# Radiatively generated fermion mass hierarchy from flavour non-universal
gauge symmetries

Gurucharan Mohanta<EMAIL_ADDRESS>Theoretical Physics Division,
Physical Research Laboratory, Navarangpura, Ahmedabad-380009, India  Indian
Institute of Technology Gandhinagar, Palaj-382355, India  Ketan M. Patel
<EMAIL_ADDRESS>Theoretical Physics Division, Physical Research Laboratory,
Navarangpura, Ahmedabad-380009, India

###### Abstract

A framework based on a class of abelian gauge symmetries is proposed in which
the masses of only the third generation quarks and leptons arise at the tree
level. The fermions of the first and second families receive their masses
through radiative corrections induced by the new gauge bosons in the loops. It
is shown that the class of abelian symmetries which can viably implement this
mechanism are flavour non-universal in nature. Taking the all-fermion
generalization of the well-known leptonic $L_{\mu}-L_{\tau}$ and
$L_{e}-L_{\mu}$ symmetries, we construct an explicit renormalizable model
based on two $U(1)$ which is shown to reproduce the observed fermion mass
spectrum of the Standard Model. The first and second generation fermion masses
are loop suppressed while the hierarchy between these two generations results
from a gap between the masses of two vector bosons of the extended gauge
symmetries. Several phenomenological aspects of the flavourful new physics are
discussed and lower limits on the masses of the vector bosons are derived.

##  I Introduction

Masses of the elementary fermions in the Standard Model (SM) are incalculable
parameters of the theory and their values are determined entirely from the
observations only. This leaves several unanswered questions about the observed
pattern of the masses and mixings of quarks and leptons Feruglio (2015). One
of the elegant ways to understand the peculiar hierarchical masses of the
charged fermions is to let only the third generation fermions become massive
at the leading order while the masses of the first two generations arise
through quantum corrections. In this way, the masses of the first and second
generation fermions may be made completely or partially calculable parameters
of the theory. Such an approach was considered in Barr and Zee (1978); Wilczek
and Zee (1979); Yanagida (1979a) soon after the inception of the SM.
Subsequent attempts in this direction include generating radiative masses
through loops involving new scalars and fermions Balakrishna (1988a);
Balakrishna  _et al._ (1988); Balakrishna (1988b); Balakrishna and Mohapatra
(1989); Babu and He (1989); Babu  _et al._ (1990); Graham and Rajendran
(2010); Chiang and Yagyu (2021); Chiang  _et al._ (2022); Baker  _et al._
(2021a, b), new scalars and top-quark Dobrescu and Fox (2008) and new gauge
bosons in the extended gauge theories Weinberg (2020); Jana  _et al._ (2022).

In this paper, we explore a framework in a similar direction but based on a
different class of gauge symmetry leading to very different results both
qualitatively and quantitatively. The extended gauge symmetry is abelian and
it ensures that only the third generation quarks and leptons get masses at the
leading order. Subsequently, the masses of the first and second generation
fermions are induced by radiative corrections through the vector bosons of
extended symmetry and heavy massive fermions in the loop and, therefore, are
suppressed. The masses of the first two generation fermions become calculable
parameters albeit they are expressed in terms of some undetermined parameters
of the theory which are not required to take arbitrarily small values. Through
general analysis, we show that the new abelian gauge symmetry must be flavour
non-universal to the basic mechanism to work viably. In turn, this leads to
flavourful new physics in both the quark and lepton sectors offering
phenomenologically rich possibilities. We construct an explicit model based on
two $U(1)$s which are generalizations of the well-known leptonic
$L_{\mu}-L_{\tau}$ and $L_{e}-L_{\mu}$ symmetries and are applied to all the
fermions of a given generation. It is shown through detailed numerical studies
that the model is capable of reproducing the realistic fermion mass spectrum
and it overcomes many inviable predictions found in a recent proposal along a
similar direction but based on a different gauge symmetry Weinberg (2020).

The rest of the paper is organized as follows. The basic mechanism is
discussed in section II. An explicit model based on it is outlined in section
III. In section IV, we give example solutions which reproduce the observed
fermion mass spectrum and discuss the various phenomenological implications in
section V. The study is summarized in section VI.

##  II General Framework

We first discuss the basic mechanism leading to the massive third generation
at the tree level and radiative masses for the lighter generations. Consider a
toy framework with three generations of chiral fermions, $f^{\prime}_{Li}$ and
$f^{\prime}_{Ri}$ ($i=1,2,3$), and a pair of vectorlike fermions
$F^{\prime}_{L,R}$. The fermions are charged under a $U(1)$ gauge symmetry
with the following interaction term:

$-{\cal L}_{\rm
gauge}=g_{X}X_{\mu}\left(q_{L\alpha}\,\overline{f}^{\prime}_{L\alpha}\gamma^{\mu}f^{\prime}_{L\alpha}+q_{R\alpha}\,\overline{f}^{\prime}_{R\alpha}\gamma^{\mu}f^{\prime}_{R\alpha}\right)\,,$
(1)

where $\alpha=1,...,4$,
$f^{\prime}_{L\alpha}=(f^{\prime}_{Li},F^{\prime}_{L})$,
$f^{\prime}_{R\alpha}=(f^{\prime}_{Ri},F^{\prime}_{R})$ and $q_{L4}=q_{R4}$.

The mass Lagrangian in this basis is written as

$-{\cal L}_{m}=\overline{f}^{\prime}_{L\alpha}\,{\cal
M}^{(0)}_{\alpha\beta}\,f^{\prime}_{R\beta}+{\rm h.c.}\,,$ (2)

where the $4\times 4$ tree-level mass matrix ${\cal M}^{(0)}$ has a particular
form

${\cal M}^{(0)}=\left(\begin{array}[]{cc}0_{3\times 3}&(\mu)_{3\times 1}\\\
(\mu^{\prime})_{1\times 3}&m_{F}\end{array}\right)\,.$ (3)

Here, $\mu=(\mu_{1},\mu_{2},\mu_{3})^{T}$ and
$\mu^{\prime}=(\mu^{\prime}_{1},\mu^{\prime}_{2},\mu^{\prime}_{3})$. The above
form of ${\cal M}$ results into a pair of massive and two massless fermions.
For heavy vectorlike fermions, i.e. $m_{F}\gg\mu_{i},\mu^{\prime}_{i}$, the
effective $3\times 3$ mass matrix can be obtained as

$M^{(0)}_{ij}=-\frac{1}{m_{F}}\mu_{i}\,\mu^{\prime}_{j}\,.$ (4)

The above matrix is of rank one and has two vanishing eigenvalues. In this
way, only the third generation is arranged to acquire a tree level mass in the
underlying framework.

To compute higher order corrections to tree-level masses, we obtain the
physical basis using

$f^{\prime}_{L,R}={\cal U}_{L,R}\,f_{L,R}\,,$ (5)

where the unprimed fields are in physical basis and ${\cal U}_{L,R}$ are
$4\times 4$ unitary matrices which can be obtained using

${\cal U}_{L}^{\dagger}\,{\cal M}^{(0)}\,{\cal U}_{R}={\cal D}\equiv{\rm
Diag.}(0,0,m_{3},m_{4})\,.$ (6)

The gauge interactions in Eq. (1), in the new basis, becomes

$-{\cal L}_{\rm gauge}=g_{X}X_{\mu}\left(({\cal
Q}_{L})_{\alpha\beta}\,\overline{f}_{L\alpha}\gamma^{\mu}f_{L\beta}+({\cal
Q}_{R})_{\alpha\beta}\,\overline{f}_{R\alpha}\gamma^{\mu}f_{R\beta}\right)\,,$
(7)

where

${\cal Q}_{L,R}={\cal U}_{L,R}^{\dagger}\,q_{L,R}\,{\cal U}_{L,R}\,,$ (8)

and $q_{L}={\rm Diag.}(q_{L1},q_{L2},...)$ and so on. The matrices ${\cal
Q}_{L,R}$ are not diagonal in general.

At 1-loop order, the fermions of the first and second generations can receive
masses through diagrams involving the gauge boson and massive fermions in the
loop. The one-particle-irreducible two-point function evaluated in the
Feynman-’t Hooft gauge gives the following results.

$\Sigma_{\alpha\beta}(p=0)=\sigma^{L}_{\alpha\beta}\,P_{L}+\sigma^{R}_{\alpha\beta}\,P_{R}\,,$
(9)

with

$\displaystyle\sigma^{L}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{4\pi^{2}}\,\sum_{\gamma}({\cal
Q}_{R})_{\alpha\gamma}\,({\cal
Q}_{L})_{\gamma\beta}\,m_{\gamma}\,B_{0}[M_{X}^{2},m_{\gamma}^{2}]\,,$
$\displaystyle\sigma^{R}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{4\pi^{2}}\,\sum_{\gamma}({\cal
Q}_{L})_{\alpha\gamma}\,({\cal
Q}_{R})_{\gamma\beta}\,m_{\gamma}\,B_{0}[M_{X}^{2},m_{\gamma}^{2}]\,,$ (10)

where

$B_{0}[M^{2},m^{2}]\equiv\Delta_{\epsilon}-\frac{M^{2}\ln M^{2}-m^{2}\ln
m^{2}}{M^{2}-m^{2}}\,,$ (11)

is Passarino-Veltmann function and

$\Delta_{\epsilon}=\frac{2}{\epsilon}+1-\gamma+\ln 4\pi\,,$ (12)

is divergent part of loop integration when $\epsilon\to 0$. The explicit
derivation of the above result is given in Appendix A.

The 1-loop corrected fermion mass matrix obtained using Eqs. (6,II) can be
written as

${\cal M}={\cal M}^{(0)}+\delta{\cal M}\,,$ (13)

where

$\delta{\cal M}={\cal U}_{L}\,\sigma^{R}\,{\cal U}_{R}^{\dagger}\,.$ (14)

In general, the loop contributions $\delta{\cal M}$ has divergent terms
proportional to $\Delta_{\epsilon}$. The renormalizability requires that the
$3\times 3$ upper-left block of $\delta{\cal M}$ should be finite as there are
no counterterms that could remove the divergences. Denoting the divergent part
of $\delta{\cal M}$ as $\delta{\cal M}_{\rm div}$, we find

$\delta{\cal M}_{\rm div}\propto{\cal U}_{L}\,{\cal Q}_{L}\,{\cal D}\,{\cal
Q}_{R}\,{\cal U}_{R}^{\dagger}=q_{L}\,{\cal M}^{(0)}\,q_{R}\,,$ (15)

where the last equality follows from Eqs. (6,8). Using ${\cal M}$ given in Eq.
(3) and since $q_{L,R}$ are diagonal matrices, one finds

$\left(\delta{\cal M}_{\rm div}\right)_{ij}=0\,.$ (16)

Therefore, the $3\times 3$ upper-left block of $\delta{\cal M}$ is finite as
it is expected from the renormalizabilty Barr and Zee (1978); Weinberg (1972).

