††institutetext: Department of Physics, Oklahoma State University, Stillwater,
OK 74078, USA

# Resolving $\bm{W}$ Boson Mass Shift and CKM Unitarity Violation in Left-
Right Symmetric Models with Universal Seesaw

K.S. Babu  Ritu Dcruz<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We explore the possibility of resolving simultaneously the $W$-boson mass
shift reported by the CDF collaboration and the apparent deviation from
unitarity in the first row of the CKM mixing matrix in a class of left-right
symmetric models. The fermion masses are generated in these models through a
universal seesaw mechanism, utilizing vector-like partners of the usual
fermions. We find a unique solution to the two anomalies where the mixing of
the down quark with a vector-like quark (VLQ) resolves the CKM unitarity
puzzle, and the mixing of the top quark with a VLQ partner explains the $W$
boson mass shift. The validity of this setup is tested against the stability
of the Higgs potential up to higher energies. We find upper bounds of
$(2,\,4,\,4)$ TeV on the masses of the down-type VLQ, the top-partner VLQ, and
a neutral scalar associated with $SU(2)_{R}$ gauge symmetry breaking,
respectively. This class of models can solve the strong CP problem via parity
symmetry without the need for an axion.

##  1 Introduction

Left-right symmetric models (LRSM) are well-motivated extensions of the
Standard Model (SM) that provide a natural understanding of the origin of
parity violation Pati:1974yy ; Mohapatra:1974gc ; Senjanovic:1975rk . In these
models, the gauge symmetry is enhanced to $SU(3)_{c}\times SU(2)_{L}\times
SU(2)_{R}\times U(1)_{B-L}$, with the left-handed quarks and leptons
transforming as doublets under $SU(2)_{L}$ while the right-handed ones are
doublets of $SU(2)_{R}$. The observed V-A structure of weak interactions is a
low energy manifestation of spontaneous breaking of parity (P) symmetry which
is well defined in LRSM. Apart from providing insight into the origin of
parity violation, these models, owing to the presence of right-handed
neutrinos to complete the gauge multiplets, also explain small neutrino masses
via the seesaw mechanism, either type-I Minkowski:1977sc ; GellMann:1980vs ;
Mohapatra:1979ia ; Yanagida:1980xy ; Glashow:1979nm , or type-II
Mohapatra:1980yp ; Schechter:1980gr ; Cheng:1980qt ; Magg:1980ut . Moreover,
they give a physical interpretation of the hypercharge quantum numbers of
fermions as a quantity arising from the $B-L$ (baryon number minus lepton
number) charges and the third components of the right-handed isospin. A class
of left-right symmetric models also provides a solution to the strong CP
problem via parity symmetry alone, without the need to introduce the axion
Babu:1989rb . It is this class models that is the focus of the present paper.

The strong CP problem could very well be called the strong P problem since the
large neutron electric dipole moment (EDM) induced by the QCD
$\theta$-parameter is odd under parity symmetry. In a left-right symmetric
framework where parity can be defined, the QCD $\theta$-parameter would vanish
owing to P. Even after spontaneous (and soft) symmetry breaking, the quark
flavor contribution to the observable $\overline{\theta}$ can be zero at the
tree-level. In the class of models studied in Ref. Babu:1989rb , it was shown
that $\overline{\theta}$ remains zero even after one-loop radiative
corrections are included. Small and finite $\overline{\theta}$ would arise
only via two-loop diagrams, which are consistent with neutron EDM limits
Babu:1988yq ; Hall:2018let . For early work on addressing the strong CP
problem via parity symmetry see Mohapatra:1978fy ; Beg:1978mt , and for
related work see Barr:1991qx .

The purpose of this paper is to investigate the possibility of explaining two
experimental anomalies, viz., the $W$-boson mass shift and the apparent
violation of unitarity in the first row of the CKM matrix, in the context of
these LRSM models, while maintaining the parity solution to the strong CP
problem.

The masses of quarks and leptons in this class of models arise through a
universal seesaw mechanism Davidson:1987mh ; Babu:1988mw . This is achieved by
introducing vector-like fermions (VLF) Berezhiani:1983hm ; Dimopoulos:1983rz
which are singlets of $SU(2)_{L}$ and $SU(2)_{R}$ gauge symmetries. The SM
fermions mix with these VLFs via Yukawa interactions involving a Higgs doublet
$\chi_{L}$ of $SU(2)_{L}$ and its parity partner $\chi_{R}$, a Higgs doublet
of $SU(2)_{R}$. The scalar sector of the model is thus very minimal,
consisting only of these two Higgs doublets. This minimality of Higgs fields
plays a crucial role in solving the strong CP problem since, via gauge
rotations, their vacuum expectation values (VEVs) can be both made real,
resulting in vanishing contributions to $\overline{\theta}$ from the flavor
sector at the tree-level. The light fermion masses, which are induced via a
generalized seesaw mechanism, are quadratically dependent on the Yukawa
couplings ($\mathcal{Y}_{i}$), allowing for the values of $\mathcal{Y}_{i}$
required to explain fermion mass hierarchy to be in the range
$\mathcal{Y}_{i}=(10^{-3}-1)$ as opposed to $\mathcal{Y}_{i}=(10^{-6}-1)$ in
the SM or in the standard LRSM. This class of models has received considerable
attention recently in the context of flavor physics and low energy
experimental signals Craig:2020bnv , gravitational waves Graf:2021xku ,
neutrino oscillations Babu:2022ikf  and cosmological baryogenesis
Harigaya:2022wzt . High-scale realizations of such models with exact parity
symmetry have been developed in Ref. Hall:2018let ; Dunsky:2019api . It is
noteworthy that the neutrinos can be naturally light Dirac particles, or
pseudo-Dirac particles in this context, with their masses arising from two-
loop diagrams Babu:1988yq ; Babu:2022ikf .

The presence of direct Yukawa couplings between VLF and SM fermions can give
rise to flavor-changing neutral current (FCNC) processes arising at the tree-
level and modify the SM charged current interactions. These deviations from
the SM could potentially be relevant to several experimental anomalies which
have come to light in recent years, prominent among them being the muon
anomalous magnetic moment $(g-2)_{\mu}$ (for a recent review see
Aoyama:2020ynm ), $R_{K^{(*)}}$ (for a review and update see Ref.
Altmannshofer:2021qrr ), $R_{D^{(*)}}$ (see Ref. Altmannshofer:2020oas ), the
CDF $W$-boson mass shift CDF:2022hxs , and the unitarity of the first row of
the CKM matrix, sometimes referred to as the Cabibbo anomaly (for reviews see
Ref. Belfatto:2019swo ; Crivellin:2022ctt ). New physics implied by models
such as LRSM could, in principle, resolve one or more of these anomalies.
However, with a fixed theoretical framework that solves the strong CP problem
and with no room to add extra particles so that the models remain minimal, it
is unsurprising that this class of LRSM models could not resolve all the
anomalies. Nevertheless, we find that the new physics contributions arising
from the vector-like quarks in these models can indeed explain the $W$-boson
mass shift and simultaneously explain the deviation from unitarity in the
first row of the CKM matrix. (For attempts to resolve the $R_{D^{(*)}}$
anomaly in this context see Ref. Babu:2018vrl .) We investigate the model
parameters resulting in a concurrent solution to these two anomalies.

One of the fundamental predictions of the standard model is the unitarity of
the Cabibbo-Kobayashi-Maskawa (CKM) matrix which parametrizes the charged
current weak interaction of the three generations of quarks. Each of the
elements of the CKM matrix is determined by combining experimental results
with theoretical calculations that take care of relevant radiative
corrections. The magnitudes of the CKM elements are well known, with enough
evidence suggesting the complex nature of the matrix. The unitarity condition
is a good consistency check on the otherwise overdetermined matrix. In recent
years, with progress in experimental precision and a better handle on the
theoretical uncertainties, the unitarity of the CKM matrix has been
questioned. Namely, there is a sizeable deviation in the unitarity of the
first row, as a result of the precise determination of $V_{ud}$. There is also
a slightly less significant deviation in the unitarity of the first column.
With a weighted average of $|V_{ud}|=0.97373\pm 0.00009$ Kirk:2020wdk  and the
PDG-recommended average values of ParticleDataGroup:2022pth  $|V_{us}|$ and
$|V_{cd}|$, the non-unitarity appears as follows:

$\displaystyle\Delta_{\text{CKM}}\equiv$ $\displaystyle
1-|V_{ud}|^{2}-|V_{us}|^{2}-|V_{ub}|^{2}$ $\displaystyle=$
$\displaystyle\leavevmode\nobreak\ (1.12\pm 0.28)\times 10^{-3}$
$\displaystyle\leavevmode\nobreak\ (\sim 3.9\,\sigma)\vspace{3mm}$ (1.1)
$\displaystyle\Delta^{\prime}_{\text{CKM}}\equiv$ $\displaystyle
1-|V_{ud}|^{2}-|V_{cd}|^{2}-|V_{td}|^{2}$ $\displaystyle=$
$\displaystyle\leavevmode\nobreak\ (3.0\pm 1.8)\times 10^{-3}$
$\displaystyle\leavevmode\nobreak\ (\sim 1.7\,\sigma).$

These discrepancies are often referred to as the Cabibbo anomaly, a nod to the
Cabibbo mixing of the two-generation model. The Cabibbo anomaly could be a
clear indication of new physics (NP) beyond the standard model (BSM). As we
shall see, the LRSM framework with universal seesaw can resolve this anomaly
when the down quark mixes with one of the VL-down-type quarks.

Another major challenge to the SM has appeared recently with a new high-
precision measurement of the $W$-boson mass. The prediction of $W$-boson mass
in SM depends solely on the mass of the $Z$ and the weak mixing angle, at the
lowest order. Dependence on the gauge couplings and the masses of the top
quark and the Higgs boson seep in as radiative electroweak corrections. These
higher-order corrections have been computed precisely in the SM, allowing a
consistency check against the measured $W$-boson mass. Recently, the CDF
collaboration CDF:2022hxs  at Fermilab has reported the most precise
measurement of the $W$ boson mass so far:

$M_{W}^{\textrm{CDF}}=(80.4335\pm 0.0094)\;\textrm{GeV}\leavevmode\nobreak\
(7\,\sigma\text{ deviation}).$ (1.2)

The CDF measurement deviates from the SM prediction Awramik:2003rn  at
$7\,\sigma$, and is also at odds with the prior PDG world average. If
confirmed, this is another piece of evidence for new physics. The LRSM
framework with universal seesaw provides a unique solution to this deviation
from the mixing of the top quark with one of the VL-up-type quarks.

