aainstitutetext: Department of Physics, Oklahoma State University, Stillwater,
OK 74078, USAbbinstitutetext: Tsung-Dao Lee Institute, and School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240,
Chinaccinstitutetext: National Center for Theoretical Sciences, and Department
of Physics, National Taiwan University, Taipei 10617, Taiwanddinstitutetext:
Department of Physics, University of Virginia, Charlottesville, Virginia
22904-4714, USA

# Naturally Light Dirac and Pseudo-Dirac Neutrinos from Left-Right Symmetry

K.S. Babu b,c Xiao-Gang He a Mingxian Su d Anil Thapa<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We develop a class of left-right symmetric theories based on the gauge group
$SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times U(1)$ with a generalized
seesaw mechanism for generating the charged fermion masses. Neutrinos are
naturally Dirac particles in this setup with their small masses arising from
two-loop quantum corrections. We evaluate these two-loop diagrams exactly and
analyze the flavor structure of the lepton sector. We find excellent fits to
neutrino oscillation data, independent of the right-handed gauge symmetry
breaking scale. We also explore the possibility that neutrinos are pseudo-
Dirac particles in this framework, with the tiny mass splittings between
active and sterile neutrinos arising from Planck-induced corrections and find
possible realizations. These models can be tested in the near future with
precision cosmological measurements of $\Delta N_{\rm eff}$ in CMB which is
predicted to be $\simeq 0.14$. This class of models allows for a solution to
the strong CP problem via parity symmetry without the need for an axion.

##  1 Introduction

One of the fundamental questions in particle physics today is whether or not
global lepton number and baryon number symmetries present in the standard
model (SM) are true symmetries. This question should be addressed in the
context of SM extensions that generate small neutrino masses needed to explain
the observed oscillations among different neutrino flavors. Such extensions
may or may not respect lepton number as a global symmetry. In the framework of
the popular seesaw mechanism Minkowski:1977sc ; Gell-Mann:1979vob ;
Mohapatra:1979ia ; Yanagida:1980xy ; Glashow:1979nm  which explains naturally
the smallness of the neutrino mass, lepton number is broken by the heavy
Majorana mass of the right-handed neutrino. However, neutrino oscillation
experiments cannot distinguish a Majorana neutrino from a Dirac neutrino. The
hallmark of lepton number violation in low energy experiments is the
observation of neutrinoless double beta decay, which has not been seen in
experiments to date (for a review on the current status see Ref.
Dolinski:2019nrj ). It is therefore important to explore theories where
neutrinos are Dirac particles, especially if there is a natural understanding
of their small masses.

The purpose of this paper is to develop a class of left-right symmetric
theories that lead naturally to light Dirac neutrinos Babu:1988yq . Unlike in
conventional left-right symmetric theories Pati:1974yy ; Mohapatra:1974gc ;
Mohapatra:1974hk ; Senjanovic:1975rk , in these theories the masses of all
charged fermions arise from a generalized seesaw mechanism Davidson:1987mh .
Interestingly, with the minimal fermionic content, the seesaw mechanism is not
effective in the neutrino sector, and the neutrinos remain as Dirac fermions
with their small masses generated via two-loop radiative corrections. While
this class of models was proposed and the neutrino masses were estimated in
Ref. Babu:1988yq , the flavor structure of the lepton sector has not been
studied thus far. Here we undertake a detailed study of the flavor structure
and confront these models with neutrino oscillation data. We evaluate exactly
the two-loop diagrams generating small Dirac neutrino masses and show with our
numerical analysis that these models are fully consistent with neutrino
oscillation phenomenology, regardless of the scale at which the right-handed
gauge symmetry breaks down spontaneously.

This class of left-right symmetric theories with naturally light Dirac
neutrinos is based on the gauge group $SU(3)_{c}\times SU(2)_{L}\times
SU(2)_{R}\times U(1)$. These theories are motivated on several grounds. First,
parity is a spontaneously broken symmetry in this framework, unlike in the SM.
The Higgs sector of the model is very simple, consisting of an $SU(2)_{L}$
doublet and an $SU(2)_{R}$ doublet. The generalized seesaw mechanism present
in these theories provides some understanding of the mass hierarchies observed
among quarks and leptons Davidson:1987mh ; Davidson:1987tr . Owing to parity
symmetry present in the theory, the strong CP problem can be solved without
resorting to the Peccei-Quinn symmetry and the resulting axion Babu:1989rb .
(For studies of low energy phenomenology of these models see Ref. Babu:2018vrl
; Hall:2018let ; Craig:2020bnv ).111For variations of these models where small
Dirac neutrino masses arise via one-loop diagrams in presence of an extended
Higgs sector, see Ref. Mohapatra:1987hh . These theories have a natural
embedding in $SU(5)_{L}\times SU(5)_{R}$ grand unification Davidson:1987mh ;
Lee:2016wiy .222While the models of Ref. Davidson:1987mh ; Davidson:1987tr
were termed “universal seesaw models”, the model of Ref. Babu:1988yq  which we
develop further here, does not fall into this class, since there is no seesaw
mechanism acting on the neutrinos.

While the discovery of neutrinoless double beta decay would be a definitive
confirmation of the Majorana nature of the neutrino Schechter:1981bd , there
are independent cosmological tests that can distinguish a Dirac neutrino from
a Majorana neutrino. The effective number of light degrees of freedom relevant
for early universe evolution, parameterized as $\Delta N_{\rm eff}$, will
receive an additional contribution of about 0.14 in the Dirac neutrino mass
scenario presented here, which is within the sensitivity reach of forthcoming
CMB measurements SPT-3G SPT-3G:2014dbx , SO SimonsObservatory:2019qwx  and
CMB-S4 Lee:2013bxd ; Abazajian:2019eic  experiments, which are expected to
have a sensitivity of $0.06$ or better in $\Delta N_{\rm eff}$.

In any scenario where the neutrinos are Dirac particles with naturally small
masses, there is a possibility that they could be pseudo-Dirac particles at a
more fundamental level Wolfenstein:1981kw ; Petcov:1982ya ; Valle:1983dk ;
Kobayashi:2000md . Quantum gravity corrections could generate ultra-small
Majorana masses for both the left-handed and the right-handed neutrinos via
higher dimensional operators suppressed by the Planck scale. These corrections
should respect all gauge symmetries present in the theory, but are not
expected to respect global symmetries such as lepton number. We explore such a
possibility in our left-right symmetric framework. We propose to gauge the
$(B-L)$ symmetry present in the theory in order to control the amount of
active-sterile mass splitting that these quantum gravity corrections would
induce. The spontaneous breaking of $(B-L)$ symmetry naturally would lead to
an unbroken $Z_{N}$ subgroup of lepton number, with the integer $N$ dependent
on the $(B-L)$ charge of the Higgs scalar that breaks the symmetry. For $N=2$,
Majorana neutrino masses are permitted, while for any other value of $N$, the
neutrino would remain strictly a Dirac particle, even with the inclusion of
quantum gravity corrections. We analyze the case of $N=2$ in some detail that
leads to pseudo-Dirac neutrinos, which have been subject of much discussion in
the context of ultra-high energy neutrino detection at neutrino telescopes
such as IceCube Beacom:2003eu ; Keranen:2003xd , as well as in supernova
neutrino signals Martinez-Soler:2021unz .

Naturally light Dirac neutrinos can arise in other contexts. In the mirror
universe scenario the gauge symmetry of the SM is duplicated in a mirror
sector Lee:1956qn ; Foot:1991py ; Berezhiani:1995yi . The mirror neutrino
$\nu^{\prime}$ from the mirror lepton doublet
$L^{\prime}=(\nu^{\prime},\,e^{\prime})$ can then be the Dirac partner of the
usual neutrino $\nu$. A Dirac mass connecting $\nu$ and $\nu^{\prime}$ would
arise from a dimension-five operator $(LH)(L^{\prime}H^{\prime})/\Lambda$
involving the SM Higgs ($H$) and the mirror Higgs ($H^{\prime}$) doublets,
leading to a Dirac seesaw mechanism Silagadze:1995tr . Other models of
naturally light Dirac neutrinos have been constructed assuming certain
discrete symmetries (see for e.g. Ref. Farzan:2012sa ) and/or by extending the
scalar sector (see for e.g. Ref. Ma:2016mwh ; Saad:2019bqf ; Jana:2019mgj ).
Smallness of neutrino masses is explained in these models as they arise
through loop-induced quantum corrections. In contrast to these models, the
model developed here is motivated on other grounds – parity as a spontaneously
broken symmetry, solution to the strong CP problem via parity symmetry, origin
from $SU(5)_{L}\times SU(5)_{R}$ unification, etc, and does not require an
extended Higgs sector.

