Gauged SU(3)F and loop induced quark and lepton masses

Abstract

A bstract
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 ≈ 16 MeV at M Z , which is ~2 . 4 σ 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.

Full text

JHEP10(2023)128
Published for SISSA by Springer
Received:August 18, 2023
Accepted:October 9, 2023
Published:October 20, 2023
Gauged SU(3)Fand loop induced quark and lepton
masses
Gurucharan Mohantaa,b and Ketan M. Patela
aTheoretical Physics Division, Physical Research Laboratory,
Navarangpura, Ahmedabad-380009, India
bIndian Institute of Technology Gandhinagar,
Palaj-382055, India
E-mail: gurucharan@prl.res.in,kmpatel@prl.res.in
Abstract: We investigate a local SU(3)Fflavour 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, ms≈16
MeV at MZ, which is ∼2.4σaway 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 103TeV or higher.
Keywords: Flavour Symmetries, Theories of Flavour, New Gauge Interactions, Vector-
Like Fermions
ArXiv ePrint: 2308.05642
Open Access,c
The Authors.
Article funded by SCOAP3.https://doi.org/10.1007/JHEP10(2023)128

JHEP10(2023)128
Contents
1 Introduction 1
2SU(3)Fand fermion mass generation 3
3 Gauge-boson mass hierarchy 7
4 Model implementation 8
4.1 SU(3)Fbreaking and gauge boson mass ordering 8
4.2 Charged fermion masses 11
4.3 Neutrino masses 12
5 Numerical solutions 12
6 Flavour violation 15
6.1 Quark sector 16
6.2 Lepton sector 17
7 Conclusion 19
ASU(3)Fgenerators 21
B Scalar potential and VEVs 21
1 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 [1–
6] 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)]5symmetry 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.
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JHEP10(2023)128
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 loops1
(a) spin-0 (see [7–23] for example) and (b) spin-1 [24–29]. In the case of the latter, SM
gauge symmetry needs to be extended to include a gauged flavour symmetry group GF
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 quanti-
ties 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 GFbeing an abelian group and
analysed it in detail in [29]. 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 GF= U(1)1×U(1)2where
the gauge bosons associated with U(1)1and U(1)2generate 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×U(1)2
is replaced by a simple group. The most advantageous feature of non-abelian GFin 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 GF= SU(3)Fand 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 [24,30] 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 GF= SO(3)L×SO(3)R, has been investigated relatively
recently in [27] 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
1It 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.
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JHEP10(2023)128
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)Fand the generation of radiative masses. Breaking of
the horizontal symmetry leading to desired gauge boson mass spectrum is presented in
section 3. Detailed implementation of the general scheme in the SM is discussed in section 4.
Viability of the model is established through example numerical solutions in section 5. We
also discuss some phenomenological aspects of the scheme in section 6before concluding
in section 7.
2SU(3)Fand fermion mass generation
Denoting the three generations (i= 1,2,3) of chiral fermions by f′
Li and f′
Ri and a pair
of vectorlike fermions by F′
L,R, the tree-level mass term in the basis f′
Lα ≡(f′
Li, F ′
L)and
f′
Rα ≡(f′
Ri, F ′
R)is arranged as
−Lm=f′
Lα M0αβ f′
Rβ + h.c. , (2.1)
with
M0=

03×3(µ)3×1
(µ′)1×3mF
.(2.2)
Here, f′
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 fdependency in M0,µand µ′.
We also consider that f′
Li and f′
Ri transform as fundamental representations of a hor-
izontal 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 µand µ′matrices in the
above mass Lagrangian break the SU(3)F. The vanishing 3×3sub-matrix can be obtained
utilizing the chiral nature of f′
Li and f′
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′
Lor F′
R, either
µor µ′is also protected by the chiral symmetry.
The matrix M0leads to two massless states which can be identified with the first and
second-generation fermions. For mF≫µi, µ′
i, the effective 3×3mass matrix takes the
form
M0≃ − 1
mF
µ µ′,(2.3)
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 quan-
tum corrections within this framework. In order to quantify these corrections, consider the
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JHEP10(2023)128
SU(3)Fgauge interactions of fermions with the gauge bosons Aa
µgiven by
−Lgauge =gF f′
LiγµAa
µλa
2ij
f′
Lj +f′
RiγµAa
µλa
2ij
f′
Rj !,(2.4)
where a= 1,...,8and λaare 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 Afor the
clarity. The above can be generalized to include the vectorlike fermions as
−Lgauge =gF
2f′
LαγµAa
µ(Λa)αβ f′
Lβ +f′
RαγµAa
µ(Λa)αβ f′
Rβ,(2.5)
where Λaare 4×4matrices given by
Λa=