Several NP models have been explored in the literature in resolving the CKM
unitarity puzzle (see Ref. Crivellin:2022ctt  for a comprehensive list of
references) as well as the CDF $W$-boson mass shift (see Ref. Dcruz:2022dao
and references therein). Among these are solutions to the two anomalies
independently with vector-like fermions Crivellin:2020ebi ; Belfatto:2021jhf ;
Crivellin:2021bkd ; Manzari:2021prf ; Manzari:2020eum ; Belfatto:2019swo ;
Kim:2022zhj ; Nagao:2022oin ; Kawamura:2022uft ; Balkin:2022glu . We explore
the simultaneous resolution of the two anomalies in the framework of LRSM with
universal seesaw, while maintaining parity symmetric solution to the strong CP
problem. This setup is more constraining compared to general VLQ framework
owing to two reasons. First, parity symmetry restricts the form of the mass
matrix, as given in Eq. (2.18) below, and second, possible mixing between the
usual quarks and the VLQs is constrained by the quark masses. We do find a
nontrivial solution with a vector-like down quark of mass below 5 TeV mixing
with the down quark (to resolve the unitarity puzzle) and a top-partner VLQ
with a mass below 4 TeV mixing with the top quark. In addition, a second
scalar field associated with $SU(2)_{R}$ gauge symmetry breaking should be
lighter than about 4 TeV for consistency of the model.

The remainder of the paper is organized as follows. In Sec. 2 we provide a
description of the model, where we recall the parity solution to the strong CP
problem. Sec. 3 is devoted to a discussion of the two anomalies. In Sec 3.1
and Sec. 3.2 we discuss the anomalies associated with $W$-boson mass
measurement and CKM unitarity to establish the parameter space that can
resolve these puzzles simultaneously. The main constraints on these parameters
of the model are discussed in Sec. 3.3, with a unique and concurrent
resolution to both the anomalies and the resulting predictions on the VLQ
masses discussed in Sec. 3.4. Finally, we conclude in Sec. 4.

##  2 Model Description

The particle spectrum of the LRSM with universal seesaw is composed of the
usual SM fermions, right-handed neutrinos, and a set of vector-like fermionic
partners for each of the light fermions denoted as
$(U_{a},D_{a},E_{a},N_{a})$, where, the index $a$ is the family index. The SM
fermions along with the right-handed neutrinos form left- or right-handed
doublets, assigned to the gauge group $SU(3)_{c}\times SU(2)_{L}\times
SU(2)_{R}\times U(1)_{B-L}$ as follows:

$\displaystyle\mathcal{Q}_{L,i}\left(3,2,1,+\frac{1}{3}\right)=\begin{pmatrix}u_{L}\\\
d_{L}\end{pmatrix}_{i},$
$\displaystyle\mathcal{Q}_{R,i}\left(3,1,2,+\frac{1}{3}\right)=\begin{pmatrix}u_{R}\\\
d_{R}\end{pmatrix}_{i},$ (2.3)
$\displaystyle\psi_{L,i}\left(1,2,1,-1\right)=\begin{pmatrix}\nu_{L}\\\
e_{L}\end{pmatrix}_{i},$
$\displaystyle\psi_{R,i}\left(1,1,2,-1\right)=\begin{pmatrix}\nu_{R}\\\
e_{R}\end{pmatrix}_{i},$

with $i=1-3$ being the family index. Here we follow the convention
$Q=T_{3L}+T_{3R}+\frac{B-L}{2}$ such that, $\frac{Y}{2}=T_{3R}+\frac{B-L}{2}$,
thereby giving the hypercharge a physical meaning in terms of the $SU(2)_{R}$
and $U(1)_{B-L}$ quantum numbers. Vector-like fermions (VLFs) which are
singlets under both $SU(2)_{L(R)}$ are introduced to generate masses for the
fermions via a generalized seesaw mechanism:

$\displaystyle U_{a}\left(3,1,1,+\frac{4}{3}\right),$ $\displaystyle
D_{a}\left(3,1,1,-\frac{2}{3}\right),$ $\displaystyle
E_{a}\left(1,1,1,-2\right),$ $\displaystyle N_{a}\left(1,1,1,0\right).$ (2.4)

The Higgs sector of the model is comprised of a left-handed doublet and its
parity partner, a right-handed doublet:

$\displaystyle\chi_{L}\left(1,2,1,+1\right)=\begin{pmatrix}\chi_{L}^{+}\\\
\chi_{L}^{0}\end{pmatrix},$
$\displaystyle\chi_{R}\left(1,1,2,+1\right)=\begin{pmatrix}\chi_{R}^{+}\\\
\chi_{R}^{0}\end{pmatrix}.$ (2.5)

The neutral component of $\chi_{R}$ acquires a vacuum expectation value
$\langle\chi_{R}^{0}\rangle\equiv\kappa_{R}$ at a high scale, breaking the
gauge symmetry down to that of the SM with the neutral component of $\chi_{L}$
acquiring a VEV $\langle\chi_{L}^{0}\rangle\equiv\kappa_{L}\simeq 174$ GeV
leading to the spontaneous breaking of the SM gauge symmetry. The Higgs
potential of the model is given by

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

The physical scalar spectrum $\\{h,H\\}$ arises from the mixing of the neutral
fields $\sigma_{L}=Re(\chi_{L}^{0})/\sqrt{2}$ and
$\sigma_{R}=Re(\chi_{R}^{0})/\sqrt{2}$ with a mass matrix given by

$\mathcal{M}^{2}_{\sigma_{L,R}}=\begin{bmatrix}2\lambda_{1_{L}}\kappa_{L}^{2}&&2\lambda_{2}\kappa_{L}\kappa_{R}\\\
2\lambda_{2}\kappa_{L}\kappa_{R}&&2\lambda_{1_{R}}\kappa_{R}^{2},\end{bmatrix},$
(2.7)

resulting in the mass eigenvalues

$\displaystyle M_{h}^{2}\simeq
2\lambda_{1_{L}}\left(1-\frac{\lambda_{2}^{2}}{\lambda_{1_{L}}\lambda_{1_{R}}}\right)\kappa_{L}^{2},$
$\displaystyle M_{H}^{2}=2\lambda_{1_{R}}\kappa_{R}^{2},$ (2.8)

where in the last step we have assumed the hierarchy
$\kappa_{R}\gg\kappa_{L}$. Here, the field $h\simeq\sigma_{L}$ is identified
as the SM-like Higgs boson of mass 125 GeV.

Under parity symmetry, which can be defined in LRSM owing to the enhanced
gauge structure, the quark and lepton fields, as well as the Higgs fields
transform as follows (with $F_{L,R}$ collectively denoting the vector-like
fermions):

$\mathcal{Q}_{L}\leftrightarrow\mathcal{Q}_{R},\,\,\psi_{L}\leftrightarrow\psi_{R},\,\,F_{L}\leftrightarrow
F_{R},\,\,\chi_{L}\leftrightarrow\chi_{R}.$ (2.9)

The gauge boson fields also transform under parity:

$W_{L}^{\pm}\leftrightarrow W_{R}^{\pm},$ (2.10)

so that the gauge couplings of the $SU(2)_{L}$ and $SU(2)_{R}$ factors obey
the relation $g_{L}=g_{R}$ above the parity breaking scale. In the parity
symmetric limit the quartic scalar couplings obey
$\lambda_{1_{L}}=\lambda_{1_{R}}\equiv\lambda_{1}$, although $\mu_{L}$ may be
different from $\mu_{R}$ in Eq. (2.6) since parity symmetry may be broken
softly by these dimension-two terms, without spoiling the solution to the
strong CP problem. This is what we shall assume in this work. Such a soft
breaking via $d=2$ terms is necessary to realize $\kappa_{R}\gg\kappa_{L}\neq
0$ in low-scale LRSM. (For the realization of high-scale LRSM with exact
parity where $\mu_{L}^{2}=\mu_{R}^{2}$, see Ref. Hall:2018let .) The
conditions for the potential to be bounded from below, with
$\lambda_{1_{L}}=\lambda_{1_{R}}=\lambda_{1}$, are

$\lambda_{1}\geq 0,\,\,\,\lambda_{2}\geq-\lambda_{1}.$ (2.11)

The charged gauge bosons are unmixed at tree-level in this framework, with
their masses given by

$\displaystyle
M^{2}_{W^{\pm}_{L\,(R)}}=\frac{1}{2}g_{L\,(R)}^{2}\kappa_{L\,(R)}^{2}.$ (2.12)

Among the neutral gauge bosons, photon field $A_{\mu}$ remains massless while
the two orthogonal fields $Z_{L}$ and $Z_{R}$ mix with a mass matrix given as
Babu:2018vrl :

$\mathcal{M}^{2}_{Z_{L}-Z_{R}}=\dfrac{1}{2}\begin{pmatrix}(g_{L}^{2}+g_{Y}^{2})\kappa_{L}^{2}&&g_{Y}^{2}\sqrt{\dfrac{g_{L}^{2}+g_{Y}^{2}}{g_{R}^{2}-g_{Y}^{2}}}\kappa_{L}^{2}\\\
g_{Y}^{2}\sqrt{\dfrac{g_{L}^{2}+g_{Y}^{2}}{g_{R}^{2}-g_{Y}^{2}}}\kappa_{L}^{2}&&\dfrac{g_{R}^{4}\kappa_{R}^{2}+g_{Y}^{4}\kappa_{L}^{2}}{g_{R}^{2}-g_{Y}^{2}}\end{pmatrix}$
(2.13)

This gives rise to the neutral gauge bosons eigenstates $Z_{1(2)}$ with masses

$\displaystyle
M^{2}_{Z_{1}}\simeq\frac{1}{2}(g_{Y}^{2}+g_{L}^{2})\kappa_{L}^{2},$
$\displaystyle\quad\quad
M^{2}_{Z_{2}}\simeq\frac{g_{L}^{4}\kappa_{R}^{2}+g_{Y}^{4}\kappa_{L}^{2}}{2(g_{R}^{2}-g_{Y}^{2})},$
(2.14)

where the gauge boson $Z_{1}$ or, in the limit of small mixing, $Z_{L}$, is
identified as the SM $Z$ boson of mass $91.18$ GeV. Here the hypercharge gauge
coupling $g_{Y}$ is related to the $B-L$ gauge coupling $g_{B}$ via

$g_{Y}^{-2}=g_{R}^{-2}+g_{B}^{-2}\leavevmode\nobreak\ .$ (2.15)

