11institutetext: Department of Physics, University of California, Berkeley,
California 94720, USA22institutetext: Theoretical Physics Group, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA33institutetext:
Center for Cosmology and Particle Physics, Department of Physics, New York
University, New York, NY 10003, USA44institutetext: Department of Physics,
University of Chicago, Chicago, IL 60637, USA55institutetext: Enrico Fermi
Institute and Kavli Institute for Cosmological Physics, University of Chicago,
Chicago, IL 60637, USA66institutetext: 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,
Japan77institutetext: Theoretical Physics Department, CERN, Geneva,
Switzerland

#  Leptogenesis in Parity Solutions to the Strong CP Problem and Standard
Model Parameters

Juanca Carrasco-Martinez 3 David I. Dunsky 1,2 Lawrence J. Hall 4,5,6,7
Keisuke Harigaya

###### Abstract

We study the simplest theories with exact spacetime parity that solve the
strong CP problem and successfully generate the cosmological baryon asymmetry
via decays of right-handed neutrinos. Lower bounds are derived for the masses
of the right-handed neutrinos and for the scale of spontaneous parity
breaking, $v_{R}$. For generic thermal leptogenesis,
$v_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}10^{12}$ GeV, unless the small
observed neutrino masses arise from fine-tuning. We compute $v_{R}$ in terms
of the top quark mass, the QCD coupling, and the Higgs boson mass and find
this bound is consistent with current data at $1\sigma$. Future precision
measurements of these parameters may provide support for the theory or, if
$v_{R}$ is determined to be below $10^{12}$ GeV, force modifications. However,
modified cosmologies do not easily allow reductions in $v_{R}$ – no reduction
is possible if leptogenesis occurs in the collisions of domain walls formed at
parity breaking, and at most a factor 10 reduction is possible with non-
thermal leptogenesis. Standard Model parameters that yield low values for
$v_{R}$ can only be accommodated by having a high degree of degeneracy among
the right-handed neutrinos involved in leptogenesis. If future precision
measurements determine $v_{R}$ to be above $10^{12}$ GeV, it is likely that
higher-dimensional operators of the theory will yield a neutron electric
dipole moment accessible to ongoing experiments. This is especially true in a
simple UV completion of the neutrino sector, involving gauge singlet fermions,
where the bound from successful leptogenesis is strengthened to
$v_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}10^{13}$ GeV.

##  1 Introduction

Leptogenesis requires neutrino masses to be Majorana and CP violation in the
lepton sector. Observations of neutrinoless double beta decay and different
oscillation rates for neutrinos and anti-neutrinos would therefore maintain
strong interest in leptogenesis. However, the SM viewed as an effective field
theory is also expected to have these properties, so Majorana neutrino masses
and CP violation in neutrino oscillations are hardly evidence for
leptogenesis. Evidence from neutrino data could emerge in the context of an
underlying theory where the number of parameters in the lepton sector is
reduced, either by flavor symmetries Barbieri:1999ma  or by assuming that some
entries in the neutrino Yukawa matrix are too small to be relevant
Frampton:2002qc ; Harigaya:2012bw . Here we try an alternative direction.

A particularly simple and motivated setting for leptogenesis is provided by
theories with an underlying parity symmetry. The new fermions required for
leptogenesis are right-handed neutrinos, required by the parity transformation
$\nu_{L}\leftrightarrow\nu_{R}$. Theories with parity are strongly motivated
by the elegance of the matter representation Pati:1974yy ; Fritzsch:1974nn
and as a solution to the strong CP problem Beg:1978mt ; Mohapatra:1978fy ;
Babu:1989rb . Minimal theories that solve the strong CP problem via an exact
parity symmetry double the weak gauge group to $SU(2)_{L}\times SU(2)_{R}$ and
have a single Higgs doublet for each $SU(2)$, $H_{L}$ and $H_{R}$. Parity and
$SU(2)_{R}$ breaking occur spontaneously, with
$\left\langle{H_{R}}\right\rangle=v_{R}$ generated radiatively by the Higgs
Parity mechanism Hall:2018let . The effective theory below $v_{R}$ is
essentially the Standard Model (SM), and $v_{R}$ can be computed as a function
of SM parameters because it is the scale where the SM Higgs quartic coupling
vanishes. There are large uncertainties due to experimental uncertainties in
the top quark mass, $m_{t}$, and the QCD coupling, $\alpha_{s}$. Currently,
$v_{R}$ lies in the range $(3\times 10^{10}-3\times 10^{13})$ GeV, at
$1\sigma$.

In these theories Majorana masses for the light neutrinos arise from both the
seesaw mechanism by the exchange of $\nu_{R}$, and from a direct dimension-5
Weinberg operator contribution. The coefficient of the Weinberg operator is
determined by $v_{R}$ and by the size of the right-handed neutrino masses
$M_{i}$, giving

$\displaystyle m_{i}^{\rm dir}=\frac{v_{L}^{2}}{v_{R}^{2}}\;M_{i}.$ (1)

Hence the thermal leptogenesis bound $M_{1}>10^{9}$ GeV from naturalness
Hambye:2003rt , and the requirement that the direct contribution to the
neutrino masses not exceed the observed masses, leads to a naturalness lower
bound on $v_{R}$. If the flavor mixing angles and CP violating phases are of
order unity, $v_{R}$ is predicted to be at the bound.

In this paper we study leptogenesis in these theories, where radiative
spontaneous breaking of exact parity symmetry solves the strong CP problem. We
show that successful leptogenesis gives rise to strong bounds on $v_{R}$
solely from forbidding fine-tunings in the neutrino mass matrix. Such bounds
on $v_{R}$ can be translated to bounds on Standard Model parameters such as
$m_{t}$ and $\alpha_{s}$ which are associated with $v_{R}$ as discussed above.
Moreover, we discuss the implications of these measurements for the estimate
of the neutron electric dipole moment that arises from a higher-dimension
operator involving $H_{R}$ and the breaking of parity.

Leptogenesis in the same class of theories was analyzed in Dunsky:2020dhn ,
coupled with the additional requirement that dark matter is composed of the
lightest right-handed neutrino. This dark matter was produced by relativistic
freeze-out with subsequent dilution from the decay of a heavier right-handed
neutrino or by freeze-in with a low reheating temperature. Here, we remove
this dark matter requirement and analyze in detail the lowest possible $v_{R}$
that allows the right-handed neutrinos to generate the observed baryon
asymmetry of the universe through thermal or non-thermal leptogenesis.

In Sec. 2 we review the two simplest models of parity restoration that solve
the strong CP problem. The Left-Right model has no quarks and charged leptons
beyond those of the SM, while in the Mirror model a set of mirror fermions is
introduced. In Sec. 3 we compute $v_{R}$, paying particular attention to the
dependence on $m_{t}$ and $\alpha_{s}$. In Sec. 4 we discuss the origin of
neutrino masses; the effective field theory for the neutrino sector is
identical in the Left-Right and Mirror models. We derive the relationship
between the mass matrix of the active neutrinos, the right-handed neutrino
masses and neutrino Yukawa couplings relevant for leptogenesis.

In Sec. 5 we compute the naturalness bound on $v_{R}$ from thermal
leptogenesis in the case that the right-handed neutrinos do not possess a high
degree of degeneracy. The bound is very strong, so in Sec. 6 we study whether
it can be weakened by the decays of a non-thermal population of right-handed
neutrinos 6.1, degeneracy of right-handed neutrinos 6.2, or in the Radiative
Singlet Model where the light neutrino masses vanish at tree-level 6.3.

Up until this point, we assume parity breaking occurs at a phase transition
before inflation, so that the resulting domain walls are inflated away. In
Sec. 7 we relax the requirement of exact parity in the Higgs sector so that
the spontaneous breaking of parity may occur after inflation, leading to an
era with a domain wall network. We investigate whether the bound on $v_{R}$
could be weakened by having leptogenesis occur during collisions of domain
walls.

A summary of each of these leptogenesis scenarios and their corresponding
limits on both the scale of the right-handed neutrino masses, $M_{i}$, and the
scale of right-handed symmetry breaking, $v_{R}$, is shown in the table below.
Conclusions are presented in Sec. 8.

Leptogenesis Scenario |  Lower Limit on $M_{i}$ |  Lower Limit on $v_{R}$
---|---|---
Thermal Right-Handed Neutrinos, $\nu_{R}$ |  $\sim 10^{9}$ GeV (5.1) |  $\sim 10^{12}$ GeV (5.2)
Non-thermal $\nu_{R}$ |  $\sim 10^{7}$ GeV (6.1) |  $\sim 10^{11}$ GeV (6.1)
Degenerate $\nu_{R}$ |  $\sim 2\times 10^{3}$ GeV (6.2) |  $\sim 10^{9}$ GeV (6.2)
Radiative Singlet Model |  $\sim 10^{11}$ GeV (6.3) |  $\sim 10^{13}$ GeV (6.3)
Domain Wall Collisions |  $\sim 10^{9}$ GeV (7) |  $\sim 10^{12}$ GeV (7)

##  2 Parity Restoration with Minimal Higgs Doublets (Higgs Parity)

A remarkable feature of the Standard Model (SM) is that spacetime parity is
broken. The central role played by parity conservation in atomic and nuclear
physics, and the apparently fundamental nature of the transformation
$\bf{r}\rightarrow-\bf{r}$, resulted in parity violation emerging in the
mid-1950s as a profound surprise Lee:1956qn ; Okun:2006eb . Left-handed
fermions appear as doublets of the weak interaction gauge group $SU(2)_{L}$,
while the right-handed fermions do not feel this weak force. For example,
$\ell_{L}=(\nu_{L},e_{L})$ represent the $SU(2)_{L}$ doublets of left-handed
neutrinos and charged leptons, while $e_{R}$ are $SU(2)_{L}$ singlets and
right-handed neutrinos have not been discovered.

Although $\nu_{L}$ and $e_{L}$ are treated symmetrically by the weak
interaction, it is hard to conceive of two particles less like each other than
the neutrino and electron. The differences emerge from the breaking of the
weak interaction symmetry by the condensate of the Higgs field $H_{L}$, which
is also a doublet of $SU(2)_{L}$ and was discovered a decade ago. The
potential for the Higgs field is extremely simple involving two real
parameters

$\displaystyle V_{\rm
SM}(H_{L})=-\mu^{2}\,H_{L}^{\dagger}H_{L}+\frac{\lambda}{2}\,(H_{L}^{\dagger}H_{L})^{2}.$
(2)

The minimal extension of the Standard Model that allows an excess of baryons
over anti-baryons to develop in the early universe is obtained by adding
Majorana fermions, $N$, that do not feel the strong or electroweak
interactions and have Lagrangian terms

$\displaystyle L_{N}=y\;\ell_{L}NH_{L}+\frac{1}{2}M_{N}\,NN,$ (3)

where generation indices are omitted. If the couplings $y$ are large enough,
$N$ were produced during an early era of the hot big bang. As the temperature
dropped below their mass, they generate a lepton asymmetry by decays

$\displaystyle
N\rightarrow\ell_{L}H_{L},\;\;\overline{\ell_{L}}\;\overline{H_{L}},$ (4)

with different decay rates to the particle and anti-particle modes. The lepton
asymmetry is processed to a baryon asymmetry by sphalerons of the $SU(2)_{L}$
weak interactions.

Parity restoration provides a simple framework for realizing this cosmological
scheme of leptogenesis, as the fermions $N$ are interpreted as right-handed
neutrinos, which parity requires. To construct a theory which is parity
invariant it is necessary that, in addition to reflecting the spatial
coordinate, the weak interaction of $SU(2)_{L}$ is transformed into another
interaction, $SU(2)_{R}$, that acts on right-handed fermions. Thus the
electroweak gauge group must contain $SU(2)_{L}\times SU(2)_{R}$, and nature
must contain $W_{R}$ gauge bosons of $SU(2)_{R}$, in addition to the observed
$W_{L}$ gauge bosons of $SU(2)_{L}$. Under parity, in addition to
$\bf{r}\rightarrow-\bf{r}$,

$\displaystyle\begin{aligned}
SU(2)_{L}\;\;&\leftrightarrow\;\;SU(2)_{R},\hskip 14.45377pt\\\
\ell_{L}(2,1)=\begin{pmatrix}\nu_{L}\\\
e_{L}\end{pmatrix}\;\;\;&\leftrightarrow\;\;\;\ell_{R}(1,2)=\begin{pmatrix}\nu_{R}\\\
\psi_{R}\end{pmatrix}.\end{aligned}$

There are two theories of parity restoration that are consistent with thermal
leptogenesis. In both theories, the gauge group is $SU(3)_{c}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)$ or $SU(4)\times SU(2)_{L}\times
SU(2)_{R}$. In the minimal version (Model A in Hall:2018let ) which we call
the left-right (LR) theory, the Parity partners $(\bar{q},\bar{\ell})$ of the
$SU(2)_{L}$ doublet SM fermions $(q,\ell)$ are $SU(2)_{R}$ doublets and are
right-handed SM fermions except for $\nu_{R}$, and the $U(1)$ generator is
$B-L$ when acting on SM fermions. In particular, $\psi_{R}$ are identified as
the right-handed charged leptons, $e_{R}$. In the other, non-minimal, theory
(Model D in Hall:2018let ) which we call the Mirror theory in this paper, all
of the Parity partners
$(q^{\prime},\ell^{\prime},\bar{u}^{\prime},\bar{d}^{\prime},\bar{e}^{\prime})$
of the SM fermions $(q,\ell,\bar{u},\bar{d},\bar{e})$ are new particles. These
mirror fermions have the same $SU(3)_{c}\times U(1)_{Y}$ charges as the SM
counterparts, and $(q^{\prime},\ell^{\prime})$ are $SU(2)_{R}$ doublets. In
both theories, the neutrinos $\nu_{L}$ and $\nu_{R}$ have the same mass terms
and the same interactions relevant for leptogenesis, so the analysis and
results of this paper apply to both the LR theory and the Mirror theory.

There are three other closely related theories of parity restoration. Two
(Models B and C in Hall:2018let ) have a gauge group $SU(3)_{c}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)$, but differ from the LR and mirror
theory in whether parity conjugates the color and hypercharge quantum numbers.
Another one, with electroweak symmetry completely mirrored, has a gauge group
$SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{Y}\times
U(1)_{Y^{\prime}}$, fermions are doubled, and $\psi_{R}$ are mirror leptons
charged under Mirror QED, $e^{\prime}_{R}$ Barr:1991qx . However, in these
three theories, the lightest extra quark is stable and necessarily produced at
the high temperatures necessary for leptogenesis. At the QCD phase transition,
these quarks hadronize with SM quarks, forming heavy fractionally charged
particles whose terrestrial abundance is highly constrained Dunsky:2018mqs .
Consequently, the cosmology of these theories is challenging to reconcile with
leptogenesis. For this reason, we hereafter mainly focus on the LR and Mirror
theories which do not have these problems. (See, however, discussion at the
end of Sec. 6.2.) The $U(1)\times U(1)^{\prime}$ symmetry of the Mirror
electroweak theory may be spontaneously broken down to $U(1)$ to obtain the LR
or Mirror theory.

Figure 1:  Overview of the two parity-restoring models with minimal Higgs
sectors. The left-panel shows the ‘Left-Right’ theory, which has gauge
symmetry $SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ at high
energies. Parity interchanges left-handed and right-handed quarks and leptons,
and is spontaneously broken at $v_{R}$ by the $H_{R}$ vev, which also breaks
the gauge group: $SU(2)_{R}\times U(1)_{B-L}\rightarrow U(1)_{Y}$. SM Yukawa
couplings arise from the dimension-five operators shown in green. The right-
panel shows the ‘Mirror’ theory which contains mirror quarks and leptons. The
strong interaction gauge group may be $SU(3)_{c}$ or $SU(3)\times
SU(3)^{\prime}\rightarrow SU(3)_{c}$. Spacetime parity maps SM fields to their
chirality-flipped mirror counterparts. Parity is broken at $v_{R}$ by the vev
of $H_{R}$, with charged mirror fermions acquiring masses $v_{R}/v$ larger
than their SM counterparts, as shown in red.