λa0
0 0 
.(2.6)
The physical basis of fermions, denoted by fL,R, can be obtained from the canonical
basis using the unitary transformations
f′
L,R =UL,R fL,R ,(2.7)
such that
U†
LM0UR=D= Diag.(0,0, m3, m4).(2.8)
Similarly, the physical gauge bosons Ba
µcan be obtained from Aa
µusing an 8×8real
orthogonal matrix Ras
Aaµ =Rab Bbµ .(2.9)
The matrix Rcan be obtained explicitly by diagonalizing the gauge-boson mass matrix
which is real and symmetric.
Substituting eqs. (2.7), (2.9) in eq. (2.5), the gauge interactions in the physical basis
of fermions and gauge bosons are obtained as
−Lgauge =gF
2fLαγµU†
LΛaULαβ fLβ +fRαγµUR†ΛaURαβ fRβ Rab Bb
µ.(2.10)
Since Λado not commute with each other for all a, the matrices U†
L,RΛaUL,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 [29], 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)Fgauge interactions at 1-loop, can
be written as
M=M0+δM,(2.11)
where
δM=ULΣ(0) U†
R.(2.12)
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JHEP10(2023)128
fLα
fRβ fRσ mσfLσ
Bb
µ
Figure 1. Diagram representing gauge boson induced fermion self-energy correction.
Using eq. (2.10), the 1-loop contribution (see figure 1) can be computed as
−i(Σ(p))αβ =Zd4k
(2π)4−igF
2Rab(U†
LΛaUL)ασγµimσ
(k+p)2−m2
σ+iϵ
−igF
2Rcb(U†
RΛcUR)σβ γν∆µν (k),(2.13)
with
∆µν (k) = −i
k2−Mb2+iϵ ηµν −(1 −ζ)kµkν
k2−ζM2
b!.(2.14)
Here Mbis the mass of the gauge boson Bb
µ. In the Feynmann-’t Hooft gauge, the evaluation
of the above integral results in
(Σf(0))αβ =gF2
16π2Rab(U†
LΛaUL)ασRcb(U†
RΛcUR)σβ mσB0[M2
b, m2
σ],(2.15)
with
B0[M2, m2] = (2πµ)ϵ
iπ2Zddk1
k2−m2+iϵ
1
k2−M2+iϵ
= ∆ϵ−M2ln M2−m2ln m2
M2−m2,(2.16)
and
∆ϵ=2
ϵ+ 1 −γ+ ln 4π . (2.17)
It is straightforward to verify from eq. (2.15) that the mσindependent contribution in
δMfrom B0[M2
b, m2
σ]vanishes identically. The divergent part of δMis given by
δMdiv =ULΣ(0)div U†
R,(2.18)
where Σ(0)div captures the terms dependent only on ∆ϵfrom eq. (2.15). Explicitly,
(δMdiv)ρκ =gF2
16π2ULραRab (U†
LΛaUL)ασRcb(U†
RΛcUR)σβ mσ∆ϵU†
Rβκ ,
=gF2∆ϵ
16π2(RRT)ac ULρα (U†
LΛaUL)ασ Dσσ (U†
RΛcUR)σβ U†
Rβκ .(2.19)
Using eq. (2.8), orthogonality of Rand unitarity of UL,R, the above expression can be
simplified to
δMdiv =gF2∆ϵ
16π2ΛaM0Λa= 0 ,(2.20)
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JHEP10(2023)128
where we use the form of M0and Λagiven in eqs. (2.2), (2.6) to get the last equality. The
vanishing of δMdiv is in accordance with the renormalizability [3,31] of the theory as there
are no corresponding counterterms to renormalise.
Further simplification of the finite contribution can be achieved for mF≫µi, µ′
i. In
this case, UL,R at the leading order in µi/mFand µ′
i/mFcan be approximated as [32]
UL,R =

UL,R −ρL,R
ρ†
L,RUL,R 1
+O(ρ2
L,R),(2.21)
where ρL=−m−1
Fµand ρR†=−m−1
Fµ′.UL,R are 3×3matrices that diagonalize M0given
in eq. (2.3) such that
U†
LM0UR= Diag.(0,0, m3).(2.22)
Using eq. (2.3), the above can also be written as
(UL)i3(U∗
R)j3m3=M0
ij =−1
mF
µiµ′
j.(2.23)
Expanding the finite part of eq. (2.15) and using eqs. (2.6), (2.21), (2.23), the 1-loop
correction to the effective 3×3mass matrix can be simplified to
(δM )ij ≃gF2
16π2RabRcb (λaM0λc)ij ∆b0[M2
b],(2.24)
where
∆b0[M2
b] = −M2
bln M2
b−m2
3ln m2
3
M2
b−m2
3
+M2
bln M2
b−m2
4ln m2
4
M2
b−m2
4
.(2.25)
We also find δM4α=δMα4= 0 by utilising eq. (2.6). The non-observations of new gauge-
bosons or vectorlike states would imply m3≪Mb, m4. In this limit, the loop function in
eq. (2.24) can be approximated to
∆b0[M2
b]≃m2
4
M2
b−m2
4
ln M2
b
m2
4!.(2.26)
The following noteworthy features of the loop-corrected fermion mass matrix can be
deduced from eq. (2.24) along with eq. (2.26) and the explicit expressions of the λagiven
in Appendinx A.
•The loop-induced masses are suppressed by the loop factor g2
F/(16π2)if m4> Mb.
An additional suppression by factor m2
4/M2
barises in case m4< Mb.
•For a generic choice of gauge boson masses and the orthogonal matrix R, eq. (2.24)
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. (2.24) 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.
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JHEP10(2023)128
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.
3 Gauge-boson mass hierarchy
Consider a two-step breaking of SU(3)Fsymmetry such that
SU(3)F⟨η1⟩
−−→ SU(2)F⟨η2⟩
−−→ nothing ,(3.1)
with ⟨η1⟩≫⟨η2⟩. With this arrangement, three of the gauge bosons corresponding to the
generators of SU(2)Fare 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 inter-
mediate SU(2)Fwith the generators λαwith α= 1,2,3. Denoting the remaining indices
with m= 4, . . . 8, the gauge boson mass term in the canonical basis Aµ
a= (Aµ
α, Aµ
m)can be
written as
−LM
GB =1
2M2
ab Aµ
aAbµ ,(3.2)
such that
M2=

