac{135\zeta_{3}}{4\pi^{4}g_{*}},$ (14)

where $s=(2\pi^{2}/45)g_{*}T^{3}$ is the entropy density with $g_{*}\approx
100$ the number of relativistic degrees of freedom, and $c_{s}=28/79$ is the
conversion factor from the lepton asymmetry to the BAU. As the upper limit of
CP asymmetry $\epsilon_{1}$ is constrained by Eq. (5), we see that the maximal
value of BAU is proportional to $M_{1}$.

###  2.3 The boosted decay products of RHNs

In the plasma frame, the $\nu_{R}^{1}$’s in new vacuum are moving along the
$-z$ direction with a typical energy $E_{1}=\gamma_{1}M_{1}=M_{1}^{2}/T_{p}$.
The decay products $\ell_{L}H/\bar{\ell}_{L}H^{*}$ share the same order of
energy and hence are also boosted. These out-of-equilibrium SM particles
interact with other SM particles in the plasma, causing cascade scatterings,
which might reduce the BAU. Following Ref. Baldes:2021vyz , we model the
energy of the particles that in the $n$-th step cascade scattering as
$E_{1}/2^{n}$. The washout effect is mainly from the possibility that the
energetic particles fuse to an on-shell RHN, i.e.
$\ell_{L}H\to\nu_{R}^{1}\to\bar{\ell}_{L}H^{*}$, and the corresponding rate
can be estimated as Baldes:2021vyz

$\Gamma_{\rm
on}\approx\frac{\Gamma_{\ell_{L}H}\Gamma_{\bar{\ell}_{L}H^{*}}}{\Gamma_{D}}\frac{M_{1}T_{p}}{E_{1}^{2}}\exp\left\\{-\frac{M_{1}^{2}}{4E_{1}T_{p}}\right\\}\approx\frac{2^{2n}T_{p}^{3}}{4M_{1}^{3}}\Gamma_{D}e^{-2^{n}/4},$
(15)

where we have approximated
$\Gamma_{\ell_{L}H}\approx\Gamma_{\bar{\ell}_{L}H^{*}}\approx\Gamma_{D}/2$. We
can see that the washout rate decreases very quickly as $n$ increases, so we
only need to account for the first step of scattering, i.e. $n=1$.

Being charged under the SM gauge groups, the boosted $\ell_{L}/\bar{\ell}_{L}$
and $H/H^{*}$ particles also thermalize via the elastic EW scattering with the
SM particles in the plasma. The thermalization rate can be estimated by
calculating the energy loss of a boosted lepton in an elastic scattering with
another SM particle in the thermal bath. The two incoming particles have
momenta

$p_{1}^{\mu}=\left(\frac{E_{1}}{2^{n}},0,0,\frac{E_{1}}{2^{n}}\right),\quad
p_{2}^{\mu}=\left(T_{p},0,0,-T_{p}\right),$ (16)

respectively, and they scatter through exchanging a $t$-channel $W/Z$ boson.
It is straightforward to show that the energy loss of the boosted lepton is
$\delta E_{1}\approx-\hat{t}/(4T_{p})$ in the plasma frame, and the scattering
cross section is

$\frac{d\sigma}{d\hat{t}}=\frac{1}{16\pi\hat{s}^{2}}|i\mathcal{M}|^{2}\approx\frac{1}{16\pi\hat{s}^{2}}\frac{g_{2}^{4}\hat{s}^{2}}{\hat{t}^{2}}=\frac{\pi\alpha_{W}^{2}}{\hat{t}^{2}},$
(17)

where $g_{2}$ is the gauge coupling of the $SU(2)_{L}$ group, and $\hat{s}$,
$\hat{t}$ are the Mandelstam variables. Therefore, the thermalization rate can
be estimated as Baldes:2021vyz

$\Gamma_{\rm th}=\frac{n_{\rm EW}^{\rm
pl}}{E_{1}/2^{n}}\int_{-\hat{s}}^{-m_{W}^{2}}d\hat{t}\frac{d\sigma}{d\hat{t}}\delta
E_{1}=\frac{\zeta_{3}g_{\rm EW}2^{n}\alpha_{W}^{2}T_{p}^{3}}{4\pi
M_{1}^{2}}\ln\frac{3M_{1}^{2}}{5\pi 2^{n}\alpha_{W}T_{p}^{2}},$ (18)

where $n_{\rm EW}^{\rm pl}$ is the number density of the particles that
participate in such EW elastic scattering, and the corresponding number of
degrees of freedom is $g_{\rm EW}=46$ including the SM fermions and gauge
bosons as well as the Higgs doublet. The upper limit of integration of
$\hat{t}$ is set to $-m_{W}^{2}$ to avoid infrared divergence, where
$m_{W}^{2}=20\pi\alpha_{W}T_{p}^{2}/3$ is the thermal mass of the $W$ boson
Weldon:1982aq . We see that $\Gamma_{\rm th}$ increases rapidly with $n$.

To avoid washout from the boosted decay products, we require

$\Gamma_{\rm th}\big{|}_{n=1}>\Gamma_{\rm on}\big{|}_{n=1},\quad\Gamma_{\rm
th}\big{|}_{n=1}>H_{p},$ (19)

where $H_{p}$ is the Hubble constant at the FOPT temperature. Note that
$H_{p}$ is not solely determined by temperature, as the vacuum energy from the
potential could dominate the energy of the Universe in the case of a
supercooling FOPT. Once these inequalities are satisfied, the boosted decay
products $\ell_{L}H/\bar{\ell}_{L}H^{*}$ thermalize very quickly, and the
washout effect is completely negligible.

###  2.4 Reheating after the FOPT completes

The latent heat released from a FOPT will reheat the Universe to a new
temperature $T_{\rm rh}=(1+\alpha)^{1/4}T_{p}$, where $\alpha$ is the ratio of
latent heat to the radiation energy density of the Universe, whose detailed
definition will be given in Section 3.3. Since our scenario needs a strong
FOPT to provide fast moving bubble walls, typically $\alpha\gg 1$, the
reheating temperature could be very high, such that the $B-L$ violating
interactions are active again, erasing the generated $B-L$ asymmetry as the
situation in the conventional thermal leptogenesis.

The first type of dangerous processes is the thermally produced RHNs. For the
inverse decay, i.e. $\ell_{L}H/\bar{\ell}_{L}H^{*}\to\nu_{R}^{i}$, the
simplified Boltzmann equation gives

$\frac{dY_{\nu_{R}^{i}}}{dt}=\frac{1}{s}\int\frac{d^{3}p}{(2\pi)^{3}2E_{i}}e^{-E_{i}/T}M_{i}\Gamma_{D,i}=\frac{\Gamma_{D,i}M_{i}^{2}T}{4\pi^{2}s}K_{1}\left(\frac{M_{i}}{T}\right),$
(20)

where $Y_{\nu_{R}^{i}}=n_{\nu_{R}^{i}}/s$ is the yield of the $i$-th
generation of RHN, $E_{i}\equiv\sqrt{|\textbf{p}|^{2}+M_{i}^{2}}$ is the on-
shell energy, $\Gamma_{D,i}$ is the decay width of $\nu_{R}^{i}$, and $K_{1}$
is the modified Bessel function of the first kind. By this, the inverse decay
rate can be estimated as

$\Gamma_{\rm ID}^{i}=\frac{\Gamma_{D,i}M_{i}^{2}T_{\rm rh}}{4\pi^{2}s_{\rm
rh}}K_{1}\left(\frac{M_{i}}{T_{\rm rh}}\right),$ (21)

where $s_{\rm rh}=s|_{T_{\rm rh}}$. We should have $\Gamma_{\rm ID}^{i}<H_{\rm
rh}$ such that the RHNs are not thermally produced after the FOPT, where the
Hubble constant $H_{\rm rh}=2\pi\sqrt{\pi g_{*}/45}T_{\rm rh}^{2}/M_{\rm Pl}$
with $M_{\rm Pl}=1.22\times 10^{19}$ GeV, as the Universe is in a radiation
era after the FOPT. The rate of RHNs being produced in pair in the plasma can
be estimated as