Experimental searches for $W_{R}$ have placed a lower limit on $v_{R}$ of 13
TeV in the LR theory ATLAS:2019lsy , where the parity partners of the left-
handed SM fermions are right-handed SM fermions. In the Mirror theory, the
mirror quarks are triplets under QCD, at least at LHC energies, so that the
$u^{\prime}$ mass must be above 2 TeV, forcing $v_{R}$ above $10^{8}$ GeV.
$SU(2)_{R}$ must be broken at a larger scale than $SU(2)_{L}$. A key question
is how the Higgs fields transform under the gauge group $SU(2)_{L}\times
SU(2)_{R}$. In this paper we explore the possibility that the Higgs system is
minimal, namely under parity

$\displaystyle H_{L}(2,1)\;\;\leftrightarrow\;\;H_{R}^{\dagger}(1,2),$ (5)

leading to a doubling of the SM Higgs potential and the addition of a quartic
coupling

$\displaystyle
V(H_{L},H_{R})=\Biggl{(}-\mu^{2}\,H_{L}^{\dagger}H_{L}+\frac{\lambda}{2}\,(H_{L}^{\dagger}H_{L})^{2}\Biggr{)}+\Biggl{(}-\mu^{2}\,H_{R}^{\dagger}H_{R}+\frac{\lambda}{2}\,(H_{R}^{\dagger}H_{R})^{2}\Biggr{)}+\xi\,H_{L}^{\dagger}H_{L}H_{R}^{\dagger}H_{R}.$
(6)

At tree-level there is no realistic vacuum solution of this potential; either
$H_{L}$ and $H_{R}$ have equal vacuum values, $v_{R}=v_{L}$, or one of them
vanishes, $v_{L}=0$ Barbieri:2005ri . However, radiative corrections allow a
realistic vacuum with parity spontaneously broken at the scale where the SM
quartic coupling passes through zero, giving a $1\sigma$ range of

$\displaystyle v_{R}\simeq(3\times 10^{10}-3\times 10^{13})\;{\rm GeV},$ (7)

as discussed in the next section. This theory of Higgs Parity Hall:2018let  is
the minimal Higgs sector with parity restoration and, unlike conventional LR
theories with $SU(2)_{L}\times SU(2)_{R}$, the parameters of the Higgs
potential are real. Parity solves the strong CP problem in the LR theory, with
radiative corrections to $\bar{\theta}$ arising at 2-loops Hall:2018let ;
Hisano:2023izx  offering the possibility of detection of a neutron electric
dipole moment in upcoming experiments Alarcon:2022ero .111A non-minimal
alternative is to introduce soft breaking of parity in the scalar potential,
$\delta V=\mu^{2}(H_{L}^{\dagger}H_{L}-H_{R}^{\dagger}H_{R})$. Realistic vacua
then result at tree-level, with any value of $v_{R}$ above the experimental
bound. Indeed, this was the first model constructed with parity solving the
strong CP problem Babu:1989rb . This soft breaking in the electroweak Higgs
potential could arise from spontaneous breaking in some other sector of the
theory. In this paper we restrict our attention to the minimal electroweak
Higgs potential of (6) with parity exact. Strong CP is also solved in the
Mirror theory, providing QCD is not mirrored Barr:1991qx ; Dunsky:2019api  or
arises from the breaking of $SU(3)\times SU(3)^{\prime}$ to the diagonal sum
Barr:1991qx ; Bonnefoy:2023afx . In these cases the radiative corrections to
$\bar{\theta}$ are negligible, as in the SM.

The theory with Higgs Parity, whether LR or mirror, is an EFT and is expected
to have higher-dimension operators that generate a non-zero value of
$\bar{\theta}$ once parity is spontaneously broken. These first arise at
dimension 6 Hall:2018let

$\displaystyle{\cal
L}_{6}=\frac{C}{M_{Pl}^{2}}(H_{L}^{\dagger}H_{L}-H_{R}^{\dagger}H_{R})\;G\tilde{G}\;+\;...$
(8)

where $G$ is the QCD field strength. There are also relevant operators that
correct the quark Yukawa couplings at order $v_{R}^{2}/M_{Pl}^{2}$. The
resulting bound on $v_{R}$ is highly uncertain, as the UV completion at the
Planck scale $M_{Pl}$ is not known. For example, with $C=1$ the bound is
$v_{R}\lesssim(10^{12}-10^{13})$ GeV for $M_{Pl}$ in the range of the Planck
or reduced Planck mass. Thus, as $v_{R}$ is increased above about
$(10^{13}-10^{14})$ GeV, a “Little strong CP problem” emerges. However, it may
be that the LR or mirror theory has a UV completion before the Planck scale
that leads to $C\ll 1$, allowing these larger values of $v_{R}$. For example,
if parity is embedded into $SO(10)\times CP$ Hall:2018let  we expect $C\sim
v_{10}/M_{Pl}$, where $v_{10}$ is the $SO(10)$ breaking scale. Similarly $C\ll
1$ if supersymmetry is encountered between $v_{R}$ and $M_{Pl}$. In the coming
decade, experiments searching for the neutron electric dipole moment will
improve sensitivity by a factor of 30 Alarcon:2022ero , thereby probing values
of $v_{R}$ higher by a factor of 5. Thus, there is a good chance of discovery
if $v_{R}$ is in the range of $(10^{12}-10^{14})$ GeV and the LR or mirror
model is UV completed at the Planck scale, and there is also a chance for
discovery if $v_{R}$ is larger and a UV completion with $C\ll 1$ comes
earlier.

When gauge singlets $N$ are added to the SM, and thermal leptogenesis proceeds
via the interactions of (3), there is a bound on the mass of the lightest $N$
involved in the process, $M_{N}\mathop{}_{\textstyle\sim}^{\textstyle>}10^{9}$
GeV as discussed in Sec. 5.1. This can be relaxed in theories where the
decaying $N$ have a non-thermal abundance or where there is degeneracy among
the $N$. In this paper we study bounds on $m_{\nu_{R}}$ and $v_{R}$ from
leptogenesis in parity symmetric theories with minimal Higgs fields. In
addition to the seesaw contribution to the light neutrino mass, which is
inversely proportional to $m_{\nu_{R}}$, there is a direct term from a
dimension-5 operator, $(v_{L}^{2}/v_{R}^{2})m_{\nu_{R}}$, proportional to
$m_{\nu_{R}}$ as discussed in Sec. 4. Forbidding a fine-tuned cancellation
between these contributions, we derive a lower bound of $v_{R}>3\times
10^{11}$ GeV, in the middle of the range (7) allowed by the Higgs Parity
theory.

##  3 The Scale of $SU(2)_{R}$ in Higgs Parity

We begin with a brief review of how the scale $v_{R}$ is correlated with the
values of the top quark mass, $m_{t}$, and the QCD coupling, $\alpha_{s}$ in
the Higgs Parity theory Hall:2018let . The minimal Higgs potential, (6), is
usefully rewritten as

$\displaystyle
V(H_{L},H_{R})=-\mu^{2}\left(H_{L}^{\dagger}H_{L}+H_{R}^{\dagger}H_{R}\right)+\frac{\lambda}{2}\left(H_{L}^{\dagger}H_{L}+H_{R}^{\dagger}H_{R}\right)^{2}+\lambda^{\prime}\;H_{L}^{\dagger}H_{L}H_{R}^{\dagger}H_{R}.$
(9)

We assume that the mass scale $\mu$ is much larger than the electroweak scale,
$v_{L}$.

With positive $\mu^{2}$, $H_{R}$ obtains a large vacuum expectation value
$\left\langle{H_{R}}\right\rangle=\mu/\lambda^{1/2}\equiv v_{R}$ and spacetime
parity is spontaneously broken. After integrating out $H_{R}$ at tree-level,
the low energy effective potential of $H_{L}$ is

$\displaystyle V_{\rm
LE}(H_{L})=\lambda^{\prime}\;v_{R}^{2}\;H_{L}^{\dagger}H_{L}-\lambda^{\prime}\left(1+\frac{\lambda^{\prime}}{2\lambda}\right)(H_{L}^{\dagger}H_{L})^{2}.$
(10)

The hierarchy $v_{L}\ll v_{R}$ is obtained only if the quadratic term is
small, which requires a small value of $\lambda^{\prime}\sim-
v_{L}^{2}/v_{R}^{2}$. The quartic coupling of the Higgs $H_{L}$, $\lambda_{\rm
SM}$, is then very small at the symmetry breaking scale $v_{R}$. The nearly
vanishing quartic coupling can be understood by an approximate global $SU(4)$
symmetry under which $(H_{L},H_{R})$ forms a fundamental representation. For
$|\lambda^{\prime}|\ll 1$ the potential in Eq. (9) becomes $SU(4)$ symmetric.
The $SU(4)$ symmetry is spontaneously broken by
$\left\langle{H_{R}}\right\rangle$ and the SM Higgs is understood as a Nambu-
Goldstone boson with vanishing potential.

At tree-level the potential still leads to
$\left\langle{H_{L}}\right\rangle=\left\langle{H_{R}}\right\rangle=v_{R}$
because of the small quartic coupling. However, for extremely small
$\lambda^{\prime}$, vacuum alignment in the $SU(4)$ space is fixed by quantum
corrections which violate the $SU(4)$ symmetry. The dominant effect is
renormalization group running from energy scale $v_{R}$ down to $v_{L}$. The
top quark contribution dominates over the gauge contribution and generates a
positive quartic coupling $\lambda_{\rm SM}(v_{L})\simeq 0.1$, and creates the
minimum of the potential at $v_{L}\ll v_{R}$. From the perspective of running
from low to high energy scales, the scale at which the SM Higgs quartic
coupling nearly vanishes is the scale $v_{R}$. Threshold corrections to
$\lambda_{\rm SM}(v_{R})$ are computed in Dunsky:2019api ; Hall:2019qwx  and
are typically $O(10^{-3})$.

The vacuum alignment can be also understood in the following way. For
$\lambda^{\prime}>0$, the minima of the potential are
$\left(\left\langle{H_{L}}\right\rangle,\left\langle{H_{R}}\right\rangle\right)=\left(v_{R},0\right)$
and $\left(0,v_{R}\right)$, where $v_{R}\equiv\mu/\lambda^{1/2}$, and the mass
of Higgses are of order $\mu$. For $\lambda^{\prime}<0$, the minima are
$\left\langle{H_{L}}\right\rangle=\left\langle{H_{R}}\right\rangle\sim v_{R}$.
None of these minima for $\lambda>0$ and $\lambda^{\prime}<0$ give rise to an
$SU(2)$ sector broken at a scale much lower than the other. To obtain a viable
vacuum, we need $\lambda^{\prime}\simeq 0$, for which the potential has an
accidental $SU(4)$ symmetry and nearly degenerate vacua with
$v_{L}^{2}=v_{R}^{2}=\mu^{2}/\lambda$. In this case, quantum corrections must
be taken into account to determine the orientation of the vacuum. The dominant
effect is given by the top quark Yukawa coupling, which leads to a Colemann-
Weinberg potential that in the limit that $\lambda^{\prime}=0$ orients the vev
entirely in the $H_{R}$ or $H_{L}$ direction. However, a small negative
$\lambda^{\prime}$ slightly destabilizes the vacuum with all the vev in
$H_{R}$ to give
$\left(\left\langle{H_{L}}\right\rangle,\left\langle{H_{R}}\right\rangle\right)=\left(v_{L},v_{R}\right)$
with $v_{L}\ll v_{R}$. There is also a physically equivalent vacuum with the L
and R labels interchanged: we define L by $v_{L}\ll v_{R}$.

Between the electroweak scale and the scale of parity restoration, $v_{R}$,
the running of the Higgs quartic coupling $\lambda_{\rm SM}$ is exactly the
same as in the SM. We follow the computation in Buttazzo:2013uya  and show the
running in the left panel of Fig. 2 for a range of values for the top quark
mass $m_{t}=(172.56\pm 0.48)$ GeV, QCD coupling constant at the $Z$ boson mass
$\alpha_{S}(m_{Z})=(0.1179\pm 0.0009)$, and Higgs mass $m_{h}=(125.25\pm
0.17)$ GeV.

Figure 2: (Left) Running of the SM quartic coupling. (Right) Predictions from
the Higgs Parity mechanism for the scale $v_{R}$ as a function of the top
quark mass, $m_{t}$. Contours of $\alpha_{S}(M_{Z})$ show how the prediction
changes with the uncertainty in the QCD coupling constant. The thickness of
each contour corresponds to the current $1\sigma$ uncertainty in $m_{h}$ of
$\pm 170$ MeV. Both panels are for the LR model.

The value of the SM quartic coupling at the scale $v_{R}$ is not exactly zero
because of the threshold correction Dunsky:2019api ,

$\displaystyle\lambda_{\rm SM}(v_{R})\simeq-\frac{3}{8\pi^{2}}y_{t}^{4}\,{\rm
ln}\frac{e}{y_{t}}+\frac{3}{128\pi^{2}}(g^{2}+{g^{\prime}}^{2})^{2}\left({\rm
ln}\frac{e\sqrt{2}}{\sqrt{g^{2}+{g^{\prime}}^{2}}}-{\rm
ln}\frac{g^{2}}{\sqrt{g^{4}-g^{\prime
4}}}\right)+\frac{3}{64\pi^{2}}g^{4}\,{\rm ln}\frac{e\sqrt{2}}{g},$ (11)

where the $\overline{\rm MS}$ scheme is assumed.222Here it is assumed that the
top Yukawa is simply given by $y_{t}H_{L}q\bar{u}+y_{t}H_{R}\bar{q}u$. In the
LR theory, it is possible that $u$ and $\bar{u}$ has a Dirac mass term. The
threshold correction for this case is different from Eq. (11), but this only
makes the prediction on $v_{R}$ for given SM parameters smaller Hall:2019qwx .
In Model D, Yukawa interactions of $H_{L}qu+H_{R}\bar{q}\bar{u}$ are allowed
by the gauge symmetry. As is shown in Appendix C, the prediction on $v_{R}$
for given SM parameters can become only smaller also in this case. The
prediction for the scale $v_{R}$ is shown in the right panel of Fig. 2 as a
function of $m_{t}$. Colored contours show how the prediction for $v_{R}$
changes when the QCD coupling constant varies by $\pm 1\sigma$ and $\pm
2\sigma$ deviations about its mean. The thickness of each curve corresponds to
the $1\sigma$ uncertainty in the measured Higgs mass, $m_{h}=(125.25\pm 0.17)$
GeV. With 2$\sigma$ uncertainties, $v_{R}$ can be as low as $3\times 10^{9}$
GeV. Future measurements of SM parameters can pin down the scale $v_{R}$ with
an accuracy of a few tens of percent Dunsky:2019api .

How accurately will $v_{R}$ be determined by future measurements of
$m_{t},\alpha_{s}$ and $m_{h}$? At the present central values, varying both
$m_{t}$ and $\alpha_{s}$ by the current $1\sigma$ uncertainties of 480 MeV and
0.0009 gives a range in $v_{R}$ of about three orders of magnitude, from about
$3\times 10^{10}$ GeV to $3\times 10^{13}$ GeV. Studies of expected reductions
in these uncertainties Schwienhorst:2022yqu ; dEnterria:2022hzv ;
Begel:2022kwp ; Dawson:2022zbb  offer the possibility of a large improvement
in the determination of $v_{R}$. Over the coming decade, improvements by a
factor 3 on $\delta m_{t}$ and 5 on $m_{h}$ from High Luminosity LHC, and a
factor 2 on $\delta\alpha_{s}$, will shrink the $1\sigma$ range for $v_{R}$ to
about one order of magnitude. However, the future allowed range for $v_{R}$
depends strongly on the results for the central values of $m_{t}$ and
$\alpha_{s}$. From the right panel of Fig. 2, it is apparent that the
determination of $v_{R}$ becomes more precise at higher values of $m_{t}$,
where high values of $v_{R}$ can be reliably excluded. In the left panel of
Fig. 3, we assume central values for $m_{t}(\alpha_{s})$ that are $1\sigma$
above (below) the current values and find that the full $1\sigma$ range for
$v_{R}$ is only about a factor of 3. In the right panel, we assume central
values for $m_{t}(\alpha_{s})$ that are $1\sigma$ below (above) the current
values and find that the full $1\sigma$ range for $v_{R}$ is still 2 orders of
magnitude.

Figure 3: Future determinations of $v_{R}$ in the LR model from improved
measurements of SM parameters. The figures are analogous to the right panel of
Fig. 2 but with improved uncertainties. The labelled colored bands and the
uncertainties on the $m_{t}$ axis are for expected improvements over the next
decade, with uncertainties of $\pm 160$ MeV for $m_{t}$, $\pm 0.00045$ for
$\alpha_{S}(M_{Z})$, and $\pm 30$ MeV for $m_{h}$. The yellow band highlights
the even finer precision expected with a next generation $e^{+}e^{-}$
collider. The 3$\sigma$ uncertainty range of $m_{t}$ from a future
$e^{+}e^{-}$ collider is roughly the same as the $1\sigma$ uncertainty range
from the High Luminosity LHC, shown as a vertical dark-gray band. Left:
Centered on $m_{t}(\alpha_{s})$ that are $1\sigma$ above (below) the current
values. Right: Centered on $m_{t}(\alpha_{s})$ that are $1\sigma$ below
(above) the current values.

At a future $e^{+}e^{-}$ collider, a top threshold scan could give a further
factor of three reduction in $\delta m_{t}$, high statistics $Z$ measurements
could reduce $\delta\alpha_{s}$ by a further factor of 4, and the uncertainty
on $m_{h}$ would be around 10 MeV. The yellow bands of Fig. 3 indicate the
uncertainty in $v_{R}$ with next generation colliders. With present central
values for these parameters, the full $1\sigma$ range for $v_{R}$ would then
be about a factor of two, whereas the ranges for other choices of the central
values can be seen in the right panel of Fig. 3.

Using an analytic approximation, we find uncertainties in $\log_{10}v_{R}$ of

$\displaystyle\delta(\log_{10}v_{R})\;=\;(\log_{10}v_{R}-7.9)\left(-0.23\;\frac{\delta
m_{t}}{0.48{\rm GeV}},\;\;0.15\;\frac{\delta_{\alpha_{s}}}{0.0009}\right)$
(12)

from uncertainties $(\delta m_{t},\delta_{\alpha_{s}})$. This is a good
approximation for $10^{9}\,{\rm GeV}<v_{R}<10^{14}\,{\rm GeV}$, and the first
factor shows how the uncertainty grows with $v_{R}$. Using this approximation,
at the current central values the $1\sigma$ uncertainties in $\log_{10}v_{R}$
are $(\pm 0.83,\pm 0.54)$ now, $(\pm 0.28,\pm 0.18)$ after High Luminosity
LHC, and $(\pm 0.09,\pm 0.07)$ after runs at a future $e^{+}e^{-}$ collider.
The latter case corresponds to $\delta v_{R}/v_{R}=(\pm 0.21,\pm 0.16)$.