The rest of the paper is organized as follows. In Sec. 2 we describe the left-
right symmetric model with seesaw mechanism for charged fermions. In Sec. 3 we
evaluate the two-loop diagrams for Dirac neutrino masses exactly. In Sec. 4 we
carry out fits to the neutrino oscillation data within the model and show its
consistency. In Sec. 5 we discuss the embedding of the model in
$SU(5)_{L}\times SU(5)_{R}$ unification. In Sec. 6 we discuss cosmological
tests of these models in $N_{\rm eff}$. In Sec. 7 we discuss the possibility
of realizing pseudo-Dirac neutrinos with the aid of quantum gravity
corrections. Finally we conclude in Sec. 8.

##  2 Left-right Symmetric Models with Seesaw Mechanism for Charged Fermion
Masses

The class of models we develop here is based on the left-right symmetric gauge
group $SU(3)_{c}\otimes SU(2)_{L}\otimes SU(2)_{R}\otimes U(1)_{X}$. The
fermions transform under the gauge group in a left-right symmetric manner
which enables parity to be defined as a symmetry of the Lagrangian:

$\displaystyle Q_{L}\ (3,2,1,1/3)$ $\displaystyle=\begin{pmatrix}u_{L}\\\
d_{L}\\\ \end{pmatrix},\hskip 14.22636ptQ_{R}\
(3,1,2,1/3)=\begin{pmatrix}u_{R}\\\ d_{R}\\\ \end{pmatrix},$
$\displaystyle\Psi_{L}\ (1,2,1,-1)$ $\displaystyle=\begin{pmatrix}\nu_{L}\\\
e_{L}\\\ \end{pmatrix},\hskip 14.22636pt\Psi_{R}\
(1,1,2,-1)=\begin{pmatrix}\nu_{R}\\\ e_{R}\\\ \end{pmatrix}.$ (2.1)

Under parity symmetry $Q_{L}\leftrightarrow
Q_{R},\,\Psi_{L}\leftrightarrow\Psi_{R}$ along with $W_{L}\leftrightarrow
W_{R}$ where $W_{L,R}$ denote the three gauge bosons of $SU(2)_{L,R}$.

The model also has three families of vector-like quarks and leptons that
transform as singlets of $SU(2)_{L}\times SU(2)_{R}$:

$P(3,1,1,4/3),~{}~{}~{}N(3,1,1,-2/3),~{}~{}~{}E(1,1,1,-2)~{}.$ (2.2)

These fermions are utilized to generate masses for quarks and leptons via a
generalized seesaw mechanism. As shown in Sec. 5, the fermion multiplets shown
in Eqs. (2.1)-(2.2) neatly fit into $\\{(10,1)+(\overline{5},1)\\}$ and
$\\{(1,10)+(1,\overline{5})\\}$ representations of $SU(5)_{L}\times SU(5)_{R}$
unification, which are the simplest anomaly-free chiral representations of
this gauge group.

Here the electric charges of all particles are given by the formula

$Q=T_{3L}+T_{3R}+\frac{X}{2}$ (2.3)

where $T_{3L}$ and $T_{3R}$ are the third components of $SU(2)_{L}$ and
$SU(2)_{R}$ respectively. Thus $(P,\,N,\,E)$ have $Q=(2/3,\,-1/3,\,-1)$, which
enables mixing of these vector-like fermions with the up-type quarks, down-
type quarks and charged leptons, respectively. Such mixings are responsible
for the generation of ordinary quark and lepton masses in this scenario. Note
the absence of an electrically neutral vector-like lepton, which is crucial
for realizing Dirac neutrinos in the framework. Such vector-like leptons with
$Q=0$ are not part of the simplest anomaly-free representations of
$SU(5)_{L}\times SU(5)_{R}$.

We have denoted the $U(1)$ factor of the gauge symmetry as $U(1)_{X}$. This
$U(1)$ may be identified as $(B-L)$, as can be seen from the charge assignment
of Eq. (2.1) of the standard fermions. However, with this identification,
$(B-L)$ would appear to be broken once the up-type quarks mix with the $P$
quark carrying a different $(B-L)$ charge (and similarly in the down-quark and
charged lepton sectors). There is an unbroken $(B-L)$ symmetry in the theory
even after spontaneous gauge symmetry breaking, under which the $(P,\,N)$
quarks have charge $1/3$ and $E$ has charge $-1$, and the Higgs scalars have
charge zero. We prefer to call this unbroken $U(1)$ as $(B-L)$, which may even
be potentially gauged, owing to its anomaly-free nature. In fact, we propose
to gauge this $(B-L)$ in order to control Planck-induced Majorana masses of
$\nu_{L}$ and $\nu_{R}$, as discussed further in Sec. 7.

The Higgs sector of the model is quite simple, consisting of an $SU(2)_{L}$
doublet and an $SU(2)_{R}$ doublet:

$\displaystyle\chi_{L}\ (1,2,1,1)$
$\displaystyle=\begin{pmatrix}\chi_{L}^{+}\\\ \chi_{L}^{0}\\\
\end{pmatrix},\hskip 14.22636pt\chi_{R}\
(1,1,2,1)=\begin{pmatrix}\chi_{R}^{+}\\\ \chi_{R}^{0}\\\ \end{pmatrix},$ (2.4)

Under parity, $\chi_{L}\leftrightarrow\chi_{R}$. Once the neutral components
of these fields acquire vacuum expectation values (VEV) denoted as

$\displaystyle\langle\chi_{L}\rangle=\begin{pmatrix}0\\\ \kappa_{L}\\\
\end{pmatrix}\,,\hskip 19.91692pt\langle\chi_{R}\rangle=\begin{pmatrix}0\\\
\kappa_{R}\\\ \end{pmatrix}\,$ (2.5)

the gauge symmetry breaks down to $SU(3)_{c}\times U(1)_{\rm em}$. A hierarchy
of VEVs is necessary with $\kappa_{L}\ll\kappa_{R}$, so that in the first
stage of symmetry breaking the left-right symmetric gauge group breaks down to
the standard model symmetry. This hierarchy can be realized with a Higgs
potential that breaks parity symmetry softly via dimension-two mass terms:

$\displaystyle
V=-(\mu_{L}^{2}\chi_{L}^{\dagger}\chi_{L}+\mu_{R}^{2}\chi_{R}^{\dagger}\chi_{R})+\frac{\lambda_{1}}{2}\left\\{(\chi_{L}^{\dagger}\chi_{L})^{2}+(\chi_{R}^{\dagger}\chi_{R})^{2}\right\\}+\lambda_{2}(\chi_{L}^{\dagger}\chi_{L})(\chi_{R}^{\dagger}\chi_{R})~{}.$
(2.6)

Soft breaking of parity symmetry occurs when $\mu_{L}^{2}\neq\mu_{R}^{2}$. We
shall assume that this is the only source of explicit $P$-violation.
$\mu_{L}^{2}\neq\mu_{R}^{2}$ is necessary for phenomenology when the
$SU(2)_{R}$ breaking scale is in the multi-TeV range, otherwise minimization
of the tree-level Higgs potential would lead to $\kappa_{L}=\kappa_{R}$, or
$\kappa_{L}\kappa_{R}=0$, neither of which is acceptable phenomenologically.
When the $SU(2)_{R}$ breaking scale is of order $10^{12}$ GeV, it has been
shown that such soft parity breaking is not required for consistent
phenomenology Hall:2018let .

Note that by separate $SU(2)_{L}$ and $SU(2)_{R}$ gauge rotations the VEVs
$\kappa_{L}$ and $\kappa_{R}$ of Eq. (2.6) can be made real, which plays in
important role in solving the strong CP problem with parity symmetry. This is
in contrast to the conventional left-right symmetric theories where the VEVs
of the bidoublet scalar used cannot be made real.

The masses of the charged $W_{L,R}^{\pm}$ gauge bosons are given by

$M_{W_{L,R}^{\pm}}^{2}=\frac{1}{2}g^{2}\kappa_{L,R}^{2}\,$ (2.7)

with no tree-level mixing among the two. Here $g_{L}=g_{R}\equiv g$ has been
assumed for the two $SU(2)$ gauge couplings, owing to parity symmetry. The
neutral gauge boson mixing matrix can be found in Ref. Babu:2018vrl , and will
not be needed for our analysis.

