Baryogenesis in a Parity Solution to the Strong CP Problem

Keisuke<EMAIL_ADDRESS>and Isaac R.
<EMAIL_ADDRESS>
$\,{}^{1}$ Department of Physics, University of Chicago, Chicago, IL 60637,
USA
$\,{}^{2}$ Enrico Fermi Institute, University of Chicago, Chicago, IL 60637,
USA
$\,{}^{3}$ Kavli Institute for Cosmological Physics, University of Chicago,
Chicago, IL 60637, USA
$\,{}^{4}$ Kavli Institute for the Physics and Mathematics of the Universe
(WPI),
The University of Tokyo Institutes for Advanced Study,
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
$\,{}^{5}$ Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
$\,{}^{6}$New High Energy Theory Center, Department of Physics and Astronomy,
Rutgers University, Piscataway, NJ 08854, USA

Space-time parity can solve the strong CP problem and introduces a
spontaneously broken $SU(2)_{R}$ gauge symmetry. We investigate the
possibility of baryogenesis from a first-order $SU(2)_{R}$ phase transition
similar to electroweak baryogenesis. We consider a model with the minimal
Higgs content, for which the strong CP problem is indeed solved without
introducing extra symmetry beyond parity. Although the parity symmetry seems
to forbid the $SU(2)_{R}$ anomaly of the $B-L$ symmetry, the structure of the
fermion masses can allow for the $SU(2)_{R}$ sphaleron process to produce non-
zero $B-L$ asymmetry of Standard Model particles so that the wash out by the
$SU(2)_{L}$ sphaleron process is avoided. The setup predicts a new hyper-
charged fermion whose mass is correlated with the $SU(2)_{R}$ symmetry
breaking scale and hence with the $SU(2)_{R}$ gauge boson mass, and depending
on the origin of CP violation, with an electron electric dipole moment. In a
setup where CP violation and the first-order phase transition are assisted by
a singlet scalar field, the singlet can be searched for at future colliders.

###### Contents

  1. 1 Introduction
  2. 2 A parity-symmetric model
    1. 2.1 Gauge symmetry breaking
    2. 2.2 Charged fermion masses
    3. 2.3 Parity and strong CP phase
    4. 2.4 Neutrino masses
      1. 2.4.1 Majorana mass from dimension-five operators
      2. 2.4.2 Dirac mass from dimension-five operators
      3. 2.4.3 Mojorana mass by radiative inverse seesaw
  3. 3 Baryogenesis from $SU(2)_{R}$ phase transition
    1. 3.1 $B-L$ asymmetry
      1. 3.1.1 Right-handed charged leptons from $SU(2)_{R}$ doublets
      2. 3.1.2 A right-handed charged lepton from an $SU(2)_{R}$ singlet
    2. 3.2 Strong first-order phase transition
      1. 3.2.1 Running quartic coupling
      2. 3.2.2 Extra scalars
    3. 3.3 Example of CPV : Local baryogenesis
  4. 4 Signals
    1. 4.1 New gauge bosons
    2. 4.2 New charged particle
    3. 4.3 New heavy neutral lepton
    4. 4.4 New scalar particle
    5. 4.5 Electric dipole moment
    6. 4.6 Gravitational waves
  5. 5 Summary and discussion
  6. A Thermal potential

##  1 Introduction

The CP phases in the quark masses, which explain the CP violation in the weak
interaction, are expected to also introduce CP violation in the strong
interaction [1, 2, 3, 4]. However, the CP-violating phase in the strong
interaction is known to be smaller than $10^{-10}$ from the non-observation of
the neutron electric dipole moment (EDM) [5]. This discrepancy is called the
strong CP problem.

The absence of the strong CP violation can be explained by a space-time parity
symmetry [6, 7], which predicts the parity partner of the $SU(2)_{L}$ gauge
symmetry called the $SU(2)_{R}$ gauge symmetry. The $SU(2)_{R}$ gauge symmetry
must be spontaneously broken at an energy scale higher than the electroweak
scale to explain the absence of $SU(2)_{R}$ gauge bosons at the electroweak
scale.

A model with an $SU(2)_{L}$-doublet Higgs $H_{L}$ and an $SU(2)_{R}$-doublet
Higgs $H_{R}$ is particularly appealing since the Higgs potential does not
contain any physical phases and the strong CP problem is indeed solved without
introducing extra symmetries [8, 9]; see [6, 7, 10, 11] for setups with a
different Higgs content and additional symmetries. Quantum corrections to the
strong CP phase are computed in [12, 13] and are found to be sufficiently
small.

In this paper, we pursue a possible cosmological role of the $SU(2)_{R}$ gauge
symmetry; production of the baryon asymmetry of the universe. The baryon
symmetry has $SU(2)_{R}$ anomaly and is violated by $SU(2)_{R}$ sphaleron
processes. As in electroweak baryogenesis [14], the baryon asymmetry of the
universe may be produced from this baryon number violation, a first-order
$SU(2)_{R}$ phase transition, and some CP violation in the early universe.

There seems to be an apparent obstacle to this idea. The Standard model (SM)
$B-L$ symmetry does not have $SU(2)_{L}$ anomaly, so the parity symmetry seems
to require that $B-L$ symmetry also does not have $SU(2)_{R}$ anomaly. The
first-order $SU(2)_{R}$ phase transition cannot create $B-L$ asymmetry, and
baryon asymmetry produced by the $SU(2)_{R}$ phase transition will be
immediately washed out by the $SU(2)_{L}$ sphaleron process. This naive
expectation assumes that the asymmetry of $SU(2)_{R}$ charged particles is
rapidly transferred into that of $SU(2)_{L}$ charged particles. This is indeed
the case in the models where the SM Higgs and right-handed fermions are
embedded into a bi-fundamental Higgs $\Phi$ and $SU(2)_{R}$ doublet fermions
$\bar{f}$ respectively, and the SM Yukawa couplings come from $f\Phi\bar{f}$,
where $f$ are SM $SU(2)_{L}$-doublet fermions. As we will see, this is not
necessarily the case in the model with $H_{L}$ and $H_{R}$, and the washout
may be avoided.

Our scenario predicts a hyper-charged fermion with a mass given by
$m_{i}v_{R}/v_{L}$, where $i$ is $e$, $\mu$, or $\tau$, and $v_{R}$ and
$v_{L}$ are $SU(2)_{R}$ and $SU(2)_{L}$ symmetry breaking scale, respectively.
The mass of the new fermion is then correlated with the masses of $SU(2)_{R}$
gauge bosons, and with the electron EDM, depending on the source of CP
violation. The case with $i=\tau$ is particularly interesting since $v_{R}$
predicted from the allowed range of the new fermion mass ($\gtrsim 100$ GeV)
overlaps with $v_{R}$ that is accessible at near future colliders and
measurements of the electron EDM.

The first-order $SU(2)_{R}$ phase transition may be achieved as in electroweak
baryogenesis, namely, by a thermal potential from a quartic coupling [15, 16,
17, 18, 19, 20, 21, 22, 23] or an (effective) tree-level trilinear coupling
[24, 25, 26, 27, 28, 29] of $H_{R}$ with a new scalar. We consider two
examples without hierarchy problems beyond that of $H_{L}$ and $H_{R}$. We
analyze a model with singlet scalar fields with the minimal coupling to
$H_{L}$ and $H_{R}$ that was analyzed for electroweak baryogenesis [30, 31].
We may also utilize the running of the Higgs quartic coupling; the quartic
coupling of $H_{L}$ becomes small at high energy scales, so if $v_{R}$ is
sufficiently high, the quartic coupling of $H_{R}$, which is equal to that of
$H_{L}$ evaluated at $v_{R}$, can be small enough for a first-order
$SU(2)_{R}$ phase transition to occur. If the running is induced only by the
SM interaction, $v_{R}$ is required to be above $10^{8}$ GeV, but extra
interactions of $H_{L}$ can lower $v_{R}$.

CP violation can be obtained in various ways. Note that the parity symmetry
does not forbid the CP phases of the theory; it only puts relations among
them. As an example, we consider CP violation from a dimension-6 coupling
between $H_{R}$ and the $SU(2)_{R}$ gauge bosons [32] and that from a
dimension-5 coupling between a singlet scalar and the $SU(2)_{R}$ gauge
bosons. Those couplings induce non-zero EDMs of SM fermions. That of an
electron is detectable by near future experiments if $v_{R}=O(10)$ TeV, for
which the new hyper-charged fermion is within the reach of near-future
colliders.

There are several past works on first-order $SU(2)_{R}$ phase transitions.
Ref. [33] investigated $SU(2)_{R}$ breaking by a triplet scalar and computed
the resultant gravitational-wave spectrum. Ref. [34] considers a model of
baryogenesis with extra chiral $SU(2)_{R}$ charged fermions and the possible
embedding of the model into a parity-symmetric theory by further extending the
gauge group at UV. See [35, 36, 37, 38, 39] for baryogenesis models from a new
non-Abelian gauge symmetry other than $SU(2)_{R}$.

This paper is organized as follows. In Sec. 2, we describe a parity-symmetric
model with $H_{L}$ and $H_{R}$. We discuss how the fermion masses, including
the neutrino masses, are obtained. In Sec. 3, we present a model of
baryogenesis from the $SU(2)_{R}$ phase transition. Experimental signals are
discussed in Sec. 4. Finally, a summary and discussion are given in Sec. 5.

##  2 A parity-symmetric model

In this section, we describe a parity-symmetric model we study.

###  2.1 Gauge symmetry breaking

The gauge symmetry at UV is $SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times
U(1)_{X}$, which is broken down to the SM gauge symmetry by the vacuum
expectation value (VEV) of $H_{R}({\bf 1},{\bf 1},{\bf 2},1/2)$. The SM gauge
symmetry is broken down to $SU(3)_{c}\times U(1)_{\rm EM}$ by the VEV of
$H_{L}({\bf 1},{\bf 2},{\bf 1},-1/2)$.

We impose a parity symmetry $H_{R}\leftrightarrow H_{L}^{\dagger}$ that solves
the strong CP problem [8, 9] as we will see. The parity-symmetric potential of
$H_{L}$ and $H_{R}$ is

$\displaystyle
V=\lambda\left(|H_{R}|^{4}+|H_{L}|^{4}\right)+\lambda_{LR}|H_{L}|^{2}|H_{R}|^{2}-m^{2}\left(|H_{R}|^{2}+|H_{L}|^{2}\right).$
(1)

The mass of the $SU(2)_{R}$ gauge boson must be much larger than that of the
electroweak gauge boson. At the tree level, however, there is no vacuum with
$v_{R}>v_{L}\neq 0$ for any choice of the parameters of potential. Such a
phenomenologically viable vacuum can be obtained by softly breaking the parity
symmetry [8, 9],

$\displaystyle\Delta V=-\Delta
m^{2}\left(|H_{R}|^{2}-|H_{L}|^{2}\right),~{}~{}\Delta m^{2}>0,$ (2)

or by quantum corrections to the Higgs potential [12]. In this paper, we
consider the former option, since the latter option leads to the production of
domain walls upon $SU(2)_{R}$ phase transition. The soft breaking may be
understood as spontaneous breaking by a field that couples to $H_{L}$ and
$H_{R}$.