##  4 Neutrino Masses with Higgs Parity

In both the LR and Mirror theories with Higgs Parity, neutrino masses arise
from operators of dimension-5

$\displaystyle{\cal L}_{\nu}$
$\displaystyle=-\frac{1}{2M}\left(\ell_{i}c^{*}_{ij}\ell_{j}\;H_{L}H_{L}+\bar{\ell}_{i}c_{ij}\bar{\ell}_{j}\;H_{R}H_{R}\right)+\frac{1}{M}\,\ell_{i}b_{ij}\bar{\ell}_{j}\;H_{L}H_{R}+{\rm
h.c.},$ (13)

where $M$ is real, $c_{ij}$ is symmetric and $b_{ij}$ is Hermitian. The
leptons, which transform under $SU(2)_{L}\times SU(2)_{R}$ as $\ell(2,1)$ and
$\bar{\ell}(1,2)$, are described by two-component left-handed Weyl fields, so
that the parity transformation is $\ell\leftrightarrow\bar{\ell}^{\dagger}$,
and $H_{L}\leftrightarrow H_{R}^{\dagger}$. Previously we called the lepton
doublets of the Mirror theory $\ell^{\prime}$; now we call them $\bar{\ell}$,
so that our analysis applies to both LR and Mirror theories. In both theories,
we write the neutral field in this doublet as $\bar{\nu}$, and describe the
corresponding state as a right handed neutrino, $\nu_{R}$.

The operators of (13) can arise from exchanges of heavy fermions, transforming
under $SU(2)_{L}\times SU(2)_{R}$ as $(1,1)$ or $(3,1)+(1,3)$ for the lepton
number violating case and $(1,1)$ or $(2,2)$ for the lepton number conserving
operator Hall:2018let ; Hall:2019qwx . In section 6.3 we explore interesting
features of the theory where both types of operator are generated by three
Majorana singlets $S(1,1)$. The operators can also arise from the exchange of
massive scalars transforming as $(3,1)+(1,3)$ ($(2,2)$) for the lepton number
violating (conserving) operators. If the masses of the heavy exchanged
particles are larger than $v_{R}$, then the effective theory (13) applies at
scales of $v_{R}$ and below. Even if these masses are below $v_{R}$, providing
they are the largest mass terms in the neutral fermion mass matrix, the
effective theory relevant for neutrino masses and leptogenesis is given by
(13) with $H_{R}$ replaced by $v_{R}$.

The effective Lagrangian of (13) leads to a $6\times 6$ neutrino mass matrix,

$\displaystyle\begin{array}[]{c}\big{(}\begin{array}[]{cc}\nu_{i}&\bar{\nu}_{i}\end{array}\big{)}\\\
{}\hfil\end{array}\begin{pmatrix}M_{ij}\,v_{L}^{2}/v_{R}^{2}\hskip
18.06749pt\,&y_{ij}v_{L}\\\
y_{ji}v_{L}&M^{*}_{ij}\end{pmatrix}\bigg{(}\begin{array}[]{c}\nu_{j}\\\
\bar{\nu}_{j}\end{array}\bigg{)}\,,$ (19)

where $M_{ij}=c_{ij}v_{R}^{2}/M$ and $y_{ij}=b_{ij}v_{R}/M$. Without loss of
generality, we can work in a basis where $c_{ij}$ is diagonal such that

$\displaystyle M_{ij}$ $\displaystyle=M_{i}\,\delta_{ij},$ (20)

with all $M_{i}$ real and positive and no summation over indices. On
integrating out the three heavy states assuming $cv_{R}^{2}\gg bv_{L}v_{R}$,
we obtain a mass matrix for the three light neutrinos:

$\displaystyle m_{ij}\,$
$\displaystyle=\,\delta_{ij}\frac{v_{L}^{2}}{v_{R}^{2}}M_{i}-y_{ik}\;\frac{1}{M_{k}}\;y^{T}_{kj}v_{L}^{2}\,\equiv\,\delta_{ij}\,m_{i}^{\rm
dir}-m_{ij}^{\rm ss}.$ (21)

We call the first term the “direct” contribution and the second the “seesaw”
contribution.

It is useful to study the direct and seesaw contributions to the light
neutrino mass matrix from each $\nu_{R_{i}}$. Each $\bar{\nu}_{i}$ field
couples to a single combination of $\ell_{j}$, which we call
$\tilde{\ell}_{i}$, so that the Yukawa coupling of $\bar{\nu}_{i}$ can be
written as

$\displaystyle{\cal L}_{y_{i}}=y_{i}\;\tilde{\ell_{i}}\,\bar{\nu}_{i}\,H+{\rm
h.c.},\qquad\tilde{\ell_{i}}=\ell_{j}\,y_{ji}/y_{i},\qquad
y_{i}^{2}\equiv\sum_{j}|y_{ji}|^{2},$ (22)

with $y_{i}$ real. Thus, each $\nu_{R_{i}}$ gives mass contributions to two
different states, a direct one for $\nu_{i}$ and a seesaw one for
$\tilde{\nu}_{i}$. (Note that $\tilde{\ell}_{i}$ are not orthogonal.) If large
leptonic mixing angles arise from the neutrino sector, $\nu_{i}$ and
$\tilde{\nu}_{i}$ are expected to be very different, and generically they are
not orthogonal. Consequently, the Lagrangian for the light neutrino masses can
be written as a sum of three such terms, one from each $\nu_{R_{i}}$

$\displaystyle{\cal L}_{\nu}=\frac{1}{2}\sum_{i}\left(m_{i}^{\rm
dir}\;\nu_{i}\nu_{i}-m_{i}^{\rm
ss}\;\tilde{\nu}_{i}\tilde{\nu}_{i}\right)+{\rm
h.c.}=\frac{1}{2}\sum_{i}\left(\frac{v_{L}^{2}}{v_{R}^{2}}\;M_{i}\;\nu_{i}\nu_{i}-\;\frac{y_{i}^{2}v_{L}^{2}}{M_{i}}\;\tilde{\nu}_{i}\tilde{\nu}_{i}\right)+{\rm
h.c.}.$ (23)

In parity-symmetric theories, Eq. (21) demonstrates that the direct
contribution to the neutrino mass, $m_{i}^{\rm
dir}=(v_{L}^{2}/v_{R}^{2})M_{i}$, always contributes to the active neutrino
mass, and may dominate over the seesaw contribution, $m_{ij}^{\rm ss}$. A
useful numerical parameterization of the direct contribution is

$\displaystyle m_{i}^{\rm dir}\simeq 10^{-1}\,{\rm
eV}\left(\frac{M_{i}}{10^{9}\,\rm GeV}\right)\left(\frac{6\times 10^{11}\,\rm
GeV}{v_{R}}\right)^{2}.$ (24)

The lower bounds on $v_{R}$ derived in this paper follow directly from the
equal magnitudes of the $\ell_{i}\ell_{j}H_{L}^{2}$ and
$\bar{\ell}_{i}\bar{\ell}_{j}H_{R}^{2}$ couplings of (13), which gives
$m_{i}^{\rm dir}=(v_{L}^{2}/v_{R}^{2})M_{i}$ and (24). In conventional LR
models, with gauge symmetry breaking from bi-doublets and triplets, the direct
contribution to neutrino masses is replaced by a type-II seesaw contribution.
While this contribution is proportional to $(v_{L}^{2}/v_{R}^{2})M_{i}$, the
proportionality constant is a free parameter, typically less than unity, so
that the bounds on $v_{R}$ are lost.

##  5 Natural Bound on $v_{R}$ from Thermal Leptogenesis

In the early universe, decays of right-handed neutrinos can generate a lepton
asymmetry, which is then processed by electroweak sphalerons to give a
cosmological baryon asymmetry. In the case that the lightest right-handed
neutrino, $\nu_{R_{1}}$, was in the thermal bath before or during the era of
decay, this mechanism is known as thermal leptogenesis. The physics of
leptogenesis is the same whether the decaying state is $N_{1}$ of the
augmented Standard Model, (SM + $N$), or $\nu_{R_{1}}$ of a theory with parity
restoration. For simplicity, in both cases we refer to the decaying state as
$\nu_{R_{1}}$.

After inflation we assume that the universe reheats to a temperature
$T_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}M_{1}$ (except in Sec. 6.1) so
that thermal production of $\nu_{R_{1}}$ is not suppressed. We assume that the
maximum temperature reached after inflation $T_{\rm max}<v_{R}$, to avoid
domain wall formation at the $SU(2)_{R}$ phase transition (we remove this
constraint in Sec. 7 where we add a small $Z_{2}$-breaking term that makes the
wall network unstable). While the EFT of (13) ensures $M_{1}<v_{R}$, in many
theories of inflation $T_{\rm max}\gg T_{RH}$ Harigaya:2013vwa ;
Mukaida:2015ria , so that $v_{R}\gg M_{1}$ allows for a wide range of
reheating scenarios. Virtual processes involving gauge bosons of mass of order
$v_{R}$ put $\nu_{R_{1}}$ in thermal equilibrium if Dror:2020jzy

$\displaystyle T_{RH}>5\times 10^{10}\,{\rm
GeV}\left(\frac{v_{R}}{10^{12}\,{\rm GeV}}\right)^{4/3}.$ (25)

When leptogenesis is close to the strong washout regime, or in strong washout
as in Sec. 6.3, interactions with the SM Higgs are sufficient to put
$\nu_{R_{1}}$ in thermal equilibrium, so only $T_{R}>M_{1}$ is required.

###  5.1 Bound on $M_{1}$

Decays of $\nu_{R_{1}}$ give a yield for the baryon asymmetry of

$\displaystyle Y_{B}=\frac{n_{B}}{s}=\frac{28}{79}\,\epsilon\,\eta\,Y_{\rm
therm}\simeq 10^{-3}\epsilon\,\eta\,\qquad\text{(Thermal Leptogenesis)},$ (26)

where $Y_{\rm therm}$ is the thermal yield of $\nu_{R_{1}}$. The efficiency
factor $\eta$ is small when $\nu_{R_{1}}$ is in thermal equilibrium at decay
(the ‘strong wash-out’ regime). At the energy scale $M_{1}$, the Yukawa
interaction of $\nu_{R_{1}}$ is $y_{1}\,\tilde{\ell}_{1}\nu_{R_{1}}H_{L}$,
where we use the tilde basis of (22). Integrating out $\nu_{R_{1}}$ leads to a
seesaw mass for $\tilde{\nu}_{1}$ of size $m_{1}^{\rm
ss}=y_{1}^{2}v_{L}^{2}/M_{1}$. The ‘strong wash-out’ regime is avoided if
$m_{1}^{\rm ss}\mathop{}_{\textstyle\sim}^{\textstyle<}10^{-3}$ eV, in which
case $0.3<\eta<1$ Giudice:2003jh . $\epsilon$ is the lepton asymmetry produced
per $\nu_{R_{1}}$ decay, and results because at 1-loop level
$\Gamma(\nu_{R_{1}}\rightarrow
H_{L}\tilde{\ell}_{1})\neq\Gamma(\nu_{R_{1}}\rightarrow
H_{L}^{\dagger}\tilde{\ell}_{1}^{\dagger})$.

We begin by assuming that the dominant contribution to the 1-loop diagram for
$\epsilon$ involves the exchange of $\nu_{R_{2}}$, while the contribution from
the exchange of $\nu_{R_{3}}$ is negligible, giving Fukugita:1986hr ;
Davidson:2008bu

$\displaystyle\epsilon=$ $\displaystyle{\rm{Br}}(\nu_{R_{1}}\rightarrow
H_{L}\ell_{1})-{\rm{Br}}(\nu_{R_{1}}\rightarrow
H_{L}^{\dagger}\ell_{1}^{\dagger})$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\sum_{j=2}\frac{\text{
Im}(\mathbf{y}^{\dagger}{\mathbf{y}})^{2}_{j1}}{(\mathbf{y}^{\dagger}{\mathbf{y}})_{11}}g\left(\frac{M_{j}^{2}}{M_{1}^{2}}\right)=\frac{1}{8\pi}\frac{(y_{11}+y_{22})^{2}}{y_{11}^{2}+|y_{12}|^{2}}{\rm
Im}(y_{12}^{2})g(x_{2}),$ (27)

where $g(x)=\sqrt{x}((1-x)^{-1}+1-(1+x)\ln(1+1/x))$ and
$x_{2}=M_{2}^{2}/M_{1}^{2}$. $\epsilon$ may be written in terms of the inner
product of $\tilde{\ell}_{1}$ and $\tilde{\ell}_{2}$,

$\displaystyle(\tilde{\ell_{1}},\tilde{\ell_{2}})=\frac{(y_{11}+y_{22})y_{12}}{\sqrt{(y_{11}^{2}+|y_{12}|^{2})(y_{22}^{2}+|y_{12}|^{2})}}\equiv{\rm
cos}\theta_{2}e^{i\phi_{2}},$ (28)

so that substituting (28) into (5.1) and using the definition of $y_{2}^{2}$
gives

$\displaystyle\epsilon=\frac{y_{2}^{2}}{8\pi}\cos^{2}\theta_{2}\sin
2\phi_{2}\;g(x_{2})\,.$ (29)

Here $\theta_{2}$ is the angle between the two vectors for $\tilde{\ell}_{1}$
and $\tilde{\ell}_{2}$ in flavor space. If they are orthogonal there is no
flavor mixing between these two generations in the neutrino sector, and CP
violation vanishes.

In the case of degeneracy, $M_{2}-M_{1}\ll M_{2}+M_{1}$, $g(x)\gg 1$, and we
defer this case to section 6.2. For $M_{2}$ of order or much greater than
$M_{1}$, a good approximation is $|g(x_{2})|\simeq(3/2)M_{1}/M_{2}$, giving

$\displaystyle\epsilon\simeq\frac{3}{16\pi}\cos^{2}\theta_{2}\sin
2\phi_{2}\;\frac{m_{2}^{\rm ss}M_{1}}{v^{2}}$ (30)

where $m_{2}^{\rm ss}=y_{2}^{2}v^{2}/M_{2}$ is the seesaw mass generated by
the exchange of $\nu_{R_{2}}$. Requiring that $\epsilon$ is large enough to
give the observed baryon asymmetry, $Y_{B}\simeq 10^{-10}$, and that the
seesaw mass from $\nu_{R_{2}}$ exchange is bounded, $m_{2}^{\rm ss}<m_{2}^{\rm
ss\,*}$, leads to a lower bound on $M_{1}$

$\displaystyle M_{1}\mathop{}_{\textstyle\sim}^{\textstyle>}\;\frac{6\times
10^{8}\;{\rm GeV}}{\eta A_{2}}\;\left(\frac{0.05{\rm eV}}{m_{2}^{\rm
ss\,*}}\right),\,\hskip 18.06749pt{\rm with}\hskip
18.06749ptA_{2}=\cos^{2}\theta_{2}\sin 2\phi_{2}.$ (31)

The bound becomes stronger as $\eta$, $\cos^{2}\theta_{2}$ or $\sin 2\phi_{2}$
are taken less than unity.

The contribution to $\epsilon$ from $\nu_{R_{3}}$ exchange takes the same form
as that from $\nu_{R_{2}}$ exchange, so that together they yield

$\displaystyle\epsilon\simeq\frac{3}{16\pi}(A_{2}m_{2}^{\rm
ss}+A_{3}m_{3}^{\rm ss})\;\frac{M_{1}}{v^{2}},$ (32)

where $m_{3}^{\rm ss}=y_{3}^{2}v^{2}/M_{3}$, $A_{3}=\cos^{2}\theta_{3}\sin
2\phi_{3}$ and $\theta_{3}$ is the angle between $\tilde{\ell}_{1}$ and
$\tilde{\ell}_{3}$. We have also taken $M_{3}$ to be of order or much greater
than $M_{1}$.

Taking $m_{2,3}^{\rm ss}\,\lesssim\,m_{2,3}^{\rm ss\,*}$, the lower bound on
$M_{1}$ becomes

$\displaystyle M_{1}\;\mathop{}_{\textstyle\sim}^{\textstyle>}\;\frac{6\times
10^{8}\;{\rm GeV}}{\eta(A_{2}+A_{3})}\;\left(\frac{0.05\,{\rm
eV}}{m_{2,3}^{\rm ss\,*}}\right)>\;3\times 10^{8}\;{\rm
GeV}\left(\frac{0.05\,{\rm eV}}{m_{2,3}^{\rm ss\,*}}\right).\,$ (33)

Can this bound on the lightest right-handed neutrino be weakened by increasing
${m_{2,3}^{\rm ss\,*}}$ above 0.05 eV? The cosmological limit on the sum of
the neutrino masses DES:2021wwk  is now so severe that the three neutrinos
cannot be made almost degenerate with masses larger than 0.05 eV. Significant
weakening requires a fine-tuning to force a cancellation among different
contributions to a light neutrino mass eigenvalue. In this paper we do not
allow such tunings and hence study the consequences of the approximate
naturalness bound

$\displaystyle M_{1}\;\mathop{}_{\textstyle\sim}^{\textstyle>}\;10^{9}\;{\rm
GeV}.\,$ (34)

This bound on $M_{1}$ applies in (SM$+N$) and in all parity symmetric theories
where neutrino masses result from the operators of (13). The precise numerical
value of the bound depends on the amount of cancellation or tuning,
parameterized by ($m_{2.3}^{\rm ss\,*}/0.05$ eV) in Eq. (33).