The most general Yukawa interactions of quark and leptons with the Higgs
fields of the model are given by

$\displaystyle\mathcal{L}$ $\displaystyle=y_{u}\
(\bar{Q}_{L}\tilde{\chi}_{L}+\bar{Q}_{R}\tilde{\chi}_{R})P+y_{d}\
(\bar{Q}_{L}\chi_{L}+\bar{Q}_{R}\chi_{R})N$ $\displaystyle+y_{\ell}\
(\bar{\Psi}_{L}\chi_{L}+\bar{\Psi}_{R}\chi_{R})E+h.c.$ (2.8)

where $\tilde{\chi}_{L,R}=i\tau_{2}\chi^{*}_{L,R}$. Here we have imposed
parity symmetry with $Q_{L}\leftrightarrow Q_{R}$,
$\Psi_{L}\leftrightarrow\Psi_{R}$, $P_{L}\leftrightarrow P_{R}$,
$N_{L}\leftrightarrow N_{R}$, $E_{L}\leftrightarrow E_{R}$ and
$\chi_{L}\leftrightarrow\chi_{R}$, which relates various Yukawa coupling
matrices. In addition, bare mass terms for the vector-like fermions are
allowed:

$\mathcal{L}_{\rm mass}=M_{p^{0}}\ \bar{P}P+M_{N^{0}}\ \bar{N}N+M_{E^{0}}\
\bar{E}E\,.$ (2.9)

Parity symmetry implies that $M_{P^{0}}$, $M_{N^{0}}$, and $M_{E^{0}}$ are all
hermitian matrices. Thus, the $6\times 6$ fermion mass matrices for the up-
and down-type quarks ($M_{u}$ and $M_{d}$), and charged leptons ($M_{\ell}$)
are given by (in a notation where $(\overline{u}_{L},~{}\overline{P}_{L}$)
multiplies the mass matrix $M_{u}$ from the left and $(u_{R},~{}P_{R})^{T}$
multiplies it from the right, and so forth):

$\displaystyle M_{u}=\begin{pmatrix}0&y_{u}\,\kappa_{L}\\\
y_{u}^{\dagger}\,\kappa_{R}&M_{p^{0}}\end{pmatrix}\,,\hskip
14.22636ptM_{d}=\begin{pmatrix}0&y_{d}\,\kappa_{L}\\\
y_{d}^{\dagger}\,\kappa_{R}&M_{N^{0}}\end{pmatrix}\,,\hskip
14.22636ptM_{\ell}=\begin{pmatrix}0&y_{\ell}\,\kappa_{L}\\\
y_{\ell}^{\dagger}\,\kappa_{R}&M_{E^{0}}\end{pmatrix}\,.$ (2.10)

This form of the quark mass matrices has the virtue that parity symmetry alone
can solve the strong CP problem, without the need for an axion. Note that
$\theta_{QCD}=0$ due to parity, and from Eq. (2.10) it follows that
Arg{Det$(M_{u}M_{d})$} = 0, which implies $\overline{\theta}=0$ at tree level.
It has been shown in Ref. Babu:1989rb  that the one-loop corrections to
$\overline{\theta}$ also vanish within this model. A small nonzero
$\overline{\theta}$ will be induced at the two-loop level, which turns out to
be sufficiently small to be consistent with the experimental limits on the
neutron electric dipole moment arising from the strong CP parameter for most
of the parameter space of the model.

The mass matrices of Eq. (2.10) can be block-diagonalized in the approximation
$M^{0}\gg y\kappa_{R}\gg y\kappa_{L}$ to obtain the following light fermion
mass matrices:

$m_{u,d}=(y_{u,d}\ M_{P^{0},N^{0}}^{-1}\
y_{u,d}^{\dagger})\,\kappa_{L}\kappa_{R}\,,\hskip
14.22636ptm_{\ell}=(y_{\ell}M_{E^{0}}^{-1}y_{\ell}^{\dagger})\,\kappa_{L}\kappa_{R}\,.$
(2.11)

This is the generalized seesaw mechanism for charged fermion masses in this
framework. If one ignores family mixing, these mass formulas become

$m_{u_{i}}\approx\frac{y_{u_{i}}^{2}\kappa_{L}\kappa_{R}}{M_{P_{i}^{0}}},~{}~{}~{}~{}m_{d_{i}}\approx\frac{y_{d_{i}}^{2}\kappa_{L}\kappa_{R}}{M_{N_{i}^{0}}},~{}~{}~{}~{}m_{\ell_{i}}\approx\frac{y_{\ell_{i}}^{2}\kappa_{L}\kappa_{R}}{M_{E_{i}^{0}}},$
(2.12)

which shows the light fermion masses depending quadratically on the Yukawa
couplings. As a result, a milder hierarchy in the Yukawa couplings
($y_{i}=10^{-3}-1)$ is sufficient to explain the observed mass hierarchies as
compared to the range $(10^{-6}-1)$ needed in the standard model. We shall use
the expressions of Eq. (2.11), rather than the crude approximations of Eq.
(2.12) in our numerical study.

The approximation $y_{i}\kappa_{R}\ll M^{0}$ is very good for all the charged
fermion masses, except for the top quark. Ignoring the mixings of the top
quark with other families, one can write more exact expressions for the
physical top-partner mass (which we shall denote simply as $M_{P}$) and the
top quark mass:

$m_{t}\,M_{P}=y_{t}^{2}\,\kappa_{L}\kappa_{R},~{}~{}~{}M_{P}=\sqrt{M_{P^{0}}^{2}+|y_{t}\kappa_{R}|^{2}}~{}.$
(2.13)

This suggests the definition of a parameter $r$ with its range $0\leq r\leq 1$
as follows:

$M_{P^{0}}=r\ M_{P},\hskip
28.45274ptr=\sqrt{1-\left|\frac{m_{t}}{M_{p}}\right|\frac{M_{W_{R}}}{M_{W_{L}}}}\,.$
(2.14)

We shall make use of Eq. (2.14) in our numerical study of the Dirac neutrino
mass within these models. In the bottom quark sector we shall denote the
bottom-partner mass to be simply $M_{N}$. (The other vector-like quark masses
do not enter our discussions of neutrino masses.)

Note that the neutrino masses remain zero at tree level in these models. There
is no vector-like neutral lepton that could have induced tree-level masses for
the neutrinos. Neutrino masses do not remain zero at the quantum level, small
Dirac masses are generated via two-loop diagrams within this framework, which
we discuss in the next section.

##  3 Dirac Neutrino Masses as Two-loop Radiative Corrections

(a) (b)

Figure 1: Left: One-loop finite Dirac neutrino mass generated via the mixing
shown in Fig. (b). Right: One-loop diagram that induces mixing between
$W_{L}^{+}-W_{R}^{+}$.

The gauge bosons $W_{L}^{\pm}$ and $W_{R}^{\pm}$ do not mix at the tree level
in this class of models which employs Higgs doublet for symmetry breaking.
However, such a mixing, parameterized by an angle $\xi$, is induced at one-
loop level as shown in the Fig. 1 (b) with internal top and bottom quarks.
This mixing angle, which is finite, is approximately given by

$\xi\approx\frac{\alpha}{4\pi\sin^{2}\theta_{W}}\frac{m_{b}m_{t}}{M_{W_{R}}^{2}}\,.$
(3.15)

Note that $\xi$ is sensitive to only the top and bottom quark masses, and not
to the lighter quark masses. Once a nonzero $\xi$ is induced, the two-loop
$W_{L}^{\pm}-W_{R}^{\pm}$ diagram of Fig. 1 (a) would induce small and finite
Dirac masses for the neutrinos. The momentum of the $W_{L}^{\pm}$ and
$W_{R}^{\pm}$ need to be kept nonzero – unlike in the estimation of the
$W_{L}^{\pm}-W_{R}^{\pm}$ mixing angle $\xi$. Keeping this in mind, the
structure of the induced Dirac neutrino mass matrix is given by Babu:1988yq

$M_{\nu^{D}}=\frac{-g^{4}}{2}y_{t}^{2}y_{b}^{2}y_{\ell}^{2}\kappa_{L}^{3}\kappa_{R}^{3}\frac{r\
M_{P}M_{N}M_{E_{\ell}}}{M_{W_{L}}^{2}M_{W_{R}}^{2}}\ I_{E_{\ell}}\,,$ (3.16)

where the parameter $r$ is defined in Eq. (2.14). Here we have worked in a
basis where the vector-like lepton mass matrix is diagonal, but we allow for
the light charged fermions to have a general mass matrix. This is a consistent
approach since we treat the light charged fermion masses perturbatively. The
flavor-dependent integrals $I_{E_{\ell}}$ of Eq. (3.16) (where $\ell=1,2,3$
denote the heavy vector-like lepton flavor) are given by Babu:1988yq

$\displaystyle I_{E_{\ell}}$
$\displaystyle=\int\int\frac{d^{4}kd^{4}p}{(2\pi)^{8}}\frac{3M_{W_{L}}^{2}M_{W_{R}}^{2}+(p^{2}-M_{W_{L}}^{2})(p^{2}-M_{W_{R}}^{2})}{k^{2}(p+k)^{2}(k^{2}-M_{N}^{2})((p+k)^{2}-M_{p}^{2})p^{2}(p^{2}-M_{E_{\ell}}^{2})(p^{2}-M_{W_{L}}^{2})(p^{2}-M_{W_{R}}^{2})}$
$\displaystyle=I_{1\ell}+I_{2\ell}$ (3.17)

with

$\displaystyle I_{1\ell}$
$\displaystyle=\int\int\frac{d^{4}kd^{4}p}{(2\pi)^{8}}\frac{3M_{W_{L}}^{2}M_{W_{R}}^{2}}{k^{2}(p+k)^{2}(k^{2}-M_{N}^{2})((p+k)^{2}-M_{p}^{2})p^{2}(p^{2}-M_{E_{\ell}}^{2})(p^{2}-M_{W_{L}}^{2})(p^{2}-M_{W_{R}}^{2})}~{},$
(3.18) $\displaystyle I_{2\ell}$
$\displaystyle=\int\int\frac{d^{4}kd^{4}p}{(2\pi)^{8}}\frac{1}{k^{2}(p+k)^{2}(k^{2}-M_{N}^{2})((p+k)^{2}-M_{p}^{2})p^{2}(p^{2}-M_{E_{\ell}}^{2})}~{}.$
(3.19)

Here $M_{P}$ and $M_{N}$ are the physical masses of the top-quark partner and
the bottom-quark partner respectively.