This theory in general has three tuning; small $m^{2}/\Lambda_{\rm UV}^{2}\sim
v_{R}^{2}/\Lambda_{\rm UV}^{2}$, small $\Delta m^{2}/\Lambda_{\rm UV}^{2}\sim
v_{R}^{2}/\Lambda_{\rm UV}^{2}$, and small $(m^{2}-\Delta m^{2})/m^{2}\sim
v_{L}^{2}/v_{R}^{2}$, where $\Lambda_{\rm UV}$ is the UV scale. One may remove
the second one by generating $\Delta m^{2}$ by dynamical transmutation. The
total fine-tuning is $v_{R}^{2}/\Lambda_{\rm UV}^{2}\times
v_{L}^{2}/v_{R}^{2}=v_{L}^{2}/\Lambda_{\rm UV}^{2}$ and is the same as the SM.
This tuning may be explained by anthropic principle [40, 41, 42]. One can also
remove the first tuning by embedding the theory into solutions to the EW
hierarchy problems with a mass scale $\sim v_{R}^{2}$, such as supersymmetric
or composite scenarios. The last tuning $v_{L}^{2}/v_{R}^{2}$, however, cannot
be removed.

###  2.2 Charged fermion masses

Table 1: The gauge charges of Higgses and fermions | $H_{L}$ | $H_{R}$ | $q_{i}$ | $\bar{q}_{i}$ | $\ell_{i}$ | $\bar{\ell}_{i}$ | $U_{i}$ | $\bar{U}_{i}$ | $D_{i}$ | $\bar{D}_{i}$ | $E_{i}$ | $\bar{E}_{i}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{c}$ | 1 | 1 | 3 | ${\bf\bar{3}}$ | 1 | 1 | 3 | ${\bf\bar{3}}$ | 3 | ${\bf\bar{3}}$ | 1 | 1
$SU(2)_{L}$ | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$SU(2)_{R}$ | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1
$U(1)_{X}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{2}{3}$ | $-\frac{2}{3}$ | $-\frac{1}{3}$ | $\frac{1}{3}$ | $-1$ | $1$

The quark and lepton masses may be given by the following Yukawa interaction
and masses,

$\displaystyle{\cal L}=$ $\displaystyle
x^{u}_{ij}q_{i}\bar{U}_{j}H_{L}^{\dagger}+\bar{x}^{u}_{ij}\bar{q}_{i}U_{j}H_{R}^{\dagger}+M_{ij}^{u}U_{i}\bar{U}_{j}$
$\displaystyle+$ $\displaystyle
x^{d}_{ij}q_{i}\bar{D}_{j}H_{L}+\bar{x}^{d}_{ij}\bar{q}_{i}D_{j}H_{R}+M_{ij}^{d}D_{i}\bar{D}_{j}$
$\displaystyle+$ $\displaystyle
x^{e}_{ij}\ell_{i}\bar{E}_{j}H_{L}+\bar{x}^{e}_{ij}\bar{\ell}_{i}E_{j}H_{R}+M_{ij}^{e}E_{i}\bar{E}_{j}+{\rm
h.c.}$ (3)

The gauge charges of the fermions are listed in Table 1. Other combinations of
fermions are possible as systematically investigated in [12], but to be
concrete, we focus on this case in this paper. The parity symmetry restricts
the form of $x$, $\bar{x}$, and $M$ as we will see. See Refs. [43, 44, 45] for
the flavor phenomenology of the setup.

In the limit of $M\gg\bar{x}v_{R}$, we may integrate out the Dirac fermions
and obtain dimension-5 operators of the form

$\displaystyle
x\frac{1}{M}\bar{x}^{t}f\bar{f}H_{L}^{({\dagger})}H_{R}^{({\dagger})},$ (4)

where $f=q,\ell$. The SM fermion Yukawa couplings are given by
$x\bar{x}v_{R}/M$. In this limit, the right-handed SM fermions
$(\bar{u},\bar{d},\bar{e})$ are dominantly from $SU(2)_{R}$ doublets $\bar{q}$
and $\bar{\ell}$. In the opposite limit $M\ll\bar{x}v_{R}$, heavy fermions
obtain masses of $\bar{x}v_{R}$ and the SM Yukawas are given by $x$. The
right-handed SM fermions are dominantly from $SU(2)_{R}$ singlets $\bar{U}$,
$\bar{D}$, and $\bar{E}$. Whether or not $M\gg\bar{x}v_{R}$ can depend on the
fermion species and generations; we consider such cases in Sec. 3 to obtain
$B-L$ asymmetry. From the collider searches on extra quarks [46, 47, 48, 49],
a large Dirac mass term $M\gg xv_{R}$ is required for the first generation
unless $v_{R}\geq 10^{8}~{}\mathrm{GeV}$.

###  2.3 Parity and strong CP phase

With the $SU(2)_{R}$ gauge symmetry, we may impose a space-time parity
symmetry. It acts on gauge fields as

$\displaystyle G_{\mu}^{a}(t,x)\rightarrow G_{\mu}^{a}(t,-x)\times
s(\mu),~{}B_{\mu}(t,x)\rightarrow B_{\mu}(t,-x)\times s(\mu),$ $\displaystyle
W_{L,\mu}^{a}(t,x)\rightarrow W_{R,\mu}^{a}(t,-x)\times
s(\mu),~{}W_{R,\mu}^{a}(t,x)\rightarrow W_{L,\mu}^{a}(t,-x)\times s(\mu),$
$\displaystyle s(\mu)=\begin{cases}1&\mu=0\\\ -1&\mu=1,2,3\end{cases}$ (5)

and forbids the CP violating phase in $\theta G\tilde{G}$. The action on
fermions is

$\displaystyle q(t,x)\rightarrow
i\sigma_{2}\bar{q}^{*}(t,-x),~{}U(t,x)\rightarrow
i\sigma_{2}\bar{U}^{*}(t,-x),~{}\cdots.$ (6)

This requires that $\bar{x}_{ij}=x_{ij}^{*}$ and $M_{ij}=M_{ji}^{*}$. As a
result, the quark mass matrix becomes

$\displaystyle\begin{pmatrix}u_{i}&U_{i}\end{pmatrix}\begin{pmatrix}0&x_{ij}v_{L}\\\
x_{ji}^{*}v_{R}&M_{ij}\end{pmatrix}\begin{pmatrix}\bar{u}_{j}\\\
\bar{U}_{j}\end{pmatrix}.$ (7)

The determinant of the mass matrix is real, and at the one-loop level, the
strong CP phase is not generated from the quark mass [8, 9]. Note also that
the Higgs VEVs do not have physical phases, since the phases of the Higgses
are gauge degrees of freedom. This is an advantage of the gauge symmetry
breaking by $H_{R}$ and $H_{L}$ in comparison with that by an $SU(2)_{R}$
triplet and $SU(2)_{R}\times SU(2)_{L}$ bi-fundamentals, where the physical
phases of Higgs VEVs must be forbidden by extra symmetries [6, 7, 10, 11].

Two-loop corrections to the phase of the determinant of the quark mass matrix
are estimated in Refs. [12, 13] for $M>xv_{R}$ for all quarks and are found to
be below the current upper bound. Refs. [8, 9] introduce soft breaking of the
parity to the Dirac mass $M_{ij}$. Although the determinant of the quark mass
matrix is real at the tree level, one-loop correction to the phase is
generically too large [13]. We assume that the soft breaking in the Dirac mass
is suppressed; this is natural given that the Dirac masses are dimension-3
operators while the Higgs masses are dimension-2 operators, and that the Dirac
mass may be protected by chiral symmetry.

###  2.4 Neutrino masses

The masses of SM and right-handed neutrinos, $\nu$ and $\bar{N}$, can be
obtained in several ways. In one of them discussed in Sec. 2.4.1, right-handed
neutrinos can be dark matter.

####  2.4.1 Majorana mass from dimension-five operators

Neutrino masses can arise from the following two dimension-5 terms,

$\displaystyle
c^{M}_{ij}\ell_{i}\ell_{j}H_{L}^{{\dagger}^{2}}+c^{M*}_{ij}\bar{\ell}_{i}\bar{\ell}_{j}H_{R}^{{\dagger}^{2}}+{\rm
h.c.},$ (8)

which can be UV-completed by the see-saw mechanism [50, 51, 52, 53]. The sum
of the right-handed neutrino masses is given by

$\displaystyle\sum_{i}m_{\nu_{i}}(\frac{v_{R}}{v_{L}})^{2}=3~{}{\rm
keV}\frac{\sum_{i}m_{\nu_{i}}}{60~{}{\rm meV}}\left(\frac{v_{R}}{40~{}{\rm
TeV}}\right)^{2}.$ (9)

For experimentally allowed $v_{R}>20$ TeV, the thermal abundance of the right-
handed neutrinos exceed the observed dark matter density. This problem can be
avoided by entropy production, and the right-handed neutrinos are good dark
matter candidates. The required amount of dilution is

$\displaystyle D=40\frac{\sum_{i}m_{\nu_{i}}}{60~{}{\rm
meV}}\left(\frac{v_{R}}{40~{}{\rm TeV}}\right)^{2}\frac{80}{g_{s}(T_{D})},$
(10)

where $T_{D}$ is the temperature when the right-handed neutrinos decouple from
the thermal bath and $g_{s}$ is the entropy degree of freedom. This entropy
production also dilutes the baryon asymmetry produced at the $SU(2)_{R}$ phase
transition, and as we will see in Sec. 3.2.2, $v_{R}$ is bounded from above.

The keV-scale right-handed neutrino dark matter is warm and constrained by the
observations of the small-scale structure. Observations of Ly-$\alpha$ forests
give a constraint $m_{N}>$ few keV [54, 55, 56, 57], which is satisfied if
$v_{R}\gtrsim 40$ TeV.

The right-handed neutrinos can decay into a SM neutrino and a photon. The
decay is induced by a one-loop diagram where $N$ splits into an off-shell
$W_{R}$ and an $SU(2)_{R}$ charged lepton, they mix with a $W_{L}$ and an
$SU(2)_{L}$ charged lepton respectively, they annihilate into $\nu$, and a
photon is attached to an electromagnetically charged particle inside the loop
[58, 59, 60]. Here the Dirac mass term of $SU(2)_{L}\times SU(2)_{R}$ singlet
fermions ($M_{ij}$ in Eq. (2.2)) is necessary for the $\ell-\bar{\ell}$ and
$W_{L}-W_{R}$ mixing to exist. If those of the third-generation fermions are
suppressed, the constraint derived in Ref. [60] is relaxed. Indeed, as we will
see in Sec. 3, the Dirac mass term for the third-generation charged lepton
should be suppressed for successful baryogenesis. This weakens the lower bound
on $v_{R}$ shown in [60] by a factor of $(m_{\tau}/m_{\mu})^{1/2}\simeq 4$,
which is already enough to avoid the constraint for $m_{N}=$ few keV and
$v_{R}\sim 40$ TeV. With a moderately suppressed Dirac mass for the third
generation up quark, the constraint is further weakened.