$\Gamma_{\rm pr}^{i}=\sum_{X}n_{\nu_{R}^{i}}^{\rm
eq}\left\langle\sigma_{\nu_{R}^{i}\nu_{R}^{i}\to
X}\right\rangle=\sum_{X}2\frac{M_{i}^{2}T_{\rm
rh}}{2\pi^{2}}K_{2}\left(\frac{M_{i}}{T_{\rm
rh}}\right)\left\langle\sigma_{\nu_{R}^{i}\nu_{R}^{i}\to X}\right\rangle,$
(22)

where $n_{\nu_{R}^{i}}^{\rm eq}$ is the equilibrium distribution of
$\nu_{R}^{i}$ in the plasma whose concrete expression is given in the second
equality with $K_{2}$ being the modified Bessel function of the second kind.
Depending on the model, the pair production channels could include
$\nu_{R}^{i}\nu_{R}^{i}\to Z^{\prime*}\to f\bar{f}$,
$\nu_{R}^{i}\nu_{R}^{i}\to Z^{\prime}Z^{\prime}$, $\nu_{R}^{i}\nu_{R}^{i}\to
Z^{\prime}\phi$, $\nu_{R}^{i}\nu_{R}^{i}\to\phi\phi$, etc.

We require

$\Gamma_{\rm ID}^{i}<H_{\rm rh},\quad\Gamma_{\rm pr}^{i}<H_{\rm rh},$ (23)

to avoid thermal bath washout after the FOPT reheating. Both $\Gamma_{\rm
ID}^{i}$ and $\Gamma_{\rm pr}^{i}$ are suppressed by the Bessel functions,
which are $K_{j}(z)\sim e^{-z}\sqrt{\pi/(2z)}$ for $z\gg 1$. Therefore,
$M_{i}/T_{\rm rh}\gg 1$ could exponentially suppress those washout effects. In
other words, we need $T_{\rm rh}=(1+\alpha)^{1/4}T_{p}$ still small compared
with the RHN masses; this is, however, in tension with the requirement of a
strong supercooling FOPT which generally leads to $\alpha\gg 1$.

Provided that Eqs. (13), (19) and (23) are satisfied, the FOPT leptogenesis
scenario is realized. Namely, the $\nu_{R}^{1}$’s that have entered the new
vacuum bubble during the FOPT will decay and generate the lepton asymmetry,
which is not washed out by the plasma. The BAU survives today would be

$Y_{B}=-c_{s}\epsilon_{1}\frac{135\zeta_{3}}{4\pi^{4}g_{*}}\left(\frac{T_{p}}{T_{\rm
rh}}\right)^{3},$ (24)

which is diluted by a factor of $(T_{p}/T_{\rm rh})^{3}$ compared to Eq. (14),
due to the entropy production of the FOPT reheating. For a successful FOPT
leptogenesis, $Y_{B}$ should reach the observed BAU, i.e. $Y_{B}^{\rm
obs}\approx 0.9\times 10^{-10}$ ParticleDataGroup:2020ssz .

In summary, in the FOPT leptogenesis scenario, the FOPT should be strong to
provide fast expanding bubbles, which sweep the RHN into the new vacuum.
Therefore, the abundant massless $\nu_{R}^{1}$ density in the old vacuum can
be directly transferred into the new vacuum, where the $\nu_{R}^{1}$’s are so
massive that their out-of-equilibrium decay can generate the BAU without the
washout effects. However, the reheating effects from the strong FOPT might
cause additional washout and dilution effects, and hence the application of
this mechanism requires a highly non-trivial tradeoff between strong FOPT and
reheating. A concrete model that succeeds to realize the FOPT leptogenesis
scenario is given in the next section.

##  3 An extended classically conformal $B-L$ model

###  3.1 The model and particle spectrum

The conventional (or say, minimal) $B-L$ model Davidson:1978pm ;
Marshak:1979fm ; Mohapatra:1980qe ; Davidson:1987mh  is defined by gauging the
$U(1)_{B-L}$ group and introducing three generation of RHNs (with $B-L$
quantum number $X=-1$) for gauge anomaly cancellation, and one complex scalar
field $\Phi=(\phi+i\eta)/\sqrt{2}$ charged as $X=2$ to break the $U(1)_{B-L}$
spontaneously. In this work, we extend the model with one more complex scalar
$S$ which has the same quantum number with $\Phi$. The relevant Lagrangian can
be written as

$\begin{split}\mathcal{L}_{B-L}=&~{}\sum_{i}\bar{\nu}_{R}^{i}i\not{D}\nu_{R}^{i}-\frac{1}{2}\sum_{i,j}\left(\lambda_{R}^{ij}\bar{\nu}_{R}^{i,c}\Phi\nu_{R}^{j}+\text{h.c.}\right)-\sum_{i,j}\left(\lambda_{D}^{ij}\bar{\ell}_{L}^{i}\tilde{H}\nu_{R}^{j}+\text{h.c.}\right)\\\
&~{}+D_{\mu}\Phi^{\dagger}D^{\mu}\Phi+D_{\mu}S^{\dagger}D^{\mu}S-V(\Phi,S)-\frac{1}{4}Z^{\prime}_{\mu\nu}Z^{\prime\mu\nu},\end{split}$
(25)

where $D_{\mu}=\partial_{\mu}-ig_{B-L}XZ_{\mu}^{\prime}$ is the $U(1)_{B-L}$
gauge covariant derivative. For simplicity, we take $\lambda_{R}^{ij}={\rm
diag}\\{\lambda_{R,1},\lambda_{R,2},\lambda_{R,3}\\}$. Note that the SM
fermions are also charged under the $U(1)_{B-L}$ group, with the quarks having
$X=1/3$ and the leptons having $X=-1$. The reason why we have to extend the
minimal $B-L$ model will be given in Section 3.3. In principle, $S$ can also
couple to RHNs via $\bar{\nu}_{R}^{i,c}S\nu_{R}^{j}$; however, as we will see,
$S$ never gets a VEV, thus it does not contribute to the RHN mass. On the
other hand, $S$ can provide extra CP violating phase to $N_{1}$ decay
LeDall:2014too ; Alanne:2018brf . We do not consider such CP asymmetry
enhancement effects here, as they are irrelevant to the core of our FOPT
leptogenesis mechanism.

As for the scalar potential $V(\Phi,S)$, we adopt the classically conformal
assumption Iso:2009ss ; Iso:2009nw ; Das:2015nwk  as it is known that this
kind of potential favors a strong supercooling FOPT Jinno:2016knw ;
Iso:2017uuu ; Marzo:2018nov ; Ellis:2019oqb ; Bian:2019szo ; Ellis:2020nnr ;
Jung:2021vap . At three level, the potential is

$V_{\rm
tree}(\Phi,S)=\lambda_{\phi}|\Phi|^{4}+\lambda_{s}|S|^{4}+\lambda_{\phi
s}|\Phi|^{2}|S|^{2},$ (26)

where only dimensionless quartic couplings are involved. The one-loop
contributions from $Z^{\prime}$ and $\nu_{R}^{i}$ Iso:2009ss ; Iso:2009nw ;
Das:2015nwk  and $S$ Haruna:2019zeu ; Hamada:2020wjh ; Hamada:2021jls ;
Kawana:2022fum  induce a Colman-Weinberg potential for $\Phi$, which in the
unitary gauge can be written as

$V(\phi)=V_{0}+\frac{B}{4}\phi^{4}\left(\ln\frac{\phi}{v_{\phi}}-\frac{1}{4}\right),$
(27)

where

$B=\frac{6}{\pi^{2}}\left(\frac{\lambda_{\phi
s}^{2}}{96}+g_{B-L}^{4}-\sum_{i}\frac{\lambda_{R,i}^{4}}{96}\right),$ (28)

is a positive constant. This potential has the a minimum at
$\left\langle\phi\right\rangle=v_{\phi}\neq 0$, which breaks the $U(1)_{B-L}$
symmetry spontaneously and provides masses for the particles in Eq. (25) as
follows

$M_{Z^{\prime}}=2g_{B-L}v_{\phi},\quad
M_{i}=\lambda_{R,i}\frac{v_{\phi}}{\sqrt{2}},\quad
M_{\phi}=\sqrt{B}v_{\phi},\quad M_{S}=\frac{1}{\sqrt{2}}\sqrt{\lambda_{\phi
s}}v_{\phi}.$ (29)

The vacuum energy is adopted as $V_{0}=Bv_{\phi}^{4}/16$ to have
$V(v_{\phi})=0$.