Now we turn to evaluating the loop integral $I_{1_{\ell}}$ and $I_{2_{\ell}}$
of Eq. (3.18) and Eq. (3.19). For this purpose we define the following
parameters which would enable us to express the integrals in terms of mass
ratios:

$r_{1_{\ell}}=\frac{M_{N}^{2}}{M_{E_{\ell}}^{2}}\,,\hskip
28.45274ptr_{2_{\ell}}=\frac{M_{P}^{2}}{M_{E_{\ell}}^{2}}\,,\hskip
28.45274ptr_{3_{\ell}}=\frac{M_{W_{R}}^{2}}{M_{E_{\ell}}^{2}}\,,\hskip
28.45274ptr_{4_{\ell}}=\frac{M_{W_{L}}^{2}}{M_{E_{\ell}}^{2}}\,.$ (3.20)

We shall drop the subscript $\ell$ in these $r_{i_{\ell}}$, for brevity, but
it should be kept in mind that these $r_{i}$ factors have flavor dependence
(although ratios of $r_{i}$ are flavor-independent). We follow the procedure
outlined in Ref. vanderBij:1983bw ; Broadhurst:1987ei ; Ghinculov:1994sd ;
Usyukina:1994eg ; McDonald:2003zj ; Babu:2020bgz  to evaluate these integrals
analytically. We first simplify these integrals by partial fractions
tHooft:1972tcz  and performs integration by parts to obtain the analytical
results. The total integral $I_{E}$ then can be written down as

$\displaystyle I_{E}$ $\displaystyle=I_{1}+I_{2}$
$\displaystyle=\frac{1}{(16\pi^{2})^{2}}\frac{1}{M_{N}^{2}M_{P}^{2}}\left[G_{1}(r_{1},r_{2},r_{3},r_{4})+G_{2}(r_{1},r_{2})\right]\,,$
(3.21)

where the functions $G_{1}$ and $G_{2}$ correspond to integrals $I_{1}$ and
$I_{2}$, respectively. The loop function $G_{1}(r_{1},r_{2},r_{3},r_{4})$ is
found to be

$\displaystyle G_{1}(r_{1},$ $\displaystyle
r_{2},r_{3},r_{4})=\frac{3}{(r_{3}-1)(r_{4}-1)(r_{4}-r_{3})}\Bigg{[}-\frac{\pi^{2}}{6}(r_{1}+r_{2})(r_{3}-1)(r_{3}-r_{4})(r_{4}-1)$
(3.22)
$\displaystyle+r_{3}r_{4}(r_{4}-r_{3})\left(r_{1}F\left[\frac{1}{r_{1}},\frac{r_{2}}{r_{1}}\right]+r_{2}F\left[\frac{1}{r_{2}},\frac{r_{1}}{r_{2}}\right]+F\left[r_{1},r_{2}\right]\right)$
$\displaystyle-(r_{4}-1)r_{4}\left(r_{1}F\left[\frac{r_{3}}{r_{1}},\frac{r_{2}}{r_{1}}\right]+r_{2}F\left[\frac{r_{3}}{r_{2}},\frac{r_{1}}{r_{2}}\right]+r_{3}F\left[\frac{r_{1}}{r_{3}},\frac{r_{2}}{r_{3}}\right]\right)$
$\displaystyle+(r_{3}-1)r_{3}\left(r_{1}F\left[\frac{r_{4}}{r_{1}},\frac{r_{2}}{r_{1}}\right]+r_{2}F\left[\frac{r_{4}}{r_{2}},\frac{r_{1}}{r_{2}}\right]+r_{4}F\left[\frac{r_{1}}{r_{4}},\frac{r_{2}}{r_{4}}\right]\right)$
$\displaystyle+(r_{3}-r_{4})(r_{3}-1)(r_{4}-1)\left(r_{2}Li_{2}\left[1-\frac{r_{1}}{r_{2}}\right]+r_{1}Li_{2}\left[1-\frac{r_{2}}{r_{1}}\right]\right)$
$\displaystyle+r_{3}r_{4}(r_{3}-r_{4})\left(Li_{2}[1-r_{1}]+Li_{2}[1-r_{2}]+r_{1}Li_{2}\left[\frac{r_{1}-1}{r_{1}}\right]+r_{2}Li_{2}\left[\frac{r_{2}-1}{r_{2}}\right]\right)$
$\displaystyle+r_{4}(r_{4}-1)\left(r_{3}Li_{2}\left[1-\frac{r_{1}}{r_{3}}\right]+r_{3}Li_{2}\left[1-\frac{r_{2}}{r_{3}}\right]+r_{1}Li_{2}[1-\frac{r_{3}}{r_{1}}]+r_{2}Li_{2}[1-\frac{r_{3}}{r_{2}}]\right)$
$\displaystyle-
r_{3}(r_{3}-1)\left(r_{4}Li_{2}\left[1-\frac{r_{1}}{r_{4}}\right]+r_{4}Li_{2}\left[1-\frac{r_{2}}{r_{4}}\right]+r_{1}Li_{2}[1-\frac{r_{4}}{r_{1}}]+r_{2}Li_{2}[1-\frac{r_{4}}{r_{2}}]\right)\Bigg{]}\,.$

And the loop function $G_{2}(r_{1},r_{2})$ is evaluated to be

$\displaystyle G_{2}(r_{1},r_{2})$
$\displaystyle=-\frac{\pi^{2}}{6}(r_{1}+r_{2}+1)-F\left[r_{1},r_{2}\right]+Li_{2}\left[1-r_{1}\right]+Li_{2}\left[1-r_{2}\right]$
$\displaystyle~{}~{}~{}+r_{1}\left(-F\left[\frac{1}{r_{1}},\frac{r_{2}}{r_{1}}\right]+Li_{2}\left[1-\frac{1}{r_{1}}\right]+Li_{2}\left[1-\frac{r_{2}}{r_{1}}\right]\right)$
$\displaystyle~{}~{}~{}+r_{2}\left(-F\left[\frac{1}{r_{2}},\frac{r_{1}}{r_{2}}\right]+Li_{2}\left[1-\frac{1}{r_{2}}\right]+Li_{2}\left[1-\frac{r_{1}}{r_{2}}\right]\right)\,.$
(3.23)

The dilogarithm function $\mathrm{Li}_{2}(x)$ is defined as

$\mathrm{Li}_{2}(x)=-\int_{0}^{x}\frac{\log(1-y)}{y}dy\,,$ (3.24)

and the function $F(a,b)$ is defined as

$\displaystyle F(a,b)=$ $\displaystyle-\frac{1}{2}\log a\log
b-\frac{1}{2}\left(\frac{a+b-1}{\sqrt{\Delta}}\right)\Bigg{\\{}\operatorname{Li}_{2}\left(\frac{-x_{2}}{y_{1}}\right)+\operatorname{Li}_{2}\left(\frac{-y_{2}}{x_{1}}\right)-\operatorname{Li}_{2}\left(\frac{-x_{1}}{y_{2}}\right)$
$\displaystyle-\operatorname{Li}_{2}\left(\frac{-y_{1}}{x_{2}}\right)+\operatorname{Li}_{2}\left(\frac{b-a}{x_{2}}\right)+\operatorname{Li}_{2}\left(\frac{a-b}{y_{2}}\right)-\operatorname{Li}_{2}\left(\frac{b-a}{x_{1}}\right)-\operatorname{Li}_{2}\left(\frac{a-b}{y_{1}}\right)\Bigg{\\}}\,,$
(3.25)

where $\Delta=\left(1-2(a+b)+(a-b)^{2}\right)$ and

$\displaystyle x_{1}=\frac{1}{2}(1+b-a+\sqrt{\Delta}),$ $\displaystyle\hskip
14.22636ptx_{2}=\frac{1}{2}(1+b-a-\sqrt{\Delta})\,,$ $\displaystyle
y_{1}=\frac{1}{2}(1+a-b+\sqrt{\Delta}),$ $\displaystyle\hskip
14.22636pt{y_{2}=\frac{1}{2}(1+a-b-\sqrt{\Delta})}\,.$ (3.26)