The finite part of $\delta{\cal M}$ is of our main interest and it can be
simplified to

$\left(\delta{\cal
M}\right)_{\alpha\beta}=\frac{g_{X}^{2}}{4\pi^{2}}q_{L\alpha}q_{R\beta}\,\sum_{\gamma}\left({\cal
U}_{L}\right)_{\alpha\gamma}\left({\cal
U}_{R}^{*}\right)_{\beta\gamma}\,m_{\gamma}\,b_{0}[M_{X}^{2},m_{\gamma}^{2}]\,,$
(17)

where

$b_{0}[M^{2},m^{2}]\equiv-\frac{M^{2}\ln M^{2}-m^{2}\ln
m^{2}}{M^{2}-m^{2}}\,.$ (18)

Further simplification is possible in the seesaw approximation,
$m_{F}\gg\mu_{i},\mu^{\prime}_{i}$. In this case, ${\cal U}_{L,R}$ can be
written in the following form Joshipura  _et al._ (2020)

${\cal U}_{L,R}\simeq\left(\begin{array}[]{cc}U_{L,R}&-\rho_{L,R}\\\
\rho_{L,R}^{\dagger}U_{L,R}&1\end{array}\right)\,,$ (19)

where $\rho_{L}=-m_{F}^{-1}\mu$ and
$\rho_{R}^{\dagger}=-m_{F}^{-1}\mu^{\prime}$ are $3\times 1$ and $1\times 3$
are matrices, respectively. $U_{L,R}$ are $3\times 3$ matrices that
diagonalize $M^{(0)}$ given in Eq. (4). Explicitly,

$U_{L}^{\dagger}\,M^{(0)}\,U_{R}={\rm Diag.}(0,0,m_{3})\,.$ (20)

Using the definition Eq. (4), the above equation can also be written as

$\left(U_{L}\right)_{i3}\left(U_{R}^{*}\right)_{j3}\,m_{3}\,=M^{(0)}_{ij}=-\frac{1}{m_{F}}\mu_{i}\mu^{\prime}_{j}\,.$
(21)

Substituting Eqs. (19,20) in Eq. (17), we find

$\displaystyle\left(\delta{\cal M}\right)_{ij}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{X}^{2}}{4\pi^{2}}q_{Li}q_{Rj}\,\left(U_{L}\right)_{i3}\left(U_{R}^{*}\right)_{j3}\,m_{3}\,\left(b_{0}[M_{X}^{2},m_{3}^{2}]-b_{0}[M_{X}^{2},m_{F}^{2}]\right)\,,$
(22) $\displaystyle=$
$\displaystyle\frac{g_{X}^{2}}{4\pi^{2}}q_{Li}q_{Rj}\,M^{(0)}_{ij}\left(b_{0}[M_{X}^{2},m_{3}^{2}]-b_{0}[M_{X}^{2},m_{F}^{2}]\right)\,,$

with $i,j=1,2,3$ and repeated indices are not summed. The second line in the
above equation follows from Eq. (21). The correction to the remaining
elements, after some straight-forward algebraic simplification, are obtained
as

$\displaystyle\left(\delta{\cal M}\right)_{i4}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{X}^{2}}{4\pi^{2}}q_{Li}q_{R4}\,\mu_{i}\,\left(b_{0}[M_{X}^{2},m_{F}^{2}]+\sum_{j}\frac{|\mu^{\prime}_{j}|^{2}}{m_{F}^{2}}\,b_{0}[M_{X}^{2},m_{3}^{2}]\right)\,,$
$\displaystyle\left(\delta{\cal M}\right)_{4i}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{X}^{2}}{4\pi^{2}}q_{L4}q_{Ri}\,\mu^{\prime}_{i}\,\left(b_{0}[M_{X}^{2},m_{F}^{2}]+\sum_{j}\frac{|\mu_{j}|^{2}}{m_{F}^{2}}\,b_{0}[M_{X}^{2},m_{3}^{2}]\right)\,,$
$\displaystyle\left(\delta{\cal M}\right)_{44}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{X}^{2}}{4\pi^{2}}q_{L4}q_{R4}\,m_{F}\,\left(b_{0}[M_{X}^{2},m_{F}^{2}]-\frac{m_{3}^{2}}{m_{F}^{2}}\,b_{0}[M_{X}^{2},m_{3}^{2}]\right)\,.$
(23)

Using the above results, the 1-loop corrected fermion mass matrix, Eq. (13),
can be parametrized as

${\cal M}=\left(\begin{array}[]{cc}(\delta M)_{3\times
3}&(\tilde{\mu})_{3\times 1}\\\ (\tilde{\mu}^{\prime})_{1\times
3}&\tilde{m}_{F}\end{array}\right)\,,$ (24)

with

$\left(\delta M\right)_{ij}=\left(\delta{\cal
M}\right)_{ij}\,,~{}\tilde{\mu}_{i}=\mu_{i}+\left(\delta{\cal
M}\right)_{i4}\,,~{}\tilde{\mu}^{\prime}_{i}=\mu^{\prime}_{i}+\left(\delta{\cal
M}\right)_{4i}\,,~{}\tilde{m}_{F}=m_{F}+\left(\delta{\cal M}\right)_{44}\,.$
(25)

In the seesaw approximation, $\delta
M_{ij}\ll\tilde{\mu}_{i},\tilde{\mu}^{\prime}_{i}\ll\tilde{m}_{F}$, the
effective $3\times 3$ mass matrix for the lighter fermions can then be written
as

$M=\delta M-\frac{1}{\tilde{m}_{F}}\tilde{\mu}\tilde{\mu}^{\prime}\,.$ (26)

Comparing the above with Eq. (4), it is noticed that the second term is
similar to $M^{(0)}$ with original elements replaced by their 1-loop corrected
values. This contribution is still of rank-1 and contributes only to the
masses of the third generation. The first term can induce masses for the first
and/or second generation fermions depending on the chosen $U(1)$ charges.

It is important to note that the flavour universal $U(1)$ symmetry cannot
induce radiative masses for the lighter generations. This can be understood as
the following. For $q_{L1}=q_{L2}=q_{L3}$ and $q_{R1}=q_{R2}=q_{R3}$, one
finds $\delta M\propto M^{(0)}$ from Eq. (22) and $\tilde{\mu}\propto\mu$,
$\tilde{\mu^{\prime}}\propto\mu^{\prime}$, $\tilde{m_{F}}\propto m_{F}$ from
Eq. (II). Altogether, this implies $M\propto M^{(0)}$ and hence the 1-loop
corrected mass matrix remains of rank one. Therefore, a flavour non-universal
$U(1)$ is necessarily required in order to generate the masses for the first
and second generation fermions. We find that this observation of ours is in
conflict with the results of Jana  _et al._ (2022) which uses flavour
universal $U(1)_{B-L}$ to induce radiative masses for the first generation
fermions.

It can be also noted that for a generic choice of $U(1)$ charges, $\delta M$
could lead to masses of a similar magnitude for the first and second
generation fermions. To generate only the second generation fermion masses at
1-loop, one can choose $q_{L1}=q_{R1}=0$. As it can be seen from Eq. (22),
this leads to a massless first generation and loop-suppressed mass for the
second generation. Masses for the first generation fermions can be induced
similarly by introducing another flavoured $U(1)$ under which the first
generation fermions are non-trivially charged. The mass hierarchy between the
first and second generations can be arranged by choosing hierarchical masses
for the gauge bosons of two $U(1)$s. For example, the loop integration factor
for $M_{X}\gg m_{F}\gg m_{3}$ can be simplified as

$b_{0}[M_{X}^{2},m_{3}^{2}]-b_{0}[M_{X}^{2},m_{F}^{2}]\simeq-\frac{m_{F}^{2}}{M_{X}^{2}}\,\ln\frac{m_{F}^{2}}{M_{X}^{2}}\,.$
(27)

Therefore, two $U(1)$s with $M_{X_{2}}\gg M_{X_{1}}$ can lead to hierarchical
masses of first and second generations even though both are generated at
1-loop. A suitable choice of $U(1)$ charges which can achieve this is
discussed in the next section.

##  III A model

As discussed in the last section, the proposed mechanism requires at least two
$U(1)$ symmetries under which the three generations of the SM quarks and
leptons are charged non-universally. Therefore, we choose
$G_{F}=U(1)_{1}\times U(1)_{2}$ as an extended gauge symmetry for the model.
In addition to three generations of the SM quarks and leptons,
$Q_{Li}\sim(3,2,\frac{1}{6})$, $u_{Ri}\sim(3,1,\frac{2}{3})$,
$d_{Ri}\sim(3,1,-\frac{1}{3})$, $L_{Li}\sim(1,2,-\frac{1}{2})$ and
$e_{Ri}\sim(1,1,-1)$, we introduce three copies of a pair of the Higgs
doublets, i.e. $H_{ui}\sim(1,2,-\frac{1}{2})$, $H_{di}\sim(1,2,\frac{1}{2})$
and SM singlets $\eta_{i}\sim(1,1,0)$. We also introduce a pair of vectorlike
fermions in each sector, namely $T_{L,R}\sim(3,1,\frac{2}{3})$,
$B_{L,R}\sim(3,1,-\frac{1}{3})$ and $E_{L,R}\sim(1,1,-1)$. In the above, the
quantities in the brackets denote the transformation properties under
$(SU(3)_{C},SU(2)_{L},U(1)_{Y})$.

Under the new $G_{F}=U(1)_{1}\times U(1)_{2}$ symmetry, all the first, second
and third generation fermions and scalars have charges $(0,1)$, $(1,-1)$ and
$(-1,0)$, respectively. In this way, $U(1)_{1}$ can be identified as “$2-3$
symmetry” and $U(1)_{2}$ as “$1-2$ symmetry”. They are generalizations of
$L_{\mu}-L_{\tau}$ and $L_{e}-L_{\mu}$ symmetries, respectively, discussed in
the literature in the context of leptons Foot (1991); He  _et al._ (1991a, b).
The vectorlike fermions are neutral under the $G_{F}$. The fermion and scalar
fields and their charges under the SM and $G_{F}$ are summarized in Table 1.
It is straightforward to check that the $G_{F}$ is non-anomalous since a pair
of families of fermions and scalars have equal and opposite charges under each
$U(1)$.

