# Gauged $SU(3)_{F}$ and loop induced quark and lepton masses

Gurucharan Mohanta<EMAIL_ADDRESS>Theoretical Physics Division,
Physical Research Laboratory, Navarangpura, Ahmedabad-380009, India  Indian
Institute of Technology Gandhinagar, Palaj-382055, India  Ketan M. Patel
<EMAIL_ADDRESS>Theoretical Physics Division, Physical Research Laboratory,
Navarangpura, Ahmedabad-380009, India

###### Abstract

We investigate a local $SU(3)_{F}$ flavour symmetry for its viability in
generating the masses for the quarks and charged leptons of the first two
families through radiative corrections. Only the third-generation fermions get
tree-level masses due to specific choice of the field content and their gauge
charges. Unprotected by symmetry, the remaining fermions acquire non-vanishing
masses through the quantum corrections induced by the gauge bosons of broken
$SU(3)_{F}$. We show that inter-generational hierarchy between the masses of
the first two families arises if the flavour symmetry is broken with an
intermediate $SU(2)$ leading to a specific ordering in the masses of the gauge
bosons. Based on this scheme, we construct an explicit and predictive model
and show its viability in reproducing the realistic charged fermion masses and
quark mixing parameters in terms of not-so-hierarchical fundamental couplings.
The model leads to the strange quark mass, $m_{s}\approx 16$ MeV at $M_{Z}$,
which is $\sim 2.4\sigma$ away from its current central value. Large flavour
violations are a generic prediction of the scheme which pushes the masses of
the new gauge bosons to $10^{3}$ TeV or higher.

##  I Introduction

The six orders of magnitude separation observed between the masses of the
elementary quarks and the charged leptons adequately support the possibility
that some of these fermion masses are induced radiatively. In the simplest
version of this idea, only the third-generation fermions are postulated to
have tree-level masses. At the same time, those of the first and second
generations obtain their masses through higher-order corrections in a
perturbation theory. It was realised from the very beginning through the early
attempts Georgi and Glashow (1973); Mohapatra (1974); Barr and Zee (1978);
Wilczek and Zee (1979); Yanagida (1979); Barbieri and Nanopoulos (1980) that
successful implementation of such an idea necessarily requires the extension
of the Standard Model (SM). With the nonzero Yukawa couplings for only the
third-generation fermions at the tree level, the SM has a global $[U(2)]^{5}$
symmetry which remains unbroken by the electroweak symmetry breaking and
prevents the corresponding two generations of fermions from getting mass
through radiative corrections. Therefore, extensions of the SM in which the
new sector breaks at least partially this global symmetry are desired in order
to give rise to non-vanishing loop-induced masses for the first and second
families of fermions.

The extended models for radiative fermion masses can mainly be categorized
into two classes based on the nature of new particle(s) propagating in the
fermion self-energy loops111It is possible that the extensions include the
scalar and vector bosons, and both can give rise to loop-induced corrections.
However, the model can be put into one of these categories depending on the
most-dominant contribution considered. (a) spin-0 (see Balakrishna (1988a);
Balakrishna  _et al._ (1988); Balakrishna (1988b); Balakrishna and Mohapatra
(1989); Babu and He (1989); Babu  _et al._ (1990); Arkani-Hamed  _et al._
(1996); Barr (2007); Graham and Rajendran (2010); Dobrescu and Fox (2008);
Crivellin  _et al._ (2011a, b); Chiang and Yagyu (2021); Chiang  _et al._
(2022); Baker  _et al._ (2021a, b); Yin (2021) for example) and (b) spin-1
Hernandez Galeana and Montes de Oca Yemha (2004); Appelquist  _et al._ (2006);
Reig  _et al._ (2018); Weinberg (2020); Jana  _et al._ (2022); Mohanta and
Patel (2022). In the case of the latter, SM gauge symmetry needs to be
extended to include a gauged flavour symmetry group $G_{F}$ and the radiative
correction can be determined in terms of the gauge couplings and masses of the
new gauge bosons. Since these couplings and masses are independently
measurable parameters, the quantum-corrected fermion masses become in-
principle calculable quantities in this class of frameworks distinguishing
them from those in category (a). Moreover, symmetry-based extensions of this
kind typically lead to fewer parameters than extensions with scalars and hence
can provide more predictive setups. Therefore, the radiative models in which
the new gauge sector primarily induces the masses of the lighter families of
the quarks and leptons can provide potentially more attractive and economical
options and require systematic investigations.

Along these lines, we have recently proposed a model for $G_{F}$ being an
abelian group and analysed it in detail in Mohanta and Patel (2022). It was
shown that the nature of the new abelian symmetry must be flavour non-
universal if the fermion masses are to be generated radiatively. The simplest
viable and anomaly-free implementation of the scheme requires
$G_{F}=U(1)_{1}\times U(1)_{2}$ where the gauge bosons associated with
$U(1)_{1}$ and $U(1)_{2}$ generate the masses of the second and first
generations at 1-loop, respectively. The inter-generational hierarchy between
the first two generations can arise either from the hierarchy between the
gauge boson masses or the difference between the strength of the gauge
couplings of the two $U(1)$s. Some of these assumptions can be alienated and a
more predictive setup can be achieved if $U(1)_{1}\times U(1)_{2}$ is replaced
by a simple group. The most advantageous feature of non-abelian $G_{F}$ in the
context of radiative mass generation is that it naturally accommodates gauge
bosons with flavour non-diagonal couplings. The same symmetry can also be
effectively utilized in order to ensure that only the third generations
receive mass at the tree level. Moreover, being a simple group it minimally
modifies the SM gauge structure and can lead to a predictive scenario.

In the present work, we investigate a scheme based on $G_{F}=SU(3)_{F}$ and
provide a concrete and realistic implementation of this scheme for radiative
mass induction for the lighter generations of the SM fermions. The horizontal
$SU(3)$ symmetry was also proposed earlier in Hernandez Galeana and Montes de
Oca Yemha (2004); Hernandez-Galeana (2015) for a similar purpose, however
systematic and comprehensive analysis of loop-induced fermion masses and
mixing parameters along with the phenomenological constraints on the flavour
symmetry breaking scale have not been carried out. Another non-abelian
alternative, namely $G_{F}=SO(3)_{L}\times SO(3)_{R}$, has been investigated
relatively recently in Weinberg (2020) and shown to lead to an inconsistent
flavour spectrum. The present work, therefore, offers a complete and realistic
model of radiatively induced quark and lepton masses based on non-abelian
flavour symmetry. We find that a viable implementation of this scheme within
the SM requires a multiplicity of the electroweak Higgs doublets and the
existence of vectorlike fermions. The latter plays an essential role in
reproducing the observed spectrum being consistent with the constraints from
the flavour violation. We show that the hierarchy between the first and
second-generation masses can naturally be induced if the flavour symmetry is
broken in a particular way. This along with improved predictivity makes the
present model less ad-hoc than the one based on abelian symmetries discussed
earlier.

The rest of the paper is structured as the following. In the next section, we
discuss the general framework of $SU(3)_{F}$ and the generation of radiative
masses. Breaking of the horizontal symmetry leading to desired gauge boson
mass spectrum is presented in section III. Detailed implementation of the
general scheme in the SM is discussed in section IV. Viability of the model is
established through example numerical solutions in section V. We also discuss
some phenomenological aspects of the scheme in section VI before concluding in
section VII.

##  II $SU(3)_{F}$ and fermion mass generation

Denoting the three generations $(i=1,2,3)$ of chiral fermions by
$f_{Li}^{\prime}$ and $f_{Ri}^{\prime}$ and a pair of vectorlike fermions by
$F^{\prime}_{L,R}$, the tree-level mass term in the basis
$f^{\prime}_{L\alpha}\equiv(f^{\prime}_{Li},F^{\prime}_{L})$ and
$f^{\prime}_{R\alpha}\equiv(f^{\prime}_{Ri},F^{\prime}_{R})$ is arranged as

$-{\cal L}_{m}=\overline{f}^{\prime}_{L\alpha}\,\left({\cal
M}^{0}\right)_{\alpha\beta}\,f^{\prime}_{R\beta}+{\rm h.c.}\,,$ (1)

with

${\cal M}^{0}=\left(\begin{array}[]{cc}0_{3\times 3}&(\mu)_{3\times 1}\\\
(\mu^{\prime})_{1\times 3}&m_{F}\end{array}\right)\,.$ (2)

Here, $f^{\prime}_{L,R}$ is used to discuss the general case in this section
and $f=u,d,e$ will be used later to apply this discussion to the up-type,
down-type quarks and charged leptons, respectively. For the brevity, we also
suppress the $f$ dependency in ${\cal M}^{0}$, $\mu$ and $\mu^{\prime}$.

We also consider that $f^{\prime}_{Li}$ and $f^{\prime}_{Ri}$ transform as
fundamental representations of a horizontal gauged symmetry $SU(3)_{F}$. At
the same time, the vectorlike fermions are taken as singlets under the same
symmetry. It can then be seen that $\mu$ and $\mu^{\prime}$ matrices in the
above mass Lagrangian break the $SU(3)_{F}$. The vanishing $3\times 3$ sub-
matrix can be obtained utilizing the chiral nature of $f^{\prime}_{Li}$ and
$f^{\prime}_{Ri}$ under the SM gauge symmetry and it can remain zero even in
the broken phase of $SU(3)_{F}$. Depending on the SM charges of
$F^{\prime}_{L}$ or $F^{\prime}_{R}$, either $\mu$ or $\mu^{\prime}$ is also
protected by the chiral symmetry.

The matrix ${\cal M}^{0}$ leads to two massless states which can be identified
with the first and second-generation fermions. For
$m_{F}\gg\mu_{i},\mu^{\prime}_{i}$, the effective $3\times 3$ mass matrix
takes the form

$M^{0}\simeq-\frac{1}{m_{F}}\,\mu\,\mu^{\prime}\,,$ (3)

at the leading order. The above matrix is of rank one and the state
corresponding to the non-vanishing eigenvalue can be assigned to the third
generation fermion.