M2
(33) M2
(35)
(M2
(35))TM2
(55) 
.(3.3)
Here, M2
(AB)are A×Bdimensional matrices sub-blocks of the gauge boson mass matrix.
The two step breaking of the horizontal symmetry implies M2
(33), M 2
(35) ≪M2
(55). The
gauge-boson mass matrix in this specific form can be diagonalized by see-saw like diago-
nalization procedure and the orthogonal matrix at the leading order can be written as
R=

R3−ρR5
ρTR3R5
+O(ρ2),(3.4)
with ρ=−M2
(35)(M2
(55))−1. Here, R3and R5are real orthogonal matrices of dimensions
3×3and 5×5, respectively.
Substituting eq. (3.4) in eq. (2.24) and considering the leading order terms in the
seesaw expansion parameter ρ, we get
(δM )ij =gF2
16π2h(R3)αβ(R3)γβ λαM0λγij ∆b0[M2
β]
+ (R3)αβ(ρTR3)mβ λαM0λm+λmM0λαij ∆b0[M2
β]
+ (R5)mn(R5)pn λmM0λpij ∆b0[M2
n]
−(R5)mn(ρR5)αn λmM0λα+λαM0λmij ∆b0[M2
n] + O(ρ2)i.(3.5)
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JHEP10(2023)128
Recall that α, β, . . . = 1,2,3while m, n, . . . = 4,...,8. The first term gives the dominant
contribution to δM as M2
α< M2
m. Since λαhas 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 Mα< m4, 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. (3.5) and it is suppressed either by M2
α/M2
mor m2
4/M2
mwith 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
M2
α≲m2
4≲M2
m.(3.6)
This implies that the scale of SU(3)Fbreaking 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 m4and Ma.
4 Model implementation
Based on the general conditions to generate fermion mass hierarchy through quantum cor-
rections, 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 genera-
tions of SM fermions transforming as fundamental representations of the horizontal gauged
symmetry SU(3)F. We additionally consider NR, 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)Ftriplets scalars, ηs, 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 2, 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)Ftransformation properties are summarised in table 1.
4.1 SU(3)Fbreaking and gauge boson mass ordering
The non-observations of SU(3)Fgauge 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 η1,2and the contributions
from the electroweak doublets are expected to be sub-dominant. With this reasoning, we
consider the breaking of SU(3)Fdriven by η1,2only.
As an example, we consider the following VEV configuration:
⟨η1⟩= (vF,0,0)T,⟨η2⟩= (0,0, ϵvF)T,(4.1)
with ϵ < 1. One of these can always be chosen in the given form using the SU(3)Frotation
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
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JHEP10(2023)128
Fields (SU(3)c×SU(2)L×U(1)Y) SU(3)F
QL(3,2,1
6)3
uR(3,1,2
3)3
dR(3,1,−1
3)3
LL(1,2,−1
2)3
eR(1,1,−1) 3
NR(1,1,0) 3
Hu(1,2,−1
2)3
Hd(1,2,1
2)3
η1,η2(1,1,0) 3
TL, TR(3,1,2
3)1
BL, BR(3,1,−1
3)1
EL, ER(1,1,−1) 1
Table 1. The SM and GFcharges of various fermions and scalars of the model.
break fully the SU(3)Fsymmetry 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 η1,2in Appendinx Band show that
the VEV configuration given in eq. (4.1) can be obtained by a suitable choice of parameters
in the scalar potential.
The kinetic terms of η1,2after the spontaneous breaking of SU(3)Fleads to the follow-
ing gauge boson mass matrix defined in eq. (3.2):
M2
αβ =g2
F
2X
s=1,2⟨ηs⟩†λα†λβ⟨ηs⟩.(4.2)
In the notation of eq. (3.3), the above mass matrix can be written as
M2
(33) =g2
Fv2
F
2Diag.ϵ2, ϵ2, ϵ2,
M2
(55) =g2
Fv2
F
2Diag.1,1,1 + ϵ2,1 + ϵ2,1
3(4 + ϵ2),
M2
(35) =g2
Fv2
F
2




0 0 0 0 0
0 0 0 0 0
0 0 0 0 −ϵ2
√3





(4.3)
The structure of the gauge boson mass matrix in this case is extremely simple and there
exists mixing between only Aµ
3and Aµ
8states which correspond to diagonal generators.
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JHEP10(2023)128
The matrix M2can be diagonalized by Ras parametrized by eq. (3.4) with the fol-
lowing explicit forms of its sub-matrices:
R3=I3×3, R5=I5×5, ρ =




0 0 0 0 0
0 0 0 0 0
0 0 0 0 −√3
4ϵ2





.(4.4)
The diagonal gauge boson mass matrix is then obtained as
D2=g2
Fv2
F
2Diag.ϵ2, ϵ2, ϵ2,1,1,1 + ϵ2,1 + ϵ2,4
3+1
3ϵ2+O(ϵ4).(4.5)
In this setup, one obtains the hierarchical gauge boson mass spectrum M2
1,2,3≪M2
4,...,8as
required to generate the mass gaps between the first and second-generation fermions.
Substituting the above in the (δM )ij , we find
16π2
g2
F
(δM )ij =
3
X
α=1 λαM0λαij ∆b0[M2
α] +
8
X
m=4 λmM0λmij ∆b0[M2
m]
+√3
4ϵ2λ3M0λ8+λ8M0λ3ij ∆b0[M2
3]−∆b0[M2
8]+O(ϵ4).(4.6)
An approximate degeneracy of some of the gauge bosons allows further simplification.
Setting
M2
1≃M2
2≃M2
3≡M2
Z1, M2
4≃. . . ≃M2
7≃3
4M2
8≡M2
Z2,(4.7)
and using ϵ2=M2
Z1/M2
Z2, we find
16π2
g2
F
δM ≃