Fields |  $(SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y})$ |  $U(1)_{1}$ |  $U(1)_{2}$
---|---|---|---
$Q_{L_{i}}$ | $(3,2,\frac{1}{6})$ | {0,1,-1} | {1,-1,0}
$u_{R_{i}}$ | $(3,1,\frac{2}{3})$ | {0,1,-1} | {1,-1,0}
$d_{R_{i}}$ | $(3,1,-\frac{1}{3})$ | {0,1,-1} | {1,-1,0}
$L_{L_{i}}$ | $(1,2,-\frac{1}{2})$ | {0,1,-1} | {1,-1,0}
$e_{R_{i}}$ | $(1,1,-1)$ | {0,1,-1} | {1,-1,0}
$H_{u_{i}}$ | $(1,2,-\frac{1}{2})$ | {0,1,-1} | {1,-1,0}
$H_{d_{i}}$ | $(1,2,\frac{1}{2})$ | {0,1,-1} | {1,-1,0}
$\eta_{i}$ | $(1,1,0)$ | {0,1,-1} | {1,-1,0}
$T_{L},T_{R}$ | $(3,1,\frac{2}{3})$ | 0 | 0
$B_{L},B_{R}$ | $(3,1,-\frac{1}{3})$ | 0 | 0
$E_{L},E_{R}$ | $(1,1,-1)$ | 0 | 0
Table 1: The SM and $G_{F}$ charges of various fermions and scalars of the
model. Here, $i=1,2,3$ denote three generations and their respective charges
under new $U(1)$ are represented as $\\{q_{1},q_{2},q_{3}\\}$.

###  III.1 Charged fermion masses

The most general renormalizable couplings between the fermions and scalars
invariant under the SM gauge symmetry and $G_{F}$ can be written as

$\displaystyle-{\cal L}_{Y}$ $\displaystyle=$
$\displaystyle{y_{u}}_{i}\,\overline{Q_{L}}_{i}\,{H_{u}}_{i}\,T_{R}\,+{y_{u}^{\prime}}_{i}\,\overline{T_{L}}\,\eta^{*}_{i}\,u_{Ri}\,+\,{y_{d}}_{i}\,\overline{Q_{L}}_{i}\,{H_{d}}_{i}\,B_{R}\,+{y_{d}^{\prime}}_{i}\,\overline{B_{L}}\,\eta^{*}_{i}\,d_{Ri}\,$
(28) $\displaystyle+$
$\displaystyle{y_{e}}_{i}\,\overline{L_{L}}_{i}\,{H_{d}}_{i}\,E_{R}\,+{y_{e}^{\prime}}_{i}\,\overline{E_{L}}\,\eta^{*}_{i}\,e_{Ri}\,+\,{\rm
h.c.}\,.$

The direct couplings between two SM fermions and Higgs are forbidden by
$G_{F}$. The masses of vectorlike fermions are parametrized as

$-{\cal
L}_{m}=m_{T}\,\overline{T_{L}}\,T_{R}\,+\,m_{B}\,\overline{B_{L}}\,B_{R}\,+\,m_{E}\,\overline{E_{L}}\,E_{R}\,+\,{\rm
h.c.}\,.$ (29)

The spontaneous breaking of $G_{F}$ through the Vacuum Expectation Values
(VEVs) of $\eta_{i}$ and ${H_{ui,di}}$ gives rise to $4\times 4$ mass matrices
for the charged fermions identical to the one given in Eq. (3). Explicitly,

${\cal M}_{u,d,e}=\left(\begin{array}[]{cc}0&\left(\mu_{u,d,e}\right)_{3\times
1}\\\ \left(\mu^{\prime}_{u,d,e}\right)_{1\times
3}&m_{T,B,E}\end{array}\right)\,,$ (30)

where

${\mu_{u}}_{i}={y_{u}}_{i}{v_{u}}_{i}\,,~{}~{}{\mu_{d}}_{i}={y_{d}}_{i}{v_{d}}_{i}\,,~{}~{}{\mu_{e}}_{i}={y_{e}}_{i}{v_{d}}_{i}\,,$
(31)
${\mu^{\prime}_{u}}_{i}={y^{\prime}_{u}}_{i}\langle\eta_{i}\rangle\,,~{}~{}{\mu^{\prime}_{d}}_{i}={y^{\prime}_{d}}_{i}\langle\eta_{i}\rangle\,,~{}~{}{\mu^{\prime}_{e}}_{i}={y^{\prime}_{e}}_{i}\langle\eta_{i}\rangle\,,$
(32)

and ${v_{u}}_{i}\equiv\langle{H_{u}}_{i}\rangle$,
${v_{d}}_{i}\equiv\langle{H_{d}}_{i}\rangle$. The repeated indices do not
imply summation in the above. The above VEVs are chosen such that they break
$G_{F}$ entirely and $SU(2)_{L}\times U(1)_{Y}$ into $U(1)_{\rm em}$. The most
general potential of the model is given in Appendix B. It is shown that there
exist a large number of parameters which may allow the desired solution for
the VEVs although we do not study the potential minimization in detail. The
effective $3\times 3$ mass matrix in each sector, analogous to Eq. (4), can be
written as

$M_{u,d,e}^{(0)}\equiv-\frac{1}{m_{T,B,E}}\,\mu_{u,d,e}\,\mu^{\prime}_{u,d,e}\,.$
(33)

The above matrices are of rank one and give masses to the third generation
charged fermions.

Following the procedure outlined in the previous section and Eq. (26), the
1-loop corrected effective $3\times 3$ mass matrices for the charged fermions
can be obtained as

$M_{f}=\delta M_{f}+M_{f}^{(0)}\,,$ (34)

where $f=u,d,e$ and the second term is written using Eq. (33). In comparison
to general result given in Eq. (26), note that the second term is same as the
tree level effective mass matrix. The loop corrections do not modify the
parameters $\mu_{i}$, $\mu^{\prime}_{i}$ and $m_{F}$ as the vectorlike
fermions are neutral under the $G_{F}$. $\delta M_{f}$ contains 1-loop
correction induced by the gauge bosons of both the $U(1)$s. Using the given
$U(1)_{1,2}$ charges and Eq. (22), we find

$\displaystyle\delta M_{f}$ $\displaystyle=$
$\displaystyle\frac{N_{f}g_{1}^{2}}{4\pi^{2}}\left(b_{0}[M_{Z_{1}}^{2},m_{f3}^{2}]-b_{0}[M_{Z_{1}}^{2},m_{F}^{2}]\right)\,\left(\begin{array}[]{ccc}0&0&0\\\
0&\left(M_{f}^{(0)}\right)_{22}&-\left(M_{f}^{(0)}\right)_{23}\\\
0&-\left(M_{f}^{(0)}\right)_{32}&\left(M_{f}^{(0)}\right)_{33}\end{array}\right)$
(38) $\displaystyle+$
$\displaystyle\frac{N_{f}g_{2}^{2}}{4\pi^{2}}\left(b_{0}[M_{Z_{2}}^{2},m_{f3}^{2}]-b_{0}[M_{Z_{2}}^{2},m_{F}^{2}]\right)\,\left(\begin{array}[]{ccc}\left(M_{f}^{(0)}\right)_{11}&-\left(M_{f}^{(0)}\right)_{12}&0\\\
-\left(M_{f}^{(0)}\right)_{21}&\left(M_{f}^{(0)}\right)_{22}&0\\\
0&0&0\end{array}\right)\,.$ (42)

Here, $g_{i}$ is the gauge coupling and $M_{Z_{i}}$ is the mass of the gauge
boson of $U(1)_{i}$. $N_{f}=3$ ($N_{f}=1$) for $f=u,d$ ($f=e$) is the factor
due to color degrees of freedom. $m_{f3}$ and $m_{F}$ are masses of the third
generation fermion and vectorlike fermion, respectively, in each sector.

The charged fermion mass matrix $M_{f}$ defined in Eq. (34) along with
expressions (33) and (38) gives the 1-loop corrected mass matrix in the model.
Each of the 1-loop generated contributions in $\delta M_{f}$ are of rank one
and induce masses for the first and second generation fermions individually.
The hierarchy among the masses of the lighter generations can be arranged by
choosing $M_{Z_{1}}\ll M_{Z_{2}}$. This suppression is common for all the
charged fermions.

Note that in determining Eq. (38), we have considered 1-loop corrections only
from the gauge boson loops. In the present model, radiative corrections also
arise from the emission and absorption of scalar bosons in the loop. Such a
contribution requires mixing between the scalar fields $\eta_{i}$ and
$H_{ui,di}$ as they exclusively couple to the right and left chiral fermions,
respectively. These corrections can be made suppressed by choosing the scalar
masses much greater than the $M_{Z_{1}}$ Weinberg (2020) and/or by assuming
small mixing between $\eta_{i}$ and $H_{ui,di}$ fields Jana  _et al._ (2022).
We assume that such suppression is arranged and do not consider the radiative
corrections from the scalar bosons.

We have also assumed that there are no kinetic mixings between different
$U(1)$. Presence of such a mixing, for example
$\epsilon{F_{1}}_{\mu\nu}{F_{2}}^{\mu\nu}$ where $F_{1,2}$ are the field
strengths of $Z_{1,2}$ bosons, can allow the first-generation fermions to
receive masses from the 1-loop diagrams involving $Z_{1}$ boson. This can be
understood from the fact that an effective coupling of order $\epsilon g_{1}$
gets induced between the first-generation fermions and $Z_{1}$ when the
kinetic mixing term is rotated away to obtain the physical gauge bosons (see
for example Babu  _et al._ (1996)). Both the first and second generations
receive mass from $Z_{1}$ loop in this case, however, the former is suppressed
by an additional factor of $\epsilon^{2}$. Therefore, the presence of kinetic
mixing is not expected to spoil the hierarchies between the first two
generations in this model if $\epsilon\ll M_{Z_{1}}/M_{Z_{2}}$.