The relatively small masses of the first two generations can be induced
through quantum corrections within this framework. In order to quantify these
corrections, consider the $SU(3)_{F}$ gauge interactions of fermions with the
gauge bosons $A_{\mu}^{a}$ given by

$-{\cal L}_{\rm
gauge}=g_{F}\left(\overline{f}^{\prime}_{Li}\gamma^{\mu}A^{a}_{\mu}\left(\frac{\lambda^{a}}{2}\right)_{ij}f^{\prime}_{Lj}+\overline{f}^{\prime}_{Ri}\gamma^{\mu}A^{a}_{\mu}\left(\frac{\lambda^{a}}{2}\right)_{ij}f^{\prime}_{Rj}\right)\,,$
(4)

where $a=1,..,8$ and $\lambda^{a}$ are the Gell-Mann matrices. For the latter,
we use the expressions in a different basis than the conventional one and they
are listed in Appendix A for the clarity. The above can be generalized to
include the vectorlike fermions as

$-{\cal L}_{\rm
gauge}=\frac{g_{F}}{2}\left(\overline{f}^{\prime}_{L\alpha}\gamma^{\mu}A^{a}_{\mu}\left(\Lambda^{a}\right)_{\alpha\beta}f^{\prime}_{L\beta}+\overline{f}^{\prime}_{R\alpha}\gamma^{\mu}A^{a}_{\mu}\left(\Lambda^{a}\right)_{\alpha\beta}f^{\prime}_{R\beta}\right)\,,$
(5)

where $\Lambda^{a}$ are $4\times 4$ matrices given by

${\Lambda}^{a}=\,\left(\begin{array}[]{cc}\lambda^{a}&0\\\
0&0\end{array}\right)\,.$ (6)

The physical basis of fermions, denoted by $f_{L,R}$, can be obtained from the
canonical basis using the unitary transformations

$f^{\prime}_{L,R}={\cal U}_{L,R}\,f_{L,R}\,,$ (7)

such that

${\cal U}^{\dagger}_{L}\,{\cal M}^{0}\,{\cal U}_{R}={\cal D}={\rm
Diag.}(0,0,m_{3},m_{4})\,.$ (8)

Similarly, the physical gauge bosons $B_{\mu}^{a}$ can be obtained from
$A_{\mu}^{a}$ using an $8\times 8$ real orthogonal matrix ${\cal R}$ as

$A_{a\mu}\,=\,{\cal R}_{ab}\,B_{b\mu}\,.$ (9)

The matrix ${\cal R}$ can be obtained explicitly by diagonalizing the gauge-
boson mass matrix which is real and symmetric.

Substituting Eqs. (7,9) in Eq. (5), the gauge interactions in the physical
basis of fermions and gauge bosons are obtained as

$-{\cal L}_{\rm
gauge}=\frac{g_{F}}{2}\left(\overline{f}_{L\alpha}\gamma^{\mu}\left({\cal
U}_{L}^{\dagger}{\Lambda^{a}}{\cal
U}_{L}\right)_{\alpha\beta}f_{L\beta}+\overline{f}_{R\alpha}\gamma^{\mu}\left({\cal
U}_{R}{}^{\dagger}{\Lambda^{a}}{\cal
U}_{R}\right)_{\alpha\beta}f_{R\beta}\right){\cal R}_{ab}B^{b}_{\mu}\,.$ (10)

Since $\Lambda^{a}$ do not commute with each other for all $a$, the matrices
${\cal U}_{L,R}^{\dagger}{\Lambda^{a}}{\cal U}_{L,R}$ cannot be made
simultaneously diagonal. Therefore, there always exists a set of gauge bosons
which has flavour-changing interactions with fermions. As noted by us
previously in Mohanta and Patel (2022), this is necessary for the generation
of masses for the first and second family fermions through radiative
corrections induced by the gauge bosons.

The fermion mass matrix, corrected by the $SU(3)_{F}$ gauge interactions at
1-loop, can be written as

${\cal M}={\cal M}^{0}+\delta{\cal M}\,,$ (11)

where

$\delta{\cal M}={\cal U}_{L}\,\Sigma(0)\,{\cal U}^{\dagger}_{R}\,.$ (12)

Figure 1: Diagram representing gauge boson induced fermion self-energy
correction.

Using Eq. (10), the 1-loop contribution (see Fig. 1) can be computed as

$\displaystyle-i(\Sigma(p))_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\left(-i\frac{g_{F}}{2}{\cal
R}_{ab}({\cal U}_{L}^{\dagger}{\Lambda^{a}}{\cal
U}_{L})_{\alpha\sigma}\right)\gamma^{\mu}\frac{im_{\sigma}}{(k+p)^{2}-m_{\sigma}^{2}+i\epsilon}\,$
(13) $\displaystyle\left(-i\frac{g_{F}}{2}{\cal R}_{cb}({\cal
U}_{R}^{\dagger}{\Lambda^{c}}{\cal
U}_{R})_{\sigma\beta}\right)\gamma^{\nu}\,\Delta_{\mu\nu}(k)\,,$

with

$\Delta_{\mu\nu}(k)=\frac{-i}{k^{2}-{M_{b}}^{2}+i\epsilon}\left(\eta_{\mu\nu}-(1-\zeta)\frac{k_{\mu}k_{\nu}}{k^{2}-\zeta
M_{b}^{2}}\right)\,.$ (14)

Here $M_{b}$ is the mass of the gauge boson $B^{b}_{\mu}$. In the Feynmann-’t
Hooft gauge, the evaluation of the above integral results in

$(\Sigma_{f}(0))_{\alpha\beta}=\frac{{g_{F}}^{2}}{16\pi^{2}}\,{\cal
R}_{ab}({\cal U}_{L}^{\dagger}{\Lambda^{a}}{\cal U}_{L})_{\alpha\sigma}{\cal
R}_{cb}({\cal U}_{R}^{\dagger}{\Lambda^{c}}{\cal
U}_{R})_{\sigma\beta}\,m_{\sigma}\,B_{0}[M_{b}^{2},m_{\sigma}^{2}]\,,$ (15)

with

$\displaystyle B_{0}[M^{2},m^{2}]$ $\displaystyle=$
$\displaystyle\frac{({2\pi\mu})^{\epsilon}}{i\pi^{2}}\int
d^{d}k\,\frac{1}{k^{2}-m^{2}+i\epsilon}\frac{1}{k^{2}-{M}^{2}+i\epsilon}\,$
(16) $\displaystyle=$ $\displaystyle\Delta_{\epsilon}-\frac{M^{2}\ln
M^{2}-m^{2}\ln m^{2}}{M^{2}-m^{2}}\,,$

and

$\Delta_{\epsilon}\,=\,\frac{2}{\epsilon}+1-\gamma+\ln{4\pi}\,.$ (17)

It is straightforward to verify from Eq. (15) that the $m_{\sigma}$
independent contribution in $\delta{\cal M}$ from
$B_{0}[M_{b}^{2},m_{\sigma}^{2}]$ vanishes identically. The divergent part of
$\delta{\cal M}$ is given by

$\delta{\cal M}_{\rm div}={\cal U}_{L}\,\Sigma(0)_{\rm div}\,{\cal
U}^{\dagger}_{R}\,,$ (18)

where $\Sigma(0)_{\rm div}$ captures the terms dependent only on
$\Delta_{\epsilon}$ from Eq. (15). Explicitly,

$\displaystyle(\delta{\cal M}_{\rm div})_{\rho\kappa}$ $\displaystyle=$
$\displaystyle\frac{{g_{F}}^{2}}{16\pi^{2}}\,{\cal U}_{L}{}_{\rho\alpha}{\cal
R}_{ab}({\cal U}_{L}^{\dagger}{\Lambda^{a}}{\cal U}_{L})_{\alpha\sigma}{\cal
R}_{cb}({\cal U}_{R}^{\dagger}{\Lambda^{c}}{\cal
U}_{R})_{\sigma\beta}\,m_{\sigma}\,\Delta_{\epsilon}\,{\cal
U}^{\dagger}_{R}{}_{\beta\kappa}\,,$ (19) $\displaystyle=$
$\displaystyle\frac{{g_{F}}^{2}\Delta_{\epsilon}}{16\pi^{2}}\,({\cal
RR^{T}})_{ac}\,{\cal U}_{L}{}_{\rho\alpha}\,({\cal
U}_{L}^{\dagger}{\Lambda^{a}}{\cal U}_{L})_{\alpha\sigma}\,{\cal
D}_{\sigma\sigma}\,({\cal U}_{R}^{\dagger}{\Lambda^{c}}{\cal
U}_{R})_{\sigma\beta}\,{\cal U}^{\dagger}_{R}{}_{\beta\kappa}\,.$

Using Eq. (8), orthogonality of ${\cal R}$ and unitarity of ${\cal U}_{L,R}$,
the above expression can be simplified to

$\delta{\cal M}_{\rm
div}=\frac{{g_{F}}^{2}\Delta_{\epsilon}}{16\pi^{2}}\,\Lambda^{a}{\cal
M}^{0}\Lambda^{a}=0\,,$ (20)

where we use the form of ${\cal M}^{0}$ and $\Lambda^{a}$ given in Eqs. (2,6)
to get the last equality. The vanishing of $\delta{\cal M}_{\rm div}$ is in
accordance with the renormalizability Barr and Zee (1978); Weinberg (1972) of
the theory as there are no corresponding counterterms to renormalise.

Further simplification of the finite contribution can be achieved for
$m_{F}\gg\mu_{i},\mu^{\prime}_{i}$. In this case, ${\cal U}_{L,R}$ at the
leading order in $\mu_{i}/m_{F}$ and $\mu^{\prime}_{i}/m_{F}$ can be
approximated as Joshipura  _et al._ (2020)

${\cal U}_{L,R}=\left(\begin{array}[]{cc}U_{L,R}&-\rho_{L,R}\\\
\rho_{L,R}^{\dagger}U_{L,R}&1\end{array}\right)+{\cal O}(\rho_{L,R}^{2})\,,$
(21)

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

$U_{L}^{\dagger}\,M^{0}\,U_{R}={\rm Diag.}(0,0,m_{3})\,.$ (22)

Using Eq. (3), the above can also be written as

$\left(U_{L}\right)_{i3}\left(U_{R}^{*}\right)_{j3}\,m_{3}\,=M^{0}_{ij}=-\frac{1}{m_{F}}\mu_{i}\mu^{\prime}_{j}\,.$
(23)

Expanding the finite part of Eq. (15) and using Eqs. (6,21,23), the 1-loop
correction to the effective $3\times 3$ mass matrix can be simplified to

$(\delta M)_{ij}\simeq\frac{{g_{F}}^{2}}{16\pi^{2}}{\cal R}_{ab}{\cal
R}_{cb}(\lambda^{a}M^{0}\lambda^{c})_{ij}\,\Delta b_{0}[M_{b}^{2}]\,,$ (24)

where

$\Delta b_{0}[M_{b}^{2}]=-\frac{M_{b}^{2}\ln M_{b}^{2}-m_{3}^{2}\ln
m_{3}^{2}}{M_{b}^{2}-m_{3}^{2}}+\frac{M_{b}^{2}\ln M_{b}^{2}-m_{4}^{2}\ln
m_{4}^{2}}{M_{b}^{2}-m_{4}^{2}}\,.$ (25)

We also find $\delta{\cal M}_{4\alpha}=\delta{\cal M}_{\alpha 4}=0$ by
utilising Eq. (6). The non-observations of new gauge-bosons or vectorlike
states would imply $m_{3}\ll M_{b},m_{4}$. In this limit, the loop function in
Eq. (24) can be approximated to

$\Delta
b_{0}[M_{b}^{2}]\simeq\frac{m_{4}^{2}}{M_{b}^{2}-m_{4}^{2}}\,\ln\left(\frac{M_{b}^{2}}{m_{4}^{2}}\right)\,.$
(26)

The following noteworthy features of the loop-corrected fermion mass matrix
can be deduced from Eq. (24) along with Eq. (26) and the explicit expressions
of the $\lambda^{a}$ given in Appendinx A.

  * •

The loop-induced masses are suppressed by the loop factor
$g_{F}^{2}/(16\pi^{2})$ if $m_{4}>M_{b}$. An additional suppression by factor
$m_{4}^{2}/M_{b}^{2}$ arises in case $m_{4}<M_{b}$.

