Gauging lepton flavor SU(3) for the muon g − 2

Gonzalo Alonso-Alvarez and James M. ClineMcGill University, Department of Physics, 3600 University St., Montreal, QC H3A2T8 Canada

Gauging a specific difference of lepton numbers such as Lµ − Lτ is a popular model-buildingoption, which gives rise to economical explanations for the muon anomalous magnetic moment.However, this choice of gauge group seems rather arbitrary, and additional physics is required toreproduce the observed neutrino masses and mixings. We address these shortcomings by embeddingLµ − Lτ in the vectorial SU(3) gauge symmetry of lepton flavor. The vacuum expectation values(VEVs) of scalar fields in the fundamental, six-dimensional and adjoint representations allow forphenomenologically viable lepton and gauge boson masses. The octet scalar gives rise to chargedlepton masses, and together with the triplet scalar generates masses for all the leptophilic gaugebosons except for the Lµ − Lτ one. The latter gets its smaller mass from the sextet VEVs, whichalso generate the neutrino masses, and are determined up to an overall scaling by the observedmasses and mixings. The model predicts three heavy neutral leptons at the GeV-TeV scale as wellas vectorlike charged lepton partners; it requires the mass of the lightest active neutrino to exceed10−4 eV, and it naturally provides a resolution of the Cabibbo angle anomaly.

1. INTRODUCTION

Leptophilic gauge bosons are popular candidates forphysics beyond the Standard Model (SM). As opposedto new gauge bosons interacting with quarks, which arestrongly constrained by LHC searches [1–4], those cou-pling solely to leptons are only subject to LEP con-straints [5] and can therefore exist below the TeV scale.As a consequence, these kind of Z ′ bosons can have im-portant phenomenological consequences for a plethora ofparticle physics experiments and observations [6].

In view of recent experimental developments, Z ′

bosons interacting with muons are of particular interest.Most importantly, they can economically solve the long-standing discrepancy between the SM prediction [7] andthe observed values [8, 9] of the muon anomalous mag-netic moment, as has been shown in [10–30]. Slight devi-ations from SM expectations in lepton flavor universality(LFU) ratios [31] and an observed deficit of unitarity inthe first row of the CKM matrix [32–35] may also be ex-plained by a new muonic force. Finally, accumulatingevidence for lepton-universality violation in b → s`+`−

transitions (see e.g. [36] and references therein) serves asa further motivation to study this kind of new physics(NP), although in this case coupling to quarks also needto be invoked. A global analysis of leptophilic gaugebosons was recently performed in [37].

In this context, the most popular phenomenologi-cal model is that of a gauge boson coupling to theLµ − Lτ lepton flavor combination [38–40]. GaugedU(1)Lµ−Lτ models are distinguished by being free fromgauge anomalies (as are other differences in baryonand/or lepton flavor numbers), and from the stringentexperimental constraints on Z ′ couplings to electrons.

One difficulty is that gauging Lµ − Lτ symmetry pre-vents the usual interactions that generate the masses andmixings of neutrinos through the dimension-5 Weinbergoperator. This means that a fully consistent model re-quires additional new physics to reproduce the observedvalues [41] of the PMNS [42, 43] matrix. Some propos-

als in this direction include extra Higgs doublets [16],soft-breaking terms [44, 45], and right-handed neutri-nos [17, 46–48]. The common element of all these modelsis the presence of (often multiple) U(1)Lµ−Lτ symmetry-breaking scalars that can significantly complicate theminimal proposal. From an aesthetic point of view, italso seems rather arbitrary that only the Lµ −Lτ differ-ence should be gauged, considering that the SM treatsall generations on an equal footing from a structural per-spective.

To address these shortcomings, in this paper we pro-pose that SU(3)` of lepton flavor is gauged in a vectorialfashion. In this setup, the Lµ − Lτ gauge boson needonly be one of eight new Z ′ states, all of which are lep-tophilic. Previous studies have gauged lepton and quarkflavors together using horizontal SU(3) family symmetry,along with additional gauge or discrete symmetries [49–52], or flavor symmetries for left- and right-handed lep-tons separately [53]. We are not aware of previous liter-ature treating the possibility of a single vectorial SU(3)`for lepton flavor.