###  3.2 FOPT and ultra-relativistic bubble walls

At finite temperature, the potential receives corrections from the one-loop
thermal integrals and daisy resummation terms

$\Delta V_{T}(\phi,T)=2\frac{T^{4}}{2\pi^{2}}J_{B}\left(\frac{\lambda_{\phi
s}\phi^{2}}{2T^{2}}\right)+3\frac{T^{4}}{2\pi^{2}}J_{B}\left(\frac{4g_{B-L}^{2}\phi^{2}}{T^{2}}\right)+2\sum_{i}\frac{T^{4}}{2\pi^{2}}J_{F}\left(\frac{\lambda_{R,i}^{2}\phi^{2}}{2T^{2}}\right)\\\
-2\frac{T}{12\pi}\frac{\lambda_{\phi
s}^{3/2}}{2^{3/2}}\left[\left(\phi^{2}+\frac{T^{2}}{12}\right)^{3/2}-\phi^{3}\right]-\frac{2g_{B-L}^{3}}{3\pi}T\left[\left(\phi^{2}+T^{2}\right)^{3/2}-\phi^{3}\right],$
(30)

where thermal integral functions are defined as

$J_{B/F}(y)=\pm\int_{0}^{\infty}dxx^{2}\ln\left(1\mp
e^{-\sqrt{x^{2}+y}}\right).$ (31)

Therefore, the complete one-loop thermal potential is

$V_{T}(\phi,T)=V(\phi)+\Delta V_{T}(\phi,T),$ (32)

which can trigger a FOPT in the early Universe. The FOPT of the minimal
classically conformal $B-L$ model has already been extensively studied
Jinno:2016knw ; Iso:2017uuu ; Marzo:2018nov ; Ellis:2019oqb ; Bian:2019szo ;
Ellis:2020nnr , and in this work we use homemade codes to derive the FOPT
dynamics of our extended $B-L$ model.

When $T$ is sufficiently high, the Universe stays in the $U(1)_{B-L}$
preserving vacuum $\phi=0$. At the critical temperature $T_{c}$, the potential
$V_{T}(\phi,T)$ develops another degenerate vacuum in $\phi=v_{c}$. When $T$
falls below $T_{c}$, the $U(1)_{B-L}$ breaking vacuum is energetically
preferred, i.e. $\left\langle\phi\right\rangle=v(T)$ becomes the true vacuum
and we have $v(T_{c})=v_{c}$ and $v(0)=v_{\phi}$. The Universe then acquires a
decay probability Linde:1981zj

$\Gamma(T)\sim T^{4}\left(\frac{S_{3}(T)}{2\pi
T}\right)^{3/2}e^{-S_{3}(T)/T},$ (33)

to the true vacuum, where $S_{3}(T)$ is the action of the bounce solution,
which we numerically resolve from $V_{T}(\phi,T)$ based on the shooting
algorithm. When the decay probability in a Hubble volume and a Hubble time
scale reaches $\mathcal{O}(1)$, new vacuum bubbles start to nucleate. Given
$\Gamma(T)$, the volume fraction of the old vacuum in the Universe is
Guth:1979bh ; Guth:1981uk

$p(T)\equiv
e^{-I(T)}=\exp\left\\{-\frac{4\pi}{3}\int_{T}^{T_{c}}dT^{\prime}\frac{\Gamma(T^{\prime})}{T^{\prime
4}H(T^{\prime})}\left[\int_{T}^{T^{\prime}}d\tilde{T}\frac{1}{H(\tilde{T})}\right]^{3}\right\\},$
(34)

where we have taken the bubble velocity $v_{w}\to 1$, and the Hubble constant
is given by the Friedmann equation

$H^{2}(T)=\frac{8\pi}{3M_{\rm
Pl}^{2}}\left(\frac{\pi^{2}}{30}g_{*}T^{4}+V_{0}\right).$ (35)

By definition $p(T_{c})=1$. When $T$ decreases, $p(T)\to 0$, and the Universe
transfers entirely to the new vacuum, completing the FOPT.

The milestone that the new vacuum bubbles form an infinite connected cluster
is called percolation, and it happens at $p(T_{p})=0.71$ rintoul1997precise ,
which defines the percolation temperature $T_{p}$ and VEV $v_{p}=v(T_{p})$. We
will calculate the leptogenesis at this temperature.333Note that the large
vacuum energy released from a supercooled FOPT can lead to a short vacuum
domination era. We have checked that the FOPT can complete via verifying
Ellis:2018mja  $H(T)\left(3+T\frac{dI(T)}{dT}\right)\Big{|}_{T_{p}}<0,$ (36)
where $I(T)$ is defined in Eq. (34). Define

$\Delta V(T)=V_{T}(0,T)-V_{T}(v(T),T),$ (37)

as the positive free energy difference between the true and false vacuum, and
let $\Delta V_{p}\equiv\Delta V(T_{p})$. The behavior of wall velocity is
determined by vacuum pressure $\Delta V_{p}$ and the leading-order (LO)
friction Bodeker:2009qy

$\mathcal{P}_{1\to 1}=\left(\lambda_{\phi
s}+12g_{B-L}^{2}+\frac{1}{2}\sum_{i}\lambda_{R,i}^{2}\right)\frac{v_{p}^{2}T_{p}^{2}}{24},$
(38)

which comes from the mass differences of $S$, $Z^{\prime}$ and RHNs between
the two sides of the bubble wall. If

$\Delta V_{p}>\mathcal{P}_{1\to 1},$ (39)

then the wall will be accelerated up to a high velocity that is very close to
the speed of light, providing necessary condition for our FOPT leptogenesis
scenario.

When $\gamma_{w}\gg 1$, the beyond LO contributions to the friction force
become important. Ref. Bodeker:2017cim  performs the first next-to-leading-
order (NLO) calculation, showing that the emission of gauge bosons when
particles cross the wall can induce a friction force scaling as
$\mathcal{P}_{1\to 2}\propto\gamma_{w}$, preventing the bubble walls from
runaway. Recently, friction force on the wall is studied in many literatures
Hoche:2020ysm ; Gouttenoire:2021kjv ; Ai:2021kak ; Cai:2020djd ; Wang:2022txy
; Laurent:2022jrs , and we take the results of Refs. Hoche:2020ysm ;
Gouttenoire:2021kjv  to calculate the evolution of bubble walls. While both
two works consider the resummation effect of the $1\to N$ emission of gauge
bosons, they obtain different friction pressures $\mathcal{P}_{1\to N}$. Ref.
Hoche:2020ysm  shows $\mathcal{P}_{1\to N}\propto\gamma_{w}^{2}$, however,
Ref. Gouttenoire:2021kjv  gives $\mathcal{P}_{1\to N}\propto\gamma_{w}$. More
concretely, applying to our model we find

$\mathcal{P}_{1\to N}^{\text{\scriptsize\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Hoche:2020ysm}{\@@citephrase{(}}{\@@citephrase{)}}}}}\approx\gamma_{w}^{2}\left[\left(4\times
2\times 3\times\frac{1}{9}+4+2\right)\times 3\right]\frac{3\zeta_{3}(2\ln
2-1)}{32\pi^{4}}g_{B-L}^{2}T_{p}^{4},$ (40)

and

$\mathcal{P}_{1\to N}^{\text{\scriptsize\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Gouttenoire:2021kjv}{\@@citephrase{(}}{\@@citephrase{)}}}}}\approx\gamma_{w}\left[\left(4\times
2\times 3\times\frac{1}{9}+4+2\right)\times
3\right]\frac{3\kappa\zeta_{3}}{8\pi^{4}}g_{B-L}^{3}v_{p}T_{p}^{3}\ln\frac{v_{p}}{T_{p}},$
(41)

where only the dominant SM fermion contributions are included, and
$\kappa\approx 4$.