The Yukawa interactions of the charged fermions and the bare masses for the
VLFs, are given in the flavor basis by the Lagrangian

$\displaystyle\mathcal{L}_{\rm Yuk}$
$\displaystyle=\mathcal{Y}_{L}^{u}\bar{\mathcal{Q}}_{L}\tilde{\chi}_{L}U_{R}+\mathcal{Y}_{R}^{u}\bar{\mathcal{Q}}_{R}\tilde{\chi}_{R}U_{L}+M_{U}\bar{U}_{L}U_{R}$
(2.16)
$\displaystyle+\mathcal{Y}_{L}^{d}\bar{\mathcal{Q}}_{L}\chi_{L}D_{R}+\mathcal{Y}_{R}^{d}\bar{\mathcal{Q}}_{R}\chi_{R}D_{L}+M_{D}\bar{D}_{L}D_{R}$
$\displaystyle+\mathcal{Y}_{L}^{e}\bar{\psi}_{L}\chi_{L}E_{R}+\mathcal{Y}_{R}^{e}\bar{\psi}_{R}\chi_{R}E_{L}+M_{E}\bar{E}_{L}E_{R}+\mathrm{h.c.}$

with $\tilde{\chi}_{L,R}=i\tau_{2}\chi^{*}_{L,R}$. Owing to parity symmetry,
the Yukawa coupling matrices and the VLF mass matrices obey the relations

$Y_{L}^{u,d,e}=Y_{R}^{u,d,e},\leavevmode\nobreak\ \leavevmode\nobreak\
M_{U,D,E}=M^{\dagger}_{U,D,E},$ (2.17)

which are crucial relations to solving the strong CP problem. The Lagrangian
given in Eq. (2.16) gives $6\times 6$ mass matrices for up-type quarks
$(u,U)$, down-type quarks $(d,D)$ and charged leptons $(e,E)$ which can be
written in the block form:

$\mathcal{M}_{F}=\begin{pmatrix}0&\mathcal{Y}_{L}^{f}\kappa_{L}\\\
\mathcal{Y}^{f\,\dagger}_{R}\kappa_{R}&M_{F}\end{pmatrix}.$ (2.18)

The strong CP problem is solved in these models via parity symmetry as
follows. The QCD $\theta$-parameter, $\theta_{QCD}$, being parity odd,
vanishes in the model. The physical parameter that contributes to the neutron
EDM is

$\overline{\theta}=\theta_{QCD}+{\rm arg}{\rm Det}({\cal M}_{U}{\cal
M}_{D})\leavevmode\nobreak\ .$ (2.19)

It is clear from the form of Eq. (2.18) that when $Y_{L}^{f}=Y_{R}^{f}$ via
parity, Det$({\cal M}_{U}{\cal M}_{D})$ is real, and thus
$\overline{\theta}=0$ at tree-level. Since the restriction from neutron EDM is
rather severe, $\overline{\theta}\leq 10^{-10}$, it is necessary to ensure
that radiative corrections do not generate a value that exceeds this limit.
Ref. Babu:1989rb  has shown that in this model the induced $\overline{\theta}$
vanishes at one-loop. Very small $\overline{\theta}$ would be induced via two-
loop diagrams, which have been estimated to be consistent with neutron EDM
limits Babu:1989rb ; Hall:2018let .

Quantum gravitational corrections are expected to violate all global
symmetries. Since parity symmetry falls under this category, one should worry
about the quality factor of the parity solution to the strong CP problem. The
leading operator that could be induced via quantum gravity that can generate
$\overline{\theta}$ is the $d=5$ operator

${\cal L}^{d=5}=\frac{1}{M_{\rm
Pl}}(\overline{Q}_{L}Q_{R})\chi_{R}^{\dagger}\,\chi_{L}.$ (2.20)

This operator would induce non-hermitian entries in the quark mass matrix.
Taking the coeffficients of these operators to be of order one, and demanding
that the complex contribution to the up-quark mass not generate
$\overline{\theta}$ larger than $10^{-10}$ would imply that $\kappa_{R}\leq
10^{5}$ GeV. This would lead to a relatively light $W_{R}$ boson, and the
associated vector-like fermions of the model, which opens up the exciting
possibility of exploring them at colliders, and possibly explaining some of
the experimental anomalies.

The mass matrices of Eq. (2.18) can be block diagonalized by bi-unitary
transformations. The light fermion mass matrices have the form

$M^{f}_{\rm ligh}\simeq-{\cal Y}_{L}^{f}M_{F}^{-1}{\cal
Y}_{R}^{f\dagger}\kappa_{L}\kappa_{R},$ (2.21)

if $|{\cal Y}_{R}^{f}\kappa_{R}|\ll|M_{F}|$ is assumed. For a single
generation this yields the light ferrmion mass as $m_{f}\simeq-{\cal
Y}^{f}_{L}{\cal Y}_{R}^{f}\kappa_{L}\kappa_{R}/M_{F}$, which is the seesaw
formula, now applied to charged fermions. It is also possible that there could
be sizable mixing between the light fermions and the vector-like fermions. It
is such mixings that enable us to explain the $W$-boson mass shift as well as
the CKM unitarity violation. As we show in the next section, the mixing of
$t$-quark with its vector-like partner can generate sufficient custodial
$SU(2)$ violation in order to induce the oblique parameter $T$ and thus
explain the $W$-boson mass shift. The CKM unitarity violation would arise via
mixing of the down quark with a vector-like quark.

##  3 Simultaneous Solution to $\mathbf{W}$-Boson Mass Shift and Cabibbo
Anomaly

In this section, we briefly discuss the anomalies associated with the
$W$-boson mass shift measured by the CDF collaboration and the apparent CKM
unitarity violation. We also explore the parameter space required to resolve
each anomaly independently in the LRSM with a universal seesaw. The flavor
structure and the mixing required in each case are summarized, along with the
potential constraints arising from the model framework. Finally, we arrive at
a unique flavor structure that can resolve the two anomalies simultaneously
within the model. It will be shown that the down quark mixing with the VL-
down-type quark and the top mixing with the VL-up-type quark provides a
concurrent solution.

###  3.1 $\bm{W}$-Boson Mass Shift

The CDF collaboration has recently updated the measurement of $W$-boson mass
(cf: Eq. (1.2)) with a very high precision that has not been achieved
previously CDF:2022hxs . This measurement is considerably at odds with the PDG
world average $M_{W}^{PDG}=(80.379\pm 0.012)$ ParticleDataGroup:2022pth  GeV,
which is a combination of the LEP ALEPH:2013dgf , Tevatron CDF:2013dpa  (CDF
CDF:2012gpf  and D0 D0:2012kms ) and LHCb LHCb:2021abm  measurements, at about
$3.6\,\sigma$. It is also at $\sim 7\,\sigma$ tension with the SM prediction
of $M_{W}^{\textrm{SM}}=(80.357\pm 0.004)\;\textrm{GeV}$ Awramik:2003rn .
Assuming that the CDF measurement will be confirmed by future experiments, the
LRSM model can address this mass shift from the perspective of VLQ-induced
corrections to the oblique parameters, $S$, $T$ and $U$ Peskin:1991sw . These
radiative corrections to the gauge boson propagators, for instance, in the SM
appear from top and bottom quarks in the loop. This could easily be enhanced
by the presence of a VL-top or -bottom quarks which have significant mixings
with the ususal quarks. The shift in the $W$-boson mass arising from the
oblique parameters is Peskin:1991sw :

$M_{W}^{2}=M^{2}_{W^{SM}}+\frac{\alpha
c^{2}}{c^{2}-s^{2}}M_{Z}^{2}\left[-\frac{1}{2}S+c^{2}T+\frac{c^{2}-s^{2}}{4s^{2}}U\right]$
(3.22)

where, $s=\sin\theta_{W}$ and $c=\cos\theta_{W}$, $\theta_{W}$ being the
Weinberg angle, and $\alpha$ is the fine structure constant. The oblique
parameters are computed in terms of the two-point functions given as (see
Appendix. B for details):

$\displaystyle\alpha S$
$\displaystyle=4c^{2}s^{2}\left[\Pi_{ZZ}^{\prime}(0)-\frac{c^{2}-s^{2}}{cs}\Pi_{\gamma
Z}^{\prime}(0)-\Pi_{\gamma\gamma}^{\prime}(0)\right]$ (3.23)
$\displaystyle\alpha T$
$\displaystyle=\frac{1}{c^{2}M_{Z}^{2}}\left[\Pi_{WW}(0)-c^{2}\Pi_{ZZ}(0)\right]$
$\displaystyle\alpha U$
$\displaystyle=4s^{2}\left[\Pi_{WW}^{\prime}(0)-c^{2}\Pi_{ZZ}^{\prime}(0)-2cs\Pi_{\gamma
Z}^{\prime}(0)-s^{2}\Pi_{\gamma\gamma}^{\prime}(0)\right].$

Violations of custodial $SU(2)$ symmetry arise in these models via the mixing
of $SU(2)_{L}$-doublet quark fields with singlet quarks. The corrections to
$S$ and $T$ from VLFs have been studied in great detail in Ref. Lavoura:1992np
; Dawson:2012di ; Chen:2017hak . We perform an independent analysis, verifying
that a positive shift to $W$ boson mass would require a correction from VL-top
contribution to the $T$ parameter.

For small $t-U$ and $d-D$ mixing angles, the corrections to $S$ and $U$ are
comparable and much smaller than $T$. To explain the $\sim 7\,\sigma$ shift in
$W$-boson mass (for a central value of $\Delta T=0.1726$ with $\Delta
S=0=\Delta U$), we find that the mixing, say, between top and VL-top quark has
to be $\sin\theta_{L}\gtrsim 0.1$. We shall address this quantitatively in
Sec. 3.4, after discussing the CKM unitarity puzzle.