  * •

For a generic choice of gauge boson masses and the orthogonal matrix ${\cal
R}$, Eq. (24) induces masses of the same order for both the first and second-
generation fermions.

  * •

A phenomenologically desired possibility would be that only the second
generation fermions become massive at 1-loop while the first generation
remains massless and receive mass at higher order. Inspecting the expression
Eq. (24) and the Gell-mann matrices, we do not find a possibility in which the
first-generation fermions can be made strictly massless at 1-loop.

The above results indicate that while the loop suppressed masses for the first
and second generations masses naturally emerge, the hierarchy between the two
requires additional arrangements. Utilizing the first feature mentioned above,
we discuss a scenario in the next section which can lead to such a hierarchy
within this framework.

##  III  Gauge-boson mass hierarchy

Consider a two-step breaking of $SU(3)_{F}$ symmetry such that

$SU(3)_{F}\,\xrightarrow{\langle\eta_{1}\rangle}\,SU(2)_{F}\,\xrightarrow{\langle\eta_{2}\rangle}\,{\rm
nothing}\,,$ (27)

with $\langle\eta_{1}\rangle\gg\langle\eta_{2}\rangle$. With this arrangement,
three of the gauge bosons corresponding to the generators of $SU(2)_{F}$ are
expected to be lighter than the remaining five gauge bosons. This hierarchy
among the gauge bosons ultimately results in the mass hierarchy between the
first and second-generation fermions as we show below.

In the choice of our basis of Gell-Mann matrices, it is convenient to identify
the intermediate $SU(2)_{F}$ with the generators $\lambda^{\alpha}$ with
$\alpha=1,2,3$. Denoting the remaining indices with $m=4,...8$, the gauge
boson mass term in the canonical basis
$A^{\mu}_{a}=\left(A^{\mu}_{\alpha},A^{\mu}_{m}\right)$ can be written as

$-{\cal L}^{M}_{\rm GB}=\frac{1}{2}{\cal M}^{2}_{ab}\,A^{\mu}_{a}A_{b\mu}\,,$
(28)

such that

${\cal M}^{2}=\left(\begin{array}[]{cc}M^{2}_{(33)}&M^{2}_{(35)}\\\
(M^{2}_{(35)})^{T}&M^{2}_{(55)}\end{array}\right)\,.$ (29)

Here, $M^{2}_{(AB)}$ are $A\times B$ dimensional matrices sub-blocks of the
gauge boson mass matrix. The two step breaking of the horizontal symmetry
implies $M^{2}_{(33)},M^{2}_{(35)}\ll M^{2}_{(55)}$. The gauge-boson mass
matrix in this specific form can be diagonalized by see-saw like
diagonalization procedure and the orthogonal matrix at the leading order can
be written as

${\cal R}=\left(\begin{array}[]{cc}R_{3}&-\rho R_{5}\\\
\rho^{T}R_{3}&R_{5}\end{array}\right)+{\cal O}(\rho^{2})\,,$ (30)

with $\rho=-M^{2}_{(35)}(M_{(55)}^{2})^{-1}$. Here, $R_{3}$ and $R_{5}$ are
real orthogonal matrices of dimensions $3\times 3$ and $5\times 5$,
respectively.

Substituting Eq. (30) in Eq. (24) and considering the leading order terms in
the seesaw expansion parameter $\rho$, we get

$\displaystyle({\delta M})_{ij}=\frac{{g_{F}}^{2}}{16\pi^{2}}$
$\displaystyle\Big{[}$
$\displaystyle(R_{3})_{\alpha\beta}(R_{3})_{\gamma\beta}\left(\lambda^{\alpha}M^{0}\lambda^{\gamma}\right)_{ij}\Delta
b_{0}[M_{\beta}^{2}]\Big{.}$ (31) $\displaystyle+$
$\displaystyle(R_{3})_{\alpha\beta}(\rho^{T}R_{3})_{m\beta}\left(\lambda^{\alpha}M^{0}\lambda^{m}+\lambda^{m}M^{0}\lambda^{\alpha}\right)_{ij}\Delta
b_{0}[M_{\beta}^{2}]$ $\displaystyle+$
$\displaystyle(R_{5})_{mn}(R_{5})_{pn}\left(\lambda^{m}M^{0}\lambda^{p}\right)_{ij}\Delta
b_{0}[M_{n}^{2}]$ $\displaystyle-$ $\displaystyle\Big{.}(R_{5})_{mn}(\rho
R_{5})_{\alpha
n}\left(\lambda^{m}M^{0}\lambda^{\alpha}+\lambda^{\alpha}M^{0}\lambda^{m}\right)_{ij}\Delta
b_{0}[M_{n}^{2}]\,+\,{\cal O}(\rho^{2})\Big{]}\,.$

Recall that $\alpha,\beta,...=1,2,3$ while $m,n,...=4,...,8$. The first term
gives the dominant contribution to $\delta M$ as $M_{\alpha}^{2}<M_{m}^{2}$.
Since $\lambda^{\alpha}$ has a vanishing first row and first column, this
contribution is of rank one and it induces only the mass for the second
generation fermion. For $M_{\alpha}<m_{4}$, this mass is suppressed by only
the loop factor in comparison to that of the third generation. The masses of
the first-generation fermions arise from the remaining terms in Eq. (31) and
it is suppressed either by $M_{\alpha}^{2}/M_{m}^{2}$ or $m_{4}^{2}/M_{m}^{2}$
with respect to the second-generation mass. In this way, the 1-loop induced
corrections can give rise to the desired hierarchy between the fermion masses
if

$M_{\alpha}^{2}\lesssim m_{4}^{2}\lesssim M_{m}^{2}\,.$ (32)

This implies that the scale of $SU(3)_{F}$ breaking and the mass scale of
vectorlike fermions are required to be close to each other. However, the
overall scale of these new states is not constrained from the pure
consideration of fermion masses as the finite corrections always come as a
ratio of the $m_{4}$ and $M_{a}$.

##  IV Model implementation

Based on the general conditions to generate fermion mass hierarchy through
quantum corrections, we now give a specific and minimal implementation of the
general framework in which all these aspects can be realised explicitly. As
anticipated, we consider three generations of SM fermions transforming as
fundamental representations of the horizontal gauged symmetry $SU(3)_{F}$. We
additionally consider $N_{R}$, triplet of three SM singlet fermions under
$SU(3)_{F}$, which is necessary for anomally cancellation. The SM Higgs is
replaced by the two Higgs doublets, each of them comes also in three copies to
form triplets of $SU(3)_{F}$. Two additional SM singlet and $SU(3)_{F}$
triplets scalars, $\eta_{s}$, are introduced for the consistent gauge symmetry
breaking and also to give rise to the desired fermion mass matrices at the
tree level. As already outlined in section II, the framework requires
vectorlike fermions which are assumed singlets under the new symmetry. The
matter and scalar fields along with their SM and $SU(3)_{F}$ transformation
properties are summarised in Table 1.

Fields |  $(SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y})$ |  $SU(3)_{F}$
---|---|---
$Q_{L}$ | $(3,2,\frac{1}{6})$ | 3
$u_{R}$ | $(3,1,\frac{2}{3})$ | 3
$d_{R}$ | $(3,1,-\frac{1}{3})$ | 3
$L_{L}$ | $(1,2,-\frac{1}{2})$ | 3
$e_{R}$ | $(1,1,-1)$ | 3
$N_{R}$ | $(1,1,0)$ | 3
$H_{u}$ | $(1,2,-\frac{1}{2})$ | 3
$H_{d}$ | $(1,2,\frac{1}{2})$ | 3
$\eta_{1}$, $\eta_{2}$ | $(1,1,0)$ | ${3}$
$T_{L},\,T_{R}$ | $(3,1,\frac{2}{3})$ | 1
$B_{L},\,B_{R}$ | $(3,1,-\frac{1}{3})$ | 1
$E_{L},\,E_{R}$ | $(1,1,-1)$ | 1
Table 1: The SM and $G_{F}$ charges of various fermions and scalars of the
model.

###  IV.1 $SU(3)_{F}$ breaking and gauge boson mass ordering

The non-observations of $SU(3)_{F}$ gauge bosons in the experiments so far
imply that the scale of the latter’s breaking is much larger than the weak
scale. Therefore, the new gauge symmetry must be primarily broken by the SM
singlet fields $\eta_{1,2}$ and the contributions from the electroweak
doublets are expected to be sub-dominant. With this reasoning, we consider the
breaking of $SU(3)_{F}$ driven by $\eta_{1,2}$ only.

As an example, we consider the following VEV configuration:

$\langle\eta_{1}\rangle=(v_{F},0,0)^{T}\,,~{}~{}\langle\eta_{2}\rangle=(0,0,\epsilon
v_{F})^{T}\,,$ (33)

with $\epsilon<1$. One of these can always be chosen in the given form using
the $SU(3)_{F}$ rotation without losing generality. Therefore, the single
field does not break the gauge symmetry completely and leaves its $SU(2)$
subgroup unbroken. This requires at least two scalars to break fully the
$SU(3)_{F}$ symmetry with VEVs in different directions. For simplicity, we
choose the other VEVs in a specific direction orthogonal to the first one. We
write down the most general gauge invariant potential involving $\eta_{1,2}$
in Appendinx B and show that the VEV configuration given in Eq. (33) can be
obtained by a suitable choice of parameters in the scalar potential.

The kinetic terms of $\eta_{1,2}$ after the spontaneous breaking of
$SU(3)_{F}$ leads to the following gauge boson mass matrix defined in Eq.
(28):

${\cal
M}^{2}_{\alpha\beta}=\frac{g_{F}^{2}}{2}\,\sum_{s=1,2}\langle\eta_{s}\rangle^{\dagger}\lambda^{\alpha\dagger}\lambda^{\beta}\langle\eta_{s}\rangle\,.$
(34)

In the notation of Eq. (29), the above mass matrix can be written as

$\displaystyle M^{2}_{(33)}$ $\displaystyle=$
$\displaystyle\frac{g_{F}^{2}v_{F}^{2}}{2}\,{\rm
Diag.}\left(\epsilon^{2},\epsilon^{2},\epsilon^{2}\right)\,,$ $\displaystyle
M^{2}_{(55)}$ $\displaystyle=$
$\displaystyle\frac{g_{F}^{2}v_{F}^{2}}{2}\,{\rm
Diag.}\left(1,1,1+\epsilon^{2},1+\epsilon^{2},\frac{1}{3}(4+\epsilon^{2})\right)\,,$
$\displaystyle M^{2}_{(35)}$ $\displaystyle=$
$\displaystyle\frac{g_{F}^{2}v_{F}^{2}}{2}\,\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&-\frac{\epsilon^{2}}{\sqrt{3}}\\\ \end{array}\right)$
(38)

The structure of the gauge boson mass matrix in this case is extremely simple
and there exists mixing between only $A^{\mu}_{3}$ and $A^{\mu}_{8}$ states
which correspond to diagonal generators.