In the ‘strong wash-out’ regime the efficiency $\eta$ is much less than unity,
strengthening this bound on $M_{1}$. The bound on $M_{1}$ in Figs. 4, 5 and
LABEL:fig:naturalnessPlotSinglet are for weak washout with $\eta\simeq 1$ in
order to show the most optimistic lower limits on $M_{1}$ and $v_{R}$ and to
demonstrate such limits are still severe. After the lepton asymmetry is
created by $\nu_{R_{1}}$ decay, removal of the asymmetry via $\nu_{R_{1}}$
production is small provided $m_{1}^{\rm
ss}\mathop{}_{\textstyle\sim}^{\textstyle<}10^{-3}\,{\rm eV}$; for larger
values of $m_{1}^{\rm ss}$, the bound on $M_{1}$ is strengthened by
approximately $(m_{1}^{\rm ss}/10^{-3}\,\mbox{eV})^{1.16}$ Giudice:2003jh .
Recall that (33) applies even if $M_{2,3}\gg M_{1}$, in which case washout by
$\nu_{R_{2,3}}$ production is negligible. Even in the case that $M_{2}$ is
comparable to $M_{1}$, and $m_{2}^{\rm ss}\simeq 0.05\,{\rm eV}$ to maximize
production of the lepton asymmetry, strong washout via $\nu_{R_{2}}$
production can be avoided by reducing $m_{1}^{\rm ss}$ well below
$10^{-3}\,{\rm eV}$ so that $\nu_{R_{1}}$ decays at a temperature well below
$M_{1}\sim M_{2}$ (but before the universe becomes matter dominated by
$\nu_{R_{1}}$). In this case, $T_{RH}$ should satisfy (25), as the Yukawa
coupling of $\nu_{R_{1}}$ is too small to put it in thermal equilibrium.
$\nu_{R_{1}}$ can also decay via $W_{R}$ exchange, but we find that this decay
mode is negligible and does not reduce the efficiency of leptogenesis.

We stress that this bound results from thermal leptogenesis without degeneracy
among the $\nu_{R_{i}}$; cases with a non-thermal initial abundance or with
degeneracy are discussed in sections 6.1 and 6.2. This naturalness bound
applies even if there are additional contributions to the light neutrino
masses coming directly from dimension-5 operators. This necessarily occurs in
the parity symmetric theories, but is also expected in (SM + $N$) which, after
all, is just an effective field theory. Finally, we note that in (SM + $N$)
this bound is not the rigorous Davidson-Ibarra bound Davidson:2002qv , derived
for $M_{2,3}\gg M_{1}$; rather, for moderate $M_{2,3}$ it can be violated if
there are unnatural cancellations in the light neutrino mass matrix
Hambye:2003rt .

A value of $M_{1}$ of order $10^{9}$ GeV only results for leptogenesis with
weak washout and if the relevant angles in $A_{2}$ or $A_{3}$ are order unity,
otherwise the naturalness bound on $M_{1}$ can be orders of magnitude more
severe.

As an example of the tuning required to avoid (34), consider the case with
only seesaw neutrino masses, which is possible in both (SM + $N$) and theories
with parity restoration with sufficiently high $v_{R}$. Leptogenesis with weak
washout, $m_{1}^{\rm ss}\mathop{}_{\textstyle\sim}^{\textstyle<}10^{-3}\,{\rm
eV}$, implies that only the seesaw masses from $\nu_{R_{2,3}}$ are relevant
for current neutrino mass observations. One neutrino can be taken as
essentially massless and the mass matrix for the two heavy states can be put
in the form

$\displaystyle{\bf m_{\nu}}=\begin{pmatrix}\sin^{2}\theta\,m_{2}^{\rm
ss}&\cos\theta\sin\theta\,m_{2}^{\rm ss}\\\ \cos\theta\sin\theta\,m_{2}^{\rm
ss}\;\;\;\;&e^{2i\phi}\,m_{3}^{\rm ss}+\cos^{2}\theta\,m_{2}^{\rm
ss}\end{pmatrix}\,$ (35)

where the angles $\theta$ and $\phi$ specify the orientation between the two
vectors for $\tilde{\ell}_{2}$ and $\tilde{\ell}_{3}$ in flavor space and are
not determined by leptogenesis. If $\theta\neq 0$ the mass eigenvalues are not
$m_{2,3}^{\rm ss}$ because $\nu_{R_{2,3}}$ couple to different combinations of
light neutrinos. If $\phi=0$, $m_{2,3}^{\rm ss}\gg 0.05$ eV is excluded: a
cancellation between them is not possible as they are both positive. On the
other hand for a range of $\phi$ a cancellation is possible. For example,
$m_{2,3}^{\rm ss}\gg 0.05$ eV is possible if $\phi=\pi/2$, giving $m_{\rm
atm}\simeq m_{3}^{\rm ss}-m_{2}^{\rm ss}$ and
$m_{\odot}\simeq\sin^{2}\theta\;(m_{2,3}^{\rm ss})^{2}/m_{\rm atm}$. Here
$m_{\rm atm}\simeq 0.05$ eV and $m_{\odot}\simeq 0.01$ eV describe the mass
splittings for atmospheric and solar oscillations. In this paper we explore
the consequences of avoiding such tunings.

###  5.2 Bound on $v_{R}$

In parity symmetric models the right-handed neutrinos are massless until
$SU(2)_{R}$ is broken, implying that
$v_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}M_{+}$, the mass of the heaviest
right-handed neutrino, which could be $M_{2}$ or $M_{3}$. Hence, thermal
leptogenesis, via (34), already implies that
$v_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}10^{9}$ GeV. However, there is a
much stronger bound. Parity invariance of the dimension-5 operators implies a
direct contribution to the light neutrino masses, $m_{i}^{\rm dir}$, which is
$v_{L}^{2}/v_{R}^{2}$ times the masses of $\nu_{R_{i}}$, as shown in Eq. (24).
In the absence of fine-tuning in the light neutrino mass matrix, $m_{+}^{\rm
dir}\mathop{}_{\textstyle\sim}^{\textstyle<}m_{+}^{{\rm dir}*}$ so that

$\displaystyle
v_{R}\;\;\mathop{}_{\textstyle\sim}^{\textstyle>}\;\;10^{12}\,{\rm
GeV}\left(\frac{M_{+}}{10^{9}\,{\rm
GeV}}\right)^{\frac{1}{2}}\left(\frac{0.05\,{\rm eV}}{m_{+}^{{\rm
dir}*}}\right)^{\frac{1}{2}}.$ (36)

Hence, the thermal leptogenesis bound of (33) leads to a numerical bound on
$v_{R}$ of

$\displaystyle
v_{R}\;\;\mathop{}_{\textstyle\sim}^{\textstyle>}\;\;10^{12}\,{\rm
GeV}\left(\frac{M_{+}}{M_{1}}\right)^{\frac{1}{2}}\left(\frac{1}{(A_{2}+A_{3})\eta}\right)^{\frac{1}{2}}\left(\frac{0.05\,{\rm
eV}}{m_{+}^{{\rm dir}*}}\right)^{\frac{1}{2}}\left(\frac{0.05\,{\rm
eV}}{m_{2,3}^{\rm ss\,*}}\right)^{\frac{1}{2}}\,.$ (37)

This bound is applicable only when there is no high degeneracy between $M_{1}$
and $M_{2}$ or $M_{3}$. This case is studied in Sec. 6.2 and is found to
weaken the bound by up to three orders of magnitude.

The lowest values for $v_{R}$ are obtained in weak washout, with $\eta\simeq
1$. Washout is negligible, for example, when $m_{1}^{\rm ss}\sim 10^{-3}$ eV,
and $M_{2,3}/M_{1}\gtrsim 4$ so that $\nu_{R_{2,3}}$ production is Boltzmann
suppressed at the time of $\nu_{R_{1}}$ decay, making $\nu_{R_{2,3}}$ back-
reactions small. The lowest value of $v_{R}$ occurs when: $M_{2,3}$ is
comparable to $M_{1}$; $m_{2,3}^{\rm ss}\simeq 0.05\,{\rm eV}$ to maximize
production of the lepton asymmetry; and strong washout via $\nu_{R_{2,3}}$
production is avoided by reducing $m_{1}^{\rm ss}$ well below $10^{-3}\,{\rm
eV}$, so that $\nu_{R_{1}}$ decays at a temperature well below $M_{1}\sim
M_{2}\sim M_{3}$. If washout is important, the bound on $v_{R}$ becomes
stronger than (37).

Remarkably, from Fig. 2 we see that the central value of the prediction for
$v_{R}$ from the Higgs Parity mechanism is of order $10^{12}$ GeV, although
the 3$\sigma$ range spans several orders of magnitude. For lower values of the
top quark mass the above bound on $v_{R}$ is easily satisfied. However, larger
values are inconsistent with this bound, and would provide a strong motivation
to modify the minimal leptogenesis scheme of the previous sub-section. We
stress that the bound on $v_{R}$ is strengthened as the right-handed neutrinos
become more hierarchical in mass; if future measurements of $m_{t}$ are high,
consistency will limit this hierarchy. Thermal leptogenesis with small angles
in $A_{2}$ or $A_{3}$, strengthens the bound on $v_{R}$, but more mildly than
for the bound on $M_{1}$. The bounds of (33) and (37) are both weakened
linearly if the relevant $m^{*}$ are taken larger than 0.05 eV. For example,
if cancellations of a factor of three occur between differing contributions to
a light neutrino mass eigenvalue, then $m^{*}=0.15$ eV and the lower limits on
$M_{1}$ and $v_{R}$ are both lowered by a factor of 3.

Figure 4:  Lower bounds on $M_{1}$ and $v_{R}$ in minimal theories where
parity solves the strong CP problem and $\nu_{R_{1}}$ decay gives successful
leptogenesis. Neutrino masses arise from the dimension-5 operators of (13)
without fine-tuning. The horizontal bound is from requiring the observed
baryon asymmetry to arise from thermal leptogenesis without degeneracy of
$\nu_{R}$. The sloped bound arises from limits on direct contributions to
light neutrino masses, for three illustrative values of $M_{+}/M_{1}$. In the
shaded blue and orange regions, fine-tuning between contributions to the
neutrino masses increases with distance from the natural unshaded region. At
values of $v_{R}$ larger than $10^{14}$ GeV the strong CP problem re-emerges
from (8), as illustrated by the gray shading and the top axis. The region
above the dashed line for $M_{1}=v_{R}$ is unphysical in the EFT of (13).

The bounds of (33) on $M_{1}$ and (36) on $v_{R}$ are shown in Fig. 4 in the
$(v_{R},M_{1})$ plane. As $v_{R}$ is increased above about $10^{14}$ GeV, the
strong CP problem re-emerges. Hence, in the unshaded region, which
successfully accounts for both neutrino masses and the cosmological baryon
asymmetry, there is a good possibility that the operator of (8) will give a
positive signal for the neutron electron dipole moment in current experiments,
providing the coefficient $C$ is not small, $C>0.01$. The unshaded region
overlaps the region of $v_{R}$ predicted by Higgs Parity, illustrated by the
vertical dashed lines showing the range of $v_{R}$ allowed by the current
$1\sigma$ uncertainty in $m_{t}$. As uncertainties on $m_{t}$ and $\alpha_{s}$
are reduced, it will be interesting to discover whether consistency between
Higgs Parity and thermal leptogenesis is maintained.

In the ‘strong wash-out’ regime, the bound (37) on $v_{R}$ strengthens, as it
is proportional to $1/\sqrt{\eta}\simeq(m_{1}^{\rm ss}/0.5\times
10^{-3}\mbox{eV})^{0.58}$ Giudice:2003jh , enlarging the blue region of Fig.
4. For example, for $m_{1}^{\rm ss}=0.003\,\mbox{eV}$ this is about a factor
3. Entering the strong washout regime removes the bound (25) on $T_{R}$ for
thermal leptogenesis – it is only necessary that
$T_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}M_{1}$. This opens up more
parameter space for the theory of reheating after inflation, which should now
satisfy $v_{R}>T_{\rm{max}}>T_{R}>M_{1}$.

The lower bound (37) on $v_{R}$ follows directly from the equal magnitudes of
the $\ell_{i}\ell_{j}H_{L}^{2}$ and $\bar{\ell}_{i}\bar{\ell}_{j}H_{R}^{2}$
couplings of (13). In conventional LR models, electroweak symmetry is broken
by bi-doublets, $\Phi$, and triplets, $\Delta_{L,R}$. The $\bar{\nu}\bar{\nu}$
mass term arises from the vev of $\Delta_{R}$, and there is a type-II seesaw
$\nu\nu$ contribution from $\Delta_{L}$ exchange. If the mass of $\Delta_{L}$,
$M_{\Delta_{L}}$, is of order $v_{R}$, the ratio of these two mass terms is
proportional to $v_{L}^{2}/v_{R}^{2}$. However, it is also proportional to
$\lambda_{LR}$, the quartic coupling for the
$\Phi^{\dagger}\Phi\Delta_{L}\Delta_{R}$ operator Dror:2020jzy

$\displaystyle
r\;=\;\left(\frac{\rm{type}\;II\;\rm{seesaw}\;\;\nu\nu\;\;\rm{mass}}{\bar{\nu}\bar{\nu}\;\;\rm{mass}}\right)\;=\;\lambda_{LR}\;\frac{v_{L}^{2}}{M_{\Delta_{L}}^{2}}.$
(38)

If $\lambda_{LR}\sim 1$, the order of magnitude of the bound survives; but if
$\lambda_{LR}\ll 1$, the bound is considerably weakened by a factor of
$\sqrt{r}$. The bound would be strengthened for $m\ll v_{R}$, but this
requires fine-tuning.

###  5.3 The $\nu_{R}$ Spectrum

In this paper we define $\nu_{R_{1}}$ to be the right-handed neutrino whose
decays yield leptogenesis. In the previous sections, for convenience, we also
took $\nu_{R_{1}}$ to be the lightest right-handed neutrino, but this is
unnecessary. For example, we can take leptogenesis to arise from virtual
$\nu_{R_{2}}$ exchange in $\nu_{R_{1}}$ decays, with $M_{2}\geq M_{1}$, and
take $M_{3}$ much larger or much smaller than $M_{1}$ as long as it does not
upset the leptogenesis mechanism. This requires that $\nu_{R_{3}}$ is much
lighter and decays while relativistic, or decays soon after it becomes non-
relativistic; in either scenario it will not washout the lepton asymmetry
previously created by $\nu_{R_{1}}$ decay. However, if $\nu_{R_{3}}$ comes to
dominate the energy density of the universe, then its decays will dilute the
lepton asymmetry, and we do not allow this case. It is even possible that
$\nu_{R_{3}}$ is in the ($\sim$2-100) keV mass range and is sufficiently
stable to be dark matter Dunsky:2020dhn . Finally, even if $W_{R}$ exchange
led to an initial thermal abundance of right-handed neutrinos, it it still
possible that $\nu_{R_{1}}$ is sufficiently heavy and long-lived to come to
dominate the energy density of the universe before decaying, thereby diluting
$\nu_{R_{3}}$ to the observed dark matter abundance.

##  6 Relaxing the Leptogenesis Bound on $v_{R}$

The lower bound on the scale of parity violation derived in the last section
from thermal leptogenesis, $v_{R}>10^{12}$ GeV, can be weakened by a factor
$1/\sqrt{E}$, when the baryon asymmetry from leptogenesis is enhanced by a
factor $E$. There are two well-known ways of enhancing leptogenesis. In the
first a cosmological mechanism is introduced to increase the yield of
$\nu_{R_{1}}$ above thermal. In the second, the lepton asymmetry per
$\nu_{R_{1}}$ decay is increased by having degeneracy among the $\nu_{R_{i}}$,
which can result from an approximate symmetry. In the first two subsections we
study how each of these affects the lower bound on $v_{R}$ in theories with
parity restoration. In the final subsection, we investigate how the bounds on
$v_{R}$ are modified when the dimension-five operators for neutrino masses
arise from a very simple theory, the Radiative Singlet Model, where seesaw and
direct contributions to light neutrino masses cancel at tree-level.

###  6.1 Non-Thermal Leptogenesis

In this subsection, we study the possibility that before decaying, the
$\nu_{R_{1}}$ are not in thermal equilibrium and have a number density greater
than the thermal value. We show that this case can relax the bound on $v_{R}$
only by about an order of magnitude.