0 0 0
0M0
22 + 2M0
33 −M0
23
0−M0
32 2M0
22 +M0
33





∆b0[M2
Z1]
+




2(M0
22 +M0
33) 0 0
0 2M0
11 0
0 0 2M0
11





∆b0[M2
Z2]
+1
3




4M0
11 −2M0
12 −2M0
13
−2M0
21 M0
22 M0
23
−2M0
31 M0
32 M0
33





∆b04
3M2
Z2
+M2
Z1
2M2
Z2





0−M0
12 M0
13
−M0
21 M0
22 0
M0
31 0−M0
33




∆b0[M2
Z1]−∆b04
3M2
Z2.(4.8)
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JHEP10(2023)128
4.2 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
−LY=yuQ′LiHi
uT′
R+ydQ′LiHi
dB′
R+yeL′
iLHi
dE′
R
+y′(s)
uT′Lη†
si u′i
R+y′(s)
dB′Lη†
si d′i
R+y′(s)
eE′Lη†
si e′i
R
+mTT′LT′
R+mBB′LB′
R+mEE′LE′
R+ h.c.(4.9)
where i= 1,2,3is an SU(3)Findex and s= 1,2denotes multiplicity of ηfields. The
primed notation is used for the fields in a flavour basis.
It is straightforward to see that after the electroweak and SU(3)Fbreaking, the Yukawa
interactions in eq. (4.9) lead to the tree-level mass matrices in the desired form of eq. (2.2)
with
µf=yfvf
1yfvf
2yfvf
3Tand µ′
f=y′(1)
fvF0y′(2)
fϵvF,(4.10)
where f=u, d, e which accounts for three type of charged fermions. Also, vu
i=⟨Hi
u⟩,
vd
i=ve
i=⟨Hi
d⟩. The VEVs of η1,2are taken from eq. (4.1). The tree-level effective mass
matrix after integrating out the heavy vectorlike states is given by
M0
u,d,e ≡ − 1
mT,B ,E
µu,d,e µ′
u,d,e .(4.11)
The matrices M0
fhas vanishing second column.
At 1-loop, the charged fermion mass matrices are given by
Mf=M0
f+δMf,(4.12)
and δMfcan be obtained using the general expression, eq. (4.8). Using the form of the
tree-level mass matrices, we get
δMf≃g2
FNf
16π2