####  2.4.2 Dirac mass from dimension-five operators

Dirac neutrino masses may arise from the following dimension-5 operator,

$\displaystyle
c^{D}_{ij}\ell_{i}\bar{\ell}_{j}H_{L}^{\dagger}H_{R}^{\dagger}+{\rm h.c.},$
(11)

which can be UV-completed, e.g., by

$\displaystyle{\cal L}=x^{\nu}\ell
H_{L}^{\dagger}\bar{S}+x^{\nu*}\bar{\ell}H_{R}^{\dagger}S+M^{\nu}S\bar{S}.$
(12)

The right-handed neutrinos have the same mass as the SM ones and behave as
dark radiation. For experimentally allowed $v_{R}>20$ TeV, right-handed
neutrinos decouple from the thermal bath before the QCD phase transition, so
$\Delta N_{\rm eff}<0.3$ and is below the current upper bound [61].

####  2.4.3 Mojorana mass by radiative inverse seesaw

The right-handed neutrinos can be heavy in the following setup,

$\displaystyle\chi_{i}\left(y^{\chi}_{ij}\ell_{j}H_{L}^{\dagger}+y^{\chi*}_{ij}\bar{\ell}_{j}H_{R}^{\dagger}\right),$
(13)

where $\chi_{i}$ are gauge-singlet fermions. After $SU(2)_{R}$ and $SU(2)_{L}$
breaking, only three linear combinations of the right-handed neutrinos and the
SM neutrinos obtain Dirac masses paired with $\chi_{i}$. Because of $v_{R}\gg
v_{L}$, the heavy ones are mostly the right-handed neutrinos with masses
$y^{\chi}v_{R}$. It can decay into a SM Higgs and a lepton doublet without
leaving any cosmological impacts.

In this setup, the naive lepton number $\ell(+1)$ and $\bar{\ell}(-1)$ is
violated by the coupling with $\chi$. This may lead to the wash-out of
asymmetry produced by the $SU(2)_{R}$ phase transition. As we will see in the
next section, however, this is not necessarily the case. Rather, the violation
helps the generation of the $B-L$ asymmetry of SM particles.

The SM neutrinos remain massless. This can be understood by a $U(1)$ symmetry
with charges $\chi(-1)$, $\ell(+1)$, and $\ell(+1)$. In the effective theory,
after integrating out heavy fields, the neutrino mass, if exists, should be
given by a Majorana mass $\nu^{2}$, but this is forbidden by the $U(1)$
symmetry. Non-zero neutrino masses can be generated by adding a Majorana mass
$m_{\chi}\chi^{2}$ that explicitly breaks the $U(1)$ symmetry. The neutrino
mass is still zero at tree level, and a non-zero neutrino mass arises at one-
loop level, as in the radiative inverse-seesaw model [62],

$\displaystyle
m_{\nu}\sim\frac{(y^{\chi})^{2}}{16\pi^{2}}\frac{m_{\chi}v_{L}^{2}}{(y^{\chi}v_{R})^{2}}=\frac{1}{16\pi^{2}}\frac{m_{\chi}v_{L}^{2}}{v_{R}^{2}}\sim
0.1~{}{\rm eV}\frac{m_{\chi}}{10~{}{\rm MeV}}\left(\frac{100~{}{\rm
TeV}}{v_{R}}\right)^{2},$ (14)

where we assume $y^{\chi}v_{R}>m_{W}$.

##  3 Baryogenesis from $SU(2)_{R}$ phase transition

In this section, we discuss how a first-order $SU(2)_{R}$ phase transition and
$SU(2)_{R}$ sphaleron processes can produce baryon asymmetry. The essential
idea is the same as electroweak baryogenesis; the phase transition proceeds
through the formation of bubbles, which expand and provide the deviation from
thermal equilibrium. The $SU(2)_{R}$ sphaleron process violates the baryon
number outside the bubble to create non-zero baryon asymmetry with an aid of
some CP violation. The $SU(2)_{R}$ sphaleron process becomes ineffective
inside the bubbles and the baryon asymmetry is frozen.

There seem to be, however, two apparent obstacles to this idea because of the
parity symmetry:

  * •

With a naive assignment of the lepton charge $-1$ to $\bar{\ell}$, $B-L$ is
anomaly-free. The $SU(2)_{R}$ sphaleron process does not produce $B-L$
asymmetry, so the baryon asymmetry produced by a first-order $SU(2)_{R}$ phase
transition would be washed-out by the $SU(2)_{L}$ sphaleron process.

  * •

The quartic coupling of the $SU(2)_{R}$-breaking Higgs and the $SU(2)_{R}$
gauge coupling would be as large as the SM counterparts, so the strong first-
order phase transition does not seem to take place.

We discuss how the two obstacles can be avoided in Secs. 3.1 and 3.2,
respectively.

CP violation can be obtained in various ways. Note that unlike CP symmetry,
parity symmetry does not forbid CP phases but only relates the CP phases of
parity partners. To be concrete, we discuss local baryogenesis [32] in Sec.
3.3.

###  3.1 $B-L$ asymmetry

If the lepton charge of $\bar{\ell}$ is $-1$, the $B-L$ symmetry does not have
$SU(2)_{R}$ anomaly, so the $SU(2)_{R}$ sphaleron process does not produce
$B-L$ asymmetry. This seems to lead to the washout of baryon asymmetry
produced during the $SU(2)_{R}$ phase transition. However, what lepton charge
to be assigned to $\bar{\ell}$ depends on how its asymmetry is transferred
into SM particles. In the following, we discuss a few examples where the
washout is avoided because of this ambiguity.

####  3.1.1 Right-handed charged leptons from $SU(2)_{R}$ doublets

We first consider the case where all of charged-lepton Yukawa interactions are
obtained from the dimension-5 operator in Eq. (4), for which right-handed SM
charged leptons $\bar{e}$ are from $SU(2)_{R}$ doublets $\bar{\ell}$. The
asymmetry in $\bar{e}$ can be then transferred into $\ell$ via the Yukawa
coupling. If the scattering between right-handed neutrinos $\bar{N}$ and
right-handed electrons $\bar{e}$ via the $W_{R}$ exchange is also effective,
the lepton charge of $\bar{e}$ and $\bar{N}$ should be $-1$, and the asymmetry
is washed out. However, if the $W_{R}$ exchange decouples before the Yukawa
interaction becomes efficient, the wash-out can be avoided.

##### Light right-handed neutrino

In the neutrino mass model in Secs. 2.4.1 and 2.4.2, the right-handed neutrino
masses are negligible in the early universe. The scattering between $\bar{e}$
and $N$ mediated by $W_{R}$ is suppressed only by the heavy mass of $W_{R}$
and decouples at a temperature

$\displaystyle T_{\rm D}\simeq 10^{8}~{}{\rm
GeV}\left(\frac{v_{R}}{10^{10}~{}{\rm GeV}}\right)^{4/3}.$ (15)

For $v_{R}>5\times 10^{7}$ GeV, the $W_{R}$ exchange decouples before the
electron Yukawa becomes effective at a temperature $8.5\times 10^{4}$ GeV
[63]. The asymmetry of $\bar{N}_{1}$ is not communicated to $\ell$, and the
total $B-L$ asymmetry of the SM particles is non-zero,

$\displaystyle\frac{(B-L)_{\rm
SM}}{s}=-\frac{n_{\bar{N}_{1}}}{s}=-\frac{1}{2}\left.\frac{n_{\bar{\ell}_{1}}}{s}\right|_{T\sim
v_{R}},$ (16)

and the wash-out is avoided [64]. For even higher $v_{R}$, the muon and tau
Yukawa interactions are also out of equilibrium when the $W_{R}$ exchange
decouples, so $B-L$ asymmetry becomes larger.

##### Heavy right-handed neutrino

In the neutrino mass model in Sec. 2.4.3, the right-handed neutrinos can be
heavy and their abundance is suppressed for $T\ll y^{\chi}v_{R}$. The
scattering rate between $\bar{N}_{1}$ and $\bar{e}_{1}$ is suppressed not only
by the large $W_{R}$ mass but also by the Boltzmann factor ${\rm
exp}(-y^{\chi}v_{R}/T)$,

$\displaystyle\Gamma\sim\frac{T^{3/2}(y^{\chi}v_{R})^{7/2}}{8\pi
v_{R}^{4}}\times{\rm exp}\left(-\frac{y^{\chi}v_{R}}{T}\right).$ (17)

For $y^{\chi}=O(1)$, the $W_{R}$ exchange decouples before the electron Yukawa
gets into equilibrium if $v_{R}\gtrsim 2\times 10^{6}$ GeV.

Since the asymmetry in $\bar{N}_{1}$ is suppressed by the Boltzmann factor
unlike the setup with light right-handed neutrinos, one may worry that the
total $B-L$ asymmetry is also exponentially suppressed; this may not the case
because of the $B-L$ violation by $y^{\chi}$, which can process the asymmetry
produced by the $SU(2)_{R}$ phase transition into non-zero $B-L$ asymmetry.
(This is analogous to the conversion of the asymmetry produced by the GUT
baryogenesis, where $B-L=0$, into non-zero $B-L$ asymmetry by lepton-number
violation [65, 66, 67, 68, 69].) The $B-L$ violation generically leads to the
complete wash-out of all asymmetries. However, if $y^{\chi}$ is nearly
diagonal in the charged lepton flavor basis,111The neutrino mixing can come
from non-diagonal $m_{\chi}$. the following symmetry, which we call
$U(1)_{L_{1}^{\prime}}$, is approximately preserved before the electron Yukawa
gets into thermal equilibrium at the classical level: $\ell_{1}(+1)$,
$\bar{\ell}_{1}(+1)$, $\chi_{1}(-1)$. $B/3-L_{1}^{\prime}$ is an approximate
symmetry without $SU(3)_{c}\times SU(2)_{L}$ gauge anomaly, but has
$SU(2)_{R}$ anomaly and can be produced by the $SU(2)_{R}$ phase transition.
Using the method in [70], one can show that the standard $B-L$ charge
($\bar{e}_{1}(+1)$, $\ell_{1}(-1)$, …) indeed becomes non-zero.
$U(1)_{L_{1}^{\prime}}$ is explicitly broken by the electron Yukawa,222The
symmetry is also explicitly broken by the Majorana mass term $m_{\chi}$, but
it is still small enough and the wash-out by this explicit breaking is
ineffective. but by the time it becomes effective, $\bar{e}_{1}$ no longer
communicates its charge with $\bar{N}_{1}$ via the $W_{R}$ exchange and the
$B-L$ violation by $y^{\chi}$ is ineffective. Because of the standard $B-L$
charge conservation, the baryon asymmetry remains non-zero.