Consider a cosmological era with the energy density of the universe dominated
by $\nu_{R_{1}}$. At the end of this era, when $\nu_{R_{1}}$ decay, we take
the $\nu_{R_{1}}$ number density to be $n_{D}$ and their momentum distribution
to be peaked around $p_{D}$, corresponding to a typical energy
$E_{D}=\sqrt{M_{1}^{2}+p_{D}^{2}}$. If the decay rate of $\nu_{R_{1}}$ at rest
is $\Gamma_{1}$, the Hubble parameter at decay is

$\displaystyle H_{D}\simeq\Gamma_{1}\frac{M_{1}}{E_{D}}.$ (39)

The decay products, $\ell,H_{L}$ and their anti-particles, reheat the universe
via gauge interactions to a temperature $T_{RH}$, given by

$\displaystyle\frac{\pi^{2}}{30}g_{*}T_{RH}^{4}\simeq\rho_{D}\simeq
n_{D}E_{D}.$ (40)

At the same time, the $\nu_{R_{1}}$ decays produce a lepton number density
$\epsilon n_{D}$ which, in the resulting thermal bath, becomes a baryon yield

$\displaystyle Y_{B,\rm non-
therm}=\frac{28}{79}\,\epsilon\eta\left(\frac{3}{4}\frac{T_{\rm
RH}}{E_{D}}\right)\,.\quad\text{(Non-Thermal Leptogenesis)}$ (41)

Comparing with the result (26) for thermal leptogenesis, there is an
enhancement factor

$\displaystyle F\equiv\frac{Y_{B,\rm non-therm}(\eta)}{Y_{B,\rm
therm}(\eta=1)}\simeq 200\frac{T_{RH}}{E_{D}}\eta,$ (42)

for equal values of $\epsilon$. This ratio can exceed unity, and hence non-
thermal leptogenesis can produce the observed baryon asymmetry with a smaller
$\epsilon$ and therefore a smaller $M_{1}$ than in the thermal case. However,
the size of this ratio and its dependence on $E_{D}$ is limited by the Pauli
exclusion principle and by washout.

The Pauli exclusion principle limits the number density of $\nu_{R_{1}}$ at
decay: $n_{D}\lesssim p_{D}^{3}/\pi^{2}$. Using this in (40) gives

$\displaystyle\frac{T_{RH}}{E_{D}}\;\lesssim\;\frac{1}{\pi}\left(\frac{p_{D}}{E_{D}}\right)^{3/4}.$
(43)

Thus the enhancement factor, $F$, is below unity in the non-relativistic
region when $p_{D}<M_{1}/250$, rises as $p_{D}$ is increased, but is no more
than $\sim 60$ in the relativistic regime.

After reheat there is the possibility that scattering of the leptons $\ell$
via the Yukawa coupling to $\nu_{R_{1}}$ will washout the lepton asymmetry.
When $p_{D}\lesssim M_{1}$, (43) gives $T_{RH}<M_{1}$, giving the weak washout
condition

$\displaystyle\Gamma_{\rm
washout}\;\simeq\;\Gamma_{1}\frac{T_{RH}}{M_{1}}\;e^{-(M_{1}/T_{RH})}\;\lesssim\;H_{D}\;\simeq\;\Gamma_{1}\frac{M_{1}}{E_{D}}.$
(44)

which is satisfied for all $p_{D}\lesssim M_{1}$. As $p_{D}$ approaches the
relativistic region, $F$ rises to near 60. In the relativistic region, with
$T_{RH}>M_{1}$, the weak washout condition is the same as for thermal
leptogenesis, $m_{1}^{\rm ss}\mathop{}_{\textstyle\sim}^{\textstyle<}10^{-3}$
eV. However, such high values for $T_{RH}$ require a rapid decay rate for
$\nu_{R_{1}}$, and are inconsistent with this weak washout condition.
Therefore, for relativistic $\nu_{R_{1}}$ decays the enhancement factor is
$F\simeq 60\eta$ with $\eta\simeq{\cal O}(10)(M_{1}/E_{D})^{3.5}$. Hence, by
the same logic as Sec. 5.2, in non-thermal leptogenesis with $p_{D}\sim
M_{1}$, the limit on $v_{R}$ can be weakened by $\sqrt{F}$ compared to thermal
leptogenesis to

$\displaystyle
v_{R}\;\;\mathop{}_{\textstyle\sim}^{\textstyle>}\;\;10^{11}\,{\rm GeV}.$ (45)

When the bound is saturated, the reheat temperature is around $10^{6}$ GeV.

Leptogenesis with nearly degenerate right-handed neutrinos can indeed be
achieved when the inflaton, $\phi$, decays dominantly by
$\phi\rightarrow\nu_{R_{1}}\nu_{R_{1}}$. If the inflaton number density is
sufficiently high when Hubble is of order the inflaton decay rate,
$\Gamma_{\phi}$, the decays are Pauli-blocked. This can be seen by noting the
inflaton energy density $\rho_{\phi}=m_{\phi}n_{\phi}=3H^{2}/8\pi G$, so that
the the faster the $\phi$ decay rate is (occurring at $\Gamma_{\phi}\simeq
H$), the greater the inflaton number density, $n_{\phi}$, is. If
$\Gamma_{\phi}\simeq H\gg m_{\phi}^{2}/4\pi M_{Pl}$, the inflaton number
density at decay is $n_{\phi}\gg m_{\phi}^{3}$, which would generate a number
density of $\nu_{R_{1}}$ much greater than the Pauli degenerate limit of
$n_{\rm PD}\simeq(m_{\phi}/2)^{3}/\pi^{2}$. This is forbidden and so $\phi$
decays must occur over a long period until $n_{\phi}$ drops below $n_{\rm
PD}$. During this period, $\phi$ produces a nearly degenerate Fermi gas of
$\nu_{R_{1}}$ which are continually red-shifted by the expansion of the
universe. The phase space of the $\nu_{R_{1}}$ is filled until the last
inflatons to decay populate states near the Fermi surface, which dominate the
energy density of the distribution. The non-thermal but relativistic
$\nu_{R_{1}}$ then redshift with the expansion until decaying. Achieving the
largest possible enhancement factor over thermal leptogenesis, $F\simeq 60$,
requires that $\nu_{R_{1}}$ be at least semi-relativistic and in the weak
washout regime. With an initial momentum of $m_{\phi}/2$ there can therefore
be no more than a factor of $(m_{\phi}/2)/(M_{1})$ in the redshift between
$\nu_{R_{1}}$ production from the inflaton and $\nu_{R_{1}}$ decay. This
maximum possible redshift implies the minimum energy density of $\nu_{R_{1}}$
at decay is $\rho_{D}\gtrsim m_{\phi}n_{\rm PD}(2M_{1}/m_{\phi})^{4}\approx
M_{1}^{4}/\pi^{2}$. Equivalently, the $\nu_{R_{1}}$ decay rate must be

$\displaystyle\Gamma_{1}\approx\frac{M_{1}^{2}}{M_{Pl}}\,.$ (46)

for the lowest limit on $v_{R}$ to be achieved, Eq. (45). See
Lazarides:1990huy ; Asaka:1999yd  for leptogenesis from the inflaton decay
with a more generic parameter space.

###  6.2 Degenerate Right-Handed Neutrinos

In this subsection, we show how increasing the lepton asymmetry per decay,
$\epsilon$, via degenerate right-handed neutrinos can relax the natural lower
bound on $v_{R}$ by three orders of magnitude to $10^{9}$ GeV.

Consider the case where at least two right-handed neutrinos are so degenerate
that their mass difference $\Delta M$, is much smaller than their mass,
$M_{1}\simeq M_{2}\simeq M$. The lepton asymmetry generated by the decay of
$\nu_{R_{1}}$ is then enhanced by the large $g(x)$ factor in Eq. (29), which
takes the form of

$\displaystyle\left|g\left(\frac{M_{2}^{2}}{M_{1}^{2}}\right)\right|\simeq\frac{1}{2}\frac{M}{|\Delta
M|}\gg 1$ (47)

for $M_{1}\simeq M_{2}\simeq M$. 333Eq. (47) breaks down when $\nu_{R_{1}}$
and $\nu_{R_{2}}$ are so degenerate that $|\Delta M|\lesssim\Gamma_{2}$, where
$\Gamma_{2}$ is the decay rate of $\nu_{R_{2}}$. If this occurs, $g(x)$ is not
as enhanced as (47) and thus the lower limit on $v_{R}$ is greater than
$10^{9}$ GeV.

Such a degeneracy between $M_{1}$ and $M_{2}$ can arise, for example, if there
exists a discrete symmetry in the lepton number violating operator of (13)

$\displaystyle\ell_{1}\leftrightarrow\ell_{2}\qquad\ell_{1}\leftrightarrow-\ell_{1}\,,$
(48)

and similarly on the $\bar{l}$ fields to maintain parity, so that $c_{ij}$ is
diagonal with $c_{11}=c_{22}$ giving $M_{1}=M_{2}=|c_{11}|^{2}v_{R}^{2}/2M$
Dunsky:2020dhn . This symmetry cannot be imposed on the lepton number
conserving operator of (13) since if $b_{ij}$ is also diagonal there is no
flavor violation for leptogenesis. Similarly, it cannot be imposed on the
operators generating charged lepton masses, as there is no degeneracy in the
charged lepton spectrum. In the underlying theory of lepton flavor, there must
be different patterns of symmetry breaking in the lepton number preserving and
lepton number violating sectors.

How small can $\Delta M/M$ be? In Appendix A, we demonstrate that the charged
fermion Yukawa operators necessarily generate a wavefunction renormalization
for $\nu_{R_{1,2}}$ and hence a mass splitting $\Delta M$ that is at least of
order $y_{\tau}^{2}/8\pi\sim 10^{-6}$. Eq. (47) implies then that $g(x)_{\rm
max}\simeq 5\times 10^{5}$ is the largest natural value in any LR theory. The
naturalness bound of (33) is greatly weakened and becomes

$\displaystyle M_{1}\;\mathop{}_{\textstyle\sim}^{\textstyle>}\;\frac{2\times
10^{3}\;{\rm GeV}}{A_{2}}\left(\frac{5\times 10^{5}}{g(x)_{\rm
max}}\right)\left(\frac{0.05\,{\rm eV}}{m_{2}^{\rm ss\,*}}\right)\hskip
18.06749pt\,$ (49)

for weak washout.444This result follows from the lepton asymmetry produced by
$\nu_{R_{1}}$ decay; a comparable lepton asymmetry from $\nu_{R_{2}}$ decay
could weaken the bound by up to a factor of 2. The horizontal orange lines in
Fig. 5 show the limit on $M_{1}$ for several values of the degeneracy $\Delta
M/M$. Note that, since $\nu_{R_{1}}$ and $\nu_{R_{2}}$ are degenerate, for
$T_{RH}\sim M_{1}$ a value of $m_{1,2}^{\rm ss}$ as large as 0.05 eV will lead
to strong washout, strengthening the bound on $M_{1}$ by a factor of 50.
However, strong washout can be avoided by taking $T_{RH}$ somewhat less than
$M_{1}$ so that washout through $\nu_{R_{2}}$ production is exponentially
suppressed.

Figure 5: Lower bounds on $M_{1}$ and $v_{R}$ in theories of parity
restoration, with minimal scalars and no fine-tuning in neutrino masses. The
horizontal bound is from requiring the observed baryon asymmetry to arise from
thermal leptogenesis; it is shown for several values of the degree of
degeneracy $\Delta M/M$ between the decaying and virtual $\nu_{R}$. The case
without degeneracy, $\Delta M/M\approx 1$, gives the bound of Fig. 4. As
$\Delta M/M$ is decreased to its smallest natural value of $10^{-6}$, the
bound on $M_{1}$ decreases linearly. As indicated by the gray shaded region,
the strong CP problem re-emerges at values of $v_{R}$ larger than $10^{14}$
GeV, as discussed around 8.

The blue curve of Fig 5 shows the lower bound on $v_{R}$ from the direct
contribution to the light neutrino mass as a function of $M_{1}$; it is (36)
with $M_{i}$ replaced by $M_{1}$. The reduction in the lower bound on $M_{1}$
from leptogenesis by a factor of order $10^{6}$, decreases the naturalness
bound of (37) on $v_{R}$ by $10^{3}$ to

$\displaystyle
v_{R}\;\;\mathop{}_{\textstyle\sim}^{\textstyle>}\;\;\frac{10^{9}\,{\rm
GeV}}{(A_{2})^{1/2}}\left(\frac{5\times 10^{5}}{g(x)_{\rm
max}}\;\;\frac{M_{+}}{M_{1}}\;\;\frac{0.05\,{\rm eV}}{m_{+}^{{\rm
dir}*}}\right)^{\frac{1}{2}}\left(\frac{0.05\,{\rm eV}}{m_{2}^{\rm
ss}}\right)^{\frac{1}{2}}$ (50)

for weak washout. Taking $M_{+}=M_{3}\sim M_{1}$, this bound corresponds to
the intersection of the blue line with the orange lines in Fig 5.

It is interesting to note that for such highly degenerate $\nu_{R_{1,2}}$ the
direct contributions to the light neutrino masses become equal, $m_{1,2}^{\rm
dir}=(v_{L}^{2}/v_{R}^{2})M_{1,2}$. 555When $\nu_{R_{2}}$ and $\nu_{R_{3}}$
are nearly degenerate, it is possible to weaken the bound on $M_{1}$ by a
factor of 2 and the bound on $v_{R}$ by a factor of $\sqrt{2}$ if the
following two conditions are met: 1) $y_{1}^{2}\simeq y_{2}^{2}$, 2) Their CP
phases are opposite, $\phi_{1}=-\phi_{2}$, so as to cancel the negative sign
picked up by $g(M_{2}^{2}/M_{1}^{2})\simeq-g(M_{1}^{2}/M_{2}^{2})$.

In the Mirror theory, a large degeneracy allows for the possibility that the
reheating temperature required for leptogenesis can be far below the mass of
the lightest mirror quark

$\displaystyle T_{\rm RH}\simeq M_{1}\ll m_{u^{\prime}}\simeq 10^{-5}v_{R}.$
(51)

In this case, there may be negligible production of $u^{\prime}$ after
inflation, allowing $u^{\prime}$ to be stable and still below terrestrial
bounds on fractionally charged particles Dunsky:2018mqs . This allows an
additional $U(1)$ in the theory so that the Mirror theory can have electroweak
gauge group $SU(2)_{L}\times U(1)_{Y}\times SU(2)_{R}\times
U(1)^{\prime}_{Y^{\prime}}$ as in Dunsky:2019api ; Bonnefoy:2023afx .

###  6.3 Radiative Singlet Model

In our parity symmetric theory, neutrino masses are described by the
dimension-5 operators of (13), in both LR and Mirror versions. The physics of
neutrino masses and leptogenesis depends on two independent flavor matrices,
$b$ and $c$. Significant cancellations between different contributions to the
light neutrino mass matrix can only arise from fine-tuning the parameters of
these matrices, leading to the naturalness bounds discussed so far in this
paper. However, cancellations between seesaw and direct contributions to
neutrino masses occur naturally in one of the simplest UV completions of this
EFT Hall:2019qwx , which we call the Radiative Singlet Model. In this theory
the $b$ and $c$ matrices are correlated, resulting in all three active
neutrino masses vanishing at tree-level. In this section, we study whether
such cancellations lead to a relaxation of the naturalness bound on $v_{R}$,
finding that we can increase the the asymmetry per decay parameter $\epsilon$,
but only if leptogenesis occurs in the strong washout regime. The reduction in
efficiency from strong washout roughly cancels the enhancement in the
asymmetry per decay so that $v_{R}$ is again difficult to drop below $\sim
10^{12}$ GeV.

Consider a theory for neutrino masses with three gauge-singlet Weyl fermions
$S_{i}$, that are parity even, $S_{i}\leftrightarrow S_{i}^{\dagger}$, coupled
to leptons via the interactions

$\displaystyle{\cal
L}(S_{i})=S_{i}\,(x^{*}_{ij}\,\ell_{j}H_{L}+x_{ij}\,\bar{\ell}_{j}H_{R})+\frac{1}{2}M_{S_{i}}\,S_{i}S_{i}+{\rm
h.c.}$ (52)

with $M_{S_{i}}$ real. We take

$\displaystyle M_{S_{i}}\gg|x_{ij}|\,v_{R},\hskip 36.135pt\mbox{all }i,j$ (53)

so that integrating out $S_{i}$ leads to the dimension-5 operators of (13).

To see that the active neutrinos are massless at tree level, introduce a
hatted basis so that the Yukawa interactions of $S_{i}$ can be written as

$\displaystyle{\cal
L}_{x_{i}}=\hat{x}_{i}\;S_{i}\,(\hat{\ell}_{i}\,H_{L}+\hat{\bar{\ell}}_{i}\,H_{R})+{\rm
h.c.},\qquad\hat{\ell_{i}}=x_{ij}\,\ell_{j}/\hat{x}_{i},\qquad\hat{x}_{i}^{2}\equiv\sum_{j}|x_{ij}|^{2}.$
(54)

Note that the $\hat{\ell}_{i}$ are not orthogonal. The EFT below $M_{S_{i}}$
is

$\displaystyle{\cal
L}_{EFT}(S_{i})\;=\;\frac{1}{2}\,\sum_{i}\frac{x_{i}^{2}}{M_{S_{i}}}\;(\hat{\ell}_{i}\,H_{L}+\hat{\bar{\ell}}_{i}\,H_{R})^{2}+{\rm
h.c.}.$ (55)

The light neutrinos are massless at tree level because each $S_{i}$ couples to
only one combination of right-handed neutrinos and neutrinos,
$v_{R}\,\hat{\nu}_{Ri}+v\,\hat{\nu}_{i}$, leaving the three orthogonal
combinations massless. For example, in a 1-generation version of the theory
the right-handed neutrino mass is $M=x^{2}v_{R}^{2}/M_{S}$ and the neutrino
Yukawa coupling is $y=x^{2}v_{R}/M_{S}$, giving the correlation $y=M/v_{R}$.
The direct and seesaw neutrino masses, defined in 23, are equal

$\displaystyle m_{\nu}^{\rm dir}=m_{\nu}^{\rm
ss}=\frac{y^{2}v^{2}}{M}=\frac{x^{2}v^{2}}{M_{S}}.$ (56)

so that the light active neutrino is massless at tree-level.