0 0 0
0 2(M0
f)33 −(M0
f)23
0 0 (M0
f)33





∆b0[M2
Z1]
+2 




(M0
f)33 0 0
0 (M0
f)11 0
0 0 (M0
f)11





∆b0[M2
Z2]
+1
3




4(M0
f)11 0−2(M0
f)13
−2(M0
f)21 0 (M0
f)23
−2(M0
f)31 0 (M0
f)33





∆b04
3M2
Z2
+1
2
M2
Z1
M2
Z2





0 0 (M0
f)13
−(M0
f)21 0 0
(M0
f)31 0−(M0
f)33




∆b0[M2
Z1]−∆b04
3M2
Z2




,(4.13)
where Nf= 3 for f=u, d and Nf= 1 for f=e. As we demonstrate in the next section, the
above Mfcan reproduce the observed charged fermion mass spectrum and quark mixing.
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JHEP10(2023)128
4.3 Neutrino masses
As noted earlier, the anomaly-free model requires the SM singlet fermions NR. With
the present field content and the symmetry of the model, it is evident that there is no
Dirac Yukawa coupling between LLand NRand no Majorana mass term for NRat the
renormalizable level. This prevents NRfrom contributing to the light neutrino masses
through the conventional type I seesaw mechanism.
The symmetry of the model, however, allows the following Weinberg operators
c1
ΛLc
L
iH∗
uiLj
LH†
uj+c2
Λλa
2k
iλa
2l
jLc
L
iH∗
ukLj
LH†
ul+c3
ΛLc
L
iLj
LH†
uiH∗
uj,
(4.14)
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 νRk, which are singlets under the full gauge symmetry and, therefore, do not give
rise to anomalies. They can couple to SU(3)Ftriplet Li
Land anti-triplet H†
ui 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. (4.14) gets generated when
νRk are integrated out from the spectrum. Similarly, the second and third operators can
be induced by heavy SU(3)Fadjoint 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 underly-
ing 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.
5 Numerical solutions
To establish the validity of the model and to understand the pattern of its various param-
eters, 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. (4.9), it can be
seen that the parameters yf,y′(1)
fand mT,B ,E can be made real. Moreover, we assume
that all the VEVs are real. This leaves only three complex parameters, namely y′(2)
u,d,e in
the model. Through eq. (4.10), this implies real (µf)i(for i= 1,2,3), real (µ′
f)1and a
complex (µ′
f)3. Moreover,
µe=ye
yd
µd≡r µd,(5.1)
where ris a real parameter. Altogether, there are 21 real parameters (real µui,µdi ,r,
µ′
u1,µ′
d1,µ′
e1,mT,mB,mE,MZ1,MZ2and complex µ′
u3,µ′
d3,µ′
e3) in the model leading
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JHEP10(2023)128
Parameters Solution 1 (S1) Solution 2 (S2) Solution 3 (S3)
MZ1104GeV 106GeV 108GeV
ϵ0.0131 0.0083 0.0108
ϵT0.3576 0.3455 0.4357
ϵB0.9838 0.9999 0.9956
ϵE0.8831 0.3881 0.7358
ϵu10.4013 −0.3903 −0.3745
ϵu20.7211 0.6237 −0.8653
ϵu3−0.8575 0.6714 −0.8504
ϵ′
u10.2970 0.3425 −0.3393
ϵ′
u30.0115 −i0.32 ×10−40.0142 + i0.08 ×10−40.0140 −i0.72 ×10−4
ϵd1−0.1160 0.2935 −0.2423
ϵd2−0.2288 −0.4345 −0.5288
ϵd30.2494 −0.5095 −0.5668
ϵ′
d10.0460 0.0227 −0.0205
ϵ′
d30.0030 + i0.95 ×10−30.0017 −i0.33 ×10−30.0011 + i0.41 ×10−3
r3.7895 −0.5749 −0.4040
ϵ′
e1−0.0023 −0.0013 −0.0079
ϵ′
e3−0.0062 −i0.28 ×10−3−0.0093 −i0.86 ×10−30.0214 + i0.42 ×10−3
Table 2. Three benchmark solutions and the optimized values of the model parameters for different
MZ1which lead to viable charged fermion masses and quark mixing.
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.
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 M2
Z1≲m2
T,B ,E ≲M2
Z2(see eq. (3.6)), we define
mT=ϵTMZ2, mB=ϵBMZ2, mE=ϵEMZ2.(5.2)
Also recall that MZ1=ϵMZ2. Moreover, we also define
µ′
f1=ϵ′
f1MZ2, µ′
f3=ϵ′
f3MZ2,(5.3)
and
µui =ϵuiv, µdi =ϵdiv, (5.4)
where v= 174 GeV. In this way, various ϵand ϵ′can preferably take values less than unity.
Taking a particular value of MZ1, we obtain the values of the remaining dimensionless
parameters using the χ2optimization technique. Our methodology for the latter is de-
scribed in detail in [33]. Three benchmark solutions obtained in this way are displayed in
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JHEP10(2023)128
Observable Value S1 S2 S3
mu[MeV] 1.27 ±0.5 1.31 1.26 1.23
mc[GeV] 0.619 ±0.084 0.567 0.662 0.612
mt[GeV] 171.7±3.0 171.7 171.6 171.6
md[MeV] 2.90 ±1.24 3.71 4.15 3.55
ms[GeV] 0.055 ±0.016 0.016 0.018 0.016
mb[GeV] 2.89 ±0.09 2.89 2.88 2.89
me[MeV] 0.487 ±0.049 0.492 0.487 0.489
mµ[GeV] 0.1027 ±0.0103 0.1007 0.099 0.1004
mτ[GeV] 1.746 ±0.174 1.784 1.786 1.777
|Vus|0.22500 ±0.00067 0.21614 0.22242 0.22226
|Vcb|0.04182 ±0.00085 0.04110 0.04270 0.04207
|Vub|0.00369 ±0.00011 0.00363 0.00378 0.00371
JCP (3.08 ±0.15) ×10−53.14 ×10−53.02 ×10−53.06 ×10−5
Table 3. The fitted values of the charged fermion masses and quark mixing parameters at the
minimum of χ2for three benchmark solutions are displayed in table 2. The second column denotes
experimentally measured value of corresponding observable extrapolated at MZthat has been used
in the χ2function.
table 2for different MZ1. The minimized values of χ2are 6.97,6.90 and 6.46 for the solu-
tions 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 3along with the corresponding
experimental values for comparison.
Some of the noteworthy features of the model that can be derived from table 2are as
the following. We obtain almost similar values of the minimized χ2for different values of
MZ1. 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 ϵfi are of O(10−1)implying the fundamental Yukawa couplings yfof the same order
and no large hierarchy between the VEVs, vu,d
i. One also finds ϵ′
f3< ϵ′
f1for f=u, d as
expected from eq. (4.10). The fitted values of ϵ′
e1, however, require two orders of separation
between the magnitudes of y′(1)
eand y′(1)
u. 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 mtand mb,τ does not follow naturally and requires
ϵT≪ϵB,E .
It can be noticed from table 3that all the observables, except ms, are fitted within ±1σ
range of their reference values for all the solutions. The fitted value of msis ∼2.4σaway
from the experimental value. Despite having a sufficiently large number of parameters than
the observable, the inability to reproduce the central value of msindicates the existence of
non-trivial correlations between the observables that result from the predictive nature of
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JHEP10(2023)128
non-abelian flavour symmetry. Remarkably, a more precise measurement of strange quark
mass can falsify the model irrespective of the scale of SU(3)Fbreaking.
6 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 MZ1≪mT,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. (2.5) in the physical basis of fermions, one finds
−Lgauge =gF
2fLiγµ˜
λa
fLij fLj +fRiγµ˜
λa
fRij fRj RabBb
µ,(6.1)
where
˜
λa
fL,R =Uf†
L,R λaUfL,R ,(6.2)
and UfL,R are unitary matrices that diagonalize the 1-loop corrected Mf. Integrating out
the gauge bosons, we find the effective dimension-6 operators as
Leff =C(ff′)LL
ijkl fLi γµfLj f′Lk γµf′
Ll +C(ff′)RR
ijkl fRi γµfRj f′Rk γµf′
Rl
+C(ff′)LR
ijkl fLi γµfLj f′Rk γµf′
Rl +C(ff ′)RL
ijkl fRi γµfRj f′Lk γµf′
Ll ,(6.3)
where
C(ff′)P P ′
ijkl =g2
F
8M2
bRabRcb ˜
λa
fPij ˜
λc
f′P′kl ,(6.4)
and P, P ′=L, R and f, f ′=u, d, e. The coefficients of the effective operators can be
further simplified for the hierarchical gauge boson mass spectrum. Using eqs. (3.3), (3.4),
we find at leading order in ρ
8
g2
F
C(ff′)P P ′
ijkl ≃1
M2
α(R3)βα(R3)γα ˜
λβ
fPij ˜
λγ
f′P′kl
+ (R3)βα(ρTR3)mα ˜
λβ
fPij ˜
λm
f′P′kl +(ρTR3)mα(R3)βα ˜
λm
fPij ˜
λβ
f′P′kl
+1
M2
m(R5)nm(R5)pm ˜
λn
fPij ˜
λp
f′P′kl
−(ρR5)αm(R5)nm ˜
λα
fPij ˜
λn
f′P′kl −(R5)nm(ρR5)αm ˜
λn
fPij ˜
λα
f′P′kl.
(6.5)
– 15 –