The matrix ${\cal M}^{2}$ can be diagonalized by ${\cal R}$ as parametrized by
Eq. (30) with the following explicit forms of its sub-matrices:

$R_{3}=\mathbb{I}_{3\times 3}\,,~{}~{}R_{5}=\mathbb{I}_{5\times
5}\,,~{}~{}\rho=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&-\frac{\sqrt{3}}{4}\epsilon^{2}\\\ \end{array}\right)\,.$ (39)

The diagonal gauge boson mass matrix is then obtained as

${\cal D}^{2}=\frac{g_{F}^{2}v_{F}^{2}}{2}\,{\rm
Diag.}\left(\epsilon^{2},\epsilon^{2},\epsilon^{2},1,1,1+\epsilon^{2},1+\epsilon^{2},\frac{4}{3}+\frac{1}{3}\epsilon^{2}\right)\,+\,{\cal
O}(\epsilon^{4})\,.$ (40)

In this setup, one obtains the hierarchical gauge boson mass spectrum
$M^{2}_{1,2,3}\ll M^{2}_{4,...,8}$ as required to generate the mass gaps
between the first and second-generation fermions.

Substituting the above in the $(\delta M)_{ij}$, we find

$\displaystyle\frac{16\pi^{2}}{g_{F}^{2}}(\delta M)_{ij}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{3}\left(\lambda^{\alpha}M^{0}\lambda^{\alpha}\right)_{ij}\Delta
b_{0}[M_{\alpha}^{2}]+\sum_{m=4}^{8}\left(\lambda^{m}M^{0}\lambda^{m}\right)_{ij}\Delta
b_{0}[M_{m}^{2}]$ (41) $\displaystyle+$
$\displaystyle\frac{\sqrt{3}}{4}\epsilon^{2}\left(\lambda^{3}M^{0}\lambda^{8}+\lambda^{8}M^{0}\lambda^{3}\right)_{ij}\left(\Delta
b_{0}[M_{3}^{2}]-\Delta b_{0}[M_{8}^{2}]\right)\,+\,{\cal O}(\epsilon^{4})\,.$

An approximate degeneracy of some of the gauge bosons allows further
simplification. Setting

$M_{1}^{2}\simeq M_{2}^{2}\simeq M_{3}^{2}\equiv
M_{Z_{1}}^{2}\,,~{}~{}M_{4}^{2}\simeq...\simeq
M_{7}^{2}\simeq\frac{3}{4}M_{8}^{2}\equiv M_{Z_{2}}^{2}\,,$ (42)

and using $\epsilon^{2}=M_{Z_{1}}^{2}/M_{Z_{2}}^{2}$, we find

$\displaystyle\frac{16\pi^{2}}{g_{F}^{2}}\,{\delta M}$ $\displaystyle\simeq$
$\displaystyle\left(\begin{array}[]{ccc}0&0&0\\\
0&M^{0}_{22}+2M^{0}_{33}&-M^{0}_{23}\\\
0&-M^{0}_{32}&2M^{0}_{22}+M^{0}_{33}\end{array}\right)\,\Delta
b_{0}[M_{Z_{1}}^{2}]$ (46) $\displaystyle+$
$\displaystyle\left(\begin{array}[]{ccc}2(M^{0}_{22}+M^{0}_{33})&0&0\\\
0&2M^{0}_{11}&0\\\ 0&0&2M^{0}_{11}\end{array}\right)\,\Delta
b_{0}[M_{Z_{2}}^{2}]$ (50) $\displaystyle+$
$\displaystyle\frac{1}{3}\left(\begin{array}[]{ccc}4M^{0}_{11}&-2M^{0}_{12}&-2M^{0}_{13}\\\
-2M^{0}_{21}&M^{0}_{22}&M^{0}_{23}\\\
-2M^{0}_{31}&M^{0}_{32}&M^{0}_{33}\end{array}\right)\,\Delta
b_{0}\Big{[}\frac{4}{3}M_{Z_{2}}^{2}\Big{]}$ (54) $\displaystyle+$
$\displaystyle\frac{M_{Z_{1}}^{2}}{2M_{Z_{2}}^{2}}\left(\begin{array}[]{ccc}0&-M^{0}_{12}&M^{0}_{13}\\\
-M^{0}_{21}&M^{0}_{22}&0\\\
M^{0}_{31}&0&-M^{0}_{33}\end{array}\right)\left(\Delta
b_{0}[M_{Z_{1}}^{2}]-\Delta
b_{0}\Big{[}\frac{4}{3}M_{Z_{2}}^{2}\Big{]}\right).$ (58)

###  IV.2 Charged fermion masses

With the set of fields and their transformation properties defined in Table 1,
the most general renormalizable Yukawa Lagrangian of the model can be written
as

$\displaystyle-{\cal L}_{Y}$ $\displaystyle=$
$\displaystyle{y}_{u}\,\overline{Q^{\prime}}_{Li}H^{i}_{u}T^{\prime}_{R}\,+{y}_{d}\,\overline{Q^{\prime}}_{Li}H_{d}^{i}B^{\prime}_{R}\,+{y}_{e}\,\overline{L^{\prime}_{i}}_{L}H_{d}^{i}E^{\prime}_{R}\,$
(59) $\displaystyle+$
$\displaystyle{y}^{\prime(s)}_{u}\,\overline{T^{\prime}}_{L}\eta^{\dagger}_{si}\,u^{\prime
i}_{R}\,+{y}^{\prime(s)}_{d}\,\overline{B^{\prime}}_{L}\eta^{\dagger}_{si}\,d^{\prime
i}_{R}\,+{y}^{\prime(s)}_{e}\,\overline{E^{\prime}}_{L}\eta^{\dagger}_{si}\,e^{\prime
i}_{R}\,$ $\displaystyle+$ $\displaystyle
m_{T}\overline{T^{\prime}}_{L}T^{\prime}_{R}+\,m_{B}\overline{B^{\prime}}_{L}B^{\prime}_{R}+\,m_{E}\overline{E^{\prime}}_{L}E^{\prime}_{R}\,+{\rm
h.c.}\,$

where $i=1,2,3$ is an $SU(3)_{F}$ index and $s=1,2$ denotes multiplicity of
$\eta$ fields. The primed notation is used for the fields in a flavour basis.

It is straightforward to see that after the electroweak and $SU(3)_{F}$
breaking, the Yukawa interactions in Eq. (59) lead to the tree-level mass
matrices in the desired form of Eq. (2) with

$\mu_{f}\,=\,\left(\begin{array}[]{ccc}y_{f}v^{f}_{1}&y_{f}v^{f}_{2}&y_{f}v^{f}_{3}\end{array}\right)^{T}\,~{}\text{and
}\,~{}\mu^{\prime}_{f}\,=\,\left(\begin{array}[]{ccc}y^{\prime(1)}_{f}\,v_{F}&0&y^{\prime(2)}_{f}\,\epsilon
v_{F}\end{array}\right)\,,$ (60)

where $f=u,d,e$ which accounts for three type of charged fermions. Also,
$v^{u}_{i}=\langle H_{u}^{i}\rangle$, $v^{d}_{i}=v^{e}_{i}=\langle
H_{d}^{i}\rangle$. The VEVs of $\eta_{1,2}$ are taken from Eq. (33). The tree-
level effective mass matrix after integrating out the heavy vectorlike states
is given by

$M_{u,d,e}^{0}\equiv-\frac{1}{m_{T,B,E}}\,\mu_{u,d,e}\,\mu^{\prime}_{u,d,e}\,.$
(61)

The matrices $M_{f}^{0}$ has vanishing second column.

At 1-loop, the charged fermion mass matrices are given by

$M_{f}=M_{f}^{0}+\delta M_{f}\,,$ (62)

and $\delta M_{f}$ can be obtained using the general expression, Eq. (46).
Using the form of the tree-level mass matrices, we get

$\displaystyle\delta M_{f}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{F}^{2}N_{f}}{16\pi^{2}}\left[\left(\begin{array}[]{ccc}0&0&0\\\
0&2(M^{0}_{f})_{33}&-(M^{0}_{f})_{23}\\\
0&0&(M^{0}_{f})_{33}\end{array}\right)\,\Delta b_{0}[M_{Z_{1}}^{2}]\right.$
(66) $\displaystyle+$ $\displaystyle
2\left(\begin{array}[]{ccc}(M^{0}_{f})_{33}&0&0\\\ 0&(M^{0}_{f})_{11}&0\\\
0&0&(M^{0}_{f})_{11}\end{array}\right)\,\Delta b_{0}[M_{Z_{2}}^{2}]$ (70)
$\displaystyle+$
$\displaystyle\frac{1}{3}\left(\begin{array}[]{ccc}4(M^{0}_{f})_{11}&0&-2(M^{0}_{f})_{13}\\\
-2(M^{0}_{f})_{21}&0&(M^{0}_{f})_{23}\\\
-2(M^{0}_{f})_{31}&0&(M^{0}_{f})_{33}\end{array}\right)\,\Delta
b_{0}\Big{[}\frac{4}{3}M_{Z_{2}}^{2}\Big{]}$ (74) $\displaystyle+$
$\displaystyle\left.\frac{1}{2}\frac{M_{Z_{1}}^{2}}{M_{Z_{2}}^{2}}\left(\begin{array}[]{ccc}0&0&(M^{0}_{f})_{13}\\\
-(M^{0}_{f})_{21}&0&0\\\
(M^{0}_{f})_{31}&0&-(M^{0}_{f})_{33}\end{array}\right)\,\left(\Delta
b_{0}[M_{Z_{1}}^{2}]-\Delta
b_{0}\Big{[}\frac{4}{3}M_{Z_{2}}^{2}\Big{]}\right)\right]\,,$ (78)

where $N_{f}=3$ for $f=u,d$ and $N_{f}=1$ for $f=e$. As we demonstrate in the
next section, the above $M_{f}$ can reproduce the observed charged fermion
mass spectrum and quark mixing.

###  IV.3 Neutrino masses

As noted earlier, the anomaly-free model requires the SM singlet fermions
$N_{R}$. With the present field content and the symmetry of the model, it is
evident that there is no Dirac Yukawa coupling between $L_{L}$ and $N_{R}$ and
no Majorana mass term for $N_{R}$ at the renormalizable level. This prevents
$N_{R}$ from contributing to the light neutrino masses through the
conventional type I seesaw mechanism.

The symmetry of the model, however, allows the following Weinberg operators

$\frac{c_{1}}{\Lambda}\left(\overline{L_{L}^{c}}^{i}H_{ui}^{*}\right)\left(L_{L}^{j}H_{uj}^{\dagger}\right)+\frac{c_{2}}{\Lambda}\left(\frac{\lambda^{a}}{2}\right)^{k}_{i}\left(\frac{\lambda^{a}}{2}\right)^{l}_{j}\left(\overline{L_{L}^{c}}^{i}H_{uk}^{*}\right)\left(L_{L}^{j}H_{ul}^{\dagger}\right)+\frac{c_{3}}{\Lambda}\left(\overline{L_{L}^{c}}^{i}L_{L}^{j}\right)\left(H_{ui}^{\dagger}H_{uj}^{*}\right)\,,$
(79)

which can induce the suppressed neutrino masses in comparison to those of the
charged fermions. It is straightforward to extend the model for an ultra-
violet completion of the above operators. For instance, the simplest
possibility is to introduce two or more fermions, namely $\nu_{Rk}$, which are
singlets under the full gauge symmetry and, therefore, do not give rise to
anomalies. They can couple to $SU(3)_{F}$ triplet $L_{L}^{i}$ and anti-triplet
$H_{ui}^{\dagger}$ giving rise to the usual Dirac Yukawa term and can also
possess Majorana mass term unrestricted by the gauge symmetry of the model.
The first operator in Eq. (79) gets generated when $\nu_{Rk}$ are integrated
out from the spectrum. Similarly, the second and third operators can be
induced by heavy $SU(3)_{F}$ adjoint fermions and sextet scalars respectively.