###  III.2 Neutrino masses

Although our main aim is to explain charged fermion mass hierarchies, we also
comment on the possibility of neutrino mass generation within this model. Most
simply, the naturally small neutrino masses can be accommodated by introducing
three RH neutrinos, singlet under both the SM gauge symmetry and $G_{F}$, and
allowing Majorana masses for them. The gauge invariant renormalizable
interactions can be parametrized as

$-{\cal
L}_{\nu}={y_{D}}_{ij}\,\overline{L_{L}}_{i}\,{H_{u}}_{i}\,{\nu_{R}}_{j}+\frac{1}{2}{M_{R}}_{ij}\,\nu_{Ri}^{T}C^{-1}\nu_{Rj}+{\rm
h.c.}\,.$ (43)

Electroweak symmetry breaking gives rise to the Dirac neutrino mass matrix,
${M_{D}}_{ij}={y_{D}}_{ij}{v_{u}}_{i}$. For $M_{R}\gg M_{D}$, the usual type I
seesaw mechanism Minkowski (1977); Yanagida (1979b); Mohapatra and Senjanovic
(1980) can be realized and the light neutrino mass matrix is given by

$M_{\nu}=-M_{D}\,M_{R}^{-1}\,M_{D}^{T}\,.$ (44)

Unlike the charged fermions, all the three neutrino masses can arise at the
tree level in general. This is a welcome feature as the inter-generation
hierarchies in the neutrino masses are not as strong as that in the charged
fermion masses. All the elements of the Dirac neutrino Yukawa coupling matrix
$y_{D}$ can be of ${\cal O}(1)$ leading to an anarchic structure for
$M_{\nu}$. This, in turn, can explain the relatively feeble hierarchy in the
neutrino masses and the large mixing in the lepton sector Hall  _et al._
(2000); de Gouvea and Murayama (2015).

##  IV Example solutions

We investigate the viability of the proposed framework in reproducing charged
fermion masses and quark mixing by finding numerical values for the parameters
$\mu_{fi}$, $\mu^{\prime}_{fi}$, $m_{T}$, $m_{B}$, $m_{E}$ and $M_{Z_{1,2}}$.
It can be noted from Eq. (28) that $y_{ui}$, $y^{\prime}_{ui}$,
$y^{\prime}_{di}$, $y_{ei}$ and $y^{\prime}_{ei}$ can be chosen real by
rotating away their phases through redefinitions of the various quarks and
lepton fields. Similarly, one of the $y_{di}$ can also be made real. An
analogous treatment in Eq. (29) leads to real $m_{T}$, $m_{B}$ and $m_{E}$.
Moreover, we assume that all the VEVs are real. Altogether, this implies 25
real parameters (real $\mu_{ui}$, $\mu^{\prime}_{ui}$, $\mu_{d3}$,
$\mu^{\prime}_{di}$, $\mu_{ei}$, $\mu^{\prime}_{ei}$, $m_{T}$, $m_{B}$,
$m_{E}$, $M_{Z_{1}}$, $M_{Z_{2}}$ and complex $\mu_{d1}$, $\mu_{d2}$) which
can be used to determine 13 observables (9 charged fermion masses, 3 angles
and a Dirac CP phase of the quark mixing matrix). The number of parameters is
greater than the number of observables and hence one expects to get viable
solutions. Nevertheless, considering that masses and mixing observables are
complex non-linear functions of the input parameters and the latter are
expected to take not-so-hierarchical values in the present model, it is not
obvious that viable solutions would exist.

We determine the 25 real parameters of the underlying framework through the
usual $\chi^{2}$ function minimization technique. The $\chi^{2}$ function (see
for example Mummidi and Patel (2021) for the definition and details) consists
of 13 observables, the mean values and standard deviations of which are listed
in Table 2.

Observable |  Value |  Observable |  Value
---|---|---|---
$m_{u}$ |  $1.27\pm 0.50$ MeV | $m_{e}$ |  $0.487\pm 0.049$ MeV
$m_{c}$ |  $0.619\pm 0.084$ GeV | $m_{\mu}$ |  $1.027\pm 0.103$ MeV
$m_{t}$ |  $171.7\pm 3.0$ GeV | $m_{\tau}$ |  $1.746\pm 0.174$ GeV
$m_{d}$ |  $2.90\pm 1.24$ MeV | $|V_{us}|$ | $0.22500\pm 0.00067$
$m_{s}$ |  $0.055\pm 0.016$ GeV | $|V_{cb}|$ | $0.04182\pm 0.00085$
$m_{b}$ |  $2.89\pm 0.09$ GeV | $|V_{ub}|$ | $0.00369\pm 0.00011$
|  | $J_{\rm CP}$ | $(3.08\pm 0.15)\times 10^{-5}$
Table 2: Values of charged fermion masses and CKM parameters extrapolated at
$M_{Z}$ used in the fits to obtain the example solutions. The values of the
charged fermion masses and quark mixing parameters are taken from Xing  _et
al._ (2008) and Workman and Others (2022), respectively.

For definiteness on the mass scale of new physics, we take three different
values for $M_{Z_{1}}$ and obtain a benchmark solution for each. The optimized
values of the remaining parameters are listed in Table 3 for each solution.

Parameters |  Solution 1 (S1) |  Solution 2 (S2) |  Solution 3 (S3)
---|---|---|---
$M_{Z_{1}}$ | $10^{4}$ | $10^{6}$ | $10^{8}$
$M_{Z_{2}}$ | $2.8708\times 10^{5}$ | $1.4470\times 10^{7}$ | $1.9110\times 10^{9}$
$m_{T}$ | $1.1000\times 10^{4}$ | $1.1000\times 10^{6}$ | $1.1003\times 10^{8}$
$m_{B}$ | $3.0754\times 10^{5}$ | $2.4839\times 10^{7}$ | $1.4015\times 10^{9}$
$m_{E}$ | $2.8128\times 10^{5}$ | $4.9462\times 10^{7}$ | $6.7820\times 10^{8}$
$\mu_{u1}$ | $1.8023\times 10^{1}$ | $-1.6702\times 10^{1}$ | $1.5410\times 10^{1}$
$\mu_{u2}$ | $3.0901\times 10^{2}$ | $3.0969\times 10^{2}$ | $-0.6860$
$\mu_{u3}$ | $1.4763$ | $-0.5133$ | $2.8599\times 10^{2}$
$\mu^{\prime}_{u1}$ | $-3.9573\times 10^{3}$ | $3.3923\times 10^{5}$ | $-3.8518\times 10^{7}$
$\mu^{\prime}_{u2}$ | $3.5446\times 10^{3}$ | $-3.9343\times 10^{5}$ | $-3.7762\times 10^{7}$
$\mu^{\prime}_{u3}$ | $-2.8507\times 10^{3}$ | $29653\times 10^{5}$ | $-3.7718\times 10^{7}$
$\mu_{d1}$ | $2.3809\times 10^{1}+i\ 6.7460$ | $-1.0402\times 10^{1}+i\ 6.7204$ | $4.6117+i\ 4.3534$
$\mu_{d2}$ | $1.4422\times 10^{2}+i\ 2.6938\times 10^{1}$ | $2.7279\times 10^{1}-i\ 1.2285\times 10^{2}$ | $-0.1868-i\ 0.8509$
$\mu_{d3}$ | $5.6345$ | $-2.1306$ | $1.0439\times 10^{2}$
$\mu^{\prime}_{d1}$ | $-6.1152\times 10^{2}$ | $8.8600\times 10^{4}$ | $-2.2724\times 10^{7}$
$\mu^{\prime}_{d2}$ | $3.7385\times 10^{3}$ | $3.4743\times 10^{5}$ | $2.4618\times 10^{7}$
$\mu^{\prime}_{d3}$ | $-7.5116\times 10^{2}$ | $1.2636\times 10^{5}$ | $-1.9748\times 10^{7}$
$\mu_{e1}$ | $8.1306\times 10^{1}$ | $-7.6362\times 10^{1}$ | $-0.9180$
$\mu_{e2}$ | $2.7874\times 10^{1}$ | $-1.5295\times 10^{2}$ | $-9.6383$
$\mu_{e3}$ | $-1.1628$ | $3.2793$ | $-2.2318\times 10^{1}$
$\mu^{\prime}_{e1}$ | $-1.2295\times 10^{3}$ | $-1.9688\times 10^{5}$ | $2.7449\times 10^{7}$
$\mu^{\prime}_{e2}$ | $-3.9625\times 10^{3}$ | $-3.8735\times 10^{5}$ | $-3.5794\times 10^{7}$
$\mu^{\prime}_{e3}$ | $3.9792\times 10^{3}$ | $2.9732\times 10^{4}$ | $2.0714\times 10^{7}$
Table 3: Optimized values of various input parameters obtained for three
example solutions. All the values are in GeV.

All the solutions provide an excellent fit to the observables and reproduce
their central values with a total $\chi^{2}\ll 1$ in all three cases. We do
not include neutrino masses and lepton mixing in the fit as they are given by
an entirely different set of parameters, see Eq. (44), which are arbitrary.
The latter can be chosen to reproduce viable neutrino masses and mixing
parameters.

The example solutions given in Table 3 indicate that the model can reproduce a
realistic charged fermion spectrum irrespective of the scales of $U(1)_{1,2}$
breaking. As discussed earlier, the mass gap between $Z_{1}$ and $Z_{2}$ is
determined by the mass hierarchies between the first and second generation
fermions and one finds $M_{Z_{2}}^{2}/M_{Z_{1}}^{2}\simeq{\cal O}(10^{2})$,
however, the absolute scale of $M_{Z_{1}}$ is not fixed. We impose
$M_{Z_{1}}\leq m_{T},m_{B},m_{E}$ while performing the numerical fits. It is
found that $m_{T}\ll m_{B},m_{E}$ is required in order to generate $m_{t}\gg
m_{b},m_{\tau}$. The parameters $\mu_{fi}$, $f=u,d,e$, are generated through
electroweak symmetry breaking, see Eq. (31), for which we impose
$|y_{fi}|<\sqrt{4\pi}$ and $v_{ui},v_{di}<174$ GeV. As a result, all the
$\mu_{fi}$ are found to be of ${\cal O}(100)$ GeV or less. On the other hand,
$\mu^{\prime}_{fi}$ are induced by the VEVs of SM singlet but $U(1)_{1,2}$
charged fields and their determined values are close to the breaking scale of
$U(1)_{1,2}$. The most noteworthy feature of each of the solutions obtained in
Table 3 is that there are no large hierarchies within the values of various
$\mu_{fi}$ or $\mu^{\prime}_{fi}$. This implies that the dimensionless
parameters of the underlying framework can be of the same magnitude. Despite
this, the observed difference of five orders of magnitude between the masses
of the first and third generation fermions is achieved through the radiative
mass generation mechanism.

##  V Phenomenological aspects

The model has rich phenomenological implications due to the presence of two
flavourful $U(1)$ gauge symmetries and additional vectorlike pair of quarks
and leptons. For all the solutions listed in the previous section, one finds
$Z_{1}$ as the lightest among all the new particles. We, therefore, discuss
various constraints on $Z_{1}$ boson arising from the processes involving
quark and lepton Flavour Changing Neutral Currents (FCNCs).