By explicit symmetrization under $a\leftrightarrow b$ and using the relations

$\displaystyle{\rm Li}_{2}(1-z)$ $\displaystyle=-\mathrm{Li}_{2}(z)-\log
z\log(1-z)+\frac{1}{6}\pi^{2}\,,$ $\displaystyle{\rm
Li}_{2}\left(\frac{1}{z}\right)$
$\displaystyle=-\mathrm{Li}_{2}(z)-\frac{1}{2}\log^{2}(-z)-\frac{1}{6}\pi^{2}\,,$
(3.27)

the function $F(a,b)$ can be simplified to:

$\displaystyle F(a,b)=-\frac{1}{2}\log a\log b$ $\displaystyle-$
$\displaystyle\left(\frac{a+b-1}{\sqrt{\Delta}}\right)\bigg{\\{}\mathrm{Li}_{2}\left(\frac{-x_{2}}{y_{1}}\right)+\mathrm{Li}_{2}\left(\frac{-y_{2}}{x_{1}}\right)+\frac{1}{4}\log^{2}\frac{x_{2}}{y_{1}}$
(3.28)
$\displaystyle+\frac{1}{4}\log^{2}\frac{y_{2}}{x_{1}}+\frac{1}{4}\log^{2}\frac{x_{1}}{y_{1}}-\frac{1}{4}\log^{2}\frac{x_{2}}{y_{2}}+\frac{\pi^{2}}{6}\bigg{\\}}\,.$

It is to be noted that function $F(a,b)$ in Eq. (3.28) can have a non-zero
imaginary part Coleman:1965xm ; McDonald:2003zj . However, these imaginary
components cancel out with judicious logarithmic branch cut choice. The real
part of Eq. (3.28) is in full agreement with Eq. (3.25).

###  3.1 Evaluation of the loop integrals

Figure 2: Left: Integral function $G_{\rm TOT}=G_{1}+G_{2}$ vs $G_{i}\
(i=1,2)$. Right: contribution from $G_{2}(r_{1},r_{2})$ as a function of mass
ratio $r_{1}$ for three different choices $r_{2}=100\ r_{1}$ (red),
$r_{2}=r_{1}$ (blue), and $r_{2}=0.01\ r_{1}$ (green).

After scanning the parameters of the model numerically imposing the
phenomenolgical constraints that $M_{W_{R}^{\pm}}\geq 4$ TeV, and $0\leq r\leq
1$ in Eq. (2.14), we find that the function $G_{2}$ dominates $G_{1}$ over the
entire the parameter space. This is shown in Fig. 2 (left panel), where we
have plotted $G_{\rm TOT}=G_{1}+G_{2}$ as a function of $G_{1}$ and $G_{2}$.
The blue and the red regions in Fig. 2 respectively represents $G_{1}$ and
$G_{2}$. For any choice of parameters the total contribution $G_{\rm TOT}$ is
approximately equal to $G_{2}$. This feature is not surprising, as $G_{2}$
arises from the exchange of longitudinal $W_{L,R}^{\pm}$, while $G_{1}$
corresponds to contribution from the transverse modes. The Goldstone boson
contributions are explicitly shown in Fig. 3. These contributions to the
neutrino mass do not decouple when the $W_{R}$ mass is taken to infinity, as
can be confirmed by the power counting of Fig. 3. This limit should be
accompanied by taking the vector-like fermion masses to infinity as well, so
that the light fermion masses remain finite. The transverse $W_{L,R}$
contributions to the neutrino masses, on the other hand, would decouple as
$W_{R}$ mass is taken to infinity, thus leading to the result $G_{1}\ll
G_{2}$. We therefore safely ignore the contributions from $G_{1}$ in our
numerical study. Fig. 2 (right panel) shows the contribution from
$G_{2}(r_{1},r_{2})$ as a function of mass ratio $r_{1}$ for three different
choices of $r_{2}$: $r_{2}=100\ r_{1}$ (red), $r_{2}=r_{1}$ (blue), and
$r_{2}=0.01\ r_{1}$ (green). We also note that our results agree with a
Feynman parametric integral representations given in Ref. Babu:1988yq , upon
numerically integrating those functions.

It is useful to find the asymptotic behavior of the function
$G_{2}(r_{1},r_{2})$. Following the approximations developed in Ref.
vanderBij:1983bw , we find the leading asymptotic behavior of $G_{2}$ in
various limits to be:

$G_{2}(r_{1},r_{2})\ \rightarrow\
\left\\{\begin{array}[]{ll}\frac{-\pi^{2}}{6}(r_{1}+r_{2}-2r_{1}r_{2})-r_{1}r_{2}(1-\log
r_{1}\log r_{2})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r_{2}Li_{2}\left[1-\frac{r_{1}}{r_{2}}\right]+r_{1}Li_{2}\left[1-\frac{r_{2}}{r_{1}}\right]&\hskip
22.76219pt\text{ for }\ r_{1},r_{2}\ll 1\,,\\\\[10.0pt]
\frac{-\pi^{2}}{6}(1+r_{2})+Li_{2}\left[1-r_{2}\right]+r_{2}\
Li_{2}\left[1-\frac{1}{r_{2}}\right]&\hskip 22.76219pt\text{ for }\ r_{2}\ll
r_{1}\gg 1\,,\\\\[10.0pt] -(2+\frac{1}{4}\log r+\frac{1}{2}\log^{2}r)&\hskip
22.76219pt\text{ for }\ r_{1}\simeq r_{2}=r\gg 1.\end{array}\right.$ (3.29)

Furthermore, in the limit of $r_{1},r_{2}\ll 1$ and $r_{2}\ll r_{1}$, the
first expression given in Eq. (3.29) further simplifies to

$G_{2}(r_{1},r_{2})\simeq
r_{2}\left(\frac{\pi^{2}}{3}(r_{1}-1)-(r_{1}+1)+r_{1}\log r_{1}\log
r_{2}+\log\frac{r_{2}}{r_{1}}-\frac{1}{2}\log^{2}r_{2})\right).$ (3.30)

For $r_{1}\ll r_{2}$, the approximation for the function can be obtained by
simply replacing $r_{1}\leftrightarrow r_{2}$. Similarly, in the limit
$r_{2}\ll r_{1}\gg 1$ and $r_{2}\ll 1$, the second expression of Eq. (3.29)
reduces to

$G_{2}(r_{1},r_{2})\simeq-r_{2}\left(1+\frac{\pi^{2}}{3}+\frac{1}{2}\log
r_{2}(-1+\log r_{2})\right)\,.$ (3.31)

Figure 3: The longitudinal $W_{L,R}^{\pm}$ gauge boson contributions to the
neutrino mass, expressed in terms of the Goldstone boson contributions.

##  4 Fits to Neutrino Oscillation Data

Oscillation |  $3\sigma$ range | Model prediction
---|---|---
parameters |  NuFit5.1 Esteban:2020cvm  | BP I (NH) | BP II (NH) | BP III (IH) | BP IV (IH)
$\Delta m_{21}^{2}(10^{-5}~{}{\rm eV}^{2}$) | 6.82 - 8.04 | 7.42 | 7.38 | 7.35 | 7.35
$\Delta m_{23}^{2}(10^{-3}~{}{\rm eV}^{2})$(IH) | 2.410 - 2.574 | - | - | 2.48 | 2.52
$\Delta m_{31}^{2}(10^{-3}~{}{\rm eV}^{2})$(NH) | 2.43 - 2.593 | 2.49 | 2.51 | - | -
$\sin^{2}{\theta_{12}}$ | 0.269 - 0.343 | 0.324 | 0.301 | 0.306 | 0.310
$\sin^{2}{\theta_{23}}$ (IH) | 0.410 - 0.613 | - | - | 0.510 | 0.550
$\sin^{2}{\theta_{23}}$ (NH) | 0.408 - 0.603 | 0.491 | 0.533 | - | -
$\sin^{2}{\theta_{13}}$ (IH) | 0.02055 - 0.02457 | - | - | 0.0219 | 0.0213
$\sin^{2}{\theta_{13}}$(NH) | 0.02060 - 0.02435 | 0.0234 | 0.0213 | - | -
$\delta_{\rm CP}$ (IH) | 192 - 361 | - | - | $236^{\circ}$ | $279^{\circ}$
$\delta_{\rm CP}$ (NH) | 105 - 405 | $199^{\circ}$ | $280^{\circ}$ | - | -
$m_{\rm light}$ $(10^{-3})$ eV | 0.66 | 2.04 | 14.1 | 8.50
$M_{E_{1}}/M_{W_{R}}$ | 917 | 45.5 | 1936 | 1990
$M_{E_{2}}/M_{W_{R}}$ | 0.650 | 0.43 | 0.12 | 0.11
$M_{E_{3}}/M_{W_{R}}$ | 0.019 | 0.029 | 0.015 | 0.012
Table I: Fits to the neutrino oscillation parameters in the model with normal and inverted hierarchy. For comparison, the $3\sigma$ allowed range for the oscillation parameters are also given. | $M_{N}/M_{W_{R}}$ | $y_{1}$ | $y_{2}$ | $y_{3}$ | $\phi_{12}$ | $\phi_{13}$ | $\phi_{23}$ | $\varphi$ |  r ($10^{-3}$ )
---|---|---|---|---|---|---|---|---|---
BP I (NH) | 50.4 | 0.0051 | 0.012 | 0.076 |  23.8∘ |  31.5∘ |  12.4∘ |  80.2∘ | 0.174
BP II (NH) | 59.9 | 0.02 | 0.005 | 0.01 |  67.2∘ |  49.0∘ |  20.3∘ |  87.1∘ | $0.264$
BP III (IH) | 41.5 | 0.052 | 0.0053 | 0.0095 |  4.59∘ |  4.25∘ |  17.2∘ |  5.79∘ | 2.17
BP IV (IH) | 41.4 | 0.053 | 0.0047 | 0.0091 |  5.2∘ |  4.8∘ |  21.1∘ |  6.28∘ | 2.14
Table II: Model parameters that give neutrino oscillation fit of Table I. Here
$M_{P}/M_{W_{R}}\simeq 2.0$.