In contrast to charged fermions, neutrinos can acquire their masses at the
tree level through dimension-5 operators without being restricted by the
constraints of underlying flavour symmetries. This property is particularly
advantageous because the inter-generational mass hierarchy among neutrinos is
comparatively weaker than that observed among charged fermions. Consequently,
the masses and mixing parameters of neutrinos remain relatively unconstrained
within the framework of the effective theory. Nevertheless, it’s worth noting
that specific constraints could emerge depending on the chosen ultra-violet
completion.

##  V Numerical Solutions

To establish the validity of the model and to understand the pattern of its
various parameters, we carry out numerical analysis to find example solutions
which can reproduce the observed values of the charged fermion masses and
quark mixing parameters. Removing the unphysical phases through redefinitions
of various matter fields in Eq. (59), it can be seen that the parameters
$y_{f}$, $y_{f}^{\prime(1)}$ and $m_{T,B,E}$ can be made real. Moreover, we
assume that all the VEVs are real. This leaves only three complex parameters,
namely $y_{u,d,e}^{\prime(2)}$ in the model. Through Eq. (60), this implies
real $(\mu_{f})_{i}$ (for $i=1,2,3$), real $(\mu_{f}^{\prime})_{1}$ and a
complex $(\mu_{f}^{\prime})_{3}$. Moreover,

$\mu_{e}=\frac{y_{e}}{y_{d}}\mu_{d}\equiv r\,\mu_{d}\,,$ (80)

where $r$ is a real parameter. Altogether, there are $21$ real parameters
(real $\mu_{ui}$, $\mu_{di}$, $r$, $\mu_{u1}^{\prime}$, $\mu_{d1}^{\prime}$,
$\mu_{e1}^{\prime}$, $m_{T}$, $m_{B}$, $m_{E}$, $M_{Z_{1}}$, $M_{Z_{2}}$ and
complex $\mu_{u3}^{\prime}$, $\mu_{d3}^{\prime}$, $\mu_{e3}^{\prime}$) in the
model leading to $13$ observables (9 charged fermion masses, 3 quark mixing
angles and a CP phase). Despite of a large number of parameters than the
observables, it is not obvious that the model can viably reproduce the latter
given various constraints and correlations among the parameters as we describe
below.

Parameters |  Solution 1 (S1) |  Solution 2 (S2) |  Solution 3 (S3)
---|---|---|---
$M_{Z_{1}}$ |  $10^{4}$ GeV |  $10^{6}$ GeV |  $10^{8}$ GeV
$\epsilon$ | $0.0131$ | $0.0083$ | $0.0108$
$\epsilon_{T}$ | $0.3576$ | $0.3455$ | $0.4357$
$\epsilon_{B}$ | $0.9838$ | $0.9999$ | $0.9956$
$\epsilon_{E}$ | $0.8831$ | $0.3881$ | $0.7358$
$\epsilon_{u1}$ | $0.4013$ | $-0.3903$ | $-0.3745$
$\epsilon_{u2}$ | $0.7211$ | $0.6237$ | $-0.8653$
$\epsilon_{u3}$ | $-0.8575$ | $0.6714$ | $-0.8504$
$\epsilon^{\prime}_{u1}$ | $0.2970$ | $0.3425$ | $-0.3393$
$\epsilon^{\prime}_{u3}$ | $0.0115-i\,0.32\times 10^{-4}$ | $0.0142+i\,0.08\times 10^{-4}$ | $0.0140-i\,0.72\times 10^{-4}$
$\epsilon_{d1}$ | $-0.1160$ | $0.2935$ | $-0.2423$
$\epsilon_{d2}$ | $-0.2288$ | $-0.4345$ | $-0.5288$
$\epsilon_{d3}$ | $0.2494$ | $-0.5095$ | $-0.5668$
$\epsilon^{\prime}_{d1}$ | $0.0460$ | $0.0227$ | $-0.0205$
$\epsilon^{\prime}_{d3}$ | $0.0030+i\,0.95\times 10^{-3}$ | $0.0017-i\,0.33\times 10^{-3}$ | $0.0011+i\,0.41\times 10^{-3}$
$r$ | $3.7895$ | $-0.5749$ | $-0.4040$
$\epsilon^{\prime}_{e1}$ | $-0.0023$ | $-0.0013$ | $-0.0079$
$\epsilon^{\prime}_{e3}$ | $-0.0062-i\,0.28\times 10^{-3}$ | $-0.0093-i\,0.86\times 10^{-3}$ | $0.0214+i\,0.42\times 10^{-3}$
Table 2: Three benchmark solutions and the optimized values of the model
parameters for different $M_{Z_{1}}$ which lead to viable charged fermion
masses and quark mixing.

For simplicity, various dimension-full parameters can be expressed in terms of
the mass scales in the model and dimension-less quantities. Considering that
viable fermion mass hierarchy would prefer $M_{Z_{1}}^{2}\lesssim
m_{T,B,E}^{2}\lesssim M_{Z_{2}}^{2}$ (see Eq. (32)), we define

$m_{T}=\epsilon_{T}M_{Z_{2}},\,m_{B}=\epsilon_{B}M_{Z_{2}},\,m_{E}=\epsilon_{E}M_{Z_{2}}.$
(81)

Also recall that $M_{Z_{1}}=\epsilon M_{Z_{2}}$. Moreover, we also define

$\mu_{f1}^{\prime}=\epsilon_{f1}^{\prime}M_{Z_{2}},\,\mu_{f3}^{\prime}=\epsilon_{f3}^{\prime}M_{Z_{2}}\,,$
(82)

and

$\mu_{ui}=\epsilon_{ui}v,\,\mu_{di}=\epsilon_{di}v,$ (83)

where $v=174$ GeV. In this way, various $\epsilon$ and $\epsilon^{\prime}$ can
preferably take values less than unity.

Taking a particular value of $M_{Z_{1}}$, we obtain the values of the
remaining dimensionless parameters using the $\chi^{2}$ optimization
technique. Our methodology for the latter is described in detail in Mummidi
and Patel (2021). Three benchmark solutions obtained in this way are displayed
in Table 2 for different $M_{Z_{1}}$. The minimized values of $\chi^{2}$ are
$6.97$, $6.90$ and $6.46$ for the solutions S1, S2 and S3, respectively. We
also list the resulting values of charged fermion masses and quark mixing
parameters for all three solutions in Table 3 along with the corresponding
experimental values for comparison.

Observable | Value |  S1 |  S2 |  S3
---|---|---|---|---
$m_{u}$ [MeV] | $1.27\pm 0.5$ | $1.31$ | $1.26$ | $1.23$
$m_{c}$ [GeV] | $0.619\pm 0.084$ | $0.567$ | $0.662$ | $0.612$
$m_{t}$ [GeV] | $171.7\pm 3.0$ | $171.7$ | $171.6$ | $171.6$
$m_{d}$ [MeV] | $2.90\pm 1.24$ | $3.71$ | $4.15$ | $3.55$
$m_{s}$ [GeV] | $0.055\pm 0.016$ | $0.016$ | $0.018$ | $0.016$
$m_{b}$ [GeV] | $2.89\pm 0.09$ | $2.89$ | $2.88$ | $2.89$
$m_{e}$ [MeV] | $0.487\pm 0.049$ | $0.492$ | $0.487$ | $0.489$
$m_{\mu}$ [GeV] | $0.1027\pm 0.0103$ | $0.1007$ | $0.099$ | $0.1004$
$m_{\tau}$ [GeV] | $1.746\pm 0.174$ | $1.784$ | $1.786$ | $1.777$
$|V_{us}|$ | $0.22500\pm 0.00067$ | $0.21614$ | $0.22242$ | $0.22226$
$|V_{cb}|$ | $0.04182\pm 0.00085$ | $0.04110$ | $0.04270$ | $0.04207$
$|V_{ub}|$ | $0.00369\pm 0.00011$ | $0.00363$ | $0.00378$ | $0.00371$
$J_{CP}$ | $(3.08\pm 0.15)\times 10^{-5}$ | $3.14\times 10^{-5}$ | $3.02\times 10^{-5}$ | $3.06\times 10^{-5}$
Table 3: The fitted values of the charged fermion masses and quark mixing
parameters at the minimum of $\chi^{2}$ for three benchmark solutions are
displayed in Table 2. The second column denotes experimentally measured value
of corresponding observable extrapolated at $M_{Z}$ that has been used in the
$\chi^{2}$ function.

Some of the noteworthy features of the model that can be derived from Table 2
are as the following. We obtain almost similar values of the minimized
$\chi^{2}$ for different values of $M_{Z_{1}}$. The ability to reproduce the
realistic flavour hierarchies, therefore, depends on the relative masses of
new gauge bosons and vectorlike states and not on the overall flavour symmetry
breaking scale. This is expected as the flavour hierarchies are technically
natural. All the $\epsilon_{fi}$ are of ${\cal O}(10^{-1})$ implying the
fundamental Yukawa couplings $y_{f}$ of the same order and no large hierarchy
between the VEVs, $v^{u,d}_{i}$. One also finds
$\epsilon^{\prime}_{f3}<\epsilon^{\prime}_{f1}$ for $f=u,d$ as expected from
Eq. (60). The fitted values of $\epsilon^{\prime}_{e1}$, however, require two
orders of separation between the magnitudes of $y_{e}^{\prime(1)}$ and
$y_{u}^{\prime(1)}$. Altogether, the values of fundamental Yukawa couplings of
the model range in just two orders of magnitude unlike in the SM where such a
range spans at least five orders. Since all the third-generation fermions
receive their masses at the tree level, the hierarchy between $m_{t}$ and
$m_{b,\tau}$ does not follow naturally and requires
$\epsilon_{T}\ll\epsilon_{B,E}$.

It can be noticed from Table 3 that all the observables, except $m_{s}$, are
fitted within $\pm 1\sigma$ range of their reference values for all the
solutions. The fitted value of $m_{s}$ is $\sim 2.4\sigma$ away from the
experimental value. Despite having a sufficiently large number of parameters
than the observable, the inability to reproduce the central value of $m_{s}$
indicates the existence of non-trivial correlations between the observables
that result from the predictive nature of non-abelian flavour symmetry.
Remarkably, a more precise measurement of strange quark mass can falsify the
model irrespective of the scale of $SU(3)_{F}$ breaking.