###  3.2 CKM Unitarity Puzzle

In recent years, with the precise determinations of $|V_{ud}|$ and $|V_{us}|$,
the unitarity of the first row of the CKM is being tested. Currently, the most
precise extraction of $|V_{ud}|$, aided by the progress in calculation of
radiative corrections Seng:2018yzq ; Czarnecki:2019mwq ; Seng:2020wjq , comes
from the super-allowed $0^{+}\to 0^{+}$ nuclear $\beta$ decays Hardy:2014qxa ;
Hardy:2018zsb . $V_{us}$ is determined from the semi-leptonic $K$ meson decays
$K\to\pi\ell\nu$ ($K_{\ell 3}$, with $\ell={e,\mu}$) which involves lattice-
QCD calculation of the $K\to\pi$ form factor. It can also be obtained by
comparing the radiative decay rates of kaon and pion (see Ref Moulson:2017ive
). A significant deficit in the first row unitarity has been reported as a
result of improved precision in the calculation of the radiative corrections
($\Delta_{R}$) to nuclear $\beta$ decay 2103.05549 ; Kirk:2020wdk . The SM
prediction of the CKM matrix leads to the unitarity in the first row:

$|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1$ (3.24)

With the contribution from $|V_{ub}|=3.94(36)\times 10^{-3}$ being negligible,
the above expression reduces to the unitarity in a two-generation model
parametrized by the Cabibbo mixing angle. Using the more precise
$|V_{ud}|=0.97373(9)$ ParticleDataGroup:2022pth ; Kirk:2020wdk  and the PDG
average for $|V_{us}|=0.2252(5)$ the deviation from unitarity in the first row
turns out to be about 3.9$\,\sigma$ Kirk:2020wdk  as shown in Eq. (1.1).
Different approaches in determining the radiative corrections and choices of
the experiments involved in estimating the CKM elements lead to inconsistency
among the moduli, however all of them show a deficit from the unitarity of
about $\sim 3.3-4\,\sigma$. There is also a slightly less significant deficit
in the unitarity of the first column. We address both of these in our work,
although the focus is primarily on the first row non-unitarity.

One possible explanation of the Cabibbo anomaly stems from the idea that the
CKM mixing is a sub-matrix of a more general, unitary matrix arising from the
mixing of SM quarks with extra undiscovered quarks. Exploiting the mixing of
SM and VL-quarks (VLQ), we explore the scenarios inducing non-unitarity in the
current model. The deviation can arise from mixing between the VLQ and light
quarks in the up-sector, down-sector, or both. For illustrative purposes, let
us assume that the major correction to the CKM matrix (or Cabibbo mixing
matrix in case of two family mixing) arises from only one flavor of SM fermion
mixing with the corresponding VLQ (referred to as VLQ-mixing, henceforth). The
Cabibbo mixing can be thought of a small corrections induced by the
diagonalization of the light quark matrix. The structure of the charge current
interaction would heavily depend on the VLQ-mixing. We can safely ignore the
case were the VLQ mixing appears from the third family as the contribution to
non-unitarity would be negligible owing to the smallness of $V_{ub}$ element.
VLQ-mixing arising from either the first or second generations can provide
large enough correction to resolve the CKM unitarity problem. For instance,
the VLQ-mixing appearing from down quark mixing with VL-down type quark can
lead to a charge current structure given by

$\begin{pmatrix}V_{ud}\leavevmode\nobreak\ c_{L}&&V_{us}\\\
V_{cd}\leavevmode\nobreak\ c_{L}&&V_{cs}\end{pmatrix}\text{ or
}\begin{pmatrix}V_{ud}\leavevmode\nobreak\ c_{L}&&V_{us}\leavevmode\nobreak\
c_{L}\\\ V_{cd}&&V_{cs}\end{pmatrix},$ (3.25)

depending on whether the Cabibbo mixing is appearing from down-sector or up-
sector, respectively. Here, $c_{L}=\cos\theta_{L}$, which parametrizes the
leakage of the left-handed up (down) quark into the VLQ sector, which being
singlets of $SU(2)_{L}$ have no direct couplings to the $W^{\pm}$ boson..
There are, in general, six different possible ways of inducing non-unitarity
in the first row. A VLQ-mixing of $\sin\theta_{L}\simeq 0.034$ arising from up
or down quark mixing with the corresponding VLQ can resolve the $\sim
3.9\,\sigma$ deficit in the first row unitarity, while $\sin\theta_{L}\simeq
0.15$ would be required if the VLQ-mixing arises from the charm or strange
quark. Of these, only the VLQ-mixing arising from the first generation will
survive the constraint from hadronic decays of the $Z$ boson (see Sec. 3.3.1).

###  3.3 Constraints on model parameters

####  3.3.1 Light fermion masses

In the simplified one family mixing formulation, where the $d$-quark mixes
with a vector-like $D$-quark, or equivalently $s$-quark mixes with its VLQ
partner, the mass of $d$-quark (or $s$-quark) can be written as
$m_{i}=(\kappa_{L}\kappa_{R}s_{L}s_{R})M_{i}$, where $s_{L}$ and $s_{R}$ are
the sines of left and right mixing angles among $d_{i}$ and $D_{i}$ quarks.
The tangent functions ($t_{L,R}$) obey the relation
$t_{R}=(\kappa_{R}/\kappa_{L})t_{L}$. Therefore, with $t_{L}\simeq s_{L}$ we
have

$s_{L}^{2}=\frac{\kappa_{L}}{\kappa_{R}}\frac{m_{d_{i}}}{M_{D_{i}}}.$ (3.26)

Since $\sin\theta_{L}$ should be of $\mathcal{O}(0.034)$ to resolve the
Cabibbo anomaly, the products $\kappa_{R}M_{D_{i}}$ are severely restricted
due to the smallness of the $d_{i}$ masses:
$\kappa_{R}M_{D_{i}}\in[325,1.9\times 10^{5}]$ GeV2 to fit the $d$\- and
$s$-quark masses. This is in direct contradiction with the experimental limits
on these parameters: $\kappa_{R}\geq 10$ TeV, corresponding to the 5 TeV lower
bound on $M_{Z_{R}}$ ATLAS:2019erb  and $M_{F}>1$ TeV, from the production and
decay of pair produced VLF partners ATLAS:2018ziw ; CMS:2018zkf . Note that
parity symmetry plays in important role to arrive at this conclusion. Indeed,
within the same model, without parity, the relation
$t_{R}=(\kappa_{R}/\kappa_{L})t_{L}$ will not hold, which would enable
achieving large $d_{L}-D_{L}$ mixing. In such an attempt, however, the parity
solution to the strong CP problem would be lost.

These constraints from light fermion masses can be evaded with parity symmetry
intact by considering the mixing of the down quark (or the strange quark) with
two VLQs. In such a setup large $d_{L}-D_{L}$ mixing can be realized even in
the limit of vanishing $d$-quark mass, as will be discussed in Sec. 3.4. Small
$d$-quark mass can be generated via perturbations to the mass matrix proposed
there. This scenario, where the light quark masses arise as perturbative
corrections, however, is not applicable in the case of top quark, since its
mass is not so small. So, we also include the fit to the mass of top quark in
our analysis.

####  3.3.2 $\bm{Z}$ Decay Width

Another major constraint on the mixing of $d_{L}$ with $D_{L}$(or $s_{L}$ with
its VLQ partner) arises from the modification of the $Z$ boson couplings to
the fermions. The left-handed fermion interaction vertex of $Z$ is modified by
the cosine of the VLQ mixing ($\cos\theta_{L}$), thereby altering the total
decay width of the $Z$ boson:

$\mathcal{L}_{Z}\supset\frac{g}{\cos\theta_{W}}\overline{f}_{L}\gamma_{\mu}\left(T_{3L}\cos\theta_{L}^{2}-Q\sin\theta_{W}^{2}\right)f_{L}Z^{\mu}$
(3.27)

For instance, resolving the Cabibbo anomaly with VLQ mixing from first
generation requires $\sin\theta_{L}=0.034$. This results in
$\Delta\Gamma_{Z}\simeq 1\leavevmode\nobreak\ (0.8)$ MeV from down (up)-quark
mixing with VLF. On the other hand, $\sin\theta_{L}\simeq 0.15$ in case of the
second family mixing, leading to $\Delta\Gamma_{Z}\simeq 20$ MeV which is
excluded by the experimental measurement of $Z$ decay with an uncertainty of
$\Delta\Gamma_{Z}\leq 2.3$ MeV at $1\,\sigma$. Therefore, the Cabibbo anomaly
can be resolved only by the VLQ-mixing with the first family. For a general
discussion of flavor non-universality with VLQ mixing with the usual quarks,
see Ref. Belfatto:2019swo .

###  3.4 Combined Solution

Figure 1: Oblique parameters $S,\,T,\,U$ as a function of the mass of the
lightest vector-like quark ($M_{U_{2}}$) for different choices of the Yukawa
coupling. The unshaded region shows the required range of $T$ parameter that
can explain the CDF $W$-boson mass shift. The vertical dotted line shows the
current experimental lower bound on VLQ mass ATLAS:2018ziw ; CMS:2018zkf .
This plot is consistent with the mass of top quark. The Yukawa coupling needs
to be at least $y\simeq 1.35$ to explain the $W$-boson mass shift while
evading the current experimental bound on the mass of VLQ. The corrections to
$S$ and $U$ parameters are seen to be much smaller than $T$.

From the discussions so far, it is clear that the Cabibbo anomaly can only be
resolved if the mixing arises from up or down quark with a corresponding VLQ.
Since the mixing angle required in this case, $\sin\theta_{L}\simeq 0.034$, is
not large enough to accommodate the explanation for $W$ boson mass shift,
which requires $\sin\theta_{L}\sim\mathcal{O}(0.1)$, and such large mixings
are subjected to constraint from $Z$ decay width (Sec. 3.3.2), the only
remaining choice is to invoke top quark mixing with VLQ to explain the shift
in the $W$-boson mass.

As mentioned previously, a simple one-family mixing between the light and VL-
quark suffers from a severe constraint arising from the smallness of the light
fermion masses (Sec. 3.3.1). To evade this, we assume that the down quark
mixes with two vector-like quarks. We can work in the limit of neglecting all
SM fermion masses, except for the top quark. It turns out that top-quark
mixing with a single VLQ cannot induce sufficiently large $T$-parameter to
explain the $W$-boson mass shift. This is again due to parity symmetry which
constrains the form of such a $2\times 2$ mass matrix. When the top quark
mixes with two VLQs, the contribution to the $T$-parameter can be sufficiently
large, which we shall adopt.