####  3.1.2 A right-handed charged lepton from an $SU(2)_{R}$ singlet

The scenarios in Sec. 3.1.1 require large $v_{R}$ and there are no light
enough particles to be produced at near future colliders. Successful scenarios
with lower $v_{R}$ exist if the charged lepton masses have the following
structure,

$\displaystyle{\cal L}=$ $\displaystyle
x^{e}_{ij}\ell_{i}\bar{E}_{j}H_{L}+x^{e*}_{ij}\bar{\ell}_{i}E_{j}H_{R}+M_{ij}^{e}E_{i}\bar{E}_{j}~{}~{}(i,j=1,2)$
$\displaystyle+y_{\tau}\ell_{3}\bar{E}_{3}H_{L}+y_{\tau}^{*}\bar{\ell}_{3}E_{3}H_{R}+{\rm
h.c.},$ (18)

which may be ensured by an approximate $U(1)$ symmetry with charges
$\ell_{3}(1)$, $\bar{\ell}_{3}(1)$, $E_{3}(-1)$, and $\bar{E}_{3}(-1)$. For
this structure, the lepton charge of $\bar{\ell}_{3}$ and $E_{3}$ are not
necessarily $-1$ and $+1$ and the wash-out may be avoided. We call the two
components in $\bar{\ell}_{3}$ as $\bar{N}_{3}$ and $\bar{\tau}^{\prime}$, and
$E_{3}$ as $\tau^{\prime}$.

Because of the parity symmetry, $\tau^{\prime}$ mass is predicted to be
$m_{\tau}v_{R}/v_{L}$. The collider signature of $\tau^{\prime}$ is discussed
in Sec. 4. It is possible that the first two generations have the structure of
a vanishing Dirac mass, for which even lighter new fermions are predicted. To
be conservative, we assume that only $\tau^{\prime}$ is light.

##### Light right-handed neutrino

For the neutrino mass models in Secs. 2.4.1 and 2.4.2, the interaction of
$\bar{\ell}$ through the neutrino mass operators is ineffective because of the
smallness of the neutrino mass. With the structure of charged leptons in Eq.
(3.1.2), among the asymmetry of $\bar{\ell}_{1,2,3}$ created by the
$SU(2)_{R}$ sphaleron process, only that of $\bar{\ell}_{1,2}$ is transferred
into $SU(2)_{L}$ charged particles. Therefore, the total $B-L$ asymmetry of
the SM particles is non-zero,

$\displaystyle\frac{(B-L)_{\rm
SM}}{s}=-\left.\frac{n_{\bar{\ell}_{3}}}{s}\right|_{T\sim v_{R}}.$ (19)

The same way of obtaining non-zero baryon asymmetry is also employed in a
model of baryogenesis from axion rotation, called axiogenesis [71], in [64].

In this scenario, it is crucial that the transfer of the asymmetries in
$\bar{\ell}_{3}$ and $E_{3}$ into other leptons is negligible, since otherwise
the wash-out of the asymmetry occurs. Let us derive the bound on the
parameters of the theory, taking the mass term $M_{3}\bar{E}_{3}E_{3}$ as an
example. Since the scattering rate via $M_{3}$ is exponentially suppressed at
$T<m_{\tau^{\prime}}$, the ratio between the scattering rate and the Hubble
expansion rate is maximized at $T\sim m_{\tau^{\prime}}$. A care must be taken
in the choice of basis. Instead of choosing the basis where the Yukawa
interaction $y_{\tau}$ is diagonal, we may rotate
$(\bar{E}_{3},\bar{\tau}^{\prime})$ to remove $M_{3}$ and define the asymmetry
in this basis. We should take the basis with a smaller transfer late [72, 73],
since it is enough to have one basis in which the charges are separated with
each other. It turns out that we shall use the basis with a diagonal Dirac
mass term when $y_{\tau}T<m_{\tau^{\prime}}$, which is indeed the case after
$SU(2)_{R}$ phase transition since $m_{\tau^{\prime}}=y_{\tau}v_{R}$. This can
be also understood from the thermal mass given by the Yukawa interaction
$y_{\tau}T$ being smaller than $m_{\tau}^{\prime}$, so that the Hamiltonian of
quasi-particles on the thermal background is closer to a diagonal one in the
basis where the Dirac mass is diagonalized. In this basis, the charge transfer
is induced by a Yukawa interaction $y_{\tau}M_{3}/m_{\tau^{\prime}}$. We
require that

$\displaystyle\alpha_{2}y_{\tau}^{2}\frac{M_{3}^{2}}{m_{\tau^{\prime}}^{2}}T<H(T=m_{\tau^{\prime}})\longrightarrow
M_{3}<20~{}{\rm MeV}\left(\frac{v_{R}}{100~{}{\rm TeV}}\right)^{3/2}.$ (20)

Even if this condition is violated, the washout is avoided if the scattering
between $\bar{\tau}^{\prime}$ and $\bar{N}_{3}$ via the $W_{R}$ exchange are
ineffective when the scattering by $M_{3}$ becomes effective. However, the
$W_{R}$ exchange is indeed effective at $T\sim m_{\tau^{\prime}}$, so the
upper bound in Eq. (20) is applicable.

##### Heavy right-handed neutrino

We next consider the neutrino mass model in Sec. 2.4.3. If the coupling
$y^{\chi}_{ij}$ is not nearly diagonal in the charged lepton mass eigenbasis,
all possible lepton symmetries are violated, and the asymmetry produced by the
$SU(2)_{R}$ phase transition is washed out.

On the other hand, if $y^{\chi}_{ij}$ is nearly diagonal, although the first
and second-generation lepton symmetry is violated, the third-generation
leptons preserve the following symmetry which we call $U(1)_{L_{3}^{\prime}}$;
$\ell_{3}(+1)$, $\bar{E}_{3}(-1)$, $\bar{\ell}_{3}(+1)$, $E_{3}(-1)$, and
$\chi_{3}(-1)$. With this charge assignment, $B/3-L_{3}^{\prime}$ has
$SU(2)_{R}$ anomaly and hence can be produced by the $SU(2)_{R}$ phase
transition. As the temperature drops below the right-handed neutrino mass
$y^{\chi}v_{R}$ and the $\tau^{\prime}$ mass, the third-generation lepton
asymmetry is stored dominantly in $\ell_{3}$ and $\bar{E}_{3}$, namely, the
tau and tau neutrino. Because of the violation of the standard $B-L$ symmetry
by $y^{\chi}$, the total $B-L$ asymmetry of the SM particles becomes non-zero.

$L_{3}^{\prime}$-breaking parameters should be sufficiently small to avoid
washout. Let us again take $M_{3}$ as an example. Note that the washout is
avoided even if the scattering or decay involving $M_{3}$ becomes effective at
a low temperature, as long as the scattering between $\tau^{\prime}$ and
$N_{3}$ has already decoupled by that time. This is because the following
standard lepton symmetry is preserved: $\ell_{3}(+1)$, $\bar{E}_{3}(-1)$
$\bar{\tau}^{\prime}(-1)$, $E_{3}(+1)$. One can show that this standard $B-L$
charge indeed becomes non-zero. At $T\ll y^{\chi}v_{R}$, the scattering rate
between $\bar{\tau}^{\prime}$ and $\bar{N}_{3}$ is suppressed not only by the
large $W_{R}$ mass but also by the Boltzmann factor ${\rm
exp}(-y^{\chi}v_{R}/T)$,

$\displaystyle\Gamma\sim\frac{T^{3/2}(y^{\chi}v_{R})^{7/2}}{8\pi
v_{R}^{4}}\times{\rm exp}\left(-\frac{y^{\chi}v_{R}}{T}\right).$ (21)

For example, for $v_{R}=20-100$ TeV and $y^{\chi}=1$, the scattering decouples
at $T_{d}=0.7-4$ TeV, which is above $m_{\tau^{\prime}}$. The upper bound on
$M_{3}$ is given by

$\displaystyle\alpha_{2}y_{\tau}^{2}\frac{M_{3}^{2}}{m_{\tau^{\prime}}^{2}}T\lesssim
H(T=T_{d})\longrightarrow M_{3}<40~{}{\rm MeV}\left(\frac{T_{d}}{4~{}{\rm
TeV}}\right)^{1/2}\left(\frac{v_{R}}{100~{}\mathrm{TeV}}\right).$ (22)

###  3.2 Strong first-order phase transition

The SM Higgs coupling is $0.13$ around the weak scale. If the quartic coupling
of $H_{R}$ is of this order and $H_{R}$ only has the Yukawa couplings in Eqs.
(1) and (2.2) and gauge interactions, strong first-order phase transition
cannot occur. Indeed, as in the SM, for $g_{R}=g_{L}\simeq 0.65$, the
$SU(2)_{R}$ Higgs quartic coupling must be smaller than $0.017$ [74, 75, 76,
77, 78, 79] for a strong first-order phase transition to occur.

In the rest of this subsection, we discuss how the $SU(2)_{R}$ phase
transition can be of strong first order. In the models we discuss, no
hierarchy problems beyond that of $H_{L}$ and $H_{R}$ are introduced. The
effective thermal potential is computed up to one-loop level, as described in
Appendix A. For marginally SFOPT, high-temperature expansion shows a great
agreement with the full form. Since we are aiming at finding the boundary of
the parameter space to achieve SFOPT, we use the high-temperature expansion
throughout this subsection to simplify the computation.

####  3.2.1 Running quartic coupling

With the hierarchy $v_{L}\ll v_{R}$, it is possible to utilize the running of
the quartic coupling. The SM Higgs quartic coupling becomes smaller at high
energy scales because of the quantum correction from Yukawa couplings. If
there are no extra particles contributing to the $\beta$-function of the
quartic coupling, the quartic coupling $\lambda_{R}$ becomes smaller than
0.015 at around $10^{8}~{}\mathrm{GeV}$. We computed the effective potential
for $H_{R}$ at different energy scales with corresponding running couplings,
and found that a SFOPT can be achieved for $v_{R}>2\times
10^{8}~{}\mathrm{GeV}$. Here and hereafter, we assume that the quartic
coupling $|H_{L}|^{2}|H_{R}|^{2}$ is negligible. If not, the threshold
correction to $|H_{L}|^{4}$ at the scale $v_{R}$ via the $H_{R}$ exchange
makes the SM Higgs quartic coupling $|H_{L}|^{4}$ smaller than $|H_{R}|^{4}$
and the lower bound on $v_{R}$ becomes stronger.