##  VI Flavour violation

One of the most common features of radiative fermion mass models is the
inherent presence of flavour-changing neutral currents. In the present
framework, they arise from (i) flavour non-universal gauge interactions and
(ii) mixing between the chiral and vectorlike fermions. Typically for
$M_{Z_{1}}\ll m_{T,B,E}$, the first provides dominant contributions over the
second and leads to a lower limit on the mass scales of the new fields. We
study them in detail in this section by first deriving the general dimension-6
effective operators and then estimating various relevant quark and lepton
flavour transitions.

Rewriting Eq. (5) in the physical basis of fermions, one finds

$-{\cal L}_{\rm
gauge}=\frac{g_{F}}{2}\left(\overline{f}_{Li}\gamma^{\mu}\left(\tilde{\lambda}^{a}_{f}{}_{L}\right)_{ij}f_{Lj}\,+\,\overline{f}_{Ri}\gamma^{\mu}\left(\tilde{\lambda}^{a}_{f}{}_{R}\right)_{ij}f_{Rj}\right)\,{\cal
R}_{ab}B^{b}_{\mu}\,,$ (84)

where

$\tilde{\lambda}^{a}_{f}{}_{L,R}=U^{f}{}^{\dagger}_{L,R}\,\lambda^{a}\,U^{f}{}_{L,R}\,,$
(85)

and $U^{f}{}_{L,R}$ are unitary matrices that diagonalize the 1-loop corrected
$M_{f}$. Integrating out the gauge bosons, we find the effective dimension-6
operators as

$\displaystyle{\cal L}_{\rm eff}=$ $\displaystyle
C^{(ff^{\prime})LL}_{ijkl}\,\overline{f}_{Li}\gamma^{\mu}f_{Lj}\,\overline{f^{\prime}}_{Lk}\gamma_{\mu}f^{\prime}_{Ll}+C^{(ff^{\prime})RR}_{ijkl}\,\overline{f}_{Ri}\gamma^{\mu}f_{Rj}\,\overline{f^{\prime}}_{Rk}\gamma_{\mu}f^{\prime}_{Rl}\,$
(86) $\displaystyle+$ $\displaystyle
C^{(ff^{\prime})LR}_{ijkl}\,\overline{f}_{Li}\gamma^{\mu}f_{Lj}\,\overline{f^{\prime}}_{Rk}\gamma_{\mu}f^{\prime}_{Rl}+C^{(ff^{\prime})RL}_{ijkl}\,\overline{f}_{Ri}\gamma^{\mu}f_{Rj}\,\overline{f^{\prime}}_{Lk}\gamma_{\mu}f^{\prime}_{Ll}\,,$

where

$C^{(ff^{\prime})PP^{\prime}}_{ijkl}=\frac{g_{F}^{2}}{8M_{b}^{2}}\,{\cal
R}_{ab}{\cal
R}_{cb}\,\left(\tilde{\lambda}^{a}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{c}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}\,,$
(87)

and $P,P^{\prime}=L,R$ and $f,f^{\prime}=u,d,e$. The coefficients of the
effective operators can be further simplified for the hierarchical gauge boson
mass spectrum. Using Eqs. (29,30), we find at leading order in $\rho$

$\displaystyle\frac{8}{g_{F}^{2}}C^{(ff^{\prime})PP^{\prime}}_{ijkl}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{M_{\alpha}^{2}}\left((R_{3})_{\beta\alpha}(R_{3})_{\gamma\alpha}\left(\tilde{\lambda}^{\beta}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{\gamma}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}\right.$
(88) $\displaystyle+$
$\displaystyle\left.(R_{3})_{\beta\alpha}(\rho^{T}R_{3})_{m\alpha}\left(\tilde{\lambda}^{\beta}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{m}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}+(\rho^{T}R_{3})_{m\alpha}(R_{3})_{\beta\alpha}\left(\tilde{\lambda}^{m}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{\beta}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{M_{m}^{2}}\left((R_{5})_{nm}(R_{5})_{pm}\left(\tilde{\lambda}^{n}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{p}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}\right.$
$\displaystyle-$ $\displaystyle\left.(\rho R_{5})_{\alpha
m}(R_{5})_{nm}\left(\tilde{\lambda}^{\alpha}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{n}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}-(R_{5})_{nm}(\rho
R_{5})_{\alpha
m}\left(\tilde{\lambda}^{n}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{\alpha}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}\right)\,.$

Further simplification can be achieved in the explicit model with the help of
Eqs. (39) and (42). Substituting them in the above, we find

$\displaystyle\frac{8M_{Z_{1}}^{2}}{g_{F}^{2}}\,C^{(ff^{\prime})PP^{\prime}}_{ijkl}$
$\displaystyle\simeq$
$\displaystyle\sum_{\alpha=1}^{3}\left(\tilde{\lambda}^{\alpha}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{\alpha}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}+\epsilon^{2}\sum_{m=4}^{7}\left(\tilde{\lambda}^{m}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{m}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}$
(89) $\displaystyle-$
$\displaystyle\frac{\sqrt{3}}{4}\epsilon^{2}\left(\left(\tilde{\lambda}^{3}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{8}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}+\left(\tilde{\lambda}^{8}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{3}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}\right)$
$\displaystyle+$
$\displaystyle\frac{3}{4}\epsilon^{2}\left(\tilde{\lambda}^{8}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{8}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}+{\cal
O}(\epsilon^{4})\,.$

At the leading order, the flavour violation is governed by the coupling
matrices $\tilde{\lambda}^{\alpha}_{f}{}_{L,R}$. Since they do not commute
with each other, all of them cannot take diagonal form for any $U_{L,R}$.
Therefore, the most dominant flavour violations in the model are captured by
the coefficients

$\displaystyle C^{(ff^{\prime})PP^{\prime}}_{ijkl}$ $\displaystyle\simeq$
$\displaystyle\frac{g_{F}^{2}}{8M_{Z_{1}}^{2}}\sum_{\alpha=1}^{3}\left(\tilde{\lambda}^{\alpha}_{f}{}_{P}\right)_{ij}\left(\tilde{\lambda}^{\alpha}_{f^{\prime}}{}_{P^{\prime}}\right)_{kl}\,.$
(90)

The above can be used to estimate the most dominant contribution to various
flavour-violating processes in the quark and lepton sectors.

###  VI.1 Quark sector

The strongest constraints on various $C^{(ff^{\prime})PP^{\prime}}_{ijkl}$
primarily arise from meson-antimeson oscillations such as
$K^{0}-\overline{K}^{0}$, $B_{d}-\overline{B}_{d}^{0}$,
$B_{s}-\overline{B}_{s}^{0}$ and $D^{0}-\overline{D}^{0}$. To quantify these
constraints, we closely follow the procedure adopted in our previous analysis
Mohanta and Patel (2022). Comparing Eq. (86) with the effective Hamiltonian,
${\cal
H}_{eff}\,=\,\sum_{i=1}^{5}C_{M}^{i}Q^{i}+\sum_{i=1}^{3}\tilde{C}_{M}^{i}\tilde{Q}_{i}$,
which parametrizes $\Delta F=2$ transitions $M-\overline{M}$ Bona  _et al._
(2008), we find the effective Wilson coefficients at $\mu=M_{Z_{1}}$ as

$\displaystyle C^{1}_{K}\,=\,-C^{(dd)LL}_{1212}$ ,
$\displaystyle\tilde{C}^{1}_{K}=-C^{(dd)RR}_{1212}$ ,
$\displaystyle{C}^{5}_{K}=-4\,C^{(dd)LR}_{1212}\,,$ (91) $\displaystyle
C^{1}_{B_{d}}\,=\,-C^{(dd)LL}_{1313}$ ,
$\displaystyle\tilde{C}^{1}_{B_{d}}=-C^{(dd)RR}_{1313}$ ,
$\displaystyle{C}^{5}_{B_{d}}=-4\,C^{(dd)LR}_{1313}\,,$ (92) $\displaystyle
C^{1}_{B_{s}}\,=\,-C^{(dd)LL}_{2323}$ ,
$\displaystyle\tilde{C}^{1}_{B_{s}}=-C^{(dd)RR}_{2323}$ ,
$\displaystyle{C}^{5}_{B_{s}}=-4\,C^{(dd)LR}_{2323}\,,$ (93) $\displaystyle
C^{1}_{D}\,=\,-C^{(uu)LL}_{1212}$ ,
$\displaystyle\tilde{C}^{1}_{D}=-C^{(uu)RR}_{1212}$ ,
$\displaystyle{C}^{5}_{D}=-4\,C^{(uu)LR}_{1212}\,.$ (94)

The remaining $C^{i}_{M}$ and $\tilde{C}^{i}_{M}$ are vanishing at this scale.

Using the renormalization group equations, we evolve all the coefficients from
$\mu=M_{Z_{1}}$ to the $\mu=2$ GeV for $K$ meson Ciuchini  _et al._ (1998),
$\mu=4.6$ GeV for $B$ mesons Becirevic  _et al._ (2002) and $\mu=2.8$ GeV for
$D$ meson Bona  _et al._ (2008). The running gives rise to non-vanishing
$C^{4}_{M}$ while $C^{2,3}_{M}$ and $\tilde{C}^{2,3}_{M}$ remains zero. The
evolved Wilson coefficients are computed using Eq. (90) for the three
benchmark solutions listed in Table 2 and are compared with the corresponding
experimental limits obtained by the UTFit collaboration Bona  _et al._ (2008).
The results are listed in Table 4. It can be seen that the strongest limits
arise from $K^{0}-\overline{K}^{0}$ mixing which disfavours $M_{Z_{1}}\leq
10^{6}$ GeV. The same limit was also observed in our previous framework based
on flavour non-universal abelian symmetries.