In the following, we identify the additional particlecontent needed to spontaneously break SU(3)` to en-gender the observed lepton and neutrino masses, whichwould otherwise be forced to be flavor universal. Fur-ther guided by experimental observations, we focus onscenarios where the Lµ − Lτ gauge boson is lighter thanthe other seven and can thus address the (g−2)µ tension.For these purposes, we find it sufficient to add a tripletof vectorlike heavy leptons Ei and three scalars in the 3,6 and 8 representations, which we denote by Φ3, Φ6, andΦ8, respectively. The usual triplet of right-handed neu-trinos Ni is also present. The vacuum expectation values(VEVs) of the scalars give rise to the required lepton andgauge boson masses. This particle content is summarizedin Table I.

A modest hierarchy in the scalar VEVs, 〈Φ3〉 ∼ 〈Φ8〉 �〈Φ6〉 is needed to get the desired spectrum of leptophilicgauge boson masses. The gauge boson coupling to theLµ − Lτ current is lighter than the rest, which allows it

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to solve the muon (g − 2) anomaly in a way consistentwith all other constraints [29, 30]. At the same time, theCabibbo angle anomaly (CAA), a 3-5σ tension in theunitarity of the top row of the CKM matrix [32–35], canbe resolved by new contributions to µ→ eνν decay fromtwo of the heavier new gauge bosons. A mild 2σ hint ofnonuniversality in τ lepton decays [31] can be similarlyameliorated.

The Φ6 VEVs further determine the mass matrix of theright-handed neutrinos, and thereby the pattern of lightneutrino masses and mixings via a type I seesaw mecha-nism [54–60]. A striking prediction of our scenario is thatthe lightest active neutrino mass must exceed ∼ 0.1 meV.We find the right-handed neutrino masses to lie in theGeV to TeV range and to be inversely proportional tothe light neutrino masses. They are therefore heavy neu-tral leptons (HNLs), whose mixings with the active statesexactly match the flavor composition of the light neutrinomass eigenstates and are therefore completely determinedby the PMNS matrix. We show that the mixings of theHNLs with active neutrinos can be in a range that isrelevant for affecting the primordial abundances of lightelements.

The paper is structured as follows. In Section 2 we in-troduce the particle content of the model and the result-ing spectra of charged leptons, heavy vectorlike leptons,light neutrinos, and new gauge bosons. Constraints onthe gauge bosons from flavor-sensitive processes are dis-cussed in Section 3, including the muon (g − 2), LEPdi-lepton searches, the Cabibbo angle anomaly, and lep-ton flavor universality limits. The detailed properties ofthe heavy neutral leptons, and their phenomenologicalimplications, are discussed in Section 4. Conclusions aregiven in Section 5, and a brief discussion of the chal-lenges associated with the construction of a potential inthe scalar sector that can give rise to the desired patternof VEVs is presented in Appendix A.

2. MODEL AND MASS SPECTRA

Our starting point is the gauging of the leptonic SU(3)flavor symmetry, under which both left- and right-handedlepton fields transform simultaneously. Clearly, ourworld does not exactly respect such a symmetry, requir-ing it to be spontaneously broken at some scale. Forthis purpose, we introduce three real scalar fields Φ3, Φ6,and Φ8 in the fundamental, symmetric two-index, andadjoint representations, respectively. Although Φ6 by it-self would be sufficient to fully break SU(3)`, this is theminimal scalar sector that we have identified as beingphenomenologically viable,1 as we will explain.

Clear observational evidence for breaking of SU(3)` is

1 Other possibilities may exist; it is not our purpose to exhaustthem but rather to construct one instance of a working model.