In the physical basis of quarks and leptons, the couplings of $Z_{1,2}$ can be
determined from Eqs. (7,19) as:

$-{\cal
L}_{Z_{1,2}}=\sum_{k=1,2}\,g_{k}\,\left(\left(X^{(k)}_{f_{L}}\right)_{ij}\,\overline{f_{Li}}\,\gamma^{\mu}f_{Lj}+\left(X^{(k)}_{f_{R}}\right)_{ij}\,\overline{f_{Ri}}\,\gamma^{\mu}f_{Rj}\right)Z_{k\mu}\,,$
(45)

where $f=u,d,e$. The $3\times 3$ coupling matrices are given by

$X^{(k)}_{f_{L}}=U_{f_{L}}^{\dagger}\,q^{(k)}_{fL}\,U_{f_{L}}\,,$ (46)

and similar expression for $X^{(k)}_{f_{R}}$ can be obtained by replacing
$L\to R$. In the present framework, we have $q^{(1)}_{fL}=q^{(1)}_{f_{R}}={\rm
Diag.}(0,1,-1)$ and $q^{(2)}_{fL}=q^{(2)}_{f_{R}}={\rm Diag.}(1,-1,0)$ for all
$f$ as specified earlier. The unitary matrices $U_{f_{L}}$ and $U_{f_{R}}$ are
obtained from diagonalizing the 1-loop corrected mass matrices $M_{f}$ such
that $U_{f_{L}}^{\dagger}M_{f}U_{f_{R}}={\rm
Diag.}(m_{f_{1}},m_{f_{2}},m_{f_{3}})$. Since the underlying $U(1)$ symmetries
are non-universal, $X^{(k)}_{f_{L}}$ and $X^{(k)}_{f_{R}}$ are non-diagonal in
general. This can lead to large flavour violations both in the quark and
lepton sectors. The numerical values of various $X^{(k)}_{f_{L}}$ and
$X^{(k)}_{f_{R}}$ for one of the benchmark solutions are given in Appendix C
for reference.

###  V.1 Quark flavour violation

Due to its non-vanishing off-diagonal couplings with the quarks, the $Z_{1}$
boson contributes to the meson-antimeson mixing at the tree level itself. To
estimate these contributions for $K^{0}-\overline{K}^{0}$,
$B_{d}^{0}-\overline{B}_{d}^{0}$, $B_{s}^{0}-\overline{B}_{s}^{0}$ and
$D^{0}-\overline{D}^{0}$, we follow the effective operator based analysis (see
for example Bona  _et al._ (2008)) and parametrize the new contributions in
terms of the well-known Wilson coefficients (WCs). Subsequently, we use the
limits on these coefficients obtained from a fit to experimental data by UTFit
collaboration Bona  _et al._ (2008) to derive constraints on the mass scale of
$Z_{1}$ boson.

For $K^{0}-\overline{K}^{0}$ mixing, the effective Hamiltonian for $\Delta
S=2$ is written as $H_{\rm
eff}=\sum_{i=1}^{5}C_{K}^{i}\,Q_{i}+\sum_{i=1}^{3}\tilde{C}_{K}^{i}\tilde{Q}_{i}$
where explicit form of operators are listed in Bona  _et al._ (2008); Ciuchini
_et al._ (1998). When $Z_{1}$ is integrated out, its contribution to the
Wilson coefficients $C_{K}^{i}$ and $\tilde{C}_{K}^{i}$ is obtained, at the
scale $\mu=M_{Z_{1}}$, as Smolkovič  _et al._ (2019)

$C_{K}^{1}=\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left[\left(X^{(1)}_{d_{L}}\right)_{12}\right]^{2}\,,~{}~{}\tilde{C}_{K}^{1}=\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left[\left(X^{(1)}_{d_{R}}\right)_{12}\right]^{2}\,,~{}~{}C_{K}^{5}=-4\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left(X^{(1)}_{d_{L}}\right)_{12}\left(X^{(1)}_{d_{R}}\right)_{12}\,.$
(47)

The remaining $C_{K}^{i}$ and $\tilde{C}_{K}^{i}$ have vanishing values at
$\mu=M_{Z_{1}}$. For a precise comparison with the experimental values, these
coefficients need to be evolved from $\mu=M_{Z_{1}}$ to $\mu=2$ GeV. We
perform such running using the Renormalization Group Equations (RGE) given in
Ciuchini  _et al._ (1998). It is found that RGE effects induce non-zero
$C_{K}^{4}$ while the $C^{2,3}_{K}$ and $\tilde{C}^{2,3}_{K}$ have vanishing
values at the low scale. The RGE evolved values are listed in Table 4 and
compared with the corresponding experimentally allowed ranges extracted from a
fit performed by the UTFit collaboration Bona  _et al._ (2008).

Wilson coefficient |  Allowed range |  S1 |  S2 |  S3
---|---|---|---|---
Re$C_{K}^{1}$ | $[-9.6,9.6]\times 10^{-13}$ | $-9.5\times 10^{-10}$ | $-5.4\times 10^{-14}$ | $6.2\times 10^{-18}$
Re$\tilde{C}_{K}^{1}$ | $[-9.6,9.6]\times 10^{-13}$ | $-1.6\times 10^{-9}$ | $-1.6\times 10^{-13}$ | $2.8\times 10^{-17}$
Re$C_{K}^{4}$ | $[-3.6,3.6]\times 10^{-15}$ | $6.2\times 10^{-9}$ | $5.0\times 10^{-13}$ | $-7.5\times 10^{-17}$
Re$C_{K}^{5}$ | $[-1.0,1.0]\times 10^{-14}$ | $5.4\times 10^{-9}$ | $4.2\times 10^{-13}$ | $-5.9\times 10^{-17}$
Im$C_{K}^{1}$ | $[-9.6,9.6]\times 10^{-13}$ | $5.9\times 10^{-25}$ | $9.5\times 10^{-30}$ | $1.7\times 10^{-33}$
Im$\tilde{C}_{K}^{1}$ | $[-9.6,9.6]\times 10^{-13}$ | $-1.0\times 10^{-24}$ | $-3.8\times 10^{-29}$ | $3.9\times 10^{-31}$
Im$C_{K}^{4}$ | $[-1.8,0.9]\times 10^{-17}$ | $9.5\times 10^{-26}$ | $1.5\times 10^{-29}$ | $-5.3\times 10^{-31}$
Im$C_{K}^{5}$ | $[-1.0,1.0]\times 10^{-14}$ | $8.3\times 10^{-26}$ | $1.3\times 10^{-29}$ | $-4.2\times 10^{-31}$
$|C_{B_{d}}^{1}|$ | $<2.3\times 10^{-11}$ | $1.6\times 10^{-12}$ | $9.9\times 10^{-18}$ | $5.8\times 10^{-22}$
$|\tilde{C}_{B_{d}}^{1}|$ | $<2.3\times 10^{-11}$ | $2.9\times 10^{-12}$ | $3.8\times 10^{-18}$ | $1.0\times 10^{-18}$
$|C_{B_{d}}^{4}|$ | $<2.1\times 10^{-13}$ | $5.1\times 10^{-12}$ | $1.6\times 10^{-17}$ | $6.7\times 10^{-20}$
$|C_{B_{d}}^{5}|$ | $<6.0\times 10^{-13}$ | $9.1\times 10^{-12}$ | $2.6\times 10^{-17}$ | $1.0\times 10^{-19}$
$|C_{B_{s}}^{1}|$ | $<1.1\times 10^{-9}$ | $8.3\times 10^{-11}$ | $2.8\times 10^{-15}$ | $3.0\times 10^{-19}$
$|\tilde{C}_{B_{s}}^{1}|$ | $<1.1\times 10^{-9}$ | $2.0\times 10^{-10}$ | $5.8\times 10^{-14}$ | $4.2\times 10^{-17}$
$|C_{B_{s}}^{4}|$ | $<1.6\times 10^{-11}$ | $3.1\times 10^{-10}$ | $3.3\times 10^{-14}$ | $9.8\times 10^{-18}$
$|C_{B_{s}}^{5}|$ | $<4.5\times 10^{-11}$ | $5.5\times 10^{-10}$ | $5.4\times 10^{-14}$ | $1.5\times 10^{-17}$
$|C_{D}^{1}|$ | $<7.2\times 10^{-13}$ | $2.0\times 10^{-10}$ | $2.9\times 10^{-15}$ | $6.5\times 10^{-19}$
$|\tilde{C}_{D}^{1}|$ | $<7.2\times 10^{-13}$ | $3.5\times 10^{-9}$ | $2.7\times 10^{-13}$ | $2.8\times 10^{-17}$
$|C_{D}^{4}|$ | $<4.8\times 10^{-14}$ | $3.2\times 10^{-9}$ | $1.1\times 10^{-13}$ | $1.7\times 10^{-17}$
$|C_{D}^{5}|$ | $<4.8\times 10^{-13}$ | $3.7\times 10^{-9}$ | $1.2\times 10^{-13}$ | $1.9\times 10^{-17}$
Table 4: Strength of various Wilson coefficients relevant for meson-antimeson
mixing estimated for three example solutions and corresponding experimentally
allowed range at $95\%$ confidence level. For the later we use the results
from Bona  _et al._ (2008). All the values are in ${\rm GeV}^{-2}$. The values
highlighted in green (red) are allowed (excluded) by experimental limits.