To study leptogenesis in the 3-generation theory, we abandon the hatted basis
in favor of a mass basis for the right-handed neutrinos. Since the $S_{i}$ are
integrated out, we find it convenient to first go to a non-canonical basis by
rescaling $S_{i}\rightarrow S_{i}\sqrt{M_{S}/M_{S_{i}}}$, where $M_{S}$ is any
convenient mass scale, so that the $S$ mass matrix is proportional to the unit
matrix. In this basis the dimension-5 operators of the EFT are

$\displaystyle{\cal L}_{\nu}$
$\displaystyle=-\frac{1}{2M_{S}}\left(\ell_{i}\,(x^{T}x)_{ij}^{*}\,\ell_{j}\;H_{L}H_{L}+\bar{\ell}_{i}\,(x^{T}x)_{ij}\,\bar{\ell}_{j}\;H_{R}H_{R}\right)+\frac{1}{M_{S}}\,\ell_{i}\,(x^{\dagger}x)_{ij}\,\bar{\ell}_{j}\;H_{L}H_{R}+{\rm
h.c.}.$ (57)

Comparing with (13), the previously independent coupling matrices $b$ and $c$
are now correlated, with $b=x^{\dagger}x$, $c=x^{T}x$ and $M=M_{S}$. Thus the
right-handed neutrino mass matrix and the Yukawa coupling matrix are also
correlated

$\displaystyle M=x^{T}x\,\frac{v_{R}^{2}}{M_{S}},\hskip
36.135pty=x^{\dagger}x\,\frac{v_{R}}{M_{S}}.$ (58)

Neutrino masses and leptogenesis thus depend on a single flavor matrix $x$. It
is a general complex matrix that can be made diagonal by a bi-unitary
transformation $x=Ux_{D}V$ where $x_{D}$ is diagonal with entries $x_{1}\leq
x_{2}\leq x_{3}$. The unitary matrix $V$ can be eliminated by a transformation
on the lepton doublets. Moreover, the unitary matrix $U$ contains three
rotation angles and three phases, since three other phases can also be removed
by transforming lepton doublet fields. A convenient form for the resulting $x$
matrix is

$\displaystyle x=R\,e^{iA}\,x_{D},\qquad A_{ij}=\epsilon_{ijk}\beta_{k}\,,$
(59)

where the matrix $A$ is real and anti-symmetric and $R$ is a real rotation
matrix, $R^{T}=R^{-1}$. This parameterization is a mass basis for $\bar{\nu}$,
with

$\displaystyle M_{ij}=\delta_{ij}\,M_{i},\hskip
21.68121ptM_{i}=x_{i}^{2}\,\frac{v_{R}^{2}}{M_{S}};\hskip
36.135pty_{ij}=x_{i}(e^{2iA})_{ij}x_{j}\,\frac{v_{R}}{M_{S}}.$ (60)

Remarkably, the three rotation angles of the $R$ matrix do not appear in
either $M_{ij}$ or $y_{ij}$ and hence do not affect neutrino physics.

The direct and seesaw masses for the active neutrinos at tree-level are also
diagonal in this basis

$\displaystyle{m_{\nu}^{\rm
dir,ss}}_{ij}\;=\;\delta_{ij}\,\;x_{i}^{2}\,\frac{v^{2}}{M_{S}}.$ (61)

Since the active neutrino mass matrix is the difference between the direct and
seesaw masses, (21), the light active neutrinos are massless at tree-level.
However, this cancellation is upset by 1-loop electroweak radiative
corrections. For the case that $M_{S_{i}}\gg v_{R}\gg M_{i}$, the masses of
the light neutrinos at leading logarithm are HHS:2023

$\displaystyle
m_{\nu_{i}}=\delta_{i}\;x_{i}^{2}\,\frac{v^{2}}{M_{S}}=\delta_{i}\;M_{i}\,\frac{v^{2}}{v_{R}^{2}},\qquad\delta_{i}=\frac{3g_{2}^{2}}{16\pi^{2}}\left(2\ln\frac{M_{S_{i}}}{v_{R}}+\ln\frac{v_{R}}{M_{i}}\right)\approx
0.006\,\ln\frac{M_{S_{i}}^{2}}{v_{R}M_{i}},$ (62)

where we take $A=0$ for simplicity. Non-zero $A$ introduces $O(1)$
corrections. The two logs correspond to running in the EFTs above and below
$v_{R}$. Since the generation dependence of the loop factor $\delta_{i}$ is
only logarithmically dependent on generation, the ratio of active neutrino
masses is close to the ratio of right-handed neutrino masses,
$m_{\nu_{i}}/m_{\nu_{j}}\sim M_{i}/M_{j}$.

To avoid tuned cancellations between terms in the matrix product $x^{T}x$ for
the right-handed neutrino masses, the “phases”, $\beta_{i}$, of the
antisymmetric matrix $A$ of (59) should not be taken larger than unity. If
$\beta_{i}\ll 1$, the lepton asymmetry generated per $\nu_{R_{1}}$ decay,
$\epsilon$, occurs first at order $\beta_{1}\beta_{2}\beta_{3}$ and is
suppressed. Hence, we estimate naturalness bounds on $M_{1}$ and $v_{R}$ by
studying the case of $\beta_{i}={\cal O}(1)$, which gives

$\displaystyle y_{ij}\simeq
x_{i}x_{j}\,\frac{v_{R}}{M_{S}}\;e^{i\phi_{ij}},\qquad\phi_{ii}=0;\;\;\phi_{ij}={\cal
O}(1),\;i\neq j$ (63)

and a lepton asymmetry per $\nu_{R_{1}}$ decay of

$\displaystyle\epsilon\simeq\frac{1}{8\pi}x_{1}^{2}x_{3}^{2}\,\frac{v_{R}^{2}}{M_{S}^{2}}\;\simeq\frac{1}{8\pi}\frac{m_{\nu_{3}}M_{1}}{\delta_{3}v^{2}}\,,$
(64)

where $\nu_{3}$ is the heaviest of the three light neutrinos. The result for
$\epsilon$ is dominated by the virtual exchange of $\nu_{R_{2}}$ and should be
compared with (32): $\epsilon$ is enhanced in the singlet model by a factor
$1/\delta_{3}$. As neutrino masses occur at 1-loop level, the Yukawa couplings
required to generate the observed neutrino masses are larger than usual,
enhancing $\epsilon$. However, larger Yukawa couplings are a concern as they
enhance the washout rate of the lepton asymmetry after production, and we now
turn to this.

In theories where the dimension-5 operators generating the matrices $M$ and
$y$ are independent, it is always possible to avoid strong washout by imposing
a small contribution to the neutrino mass matrix from the seesaw exchange of
$\nu_{R_{1}}$: $m_{\nu_{1}}^{\rm
ss}\mathop{}_{\textstyle\sim}^{\textstyle<}10^{-3}$ eV. However, when $M$ and
$y$ matrices are correlated as in (58), and the $\beta$ parameters are of
order unity, the condition for weak washout becomes
$m_{\nu_{3}}/\delta_{3}\;\mathop{}_{\textstyle\sim}^{\textstyle<}\;10^{-3}\mbox{eV}$.
Since $m_{\nu_{3}}\simeq 0.05$ eV, leptogenesis is firmly in the strong
washout regime, with efficiency parameter (see (26))

$\displaystyle\eta\simeq 10^{-2}\delta_{3}^{1.16}.$ (65)

Using (64) and (65) in (26), leads to a lower bound on the mass of
$\nu_{R_{1}}$ that is insensitive to $\delta_{3}$

$\displaystyle
M_{1}\mathop{}_{\textstyle\sim}^{\textstyle>}\frac{10^{11}\;\mbox{GeV}}{\delta_{3}^{0.16}}\;.$
(66)

Using this result in (62), the bound on $v_{R}$ depends on the mass of the
lightest active neutrino

$\displaystyle v_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}1.7\times
10^{13}\;\mbox{GeV}\;\frac{\delta_{1}^{0.5}}{\delta_{3}^{0.08}}\left(\frac{0.01\,\mbox{eV}}{m_{\nu_{1}}}\right)^{0.5}.$
(67)

In (66) and (67) the equality sign holds if $\beta_{i}\simeq 1$. Therefore, in
the Radiative Singlet Model, the lower bound on $v_{R}$ is stronger than the
naturalness limit from thermal leptogenesis with generic dimension-5 operators
(36). Furthermore, $v_{R}$ becomes larger as $\nu_{1}$ is made lighter.666 In
the case that $\beta_{i}\simeq 1$, (66) and (67) with equality signs become
order of magnitude predictions for $M_{1}$ and $v_{R}$. Leptogenesis and the
three light neutrino mass eigenvalues can then be used to determine $v_{R}$
and $x_{i}$. For example, with $\delta_{i}=0.1$, neutrino masses of
$m_{\nu_{i}}\simeq(0.005,0.01,0.05)$ eV give $x_{i}\simeq 0.14,0.2,0.45$, and
$v_{R}\simeq 10^{13}$ GeV, where the arbitrary scale $M_{S}$ appearing in (57)
- (64) has been set equal to $v_{R}$.

Could degeneracy among the right-handed neutrinos play a role in increasing
$\epsilon$ and loosening the bound of (67) on $v_{R}$? Since $\delta_{i}$ is
only logarithmically dependent on generation, degeneracy among $\nu_{R}$
leads, via (62), to degeneracy among the active neutrinos. Since we require

$m_{\nu_{2}}^{2}-m_{\nu_{1}}^{2}=7\times 10^{-5}\,\text{eV}^{2},\quad
m_{\nu_{3}}^{2}-m_{\nu_{2}}^{2}=2.4\times 10^{-3}\,\text{eV}^{2},$ (68)

increasing the degeneracy of $\nu_{1}$ and $\nu_{2}$ requires increasing
$m_{\nu_{1,2}}$, which is limited by the cosmological constraint on the sum of
the neutrino masses,

$m_{\nu_{1}}+m_{\nu_{2}}+m_{\nu_{3}}\leq 0.12\,\text{eV}.$ (69)

This leads to a limit on the degeneracy and therefore a limit on the
degeneracy factor $g(x)$ relevant for leptogenesis in Eq. (29) of $g(x)\simeq
m_{\nu_{1}}/2(m_{\nu_{2}}-m_{\nu_{1}})\lesssim 13$. This relaxes the
constraint (66) on $M_{1}$ by a factor 13 and the constraint (67) on $v_{R}$
by a factor $\sqrt{13}$. Furthermore, there is no symmetry in the Radiative
Singlet Model that guarantees this degeneracy, so any weakening of (66) and
(67) is accidental. Thus the Radiative Singlet Model is unable to weaken the
bound on $v_{R}$ that results from the case of general dimension-5 neutrino
mass operators.

In the non-thermal leptogenesis scheme of Sec. 6.1, the reheat temperature
after decay of a degenerate gas of $\nu_{R_{1}}$ is below $M_{1}$ for
$p_{D}\leq M_{1}$, as shown in (43). This can lead to weak washout, even if
the typical $\nu_{R_{1}}$ momentum at decay, $p_{D}$, is close to $M_{1}$, so
that the enhancement factor of (42) approaches 60. In addition, $\epsilon$ of
(64) is enhanced by $1/\delta_{3}$, so that the lepton asymmetry is enhanced
by a factor as large as $60/\delta_{3}$ relative to thermal leptogenesis. For
$\delta_{3}\sim 0.1$, the bound of (67) on $v_{R}$ is then loosened to about
$4\times 10^{10}$ GeV. While this approaches the lowest bound on $v_{R}$ with
$\nu_{R}$ degeneracy, of (50), it requires a particular setup for reheating
after inflation, and a particular $\nu_{R_{1}}$ decay rate.

##  7 Domain Wall Leptogenesis

Domain walls offer another mechanism to produce right-handed neutrinos non-
thermally, which can potentially enhance $Y_{1}$ above the thermal value and
relax the naturalness bound on $v_{R}$. In this section we show that non-
thermal leptogenesis produced by the production of $\nu_{R_{1}}$ from the
decay of domain walls requires $v_{R}\gtrsim 2\times 10^{12}$ GeV and thus
cannot beat the naturalness bound of $v_{R}\gtrsim 10^{12}\,\rm GeV$ (36).

Domain walls are interesting to LR models since they are naturally inherent to
the theory; the breaking of the discrete symmetry associated with the LR
symmetry generates domain walls at the scale $v_{R}$ Kibble:1976sj ;
everett1982left ; Kibble:1982dd . Oftentimes, the default assumption is that
the reheat temperature of the universe after inflation is below the scale
$v_{R}$ so that the domain walls are effectively diluted away and can never
come to dominate the universe - a generic problem of domain walls known as the
‘domain wall problem’. Nevertheless, this assumption need not be true and it
is possible that the reheat temperature after inflation was above the scale
$v_{R}$ so that domain walls formed in our early universe after inflation.
Compatibility with our present-day universe then requires that the walls decay
before dominating the universe777Domain walls experience repulsive
gravitational accelerations of order $G\sigma$ Ipser:1983db ; Sikivie:2006ni .
This repulsive pressure, $\sim G\sigma^{2}$ always dominates over any vacuum
pressure difference on the walls if the vacuum pressure is weak enough to
allow the walls to dominate the energy density of the universe in the first
place., which can result from a small parity breaking term in the theory or,
depending on the GUT completion, from becoming attached to cosmic strings,
causing the wall-bounded string system to decay via gravitational waves
everett1982left ; Martin:1996ea ; Dunsky:2021tih .

We now proceed with a few simple arguments to show that leptogenesis via non-
thermal $\nu_{R}$ production from domain wall annihilation does not allow
values of $v_{R}$ below $10^{12}$ GeV, the lower bound from thermal
leptogenesis.

Without loss of generality, let $\nu_{R_{1}}$ be the right-handed neutrino
whose decays are responsible for leptogenesis. The non-thermal yield of
$\nu_{R_{1}}$ from wall annihilation can be written as

$\displaystyle Y_{\nu_{R_{1}}}=\mathcal{F}\;\frac{\rho_{\rm
DW}(t_{\Gamma})}{M_{1}s(t_{\Gamma})}$ (70)

where $\rho_{\rm DW}(t_{\Gamma})$ is the energy density of walls at the wall
annihilation time, $t_{\Gamma}$, $s(t_{\Gamma})$ is the entropy density at
$t_{\Gamma}$. The efficiency factor $\eta\leq 1$ parameterizes how effectively
the wall energy is transmitted to right-handed neutrinos; it is at most unity
when all the energy density of the walls is transmuted to non-relativistic
$\nu_{R_{1}}$. In general, $\mathcal{F}$ is the fraction of $h$ that end up
decaying to $\nu_{R_{1}}$. As discussed in Appendix B, $\mathcal{F}$ depends
on the product of the branching ratio of $h\rightarrow\nu_{R_{1}}l_{1}$ and
the fraction of decays that occur in the time when $h$ can kinematically decay
into $\nu_{R_{1}}$. We estimate $\mathcal{F}$ in Appendix B and find it to be
much less than $10^{-3}$ for $v_{R}<10^{12}$ GeV. The left panel of Fig. 6
shows the dependence of $\mathcal{F}$ on $M_{1}/v_{R}$ for a variety of
$v_{R}$.

Figure 6: Domain Wall Leptogenesis. Left: Efficiency parameter $\mathcal{F}$
(fraction of $h$ that decay to $\nu_{R_{1}}$) as a function of $M_{1}/v_{R}$.
For small $M_{1}/v_{R}$, $\mathcal{F}$ is set by the branching ratio of
$h\rightarrow\nu_{R_{1}}l_{1}$, corresponding to the first argument of the Min
function in Eq. (84); the branching ratio increases linearly with $M_{1}$ and
hence $\mathcal{F}$ rises linearly. For larger $M_{1}/v_{R}$, $\mathcal{F}$ is
set by the fraction of decays that occur when $h$ can kinematically decay to
$\nu_{R_{1}}$, which occurs at times much earlier than the lifetime of $h$,
corresponding to the second argument of the Min function in Eq. (84); the
kinematically allowed fraction of decays decreases linearly with $M_{1}$ and
hence $\mathcal{F}$ decreases linearly. Right: Domain wall leptogenesis can
generate the observed baryon asymmetry, using the efficiency $\mathcal{F}$ of
the left panel, in the unshaded region of the $M_{1}-v_{R}$ plane, requiring
$v_{R}\gtrsim 10^{12}$ GeV.