The wall stops accelerating when the friction force balances the vacuum
pressure, i.e. $\Delta V_{p}=\mathcal{P}_{1\to 1}+\mathcal{P}_{1\to N}$.
Therefore, given the resummed friction force $\mathcal{P}_{1\to N}$, we are
able to derive the terminal wall velocity

$\gamma_{\rm eq}^{\text{\scriptsize\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Hoche:2020ysm}{\@@citephrase{(}}{\@@citephrase{)}}}}}=\sqrt{\frac{\Delta
V_{p}-\mathcal{P}_{1\to 1}}{\mathcal{P}_{1\to
N}^{\text{\scriptsize\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Hoche:2020ysm}{\@@citephrase{(}}{\@@citephrase{)}}}}}/\gamma_{w}^{2}}};\quad\gamma_{\rm
eq}^{\text{\scriptsize\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Gouttenoire:2021kjv}{\@@citephrase{(}}{\@@citephrase{)}}}}}=\frac{\Delta
V_{p}-\mathcal{P}_{1\to 1}}{\mathcal{P}_{1\to
N}^{\text{\scriptsize\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Gouttenoire:2021kjv}{\@@citephrase{(}}{\@@citephrase{)}}}}}/\gamma_{w}}.$
(42)

However, the wall might have not yet reached the terminal velocity at $T_{p}$.
We use the method from Refs. Ellis:2019oqb ; Ellis:2020nnr  to evaluate the
evolution of the wall velocity and confirm that for the parameter space of
interest $\gamma_{w}\big{|}_{T_{p}}\gg M_{1}/T_{p}$ is indeed satisfied for
either choice of $\mathcal{P}_{1\to N}$, and hence all the $\nu_{R}^{1}$’s can
penetrate the wall, which is the necessary condition for the FOPT
leptogenesis.

###  3.3 FOPT leptogenesis and the GW signals

Given the FOPT environment with ultra-relativistic bubble walls that can sweep
all the RHNs into the new vacuum, we apply Eqs. (13), (19) and (23) in Section
2 to our model Eq. (25) to ensure that the penetrated $\nu_{R}^{1}$’s indeed
decay before annihilating and scattering, and the boosted decay products
thermalize instead of erasing the $B-L$ asymmetry, and the reheating
temperature is still significantly below the RHN masses so that the thermal
washout processes are Boltzmann suppressed.

As for the $\nu_{R}^{1}$’s right after penetration, the possible annihilation
channels include $\nu_{R}^{1}\nu_{R}^{1}\to\ell_{L}\bar{\ell}_{L}/HH^{*}$,
$\nu_{R}^{1}\nu_{R}^{1}\to Z^{\prime*}\to f\bar{f}$ with $f$ being the SM
fermions, and $\nu_{R}^{1}\nu_{R}^{1}\to
Z^{\prime}Z^{\prime}/Z^{\prime}\phi/\phi\phi$. We calculate the corresponding
annihilation rates using the FeynCalc package Mertig:1990an ;
Shtabovenko:2016sxi ; Shtabovenko:2020gxv  and check that they are consistent
with those in Refs. Blanchet:2009bu ; Heeck:2016oda ; Duerr:2016tmh . The
$\nu_{R}^{1}$ scattering processes include $\nu_{R}^{1}\ell_{L}\to
q_{L}\bar{t}_{R}$, $\nu_{R}^{1}t_{R}\to q_{L}\bar{\ell}_{L}$ and their charge
conjugations and crossing diagrams. $\Gamma_{D}>\Gamma_{\rm ann}$ and
$\Gamma_{D}>\Gamma_{\rm sca}$ are required for the fast decay of
$\nu_{R}^{1}$, see Eq. (13). After decay, $\Gamma_{\rm th}>H_{p}$ and
$\Gamma_{\rm th}>\Gamma_{\rm on}$ are needed for the decay products to
thermalize quickly and do not reduce the generated $B-L$ asymmetry, see Eq.
(19). The reheating temperature is $T_{\rm rh}=(1+\alpha)^{1/4}T_{p}$, where

$\alpha=\frac{1}{g_{*}\pi^{2}T_{p}^{4}/30}\left(\Delta
V(T)-\frac{T}{4}\frac{\partial\Delta V(T)}{\partial T}\right)\Big{|}_{T_{p}},$
(43)

is the ratio of the FOPT latent heat to the radiation energy density. After
confirming that the washout effects are suppressed even after reheating, i.e.
Eq. (23), the eventual generated BAU is given by Eq. (24).

As stated in the Introduction, it is challenging to strike a balance between a
strong FOPT and a not-so-strong reheating. A supercooling FOPT can provide
fast-moving bubble walls, but the resultant large latent heat will push the
Universe to a high $T_{\rm rh}$ that the thermal washout processes are active
again, reducing any $B-L$ asymmetry generated during the FOPT. Especially, we
find that it is in general difficult for the minimal classically conformal
$B-L$ model Iso:2009ss ; Iso:2009nw  to realize the mechanism. In that model,
the coefficient $B$ in Eq. (28) is determined only by $g_{B-L}$ and
$\lambda_{R,i}$ that

$B\xrightarrow[B-L]{\rm
Minimal}\frac{6}{\pi^{2}}\left(g_{B-L}^{4}-\sum_{i}\frac{\lambda_{R,i}^{4}}{96}\right)=\frac{3}{8\pi^{2}v_{\phi}^{4}}\left(M_{Z^{\prime}}^{4}-\sum_{i}\frac{2M_{i}^{4}}{3}\right).$
(44)

Therefore, $M_{Z^{\prime}}\gtrsim M_{i}$ is required for a positive $B$ to
ensure the vacuum stability. In addition, the FOPT requires a sizable $B$ to
generate the potential barrier, and this implies a sizable $g_{B-L}$ and hence
$M_{Z^{\prime}}$ dominates Eq. (44). On the other hand, for a supercooling
FOPT, the cosmic energy density is dominated by the vacuum energy and hence
$T_{\rm rh}\sim V_{0}^{1/4}\sim B^{1/4}v_{\phi}\sim g_{B-L}v_{\phi}\sim
M_{Z^{\prime}}$. To have $M_{1}/T_{\rm rh}\gg 1$ after reheating, we must have
$M_{1}/M_{Z^{\prime}}\gg 1$, which is in contrast with the vacuum stability
and FOPT conditions. We confirm this qualitative argument by a detailed
numerical scan. Therefore, we extend the model with one extra scalar $S$, as
we did in Eq. (25). In this new model, the contribution to $B$ can be
dominated by the scalar portal coupling $\lambda_{\phi s}$, and the reheating
temperature is no longer directly related to $M_{Z^{\prime}}$.

Figure 2: The allowed parameter space of the FOPT leptogenesis scenario is
shown in white region, for $M_{1}=2.5\times 10^{11}$ GeV, $\lambda_{R,1}=0.3$
and $\lambda_{R,2}=\lambda_{R,3}=4\lambda_{R,1}$. The blue and orange shaded
regions are excluded by thermal washout and dilution effects after the FOPT
reheating, respectively. The $M_{1}/T_{p}$ and $\alpha$ contours are shown in
the left and right panels, respectively. The green star is the benchmark
adopted for GW calculation, see Fig. 3 for details.