In this section, we show that the model can correctly reproduce the neutrino
oscillation data by fitting the model parameters to the observables ($\Delta
m_{21}^{2},\,\Delta
m_{31}^{2},\,\sin^{2}\theta_{13},\,\sin^{2}\theta_{23},\,\sin^{2}\theta_{12}$).
We find good fits to the normal ordering of neutrino masses as well as
inverted ordering. Furthermore, the model does not place any restriction of
the CP violating phase in neutrino oscillation, as we find fits for any value
of $\delta_{CP}$.

The Dirac neutrino mass matrix arising from the two-loop diagrams has a
structure given in Eq. (3.16). Upon evaluation of the loop integral $I_{E}$,
this matrix can be written as

$M_{\nu^{D}}=y_{\ell}\ M_{E}I_{E}\ y_{\ell}^{\dagger}\,,$ (4.32)

where $I_{E}$ is a dimensionless function that contains all the relevant
factors and the loop functions:

$I_{E}=-\frac{g^{2}}{(16\pi^{2})^{2}}\frac{m_{t}m_{b}}{M_{W_{L}}M_{W_{R}}}\ r\
\left[G_{1}(r_{1},r_{2},r_{3},r_{4})+G_{2}(r_{1},r_{2})\right]\,,$ (4.33)

In obtaining the form of Eq. (4.33) we have used Eq. (2.7), Eq. (2.11), as
well as Eq. (2.14).

The charged lepton mass matrix is given by

$\displaystyle m_{\ell}$
$\displaystyle=(y_{\ell}\,M_{E^{0}}^{-1}\,y_{\ell}^{\dagger})\,\kappa_{L}\kappa_{R}\,.$
(4.34)

We wish to stay in a basis where the charged lepton mass matrix is diagonal.
For this purpose, we note that the heavy lepton mass matrix $M_{E^{0}}$ can be
taken to be diagonal and real without loss of generality. In the seesaw
approximation, the matrix is very nearly equal to the physical vector-like
lepton mass matrix $M_{E}={\rm diag}(M_{E_{1}},\,M_{E_{2}},\,M_{E_{3}})$. We
can also parametrize $y_{\ell}$ in the same basis as

$y_{\ell}=y_{D}\,V^{\dagger}$ (4.35)

where $V$ is some unitary matrix and $y_{D}={\rm
diag}(y_{1},\,y_{2},\,y_{3})$. The hermitian matrix $m_{\ell}$ of Eq. (4.34)
can be diagonalized to obtain the physical charged lepton mass matrix
$M_{\ell}$ as

$m_{\ell}=UM_{\ell}U^{\dagger}$ (4.36)

with $M_{\ell}={\rm diag}(m_{e},\,m_{\mu},\,m_{\tau})$.

We can now invert Eq. (4.34) and solve for the matrix $M_{E}$ (which at the
leading order is the same as $M_{E^{0}}$). Doing so we obtain

$M_{E}=\frac{2M_{W_{L}}M_{W_{R}}}{g^{2}}\left(Vy_{D}UM_{\ell}^{-1}U^{\dagger}y_{D}V^{\dagger}\right)\,.$
(4.37)

For the matrix $M_{E}$ to be diagonal, the unitary matrix $V$ appearing in Eq.
(4.37) will be determined once the other entries of this equation are
specified. In addition to $y_{D}={\rm diag}(y_{1},\,y_{2},\,y_{3})$, we must
specify the elements of the unitary matrix $U$. If we write $U=P.\hat{U}.Q$,
where $P$ and $Q$ are diagonal phase matrices, $P$ can be absorbed into $V$
and $Q$ disappears from Eq. (4.37). This suggests that $U$ in Eq. (4.37) can
be parametrized in terms of three mixing angles and one phase, very much like
the standard form of the CKM matrix. We denote the three angles of $U$ to be
($\phi_{12}$, $\phi_{13}$, $\phi_{23}$) and the single phase of $U$ to be
$\varphi$ in the standard CKM-like parametrization of $U$. Thus, in a basis
where the charged lepton mass matrix is diagonal, the Dirac neutrino mass
matrix of Eq. (4.32) takes the form

$M_{\nu^{D}}=-\frac{2m_{t}m_{b}}{(16\pi^{2})^{2}}\
r\left(U^{\dagger}y_{D}^{2}UM_{\ell}^{-1}U^{\dagger}y_{D}V^{\dagger}G_{2}(r_{1_{\ell}},r_{2_{\ell}})Vy_{D}U\right)\,.$
(4.38)

The neutrino mass matrix given by Eq. (4.38) is diagonalized by a unitary
transformation

$U^{T}_{\text{PMNS}}M_{\nu}U_{\text{PNMS}}\ =\ \widehat{M}_{\nu}\,,$ (4.39)

where $\widehat{M}_{\nu}$ is the diagonal mass matrix and $U_{\text{PMNS}}$ is
the $3\times 3$ PMNS lepton mixing matrix.

To summarize, we work in a basis where the charged lepton mass matrix is
diagonal. The two-loop neutrino mass diagram is evaluated in a basis where the
vector-like lepton mass matrix is diagonal. The unitary matrix $U$ of Eq.
(4.38) is a function of $(\phi_{12},\,\phi_{13},\,\phi_{23},\,\varphi)$. For
any given choice of $y_{D}={\rm diag}(y_{1},\,y_{2},\,y_{3})$, $U$ and
$M_{W_{R}}$, the eigenvalues of $M_{E}$ are determined via Eq. (4.37). The
unitary matrix $V$ is also determined as the diagonalizing matrix for $M_{E}$,
since we work in a basis where the vector-like lepton mass matrix is diagonal.
Once $M_{E}$ eigenvalues are known, and once $M_{P}$ and $M_{N}$ are
specified, the loop function $G_{2}(r_{1},r_{2})$ is also fixed. Thus the
neutrino mass matrix is determined via Eq. (4.38). The variables of the
neutrino fit are three Yukawa couplings $(y_{1},\,y_{2},\,y_{3})$, three
mixing angles ($\phi_{12},\,\phi_{13},\,\phi_{23})$ and a phase $\varphi$ that
parametrize the unitary matrix $U$, the parameter $r$, and the masses
$M_{P},\,M_{N}$. Note that $M_{W_{R}}$ is not counted as a free parameter in
this set, since it is fixed in terms of the parameter $r$, see Eq. (2.14).
This is a total of 10 parameters that we shall vary in our numerical fits.

We then perform a numerical scan over nine of these ten input parameters,
keeping $M_{P}/M_{W_{R}}\simeq 2.0$ fixed. The best fit values of these
parameters that give consistent fits to neutrino oscillations are given in
Table. II for four benchmark points. We have found good fits for both normal
(NH) and inverted (IH) hierarchy of neutrino masses with the constrained
minimization where the observables are required to be within $3\sigma$ of
their experimentally measured values. Fits to both the NH and the IH are given
in Table I, corresponding to the input parameter given in Table II. These fits
are in excellent agreement with the observed experimental values.