JHEP10(2023)128
Further simplification can be achieved in the explicit model with the help of eqs. (4.4)
and (4.7). Substituting them in the above, we find
8M2
Z1
g2
F
C(ff′)P P ′
ijkl ≃
3
X
α=1 ˜
λα
fPij ˜
λα
f′P′kl +ϵ2
7
X
m=4 ˜
λm
fPij ˜
λm
f′P′kl
−√3
4ϵ2˜
λ3
fPij ˜
λ8
f′P′kl +˜
λ8
fPij ˜
λ3
f′P′kl
+3
4ϵ2˜
λ8
fPij ˜
λ8
f′P′kl +O(ϵ4).(6.6)
At the leading order, the flavour violation is governed by the coupling matrices ˜
λα
fL,R.
Since they do not commute with each other, all of them cannot take diagonal form for any
UL,R. Therefore, the most dominant flavour violations in the model are captured by the
coefficients
C(ff′)P P ′
ijkl ≃g2
F
8M2
Z1
3
X
α=1 ˜
λα
fPij ˜
λα
f′P′kl .(6.7)
The above can be used to estimate the most dominant contribution to various flavour-
violating processes in the quark and lepton sectors.
6.1 Quark sector
The strongest constraints on various C(f f ′)P P ′
ijkl primarily arise from meson-antimeson oscil-
lations such as K0−K0,Bd−B0
d,Bs−B0
sand D0−D0. To quantify these constraints, we
closely follow the procedure adopted in our previous analysis [29]. Comparing eq. (6.3) with
the effective Hamiltonian, Heff =P5
i=1 Ci
MQi+P3
i=1 ˜
Ci
M˜
Qi, which parametrizes ∆F= 2
transitions M−M[34], we find the effective Wilson coefficients at µ=MZ1as
C1
K=−C(dd)LL
1212 ,˜
C1
K=−C(dd)RR
1212 , C5
K=−4C(dd)LR
1212 ,(6.8)
C1
Bd=−C(dd)LL
1313 ,˜
C1
Bd=−C(dd)RR
1313 , C5
Bd=−4C(dd)LR
1313 ,(6.9)
C1
Bs=−C(dd)LL
2323 ,˜
C1
Bs=−C(dd)RR
2323 , C5
Bs=−4C(dd)LR
2323 ,(6.10)
C1
D=−C(uu)LL
1212 ,˜
C1
D=−C(uu)RR
1212 , C5
D=−4C(uu)LR
1212 .(6.11)
The remaining Ci
Mand ˜
Ci
Mare vanishing at this scale.
Using the renormalization group equations, we evolve all the coefficients from µ=MZ1
to the µ= 2 GeV for Kmeson [35], µ= 4.6GeV for Bmesons [36] and µ= 2.8GeV for
Dmeson [34]. The running gives rise to non-vanishing C4
Mwhile C2,3
Mand ˜
C2,3
Mremains
zero. The evolved Wilson coefficients are computed using eq. (6.7) for the three benchmark
solutions listed in table 2and are compared with the corresponding experimental limits
obtained by the UTFit collaboration [34]. The results are listed in table 4. It can be seen
that the strongest limits arise from K0−K0mixing which disfavours MZ1≤106GeV. The
same limit was also observed in our previous framework based on flavour non-universal
abelian symmetries.
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JHEP10(2023)128
Wilson coefficient Allowed range S1 S2 S3
ReC1
K[−9.6,9.6] ×10−13 6.0×10−11 −1.0×10−14 −5.2×10−19
Re ˜
C1
K[−9.6,9.6] ×10−13 −4.8×10−16 −1.7×10−19 4.0×10−24
ReC4
K[−3.6,3.6] ×10−15 3.2×10−10 −1.0×10−13 −2.5×10−18
ReC5
K[−1.0,1.0] ×10−14 2.8×10−10 −8.3×10−14 −2.0×10−18
ImC1
K[−9.6,9.6] ×10−13 4.4×10−11 4.3×10−15 −3.8×10−19
Im ˜
C1
K[−9.6,9.6] ×10−13 −3.4×10−15 −5.8×10−19 1.8×10−23
ImC4
K[−1.8,0.9] ×10−17 −1.6×10−10 −4.3×10−14 1.4×10−18
ImC5
K[−1.0,1.0] ×10−14 −1.4×10−10 −3.6×10−14 1.1×10−18
|C1
Bd|<2.3×10−11 9.2×10−11 1.2×10−14 8.0×10−19
|˜
C1
Bd|<2.3×10−11 2.2×10−12 2.8×10−16 1.6×10−20
|C4
Bd|<2.1×10−13 9.2×10−11 1.4×10−14 8.1×10−19
|C5
Bd|<6.0×10−13 1.6×10−10 2.4×10−14 1.3×10−18
|C1
Bs|<1.1×10−91.1×10−11 2.9×10−15 7.7×10−20
|˜
C1
Bs|<1.1×10−91.7×10−12 2.3×10−16 1.3×10−20
|C4
Bs|<1.6×10−11 1.3×10−11 2.9×10−15 1.1×10−19
|C5
Bs|<4.5×10−11 2.4×10−11 4.8×10−15 1.8×10−19
|C1
D|<7.2×10−13 5.5×10−11 7.5×10−15 6.5×10−19
|˜
C1
D|<7.2×10−13 4.4×10−16 5.6×10−20 5.4×10−24
|C4
D|<4.8×10−14 1.9×10−10 3.6×10−14 2.1×10−18
|C5
D|<4.8×10−13 2.2×10−10 4.1×10−14 2.2×10−18
Table 4. Numerical values of various Wilson coefficients (in GeV−2unit) of the operators leading
to meson-antimeson oscillations estimated for three example solutions. The experimentally allowed
ranges are taken from [34]. The values highlighted in red are excluded by the respective limits.
6.2 Lepton sector
As noted in the previous section, the dominant contribution to the flavour violation process
is governed by the first three gauge bosons Bα
µ. The exchange of Bα
µmediate lepton flavour
violating process like µ→econversion in nuclei, li→3ljand li→ljγ. The first two
processes arise at the tree level whereas the latter is at the one-loop level in the present
model.
The µ→econversion in the field of the nucleus is strongly constrained by the SIN-
DRUM II experiment [37] which uses 197Au nucleus. The relevant branching ratio estimated
in [38] is given by
BR[µ→e] = 2G2
F
ωcapt
(V(p))2|g(p)
LV |2+|g(p)
RV |2.(6.12)
Here, V(p)= 0.0974 m5/2
µis an integral involving proton distribution and ωcapt = 13.07 ×
106s−1is muon capture rate for 197Au [38]. g(p)
LV,RV depend on the flavour-violating cou-
– 17 –