If $H_{L}$ has $O(1)$ Yukawa couplings to new fermions, the running becomes
faster, and a strong first-order phase transition can occur for much smaller
$v_{R}$. As an example, we add extra fermions

$\displaystyle\psi=({\bf 1},{\bf 2},{\bf 1},-\frac{1}{2}),~{}\bar{\psi}=({\bf
1},{\bf 2},{\bf 1},\frac{1}{2}),~{}\Psi=({\bf 1},{\bf 1},{\bf 1},0),$ (23)

with Yukawa couplings $y_{\psi}H_{L}^{\dagger}\psi\Psi$ and
$\bar{y}_{\psi}H_{L}\bar{\psi}\Psi$ and mass terms $m_{\psi}\psi\bar{\psi}$
and $m_{\Psi}\Psi^{2}$. The corresponding $SU(2)_{R}$ charged partners and
their interactions are also introduced. This extra Yukawa coupling quickly
makes the quartic coupling small enough to achieve SFOPT. The required Yukawa
coupling $y_{\psi}$ to obtain a small enough $\lambda_{R}$ depends on the
masses of these extra fermions. Their lightness does not introduce extra
hierarchy problems because of the protection of fermion masses by chiral
symmetry.

Figure 1: Required scale of new physics $\Lambda_{\rm UV}$ to avoid the
instability of the Higgs potential for extra leptons with a mass of 1 TeV.
Here the extra Yukawa couplings are chosen to be the minimal value to achieve
the SFOPT.

On the other hand, a large extra Yukawa coupling may introduce the problem of
the instability of the potential. With a large extra Yukawa coupling, the
quartic coupling at a large Higgs field value quickly becomes negative. The
tunneling rate from the metastable point $\langle H_{R}\rangle=v_{R}$ to
infinity can be computed via the pseudo-bounce action method [80]. We use the
zero-temperature effective potential $V_{T=0}$ in Appendix A to compute the
tunneling action. For an escape point with a large field value, the pseudo-
bounce action becomes smaller, making the tunneling more likely to happen. A
large enough escape point thus always leads to instability. The value of the
escape point that gives a small enough bounce action is regarded as a UV scale
$\Lambda_{\rm UV}$ below which the modification of the Higgs potential by new
physics is required. In Fig. 1, we show $\Lambda_{\rm UV}/v_{R}$ as a function
of $v_{R}$, with the extra Yukawa coupling chosen to be the minimal value
achieving SFOPT. The extra fermion assisting the running is assumed to have a
mass of $1$ TeV.

We cannot argue that SFOPT can be achieved for $v_{R}<10^{5}$ GeV. Even for
the TeV-scale extra leptons, the required Yukawa coupling $y_{\psi}$ is order
one, which is close to that of the top quark at the EW scale. For such a large
Yukawa coupling, similar to the SM EW phase transition analyzed in Ref [81],
the result of 1-loop perturbative computation highly depends on approximation
methods and renormalization schemes/scales. Some of them deviate from the
result of lattice simulation significantly. More advanced computation
techniques such as dimension reduction, and computation up to higher-order
loops are required in such a case [82, 83, 84, 85], which is beyond the scope
of this paper.

We comment on the uncertainty from the top quark mass, to which the running is
sensitive. We used the central value $m_{t}=172.69~{}\mathrm{GeV}$ [86]. If
the top quark mass shifts by $1$ GeV, the minimal $v_{R}$ achieving SFOPT in
the minimal model changes by a factor of $10$. The prediction for
$\Lambda_{\rm UV}/v_{R}$ in Fig. 1, however, does not change, since the shift
of the top quark mass can be absorbed into the shift of the free parameters
$y_{\psi}$ and $\bar{y}_{\psi}$.

####  3.2.2 Extra scalars

A strong first-order phase transition may be also achieved via extra scalar
fields that couple to Higgses, which have been intensively investigated in the
literature in the context of electroweak phase transition. Those models can be
categorized into two types: the couplings may enhance the cubic term (see
Appendix A) via loop effects, or modify the (effective) tree-level quartic
coupling directly via mixing with the Higgs. The former includes, for example,
the MSSM [18, 19, 20, 21, 22, 23], a real $Z_{2}$-even singlet scalar [16,
17], and so forth. The latter includes the 2HDM model [26, 27, 28], a real
$Z_{2}$-odd singlet scalar [25], the NMSSM [24], a complex singlet scalar
[29], and so forth. For a review, see [87]. Here we analyze an example of real
singlet scalars investigated in [30, 31], where the singlet scalars are
naturally light and no extra hierarchy problem is introduced.

We introduce two extra scalar singlets $S_{L}$ and $S_{R}$. (One singlet case
does not work as we explain later.) The tree-level potential is

$\displaystyle V_{0}=$
$\displaystyle-\frac{1}{2}\mu_{H_{L}}^{2}h_{L}^{2}-\frac{1}{2}\mu_{H_{R}}^{2}h_{R}^{2}+\frac{1}{4}\lambda(h_{L}^{4}+h_{R}^{4})+\frac{1}{4}\lambda_{\rm
LR}h_{L}^{2}h_{R}^{2}$
$\displaystyle+\frac{1}{2}\mu_{S}^{2}(S_{L}^{2}+S_{R}^{2})+\frac{1}{2}AS_{L}(h_{L}^{2}-2v_{L}^{2})+\frac{1}{2}AS_{R}(h_{R}^{2}-2v_{R}^{2})$
$\displaystyle+\frac{1}{2}A^{\prime}S_{L}(h_{R}^{2}-2v_{R}^{2})+\frac{1}{2}A^{\prime}S_{R}(h_{L}^{2}-2v_{L}^{2}).$
(24)

$S_{L}$ and $S_{R}$ enjoy a shift symmetry $S_{L,R}\rightarrow
S_{L,R}+\delta_{L,R}$ that is softly broken by the mass terms and the
trilinear couplings. As long as $A,A^{\prime}<\mu_{S}$, the lightness of
$S_{L,R}$ is natural.

Let us for now take $A^{\prime}$ to be zero and discuss the effect of non-zero
$A^{\prime}$ later. For given field values of $h_{L}$ and $h_{R}$, the
potential is minimized at

$\displaystyle\langle
S_{{L,R}}\rangle=\frac{A}{2\mu_{S}^{2}}(h_{L,R}^{2}-2v_{L,R}^{2}).$ (25)

The potential along this trajectory is

$\displaystyle V_{0}(h_{L},h_{R},\langle S_{L}\rangle,\langle
S_{R}\rangle)=-\frac{1}{2}\mu_{H_{L}}^{2}h_{L}^{2}-\frac{1}{2}\mu_{H_{R}}^{2}h_{R}^{2}+\frac{1}{4}(\lambda-\frac{A^{2}}{2\mu_{S}^{2}})(h_{L}^{4}+h_{R}^{4})+\frac{1}{4}\lambda_{\rm
LR}h_{L}^{2}h_{R}^{2}.$ (26)

One can see that the effective quartic coupling $\lambda_{\rm eff}$ receives a
tree-level modification, $\lambda_{\rm
eff}=\lambda(v_{R})-A^{2}/(2\mu_{S}^{2})$, and the phase transition strength
is enhanced.

The lower bound on $A$ to achieve a SFOPT is translated into a lower bound on
$S_{L}-h_{L}$ mixing, which is shown by the lower blue-shaded regions in Fig.
2. We take into account the running of $A$ and $\lambda$ from the EW scale to
$v_{R}$. We take $\lambda_{\rm LR}$ to be zero; non-zero $\lambda_{\rm LR}$
gives a tree-level threshold correction to the quartic coupling of $H_{L}$ via
the $H_{R}$ exchange and the required $A$ becomes larger. $\lambda_{\rm LR}$
induced by $U(1)_{X}$ interaction is negligible. Since the quartic coupling
$\lambda$ is smaller at $v_{R}$ than at the EW scale, the required magnitude
of $A$ to achieve the SFOPT becomes smaller and thus mixing angle is smaller
than that in Ref. [30, 31]. In the upper blue-shaded region,
$\lambda-A^{2}/(2\mu_{S}^{2})$ at $v_{R}$ is negative, and the potential is
unstable. Here we again assume $m_{t}=172.69~{}\mathrm{GeV}$ [86]. We found
that the prediction for the mixing angle shifts by $5\%$ for the shift of the
top quark mass by $1$ GeV.

Here it is crucial that the mass of $S$ is below the EW scale and cannot be
integrated out. Otherwise, the effective coupling of $H_{L}$ corrected by $A$
is no longer an effective one. Rather, it is the actual Higgs coupling in the
effective theory after integrating out $S$ that determines the SM Higgs mass.
As a result, the quartic coupling at the energy scale above the $S$ mass
becomes larger and the $SU(2)_{R}$ phase transition cannot be a SFOPT unless
$v_{R}$ is above $10^{8}$ GeV.

The model is further constrained by direct searches for the singlet scalars.
The mixing between $S_{R}$ and the SM Higgs is suppressed by $\lambda_{\rm
LR}$, so we focus on $S_{L}$ that mixes with the SM Higgs and can be probed in
various ways. $S_{L}$ heavier than a few GeV can be probed in collider
experiments by direct production of $S_{L}$. A search up to $100$ GeV was
performed assuming that $S_{L}$ decays into SM fermions, with
$S_{L}\rightarrow b\bar{b}$ providing the most stringent bound [88]. A search
independent of the decay mode of $S_{L}$ via $e^{+}e^{-}\rightarrow ZS_{L}$ is
also performed [89, 90]. (This search is also applicable even if $S_{L}$
dominantly decays into a dark sector.) We compare all the bounds mentioned
above and choose the most stringent one for each mass, leading to the magenta-
shaded region of Fig. 2.

This scalar can be also searched via an extra decay channel $h\rightarrow
S_{L}S_{L}$ of the SM Higgs. This decay channel was searched at LHC for
various final states. Ref. [91] summarizes all current searches and derives a
combined exclusion curve, using a proper branching ratio for $S_{L}$ decaying
into SM fermions. In Fig. 2, the current exclusion from the LHC is shown by
the orange-shaded region. The prospect of future search is discussed in Sec.
4.4.

$S_{L}$ with a mass below a few GeV can be probed by rare meson decay (see
[92] for review, and see [93] for recent updates.) LHCb performed a search
with $B^{+}\rightarrow K^{+}S_{L}(\mu^{+}\mu^{-})$ [94] and $B^{0}\rightarrow
K^{0}S_{L}(\mu^{+}\mu^{-})$ [95] for $200~{}\mathrm{MeV}\leq m_{S}\leq
4~{}\mathrm{GeV}$. The scalar mass at the MeV scale is not excluded by the
direct searches for $S_{L}$, but is excluded by the cosmological problem of
$S_{R}$ as explained later.