In case of $B_{q}-\overline{B}_{q}^{0}$ $(q=d,s)$ mixing, the relevant WCs at
$\mu=M_{Z_{1}}$ are Smolkovič  _et al._ (2019)

$C_{B_{d}}^{1}=\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left[\left(X^{(1)}_{d_{L}}\right)_{13}\right]^{2}\,,~{}~{}\tilde{C}_{B_{d}}^{1}=\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left[\left(X^{(1)}_{d_{R}}\right)_{13}\right]^{2}\,,~{}~{}C_{B_{d}}^{5}=-4\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left(X^{(1)}_{d_{L}}\right)_{13}\left(X^{(1)}_{d_{R}}\right)_{13}\,,$
(48)

and

$C_{B_{s}}^{1}=\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left[\left(X^{(1)}_{d_{L}}\right)_{23}\right]^{2}\,,~{}~{}\tilde{C}_{B_{s}}^{1}=\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left[\left(X^{(1)}_{d_{R}}\right)_{23}\right]^{2}\,,~{}~{}C_{B_{s}}^{5}=-4\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left(X^{(1)}_{d_{L}}\right)_{23}\left(X^{(1)}_{d_{R}}\right)_{23}\,.$
(49)

These coefficients are run down to $\mu=M_{b}=4.6$ GeV following Becirevic
_et al._ (2002). Similarly, for the charm mixing governing the
$D^{0}-\overline{D}^{0}$ oscillations, the WCs at the scale $M_{Z_{1}}$ are
given by

$C_{D}^{1}=\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left[\left(X^{(1)}_{u_{L}}\right)_{12}\right]^{2}\,,~{}~{}\tilde{C}_{D}^{1}=\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left[\left(X^{(1)}_{u_{R}}\right)_{12}\right]^{2}\,,~{}~{}C_{D}^{5}=-4\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\left(X^{(1)}_{u_{L}}\right)_{12}\left(X^{(1)}_{u_{R}}\right)_{12}\,.$
(50)

They are also evolved to the relevant low scale $\mu=2.8$ GeV using the RGE
equations given in Bona  _et al._ (2008).

We list the values of all the WCs at their relevant hadronic scales for the
three benchmark solutions and compare them with the corresponding experimental
limits in Table 4. It is noticed that constraints from meson-antimeson mixings
put a strong limit on the mass of $Z_{1}$ boson since the latter typically has
${\cal O}(1)$ off-diagonal couplings with the quark flavours, see Appendix C
for example. It can be seen that both the solutions S1 and S2 are disfavoured
implying $M_{Z_{1}}>10^{3}$ TeV for phenomenologically consistent solutions.
This also implies that the model cannot account for the neutral current
$B$-anomalies which generically require $M_{Z_{1}}\leq 2$ TeV Di Luzio  _et
al._ (2018); Allanach  _et al._ (2019).

###  V.2 Lepton flavour violation

The flavourful $Z_{1}$ mediate charged lepton flavour violating processes like
$\mu$ to $e$ conversion in nuclei and $l_{i}\to 3l_{j}$ at tree level. The
processes like $l_{i}\to l_{j}\gamma$ arise at 1-loop through $Z_{1}$ and the
charged leptons in the loop. In this subsection, we estimate the constraints
on $Z_{1}$ from all these processes.

In the field of nucleus, the muons can undergo transition to electrons through
flavour violating coupling of $\mu$ and $e$ with $Z_{1}$ boson. The strongest
limit on such a process has been obtained by SINDRUM II experiment which uses
197Au nucleus Bertl  _et al._ (2006). The branching ratio for this process
computed in Kitano  _et al._ (2002) is given by

${\rm BR}[\mu\to e]=\frac{2G_{F}^{2}}{\omega_{\rm
capt}}\,(V^{(p)})^{2}\,\left(|g^{(p)}_{LV}|^{2}+|g^{(p)}_{RV}|^{2}\right)\,,$
(51)

where $V^{(p)}$ is an integral involving proton distribution in a given
nucleus, $\omega_{\rm capt}$ is muon capture rate by the nucleus and

$g^{(p)}_{LV,RV}=2g^{(u)}_{LV,RV}+g^{(d)}_{LV,RV}\,.$ (52)

For $Z_{1}$ mediated contributions and $M_{Z_{1}}\gg m_{\mu}$, the above
couplings are given by Smolkovič  _et al._ (2019)

$g^{(f)}_{LV}\sim\frac{\sqrt{2}}{G_{F}}\frac{g_{1}^{2}}{M_{Z_{1}}^{2}}\,\left(X^{(1)}_{e_{L}}\right)_{12}\frac{1}{2}\left[\left(X^{(1)}_{f_{L}}\right)_{11}+\left(X^{(1)}_{f_{R}}\right)_{11}\right]\,$
(53)

with $f=u,d$. Similarly, $g^{(f)}_{RV}$ is given by replacement
$L\leftrightarrow R$ in the above expression. Substituting Eqs. (53,52) in
(51) and using $V^{(p)}=0.0974\,m_{\mu}^{5/2}$, $\omega_{\rm capt}=13.07\times
10^{6}\,{\rm s}^{-1}$ for 197Au from Kitano  _et al._ (2002), we estimate
${\rm BR}[\mu\to e]$ for the obtained solutions and list them in Table 5. We
also give the latest experimental limit on ${\rm BR}[\mu\to e]$ in the same
table for comparison.

LFV observable |  Limit |  S1 |  S2 |  S3
---|---|---|---|---
${\rm BR}[\mu\to e]$ | $<7.0\times 10^{-13}$ | $7.2\times 10^{-7}$ | $4.0\times 10^{-15}$ | $3.7\times 10^{-22}$
${\rm BR}[\mu\to 3e]$ | $<1.0\times 10^{-12}$ | $7.9\times 10^{-9}$ | $6.0\times 10^{-17}$ | $2.5\times 10^{-25}$
${\rm BR}[\tau\to 3\mu]$ | $<2.1\times 10^{-8}$ | $2.3\times 10^{-8}$ | $1.7\times 10^{-18}$ | $1.1\times 10^{-24}$
${\rm BR}[\tau\to 3e]$ | $<2.7\times 10^{-8}$ | $9.2\times 10^{-11}$ | $6.5\times 10^{-19}$ | $4.2\times 10^{-28}$
${\rm BR}[\mu\to e\gamma]$ | $<4.2\times 10^{-13}$ | $2.0\times 10^{-11}$ | $1.3\times 10^{-19}$ | $3.5\times 10^{-27}$
${\rm BR}[\tau\to\mu\gamma]$ | $<4.4\times 10^{-8}$ | $9.3\times 10^{-13}$ | $7.1\times 10^{-20}$ | $3.1\times 10^{-27}$
${\rm BR}[\tau\to e\gamma]$ | $<3.3\times 10^{-8}$ | $3.9\times 10^{-13}$ | $2.1\times 10^{-21}$ | $4.0\times 10^{-29}$
Table 5: Estimated values of various charged lepton flavour violating
observables for the three benchmark solutions and the present experimental
limits at $90\%$ confidence level. The latter are taken from Calibbi and
Signorelli (2018). The values highlighted in green (red) are allowed
(excluded) by experimental limits.

Next, we estimate the branching ratios of $\mu\to 3e$, $\tau\to 3\mu$ and
$\tau\to 3e$ following Heeck (2016); Smolkovič  _et al._ (2019). The relevant
decay width, estimated neglecting sub-leading terms proportional to
$m_{l_{j}}$, is given by

$\displaystyle\Gamma[l_{i}\to 3l_{j}]$ $\displaystyle\simeq$
$\displaystyle\frac{g_{1}^{4}m_{l_{i}}^{5}}{768\pi^{3}M_{Z_{1}}^{4}}\,\left[4{\rm
Re}\left(\left(X_{eV}\right)_{ji}\left(X_{eA}\right)_{ji}\left(X_{eV}\right)^{*}_{jj}\left(X_{eA}\right)^{*}_{jj}\right)\right.$
(54) $\displaystyle+$
$\displaystyle\left.3\left(\left|\left(X_{eV}\right)_{ji}\right|^{2}+\left|\left(X_{eA}\right)_{ji}\right|^{2}\right)\left(\left|\left(X_{eV}\right)_{jj}\right|^{2}+\left|\left(X_{eA}\right)_{jj}\right|^{2}\right)\right]\,,$

where

$X_{eV,eA}=\frac{1}{2}\left(X^{(1)}_{e_{L}}\pm X^{(1)}_{e_{R}}\right)\,,$ (55)

are couplings for vector and axial-vector currents, respectively. Using the
above expression, the evaluated numbers for ${\rm BR}[l_{i}\to 3l_{j}]$ are
given in Table 5 along with the latest limits from experiments.

Unlike the previous decays, the decays like $l_{i}\to l_{j}\gamma$ arise at
1-loop level. Nevertheless, we estimate these decays considering relatively
strong limits on ${\rm BR}[\mu\to e\gamma]$. The corresponding decay width is
given by Lavoura (2003)

$\Gamma[l_{i}\to l_{j}\gamma]=\frac{\alpha
g_{1}^{4}}{4\pi}\,\left(1-\frac{m_{l_{j}}^{2}}{m_{l_{i}}^{2}}\right)^{3}\,\frac{m_{l_{i}}^{4}}{M_{Z_{1}}^{4}}\,m_{l_{i}}\,\left(|c^{\gamma}_{L}|^{2}+|c_{R}^{\gamma}|^{2}\right)\,.$
(56)

Here,

$\displaystyle c_{L}^{\gamma}$ $\displaystyle=$
$\displaystyle\sum_{k}Q_{k}\left[\left(X^{(1)}_{e_{R}}\right)^{*}_{jk}\left(X^{(1)}_{e_{R}}\right)_{ik}y_{RR}+\left(X^{(1)}_{e_{L}}\right)^{*}_{jk}\left(X^{(1)}_{e_{L}}\right)_{ik}y_{LL}\right.$
(57) $\displaystyle+$
$\displaystyle\left.\left(X^{(1)}_{e_{R}}\right)^{*}_{jk}\left(X^{(1)}_{e_{L}}\right)_{ik}y_{RL}+\left(X^{(1)}_{e_{L}}\right)^{*}_{jk}\left(X^{(1)}_{e_{R}}\right)_{ik}y_{LR}\right]\,,$

and $c_{R}^{\gamma}$ can be obtained with replacement $L\leftrightarrow R$ in
the coupling matrices appearing in the above expression. $Q_{k}$ denotes
electric charge of $l_{k}$ lepton. The explicit expressions of the loop
functions $y_{LL}$, $y_{RR}$, $y_{LR}$ and $y_{RL}$ can be found in Lavoura
(2003). Numbers estimated for ${\rm BR}[\mu\to e\gamma]$, ${\rm
BR}[\tau\to\mu\gamma]$ and ${\rm BR}[\tau\to e\gamma]$ using the above
expressions are listed in Table 5 for three example solutions.

It can be seen from Table 5 that the strongest constraints on the flavourful
$Z_{1}$ interactions arise from $\mu$ to $e$ transition and processes like
$l_{i}\to 3l_{j}$ as they arise at the tree level. The process $\mu\to
e\gamma$ also puts a comparable limit on $M_{Z_{1}}$. It is seen that
$M_{Z_{1}}$ up to 10 TeV is ruled out by these LFV processes disfavouring the
benchmark solution S1. A comparison between the various numbers in Table 4 and
5 indicates that the LFV constraints are less stringent than those arising
from meson-antimeson oscillations.