Since there are three up-type VLQs in the model, and since top quark mixes
with two of them, there is no room for the up quark to mix significantly with
any VLQs in order to explain the CKM unitarity puzzle. We are then cornered to
a unique solution that simultaneously explains the $W$-boson mass shift and
the CKM unitarity puzzle: The top quark mixes significantly with two up-type
VLQs, and the down quark mixes with two down-type VLQs. The specific
structures of the mass matrices in these sectors are uniquely determined and
are given below:

$\displaystyle\mathcal{M}_{d}=\begin{pmatrix}0&&0&&y_{d}\kappa_{L}\\\
0&&0&&M_{1d}\\\ y_{d}\kappa_{R}&&M_{1d}&&M_{2d}\end{pmatrix}\text{ in
$\\{d\leavevmode\nobreak\ D_{2}\leavevmode\nobreak\ D_{3}\\}$ basis}$ (3.28)
$\displaystyle\mathcal{M}_{t}=\begin{pmatrix}0&&0&&y_{u}\kappa_{L}\\\
0&&M_{3u}&&M_{1u}\\\ y_{u}\kappa_{R}&&M_{1u}&&M_{2u}\end{pmatrix}\text{ in
$\\{t\leavevmode\nobreak\ U_{2}\leavevmode\nobreak\ U_{3}\\}$
basis}\leavevmode\nobreak\ .$

Details of the diagonalization of ${\cal M}_{d}$ are provided in Appendix. A.
The up-sector diagonalization has been done numerically, with the $3\times 3$
unitary matrix represented using the standard parametrization of the CKM
matrix (Eq. (A.36)) (where the CP violating phase is set to zero for
simplicity). Although the top-mixing also contributes to the charged current
interactions, it is the the down-sector mixing that resolves the Cabibbo
anomaly with $\Delta_{CKM}=s_{1d}^{2}$ with $s_{1d}\simeq 0.034$, assuming
that the CKM mixing receives contribution from down sector. This can be seen
from the extended CKM matrix with
$\left(\overline{u}_{L}\,\overline{c}_{L}\,\overline{t}_{L}\,\overline{U}_{2_{L}}\,\overline{U}_{3_{L}}\right)$
multiplied from the left and
$\left(d_{L}\,s_{L}\,b_{L}\,D_{2_{L}}\,D_{3_{L}}\right)^{T}$ from the right
(see Appendix A for details):

$\begin{pmatrix}{\color[rgb]{0.0,0.5,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.5,0.0}c_{{1d}}V_{ud}}&&{\color[rgb]{0.0,0.5,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.5,0.0}c_{{1d}}V_{us}}&&{\color[rgb]{0.0,0.5,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.5,0.0}c_{{1d}}V_{ub}}&&-c_{{2d}}s_{{1d}}&&-s_{{1d}}s_{{2d}}\vspace{1mm}\\\
{\color[rgb]{0.0,0.5,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.5,0.0}V_{cd}}&&V_{cs}&&V_{cb}&&0&&0\vspace{1mm}\\\
{\color[rgb]{0.0,0.5,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.5,0.0}u_{11}V_{td}}&&u_{11}V_{ts}&&u_{11}V_{tb}&&0&&0\vspace{1mm}\\\
u_{21}V_{td}&&u_{21}V_{ts}&&u_{21}V_{tb}&&0&&0\vspace{1mm}\\\
u_{31}V_{td}&&u_{31}V_{ts}&&u_{31}V_{tb}&&0&&0\\\ \end{pmatrix}$ (3.29)

From Eq. (3.29) one can also see that a combination of
$u_{11}=\hat{s}_{12}^{u}$ when $\hat{s}_{13}^{u}\ll 1$ (Eq. (A.36)) and
$s_{1d}$ can explain the non-unitarity of the CKM matrix in the first column,
with the main correction appearing from the down-sector owing to the smallness
of $V_{td}$. The $Z$ interaction with left-handed light fermions gets modified
owing to the mixing of $d$ with $D$ and $t$ with $U$. New interactions
involving $\overline{t}U_{i}Z$ and $\overline{d}D_{i}Z$ arise in the model
proportional to these VLQ mixings. The remaining left-handed quark vertices
with the $Z$ boson, and all the right-handed quark vertices with $Z$, however,
remain unchanged. These new interactions by themselves do not cause any tree-
level flavor changing processes.

Both up- and down-sectors contribute to the corrections to the oblique
parameters, but the effect from down-sector is negligible compared to that of
top-mixing, since the mixing of only $s_{1d}\simeq 0.034$ is required resolve
the Cabibbo anomaly. The expressions for $S$, $T$ and $U$ arising from $t-U$
mixing has been verified against Ref. Dawson:2012di ; Chen:2017hak . The exact
expressions in our framework are given in Appendix B (Eqs. B.45, B.46 and
B.47). We find that the corrections to $S$ and $U$ are negligible compared to
the $T$ parameter.

The behavior of the $T$ parameter as a function of the lightest VLQ mass
($M_{U_{2}}$) is shown in Fig. 1, for different choices of the Yukawa coupling
$y$, imposing the fit for top quark mass. It becomes evident that the Yukawa
coupling $y_{u}$ has to be at least 1.35 to explain the positive shift in
$W$-boson mass while evading the LHC constraint on VLQ mass. The fit to the
CDF $W$-boson mass shift using the full expressions of oblique parameters
(Eqs. B.45, B.46 and B.47) is shown in Fig. 2. The region shown in the plot is
also consistent with $1\text{ and }2\leavevmode\nobreak\ \,\sigma$ allowed
region from resolving the CKM unitarity puzzle from down-sector mixing,
although this alone is not enough to explain the shift in $W$-boson mass and
CKM non-unitarity simultaneously. In Fig. 2 we have also indicated the
excluded region from $|V_{tb}|$ measurement as a shaded region on the top. In
resolving the Cabibbo anomaly, it is essential to understand that the new
elements of the CKM matrix are identified with those determined
experimentally, i.e., $|c_{1d}V_{us}|\equiv|V_{us}^{PDG}|$, etc. These
redefinitions, however, do not impose any constraints on these essentially
free parameters.

With only the down quark and the top quark mixing with VLQs significantly,
there won’t be large FCNC mediated by the $Z$ boson or the Higgs boson.
However, once all the CKM mixing angles are induced there could be such
effects. To control these, we note that $V_{us}$ can arise from the up-sector,
while $V_{cb}$ arises from the down-sector. These have no bearing on the
light-heavy mixing. The smaller $V_{ub}$ may arise from either sector,
involving admixture of the $(u-c)$ sector in the mass matrix ${\cal M}_{t}$ of
Eq. (3.28), which is not expected to lead to large FCNC effects, owing to the
smallness of $V_{ub}$.

###  3.5 Stability of the Higgs Potential

The resolution to the $W$-boson mass shift, along with the necessary fit to
the top quark mass requires Yukawa coupling $y_{u}$ to be larger than about
1.35. Although such large Yukawa couplings are not excluded by the
perturbative unitarity bounds Chanowitz:1978uj ; Marciano:1989ns ;
Whisnant:1994fh ; Allwicher:2021rtd , the Higgs potential is at risk of
becoming unstable as the quartic coupling $\lambda_{1_{L}}$ Eq. (2.6) could
turn negative when extrapolated to higher energies. Partial wave unitarity
would require $y_{u}^{2}<8\pi/3$ Marciano:1989ns , obtained from the condition
Re($a_{0})\leq\frac{1}{2}$, for the elastic fermion-antifermion scattering in
the color singlet channel, at center of mass energies much above the fermion
mass. While this condition was derived for a chiral fermion of the SM such as
the top quark, it would equally apply to the scattering of vector-like top
quark at energies well above its bare mass. It should be noted that the
heavier VLQ with order one Yukawa coupling obtains its mass primarily via the
Yukawa coupling, rather than through its bare mass.

The stability of the Higgs potential is especially of concern since the
heavier VLQ with order one Yukawa coupling has a mass of order 20 TeV in the
model. This requires that $\lambda_{1_{L}}$ should remain positive up to this
energy scale. Here, we show that the renormalization group flow of the quartic
couplings in the momentum range $(4-20)$ TeV would keep $\lambda_{1_{L}}$
positive, which would guarantee the stability of the Higgs potential up to
this energy scale.

Suppose that the scalar field $\sigma_{R}$ associated with the $SU(2)_{R}$
symmetry breaking has a mass of order 20 TeV. In this case only the quartic
coupling $\lambda_{1_{L}}$ is active below 20 TeV, which by virtue of the
Yukawa coupling $y_{u}$ would turn negative in going from 4 TeV to this scale.
On the other hand, if $\sigma_{R}$ has a mass comparable to the lighter VLQ,
of order 4 TeV, in the momentum range $(4-20)$ TeV both $\lambda_{1_{L}}$ and
$\lambda_{1_{R}}$ are active. $\lambda_{1_{L}}$ would decrease in going to
higher energies driven by the Yukawa coupling of $\sigma_{L}$, while
$\lambda_{1_{R}}$ will not feel the effect of the Yukawa coupling. This is
because the Yukawa coupling of $\sigma_{R}$ involves heavier fields. This is
the scenario that is consistent with our framework. Naturally, this scheme
would predict that the $\sigma_{R}$ field should have a mass of the same order
as the $U_{2}$ field, which is of order 4 TeV or lower.

To test the validity of our scenario, we have computed the RG evolution of the
quartic coupling and shown that the coupling $\lambda_{1_{L}}\geq 0$ at least
up to 20 TeV. We have presented the one-loop renormalization group equations
for the dimensionless parameters of the model in Appendix C. These are
generalizations of equations given in Ref. Koide:2000ks ; Mohapatra:2014qva
that can be adopted to the regime below 20 TeV, where parity symmetry is not
exact. From the RG evolution of the Yukawa couplings, given in Eq. (C.51), we
find that for $y_{u,d}\simeq 1.4$, there is no significant running in
$y_{u,d}$ in the momentum range $(4-20)$ TeV. Thus it is a good approximation
to take $y_{u,d}$ to be constants in this regime. With this approximation we
can compute the evolution of the quartic scalar couplings analytically.