Now let us discuss the effect of $A^{\prime}$. Without loss of generality we
take $A^{\prime}<A$, otherwise we can rename $S_{L,R}$ as $S_{R,L}$. The most
important effect of this coupling is to give a tree-level threshold correction
to the mass of $S_{L}$ via $h_{R}$ exchange;

$\displaystyle\mu_{S}^{2}$
$\displaystyle\rightarrow\mu_{S_{L}}^{2}=\mu_{S}^{2}-\frac{A^{\prime
2}}{2\lambda(v_{R})}$ $\displaystyle\mu_{S}^{2}$
$\displaystyle\rightarrow\mu_{S_{R}}^{2}=\mu_{S}^{2}-\frac{A^{2}}{2\lambda(v_{R})}=\mu_{S}^{2}\frac{\lambda_{\rm
eff}}{\lambda(v_{R})}$ (27)

That means that for a given $\mu_{S_{L}}^{2}$, $\mu_{S}^{2}$ is larger by
$A^{\prime}/(2\lambda(v_{R}))$. To achieve SFOPT, a larger $A$ is required to
enhance the tree-level correction to $\lambda_{\rm eff}$, i.e.,
$-A^{2}/(2\mu_{S}^{2})$. The allowed parameter space in $(m_{S},\theta)$ for
SFOPT is shifted above. For example, if $A^{\prime}$ is not smaller than $A$
by a factor of 3, the allowed parameter space for SFOPT is excluded by
collider experiments. This is also the reason why the model with a single $S$
and a coupling $AS(h_{L}^{2}+h_{R}^{2})$ does not work.

Figure 2: Parameter space for the scalar extension in Eq. (3.2.2). Blue-shaded
region: excluded by the SFOPT condition and stability requirement. Magenta-
shaded region: LEP constraint from direct production. Orange-shaded region:
LHC constraint from Higgs exotic decay. Gray shaded region: LHCb constraint
from a rare B-meson decay. White space: allowed parameter space. Orange dashed
line: HL-LHC prospect for extra scalar search via Higgs exotic decay. Purple
dashed line: Higgs factory prospects for extra scalar search via Higgs exotic
decay.

The MeV-scale mass is probed by rare kaon decay [96, 97] and negative $\Delta
N_{\rm eff}$ [61, 98]. Large mixing with $h_{L}$ is excluded by the rare kaon
decay because of the large branching fraction of $K\rightarrow\pi S$, while
small mixing is excluded by the decay of $S$ into electrons after neutrinos
decouple, which makes neutrinos relatively cooler. These constraints on
$S_{L}$ are satisfied for $m_{S_{L}}<20$ MeV, but not once the cosmology of
$S_{R}$ is taken into account. $S_{R}$ couples with $H_{R}$ and is thermalized
when $T\sim v_{R}$. If $A^{\prime}$ is small, the mixing of $S_{R}$ with
$h_{L}$ is small, so $S_{R}$ decouples from the bath when it is relativistic
and decays much after neutrinos decouple and $S_{R}$ becomes non-relativistic.
This leads to a too large negative $\Delta N_{\rm eff}$. If $A^{\prime}$ is of
the same order as $A$, $S_{R}$ mixes with $h_{L}$, and the $\Delta N_{\rm
eff}$ constraint can be evaded. However, the threshold correction to
$\mu^{2}_{S_{L}}$ discussed above shifts the allowed parameter space for SFOPT
above, and the constraint from the rare kaon decay excludes it. Combining all
these requirements, we found no allowed parameter.

###  3.3 Example of CPV : Local baryogenesis

In this paper, we consider local baryogenesis, following the proposal in [32]
for the electroweak phase transition. We first review the local baryogenesis
in a general way for the thick-wall regime where the wall is thicker than the
mean free-path of particles $\sim(\alpha_{R}T)^{-1}$. Indeed, for the models
using the running quartic or the extra singlet scalar, we find that the wall
thickness is around $100/T$. We then discuss two specific models.

If $H_{R}$ couples to heavy $SU(2)_{R}$ charged fermions directly or
indirectly and CP is violated by the coupling or mass of the fermions, the
$SU(2)_{R}$ theta term may depend on $H_{R}$,

$\displaystyle\frac{g^{2}}{32\pi^{2}}\theta(H_{R})W_{R}\tilde{W}_{R}.$ (28)

We follow the method in [32] to calculate the baryon asymmetry in the thick-
wall regime. The change of the field value of $H_{R}$ across the bubble wall
gives a time-dependent $\theta$ term and hence a bias between the $SU(2)_{R}$
sphaleron transition increasing asymmetry and that decreasing asymmetry. The
produced asymmetry gives an opposite bias, but it is negligible unless the
asymmetry reaches an equilibrium value. As discussed in Sec. 3.1, the final
baryon number is determined by the lepton number carried by the new charged
lepton $\bar{\ell}_{3}$ and $E_{3}$. The $SU(2)_{R}$ sphaleron process
produces the asymmetry of $\bar{\ell}_{3}$,

$\displaystyle\dot{n}_{\bar{\ell}_{3}}=\frac{\Gamma_{\rm
Rs}}{T^{3}}\dot{\theta}T^{2},$ (29)

where $\Gamma_{\rm Rs}$ is the $SU(2)_{R}$ sphaleron transition rate. This
equation can be solved by performing an integration over time on the right-
hand side along the wall profile. When the Higgs field value is sufficiently
small, the sphaleron transition rate is given by [99]

$\displaystyle\Gamma_{\rm Rs}=\kappa\alpha_{R}^{5}T^{4},$ (30)

where $\kappa\simeq 20$. The sphaleron rate receives an exponential
suppression when the field value of $H_{R}$ becomes large enough. By the time
this occurs, $\theta(H_{R})$ changes by $\delta\theta$. Then the baryon
asymmetry produced by the $SU(2)_{R}$ phase transition normalized by the
entropy density is given by

$\displaystyle
Y_{B}=\frac{n_{B}}{s}=\frac{28}{79}\frac{1}{s}\kappa\alpha_{R}^{5}T^{3}\delta\theta=8.6\times
10^{-11}\frac{\delta\theta}{0.02}.$ (31)

One can see that the observed baryon asymmetry can be obtained even with
$\delta\theta\ll 1$.

In the neutrino mass model in Sec. 2.4.1, right-handed neutrinos can be dark
matter if entropy production given by Eq. (10) occurs. The entropy production
also dilutes baryon asymmetry, so larger $\delta\theta$ is required. This
gives an upper bound on the sum of right-handed neutrino mass,

$\displaystyle\sum_{i}m_{N_{i}}<10~{}{\rm keV}\times\frac{\delta\theta_{\rm
max}}{\pi}.$ (32)

Here we take the maximal $\delta\theta$, $\delta\theta_{\rm max}$, to be
$\pi$. If there are multiple heavy $SU(2)_{R}$ charged fermions in the UV
completion of the local operator in Eq. (28), $\delta\theta_{\rm max}$ can be
larger. Unless the number of those heavy fermions is large, the warmness of
dark matter can be observed by future observations of 21cm lines [100]. The
upper bound on the masses of right-handed neutrinos also gives an upper bound
on $v_{R}$,

$\displaystyle v_{R}\leq
70~{}\mathrm{TeV}\left(\frac{60\mathrm{meV}}{\sum_{i}m_{\nu_{i}}}\right)^{1/2}\left(\frac{g_{s}(T_{D})}{80}\right)^{1/2}\left(\frac{\delta\theta_{\rm
max}}{\pi}\right)^{1/2}.$ (33)

Now we discuss more specific models, which may be subject to stronger
constraints. We consider the following dimension-6 CP-odd operator [32],

$\displaystyle\frac{g^{2}}{32\pi^{2}M^{2}}|H_{R}|^{2}W_{R}\tilde{W}_{R}.$ (34)

The parity symmetry does not forbid this CP-odd operator and only requires
another dimension-6 operator composed of $H_{L}$ and $W_{L}$ with the same
strength. The operator may be UV-completed by, e.g., the following parity-
symmetric interactions and masses,

$\displaystyle yH_{R}L^{\prime}\eta+\lambda
H_{R}^{\dagger}\bar{L}^{\prime}\eta+m_{L}L^{\prime}\bar{L}^{\prime}+\frac{1}{2}m_{\eta}\eta^{2}$
$\displaystyle+y^{*}H_{L}\bar{L}\eta+\lambda^{*}H_{L}^{\dagger}L\eta+m_{L}^{*}L\bar{L}+{\rm
h.c.},$ (35)

with the $SU(2)_{L}\times SU(2)_{R}\times U(1)_{X}$ charges of the fermions
given by $L^{\prime}({\bf 1},{\bf 2},-1/2)$, $\bar{L}^{\prime}({\bf 1},{\bf
2},-1/2)$, $\eta({\bf 1},{\bf 1},0)$, $L({\bf 2},{\bf 1},1/2)$, and
$\bar{L}({\bf 2},{\bf 1},1/2)$. The parity symmetry requires that $m_{\eta}$
is real, but the physical CP phase ${\rm arg}(y\lambda/(m_{L}m_{\eta}))$ is in
general non-zero.

We make an approximation that the sphaleron rate is given by Eq. (30) when
$m_{W_{R}}(H_{R})<\sigma\alpha_{R}T$ and zero for
$m_{W_{R}}(H_{R})>\sigma\alpha_{R}T$. Here $\sigma$ is a constant that can be
estimated from the simulation in Ref [99], and we obtain $\sigma\simeq 3$.
Putting all information together, the final baryon-to-entropy ratio is given
by

$\displaystyle
Y_{B}=\frac{n_{B}}{s}=\left.\frac{28}{79}\frac{1}{s}\frac{4\Gamma_{\rm
sph}\sigma^{2}\alpha_{R}^{2}T}{M^{2}g^{2}}\right|_{T=T_{n}}\simeq 8.9\times
10^{-11}\left(\frac{1.7T_{n}}{M}\right)^{2}.$ (36)

The observed baryon asymmetry $Y_{B}\simeq 8.7\times 10^{-11}$ can be achieved
for $M$ slightly above $T_{n}$. This justifies the use of the dimension-6
operator; the masses or heavy particles in UV-completion of the operator
($\bar{L}$, $\bar{L}^{\prime}$, and $\eta$ in the above example) can be above
$T_{n}$ so that they may be indeed integrated out. The setup, however, cannot
accommodate the dilution necessary for the model with right-handed neutrino
dark matter while satisfying $M>T_{n}$.

In the model with extra scalars, the following dimension-5 operator can
achieve local baryogenesis [31],

$\displaystyle\mathcal{L}=\frac{\alpha_{R}}{8\pi}\frac{S_{R}W_{R}\tilde{W_{R}}}{M}.$
(37)

This operator may be UV-completed by identifying $S$ with a pseudo Nambu-
Goldstone boson with a decay constant $\sim M$ and weak anomaly. Since the
shift of the field value of $S_{R}$ is around $\Delta S_{R}\simeq A\Delta
h_{R}^{2}/(2\mu_{S}^{2})$, the baryon asymmetry is given by Eq. (36)
multiplied by $MA/(2\mu_{S}^{2})$. Around the allowed parameter space in Fig.
2, we always have $A/\mu_{S}\simeq 0.3$. Thus we obtain

$\displaystyle Y_{B}\simeq 8.7\times
10^{-11}\left(\frac{v_{R}}{20~{}\mathrm{TeV}}\right)\left(\frac{T_{n}}{0.2v_{R}}\right)^{2}\left(\frac{40v_{R}}{M}\right)\left(\frac{10~{}\mathrm{GeV}}{\mu_{S}}\right).$
(38)

Here we normalized $T_{n}$ by the value we find by computing the bounce
action, $T_{n}\simeq 0.2v_{R}$. The smallness of $\mu_{S}$ causes the large
shift of the field value of $S_{R}$ during the phase transition and strongly
enhances the generated baryon asymmetry. As a result, $M$ may be much above
$T_{n}$.