###  V.3 Direct and electroweak constraints

The quark and lepton flavour violations put strong lower bounds on the masses
of new particles which more or less supersede the direct search constraints.
For example, the latest results from the LHC lead to $M_{Z_{1}}>5.15$ TeV for
$Z_{1}$ with ${\cal O}(1)$ flavour diagonal couplings with the SM fermions
Sirunyan  _et al._ (2021). The limit increases to $M_{Z_{1}}>7.20$ TeV if
$Z_{1}$ has generic diquark couplings Sirunyan  _et al._ (2018). Similarly,
the current direct search constraints on the vectorlike fermions imply
$m_{B}>1.57$ TeV Sirunyan  _et al._ (2020); Aaboud  _et al._ (2018a) and
$m_{T}>1.31$ TeV Sirunyan  _et al._ (2019); Aaboud  _et al._ (2018b). It can
be seen from Table 3 and results of the previous subsections that these
constraints are much weaker than the ones imposed by FCNCs.

Another class of constraints arise in the model due to $Z-Z_{1,2}$ mixing as
the Higgses are charged under both the SM and extended gauge symmetries.
Analogous to Babu  _et al._ (1998), this mixing can be parametrized by mixing
angles

$\sin\theta_{1,2}=\frac{g_{1,2}}{\sqrt{g^{2}+g^{\prime
2}}}\,\left(\frac{M_{Z}}{M_{Z_{1,2}}}\right)^{2}\,,$ (58)

where $g$ and $g^{\prime}$ are the strengths of $SU(2)_{L}$ and $U(1)_{Y}$
gauge interactions, respectively. In the present framework, the fermion mass
hierarchy implies $\theta_{2}\ll\theta_{1}\ll 1$ and the dominant effects
arise due to $Z-Z_{1}$ mixing. The latter leads to the flavour non-universal
couplings to the SM fermions for $Z$ boson also. However, these couplings are
suppressed by a factor of $M_{Z}^{2}/M_{Z_{1}}^{2}$ in comparison to those of
$Z_{1}$.

The $Z-Z_{1}$ mixing modifies the $\rho$ parameter which is precisely measured
along with the other electroweak observables. At the leading order in
$\theta_{1}$, the shift in the $\rho$ parameter can be obtained as Allanach
_et al._ (2021)

$\Delta\rho=\frac{g_{1}^{2}}{g^{2}+g^{\prime
2}}\,\left(\frac{M_{Z}}{M_{Z_{1,2}}}\right)^{2}\,.$ (59)

A global fit result, $\rho=1.00039\pm 0.00019$ Workman and Others (2022), then
implies $M_{Z_{1}}\geq 4.5$ TeV for $g_{1}=1$. Non-zero $Z-Z_{1}$ mixing also
modifies the couplings of $Z$ with neutrinos which can be constrained by the
invisible decay width of $Z$ boson. This constraint translates to
$M_{Z_{1}}/g_{1}\geq 0.95$ TeV Davighi (2021). The flavour non-universal
couplings of $Z$ to leptons induced by $Z-Z_{1}$ mixing give rise to lepton
flavour universality violation in $Z$ decays. The latter is severely
constrained by LEP measurements which implies $R=0.999\pm 0.003$ Workman and
Others (2022) where $R$ is a ratio of partial decay widths of $Z$ decaying
into a pair of electrons and muons. At the leading order in $\theta_{1}$, the
shift in $R$ from unity due to the new physics contributions is given by
Davighi (2021)

$\Delta R\simeq
4g_{1}\,\sin\theta_{1}\,\frac{g\cos\theta_{W}-3g^{\prime}\sin\theta_{W}}{\left(g\cos\theta_{W}-g^{\prime}\sin\theta_{W}\right)^{2}+4g^{\prime
2}\,\sin^{2}\theta_{W}}\,.$ (60)

The LEP constraint then leads to $M_{Z_{1}}/g_{1}\geq 1.3$ TeV.

In summary, the constraints from the direct searches and electroweak precision
observables are at least two orders of magnitude weaker than those from quark
and lepton flavour violating interactions. Various limits discussed in this
section suggest a lower bound, $M_{Z_{1}}/g_{1}>10^{3}$ TeV, for the generic
viable solutions obtained in the present model.

##  VI Summary and outlook

We explore a mechanism for the radiative induction of the masses for the first
and second generation charged fermions. It uses extended abelian gauge
symmetry which prevents tree-level masses for all the SM fermions. The third
generation fermions can acquire masses with the help of an additional
vectorlike family through the seesaw-like mechanism. Subsequently, the
radiative corrections induced by spontaneous breaking of extended gauge
symmetry can give rise to masses for the remaining fermions explaining their
hierarchical spectrum. It is shown that, for the underlying mechanism to work
viably, the SM fermions should have flavour non-universal charges under the
new symmetry.

Using this general setup, we give an explicit model based on $U(1)_{1}\times
U(1)_{2}$ symmetry which is a generalization of well-known $L_{\mu}-L_{\tau}$
and $L_{e}-L_{\mu}$ symmetries, respectively. The breaking of $U(1)_{1}$
($U(1)_{2}$) induces radiative masses for the second (first) generation
fermions and the hierarchy among the masses of the first two families can be
attributed to the hierarchy between the breaking scales of two $U(1)$s. We
give three example numerical solutions which reproduce the observed charged
fermion masses and quark mixing parameters and discuss various constraints
from the quark and lepton flavour violations, direct searches and electroweak
precision observables on the obtained solutions. Although the radiative mass
generation mechanism does not fix unambiguously the scale of new physics, the
current constraints imply that the new particles must be heavier than $10^{3}$
TeV. The requirement of reproducing viable fermion mass spectrum more or less
fixes the relative mass scales of new vector bosons and vectorlike fermions.

Although the main motivation for the radiative mass generation mechanism is to
make the masses of fundamental fermions calculable parameters of the theory,
the results obtained in this paper still involve a large number of free
parameters. Unlike in the SM, the fundamental parameters of the model do not
span a wide range of magnitude. However, their large number and non-unique
values make the model less predictive. One way to improve upon this is to
accommodate the $U(1)_{1}\times U(1)_{2}$ symmetry in a larger flavour
symmetry based on a single gauge group or to use only one $U(1)$ with
appropriate flavour non-universal charges. For the latter, a systematic scan
of the fermion spectrum based on anomaly-free charges, similar to the one
performed recently in Patel (2021) in the context of 5D models, would be
required. Another interesting possibility is to restrict vertically by
unifying various SM quarks and leptons in some irreducible representations of
a grand unified theory. Both these approaches can lead to a reduction in the
number of free parameters and may provide more predictive models. These
alternatives shall be explored in our future works.

## Acknowledgements

This work is partially supported under MATRICS project (MTR/2021/000049) by
the Science & Engineering Research Board (SERB), Department of Science and
Technology (DST), Government of India.

##  Appendix A Calculation of 1-loop self-energy correction to the fermion
masses

The gauge interactions in Eq. (7) can be written as

$\displaystyle-{\cal L}_{\rm gauge}$ $\displaystyle=$ $\displaystyle
g_{X}\,X_{\mu}\overline{f}_{\alpha}\gamma^{\mu}\,{\cal
C}_{\alpha\beta}\,f_{\beta},$ (61)

with the coupling defined as

${\cal C}_{\alpha\beta}=({\cal Q}_{L})_{\alpha\beta}\,P_{L}+({\cal
Q}_{R})_{\alpha\beta}P_{R}\ .$ (62)

The 1-loop fermion self energy correction induced by the gauge boson in the
loop is shown in Fig. 1. The amplitude of this diagram is given by

$\displaystyle-i\Sigma_{\alpha\beta}(p)$ $\displaystyle=$
$\displaystyle\sum_{\gamma}\int\frac{d^{4}k}{(2\pi)^{4}}(-ig_{X}\gamma^{\mu}{\cal
C}^{\dagger}_{\alpha\gamma})\frac{i(\not{k}+\not{p}+m_{\gamma})}{[(k+p)^{2}-m_{\gamma}^{2}+i\epsilon]}(-ig_{X}\gamma^{\nu}{\cal
C}_{\gamma\beta})\Delta_{\mu\nu}(k)\,,$ (63)

with

$\Delta_{\mu\nu}(k)=\frac{-i}{k^{2}-{M_{X}}^{2}+i\epsilon}\left(\eta_{\mu\nu}-(1-\zeta)\frac{k_{\mu}k_{\nu}}{k^{2}-\zeta
M_{X}^{2}}\right)\,.$ (64)

Figure 1: Gauge boson induced fermion self-energy correction at 1-loop

We set $p=0$ in order to go on the mass shell for the massless fermion and
compute the loop contribution in the Feynman-’t Hooft gauge ($\zeta=1$) in
dimensional regularization scheme. As the denominator is an even function of
$k$, the terms proportional to odd number of $k^{\prime}$s in numerator
vanishes. Therefore,

$\displaystyle\Sigma_{\alpha\beta}(0)=\ -ig_{X}^{2}\mu^{\epsilon}\ $
$\displaystyle\sum_{\gamma}\left[({{\cal
Q}_{L}})^{\dagger}_{\alpha\gamma}P_{R}+({{\cal
Q}_{R}})^{\dagger}_{\alpha\gamma}P_{L}\right]$
$\displaystyle\times\int\frac{d^{d}k}{(2\pi)^{d}}\frac{d\
m_{\gamma}}{k^{2}-m_{\gamma}^{2}+i\epsilon}\frac{1}{k^{2}-{M_{X}}^{2}+i\epsilon}\
{{\cal C}}_{\gamma\beta}$ (65)

where, $d=4-\epsilon$ and we have used the following relations in order to
obtain Eq. (65).