Wilson coefficient |  Allowed range |  S1 |  S2 |  S3
---|---|---|---|---
Re$C_{K}^{1}$ | $[-9.6,9.6]\times 10^{-13}$ | $6.0\times 10^{-11}$ | $-1.0\times 10^{-14}$ | $-5.2\times 10^{-19}$
Re$\tilde{C}_{K}^{1}$ | $[-9.6,9.6]\times 10^{-13}$ | $-4.8\times 10^{-16}$ | $-1.7\times 10^{-19}$ | $4.0\times 10^{-24}$
Re$C_{K}^{4}$ | $[-3.6,3.6]\times 10^{-15}$ | $3.2\times 10^{-10}$ | $-1.0\times 10^{-13}$ | $-2.5\times 10^{-18}$
Re$C_{K}^{5}$ | $[-1.0,1.0]\times 10^{-14}$ | $2.8\times 10^{-10}$ | $-8.3\times 10^{-14}$ | $-2.0\times 10^{-18}$
Im$C_{K}^{1}$ | $[-9.6,9.6]\times 10^{-13}$ | $4.4\times 10^{-11}$ | $4.3\times 10^{-15}$ | $-3.8\times 10^{-19}$
Im$\tilde{C}_{K}^{1}$ | $[-9.6,9.6]\times 10^{-13}$ | $-3.4\times 10^{-15}$ | $-5.8\times 10^{-19}$ | $1.8\times 10^{-23}$
Im$C_{K}^{4}$ | $[-1.8,0.9]\times 10^{-17}$ | $-1.6\times 10^{-10}$ | $-4.3\times 10^{-14}$ | $1.4\times 10^{-18}$
Im$C_{K}^{5}$ | $[-1.0,1.0]\times 10^{-14}$ | $-1.4\times 10^{-10}$ | $-3.6\times 10^{-14}$ | $1.1\times 10^{-18}$
$|C_{B_{d}}^{1}|$ | $<2.3\times 10^{-11}$ | $9.2\times 10^{-11}$ | $1.2\times 10^{-14}$ | $8.0\times 10^{-19}$
$|\tilde{C}_{B_{d}}^{1}|$ | $<2.3\times 10^{-11}$ | $2.2\times 10^{-12}$ | $2.8\times 10^{-16}$ | $1.6\times 10^{-20}$
$|C_{B_{d}}^{4}|$ | $<2.1\times 10^{-13}$ | $9.2\times 10^{-11}$ | $1.4\times 10^{-14}$ | $8.1\times 10^{-19}$
$|C_{B_{d}}^{5}|$ | $<6.0\times 10^{-13}$ | $1.6\times 10^{-10}$ | $2.4\times 10^{-14}$ | $1.3\times 10^{-18}$
$|C_{B_{s}}^{1}|$ | $<1.1\times 10^{-9}$ | $1.1\times 10^{-11}$ | $2.9\times 10^{-15}$ | $7.7\times 10^{-20}$
$|\tilde{C}_{B_{s}}^{1}|$ | $<1.1\times 10^{-9}$ | $1.7\times 10^{-12}$ | $2.3\times 10^{-16}$ | $1.3\times 10^{-20}$
$|C_{B_{s}}^{4}|$ | $<1.6\times 10^{-11}$ | $1.3\times 10^{-11}$ | $2.9\times 10^{-15}$ | $1.1\times 10^{-19}$
$|C_{B_{s}}^{5}|$ | $<4.5\times 10^{-11}$ | $2.4\times 10^{-11}$ | $4.8\times 10^{-15}$ | $1.8\times 10^{-19}$
$|C_{D}^{1}|$ | $<7.2\times 10^{-13}$ | $5.5\times 10^{-11}$ | $7.5\times 10^{-15}$ | $6.5\times 10^{-19}$
$|\tilde{C}_{D}^{1}|$ | $<7.2\times 10^{-13}$ | $4.4\times 10^{-16}$ | $5.6\times 10^{-20}$ | $5.4\times 10^{-24}$
$|C_{D}^{4}|$ | $<4.8\times 10^{-14}$ | $1.9\times 10^{-10}$ | $3.6\times 10^{-14}$ | $2.1\times 10^{-18}$
$|C_{D}^{5}|$ | $<4.8\times 10^{-13}$ | $2.2\times 10^{-10}$ | $4.1\times 10^{-14}$ | $2.2\times 10^{-18}$
Table 4: Numerical values of various Wilson coefficients (in ${\rm GeV}^{-2}$
unit) of the operators leading to meson-antimeson oscillations estimated for
three example solutions. The experimentally allowed ranges are taken from Bona
_et al._ (2008). The values highlighted in red are excluded by the respective
limits.

###  VI.2 Lepton sector

As noted in the previous section, the dominant contribution to the flavour
violation process is governed by the first three gauge bosons
$B^{\alpha}_{\mu}$. The exchange of $B^{\alpha}_{\mu}$ mediate lepton flavour
violating process like $\mu\to e$ conversion in nuclei, $l_{i}\to 3l_{j}$ and
$l_{i}\to\l_{j}\gamma$. The first two processes arise at the tree level
whereas the latter is at the one-loop level in the present model.

The $\mu\to e$ conversion in the field of the nucleus is strongly constrained
by the SINDRUM II experiment Bertl  _et al._ (2006) which uses ${}^{197}Au$
nucleus. The relevant branching ratio estimated in Kitano  _et al._ (2002) is
given by

${\rm BR}[\mu\to e]=\frac{2G_{F}^{2}}{\omega_{\rm
capt}}\,(V^{(p)})^{2}\,\left(|g^{(p)}_{LV}|^{2}+|g^{(p)}_{RV}|^{2}\right)\,.$
(95)

Here, $V^{(p)}=0.0974\,m_{\mu}^{5/2}$ is an integral involving proton
distribution and ${\omega_{\rm capt}}=13.07\times 10^{6}\,{\rm s}^{-1}$ is
muon capture rate for ${}^{197}Au$ Kitano  _et al._ (2002). $g^{(p)}_{LV,RV}$
depend on the flavour-violating couplings and they are parametrized as

$g^{(p)}_{LV,RV}=2g^{(u)}_{LV,RV}+g^{(d)}_{LV,RV}\,.$ (96)

The expressions for $g^{(q)}_{LV,RV}$ ($q=u,d$) can be obtained using Eq. (86)
and Smolkovič  _et al._ (2019). For the present model, we find

$\displaystyle g^{(q)}_{LV}$ $\displaystyle\sim$
$\displaystyle\frac{\sqrt{2}}{G_{F}}\,\frac{1}{2}\left[C^{(eq)LL}_{2111}\,+\,C^{(eq)LR}_{2111}\right]\,$
$\displaystyle g^{(q)}_{RV}$ $\displaystyle\sim$
$\displaystyle\frac{\sqrt{2}}{G_{F}}\,\frac{1}{2}\left[C^{(eq)RR}_{2111}\,-\,C^{(eq)RL}_{2111}\right]\,.$
(97)

The branching ratios for $\mu$ to $e$ conversion, computed using Eqs. (95, 96)
and (VI.2), for the three benchmark solutions are given in Table 5. The
present experimental limit disfavours S1 in this case.

LFV observable |  Limit |  S1 |  S2 |  S3
---|---|---|---|---
${\rm BR}[\mu\to e]$ | $<7.0\times 10^{-13}$ | $1.0\times 10^{-8}$ | $1.4\times 10^{-16}$ | $1.4\times 10^{-24}$
${\rm BR}[\mu\to 3e]$ | $<1.0\times 10^{-12}$ | $2.4\times 10^{-11}$ | $5.0\times 10^{-19}$ | $1.6\times 10^{-27}$
${\rm BR}[\tau\to 3\mu]$ | $<2.1\times 10^{-8}$ | $2.3\times 10^{-11}$ | $4.2\times 10^{-19}$ | $1.7\times 10^{-27}$
${\rm BR}[\tau\to 3e]$ | $<2.7\times 10^{-8}$ | $9.4\times 10^{-12}$ | $2.7\times 10^{-19}$ | $6.0\times 10^{-28}$
${\rm BR}[\mu\to e\gamma]$ | $<4.2\times 10^{-13}$ | $7.0\times 10^{-9}$ | $4.9\times 10^{-17}$ | $6.8\times 10^{-25}$
${\rm BR}[\tau\to\mu\gamma]$ | $<4.4\times 10^{-8}$ | $2.1\times 10^{-11}$ | $1.6\times 10^{-19}$ | $2.2\times 10^{-27}$
${\rm BR}[\tau\to e\gamma]$ | $<3.3\times 10^{-8}$ | $1.3\times 10^{-12}$ | $9.3\times 10^{-21}$ | $1.3\times 10^{-28}$
Table 5: Branching ratios evaluated for various charged lepton flavour
violating processes for the three benchmark solutions listed in Table 2. The
corresponding experimental limits are extracted from Calibbi and Signorelli
(2018). The values excluded by the limits are highlighted in red.

The trilepton decay, $l_{i}\to 3l_{j}$, is mediated by the new gauge bosons at
the tree-level. The decay width for this process can be estimated following
Ramond (1999). For the present model and in the limit $m_{i}\gg m_{j}$, we
find

$\displaystyle\Gamma[l_{i}\to 3l_{j}]$ $\displaystyle=$
$\displaystyle\frac{4\,m_{i}^{5}}{1536}\,\sum_{P,P^{\prime}}\left|C^{(ee)PP^{\prime}}_{jijj}\right|^{2}\,,$
(98)

which, using Eq. (90), takes the following form

$\displaystyle\Gamma[l_{i}\to
3l_{j}]\,=\,\frac{g_{F}^{4}}{16}\frac{m_{i}^{5}}{1536}\,\sum_{\alpha,\beta=1}^{3}$
$\displaystyle\left[\left(\tilde{\lambda}^{\alpha}_{e}{}_{L}\right)_{ji}\left(\tilde{\lambda}^{\beta}_{e}{}_{L}\right)_{ji}+\left(\tilde{\lambda}^{\alpha}_{e}{}_{R}\right)_{ji}\left(\tilde{\lambda}^{\beta}_{e}{}_{R}\right)_{ji}\right]\,$
(99)
$\displaystyle\times\left[\left(\tilde{\lambda}^{\alpha}_{e}{}_{L}\right)_{jj}\left(\tilde{\lambda}^{\beta}_{e}{}_{L}\right)_{jj}+\left(\tilde{\lambda}^{\alpha}_{e}{}_{R}\right)_{jj}\left(\tilde{\lambda}^{\beta}_{e}{}_{R}\right)_{jj}\right]\,.$

The branching ratios for $\mu\to 3e$, $\tau\to 3e$ and $\tau\to 3\mu$
evaluated using the above expression are given in Table 5 for three solutions
along with their corresponding experimental limits.

To estimate the branching ratios for $l_{i}\to\l_{j}\gamma$, we follow Lavoura
(2003) and compute the decay width in the approximation $M_{Z_{1}}\gg
m_{i},m_{j}$. The result can be parametrized as

$\Gamma[l_{i}\to\l_{j}\gamma]\,\simeq\,\frac{\alpha{g_{F}}^{4}}{64}\left(1-\frac{m_{j}^{2}}{m_{i}^{2}}\right)^{3}\,{m_{i}^{5}}\,\left(\left|\sigma_{L}\right|^{2}\,+\,\left|\sigma_{R}\right|^{2}\right)\,,$
(100)

where

$\sigma_{L}\,=\,\sum_{\alpha=1}^{3}\sum_{k=1}^{3}\left[Y_{1}\,\frac{m_{j}}{m_{i}}\left(\tilde{\lambda}^{\alpha}_{L}\right)_{jk}\left(\tilde{\lambda}^{\alpha}_{L}\right)_{ki}\,+\,Y_{2}\,\left(\tilde{\lambda}^{\alpha}_{R}\right)_{jk}\left(\tilde{\lambda}^{\alpha}_{R}\right)_{ki}\,-4Y_{3}\,\frac{m_{k}}{m_{i}}\,\left(\tilde{\lambda}^{\alpha}_{R}\right)_{jk}\left(\tilde{\lambda}^{\alpha}_{L}\right)_{ki}\right]\,.$
(101)

Similarly, $\sigma_{R}$ can be obtained by replacing $L\leftrightarrow R$ in
the above expression. $Y_{1},Y_{2}$ and $Y_{3}$ are loop functions and they
are given by

$Y_{1}=Y_{2}=2a+6c+3d\,~{}~{}\,~{}~{}Y_{3}=a+2c\,,$ (102)

and the explicit expressions of $a,c$, and $d$ are given in Lavoura (2003).
Using Eq. (100), we estimate the branching ratios for $\mu\to e\gamma$,
$\tau\to\mu\gamma$ and $\tau\to e\gamma$ for three benchmark solutions and
list them in Table 5.