JHEP10(2023)128
LFV observable Limit S1 S2 S3
BR[µ→e]<7.0×10−13 1.0×10−81.4×10−16 1.4×10−24
BR[µ→3e]<1.0×10−12 2.4×10−11 5.0×10−19 1.6×10−27
BR[τ→3µ]<2.1×10−82.3×10−11 4.2×10−19 1.7×10−27
BR[τ→3e]<2.7×10−89.4×10−12 2.7×10−19 6.0×10−28
BR[µ→eγ]<4.2×10−13 7.0×10−94.9×10−17 6.8×10−25
BR[τ→µγ]<4.4×10−82.1×10−11 1.6×10−19 2.2×10−27
BR[τ→eγ]<3.3×10−81.3×10−12 9.3×10−21 1.3×10−28
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 [40]. The values excluded by the limits are highlighted in red.
plings and they are parametrized as
g(p)
LV,RV = 2g(u)
LV,RV +g(d)
LV,RV .(6.13)
The expressions for g(q)
LV,RV (q=u, d) can be obtained using eq. (6.3) and [39]. For the
present model, we find
g(q)
LV ∼√2
GF
1
2hC(eq)LL
2111 +C(eq)LR
2111 i
g(q)
RV ∼√2
GF
1
2hC(eq)RR
2111 −C(eq)RL
2111 i.(6.14)
The branching ratios for µto econversion, computed using eqs. (6.12), (6.13) and (6.14),
for the three benchmark solutions are given in table 5. The present experimental limit
disfavours S1 in this case.
The trilepton decay, li→3lj, is mediated by the new gauge bosons at the tree-level.
The decay width for this process can be estimated following [41]. For the present model
and in the limit mi≫mj, we find
Γ[li→3lj] = 4m5
i
1536 X
P,P ′C(ee)P P ′
jijj 
2,(6.15)
which, using eq. (6.7), takes the following form
Γ[li→3lj] = g4
F
16
m5
i
1536
3
X
α,β=1 ˜
λα
eLji ˜
λβ
eLji +˜
λα
eRji ˜
λβ
eRji
×˜
λα
eLjj ˜
λβ
eLjj +˜
λα
eRjj ˜
λβ
eRjj .(6.16)
The branching ratios for µ→3e,τ→3eand τ→3µevaluated using the above expression
are given in table 5for three solutions along with their corresponding experimental limits.
– 18 –