As an independent check of the procedure we adopted, we show that the
benchmark point BPI (and similarly the other benchmark points) correctly
reproduces the SM-charged lepton masses ($m_{e},m_{\mu},m_{\tau}$) and
neutrino masses and mixings given in Table I. Here we fix $M_{W_{R}}=30$ TeV,
$M_{N}=1.51\times 10^{3}$ TeV, and $M_{P}=59.7$ TeV. This leads to the
following values for the integral $I_{E}$, mass matrix $M_{E}$, and Yukawa
matrix $y_{\ell}$:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}I_{E}=-{\rm diag}\ (1.25\times
10^{-16},7.36\times 10^{-12},4.80\times 10^{-11})\,,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}M_{E}={\rm diag}\ (2.75\times
10^{4},19.5,0.57)\ {\rm TeV}\,,$ $\displaystyle y_{\ell}$
$\displaystyle=10^{-2}\begin{pmatrix}0.058i&~{}~{}-0.063-0.041\
i&~{}~{}-0.084+0.497i\\\ 0.036-0.162\ i&~{}~{}-0.37+1.10\ i&~{}~{}-0.18+0.033\
i\\\ 7.48&~{}~{}1.12&~{}~{}-0.77\\\ \end{pmatrix}\,.$ (4.40)

From these values, and using Eq. (2.10) we obtain the correct charged lepton
masses and $6\times 6$ unitary matrix that diagonalizes $M_{\ell}$ of Eq.
(2.10). Furthermore, the Dirac neutrino mass matrix of Eq. (4.32) for this
choice of input parameters is given by

$M_{\nu^{D}}=10^{-1}\ \begin{pmatrix}0.0077&~{}~{}-0.0024+0.0098\
i&~{}~{}-0.0084-0.0156\ i\\\ -0.0024-0.0098\ i&~{}~{}0.194&~{}~{}-0.055+0.173\
i\\\ -0.0084+0.0156\ i&~{}~{}-0.055-0.173\ i&~{}~{}0.39\\\ \end{pmatrix}\,{\rm
eV}.$ (4.41)

The $3\times 3$ sub-block matrix, $U_{\ell}$, of the $6\times 6$ unitary
matrix that diagonalizes $M_{\ell}$, together with the unitary matrix that
diagonalizes $M_{\nu^{D}}$ of Eq. (4.41) gives the desired PMNS matrix,
$U_{\rm PMNS}=U_{\ell}^{*}U_{\nu}$, from which one obtains the mixing angles
identical to those given in Table I corresponding to BPI.

We conclude that the model provides excellent fits to neutrino oscillation
data. Both normal ordering and inverted ordering of neutrino masses are
permitted within the model. The CP violating phase can take a large range of
values, as shown in Table I. We have verified that $\delta_{\rm CP}$ in all
four quadrants are admitted within this framework, both for NH and IH.

##  5 Origin from $SU(5)_{L}\times SU(5)_{R}$

As noted earlier, the fermion spectrum of the model has a natural embedding in
$SU(5)_{L}\times SU(5)_{R}$ unification Davidson:1987mh ; Lee:2016wiy . Under
this symmetry, all left-handed fermions of the SM fit into a ${\bf
10}+{\bf\overline{5}}$ of $SU(5)_{L}$ and all right-handed fermions of the SM
fit into ${\bf 10}+{\bf\overline{5}}$ of $SU(5)_{R}$. The vector-like quarks
and leptons fill these multiplets, with no additional degrees of freedom
allowed. Specifically, if we denote the vector-like fermions as $(U,\,D,\,E)$,
their grouping is given by

$F_{L,R}=\left[\begin{array}[]{c}D_{1}^{c}\\\ D_{2}^{c}\\\ D_{3}^{c}\\\ e\\\
-\nu\end{array}\right]_{L,R}\,\hskip
28.45274ptT_{L,R}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccccc}0&U_{3}^{c}&-U_{2}^{c}&u_{1}&d_{1}\\\
-U_{3}^{c}&0&U_{1}^{c}&u_{2}&d_{2}\\\ U_{2}^{c}&-U_{1}^{c}&0&u_{3}&d_{3}\\\
-u_{1}&-u_{2}&-u_{3}&0&E^{c}\\\
-d_{1}&-d_{2}&-d_{3}&-E^{c}&0\end{array}\right]_{L,R}\,.$ (5.42)

Here the group transformation properties are $F_{L}(\overline{5},1)$,
$F_{R}(1,\overline{5})$, $T_{L}(10,1)$ and $T_{R}(1,10)$. Parity symmetry can
be imposed on these multiplets under which $F_{L}\leftrightarrow F_{R}$ and
$T_{L}\leftrightarrow T_{R}$. Note that there is no vector-like neutral
lepton, which could have mixed with the neutrino. This embedding shows the
naturalness of Dirac neutrinos in the left-right symmetric models that we have
developed here.

Although we shall not build full models based on $SU(5)_{L}\times SU(5)_{R}$
unification here, it is worth noting that the left-right symmetric models we
have studied here arise as a natural intermediate symmetry in these unified
theories. The symmetry breaking chain can proceed as follows:

$\begin{gathered}SU(5)_{L}\times SU(5)_{R}\\\ \downarrow\Lambda\\\
SU(3)_{L}\times SU(2)_{L}\times U(1)_{L}\times SU(3)_{R}\times SU(2)_{R}\times
U(1)_{R}\\\ \downarrow\Lambda_{LR}\\\ SU(3)_{c}\times SU(2)_{L}\times
SU(2)_{R}\times U(1)_{L+R}\\\ \downarrow\kappa_{R}\\\ SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}\\\ \downarrow\kappa_{L}\\\ SU(3)_{c}\times U(1)_{\rm
em}\end{gathered}$ (5.43)

The gauge symmetry below the scale $\Lambda_{LR}$ can be identified as the
left-right symmetry that we have studied. We also note that the $U(1)$
symmetry realized at this scale is $U(1)_{L+R}$, which we have denoted as
$U(1)_{X}$. In principle, without enlarging the fermion spectrum, one could
also gauge the orthogonal $U(1)$ in the left-right symmetric models, which
would commute with the rest of the gauge symmetry.

##  6 Cosmological Tests of the Model in $N_{\rm eff}$

Figure 4: Contributions of three right-handed neutrinos to $\Delta N_{\rm
eff}$ as a function of decoupling temperature $T_{\rm dec}$ (lower horizontal
scale) as well as the $W_{R}$ mass (upper scale). The solid horizontal lines
represents the current $2\sigma$ limit from Planck+ BAO Planck:2018vyg , The
future sensitivity reach of SPT-3G SPT-3G:2014dbx , SO
SimonsObservatory:2019qwx  and CMB-S4 Lee:2013bxd ; Abazajian:2019eic  are
indicated.

If neutrinoless double beta decay is observed experimentally, it would
establish lepton number violation by two units, and in turn that the neutrinos
are Majorana particles Schechter:1981bd . There is another cosmological
observation which can distinguish a Dirac neutrino from a Majorana neutrino in
a setup such as the left-right symmetric theory. The anisotropy in the cosmic
microwave background (CMB) is sensitive to extra radiation density arising
from new light degrees of freedom that were in thermal equilibrium at some
epoch with the standard model plasma. The effect of such light particles is
parametrized as $\Delta N_{\rm eff}$, and is measured in units of extra
neutrino degrees of freedom. The Planck experiment has measured $N_{\rm
eff}=2.99\pm 0.17$ which includes baryon acoustic oscillation data. In the
near future, CMB-S4 Lee:2013bxd ; Abazajian:2019eic  and Simon South Pole
Observatory SimonsObservatory:2019qwx  will explore $N_{\rm eff}$ with a
sensitivity of 0.12, while the proposed SPT-3G SPT-3G:2014dbx  will have a 3
sigma sensitivity of 0.06. Such a precision could be used as a test of Dirac
neutrino masses in contexts such as the left-right symmetric theories
Abazajian:2019oqj .

A relativistic species that decouples from the plasma at a temperature $T_{\rm
dec}$ would contribute to $\Delta N_{\rm eff}$ as

$\Delta N_{\mathrm{eff}}\simeq
0.027\left(\frac{106.75}{g_{\star}\left(T_{\mathrm{dec}}\right)}\right)^{4/3}g_{\rm
eff}$ (6.44)

where $g_{\rm eff}=(7/8)\times(2)\times(3)=21/4$ accounts for the spin degree
of the three $\nu_{R}$ states. Now, as long as the reheat temperature after
inflation is not too low compared to the $W_{R}$ gauge boson mass, the three
$\nu_{R}$ fields would be in thermal equilibrium at a high enough temperature.
The decoupling temperature of the $\nu_{R}$’s can be estimated by equating
their interaction rate with the Hubble expansion rate Nemevsek:2012cd :

$G_{F}^{2}\left(\frac{M_{W_{L}}}{M_{W_{R}}}\right)^{4}T_{\rm
dec}^{5}\approx\sqrt{g^{*}(T_{\rm dec})}\,\frac{T_{\rm dec}^{2}}{M_{\rm Pl}},$
(6.45)

while yields

$T_{\rm dec}\simeq 400~{}\mathrm{MeV}\left(\frac{g_{*}\left(T_{\rm
dec}\right)}{70}\right)^{1/6}\left(\frac{M_{W_{R}}}{5~{}\mathrm{TeV}}\right)^{4/3}~{}.$
(6.46)

We have plotted in Fig. 4 new contributions from the three $\nu_{R}$ fields to
$\Delta N_{\rm efff}$ as a function of $T_{\rm dec}$. Also shown in the figure
is the dependence on the $W_{R}$ gauge boson mass. The solid blue line is a
more exact form of Eq. (6.44) obtained from the numerical results given in
Ref. Abazajian:2019eic . We see that with a sensitivity of 0.06 in $\Delta
N_{\rm eff}$ the Dirac neutrino contributions can be measured or excluded,
under the assumption that the reheat temperature after inflation is not orders
of magnitude below the $W_{R}$ mass scale.