At the momentum scale of order $M_{U_{3}}$, where parity symmetry is exact, we
have $\lambda_{1_{L}}=\lambda_{1_{R}}=\lambda_{1}$. At lower energies
$\lambda_{1_{L}}$ will be larger than $\lambda_{1_{R}}$, owing to the Yukawa
contributions to their evolution. Note that $\sigma_{R}$ couples to the
heavier VLQ, while $\sigma_{L}$ couples to the lighter VLQ in our fits.
Therefore below the parity restoration scale $\lambda_{1_{R}}$ does not feel
the effects of its large Yukawa couplings. The difference
$\delta_{\lambda}=\lambda_{1_{L}}-\lambda_{1_{R}}$, defined at the lower
scale, is obtained by solving the relevant renormalization group equations,
Eq. (C.53), applicable to the momentum range $(4-20)$ TeV. The solution can be
written down approximately as

$\delta_{\lambda}=\frac{3}{16\pi^{2}}\left(y_{u}^{4}\ln{\left[\frac{M_{U_{3}}}{M_{U_{2}}}\right]}+y_{d}^{4}\ln{\left[\frac{M_{U_{3}}}{M_{D_{2}}}\right]}\right).$
(3.30)

Only the Yukawa couplings cause significant split in the $\lambda_{1_{L}}$ and
$\lambda_{1_{R}}$, since the gauge couplings$g_{L}$ and $g_{R}$ run
identically below 20 TeV. We minimize the Higgs potential at a momentum scale
of order 4 TeV, which roughly corresponds to the $\sigma_{R}$ scalar mass. At
this scale $\lambda_{1_{R}}$ is taken to be a free parameter.
$\lambda_{1_{L}}$ obtained at the low momentum scale using SM RG evolution
Arason:1991ic  of the Higgs quartic coupling is used to compute the SM Higgs
mass $m_{h}$ at that scale. To ensure that the Yukawa couplings do not cross
the perturbative unitarity bound of $|y|\leq\sqrt{8\pi/3}$ Marciano:1989ns
upon RG evolution Arason:1991ic , the coupling was chosen to $|y_{u}|\lesssim
2.25$, for $|y_{d}|<1$ for the entire momentum range. A few benchmark points
illustrating that $\lambda_{1_{L}}$ remains positive are given in Table. I.

| $y_{u}$ | $y_{d}$ |  | {$M_{1u},M_{2u},M_{3u}$}
---
{$M_{1d},M_{2d}$}
$M_{D_{2}}$ | $M_{D_{3}}$ | $M_{U_{2}}$ | $M_{U_{3}}$ | $|s_{1d}|$ | $|s_{2d}|$ | $|\hat{s}^{u}_{12}|$ | $\lambda_{1_{R}}$ | $\lambda_{2}$ | $\lambda_{1_{L}}$ | $m_{H}$ | $T$
BM-1 | 2.19 | 0.28 |  | {2.9,3.0,1.3}
---
{1.5,0.2}
1.5 | 3.2 | 3.2 | 22.4 | 0.033 | 0.034 | 0.11 | 0.01 | 0.09 | 0.17 | 1.5 | 0.19
BM-2 | 1.97 | 0.19 |  | {2.8, 2.7,1.4}
---
{1.4,0.3}
1.3 | 2.4 | 3.0 | 20.0 | 0.026 | 0.093 | 0.099 | 0.008 | 0.056 | 0.17 | 1.3 | 0.15
BM-3 | 2.06 | 0.64 |  | {2.6, 3.6,1.2}
---
{4.6,3.8}
4.0 | 9.1 | 2.9 | 21.0 | 0.024 | 0.279 | 0.11 | 0.04 | 0.15 | 0.17 | 2.9 | 0.19
BM-4 | 1.92 | 0.56 |  | {2.3, 2.7,1.2}
---
{3.8,3.8}
3.2 | 8.0 | 2.5 | 19.6 | 0.025 | 0.284 | 0.12 | 0.031 | 0.11 | 0.17 | 2.5 | 0.19
BM-5 | 2.00 | 0.24 |  | {2.8, 3.0, 1.3}
---
{1.2,0.02}
1.2 | 2.7 | 3.0 | 20.4 | 0.035 | 0.005 | 0.10 | 0.007 | 0.052 | 0.19 | 1.2 | 0.16

Table I: Benchmark points showing that $\lambda_{1}$ remain positive up to the
momentum scale of the heaviest VLQ mass $M_{U_{3}}$, for Yukawa couplings
$y_{u}$ and $y_{d}$ which are chosen at the low momentum scale equivalent to
the lightest of the VLQ masses. All the masses are quoted in TeV.
$M_{U_{2(3)}}$ and $M_{D_{2(3)}}$ are the light (heavy) VLQ masses in the up-
and down-sectors, respectively. The heavy Higgs mass, $m_{H}$ is chosen to be
the lowest VLQ mass to obtain $\lambda_{1}$ and $\lambda_{2}$. The value of
$T$ parameter is given for each benchmark point.

On applying all these constraints, we find that the largest allowed mass of
up-type VLQ is $M_{U_{2}}\lesssim 3.8\leavevmode\nobreak\ (4.2)$ TeV whereas
the down-type mass can be $M_{D_{2}}\lesssim 4.2\leavevmode\nobreak\ (5.2)$
TeV, to resolve the anomalies within $1\,(2)\,\sigma$. In getting these limits
we have allowed $|y_{d}|=2.25$, its perturbative unitarity limit. However,
stability of the Higgs potential is not easy to guarantee when both $|y_{u}|$
and $|y_{d}|$ are of order 1.5 or greater. If $|y_{d}|<1$ is imposed, while
allowing for larger $|y_{u}|$, the mass of $D_{2}$ would be $M_{D_{2}}\lesssim
1.8\leavevmode\nobreak\ (2.2)$ TeV, to resolve the Cabibbo anomaly within
$1\,(2)\,\sigma$. These limits are the more conservative ones, consistent with
the Higgs potential stability.

Figure 2: 1- and 2-$\sigma$ regions required to explain CDF-$W$ mass and
Cabibbo anomaly plotted as a function of sine of the mixing angle
$\hat{s}_{12}^{u}$ (Eq. (A.36)), consistent with the stability of Higgs
potential. The green shaded region is the exclusion on VLQ mass ATLAS:2018ziw
; CMS:2018zkf  whereas the gray shaded region corresponds to the limit on
mixing angle from the determination of $|V_{tb}|$ ParticleDataGroup:2022pth
element of the CKM matrix.

##  4 Conclusion

In light of the recent CDF measurement of the $W$-boson mass reporting a
possible $7\,\sigma$ shift with the SM prediction, and the improvements in
determination of the CKM matrix elements resulting in an apparent violation of
unitarity in the first-row as large as $\sim 3.9\,\sigma$, we have
investigated new physics beyond the SM. In this paper, we focus on a
particular class of left-right symmetric models wherein the fermion masses are
generated by a universal seesaw mechanism aided by the presence of heavy
vector-like fermionic partners having Yukawa couplings to the light SM
fermions. We explore the UV-complete model to provide a successful explanation
of the CKM unitarity puzzle and the $W$-boson mass shift in a well-constrained
parameter space. We have insisted that parity symmetry is broken only by
dimension-two soft terms in the scalar potential, which would provide a
solution to the strong CP problme without the need for an axion. We find the
simplest, unique flavor structure that can explain the two anomalies where one
of the SM quarks mixes with two VLQs. The CKM unitarity puzzle can be
explained by the mixing of either or both of the first-generation quarks with
VLQs while the $W$-boson mass shift requires top-quark mixing with VLQs. For a
concurrent explanation, invoking the top-quark mixing eliminates the
possibility of up-quark mixing with two VLQs simultaneously. This leads to a
unique solution where the up-sector (top-quark mixing) and down-sector (down-
quark mixing) with two VLQs each resolve the $W$ mass shift and the CKM
unitarity puzzple, respectively. Since this framework requires
$\mathcal{O}(1)$ Yukawa coupling, we have also analyzed the stability of the
Higgs potential. Ensuring that the potential remains stable till at least up
to the mass of the second VLQ, this model predicts an upper bound on the VL
up-type quark mass of $\sim 4.2$ TeV and the VL down-type mass of $\sim 5.2$
TeV.

##  Appendix A Mass Matrix Diagonalization

The mass structures in Eq. (3.28) are diagonalized using a bi-unitary
transformation of the form $U_{L}\mathcal{M}U_{R}^{\dagger}$. Here, $U_{L(R)}$
are parameterized by $\theta_{L(R)_{1,2}}$. Since we are interested only in
the left-handed mixing angles, $U_{L}=U$ with $\theta_{L_{1,2}}=\theta_{1,2}$.
The down-sector left-handed fields are diagonalized by
$U_{d}\mathcal{M}_{d}\mathcal{M}_{d}^{T}U_{d}^{T}$, assuming all the couplings
to be real.

$\displaystyle\mathcal{M}_{d}\Rightarrow\text{Diag}(0,\,M_{D_{2}},\,M_{D_{3}}).$
(A.31) $U_{d}=\begin{pmatrix}1&0&0\\\ 0&c_{{2d}}&-s_{{2d}}\\\
0&s_{{2d}}&c_{{2d}}\end{pmatrix}\begin{pmatrix}c_{{1d}}&s_{{1d}}&0\\\
-s_{{1d}}&c_{{1d}}&0\\\ 0&0&1\par\end{pmatrix}.$ (A.32)

$c(s)$ stands for $\cos(\sin)$ where, $\theta_{(1,2)d}$ are the left-mixing
angles. The mixing angles are

$\displaystyle\theta_{1d}=\arctan(\frac{-y_{d}\kappa_{L}}{M_{1d}}),$
$\displaystyle\theta_{2d}=\frac{1}{2}\arctan\left(\frac{2M_{2d}\sqrt{M_{1d}^{2}+y_{d}^{2}\kappa_{L}^{2}}}{M_{2d}^{2}+(\kappa_{R}^{2}-\kappa_{L}^{2})y_{d}^{2}}\right),$
(A.33)

and the heavy VLQ mass eigenvalues are

$\displaystyle M_{D_{3,2}}=$
$\displaystyle\frac{1}{2}\left(2M_{1d}^{2}+M_{2d}^{2}+(\kappa_{L}^{2}+\kappa_{R}^{2})y_{d}^{2}\pm\sqrt{4M_{2d}^{2}(M_{1d}^{2}+\kappa_{L}^{2}y_{d}^{2})+\left(M_{2d}^{2}+(\kappa_{R}^{2}-\kappa_{L}^{2})y_{d}^{2}\right)^{2}}\right).$
(A.34)

For the up-sector, we can represent a general unitary matrix under the
standard CKM-like parametrization (with the CP phase set to zero for
simplicity) as

$\displaystyle U_{u}$ $\displaystyle=\begin{pmatrix}u_{11}&u_{12}&u_{13}\\\
u_{21}&u_{22}&u_{23}\\\ u_{31}&u_{32}&u_{33}\end{pmatrix}$ (A.35)
$\displaystyle=\left(\begin{array}[]{ccc}\hat{c}_{12}\hat{c}_{13}&\hat{s}_{12}\hat{c}_{13}&\hat{s}_{13}\\\
-\hat{s}_{12}\hat{c}_{23}-\hat{c}_{12}\hat{s}_{23}\hat{s}_{13}&\hat{c}_{12}\hat{c}_{23}-\hat{s}_{12}\hat{s}_{23}\hat{s}_{13}&\hat{s}_{23}\hat{c}_{13}\\\
\hat{s}_{12}\hat{s}_{23}-\hat{c}_{12}\hat{c}_{23}\hat{s}_{13}&-\hat{c}_{12}\hat{s}_{23}-\hat{s}_{12}\hat{c}_{23}\hat{s}_{13}&\hat{c}_{23}\hat{c}_{13}\end{array}\right)$

such that

$\mathcal{M}_{t}\Rightarrow\text{Diag}(m_{t},\,M_{U_{2}},\,M_{U_{3}}),$ (A.36)