In deriving the parameter space achieving SFOPT by computing the bounce
action, we assume $S^{2}$ and $S|H|^{2}$ terms and neglect higher-order terms.
This requires that the shift of $S$ from the $SU(2)_{R}$ symmetric point to
the escape point in the bounce solution be smaller than $M$, since otherwise
higher order terms in general become comparable to the terms we consider. We
find that the shift is about $T_{n}/4$, which is indeed much smaller than $M$.

##  4 Signals

In this section, we discuss the experimental signals of the model.

###  4.1 New gauge bosons

The left-right symmetric model predicts a new $W_{R}$ gauge boson and a
$Z^{\prime}$ gauge boson. Their masses are given by

$\displaystyle m_{W_{R}}$
$\displaystyle=m_{W}\frac{v_{R}}{v_{L}}=6.5~{}\mathrm{TeV}\frac{v_{R}}{15~{}\mathrm{TeV}},$
$\displaystyle m_{Z^{\prime}}$
$\displaystyle=m_{Z}\frac{v_{R}}{v_{L}}=7.4~{}\mathrm{TeV}\frac{v_{R}}{15~{}\mathrm{TeV}}.$
(39)

The new gauge bosons couple with SM leptons and quarks, and thus can be
searched for by collider experiments for various final states, including
lepton+MET, jets, di-lepton, etc. The current limit from the $W_{R}$ search is
more stringent than that of $Z^{\prime}$. The $W_{R}$ boson with a mass below
$6.0~{}\mathrm{TeV}$ is excluded by the search for di-electron final states
[101], corresponding to $v_{R}=14.1~{}\mathrm{TeV}$. The high-luminosity LHC
(HL-LHC) can detect the $W_{R}$ boson with a mass up to $7.9~{}\mathrm{TeV}$
[102], corresponding to $v_{R}=18.6~{}\mathrm{TeV}$.

###  4.2 New charged particle

As is discussed in Sec. 3.1, a non-zero $B-L$ asymmetry can be produced if
$v_{R}>10^{6}~{}\mathrm{GeV}$ or the charged lepton mass has the structure in
Eq. (3.1.2). While the former scenario has no collider signatures, the latter
predicts a parity partner of the tau lepton with a mass

$\displaystyle m_{\tau^{\prime}}$
$\displaystyle=m_{\tau}\frac{v_{R}}{v_{L}}\simeq 150~{}{\rm
GeV}\frac{v_{R}}{15~{}\mathrm{TeV}}$ (40)

and a hypercharge of unity. The mass is correlated with the masses of the new
gauge bosons $W_{R}$ and $Z^{\prime}$. It is possible that the other two
generations of leptons have the structure of the vanishing Dirac mass term
$M_{ij}$, for which case even lighter charged leptons are predicted. To be
conservative, we consider the case where only $\tau^{\prime}$ is light and
discuss how to probe it in collider experiments.

$\tau^{\prime}$ is pair-produced by the hypercharge gauge interaction at
colliders, whose cross section is computed in, e.g., [103, 104]. The decay
channel of $\tau^{\prime}$ depends on how the right-handed neutrino masses are
obtained. When they are obtained via a dimension-5 Majorana or Dirac mass
operator, $N$ is light so $\tau^{\prime}$ promptly decays into $N$ via an off-
shell $W_{R}$ boson. The collider signals are then $e^{+}e^{-}$,
$\mu^{+}\mu^{-}$, $e^{+}\mu^{-}$, $e^{-}\mu^{+}$, $e+2$ jets, $\mu+2$ jets, or
4 jets, all accompanied by missing energy from right-handed neutrinos. Note
that $\tau$ final states are absent since the SM right-handed tau is
$SU(2)_{R}$ singlet. The signal resembles that of pair-production of charginos
decaying into a neutralino and $W$. Comparing the upper bound on the cross
section provided in the HEPData version of [105] with the production cross
section of hyper-charged fermions, we find that there is no LHC constraint.
Only the LEP bound of $m_{\tau^{\prime}}>100$ GeV [106] is applicable.

When the right-handed neutrino mass is generated by Eq. (13), the decay
channel depends on whether the right-handed neutrino is heavier than
$\tau^{\prime}$. If $N_{3}$ is lighter than $\tau^{\prime}$, $\tau^{\prime}$
will decay to $N_{3}$ and emits an off-shell $W_{R}$ gauge boson. For
$m_{N_{3}}$ below the EW scale, $N_{3}$ decays into $\tau$ and an off-shell
$W_{L}$ via the $\nu-N$ mixing $\sim v_{L}/v_{R}$ with a typical decay length
in the lab frame

$\displaystyle\left(\frac{3m_{N_{3}}^{5}}{512\pi^{3}v_{L}^{2}v_{R}^{2}}\frac{m_{N_{3}}}{m_{\tau^{\prime}}/3}\right)^{-1}\simeq
1~{}{\rm
mm}\left(\frac{10~{}\mathrm{GeV}}{m_{N_{3}}}\right)^{6}\left(\frac{m_{\tau^{\prime}}}{200\mathrm{GeV}}\right)^{3}.$
(41)

For a range of $m_{N_{3}}$, the decay length is $O(1-10^{3})$ mm and $N_{3}$
can be observed as a displaced vertex. However, since $N_{3}$ is highly
boosted, the decay products of it are collimated and the efficiency of the
reconstruction of the displaced vertex may be degraded. The estimation of the
bound on $m_{\tau^{\prime}}$ for this case is beyond the scope of this paper.
For sufficiently small $m_{N_{3}}$, the typical decay length exceeds $O(1)$ m
and $N$ is observed as missing energy. The signal is the same as that in the
previous paragraph. For $m_{N_{3}}$ above the EW scale, $N_{3}$ promptly
decays into $\tau+W$ or $\nu+Z/h$. See Sec. 4.3 for the comment on the search
for singlet leptons.

If $N_{3}$ is heavier than $\tau^{\prime}$, $\tau^{\prime}$ decays into an
off-shell $N_{3}$ via $W_{R}$ exchange, or into an SM fermion and a boson via
the Dirac mass term $M_{3}E_{3}\bar{E}_{3}$. We first discuss the former decay
mode. The off-shell $W_{R}$ at least decays into $ud$ and $sc$. The decay into
$tb$ is possible, but this channel is suppressed unless the Dirac mass term
for the third generation up quark, $M_{33}^{u}$, is comparable to $v_{R}$ so
that the SM right-handed top contains a significant fraction of $SU(2)_{R}$
doublet. If $N_{1,2}$ are lighter than $\tau^{\prime}$, decay into $eN_{1}$
and $\mu N_{2}$ is possible, and $N_{1,2}$ decays into SM particles as in the
previous paragraph. The off-shell $N_{3}$ decays into either $\nu$ via $v_{L}$
insertion, or into $\nu+Z/h$ or $\tau+W_{L}$, with $Z/h/W_{L}$ dominantly
decaying into jets or leptons. For $m_{\tau^{\prime}}<$ TeV, the former
dominates and the decay length of $\tau^{\prime}$ is given by

$\displaystyle\left(\frac{1}{1536\pi^{3}}\frac{m_{\tau^{\prime}}^{5}v_{L}^{2}}{v_{R}^{6}}N_{f}\right)^{-1}\simeq
8~{}{\rm mm}\times\frac{m_{\tau^{\prime}}}{200~{}{\rm GeV}}\frac{6}{N_{f}},$
(42)

where $N_{f}$ is the number of final states. For example, if only $ud$ and
$sc$ are available, $N_{f}=6$. The decay length of $\tau^{\prime}$ is above mm
so $\tau^{\prime}$ can be observed as di-jets from a displaced vertex. The SM
neutrino from the off-shell $N$ leads to missing energy. The most recent LHC
search for displaced vertices provides upper limit for the production cross
section [107] as a function of a decay length above $1$ mm. Comparing it with
the production cross section of hyper-charged fermions [103, 104], we obtain a
bound $m_{\tau^{\prime}}>1~{}\mathrm{TeV}$.

On the other hand, $M_{3}$ induces the decay of $\tau^{\prime}$ into
$\tau+Z/h$ or $\nu+W$, and the decay length is

$\displaystyle\left(\frac{1}{4\pi}\frac{y_{\tau}^{2}M_{3}^{2}}{m_{\tau^{\prime}}}\right)^{-1}\simeq
0.1~{}\mathrm{mm}\left(\frac{3~{}\mathrm{MeV}}{M_{3}}\right)^{2}\frac{m_{\tau^{\prime}}}{200~{}\mathrm{GeV}}.$
(43)

For $M_{3}$ that saturates the upper bound in Eq. (22), the decay rate is
larger than that given by the $W_{R}$ exchange discussed above and the decay
length is below 1 mm; the decay is prompt and only the LEP bound
$m_{\tau^{\prime}}>100$ GeV is applicable. For $M_{3}$ below the MeV scale,
the decay length can be long enough that $\tau^{\prime}$ can be observed as a
displaced vertex, for which $m_{\tau^{\prime}}>1$ TeV is required.

###  4.3 New heavy neutral lepton

In the neutrino mass model in Sec. 2.4.3, right-handed neutrinos mix with
standard model neutrinos with the square of the angle

$\displaystyle\theta_{\nu
N}^{2}\simeq\left(\frac{v_{L}}{v_{R}}\right)^{2}=3.3\times
10^{-5}\left(\frac{30~{}{\rm TeV}}{v_{R}}\right)^{2}.$ (44)

If the masses of them are below 100 GeV, right-handed neutrinos are subject to
constraints from direct searches. See [108] for the overview of constraints
and prospects. This is complementary to the search for a new charged lepton,
whose prospects become worse if the right-handed tau neutrino $N_{3}$ is
lighter than $m_{\tau^{\prime}}$. Note that $N_{3}$ is also produced from the
decay of hyper-charged $\tau^{\prime}$; this may help the search for right-
handed neutrinos.

###  4.4 New scalar particle

In Sec. 3.2.2, we introduced a model with two singlet scalar particles,
$S_{L}$ and $S_{R}$, and discussed the current experimental limit. While
$S_{R}$ is difficult to probe, $S_{L}$ can be probed via exotic decay of the
SM Higgs at future colliders. In Fig. 2, we show the future prospects of HL-
LHC and Higgs factory derived in Ref. [91]. The majority of the parameter
region can be probed by Higgs factories.