For our extended $B-L$ model, we start from $M_{1}=10^{9}$ GeV and gradually
increase it to seek for viable parameter space for the FOPT leptogenesis. The
most stringent constraints for the scenario come from the washout effects
after reheating, especially $\nu_{R}^{i}\nu_{R}^{i}\to Z^{\prime}\phi$ and
$\nu_{R}^{i}\nu_{R}^{i}\to Z^{\prime*}\to f\bar{f}$. Even in case that the
reheating washout effects are suppressed, the BAU is usually diluted by the
large $\alpha$ to be lower than the experimentally observed value. Therefore,
we have to increase $M_{1}$ to $M_{1}\gtrsim 10^{11}$ GeV to generate a large
BAU. An example is shown in Fig. 2 with

$M_{1}=2.5\times 10^{11}~{}{\rm
GeV},\quad\lambda_{R,1}=0.3,\quad\lambda_{R,2}=\lambda_{R,3}=4\lambda_{R,1},$
(45)

fixed, and scanning over $\lambda_{\phi s}$ and $g_{B-L}$. The parameter space
with successful FOPT leptogenesis, i.e. can provide $Y_{B}\geqslant Y_{B}^{\rm
obs}$ for the $\epsilon_{1}$ within the Davidson-Ibarra bound, is plotted as
the white region covered by the $M_{1}/T_{p}$ (left panel) and $\alpha$ (right
panel) contours. We see $\alpha\gg 1$ for most of the parameter space,
implying a strong FOPT with vacuum energy dominance. The blue shaded region
cannot realize FOPT leptogenesis because the thermal washout processes are
active after reheating, where the $g_{B-L}\gtrsim 0.1$ region is ruled out by
the $\nu_{R}^{i}\nu_{R}^{i}\to Z^{\prime*}\to f\bar{f}$ annihilation, while
the $\lambda_{\phi s}\gtrsim 3.9$ region is excluded by the
$\nu_{R}^{i}\nu_{R}^{i}\to Z^{\prime}\phi$ annihilation. If $\lambda_{\phi s}$
is too small, the FOPT strength is so strong that the entropy production
during reheating dilutes the BAU to an unacceptable low value, as covered by
the orange shaded region in the figure. We have checked that, without the
FOPT, the same parameter space in Fig. 2 cannot realize a conventional thermal
leptogenesis in the $B-L$ model, which typically requires a CP asymmetry
$\mathcal{O}(30)$ larger than the Davidson-Ibarra bound due to the large
thermal washout effects from processes involving $Z^{\prime}$ and $\phi$.
Therefore, our model has opened up new parameter space for a novel kind of
leptogenesis.

In this scenario, the relevant energy scale is about $10^{11}$ GeV, which is
not accessible at any current or near-future colliders. However, the GWs as
byproducts of the $U(1)_{B-L}$ breaking may help to probe the scenario,
although those signals could not serve as smoking guns for this specific
mechanism. Thus, we briefly comment on the possible signals. In our scenario,
there are two sources of the GWs: first, the $U(1)_{B-L}$ FOPT itself
generates GWs via vacuum bubble collision, sound waves and magneto-
hydrodynamics (MHD) turbulence in the plasma Jinno:2016knw ; Iso:2017uuu ;
Marzo:2018nov ; Ellis:2019oqb ; Bian:2019szo ; Ellis:2020nnr ; second, the
cosmic strings forming after the $U(1)_{B-L}$ breaking keep emitting GWs
during the evolution of the Universe Buchmuller:2013lra ; Dror:2019syi ;
Fornal:2020esl ; Samanta:2020cdk ; Masoud:2021prr ; Bian:2021vmi ;
Buchmuller:2021mbb .

As an illustration, we adopt $\lambda_{\phi s}=3.5$ and $g_{B-L}=0.05$ as a
benchmark (shown as the green star in Fig. 2) to calculate the GW spectrum
today after the cosmological redshift. For the FOPT GWs, $T_{p}=6.1\times
10^{10}$ GeV, and the energy budget depends on the evolution of the wall
velocity, thus we tried both schemes from Ref. Hoche:2020ysm  (with
$\mathcal{P}_{1\to N}\propto\gamma_{w}^{2}$) and Ref. Gouttenoire:2021kjv
(with $\mathcal{P}_{1\to N}\propto\gamma_{w}$). For the former case, as the
friction increases rapidly with $\gamma_{w}$, the bubble walls have reached
the terminal velocity at $T_{p}$, thus the sound wave and MHD contributions
dominate Ellis:2018mja , and we make use of the efficiency factor $\kappa_{V}$
derived in Ref. Espinosa:2010hh ; while for the latter case, the walls are
still accelerating at $T_{p}$, and hence the bubble collision contribution
dominates, and we adopt the method in Refs. Ellis:2019oqb ; Ellis:2020nnr  to
obtain the efficiency factor $\kappa_{\rm col}$. With the efficiency
coefficients in hand, the FOPT GW spectra are evaluated using the numerical
formulae in Refs. Caprini:2015zlo ; Caprini:2019egz .444In the sound wave
dominant case, the extra suppression factor from the finite duration of sound
wave period is taken into account Ellis:2018mja ; Ellis:2019oqb ; Guo:2020grp
. For the cosmic strings GWs, the spectrum is determined by the dimensionless
combination $G\mu$, where $G=1/M_{\rm Pl}^{2}$ is the Newton’s constant of
gravitation, and $\mu\sim v_{\phi}^{2}$ is the tension of the strings. For our
benchmark, $G\mu\approx 10^{-14}$, and we use the numerical results in Refs.
Auclair:2019wcv ; Blanco-Pillado:2017oxo ; Binetruy:2012ze ; Blanco-
Pillado:2013qja  to derive the GW spectrum.555See Refs. Bian:2022tju ;
Zhao:2022cnn  for recent research on cosmic string GW simulations and
experimental constraints.

Figure 3: The GW spectra for the benchmark $\lambda_{\phi s}=3.5$ and
$g_{B-L}=0.05$ (marked as green star in Fig. 2). Both the spectra from cosmic
strings and FOPT are shown, and in the latter case both two possibilities of
$\mathcal{P}_{1\to N}\propto\gamma_{w}^{2}$ (sound wave dominant) and
$\mathcal{P}_{1\to N}\propto\gamma_{w}$ (bubble collision dominant) are
considered.

The GW spectra for our benchmark point are given in Fig. 3, where the expected
sensitivity curves for the space-based laser interferometers LISA LISA:2017pwj
, TianQin TianQin:2015yph ; Hu:2017yoc ; TianQin:2020hid , Taiji Hu:2017mde ;
Ruan:2018tsw , BBO Crowder:2005nr  and DECIGO Kawamura:2011zz , and the
ground-based interferometers LIGO LIGOScientific:2014qfs ;
LIGOScientific:2019vic , CE Reitze:2019iox  and ET Punturo:2010zz ;
Hild:2010id ; Sathyaprakash:2012jk  are also shown. We first see that the FOPT
GWs spectra peak at $\sim 10^{5}$ Hz, which is too high to be detected by the
near-future instruments. For heavier RHNs and hence higher FOPT scales, the
typical peak frequency is even higher and hence more difficult to probe.
However, the cosmic string GW spectrum is rather flat and could be reached by
quite a few future detectors such as BBO, DECIGO, CE and ET. For heavier RHNs,
$G\mu$ is larger, and the signal strength becomes stronger that LISA, TianQin
and Taiji can also probe the scenario. Therefore, we conclude that the cosmic
strings induced GWs are hopeful to be seen at the future detectors, although
this is a general feature of all the high-scale $U(1)$ breaking new physics
models, not specifically for our extended $B-L$ model.