The baryon yield eventually generated by the $\nu_{R_{1}}$ originating from
domain walls is

$\displaystyle
Y_{B}=\frac{28}{79}Y_{\nu_{R_{1}}}\epsilon_{1}\leq\frac{28}{79}Y_{\nu_{R_{1}}}\epsilon_{\rm
Max}\,,$ (71)

where $\epsilon_{\rm Max}\approx\frac{3}{16\pi}{M_{1}\Delta
m_{\nu}}/v_{L}^{2}$ is the maximum asymmetry per decay Hamaguchi:2001gw ;
Davidson:2002qv ; Davidson:2008bu  when the right-handed neutrinos are not
highly degenerate Hambye:2003rt . Inserting (70) into (71) implies the maximum
baryon asymmetry is independent of the right-handed neutrino mass, $M_{1}$,
such that

$\displaystyle Y_{B}\lesssim\mathcal{F}\frac{0.05}{g_{*s}}\frac{\Delta
m_{\nu}}{v_{L}^{2}}\frac{\rho_{\rm DW}(t_{\Gamma})}{T_{\Gamma}^{3}}\,.$ (72)

The value for $\rho_{\rm DW}/{T_{\Gamma}}^{3}$ depends on the cosmological
state of the walls at decay,

$\displaystyle\frac{\rho_{\rm DW}(t_{\Gamma})}{T_{\Gamma}^{3}}\approx
v_{R}\times\begin{dcases}\left(\frac{v_{R}}{M_{\rm
Pl}}\right)^{\frac{1}{3}}\lambda^{\frac{1}{2}}&\text{(Wall formation, Kibble-
Zurek)}\\\ \left(\frac{v_{R}}{M_{\rm
Pl}}\right)^{\frac{1}{2}}(2C\beta\lambda^{\frac{1}{2}})^{\frac{1}{2}}&\text{(Friction
Regime)}\\\ \left(\frac{v_{R}}{M_{\rm
Pl}}\right)^{\frac{1}{2}}(8C^{3}\lambda^{\frac{1}{2}})^{\frac{1}{2}}\left(\frac{t}{t_{\rm
dom}}\right)^{\frac{1}{2}}&\text{(Scaling Regime)}\end{dcases}$ (73)

Above, we have written the wall tension as $\sigma=\lambda^{1/2}v_{R}^{3}$,
where $\lambda$ is the Higgs quartic coupling. The variable $\beta$
parameterizes the number of particles scattering with the wall and is always
above unity due to the interactions of the wall scalar field with the thermal
bath. Last, $C^{2}=8\pi^{3}g_{*}/90$ and $t_{\rm dom}\equiv M_{\rm
Pl}^{2}/\sigma$ is the time at which the walls dominate the density of the
universe.

We see that Eq. (73) is maximized when the walls decay soon after formation
due to the enhanced Kibble-Zurek initial abundance. However, even with this
maximum possible $Y_{B}$, there remains a strong lower bound on $v_{R}$ in
order to achieve a sufficient baryon asymmetry; namely, to match the observed
baryon abundance, $Y_{B}^{(\rm obs)}\simeq 8\times 10^{-11}$, $v_{R}$ must be
greater than $\sim 6\times 10^{10}$ GeV assuming an unrealistic maximal
efficiency $\mathcal{F}=1$. In reality, $\mathcal{F}$ decreases dramatically
for lower $v_{R}$ since the efficiency is maximized when the mass of
$\nu_{R_{1}}$ matches that of decaying Higgs field so it as non-relativistic
as possible. Consequently, when incorporating the reduced efficiency of
$Y_{\nu_{R_{1}}}$ as shown in the left panel of Fig. 6, the lower limit on
$v_{R}$ increases to $\sim 2\times 10^{12}$ GeV, as shown by the solid blue
region in the right panel of Fig. 6. This lower limit for $v_{R}$ is
comparable to the naturalness limit from thermal leptogenesis.

##  8 Summary

In theories where the electroweak sector includes $SU(2)_{L}\times SU(2)_{R}$
with minimal Higgs doublets $H_{L}(2,1)$ and $H_{R}(1,2)$, an approximate
parity symmetry can solve the strong CP problem over a wide range of
$\left\langle{H_{R}}\right\rangle=v_{R}$. When parity is exact, spontaneous
parity breaking occurs via the radiative Higgs Parity mechanism, and the
observed values of $m_{t}$ and $\alpha_{s}$ require $v_{R}>\,10^{9}\,{\rm
GeV}$. Neutrino masses, in both LR and Mirror versions of the theory, arise
from an effective theory with dimension-5 operators restricted by parity. This
correlates the light neutrino masses with the right-handed neutrino masses and
neutrino Yukawa couplings. Requiring that the lightness of the observed
neutrinos does not involve fine-tuning, we have shown that thermal
leptogenesis requires $v_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}10^{12}$
GeV, provided there is no degeneracy among the right-handed neutrinos.888The
theory does have fine-tuning in the SM Higgs boson mass. However, the
electroweak hierarchy may originate from some mechanism that is not available
in the neutrino sector; for example, environmental selection in a multiverse.

This bound on $v_{R}$ leads to an indirect probe of leptogenesis in this
theory. The large values of $v_{R}$ required for leptogenesis are consistent
with the observed Higgs mass only for low values of $m_{t}$ and/or high values
of $\alpha_{s}$, as shown in the right panel of Figure 2. More accurate
determinations of $m_{t}$ and $\alpha_{s}$ will greatly reduce the
uncertainties in the prediction for $v_{R}$, as shown in Fig. 3 for the LR
model and Fig. 12 for the Mirror model. If $v_{R}$ is determined to be below
$10^{12}$ GeV, thermal leptogenesis would be generically excluded. On the
other hand, a determination of $v_{R}$ above $10^{12}$ GeV would lend support
to thermal leptogenesis. Furthermore, a neutron electric dipole moment becomes
more likely as $v_{R}$ increases, generated by the operator of (8). Over the
coming decade, or beyond, a determination of a large value for $v_{R}$ and a
discovery of an electric dipole moment of the neutron would together provide
significant indirect evidence that the cosmological baryon asymmetry was
created via thermal leptogenesis in this theory.

The LR theory may be embedded into $SO(10)$ unified theories Hall:2019qwx .
The preferred $v_{R}$ and unification scale from precise gauge coupling
unification depend on the mass spectrum of heavy gauge bosons which is
determined by the representations of the $SO(10)$-breaking Higgses. If
$SO(10)$ is broken only by a ${\bf 45}$ Higgs, precise gauge coupling
unification prefers $v_{R}$ and the unification scale to be around $10^{11}$
GeV and $10^{17}$ GeV, respectively. If a ${\bf 54}$ Higgs also obtains a vev,
which helps stabilize the desired vacuum Babu:1984mz , the preferred $v_{R}$
increases to $10^{12-13}$ GeV and is consistent with the requirement from
successful leptogenesis. At the same time, the preferred unification scale
decreases so that future observations of proton decay are more likely.

If future experiments find values of $m_{t}$ and $\alpha_{s}$ such that
$v_{R}$ is determined to be less than $10^{12}$ GeV, then there are four
possibilities for retaining successful leptogenesis:

  * •

Adding extra scalars to the theory below $v_{R}$. A mixed quartic coupling
with the SM Higgs affects the running of the SM quartic coupling, increasing
$v_{R}$. This extra light scalar increases the fine-tuning of the theory.999We
note that the Higgs Parity theory has the same amount of fine-tuning in the
Higgs potential as in the SM. While it does not address the hierarchy problem,
it does not make it worse.

  * •

Adding soft breaking of parity in the potential for $H_{L}$ and $H_{R}$. This
allows a tree-level vacuum with a large hierarchy of vevs, so that the SM
quartic is unconstrained. However, the soft breaking is large, typically with
a scale of order $v_{R}$, and its origin requires a complication of the
theory.

  * •

Leptogenesis may be non-thermal, for example resulting from the decays of a
degenerate gas of $\nu_{R_{1}}$ after inflation. However, this still requires
$v_{R}>10^{11}$ GeV.

  * •

Degeneracy can be imposed in the right-handed neutrino spectrum. Degeneracy at
the level of $10^{-6}$ may naturally result from approximate symmetries and
can lead to reductions in $M_{1}$ to below $10^{4}$ GeV and $v_{R}$ to near
$10^{9}$ GeV. This would be consistent with values of $m_{t}(\alpha_{s})$ that
are 3$\sigma$ above (below) the current central values.

The first two options significantly weaken the simplicity of the theory; only
$\nu_{R}$ degeneracy easily allows much lower values of $v_{R}$.

If parity is softly broken, even by a small amount, parity may break after
inflation producing a domain wall network. We find that leptogenesis during
collisions of these domain walls does not avoid the bound of
$v_{R}\mathop{}_{\textstyle\sim}^{\textstyle>}10^{12}$ GeV.

The dimension-5 neutrino mass operators can result from the exchange of
fermions that are gauge singlets. This Radiative Singlet Model is particularly
simple, and correlates parameters so that the direct and seesaw contributions
to the light neutrino masses cancel. Neutrino masses are radiative, and hence
require larger Yukawa couplings. While this enhances the creation of the
lepton asymmetry, it forces thermal leptogenesis into the strong washout
domain; the latter dominates, so that the natural bound on $v_{R}$ is
strengthened. In this model, the prospect for discovery of a neutron electric
dipole moment is excellent. In the special case that the Radiative Singlet
Model yields a lepton asymmetry via non-thermal leptogenesis, strong washout
can be avoided. The bound on $v_{R}$ in the third bullet above can then be
weakened to $v_{R}>4\times 10^{10}$ GeV.

Figure 7: Lower bounds on $v_{R}$ (horizontal axis) and $M_{1}$ (vertical
axis) from thermal leptogenesis in theories with exact parity and minimal
Higgs scalars, as a function of the degeneracy $\Delta M/M$ of the right-
handed neutrinos involved in leptogenesis. In the left of the figure, in the
region where there is a degree of degeneracy between $\nu_{R_{1}}$ and
$\nu_{R_{2}}$, we take $M_{3}\sim M_{1}$; to the right of the figure, where
$M_{2}\gg M_{1}$, we take $M_{3}\lesssim M_{2}$. Other choices for $M_{3}$
give stronger lower bounds on $v_{R}$. The curve traces out the intersection
of the blue and orange lines in Figs. 4 and 5 in the non-degenerate and
degenerate regime, respectively. The contour is dashed where the high degree
of right-handed neutrino degeneracy can only be obtained by fine-tuning. The
dashed vertical lines show the predicted range for $v_{R}$ for
$m_{t}=(172.56\pm 0.48)\;{\rm GeV}$. As $v_{R}$ is increased, the dimension-6
operator of (8) gives a contribution to $\theta$ proportional to $v_{R}^{2}$,
as indicated by the top axis. Increasing $v_{R}$ from $10^{12}\,{\rm GeV}$ to
$10^{14}\,{\rm GeV}$, current experiments searching for the neutron electric
dipole moment are progressively more likely to see a signal. Increasing
$v_{R}$ above $10^{14}$ GeV, the strong CP problem reappears, as show by the
gray shading.

Fig. 7 summarizes our results for thermal leptogenesis in the minimal Higgs
Parity theories that solve the strong CP problem via an exact parity, with
neutrino masses arising from the dimension-5 effective theory of (13). The
blue curve shows the lowest possible value of $v_{R}$ and $M_{1}$ as a
function of $\Delta M/M=(M_{2}-M_{1})/M_{1}$ which, when small, is a measure
of the degeneracy between the decaying and virtual $\nu_{R}$. The orange dots
show representative values of $\Delta M/M$ so that degeneracy increases from
right to left. The horizontal (diagonal) segment represents the non-degenerate
(degenerate) region and corresponds to the intersection of the blue and orange
lines of Fig. 4 (5), which represent the lowest possible values of $v_{R}$ and
$M_{1}$ for a given $\Delta M/M$.

Fig. 7 demonstrates that the lowest possible $v_{R}$ and $M_{1}$ require the
most degenerate $\nu_{R}$. However, flavor breaking in the charged leptons
limits the natural degeneracy of right-handed neutrinos to the level of
$10^{-6}$ as shown by the dashed blue curve, which corresponds to
$(v_{R},M_{1})$ below $(10^{9}\,{\rm GeV},10^{3}\,{\rm GeV})$. On the other
hand, for large $v_{R}$, the discovery of a neutron electric dipole moment
becomes more likely as it can be generated by the operator of (8). A signal is
likely if $v_{R}$ is of order $10^{12}$ GeV, the lower bound from leptogenesis
without degeneracy, and is strongly expected for $v_{R}$ above $10^{13}$ GeV.
As $v_{R}$ increases above $10^{14}$ GeV, the strong CP problem begins to re-
emerge, as shown by the gray shading. When parity is exact, the scale of its
spontaneously breaking, $v_{R}$, can be computed from SM parameters. For
$\alpha_{s}(M_{Z})=0.1179$ and the 1$\sigma$ range of $m_{t}=(172.56\pm
0.48){\rm GeV}$, the predicted range for $v_{R}$ is shown by the vertical
dotted lines. Of course, including uncertainties in $\alpha_{s}$ and more than
$1\sigma$ in $m_{t}$ the allowed range is much wider, but future measurements
offer the prospect of a much more accurate prediction, with important
implications for leptogenesis and neutrino masses.

###### Acknowledgements.

We thank the CERN Theory Division where part of this work was performed. DD is
supported by the James Arthur Postdoctoral Fellowship. The work of JCCM and LH
was supported by the Director, Office of Science, Office of High Energy
Physics of the U.S. Department of Energy under the Contract No. DE-
AC02-05CH11231 and by the NSF grants PHY-1915314 and PHY-2210390. The work of
KH was partly supported by Grant-in-Aid for Scientific Research from the
Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan
(20H01895) and by World Premier International Research Center Initiative
(WPI), MEXT, Japan (Kavli IPMU).

##  Appendix A Naturalness Limit on $\nu_{R}$ Degeneracy

In a natural theory, the degeneracy between $\nu_{R_{1}}$ and $\nu_{R_{2}}$
should be greater than any quantum corrections that generate a mass difference
between them. In LR models, the operators which generate the masses for the
charged leptons also generate unavoidable corrections to $M_{1}$ and $M_{2}$;
that is, they break the symmetry that made $\nu_{R_{1}}$ and $\nu_{R_{2}}$
degenerate. For example, note that the operators that describe the neutrinos
Eq. (13), and the charged fermions,

$\displaystyle\mathcal{L}_{e,u,d}=\frac{c_{ij}^{u}}{M}q_{i}\bar{q}_{j}H_{L}H_{R}+\frac{c_{ij}^{d}}{M}q_{i}\bar{q}_{j}H_{L}^{\dagger}H_{R}^{\dagger}+\frac{c_{ij}^{e}}{M}l_{i}\bar{l}_{j}H_{L}^{\dagger}H_{R}^{\dagger}+\text{h.c.}\,,$
(74)

are necessarily described by an effective field theory; the dimension-5
operators given in Eq. (74) must be generated by other physics. There are two
possible ways to generate such operators: by integrating out a heavy scalar
field or by integrating out a heavy fermion field.

First, consider the case where the charged lepton Yukawas arise from
integrating out a bifundamental scalar in the UV completion

$\displaystyle{\mathcal{L}}=-m_{\Phi}^{2}|\Phi|^{2}+(x_{ij}\Phi\ell_{i}\bar{\ell}_{j}-A\Phi^{\dagger}H_{L}^{\dagger}H_{R}^{\dagger}+{\rm
h.c.})\,,$ (75)

as shown in the left panel of Fig. 8. This generates the charged lepton Yukawa
coupling $y^{e}_{ij}=Ax_{ij}v_{R}/m_{\Phi}^{2}$. It also also generates non-
canonically normalized kinetic terms for the $\nu_{R}$,

Figure 8: Left: Feynman diagram showing how the charged lepton Yukawa
couplings are generated in a UV completion involving a bifundamental scalar.
Right: The same interaction also necessarily generates a mass splitting for
the right-handed neutrinos, limiting the smallest natural degeneracy between
$\nu_{R}$’s to be of order $\Delta M/M\sim y_{\tau}^{2}/8\pi\sim 10^{-6}$.

$\displaystyle{\cal L}=$ $\displaystyle(1+\delta
Z_{11})\;\bar{\nu}_{1}^{\dagger}\bar{\sigma}\partial\bar{\nu}_{1}+(1+\delta
Z_{22})\;\bar{\nu}_{2}^{\dagger}\bar{\sigma}\partial\bar{\nu}_{2}+\left(\delta
Z_{12}\,\bar{\nu}_{1}^{\dagger}\bar{\sigma}\partial\bar{\nu}_{2}+{\rm
h.c.}\right),$ (76)

where $\delta Z_{ij}\simeq x_{ki}x^{*}_{kj}/8\pi^{2}$, as shown by the diagram
on the right panel of Fig. 8. This wave function renormalization induces a
mass splitting of Dunsky:2020dhn

$\displaystyle\frac{\Delta M}{M}\simeq\sqrt{\left(\delta Z_{11}-\delta
Z_{22}\right)^{2}+\left(\delta Z_{12}+\delta
Z_{12}^{*}\right)^{2}}\gtrsim\frac{y_{\tau}^{2}}{8\pi^{2}}\simeq 10^{-6}\,,$
(77)

where $y_{\tau}\simeq 10^{-2}$ is the $\tau$ Yukawa coupling.

A similar constraint occurs if instead the charged lepton Yukawas arise from
integrating out heavy fermions, $E,\bar{E}$, in the UV completion

$\displaystyle{\mathcal{L}}=z_{ia}^{e}\ell_{i}\bar{E}_{a}H^{\dagger}_{L}+(z^{e}_{ia})^{*}\bar{\ell}_{i}E_{a}H^{\dagger}_{R}+M_{E,a}E_{a}\bar{E}_{a}\;+\text{h.c.},$
(78)

as shown in the left panel of Fig. 9. This generates the charged lepton Yukawa
coupling $y_{ij}^{e}=z^{e}_{ia}(z_{aj}^{e})^{*}v_{R}/M_{E,a}$ if
$M_{E,a}>z_{e}v_{R}$ or $y_{ij}^{e}=z_{e}$ if $M_{E,a}<z_{e}v_{R}$. Again, the
same operators in (78) generate non-canonically normalized kinetic terms as in
Eq. (76) now with $\delta Z_{ij}\simeq z_{ia}^{e}(z_{ja})^{*}/8\pi^{2}$.
Wavefunction renormalization thus gives the same limit on $\Delta M/M$ as Eq.
(77).