###  4.5 Electric dipole moment

The CP violation to generate baryon asymmetry in general induces electric
dipole moments (EDMs) of SM fermions. In the local baryogenesis model in Eq.
(34), this is dominated by the correction from the parity partner of the
operator $\propto|H_{L}^{2}|W_{L}\tilde{W}_{L}$. The correction is
logarithmically enhanced by the renormalization group effect and is given by
[109]

$\displaystyle\frac{d_{e}}{e}\simeq$ $\displaystyle 1\times 10^{-30}{\rm
cm}\left(\frac{20~{}\mathrm{TeV}}{M}\right)^{2}\frac{{\rm
ln}(M^{2}/m_{h}^{2})}{8}$ $\displaystyle\simeq$ $\displaystyle 1\times
10^{-30}{\rm cm}\left(\frac{Y_{B}}{8.9\times
10^{-11}}\right)\left(\frac{20~{}\mathrm{TeV}}{v_{R}}\right)^{2}\left(\frac{v_{R}}{1.7T_{n}}\right)^{2}\frac{\ln(M^{2}/m_{h}^{2})}{8}.$
(45)

The current bound $d_{e}/e<1.1\times 10^{-29}$ cm [110] is satisfied for
$v_{R}$ above the collider bound from $W_{R}$ search. Near-future detection of
the electron EDM means $v_{R}=O(10)$ TeV, and $\tau^{\prime}$ and/or
$SU(2)_{R}$ gauge bosons are predicted to be within the reach of near future
collider experiments.

The EDM in the extra scalar model induced by the operator in Eq. (37) is [31]

$\displaystyle\frac{d_{e}}{e}\simeq$ $\displaystyle 1\times
10^{-31}\mathrm{cm}\left(\frac{20~{}\mathrm{TeV}}{v_{R}}\right)\left(\frac{40v_{R}}{M}\right)\left(\frac{\mu_{S}}{10~{}\mathrm{GeV}}\right)$
$\displaystyle\simeq$ $\displaystyle 1\times
10^{-31}\mathrm{cm}\left(\frac{Y_{B}}{8.7\times
10^{-11}}\right)\left(\frac{20~{}\mathrm{TeV}}{v_{R}}\right)^{2}\left(\frac{0.2v_{R}}{T_{n}}\right)^{2}\left(\frac{\mu_{S}}{10~{}\mathrm{GeV}}\right)^{2},$
(46)

where we use Eq. (38) in the second equality. The viable parameter region with
$\mu_{S}\sim 10$ GeV can be probed by future electron EDM measurements [111]
if $v_{R}=O(10)$ TeV.

In the model with right-handed neutrino dark matter, with the dilution in Eq.
(10), the required $M$ becomes smaller and the prediction on the EDM becomes
larger,

$\displaystyle\frac{d_{e}}{e}\simeq 1\times
10^{-30}~{}\mathrm{cm}\left(\frac{\mu_{S}}{10~{}\mathrm{GeV}}\right)^{2}\left(\frac{\sum_{i}m_{\nu_{i}}}{60~{}\mathrm{meV}}\right)\left(\frac{80}{g_{s}(T_{D})}\right)\left(\frac{Y_{B}}{8.7\times
10^{-11}}\right)\left(\frac{0.2v_{R}}{T_{n}}\right)^{2}.$ (47)

Interestingly, the prediction is independent of $v_{R}$ and is within the
reach of near-future measurements.

###  4.6 Gravitational waves

In order for the gravitational waves (GWs) produced by the $SU(2)_{R}$ phase
transition to be observable, a very strong phase transition is required. A
marginally SFOPT with $\langle H_{R}\rangle\sim T$ at the nucleation
temperature is not enough, and high-temperature expansion is not justified for
the analysis of very SFOPT. We thus use the full form in Appendix A to
numerically compute the gravitational wave signal. The numerical computation
shows that the GW signals are suppressed in the models using the running
quartic coupling or extra light singlet scalars discussed in the previous
sections. Here we make the following simple arguments instead of showing a
detailed computation.

  * •

For the minimal setup using the running quartic coupling, the SFOPT is
achieved only for $v_{R}\simeq 2\times 10^{8}~{}\mathrm{GeV}$. Such a high
$v_{R}$ will make the GW-signal peak at a very high frequency, which is beyond
the sensitivity of future observations.

  * •

For the running of the quartic coupling assisted by extra leptons, the SFOPT
can be achieved either by a relatively higher $v_{R}$ or a smaller $v_{R}$ but
with a large Yukawa coupling with the extra lepton to speed up the running.
For the former case, the typical frequency of GWs is too high. For the latter,
the large Yukawa coupling itself also suppresses the strength of the phase
transition and makes it only marginally SFOPT. The resultant GW signal is weak
and cannot be detected.

  * •

For the model with a light singlet scalar, the scalar is very light compared
with the $h_{R}$ field. This leads to a huge kinetic energy of $S_{R}$ during
the $SU(2)_{R}$ phase transition. Such a huge kinetic energy makes the
duration of the phase transition short (i.e., a large $\beta/H$ parameter,)
and thus suppresses the GW signal [31].

In other models to realize a SFOPT, the GW signal may be observable. In such a
model, the GW signal will be correlated with $v_{R}$, and hence with the new
gauge boson and fermion masses and the EDM.

##  5 Summary and discussion

The parity solution to the strong CP problem introduces the $SU(2)_{R}$ gauge
symmetry. In this paper, we proposed a model of baryogenesis from a first-
order $SU(2)_{R}$ phase transition.

The key ingredient of the model is how to obtain a non-zero $B-L$ asymmetry of
the SM particles. Although the $SU(2)_{R}$ sphaleron process does not produce
a non-zero charge of the naive $B-L$ symmetry with $\bar{q}(-1/3)$ and
$\bar{\ell}(+1)$, the washout by the $SU(2)_{L}$ sphaleron process is avoided
if some of the asymmetry of $SU(2)_{R}$-charged particles is (temporarily) not
transferred into that of $SU(2)_{L}$-charged particles. This is indeed
possible in the model with the minimal Higgs content $H_{L}$ and $H_{R}$,
where the strong CP problem is solved without introducing extra symmetries.
The scheme predicts a new hyper-charged fermion whose mass is correlated with
the $SU(2)_{R}$ symmetry-breaking scale $v_{R}$.

First-order $SU(2)_{R}$ phase transition can be realized by the coupling of
Higgses to other fields, as in electroweak baryogenesis. We studied two models
that do not introduce extra hierarchy problems beyond that of $H_{R}$ and
$H_{L}$. We considered the running of the quartic coupling. If the running of
the SM Higgs coupling is the SM one, $v_{R}>10^{8}$ GeV is required. With
Yukawa couplings to extra fermions, $v_{R}$ can be much lower. We also
considered a coupling of the Higgses to singlet scalar fields. $v_{R}$ can be
as low as the experimental lower bound from $W_{R}$ search. The scalar mass is
predicted to be around $10$ GeV and the scalar can be probed by future
colliders via its mixing with the SM Higgs.

CP violation may be introduced in various ways. We applied the idea of local
electroweak baryogenesis to $SU(2)_{R}$ phase transition and studied the
dimension-6 coupling of the $SU(2)_{R}$ gauge field with $H_{R}$ and the
dimension-5 coupling with a singlet scalar. The higher dimensional operators
generate a non-zero electron EDM that is observable in near future experiments
if $v_{R}=O(10)$ TeV. It will be interesting to investigate other realizations
of CP violation and the associated predictions on EDMs.

An obvious drawback of the model in comparison with the electroweak
baryogenesis is that we generically cannot predict the scale of new physics.
Indeed, the model works even if $v_{R}\gg 100$ TeV, for which observable
signals of the model in near future experiments are not guaranteed. However,
the model still has several correlated observable signals if $v_{R}$ is small
enough and hence maintains predictability in a loose sense. Also, in the model
with right-handed neutrino dark matter, $v_{R}$ is bounded from above and the
mass of a new hyper-charged fermion is predicted to be below $1$ TeV. It will
be worth considering other models with dark matter candidates and making
predictions on the energy scale of parity-symmetric models.

## Acknowledgement

I.R.W is supported by DOE grant DOE-SC0010008.

##  Appendix A Thermal potential

In this appendix, we discuss the details of the thermal potential, from which
we can compute the wall thickness and the phase-transition strength. We first
write down the effective potential of the scalar field
$H_{R}=(0,h_{R})^{t}/\sqrt{2}$ where the dynamical field $h_{R}$ obtains a
non-zero VEV $\sqrt{2}v_{R}$. Quantum corrections are computed up to one-loop
level.

We use the on-shell renormalization scheme where the VEV of $H_{R}$, $v_{R}$,
is unchanged by quantum corrections. The effective potential is

$\displaystyle V(h_{R},T)=V_{T=0}+V_{\rm FT},$ (48)

where

$\displaystyle V_{T=0}$
$\displaystyle=-\frac{1}{2}\mu_{h_{R}}^{2}h_{R}^{2}+\frac{1}{4}\lambda
h_{R}^{4}+4Bv_{R}^{2}h_{R}^{2}-\frac{3}{2}Bh^{4}+Bh^{4}\ln\left(\frac{h_{R}^{2}}{2v_{R}^{2}}\right),$
$\displaystyle B$
$\displaystyle=\frac{1}{1024\pi^{2}}\left(6g_{R}^{4}+3(g_{R}^{2}+g_{X}^{2})^{2}-4\sum_{f}n_{f}y_{f}^{4}\right),$
$\displaystyle V_{\rm FT}$
$\displaystyle=\frac{T^{4}}{2\pi}\left(4J_{B}(\frac{m_{W_{R}}}{T})+2J_{B}(\frac{m_{Z}}{T})-\sum_{f}n_{f}J_{F}(\frac{m_{f}}{T})\right).$
(49)

We apply a factor $2/3$ on the gauge-boson contributions to the $V_{\rm FT}$
to screen out the longitudinal mode. One may instead perform the thermal
resummation while including the longitudinal mode [81, 112]. All couplings
should be defined at $v_{R}$. Scalar contributions are neglected since scalar
couplings are weak.

For marginally SFOPT, where $v_{R}(T_{n})\simeq T_{n}$, we can use the high-
temperature expansion for the finite temperature contribution $V_{\rm FT}$. As
discussed in Ref [31], this computation scheme is expected to show the best
agreement with the lattice simulation. Under the high-temperature expansion,
the potential can be written as

$\displaystyle
V=D(T^{2}-T_{0}^{2})h_{R}^{2}-ETh_{R}^{3}+\frac{1}{4}h_{R}^{4},$ (50)

where

$\displaystyle D$
$\displaystyle=\frac{1}{16}\left(3g_{R}^{2}+g_{X}^{2}+\frac{1}{3}\sum_{f}n_{f}y_{f}^{2}\right),$
$\displaystyle E$
$\displaystyle=\frac{2}{3}\frac{1}{32\pi}\left(2g_{R}^{3}+(g_{R}^{2}+g_{X}^{2})^{3/2}\right),$
$\displaystyle T_{0}^{2}$ $\displaystyle=\frac{1}{2D}\mu_{H_{R}}^{2}.$ (51)

The phase transition strength can be simply expressed by $2E/\lambda$ under
the high-temperature expansion.

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