##  4 Conclusion

In this article, we apply the mechanism of baryogenesis induced by ultra-
relativistic bubble walls to the leptogenesis case. After giving a general
discussion on the dynamics of such a scenario, we build an extended $B-L$
model to demonstrate the viable parameter space realizing the mechanism. We
have shown that the mechanism requires a trade-off between the strength of
FOPT and the level of reheating, and the successful FOPT leptogenesis requires
RHN mass $\gtrsim 10^{11}$ GeV assuming the Davidson-Ibarra bound. Meanwhile,
we verify that the same parameter space cannot generate sufficient BAU within
the conventional thermal leptogenesis mechanism. Therefore, our research
provides a novel approach to realize leptogenesis. While the frequency of GW
signals from FOPT is too high to be probed at the detectors, the GWs emitted
by the cosmic strings from $U(1)_{B-L}$ breaking might be seen at the near-
future detectors such as LISA, TianQin, Taiji, CE and ET.

Note added. Soon after the completion of this manuscript, Ref.
Dasgupta:2022isg  appears, which applies the same mechanism to the minimal
classically conformal $B-L$ model in the resonant leptogenesis regime.

###### Acknowledgements.

We would like to thank Iason Baldes, Ville Vaskonen and Shao-Jiang Wang for
the very useful and inspiring discussions. This work is supported by the
National Science Foundation under grant number PHY-1820891, and PHY-2112680,
and the University of Nebraska Foundation.