JHEP10(2023)128
To estimate the branching ratios for li→ljγ, we follow [42] and compute the decay
width in the approximation MZ1≫mi, mj. The result can be parametrized as
Γ[li→ljγ]≃αgF4
64 1−m2
j
m2
i!3
m5
i|σL|2+|σR|2,(6.17)
where
σL=
3
X
α=1
3
X
k=1 Y1
mj
mi˜
λα
Ljk ˜
λα
Lki +Y2˜
λα
Rjk ˜
λα
Rki −4Y3
mk
mi˜
λα
Rjk ˜
λα
Lki.
(6.18)
Similarly, σRcan be obtained by replacing L↔Rin the above expression. Y1, Y2and Y3
are loop functions and they are given by
Y1=Y2= 2a+ 6c+ 3d Y3=a+ 2c , (6.19)
and the explicit expressions of a, c, and dare given in [42]. Using eq. (6.17), we estimate
the branching ratios for µ→eγ,τ→µγ and τ→eγ for three benchmark solutions and
list them in table 5.
Comparing the estimated magnitudes of various charged lepton flavour violating ob-
servables given in table 5with the corresponding experimental limits, we find that MZ1≤
104GeV 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)Fbreaking comes from K-Kmixing which disfavours MZ1≤103TeV.
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 [29]. For example, the strongest limit from the direct
searches at the LHC implies MZ1>7.20 TeV [43]. Similar constraints on the vectorlike
quarks, mB>1.57 TeV [44,45] and mT>1.31 TeV [46,47], are even more weaker. The Z1,2
bosons can mix with the SM Zboson through VEVs of Hui and Hdi in the present model
which in turn contributes to the electroweak observables. The most stringent limits in this
case also imply MZ1≥4.5TeV [29] making all these constraints irrelevant in comparison
to the ones originating from the quark and lepton flavour violations.
7 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)Fsymmetry.
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)Fgauge 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
– 19 –

JHEP10(2023)128
is broken in two steps with an intermediate SU(2) symmetry. This leads to a little hi-
erarchy 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) ×U(1) symmetry was proposed by
us earlier in [29]. The framework presented in this paper replaces the pair of non-abelian
symmetries with SU(3)Fleading to two important improvements. The gauge boson mass
hierarchy, which was an ad-hoc assumption in the case of U(1) ×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 [29] 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 2.4σsmaller than the present
experimental value.
Phenomenologically, the SU(3)Fbreaking scale is constrained from below entirely from
the flavour violation. The new gauge bosons have O(1) flavour-changing couplings with
fermions leading to large rates for flavour-violating processes. This feature seems to be in-
herently present in the frameworks of radiative mass generation mechanisms. We estimate
various quark and lepton flavour-violating observables and find that the lightest gauge bo-
son of SU(3)Fis required to be heavier than 103TeV as implied by the present limits. This
makes it impossible to verify such a framework in the direct search experiments. Never-
theless, the specific model can still be probed indirectly through precision measurements
of fermion masses and mixing parameters and flavour-violating observables.
Acknowledgments
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.
– 20 –

JHEP10(2023)128
ASU(3)Fgenerators
An explicit form of the SU(3)Fgenerators λa(with a= 1,...,8) that we use in the present
work is
λ1=




0 0 0
0 0 1
0 1 0





, λ2=




0 0 0
0 0 −i
0i0





, λ3=




0 0 0
0 1 0
0 0 −1





, λ4=




0 1 0
1 0 0
0 0 0





,
λ5=




0i0
−i0 0
0 0 0





, λ6=




0 0 1
0 0 0
1 0 0





, λ7=




0 0 i
0 0 0
−i0 0





, λ8=1
√3





−2 0 0
0 1 0
0 0 1





.
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 re-
mains unbroken by the VEV of η1in eq. (4.1).
B Scalar potential and VEVs
In this appendix, we demostrate the conditions which lead to the VEV configurations of
η1,2fields considered in eq. (4.1). 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 Hi
u,d in the SU(3)Fbreaking. The most general and renormalizable
potential involving η1,2can be written as
V(η1, η2) = m2
11 η†
1η1+m2
22 η†
2η2−nm2
12 η†
1η2+ h.c.o
+ξ1
2(η†
1η1)2+ξ2
2(η†
2η2)2+ξ3(η†
1η1)(η†
2η2) + ξ4(η†
1η2)(η†
2η1)
+ξ5
2(η†
1η2)2+ξ6(η†
1η1)(η†
1η2) + ξ7(η†
2η2)(η†
1η2)+h.c..(B.1)
Here, all the parameters except ξ5,6,7and m2
12 are real.
For the VEV configuration of η1,2given in eq. (4.1), the minimization of the potential
implies
v(m2
11 +v2ξ1+ϵ2v2ξ3) = 0 ,
ϵv (m2
22 +ϵ2v2ξ2+v2ξ3) = 0 .(B.2)
The non-trivial solution of the above equations corresponds to
v2=−m2
11ξ2+m2
22ξ3
ξ1ξ2−ξ2
3
,(ϵv)2=−m2
22ξ1+m2
11ξ3
ξ1ξ2−ξ2
3
.(B.3)
The VEVs are obtained in terms of real parameters m2
11,m2
22 and ξ1,2,3. The latter are
also constrained by the stability of potential:
ξ1,2≥0, ξ3≥ −pξ1ξ2.(B.4)
– 21 –

JHEP10(2023)128
For 0> ξ3≥ −√ξ1ξ2, one finds ξ1ξ2−ξ2
3≥0. Further assuming |m2
22|≪|m2
11|,
ξ1≪ξ2and m2
11 <0, it can be seen that the VEVs in eq. (B.3) are real. Their ratio is
then determined as
ϵ2≈ −ξ3
ξ2≤sξ1
ξ2≪1.(B.5)
Moreover, for ξ3≈ −√ξ1ξ2, the VEVs obtained eq. (B.3) turn out to be the global minima
of the potential among the available solutions offered by eq. (B.2). In summary, one can
obtain the desired VEV configurations for η1,2consistent with stability constraints for a
specific choice of parameters.
As mentioned in the beginning, we have neglected contribution to the SU(3)Fbreaking
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 η†
aηaH†
u,dHu,d would induce large bare mass
terms for Hu,d when SU(3)Fgets broken, the VEVs of Hu,d would naturally tend to stay
close to the flavour symmetry breaking scale. Also, as the terms like η†
aηaH†
u,dHu,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.
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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