Comparing the estimated magnitudes of various charged lepton flavour violating
observables given in Table 5 with the corresponding experimental limits, we
find that $M_{Z_{1}}\leq 10^{4}$ GeV are excluded. However, this seems to be a
much weaker constraint compared to the one arising from the quark flavour-
violating process. Altogether the strongest limit on the scale of $SU(3)_{F}$
breaking comes from $K$-$\overline{K}$ mixing which disfavours $M_{Z_{1}}\leq
10^{3}$ TeV.

The flavour constraints on the new physics in this class of models are
dominant and they supersede the other limits put by direct searches or
precision electroweak observables as shown by us in our previous work Mohanta
and Patel (2022). For example, the strongest limit from the direct searches at
the LHC implies $M_{Z_{1}}>7.20$ TeV Sirunyan  _et al._ (2018). Similar
constraints on the vectorlike quarks, $m_{B}>1.57$ TeV Aaboud  _et al._
(2018a); Sirunyan  _et al._ (2020) and $m_{T}>1.31$ TeV Sirunyan  _et al._
(2019); Aaboud  _et al._ (2018b), are even more weaker. The $Z_{1,2}$ bosons
can mix with the SM $Z$ boson through VEVs of $H_{ui}$ and $H_{di}$ in the
present model which in turn contributes to the electroweak observables. The
most stringent limits in this case also imply $M_{Z_{1}}\geq 4.5$ TeV Mohanta
and Patel (2022) making all these constraints irrelevant in comparison to the
ones originating from the quark and lepton flavour violations.

##  VII Conclusion

We have discussed a mechanism for generating loop-induced masses for the first
and second generations of quarks and charged leptons using a gauged horizontal
$SU(3)_{F}$ symmetry. The field content of the theory ensures that the Yukawa
sector has an accidental global symmetry leading to vanishing masses for
lighter generations of fermions. This symmetry is broken by the $SU(3)_{F}$
gauge interactions which then radiatively induces the masses for the otherwise
massless fermions. We find that the radiative corrections typically generate
masses for both the second and first-generation fermions at 1-loop. The
hierarchy between the two, therefore, requires a separate explanation. We show
that this is possible if $SU(3)_{F}$ is broken in two steps with an
intermediate $SU(2)$ symmetry. This leads to a little hierarchy among the
gauge bosons of the local flavour group which is then transferred to the
fermion sector through quantum corrections. We construct an explicit model
based on this mechanism and show how the hierarchical quark and lepton masses
can be viably reproduced.

A similar setup based on flavour non-universal $U(1)\times U(1)$ symmetry was
proposed by us earlier in Mohanta and Patel (2022). The framework presented in
this paper replaces the pair of non-abelian symmetries with $SU(3)_{F}$
leading to two important improvements. The gauge boson mass hierarchy, which
was an ad-hoc assumption in the case of $U(1)\times U(1)$, now emerges from a
sequential breaking of single gauge group $SU(3)_{F}$. Secondly, the non-
abelian single flavour group leads to a more predictive framework in terms of
the number of Yukawa couplings in the theory. The number of free yukawa
couplings after removing the unphysical phases reduces from 20 in Mohanta and
Patel (2022) to 12 in the present model. This implies correlations among the
masses of various quarks and charged leptons. An example of this is seen in
the specific model which favours the value of strange quark mass $2.4\sigma$
smaller than the present experimental value.

Phenomenologically, the $SU(3)_{F}$ breaking scale is constrained from below
entirely from the flavour violation. The new gauge bosons have ${\cal O}(1)$
flavour-changing couplings with fermions leading to large rates for flavour-
violating processes. This feature seems to be inherently present in the
frameworks of radiative mass generation mechanisms. We estimate various quark
and lepton flavour-violating observables and find that the lightest gauge
boson of $SU(3)_{F}$ is required to be heavier than $10^{3}$ TeV as implied by
the present limits. This makes it impossible to verify such a framework in the
direct search experiments. Nevertheless, the specific model can still be
probed indirectly through precision measurements of fermion masses and mixing
parameters and flavour-violating observables.

## Acknowledgements

This work is partially supported under the MATRICS project (MTR/2021/000049)
by the Science & Engineering Research Board (SERB), Department of Science and
Technology (DST), Government of India. KMP acknowledges support from the ICTP
through the Associates Programme (2023-2028) where part of this work was
completed and preliminary results were presented.

##  Appendix A $SU(3)_{F}$ generators

An explicit form of the $SU(3)_{F}$ generators $\lambda^{a}$ (with
$a=1,...,8$) that we use in the present work is

$\displaystyle\lambda^{1}$ $\displaystyle=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\
0&1&0\end{array}\right)\,,\,\lambda^{2}\,=\,\left(\begin{array}[]{ccc}0&0&0\\\
0&0&-i\\\ 0&i&0\end{array}\right)\,,\,$
$\displaystyle\lambda^{3}\,=\,\left(\begin{array}[]{ccc}0&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right)\,,\,$
$\displaystyle~{}~{}\lambda^{4}\,=\,\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&0\end{array}\right)\,,$ $\displaystyle\lambda^{5}\,$
$\displaystyle=\,\left(\begin{array}[]{ccc}0&i&0\\\ -i&0&0\\\
0&0&0\end{array}\right)\,,\,\lambda^{6}\,=\,\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 1&0&0\end{array}\right)\,,\,$
$\displaystyle\lambda^{7}\,=\,\left(\begin{array}[]{ccc}0&0&i\\\ 0&0&0\\\
-i&0&0\end{array}\right)\,,\,$
$\displaystyle~{}\lambda^{8}\,=\,\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}-2&0&0\\\
0&1&0\\\ 0&0&1\end{array}\right)\,.$

These are written in the basis such that the first three generators correspond
to the gauge bosons which do not couple to the first generation in the
canonical basis. An $SU(2)$ subgroup corresponding to these three generators
of the full flavour symmetry group remains unbroken by the VEV of $\eta_{1}$
in Eq. (33).

##  Appendix B Scalar Potential and VEVs

In this Appendix, we demostrate the conditions which lead to the VEV
configurations of $\eta_{1,2}$ fields considered in Eq. (33). Since the
flavour symmetry breaking scale is required to be much larger than the
electroweak scale, we neglect the small contribution that may arise from the
VEVs of $H_{u,d}^{i}$ in the $SU(3)_{F}$ breaking. The most general and
renormalizable potential involving $\eta_{1,2}$ can be written as

$\displaystyle V(\eta_{1},\eta_{2})$ $\displaystyle=$ $\displaystyle
m_{11}^{2}\,\eta_{1}^{\dagger}\eta_{1}+m_{22}^{2}\,\eta_{2}^{\dagger}\eta_{2}-\left\\{m_{12}^{2}\,\eta_{1}^{\dagger}\eta_{2}+{\rm
h.c.}\right\\}$ (103) $\displaystyle+$
$\displaystyle\frac{\xi_{1}}{2}\,(\eta_{1}^{\dagger}\eta_{1})^{2}+\frac{\xi_{2}}{2}\,(\eta_{2}^{\dagger}\eta_{2})^{2}+\xi_{3}\,(\eta_{1}^{\dagger}\eta_{1})(\eta_{2}^{\dagger}\eta_{2})+\xi_{4}\,(\eta_{1}^{\dagger}\eta_{2})(\eta_{2}^{\dagger}\eta_{1})$
$\displaystyle+$
$\displaystyle\left\\{\frac{\xi_{5}}{2}(\eta_{1}^{\dagger}\eta_{2})^{2}+\xi_{6}\,(\eta_{1}^{\dagger}\eta_{1})(\eta_{1}^{\dagger}\eta_{2})+\xi_{7}\,(\eta_{2}^{\dagger}\eta_{2})(\eta_{1}^{\dagger}\eta_{2})+{\rm
h.c.}\right\\}\,.$

Here, all the parameters except $\xi_{5,6,7}$ and $m_{12}^{2}$ are real.

For the VEV configuration of $\eta_{1,2}$ given in Eq. (33), the minimization
of the potential implies

$\displaystyle v\,(m_{11}^{2}+v^{2}\xi_{1}+\epsilon^{2}v^{2}\xi_{3})$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle\epsilon
v\,(m_{22}^{2}+\epsilon^{2}v^{2}\xi_{2}+v^{2}\xi_{3})$ $\displaystyle=$
$\displaystyle 0\,.$ (104)

The non-trivial solution of the above equations corresponds to

$v^{2}=\frac{-m_{11}^{2}\xi_{2}+m_{22}^{2}\xi_{3}}{\xi_{1}\xi_{2}-\xi_{3}^{2}}\,,~{}~{}~{}(\epsilon
v)^{2}=\frac{-m_{22}^{2}\xi_{1}+m_{11}^{2}\xi_{3}}{\xi_{1}\xi_{2}-\xi_{3}^{2}}\,.$
(105)

The VEVs are obtained in terms of real parameters $m_{11}^{2}$, $m_{22}^{2}$
and $\xi_{1,2,3}$. The latter are also constrained by the stability of
potential:

$\xi_{1,2}\geq 0\,,~{}~{}\xi_{3}\geq-\sqrt{\xi_{1}\xi_{2}}\,.$ (106)

For $0>\xi_{3}\geq-\sqrt{\xi_{1}\xi_{2}}$, one finds
$\xi_{1}\xi_{2}-\xi_{3}^{2}\geq 0$. Further assuming
$|m_{22}^{2}|\ll|m_{11}^{2}|$, $\xi_{1}\ll\xi_{2}$ and $m_{11}^{2}<0$, it can
be seen that the VEVs in Eq. (105) are real. Their ratio is then determined as

$\epsilon^{2}\approx-\frac{\xi_{3}}{\xi_{2}}\leq\sqrt{\frac{\xi_{1}}{\xi_{2}}}\ll
1\,.$ (107)

Moreover, for $\xi_{3}\approx-\sqrt{\xi_{1}\xi_{2}}$, the VEVs obtained Eq.
(105) turn out to be the global minima of the potential among the available
solutions offered by Eq. (B). In summary, one can obtain the desired VEV
configurations for $\eta_{1,2}$ consistent with stability constraints for a
specific choice of parameters.

As mentioned in the beginning, we have neglected contribution to the
$SU(3)_{F}$ breaking from the scalars charged also under the electroweak
symmetry since the viable generation of fermion mass hierarchy along with the
flavour constraints require at least four orders of magnitude separation
between the two scales. This hierarchy among the scales, however, requires a
fine-tuning. Since the terms like
$\eta_{a}^{\dagger}\eta_{a}H_{u,d}^{\dagger}H_{u,d}$ would induce large bare
mass terms for $H_{u,d}$ when $SU(3)_{F}$ gets broken, the VEVs of $H_{u,d}$
would naturally tend to stay close to the flavour symmetry breaking scale.
Also, as the terms like $\eta_{a}^{\dagger}\eta_{a}H_{u,d}^{\dagger}H_{u,d}$
cannot be forbidden by any gauge or global symmetries within this non-
supersymmetric framework, the seperation between the two scales is not stable
under the quantum corrections and technically unnatural. This is similar to
the usual gauge hierarchy problem and requires fine-tuning.

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