## References

  * (1) M. Fukugita and T. Yanagida, “Baryogenesis Without Grand Unification,” _Phys. Lett. B_ 174 (1986) 45–47.
  * (2) M. A. Luty, “Baryogenesis via leptogenesis,” _Phys. Rev. D_ 45 (1992) 455–465.
  * (3) S. Davidson, E. Nardi and Y. Nir, “Leptogenesis,” _Phys. Rept._ 466 (2008) 105–177, [0802.2962].
  * (4) R. Davis, “A review of the Homestake solar neutrino experiment,” _Prog. Part. Nucl. Phys._ 32 (1994) 13–32.
  * (5) Super-Kamiokande collaboration, Y. Fukuda et al., “Evidence for oscillation of atmospheric neutrinos,” _Phys. Rev. Lett._ 81 (1998) 1562–1567, [hep-ex/9807003].
  * (6) KamLAND collaboration, K. Eguchi et al., “First results from KamLAND: Evidence for reactor anti-neutrino disappearance,” _Phys. Rev. Lett._ 90 (2003) 021802, [hep-ex/0212021].
  * (7) P. Minkowski, “$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?,” _Phys. Lett. B_ 67 (1977) 421–428.
  * (8) W. Buchmuller, R. D. Peccei and T. Yanagida, “Leptogenesis as the origin of matter,” _Ann. Rev. Nucl. Part. Sci._ 55 (2005) 311–355, [hep-ph/0502169].
  * (9) S. Davidson and A. Ibarra, “A Lower bound on the right-handed neutrino mass from leptogenesis,” _Phys. Lett. B_ 535 (2002) 25–32, [hep-ph/0202239].
  * (10) M. Flanz, E. A. Paschos, U. Sarkar and J. Weiss, “Baryogenesis through mixing of heavy Majorana neutrinos,” _Phys. Lett. B_ 389 (1996) 693–699, [hep-ph/9607310].
  * (11) A. Pilaftsis, “CP violation and baryogenesis due to heavy Majorana neutrinos,” _Phys. Rev. D_ 56 (1997) 5431–5451, [hep-ph/9707235].
  * (12) A. Pilaftsis and T. E. J. Underwood, “Resonant leptogenesis,” _Nucl. Phys. B_ 692 (2004) 303–345, [hep-ph/0309342].
  * (13) B. Dev, M. Garny, J. Klaric, P. Millington and D. Teresi, “Resonant enhancement in leptogenesis,” _Int. J. Mod. Phys. A_ 33 (2018) 1842003, [1711.02863].
  * (14) E. K. Akhmedov, V. A. Rubakov and A. Y. Smirnov, “Baryogenesis via neutrino oscillations,” _Phys. Rev. Lett._ 81 (1998) 1359–1362, [hep-ph/9803255].
  * (15) M. Drewes, B. Garbrecht, P. Hernandez, M. Kekic, J. Lopez-Pavon, J. Racker et al., “ARS Leptogenesis,” _Int. J. Mod. Phys. A_ 33 (2018) 1842002, [1711.02862].
  * (16) I. Baldes, S. Blasi, A. Mariotti, A. Sevrin and K. Turbang, “Baryogenesis via relativistic bubble expansion,” _Phys. Rev. D_ 104 (2021) 115029, [2106.15602].
  * (17) D. J. H. Chung, B. Garbrecht, M. J. Ramsey-Musolf and S. Tulin, “Lepton-mediated electroweak baryogenesis,” _Phys. Rev. D_ 81 (2010) 063506, [0905.4509].
  * (18) C.-W. Chiang, K. Fuyuto and E. Senaha, “Electroweak Baryogenesis with Lepton Flavor Violation,” _Phys. Lett. B_ 762 (2016) 315–320, [1607.07316].
  * (19) H.-K. Guo, Y.-Y. Li, T. Liu, M. Ramsey-Musolf and J. Shu, “Lepton-Flavored Electroweak Baryogenesis,” _Phys. Rev. D_ 96 (2017) 115034, [1609.09849].
  * (20) J. De Vries, M. Postma and J. van de Vis, “The role of leptons in electroweak baryogenesis,” _JHEP_ 04 (2019) 024, [1811.11104].
  * (21) K.-P. Xie, “Lepton-mediated electroweak baryogenesis, gravitational waves and the $4\tau$ final state at the collider,” _JHEP_ 02 (2021) 090, [2011.04821].
  * (22) J. M. Cline and B. Laurent, “Electroweak baryogenesis from light fermion sources: A critical study,” _Phys. Rev. D_ 104 (2021) 083507, [2108.04249].
  * (23) S. Pascoli, J. Turner and Y.-L. Zhou, “Baryogenesis via leptonic CP-violating phase transition,” _Phys. Lett. B_ 780 (2018) 313–318, [1609.07969].
  * (24) A. J. Long, A. Tesi and L.-T. Wang, “Baryogenesis at a Lepton-Number-Breaking Phase Transition,” _JHEP_ 10 (2017) 095, [1703.04902].
  * (25) A. Pilaftsis, “Electroweak Resonant Leptogenesis in the Singlet Majoron Model,” _Phys. Rev. D_ 78 (2008) 013008, [0805.1677].
  * (26) B. Shuve and C. Tamarit, “Phase Transitions and Baryogenesis From Decays,” _JHEP_ 10 (2017) 122, [1704.01979].
  * (27) J.-P. Hong, S. Jung and K.-P. Xie, “Fermi-ball dark matter from a first-order phase transition,” _Phys. Rev. D_ 102 (2020) 075028, [2008.04430].
  * (28) M. J. Baker, M. Breitbach, J. Kopp and L. Mittnacht, “Primordial Black Holes from First-Order Cosmological Phase Transitions,” 2105.07481.
  * (29) K. Kawana and K.-P. Xie, “Primordial black holes from a cosmic phase transition: The collapse of Fermi-balls,” _Phys. Lett. B_ 824 (2022) 136791, [2106.00111].
  * (30) M. J. Baker, M. Breitbach, J. Kopp and L. Mittnacht, “Detailed Calculation of Primordial Black Hole Formation During First-Order Cosmological Phase Transitions,” 2110.00005.
  * (31) J. Arakawa, A. Rajaraman and T. M. P. Tait, “Annihilogenesis,” _JHEP_ 08 (2022) 078, [2109.13941].
  * (32) M. J. Baker, J. Kopp and A. J. Long, “Filtered Dark Matter at a First Order Phase Transition,” _Phys. Rev. Lett._ 125 (2020) 151102, [1912.02830].
  * (33) D. Chway, T. H. Jung and C. S. Shin, “Dark matter filtering-out effect during a first-order phase transition,” _Phys. Rev. D_ 101 (2020) 095019, [1912.04238].
  * (34) W. Chao, X.-F. Li and L. Wang, “Filtered pseudo-scalar dark matter and gravitational waves from first order phase transition,” _JCAP_ 06 (2021) 038, [2012.15113].
  * (35) M. J. Baker, M. Breitbach, J. Kopp, L. Mittnacht and Y. Soreq, “Filtered Baryogenesis,” 2112.08987.
  * (36) M. Ahmadvand, “Filtered asymmetric dark matter during the Peccei-Quinn phase transition,” _JHEP_ 10 (2021) 109, [2108.00958].
  * (37) A. Katz and A. Riotto, “Baryogenesis and Gravitational Waves from Runaway Bubble Collisions,” _JCAP_ 11 (2016) 011, [1608.00583].
  * (38) A. Azatov and M. Vanvlasselaer, “Bubble wall velocity: heavy physics effects,” _JCAP_ 01 (2021) 058, [2010.02590].
  * (39) A. Azatov, M. Vanvlasselaer and W. Yin, “Dark Matter production from relativistic bubble walls,” _JHEP_ 03 (2021) 288, [2101.05721].
  * (40) A. Azatov, M. Vanvlasselaer and W. Yin, “Baryogenesis via relativistic bubble walls,” _JHEP_ 10 (2021) 043, [2106.14913].
  * (41) H. A. Weldon, “Covariant Calculations at Finite Temperature: The Relativistic Plasma,” _Phys. Rev. D_ 26 (1982) 1394.
  * (42) Particle Data Group collaboration, P. A. Zyla et al., “Review of Particle Physics,” _PTEP_ 2020 (2020) 083C01.
  * (43) A. Davidson, “$B-L$ as the fourth color within an $\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{R}\times\mathrm{U}(1)$ model,” _Phys. Rev. D_ 20 (1979) 776.
  * (44) R. E. Marshak and R. N. Mohapatra, “Quark - Lepton Symmetry and B-L as the U(1) Generator of the Electroweak Symmetry Group,” _Phys. Lett. B_ 91 (1980) 222–224.
  * (45) R. N. Mohapatra and R. E. Marshak, “Local B-L Symmetry of Electroweak Interactions, Majorana Neutrinos and Neutron Oscillations,” _Phys. Rev. Lett._ 44 (1980) 1316–1319.
  * (46) A. Davidson and K. C. Wali, “Universal Seesaw Mechanism?,” _Phys. Rev. Lett._ 59 (1987) 393.
  * (47) M. Le Dall and A. Ritz, “Leptogenesis and the Higgs Portal,” _Phys. Rev. D_ 90 (2014) 096002, [1408.2498].
  * (48) T. Alanne, T. Hugle, M. Platscher and K. Schmitz, “Low-scale leptogenesis assisted by a real scalar singlet,” _JCAP_ 03 (2019) 037, [1812.04421].
  * (49) S. Iso, N. Okada and Y. Orikasa, “Classically conformal $B^{-}$ L extended Standard Model,” _Phys. Lett. B_ 676 (2009) 81–87, [0902.4050].
  * (50) S. Iso, N. Okada and Y. Orikasa, “The minimal B-L model naturally realized at TeV scale,” _Phys. Rev. D_ 80 (2009) 115007, [0909.0128].
  * (51) A. Das, N. Okada and N. Papapietro, “Electroweak vacuum stability in classically conformal B-L extension of the Standard Model,” _Eur. Phys. J. C_ 77 (2017) 122, [1509.01466].
  * (52) R. Jinno and M. Takimoto, “Probing a classically conformal B-L model with gravitational waves,” _Phys. Rev. D_ 95 (2017) 015020, [1604.05035].
  * (53) S. Iso, P. D. Serpico and K. Shimada, “QCD-Electroweak First-Order Phase Transition in a Supercooled Universe,” _Phys. Rev. Lett._ 119 (2017) 141301, [1704.04955].
  * (54) C. Marzo, L. Marzola and V. Vaskonen, “Phase transition and vacuum stability in the classically conformal B–L model,” _Eur. Phys. J. C_ 79 (2019) 601, [1811.11169].
  * (55) J. Ellis, M. Lewicki, J. M. No and V. Vaskonen, “Gravitational wave energy budget in strongly supercooled phase transitions,” _JCAP_ 06 (2019) 024, [1903.09642].
  * (56) L. Bian, W. Cheng, H.-K. Guo and Y. Zhang, “Cosmological implications of a B $-$ L charged hidden scalar: leptogenesis and gravitational waves,” _Chin. Phys. C_ 45 (2021) 113104, [1907.13589].
  * (57) J. Ellis, M. Lewicki and V. Vaskonen, “Updated predictions for gravitational waves produced in a strongly supercooled phase transition,” _JCAP_ 11 (2020) 020, [2007.15586].