$\displaystyle\gamma^{\mu}{\cal C}^{\dagger}_{\alpha\gamma}$
$\displaystyle=\left[({{\cal Q}_{L}})^{\dagger}_{\alpha\gamma}P_{R}+({{\cal
Q}_{R}})^{\dagger}_{\alpha\gamma}P_{L}\right]\gamma^{\mu}\,,$ (66)
$\displaystyle\gamma^{\mu}\gamma_{\mu}$ $\displaystyle=d\,.$ (67)

Using ${\cal Q}_{L,R}^{\dagger}={\cal Q}_{L,R}$ (see Eq. (8)), Eq. (65) can be
simplified to

$\displaystyle\Sigma_{\alpha\beta}(0)=\ \frac{dg_{X}^{2}}{16\pi^{2}}\ $
$\displaystyle\sum_{\gamma}\ m_{\gamma}\ \left[({{\cal
Q}_{L}})_{\alpha\gamma}({{\cal Q}_{R}})_{\gamma\beta}P_{R}+({{\cal
Q}_{R}})_{\alpha\gamma}({{\cal Q}_{L}})_{\gamma\beta}P_{L}\right]$
$\displaystyle\times\frac{({2\pi\mu})^{\epsilon}}{i\pi^{2}}\int
d^{d}k\frac{1}{k^{2}-m_{\gamma}^{2}+i\epsilon}\frac{1}{k^{2}-{M_{X}}^{2}+i\epsilon}$
(68)

The integration in the above can be evaluated and expressed in terms of
Passarino-Veltmann function $B_{0}$ Passarino and Veltman (1979) leading to a
final expression

$\Sigma_{\alpha\beta}(0)=\ \frac{g_{X}^{2}}{4\pi^{2}}\ \sum_{\gamma}\
m_{\gamma}\ \left[({{\cal Q}_{L}})_{\alpha\gamma}({{\cal
Q}_{R}})_{\gamma\beta}P_{R}+({{\cal Q}_{R}})_{\alpha\gamma}({{\cal
Q}_{L}})_{\gamma\beta}P_{L}\right]\ B_{0}[M^{2}_{X},m_{\gamma}^{2}]$ (69)

with

$\displaystyle B_{0}[M^{2}_{X},m_{\gamma}^{2}]$ $\displaystyle=$
$\displaystyle\frac{({2\pi\mu})^{\epsilon}}{i\pi^{2}}\int
d^{d}k\,\frac{1}{k^{2}-m_{\gamma}^{2}+i\epsilon}\frac{1}{k^{2}-{M_{X}}^{2}+i\epsilon}\,$
(70) $\displaystyle=$
$\displaystyle\frac{2}{\epsilon}+1-\gamma+\ln{4\pi}-\frac{M_{X}^{2}\ln
M_{X}^{2}-m_{\gamma}^{2}\ln m_{\gamma}^{2}}{M_{X}^{2}-m_{\gamma}^{2}}\,.$

The above result is used in Eq. (9) to determine explicitly the radiative
contributions to the fermion mass matrices.

##  Appendix B Scalar potential

The most general renormalizable scalar potential of the model, invariant under
the SM gauge symmetry and $G_{F}$, is written as:

$\displaystyle V$ $\displaystyle=$ $\displaystyle
m_{ui}^{2}\,H_{ui}^{\dagger}H_{ui}\,+\,m_{di}^{2}\,H_{di}^{\dagger}H_{di}\,+\,m_{\eta
i}^{2}\,\eta_{i}^{\dagger}\eta_{i}\,$ (71) $\displaystyle+$
$\displaystyle\left\\{(m_{ud\eta})_{ijk}\,\epsilon_{ijk}\,\eta_{i}H_{uj}H_{dk}\,+\,(m_{\eta})_{ijk}\,\epsilon_{ijk}\,\eta_{i}\eta_{j}\eta_{k}+{\rm
h.c.}\right\\}\,$ $\displaystyle+$
$\displaystyle(\lambda_{u})_{ij}\,H_{ui}^{\dagger}H_{ui}H_{uj}^{\dagger}H_{uj}\,+\,(\lambda_{d})_{ij}\,H_{di}^{\dagger}H_{di}H_{dj}^{\dagger}H_{dj}\,+\,(\lambda_{\eta})_{ij}\,\eta_{i}^{\dagger}\eta_{i}\eta_{j}^{\dagger}\eta_{j}\,$
$\displaystyle+$
$\displaystyle(\lambda_{ud})_{ij}\,H_{ui}^{\dagger}H_{ui}H_{dj}^{\dagger}H_{dj}\,+\,(\lambda_{u\eta})_{ij}\,H_{ui}^{\dagger}H_{ui}\eta_{j}^{\dagger}\eta_{j}\,+\,(\lambda_{d\eta})_{ij}\,H_{di}^{\dagger}H_{di}\eta_{j}^{\dagger}\eta_{j}\,$
$\displaystyle+$
$\displaystyle(\tilde{\lambda}_{u})_{ij}\,H_{ui}^{\dagger}H_{uj}H_{uj}^{\dagger}H_{ui}\,+\,(\tilde{\lambda}_{d})_{ij}\,H_{di}^{\dagger}H_{dj}H_{dj}^{\dagger}H_{di}\,$
$\displaystyle+$
$\displaystyle(\tilde{\lambda}_{ud})_{ij}\,H_{ui}^{\dagger}H_{uj}H_{dj}^{\dagger}H_{di}\,+\,(\tilde{\lambda}_{u\eta})_{ij}\,H_{ui}^{\dagger}H_{uj}\eta_{j}^{\dagger}\eta_{i}\,+\,(\tilde{\lambda}_{d\eta})_{ij}\,H_{di}^{\dagger}H_{dj}\eta_{j}^{\dagger}\eta_{i}\,$
$\displaystyle+$
$\displaystyle\left\\{(\lambda_{ud\eta})_{ij}\,\eta_{i}^{\dagger}H_{ui}\eta_{j}^{\dagger}H_{dj}+(\tilde{\lambda}_{ud\eta})_{ij}\,\eta_{i}^{\dagger}H_{uj}\eta_{j}^{\dagger}H_{di}+{\rm
h.c.}\right\\}\,,$

where, $i,j,k=1,2,3$ are flavour indices. The diagonal elements of all the
$\tilde{\lambda}$ matrices can be chosen zero without loss of generality. We
assume the general VEVs for various fields, as parametrized in Eqs. (31,32),
which breaks all symmetries except $U(1)$ corresponding to electromagnetism.
Given a large number of parameters in Eq. (71), we assume that such minima
exist for a suitable choice of their values.

It can be seen that the potential does not possess any enhanced global
symmetry in its most general form. Therefore, one does not find any new
Goldstone bosons other than the ones corresponding to the spontaneous breaking
of the SM and $G_{F}$ symmetries which are eaten by the massive $W^{\pm}$, $Z$
and $Z_{1,2}$ bosons. The potential has an $SU(3)$ global symmetry if all the
quadratic, cubic and quartic couplings are assumed flavour universal. This
symmetry corresponds to an invariance under a rotation $\Phi_{i}\to
U_{ij}\Phi_{j}$ with $\Phi=H_{u},H_{d}$ and $\eta$. Moreover, for vanishing
$\tilde{\lambda}_{ud}$, $\tilde{\lambda}_{u\eta}$, $\tilde{\lambda}_{d\eta}$,
$\lambda_{ud\eta}$, $\tilde{\lambda}_{ud\eta}$, $m_{ud\eta}$ and $m_{\eta}$,
the scalar potential can possess an enhanced $[U(3)]^{3}$ symmetry
corresponding to separate rotations for $H_{u}$, $H_{d}$ and $\eta$.

##  Appendix C Computed values of flavour violating couplings

In this Appendix, we give the numerical values of various coupling matrices
$X^{(k)}_{f_{L}}$ and $X^{(k)}_{f_{R}}$ for benchmark solution 2 (S2).

$\displaystyle X^{(1)}_{u_{L}}$
$\displaystyle=\left(\begin{array}[]{ccc}-0.9964&0.0597i&0\\\
-0.0597i&-0.0008&-0.0528\\\ 0&-0.0528&0.9972\\\ \end{array}\right);$
$\displaystyle
X^{(1)}_{u_{R}}=\left(\begin{array}[]{ccc}-0.1459&0.5747i&-0.0023i\\\
-0.5747i&-0.1466&-0.7781\\\ 0.0023i&-0.7781&0.2925\\\ \end{array}\right)$ (78)
$\displaystyle X^{(1)}_{d_{L}}$
$\displaystyle=\left(\begin{array}[]{ccc}-0.9262&0.2618i&-0.0034i\\\
-0.2618i&-0.0706&-0.0576\\\ 0.0034i&-0.0576&0.9967\\\ \end{array}\right);$
$\displaystyle
X^{(1)}_{d_{R}}=\left(\begin{array}[]{ccc}-0.2665&0.4497i&0.0021i\\\
-0.4497i&-0.6887&-0.2625\\\ -0.0021i&-0.2625&0.9552\\\ \end{array}\right)$
(85) $\displaystyle X^{(1)}_{e_{L}}$
$\displaystyle=\left(\begin{array}[]{ccc}-0.9978&0.0460&-0.0105\\\
0.0460&0.1330&0.3423\\\ -0.0105&0.3423&0.8648\\\ \end{array}\right);$
$\displaystyle
X^{(1)}_{e_{R}}=\left(\begin{array}[]{ccc}-0.9741&0.1551&-0.0375\\\
0.1551&0.1169&0.3547\\\ -0.0375&0.3547&0.8572\\\ \end{array}\right)$ (92)
$\displaystyle X^{(2)}_{u_{L}}$
$\displaystyle=\left(\begin{array}[]{ccc}0.0036&0.0594i&0.0047i\\\
-0.0594i&0.9909&0.1053\\\ -0.0047i&0.1053&-0.9944\\\
\end{array}\right);X^{(2)}_{u_{R}}=\left(\begin{array}[]{ccc}0.5456&-0.0378i&0.6724i\\\
0.0378i&-0.3627&0.5613\\\ -0.6724i&0.5613&-0.1829\\\ \end{array}\right)$ (99)
$\displaystyle X^{(2)}_{d_{L}}$
$\displaystyle=\left(\begin{array}[]{ccc}0.0736&0.2603i&0.0247i\\\
-0.2603i&0.9200&0.1102\\\ -0.0247i&0.1102&-0.9936\\\
\end{array}\right);X^{(2)}_{d_{R}}=\left(\begin{array}[]{ccc}0.7233&0.4203i&0.1738i\\\
-0.4203i&0.2262&0.2348\\\ -0.1738i&0.2348&-0.9495\\\ \end{array}\right)$ (106)
$\displaystyle X^{(2)}_{e_{L}}$
$\displaystyle=\left(\begin{array}[]{ccc}0.0022&0.0420&-0.0205\\\
0.0420&0.7281&-0.6828\\\ -0.0205&-0.6828&-0.7302\\\
\end{array}\right);X^{(2)}_{e_{R}}=\left(\begin{array}[]{ccc}0.0254&0.1418&-0.0704\\\
0.1418&0.6962&-0.6869\\\ -0.0704&-0.6869&-0.7216\\\ \end{array}\right)$ (113)

It can be seen that both the diagonal and off-diagonal elements are of similar
magnitude and no particular pattern or hierarchies in values is seen. We find
similar values for the other two solutions.

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