It should be noted that the mass matrices in Eq. 3.28 need to be modified by
small corrections to fit the light fermion masses. The exact mass matrix
diagonalization can proceed through a bi-unitary transformation with the
unitary matrices $\mathcal{U}_{L(R)}$ parameterized by mixing angles
$\rho_{L(R)}\ll 1$. For instance, after the diagonalization of the large
couplings in the down sector, the mass matrix can take the following
structure:

$\mathcal{M}^{\prime}_{d}=\begin{pmatrix}x&&\epsilon\kappa_{L}\\\
\epsilon^{\prime}\kappa_{R}&&X\end{pmatrix},$ (A.37)

where $x,\epsilon,\epsilon^{\prime}\text{ and}X$ are functions of the
perturbative couplings and the large mixing angles
$\theta_{{L\,(R)}_{1d\,(2d)}}$. $\epsilon^{\prime}=\epsilon^{\dagger}$ under
$\\{c_{L_{1d}}\to c_{R_{1d}}\,c_{L_{2d}}\to c_{R_{2d}}\,s_{L_{2d}}\to
s_{R_{2d}},\kappa_{L}\to\kappa_{R}\\}$. This matrix structure can be block-
diagonalized as follows:

$\mathcal{U}_{L}\mathcal{M}^{\prime}_{d}\mathcal{U}_{R}^{\dagger}=\begin{pmatrix}\mathds{1}-\frac{1}{2}\rho_{L}\rho^{\dagger}_{L}&&\rho_{L}\\\
-\rho^{\dagger}_{L}&&\mathds{1}-\frac{1}{2}\rho^{\dagger}_{L}\rho_{L}\end{pmatrix}\begin{pmatrix}x&&\epsilon\kappa_{L}\\\
\epsilon^{\prime}\kappa_{R}&&X\end{pmatrix}\begin{pmatrix}\mathds{1}-\frac{1}{2}\rho_{R}\rho^{\dagger}_{R}&&-\rho_{R}\\\
\rho^{\dagger}_{R}&&\mathds{1}-\frac{1}{2}\rho^{\dagger}_{R}\rho_{R}\end{pmatrix},$
(A.38)

giving rise to SM fermion mass matrix of the form

$\hat{m}=x-\kappa_{L}\kappa_{R}\epsilon
M^{-1}\epsilon^{\prime}\,{}^{\dagger},$ (A.39)

where $M=\text{Diag}(M_{1},M_{2},M_{3})$ are the heavy VLQ masses.

$\rho_{L}=\kappa_{L}\epsilon
M^{-1},\,\,\qquad\qquad\rho_{R}=\kappa_{R}\,\epsilon^{\prime}\,{}^{\dagger}M^{-1},$
(A.40)

##  Appendix B Oblique Parameters

The expressions for oblique parameters in Eq. (3.23) are obtained from
Peskin:1991sw

$\displaystyle\alpha S$ $\displaystyle\equiv
4e^{2}\left[\Pi_{33}^{\prime}(0)-\Pi_{3Q}^{\prime}(0)\right]$ (B.41)
$\displaystyle\alpha T$
$\displaystyle\equiv\frac{e^{2}}{s^{2}c^{2}M_{Z}^{2}}\left[\Pi_{11}(0)-\Pi_{33}(0)\right]$
$\displaystyle\alpha U$ $\displaystyle\equiv
4e^{2}\left[\Pi_{11}^{\prime}(0)-\Pi_{33}^{\prime}(0)\right]$

and converting these to the gauge boson basis using Kundu:1996ah :

$\displaystyle\Pi_{\gamma\gamma}$
$\displaystyle=e^{2}\Pi_{QQ},\,\,\,\Pi_{WW}=\frac{e^{2}}{s^{2}}\Pi_{11},$
(B.42) $\displaystyle\Pi_{\gamma Z}$
$\displaystyle=\frac{e^{2}}{sc}(\Pi_{3Q}-s^{2}\Pi_{QQ}),$
$\displaystyle\Pi_{ZZ}$
$\displaystyle=\frac{e^{2}}{c^{2}s^{2}}(\Pi_{33}-2s^{2}\Pi_{3Q}+s^{4}\Pi_{QQ})$

Here, $s=\sin\theta_{W}$ and $c=\cos\theta_{W}$, $\theta_{W}$ being the
Weinberg angle, and $\alpha$ is the electromagnetic coupling strength. In
deriving Eq. (3.23) we have used $\Pi_{\gamma\gamma}(0)=0=\Pi_{\gamma Z}(0)$.
Other variations of these expressions can be found in Holdom:1990xp ;
Bhattacharyya:1991yw ; Ma:1992uc ; Lavoura:1992np ; Kundu:1996ah ;
Grimus:2008nb  among others. The difference between the expressions which are
written in terms of slopes and derivatives are negligible under the
approximation that $\Pi_{ij}$ are linear functions of $q^{2}$ near $q^{2}=0$,
i.e.,

$\Pi_{ij}(q^{2})=\Pi_{ij}(0)+q^{2}\Pi^{\prime}_{ij}(0)$ (B.43)

See Ref. Kundu:1996ah  for a detailed discussion on the oblique parameters.
Some expressions may be in terms of

$\Pi_{3Y}=2(\Pi_{3Q}-\Pi_{33}).$ (B.44)

Wherever the definitions of $U$ differ in terms of $M_{W}$ and $M_{Z}$, care
must be taken to use the relevant expression for $W$ mass shift. The complete
expressions for the oblique parameters in our framework are (using the short-
hand $\sin\theta_{if}=s_{if}$ and $\cos\theta_{if}=c_{if}$):

$\displaystyle T$ $\displaystyle=\frac{N_{c}}{16\pi
s^{2}M_{W}^{2}}\Bigg{[}\left(u_{11}^{4}-1\right)m_{t}^{2}+\left(u_{21}^{4}M_{U_{2}}^{2}+u_{31}^{4}M_{U_{3}}^{2}\right)$
$\displaystyle+2\left\\{u_{11}^{4}-1+m_{t}^{2}\left(\frac{1-u_{11}^{2}}{m_{t}^{2}-m_{b}^{2}}-\frac{u_{11}^{2}u_{21}^{2}}{M_{U_{2}}^{2}-m_{t}^{2}}-\frac{u_{11}^{2}u_{31}^{2}}{M_{U_{3}}^{2}-m_{t}^{2}}\right)\right\\}m_{t}^{2}\ln{\left[\frac{m_{t}^{2}}{M_{U_{3}}^{2}}\right]}$
$\displaystyle+2u_{21}^{2}\left\\{u_{21}^{2}+M_{U_{2}}^{2}\left(-\frac{1}{M_{U_{2}}^{2}-m_{b}^{2}}+\frac{u_{11}^{2}}{M_{U_{2}}^{2}-m_{t}^{2}}-\frac{u_{31}^{2}}{M_{U_{3}}^{2}-M_{U_{2}}^{2}}\right)\right\\}M_{U_{2}}^{2}\ln{\left[\frac{M_{U_{2}}^{2}}{M_{U_{3}}^{2}}\right]}$
$\displaystyle+2\left\\{\frac{u_{21}^{2}+u_{31}^{2}}{m_{t}^{2}-m_{b}^{2}}-\frac{u_{21}^{2}}{M_{U_{2}}^{2}-m_{b}^{2}}-\frac{u_{31}^{2}}{M_{U_{3}}^{2}-m_{b}^{2}}\right\\}m_{b}^{4}\ln{\left[\frac{m_{b}^{2}}{M_{U_{3}}^{2}}\right]}\Bigg{]},$
(B.45)

$\displaystyle S$
$\displaystyle=\frac{N_{c}}{18\pi}\Bigg{[}5\left(-u_{11}^{2}+u_{11}^{4}-u_{21}^{2}u_{31}^{2}\right)$
$\displaystyle-\left\\{1+2u_{11}^{2}-3u_{11}^{4}-6u_{11}^{2}m_{t}^{4}\left(u_{21}^{2}\frac{3M_{U_{2}}^{2}-m_{t}^{2}}{\left(M_{U_{2}}^{2}-m_{t}^{2}\right)^{3}}+u_{31}^{2}\frac{3M_{U_{3}}^{2}-m_{t}^{2}}{\left(M_{U_{3}}^{2}-m_{t}^{2}\right)^{3}}\right)\right\\}\ln{\left[\frac{m_{t}^{2}}{M_{U_{3}}^{2}}\right]}$
$\displaystyle+u_{21}^{2}\left\\{3u_{21}^{2}-4+6M_{U_{2}}^{4}\left(u_{31}^{2}\frac{3M_{U_{3}}^{2}-M_{U_{2}}^{2}}{\left(M_{U_{3}}^{2}-M_{U_{2}}^{2}\right)^{3}}+u_{11}^{2}\frac{M_{U_{2}}^{2}-3m_{t}^{2}}{\left(M_{U_{2}}^{2}-m_{t}^{2}\right)^{3}}\right)\right\\}\ln{\left[\frac{M_{U_{2}}^{2}}{M_{U_{3}}^{2}}\right]}$
$\displaystyle+12\left\\{u_{11}^{2}u_{21}^{2}\frac{m_{t}^{2}M_{U_{2}}^{2}}{\left(M_{U_{2}}^{2}-m_{t}^{2}\right)^{2}}+u_{11}^{2}u_{31}^{2}\frac{m_{t}^{2}M_{U_{3}}^{2}}{\left(M_{U_{3}}^{2}-m_{t}^{2}\right)^{2}}+u_{21}^{2}u_{31}^{2}\frac{M_{U_{2}}^{2}M_{U_{3}}^{2}}{\left(M_{U_{3}}^{2}-M_{U_{2}}^{2}\right)^{2}}\right\\}\Bigg{]},$
(B.46)