  * (58) S. Jung and K. Kawana, “Low-energy probes of the small cosmic microwave background amplitude in models of the radiative Higgs mechanism,” _PTEP_ 2022 (2022) 033B11, [2105.01217].
  * (59) J. Haruna and H. Kawai, “Weak scale from Planck scale: Mass scale generation in a classically conformal two-scalar system,” _PTEP_ 2020 (2020) 033B01, [1905.05656].
  * (60) Y. Hamada, H. Kawai, K.-y. Oda and K. Yagyu, “Dark matter in minimal dimensional transmutation with multicritical-point principle,” _JHEP_ 01 (2021) 087, [2008.08700].
  * (61) Y. Hamada, H. Kawai, K. Kawana, K.-y. Oda and K. Yagyu, “Minimal scenario of criticality for electroweak scale, neutrino masses, dark matter, and inflation,” _Eur. Phys. J. C_ 81 (2021) 962, [2102.04617].
  * (62) K. Kawana, “Cosmology of a supercooled universe,” _Phys. Rev. D_ 105 (2022) 103515, [2201.00560].
  * (63) A. D. Linde, “Decay of the False Vacuum at Finite Temperature,” _Nucl. Phys. B_ 216 (1983) 421.
  * (64) A. H. Guth and S. H. H. Tye, “Phase Transitions and Magnetic Monopole Production in the Very Early Universe,” _Phys. Rev. Lett._ 44 (1980) 631.
  * (65) A. H. Guth and E. J. Weinberg, “Cosmological Consequences of a First Order Phase Transition in the SU(5) Grand Unified Model,” _Phys. Rev. D_ 23 (1981) 876.
  * (66) M. D. Rintoul and S. Torquato, “Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model,” _Journal of physics a: mathematical and general_ 30 (1997) L585.
  * (67) J. Ellis, M. Lewicki and J. M. No, “On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal,” _JCAP_ 04 (2019) 003, [1809.08242].
  * (68) D. Bodeker and G. D. Moore, “Can electroweak bubble walls run away?,” _JCAP_ 05 (2009) 009, [0903.4099].
  * (69) D. Bodeker and G. D. Moore, “Electroweak Bubble Wall Speed Limit,” _JCAP_ 05 (2017) 025, [1703.08215].
  * (70) S. Höche, J. Kozaczuk, A. J. Long, J. Turner and Y. Wang, “Towards an all-orders calculation of the electroweak bubble wall velocity,” _JCAP_ 03 (2021) 009, [2007.10343].
  * (71) Y. Gouttenoire, R. Jinno and F. Sala, “Friction pressure on relativistic bubble walls,” _JHEP_ 05 (2022) 004, [2112.07686].
  * (72) W.-Y. Ai, B. Garbrecht and C. Tamarit, “Bubble wall velocities in local equilibrium,” 2109.13710.
  * (73) R.-G. Cai and S.-J. Wang, “Effective picture of bubble expansion,” _JCAP_ 03 (2021) 096, [2011.11451].
  * (74) S.-J. Wang and Z.-Y. Yuwen, “Hydrodynamic backreaction force of cosmological bubble expansion,” 2205.02492.
  * (75) B. Laurent and J. M. Cline, “First principles determination of bubble wall velocity,” 2204.13120.
  * (76) R. Mertig, M. Bohm and A. Denner, “FEYN CALC: Computer algebraic calculation of Feynman amplitudes,” _Comput. Phys. Commun._ 64 (1991) 345–359.
  * (77) V. Shtabovenko, R. Mertig and F. Orellana, “New Developments in FeynCalc 9.0,” _Comput. Phys. Commun._ 207 (2016) 432–444, [1601.01167].
  * (78) V. Shtabovenko, R. Mertig and F. Orellana, “FeynCalc 9.3: New features and improvements,” _Comput. Phys. Commun._ 256 (2020) 107478, [2001.04407].
  * (79) S. Blanchet, Z. Chacko, S. S. Granor and R. N. Mohapatra, “Probing Resonant Leptogenesis at the LHC,” _Phys. Rev. D_ 82 (2010) 076008, [0904.2174].
  * (80) J. Heeck and D. Teresi, “Leptogenesis and neutral gauge bosons,” _Phys. Rev. D_ 94 (2016) 095024, [1609.03594].
  * (81) M. Duerr, F. Kahlhoefer, K. Schmidt-Hoberg, T. Schwetz and S. Vogl, “How to save the WIMP: global analysis of a dark matter model with two s-channel mediators,” _JHEP_ 09 (2016) 042, [1606.07609].
  * (82) W. Buchmüller, V. Domcke, K. Kamada and K. Schmitz, “The Gravitational Wave Spectrum from Cosmological $B-L$ Breaking,” _JCAP_ 10 (2013) 003, [1305.3392].
  * (83) J. A. Dror, T. Hiramatsu, K. Kohri, H. Murayama and G. White, “Testing the Seesaw Mechanism and Leptogenesis with Gravitational Waves,” _Phys. Rev. Lett._ 124 (2020) 041804, [1908.03227].
  * (84) B. Fornal and B. Shams Es Haghi, “Baryon and Lepton Number Violation from Gravitational Waves,” _Phys. Rev. D_ 102 (2020) 115037, [2008.05111].
  * (85) R. Samanta and S. Datta, “Gravitational wave complementarity and impact of NANOGrav data on gravitational leptogenesis,” _JHEP_ 05 (2021) 211, [2009.13452].
  * (86) M. A. Masoud, M. U. Rehman and Q. Shafi, “Sneutrino tribrid inflation, metastable cosmic strings and gravitational waves,” _JCAP_ 11 (2021) 022, [2107.09689].
  * (87) L. Bian, X. Liu and K.-P. Xie, “Probing superheavy dark matter with gravitational waves,” _JHEP_ 11 (2021) 175, [2107.13112].
  * (88) W. Buchmuller, V. Domcke and K. Schmitz, “Stochastic gravitational-wave background from metastable cosmic strings,” _JCAP_ 12 (2021) 006, [2107.04578].
  * (89) J. R. Espinosa, T. Konstandin, J. M. No and G. Servant, “Energy Budget of Cosmological First-order Phase Transitions,” _JCAP_ 06 (2010) 028, [1004.4187].
  * (90) C. Caprini et al., “Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions,” _JCAP_ 04 (2016) 001, [1512.06239].
  * (91) C. Caprini et al., “Detecting gravitational waves from cosmological phase transitions with LISA: an update,” _JCAP_ 03 (2020) 024, [1910.13125].
  * (92) H.-K. Guo, K. Sinha, D. Vagie and G. White, “Phase Transitions in an Expanding Universe: Stochastic Gravitational Waves in Standard and Non-Standard Histories,” _JCAP_ 01 (2021) 001, [2007.08537].
  * (93) P. Auclair et al., “Probing the gravitational wave background from cosmic strings with LISA,” _JCAP_ 04 (2020) 034, [1909.00819].
  * (94) J. J. Blanco-Pillado and K. D. Olum, “Stochastic gravitational wave background from smoothed cosmic string loops,” _Phys. Rev. D_ 96 (2017) 104046, [1709.02693].
  * (95) P. Binetruy, A. Bohe, C. Caprini and J.-F. Dufaux, “Cosmological Backgrounds of Gravitational Waves and eLISA/NGO: Phase Transitions, Cosmic Strings and Other Sources,” _JCAP_ 06 (2012) 027, [1201.0983].
  * (96) J. J. Blanco-Pillado, K. D. Olum and B. Shlaer, “The number of cosmic string loops,” _Phys. Rev. D_ 89 (2014) 023512, [1309.6637].
  * (97) L. Bian, J. Shu, B. Wang, Q. Yuan and J. Zong, “Searching for cosmic string induced stochastic gravitational wave background with the Parkes Pulsar Timing Array,” 2205.07293.
  * (98) Z. Zhao, Y. Di, L. Bian and R.-G. Cai, “Probing the electroweak symmetry breaking history with Gravitational waves,” 2204.04427.
  * (99) LISA collaboration, P. Amaro-Seoane et al., “Laser Interferometer Space Antenna,” 1702.00786.
  * (100) TianQin collaboration, J. Luo et al., “TianQin: a space-borne gravitational wave detector,” _Class. Quant. Grav._ 33 (2016) 035010, [1512.02076].
  * (101) Y.-M. Hu, J. Mei and J. Luo, “Science prospects for space-borne gravitational-wave missions,” _Natl. Sci. Rev._ 4 (2017) 683–684.
  * (102) TianQin collaboration, J. Mei et al., “The TianQin project: current progress on science and technology,” 2008.10332.
  * (103) W.-R. Hu and Y.-L. Wu, “The Taiji Program in Space for gravitational wave physics and the nature of gravity,” _Natl. Sci. Rev._ 4 (2017) 685–686.
  * (104) W.-H. Ruan, Z.-K. Guo, R.-G. Cai and Y.-Z. Zhang, “Taiji program: Gravitational-wave sources,” _Int. J. Mod. Phys. A_ 35 (2020) 2050075, [1807.09495].
  * (105) J. Crowder and N. J. Cornish, “Beyond LISA: Exploring future gravitational wave missions,” _Phys. Rev. D_ 72 (2005) 083005, [gr-qc/0506015].
  * (106) S. Kawamura et al., “The Japanese space gravitational wave antenna: DECIGO,” _Class. Quant. Grav._ 28 (2011) 094011.
  * (107) LIGO Scientific, VIRGO collaboration, J. Aasi et al., “Characterization of the LIGO detectors during their sixth science run,” _Class. Quant. Grav._ 32 (2015) 115012, [1410.7764].
  * (108) LIGO Scientific, Virgo collaboration, B. P. Abbott et al., “Search for the isotropic stochastic background using data from Advanced LIGO’s second observing run,” _Phys. Rev. D_ 100 (2019) 061101, [1903.02886].
  * (109) D. Reitze et al., “Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO,” _Bull. Am. Astron. Soc._ 51 (2019) 035, [1907.04833].
  * (110) M. Punturo et al., “The Einstein Telescope: A third-generation gravitational wave observatory,” _Class. Quant. Grav._ 27 (2010) 194002.
  * (111) S. Hild et al., “Sensitivity Studies for Third-Generation Gravitational Wave Observatories,” _Class. Quant. Grav._ 28 (2011) 094013, [1012.0908].
  * (112) B. Sathyaprakash et al., “Scientific Objectives of Einstein Telescope,” _Class. Quant. Grav._ 29 (2012) 124013, [1206.0331].
  * (113) A. Dasgupta, P. S. B. Dev, A. Ghoshal and A. Mazumdar, “Gravitational Wave Pathway to Testable Leptogenesis,” 2206.07032.