It is worth noting that the idea of Dirac leptogenesis can be adopted in
scenarios with Dirac neutrinos for explaining the baryon asymmetry of the
universe Dick:1999je . Here, even though there is no net lepton asymmetry
induced, heavy particle decays or scatterings could induce equal and opposite
left-handed and right-handed lepton asymmetry. The electroweak sphalerons
would convert part of the left-handed lepton asymmetry, but not the right-
handed lepton asymmetry, to baryon asymmetry. This would lead to a net baryon
asymmetry. Of course, if neutrinos are Dirac particles, baryon number could
still be violated in processes involving only quarks, which could also explain
the observed baryon asymmetry of the universe.

##  7 Realizing Pseudo-Dirac Neutrinos

In any context where the neutrino is naturally a Dirac fermion, there is a
possibility that at a more fundamental level it is a pseudo-Dirac fermion.
This could happen if quantum gravitational corrections, which are expected to
conserve only gauge symmetries, generate tiny Majorana masses for the left-
handed and/or the right-handed neutrinos. In the model presented here, the
leading operators of this type are

$\frac{(\Psi_{L}\Psi_{L}\chi_{L}\chi_{L})}{M_{\rm
Pl}},~{}~{}\frac{(\Psi_{R}\Psi_{R}\chi_{R}\chi_{R})}{M_{\rm Pl}}~{}$ (7.47)

which conserve all gauge quantum numbers and could be induced by quantum
gravity. If the coefficients of these operators are of order one, that would
lead to a Majorana mass for the $\nu_{R}$ of order $10^{-2}$ eV (for
$\kappa_{R}=10$ TeV) and for $\nu_{L}$ of order $10^{-6}$ eV. Such a value
would result in a $\Delta m^{2}$ of order $10^{-3}~{}{\rm eV}^{2}$ for the
squared mass splitting between active and sterile neutrinos. Such values are
excluded from cosmology Barbieri:1989ti ; Enqvist:1990ek  as well as from
solar neutrino data deGouvea:2009fp , which require such mass splittings to be
less than about $10^{-11}~{}{\rm eV}^{2}$. Here we propose a resolution to
this potential problem.

| $Q_{L}$ | $Q_{R}$ | $\psi_{L}$ | $\psi_{R}$ | $P$ | $N$ | $E$ | $\chi_{L}$ | $\chi_{R}$ | $\varphi$
---|---|---|---|---|---|---|---|---|---|---
$U(1)_{X}$ | 1/3 | 1/3 | $-1$ | $-1$ | $4/3$ | $-2/3$ | $-2$ | $1$ | $1$ | 0
$U(1)_{B-L}$ | 1/3 | 1/3 | $-1$ | $-1$ | 1/3 | 1/3 | $-1$ | 0 | 0 | $q$
Table III: Charge assignment for fermions and scalars under $U(1)_{X}$ and
$U(1)_{B-L}$.

As noted in Sec. 2, the model presented has an anomaly-free $U(1)_{B-L}$
symmetry that may be gauged without adding any new fermions. Since quantum
gravity corrections should respect this gauge symmetry, operators given in
Eq.(7.47) will not be induced, as this would violate $(B-L)$. Note that under
the $(B-L)$ symmetry, lepton fields have charge $-1$, while the Higgs doublets
have charge zero. The vector-like quarks have $(B-L)$ charge of $1/3$, while
vector-like leptons have charge $-1$. The full list of $(B-L)$ charges, along
with the $U(1)_{X}$ charges of fermions and scalars are listed in Table III.
Note that with this charge assignment the Yukawa structure of Eq. (2.10)
remains intact even with the gauging of $(B-L)$ symmetry.

If the $(B-L)$ symmetry is gauged and remains unbroken, the neutrinos will be
strictly Dirac fermions. In this case, however, there are stringent limits on
the $(B-L)$ gauge coupling arising from long-range forces. It is preferable
perhaps to break the symmetry spontaneously by the VEV of a singlet scalar
$\varphi$. The $(B_{L})$ charge of $\varphi$ is denoted as $q$, which is a
rational number, $q\equiv m/n$, with $m,\,n$ being integers. $\varphi$ carries
no $U(1)_{X}$ charge, as indicated in Table III.

Once the scalar field $\varphi$ acquires a nonzero VEV, $U(1)_{B-L}$ would
break down to a discrete $Z_{3m}$ subgroup. This can be inferred by making all
the charges under $(B-L)$ to be integers, which can be achieved by multiplying
the charges listed in Table III by $3n$, where the definition $q\equiv m/n$ is
used. With this normalization, the $(B-L)$ charge of $\varphi$ is $3m$, and
consequently $Z_{3m}$ would remain unbroken once $\varphi$ acquires a VEV.
While the operators of Eq. (7.47) would not be induced by quantum
gravitational corrections, operators of the type

$\frac{(\Psi_{L}\Psi_{L}\chi_{L}\chi_{L}\varphi^{k})}{M^{k+1}_{\rm
Pl}},~{}~{}\frac{(\Psi_{R}\Psi_{R}\chi_{R}\chi_{R}\varphi^{k})}{M^{k+1}_{\rm
Pl}}~{}$ (7.48)

could potentially be induced for some integer $k$. However, this would require
the following condition to be satisfied:

$-2+kq=0,~{}~{}{\rm or}~{}~{}\frac{m}{n}=\frac{2}{k}~{}.$ (7.49)

Thus, if the charge of $\varphi$ is not of the form $q=2/k$, no operator that
violates lepton number by two units could be induced by Planck-scale
corrections, and the neutrino would remain strictly a Dirac particle.

It is of interest to inquire possibilities when the $(B-L)$ charge of
$\varphi$ is of the form $q=2/k$. For $k=1$, the operator
$(\Psi_{R}\Psi_{R}\chi_{R}\chi_{R}\varphi/M^{2}_{\rm Pl})$ would be allowed.
For other integer values of $k$, such as $k=2,\,3$, etc, the induced Majorana
mass terms would be more suppressed. The scale of $(B-L)$ symmetry breaking
can be adjusted to obtain any desired mass-splitting between active and
sterile neutrinos. If $\Delta m^{2}$ is in the range of
$(10^{-12}-10^{-20})~{}{\rm eV}^{2}$, such pseudo-Dirac neutrinos may be
tested in ultra high energy neutrinos at IceCube Keranen:2003xd ;
Beacom:2003eu  or in supernova neutrino signals Martinez-Soler:2021unz . The
framework presented therefore provides a natural setup for pseudo-Dirac
neutrinos as well.

##  8 Conclusions

In this paper we have developed further a class of left-right symmetric models
where the neutrinos are Dirac particles with naturally small masses
Babu:1988yq . The gauge symmetry of these models is $SU(3)_{c}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)$ with the particles assigned to the gauge
group in a left-right symmetric fashion. This setup allows to identify parity
as a good symmetry, which is only broken spontaneously and softly. The charged
fermion masses arise in this framework via their mixing with vector-like
quarks and leptons, while the neutrino has no partner to mix with and thus
remains massless at the tree-level. Quantum corrections induce small Dirac
masses for the neutrinos. We have evaluated the leading two-loop diagrams for
the Dirac neutrino masses and shown the consistency of the framework with
neutrino oscillation data. Both normal and inverted ordering of neutrino
masses can be realized and the CP violating phase in neutrino oscillations is
arbitrary.

The structure of the quark mass matrices in this framework is such that it
provides a solution to the strong CP problem based on parity symmetry, without
the need for an axion. The fermions of the model can arise naturally from
$SU(5)_{L}\times SU(5)_{R}$ unification in its minimal version.

We have shown how to preserve lepton number as a good symmetry in the neutrino
mass matrix by gauging $(B-L)$ symmetry that is already anomaly free within
the framework. Upon spontaneous symmetry breaking, a discrete $Z_{N}$ subgroup
of $(B-L)$ persists that prevents any $\Delta L=2$ Planck-induced higher
dimensional operators. For certain values of $N$ of the discrete subgroup
$Z_{N}$, the neutrino may be a pseudo-Dirac particle. The amount of mass
splitting between the active and sterile components of the neutrino is
controlled by the scale of $(B-L)$ symmetry breaking. We have also noted that
these models can be tested in cosmological measurements, especially with high
precision measurement of $\Delta N_{\rm eff}$, which is predicted to be
$\Delta N_{\rm eff}\simeq 0.14$.

## Acknowledgements

We thank Julian Heeck for useful discussions. The work of KSB is supported in
part by the US Department of Energy grant No. DE-SC 0016013. The work of XGH
is supported in part by the MOST grant No. MOST 109-2112-M-002-017-MY3. KSB
thanks KITP for the hospitality where part of this work was done. The research
at KITP was supported in part by the National Science Foundation under Grant
No. NSF PHY-1748958.

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