Figure 9: Same as Fig. 8 but when the charged lepton Yukawa couplings are
generated by heavy fermions $E,\bar{E}$. The naturalness limit on the
radiative mass splitting is also $\Delta M/M\sim y_{\tau}^{2}/8\pi\sim
10^{-6}$. Figure 10:  In the effective field theory, radiative corrections
generate dimension-5 operators for neutrino masses from the dimension-5
interaction that leads to the charged lepton Yukawa couplings.

Last, even without reference to any UV completion, in the LR effective field
theory of Eq. (74), the dimension-5 operators for the charged lepton masses
necessarily generate a mass splitting for $\nu_{R,i}$. This is demonstrated by
the diagram of Fig. 10 which generates a wave function renormalization
parametrically similar to that generated in the two aforementioned UV
completions,

$\displaystyle\frac{|\Delta
M|}{M}\sim\frac{1}{(16\pi^{2})^{2}}y_{\tau}^{2}\left(\frac{\Lambda}{v_{R}}\right)^{2}\sim
5\times 10^{-9}\left(\frac{\Lambda}{v_{R}}\right)^{2}$ (79)

which is roughly one-loop factor smaller than Eq. (77), though it can be
larger for $\Lambda\gg v_{R}$.

##  Appendix B Efficiency of Domain Wall Leptogenesis

In this section, we elaborate on the production of $\nu_{R}$ from the
annihilation of domain walls and estimate the efficiency parameter, $\eta$, of
Eq. (70), showing that it is highly suppressed for $v_{R}\lesssim 10^{12}$ GeV
and becomes $\mathcal{O}(1)$ only for $v_{R}\gg 10^{12}$ GeV. Consequently,
domain wall leptogenesis is not a promising way to lower the naturalness bound
on $v_{R}$ below $10^{12}$ GeV.

Consider two domain walls immediately before coming into contact and
annihilating into Higgs particles. Sufficiently close to the walls, the
surfaces are roughly plane parallel patches of area $A$ as shown in Fig. 11.
The energy in each wall patch is

$\displaystyle E_{w}$ $\displaystyle\simeq\gamma(v_{w})\sigma
A,\quad\gamma(v_{w})=\frac{1}{\sqrt{1-v_{w}^{2}}}\ .$ (80)

The Lorentz factor of the walls is roughly unity since friction between the
wall surface and the background plasma impedes wall motion even with the false
vacuum accelerating the walls together.

Figure 11: Left: Visualization of vacuum configuration in field space for
$H_{L}$ and $H_{R}$. Because the Coleman-Weinberg potential and the
$Z_{2}$-breaking piece are small (see Eq. (87)), the potential is nearly
$SU(4)$ symmetric with the the angular excitations $h$ being nearly massless.
The curvature of the Coleman-Weinberg potential generates a small mass for the
$H_{L}$ vacuum while the $Z_{2}$-breaking piece ensures that this minimum is
the absolute minimum of the potential, causing the domain walls to annihilate.
Right: Simplified illustration of leptogenesis from domain wall annihilation.
If there exists a small parity breaking term in the theory, a pressure
difference is generated between the walls, eventually causing them to collide
and annihilate. After annihilation, excitations of the angular field $h$
emanate from the surface of contact of the walls, representing excitation of
Higgs quanta. These Higgs quanta initially have large field values and can
thus kinematically decay to $\nu_{R}$, potentially generating a large non-
thermal abundance of $\nu_{R}$ whose own decays eventually generate a lepton
asymmetry.

After wall annihilation, Higgs particles are produced. We have performed 1D
simulations of planar walls annihilating and find that after a time $t$, the
Higgs field approximately spreads out uniformly in space out to a distance
$L\approx t$. The average energy density of the Higgs field $h$ (the angular
excitation between $H_{L}$ and $H_{R}$ as shown in Fig. 11) within a distance
$L\ll t_{\Gamma}$ from the point of annihilation is then

$\displaystyle\rho_{h}\simeq\frac{E_{w}}{A\times
L}=\frac{\gamma(v_{w})\sigma}{L}\approx\lambda h^{4}.$ (81)

Since the wall tension of the Higgs field is approximately
$\sigma\simeq\sqrt{\lambda}v_{R}^{3}$ and the Higgs fields propagates
relativistically, the value of the Higgs field a time $t\simeq L$ after
annihilation is

$\displaystyle h\approx\left(\frac{v_{R}^{3}}{\lambda^{1/2}t}\right)^{1/4}\,.$
(82)

We see that at early times after wall annihilation, the Higgs field is
largest. Kinematic production of $\nu_{R_{1}}$ requires that the effective
Higgs mass, $m_{{\rm eff},h}\approx\sqrt{\lambda}h$, is greater than $M_{1}$.
This occurs at a time less than

$\displaystyle t_{\rm max}=\lambda^{3/2}\frac{v_{R}^{3}}{M_{1}^{4}}\,.$ (83)

When $t_{\rm max}<\Gamma_{h\rightarrow\nu_{R_{1}}l_{1}}^{-1}$, the production
of $\nu_{R_{1}}$ is suppressed since only a small fraction of decays occur
while $h$ is large enough to permit the reaction
$h\rightarrow\nu_{R_{1}}l_{1}$. Quantitatively, the fraction of $\nu_{R_{1}}$
ultimately produced – equivalently, the efficiency factor $\mathcal{F}$ – is

$\displaystyle\mathcal{F}=\text{Br}(h\rightarrow\nu_{R_{1}}l_{1})\times{\text{Min}}\left\\{1,\,\Gamma_{h}t_{\rm
max}\right\\}\simeq\frac{y_{1}^{2}}{y_{b}^{2}+y_{1}^{2}}\times{\text{Min}}\left\\{1,\,\frac{\lambda}{8\pi}(y_{1}^{2}+y_{b}^{2})\frac{v_{R}^{2}}{M_{1}^{2}}\right\\}\,.$
(84)

Here, $\Gamma_{h}\simeq(y_{1}^{2}+y_{b}^{2})/(8\pi)\sqrt{\lambda}h(h/v_{R})$
is the total decay rate of the Higgs particles emanating from the wall,
$y_{b}$ the bottom quark Yukawa coupling, and $h/v_{R}$ is the time dilation
factor of the Higgs particles escaping from the wall which have momenta
$k\approx\sqrt{\lambda}v_{R}$ and mass $m_{{\rm
eff},h}\approx\sqrt{\lambda}h$. Note that since the effective top quark mass
$m_{{\rm eff},t}\approx y_{t}h$ is greater than $m_{{\rm eff},h}$, $h$ cannot
decay to two top quarks. Instead, the dominant $h$ decay out of
$\nu_{R_{1}}l_{1}$ is to two bottom quarks. For $y_{1}<y_{b}$, the efficiency
is suppressed by the small branching ratio into $\nu_{R_{1}}$ compared to
bottom quarks. We take the renormalized value of $y_{b}$ above $v_{R}>10^{10}$
GeV to be around $0.008$.

We show the value of $\mathcal{F}$ as a function of $M_{1}/v_{R}$ in the left
panel of Fig. 6. Here, we take $y_{1}^{2}=M_{1}\tilde{m}/v^{2}$ where
$\tilde{m}=0.1$ eV is the largest natural value for $\tilde{m}$ without
needing fine-tuned cancellations between the direct and seesaw mass
contributions of $\nu_{1}$. Note that since domain wall leptogenesis is non-
thermal, it is possible to take $y_{1}>y_{b}$ without causing washout which
can occur for such large $\tilde{m}$. We also take $\lambda=0.1$ which is an
optimistically large value of the Higgs quartic near the scale $v_{R}$. The
right panel of Fig. 6 shows the region where the baryon asymmetry given by Eq.
(70), and using the efficiency of Eq. (84), is unable to match the observed
baryon $Y_{B}\simeq 8\times 10^{-11}$. We see that domain wall leptogenesis
requires $v_{R}\gtrsim 5\times 10^{12}$ GeV, comparable to the bound from
natural thermal leptogenesis.

##  Appendix C Model-dependence of Parity Breaking Scale

In this appendix, we discuss the model dependence of the Parity breaking scale
arising from the threshold correction at the parity breaking scale and the
running of the quartic coupling. We discuss Models A and D of Ref.
Hall:2018let , which have no additional stable charged particles and are
consistent with cosmological evolution with high enough reheat temperature to
achieve thermal leptogenesis. The fermion contents of Models A and D are shown
in Tables 1 and 2. Model A with large Dirac masses,
$M_{U_{i}}U_{i}\bar{U}_{i}+M_{D_{i}}D_{i}\bar{D}_{i}+M_{E_{i}}E_{i}\bar{E}_{i}$,
gives the LR theory. Integrating out the heavy fermions generates dimension-5
operators leading to SM quark and charged lepton masses after electroweak
symmetry breaking. Model $D$ is the Mirror model.

Table 1: The gauge charges of Higgses and fermions in the LR theory (Model A). | $H_{L}$ | $H_{R}$ | $q_{i}$ | $\bar{q}_{i}$ | $\ell_{i}$ | $\bar{\ell}_{i}$ | $U_{i}$ | $\bar{U}_{i}$ | $D_{i}$ | $\bar{D}_{i}$ | $E_{i}$ | $\bar{E}_{i}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{c}$ | 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
$U(1)_{X}$ | $\frac{1}{2}$ | $-\frac{1}{2}$ | $\frac{1}{6}$ | $-\frac{1}{6}$ | $-\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{2}{3}$ | $-\frac{2}{3}$ | $-\frac{1}{3}$ | $\frac{1}{3}$ | $-1$ | $1$
Table 2: The gauge charges of Higgses and fermions in the Mirror theory (Model D). | $H_{L}$ | $H_{R}$ | $q_{i}$ | $q^{\prime}_{i}$ | $\ell_{i}$ | $\ell^{\prime}_{i}$ | $\bar{u}_{i}$ | $\bar{u}^{\prime}_{i}$ | $\bar{d}_{i}$ | $\bar{d}^{\prime}_{i}$ | $\bar{e}_{i}$ | $\bar{e}^{\prime}_{i}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{c}$ | ${\bf 1}$ | 1 | ${\bf 3}$ | ${\bf 3}$ | 1 | 1 | ${\bf\bar{3}}$ | ${\bf\bar{3}}$ | ${\bf\bar{3}}$ | ${\bf\bar{3}}$ | 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
$U(1)_{X}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{2}{3}$ | $-\frac{2}{3}$ | $\frac{1}{3}$ | $\frac{1}{3}$ | $1$ | $1$

###  C.1 Threshold correction

Model-dependence arises from threshold corrections from the interactions that
lead to the top quark Yukawa coupling. In Model A, these interactions are

$\displaystyle{\cal L}=-x\,q\bar{U}H_{L}-x\,\bar{q}UH_{R}-M\bar{U}U.$ (85)

After $SU(2)_{R}$ symmetry breaking, a linear combination of
$\bar{u}\supset\bar{q}$ and $\bar{U}$ obtains a mass
$\sqrt{x^{2}v_{R}^{2}+M^{2}}$ and the other linear combination remains
massless to be identified with the right-handed top quark. The top Yukawa is

$\displaystyle y_{t}=\frac{x^{2}v_{R}}{\sqrt{x^{2}v_{R}^{2}+M^{2}}}.$ (86)

The threshold correction to $\lambda_{\rm SM}(v_{R})$ can be computed in the
following way.

  1. 1.

Compute the Coleman-Weinberg potential of $H_{L}$ and $H_{R}$ from the top
Yukawa sector and add it to the tree-level potential,

$\displaystyle
V=\lambda(|H_{L}|^{2}+|H_{R}|^{2})^{2}-m_{H}^{2}(|H_{L}|^{2}+|H_{R}|^{2})+\epsilon|H_{L}|^{2}|H_{R}|^{2}+V_{\rm
CW}(H_{L},H_{R}).$ (87)

  2. 2.

Minimize the potential with respect to $H_{R}$ while taking $H_{L}=0$ to fix
the parameter $m_{H}^{2}$ so that $\left\langle{H_{R}}\right\rangle=v_{R}$.

  3. 3.

Integrating out $H_{R}$ to obtain the effective potential $V_{\rm eff}(H_{L})$
and fix the parameter $\epsilon$ to take the quadratic term of $H_{L}$ to be
zero, which corresponds to electroweak fine-tuning.

  4. 4.

The coefficient of $|H_{L}|^{4}$ is of the form

$\displaystyle V_{\rm eff}/|H_{L}|^{4}=a-\frac{3}{16\pi^{2}}y_{t}^{4}\,{\rm
ln}\frac{|H_{L}^{2}|}{v_{R}^{2}},$ (88)

where the constant term $a$ depends on the couplings of the UV theory.

  5. 5.

Comparing this with the CW potential of the SM in the $\overline{\rm MS}$
scheme,

$\displaystyle V_{\rm CW,SM}/|H_{L}|^{4}=\lambda_{\rm
SM}(\mu)-\frac{3}{16\pi^{2}}\,y_{t}^{4}\left({\rm
ln}\frac{y_{t}^{2}|H_{L}|^{2}}{\mu^{2}}-\frac{3}{2}\right),$ (89)

we obtain

$\displaystyle\lambda_{\rm SM}(v_{R})=a+\frac{3}{16\pi^{2}}y_{t}^{4}\left({\rm
ln}y_{t}^{2}-\frac{3}{2}\right).$ (90)

Applying this procedure to the Lagrangian in Eq. (85), we find

$\displaystyle\lambda_{\rm
SM}(v_{R})=\frac{3}{16\pi^{2}}y_{t}^{4}\left(-2+{\rm ln}y_{t}^{2}+2{\rm
ln}\frac{x^{2}}{y_{t}^{2}}\right)\geq\frac{3}{16\pi^{2}}y_{t}^{4}\left(-2+{\rm
ln}y_{t}^{2}\right),$ (91)

where the inequality is saturated when $x=y_{t}$, i.e., $M=0$. The prediction
on $v_{R}$ for $M\neq 0$ is always smaller than that for $M=0$.

In Model D, the top Yukawa is given by the Lagrangian

$\displaystyle{\cal
L}=-xq\bar{u}H_{L}-xq^{\prime}\bar{u}^{\prime}H_{R}-zq\bar{u}^{\prime}H_{L}-zq^{\prime}\bar{u}H_{R}.$
(92)

After Parity breaking, a linear combination of $\bar{u}$ and
$\bar{u}^{\prime}$ obtains a mass $\sqrt{x^{2}+z^{2}}\,v_{R}$. Another linear
combination remains massless and is identified as the right-handed top quark.
The top Yukawa is

$\displaystyle y_{t}=\frac{x^{2}-z^{2}}{\sqrt{x^{2}+z^{2}}}.$ (93)

For $z\ll x$ and $x\ll z$, $y_{t}=x$ and $z$, respectively. We find that the
threshold correction to $\lambda_{\rm SM}(v_{R})$ is

$\displaystyle\lambda_{\rm
SM}(v_{R})=\frac{3}{16\pi^{2}}y_{t}^{4}\left(-2+{\rm ln}y_{t}^{2}+{\rm
ln}\frac{x^{2}+z^{2}}{y_{t}^{2}}\right)\geq\frac{3}{16\pi^{2}}y_{t}^{4}\left(-2+{\rm
ln}y_{t}^{2}\right),$ (94)

where the inequality is saturated when $z=0$ or $x=0$. The prediction on
$v_{R}$ for $z\neq 0$ or $x\neq 0$ is always smaller than that for $z=0$ or
$x=0$.

###  C.2 Running of quartic

In Model A where the right-handed SM fermions are mostly the Parity partners
of the left-handed SM fermion, the RGE running of the Higgs quartic coupling
is the same as the SM. If we take the Dirac masses analogous to $M$ in Eq. 85
to be zero, the Parity partners of the SM fermions have masses $m_{\rm
SM}v_{R}/v_{L}$ and the running of the $SU(3)_{c}\times U(1)_{Y}$ gauge
couplings is modified. In Model D, where the parity partners of the SM fermion
are new particles, the Parity partners may be also as light as $m_{\rm
SM}v_{R}/v_{L}$.

Figure 12: Future determinations of $v_{R}$ from improved measurements of SM
parameters. The figures are analogous to Fig. 3 but for the Mirror Theory. The
two modifications are the the running of $\alpha_{s}$ from mirror quarks which
are charged under the Standard Model $SU(3)$, as well as the modified
threshold correction at $v_{R}$. Left: Centered on $m_{t}(\alpha_{s})$ that
are $1\sigma$ above (below) the current values. Right: Centered on
$m_{t}(\alpha_{s})$ that are $1\sigma$ below (above) the current values.

Thus, the two main modifications to determining the running of the quartic in
Model D are: 1) The incorporation of the mirror quarks to the running of
$\alpha_{s}$ and 2) the incorporation of all mirror particles that possess
hypercharge to the running of $\alpha_{Y}$. In both cases, we manually
increment the number of light mirror fermions to the relevant beta functions
at the scale $\mu=y_{i}^{\prime}v_{R}$, where $y_{i}^{\prime}$ are the mirror
fermion Yukawa couplings. Note in the Mirror theory, the threshold correction
Eq. (11) to $\lambda$ at the scale $v_{R}$ is the same as in the Left-Right
theory. We numerically determine the renormalization scale at which
$\lambda_{\rm SM}$ equals (11) and show the results in Fig. 12, which is
analogous to Fig. 3 for the LR theory. In general, the incorporation of mirror
quarks reduces the running of $\alpha_{s}$ at high scales, shifting the
colored bands slightly up relative to Fig. 3.

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