$\displaystyle U$
$\displaystyle=\frac{N_{c}}{18\pi}\Bigg{[}\frac{7}{2}\left(1-u_{11}^{4}\right)-\frac{3}{2}\left(u_{21}^{4}+u_{31}^{4}\right)-2\left(u_{21}^{2}+u_{31}^{2}-u_{21}^{2}u_{31}^{2}\right)$
$\displaystyle+12\left\\{m_{b}^{2}\left(\left(u_{11}^{2}-1\right)\frac{m_{t}^{2}}{\left(m_{t}^{2}-m_{b}^{2}\right)^{2}}+u_{21}^{2}\frac{M_{U_{2}}^{2}}{\left(M_{U_{2}}^{2}-m_{b}^{2}\right)^{2}}+u_{31}^{2}\frac{M_{U_{3}}^{2}}{\left(M_{U_{3}}^{2}-m_{b}^{2}\right)^{2}}\right)\right.$
$\displaystyle\left.+m_{t}^{2}u_{11}^{2}\left(u_{21}^{2}\frac{M_{U_{2}}^{2}}{\left(M_{U_{2}}^{2}-m_{t}^{2}\right)^{2}}-u_{31}^{2}\frac{M_{U_{3}}^{2}}{\left(M_{U_{3}}^{2}-m_{t}^{2}\right)^{2}}\right)-u_{21}^{2}u_{31}^{2}\frac{M_{U_{2}}^{2}M_{U_{3}}^{2}}{\left(M_{U_{3}}^{2}-M_{U_{2}}^{2}\right)^{2}}\right\\}$
$\displaystyle+6\left\\{\left(u_{11}^{2}-1\right)\frac{3m_{t}^{2}-m_{b}^{2}}{\left(m_{t}^{2}-m_{b}^{2}\right)^{3}}+u_{21}^{2}\frac{3M_{U_{2}}^{2}-m_{b}^{2}}{\left(M_{U_{2}}^{2}-m_{b}^{2}\right)^{3}}+u_{31}^{2}\frac{3M_{U_{3}}^{2}-m_{b}^{2}}{\left(M_{U_{3}}^{2}-m_{b}^{2}\right)^{3}}\right\\}m_{b}^{4}\ln{\left[\frac{m_{b}^{2}}{M_{U_{3}}^{2}}\right]}$
$\displaystyle-6\left\\{\left(u_{11}^{2}-1\right)\frac{m_{t}^{2}-3m_{b}^{2}}{\left(m_{t}^{2}-m_{b}^{2}\right)^{3}}+u_{11}^{2}u_{21}^{2}\frac{3M_{U_{2}}^{2}-m_{t}^{2}}{\left(M_{U_{2}}^{2}-m_{t}^{2}\right)^{3}}+u_{11}^{2}u_{31}^{2}\frac{3M_{U_{3}}^{2}-m_{t}^{2}}{\left(M_{U_{3}}^{2}-m_{t}^{2}\right)^{3}}\right\\}m_{t}^{4}\ln{\left[\frac{m_{t}^{2}}{M_{U_{3}}^{2}}\right]}$
$\displaystyle+6u_{21}^{2}\left\\{\frac{M_{U_{2}}^{2}-3m_{b}^{2}}{\left(M_{U_{2}}^{2}-m_{b}^{2}\right)^{3}}-u_{11}^{2}\frac{M_{U_{2}}^{2}-3m_{t}^{2}}{\left(M_{U_{2}}^{2}-m_{t}^{2}\right)^{3}}-u_{31}^{2}\frac{3M_{U_{3}}^{2}-M_{U_{2}}^{2}}{\left(M_{U_{3}}^{2}-M_{U_{2}}^{2}\right)^{3}}\right\\}M_{U_{2}}^{4}\ln{\left[\frac{M_{U_{2}}^{2}}{M_{U_{3}}^{2}}\right]}$
$\displaystyle+3\left(1-u_{11}^{4}\right)\ln{\left[\frac{m_{t}^{2}}{M_{U_{3}}^{2}}\right]}-3u_{21}^{2}u_{21}^{2}\ln{\left[\frac{M_{U_{2}}^{2}}{M_{U_{3}}^{2}}\right]}\Bigg{]}.$
(B.47)

The contributions from down sector mixing can be obtained under the
transformation

$\begin{pmatrix}u_{11}&u_{12}&u_{13}\\\ u_{21}&u_{22}&u_{23}\\\
u_{31}&u_{32}&u_{33}\end{pmatrix}\to\begin{pmatrix}c_{1d}&&s_{1d}&&0\\\
-c_{2d}s_{1d}&&c_{1d}c_{2d}&&-s_{2d}\\\
-s_{1d}s_{2d}&&c_{1d}s_{2d}&&c_{2d}\end{pmatrix}$ (B.48)

with $\\{M_{U_{2}}\to M_{D_{2}},\,M_{U_{3}}\to M_{D_{3}},\,m_{t}\to
m_{d},\,\text{and }m_{b}\to m_{u}\\}$

##  Appendix C One-loop Renormalization Group Equations

Here we present the full set of one-loop RGE for the dimensionless parameters
of the LRSM with universal seesaw. These generalize the equations givein in
Ref. Koide:2000ks ; Mohapatra:2014qva .

The gauge couplings $g_{3},\,g_{L},\,g_{R},$ and $g_{B}$ evolve with momentum
according to the renormalization group equations (with $t={\rm ln}\mu$)

$16\pi^{2}\frac{dg_{i}}{dt}=b_{i}g_{i}^{3}$ (C.49)

where $b_{i}=(-3,\,-\frac{19}{6},\,-\frac{19}{6},\,\frac{41}{2})$ for
$i=(3,\,L,\,R,\,B)$.

The Yukawa coupling matrices of Eq. (2.16) evolve according to the equations

$\displaystyle 16\pi^{2}\frac{d{\cal
Y}_{A}^{f}}{dt}=(T_{A}^{f}-G_{A}^{f}+H_{A}^{f})\,Y_{A}^{f},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ A=L,R,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ f=u,d,e.$ (C.50)

Here $T_{A}^{f}$ arise from fermion loops, $G_{A}^{f}$ from gauge boson loops
and $H_{A}^{f}$ from Higgs boson loops. The functions $T_{A}^{f}$ and
$G_{A}^{f}$ are proportional to the unit matrix, which $H_{A}^{f}$ is flavor-
dependent. The explicit forms of these functions are given by Koide:2000ks

$\displaystyle T_{A}^{u}$ $\displaystyle=$ $\displaystyle
T_{A}^{d}=T_{A}^{e}=3[{\rm Tr}(Y^{u}_{A}Y^{u\dagger}_{A})+{\rm
Tr}(Y^{d}_{A}Y^{d\dagger}_{A})]+{\rm Tr}(Y^{e}_{A}Y^{e\dagger}_{A})$
$\displaystyle G_{A}^{u}$ $\displaystyle=$
$\displaystyle\frac{17}{8}g_{B}^{2}+\frac{9}{4}g_{2A}^{2}+8g_{3}^{2}$
$\displaystyle G_{A}^{d}$ $\displaystyle=$
$\displaystyle\frac{5}{8}g_{B}^{2}+\frac{9}{4}g_{2A}^{2}+8g_{3}^{2}$
$\displaystyle G_{A}^{e}$ $\displaystyle=$
$\displaystyle\frac{45}{8}g_{B}^{2}+\frac{9}{4}g_{2A}^{2}$ $\displaystyle
H_{A}^{u}$ $\displaystyle=$ $\displaystyle-
H_{A}^{d}=\frac{3}{2}(Y_{A}^{u}Y_{A}^{u\dagger}-Y_{A}^{d}Y_{A}^{d\dagger})$
$\displaystyle H_{A}^{e}$ $\displaystyle=$
$\displaystyle\frac{3}{2}Y_{A}^{e}Y_{A}^{e\dagger}\leavevmode\nobreak\ .$
(C.51)

The quartic scalar couplings of Eq. (2.6) evolve with momentum according to
the equations

$\displaystyle 16\pi^{2}\frac{\lambda_{1_{L}}}{dt}$ $\displaystyle=$
$\displaystyle
12\lambda_{1_{L}}^{2}+4\lambda_{2}^{2}+\frac{9}{4}(g_{L}^{4}+g_{L}^{2}g_{B}^{2}+\frac{3}{4}g_{B}^{4})-\lambda_{1_{L}}(9g_{L}^{2}+\frac{9}{2}g_{B}^{2})$
(C.52) $\displaystyle+$ $\displaystyle 4\lambda_{1_{L}}\\{3{\rm
Tr}(Y_{L}^{u}Y_{L}^{u\dagger})+3{\rm Tr}(Y_{L}^{d}Y_{L}^{d\dagger})+{\rm
Tr}(Y_{L}^{e}Y_{L}^{e\dagger})\\}$ $\displaystyle-$ $\displaystyle
4\left\\{3\left({\rm Tr}(Y_{L}^{u}Y_{L}^{u\dagger}\right)^{2}+3\left({\rm
Tr}(Y_{L}^{d}Y_{L}^{d\dagger})\right)^{2}+\left({\rm
Tr}(Y_{L}^{e}Y_{L}^{e\dagger}\right)^{2}\right\\}$ $\displaystyle
16\pi^{2}\frac{\lambda_{1_{R}}}{dt}$ $\displaystyle=$ $\displaystyle
12\lambda_{1_{R}}^{2}+4\lambda_{2}^{2}+\frac{9}{4}(g_{R}^{4}+g_{R}^{2}g_{B}^{2}+\frac{3}{4}g_{B}^{4})-\lambda_{1_{R}}(9g_{R}^{2}+\frac{9}{2}g_{B}^{2})$
(C.53) $\displaystyle+$ $\displaystyle 4\lambda_{1_{R}}\\{3{\rm
Tr}(Y_{R}^{u}Y_{R}^{u\dagger})+3{\rm Tr}(Y_{R}^{d}Y_{R}^{d\dagger})+{\rm
Tr}(Y_{R}^{e}Y_{R}^{e\dagger})\\}$ $\displaystyle-$ $\displaystyle
4\left\\{3\left({\rm Tr}(Y_{R}^{u}Y_{R}^{u\dagger}\right)^{2}+3\left({\rm
Tr}(Y_{R}^{d}Y_{R}^{d\dagger})\right)^{2}+\left({\rm
Tr}(Y_{R}^{e}Y_{R}^{e\dagger}\right)^{2}\right\\}\leavevmode\nobreak\ .$

## Acknowledgements

This work is supported by the U.S. Department of Energy under grant number DE-
SC0016013. Some computing for this project was performed at the High-
Performance Computing Center at Oklahoma State University, supported in part
through the National Science Foundation grant OAC-1531128.

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