SU(3)` SU(2)L U(1)y

Li 3 2 −1/2

ec,i 3 1 +1

EL,i 3 1 −1

Ec,iR 3 1 +1

Nc,i 3 1 0

Φ3,i 3 1 0

Φ6,ij 6 1 0

Φi8,j 8 1 0

TABLE I: Field content of the gauged SU(3)` model alongwith the corresponding gauge charges. The fermionic fieldsare all listed as left-handed Weyl states. Li and ei denote theSM lepton doublets and charged singlet leptons, respectively.

provided by the hierarchy of lepton masses. Indeed, withonly the SM particle content, the gauge-invariant Higgscoupling LiHei (where i is the SU(3)` index) only al-lows for a universal charged lepton mass. To obtain theobserved mass splittings through spontaneous breakingof the gauge symmetry in a renormalizable way, we adda triplet of vectorlike charged lepton partners, Ei. Ad-ditionally, to generate neutrino masses, a right-handedneutrino triplet Ni is included, which completes the par-ticle content of the model. The list of particles and gaugecharges are shown in Table I.

The addition of the right-handed neutrino triplet Nimakes SU(3)` anomaly free, since all leptons now comein chiral pairs. Mixed anomalies involving non-abelianfactors likewise cancel because of the tracelessness of thegenerators, and the SU(3)2`×U(1)y one vanishes due tothe hypercharges of the SU(3)` triplets adding up to zero.

With the field content listed in Table I, the most gen-eral renormalizable Lagrangian that can be constructed(omitting kinetic terms) is

L = − EiL(µEEδji + yEEΦj8,i)ER,j

− EiL(µEeδji + yEeΦ

j8,i)ej

− LiH(yLeei + yLEER,i)− yLN LiHNi− yN

2N iΦ6,ijN

c,j + h.c.

− V (H, Φ3, Φ6, Φ8), (1)

where V denotes the scalar potential, µi are constantswith dimensions of mass, and yi are dimensionless cou-plings. In the following sections we describe the conse-quences of nonvanishing vacuum expectation values forthe scalar fields. The required values of the VEVs areconstructed in a bottom-up fashion, making use of exper-imental measurements and constraints as guiding princi-ples. Obtaining the desired symmetry breaking patternfrom a specific scalar potential is likely to be a challeng-ing task, beyond the scope of the present work. We how-ever take some preliminary steps in this direction in Ap-pendix A.

3

eR

EL ER

µEe

∼ µEE

∼

L

H

yLE

ξ

FIG. 1: Extra contribution to the charged lepton masses fromintegrating out the heavy vectorlike lepton, that splits themrelative to the flavor-independent mass yLev.

In the following, it will be convenient to order thegauge indices associated with the lepton flavors as(1, 2, 3) = (τ, µ, e). This allows the µ − τ gauge bosonto be associated with the third generator T3 of SU(3)`(using the standard form of the Gell-Mann matrices) in-stead of a linear combination of T3 and T8.

2.1. Charged lepton masses

In the absence of spontaneous breaking of SU(3)`,charged leptons get a universal mass from electroweaksymmetry breaking. We will use the Φ8 scalar field VEV,in conjunction with the vectorlike lepton partners Ei, tosplit masses between the different lepton generations.2

The Lagrangian (1) gives rise to mass mixing betweenthe SM leptons and the heavy Ei partners. For simplic-ity, we take 〈Φ8〉 to be diagonal, in which case the 6× 6Dirac mass matrix for the charged leptons becomes blockdiagonal (no flavor mixing), with three blocks of the form

(eiL, EiL)

(yLev

µiEe

yLE v

µiEE

)(eR,iER,i

), (2)

where v ∼= 174 GeV is the complex Higgs VEV and wedefine

µiEe = µEe + yEe〈Φ8〉ii ,µiEE = µEE + yEE〈Φ8〉ii . (3)

Diagonalizing each of the 2 × 2 blocks separately underthe assumption yLev, yLE v � µiEE or µiEe leads to threeheavy and three light mass eigenstates with masses

(miE)2 = (µiEe)

2 + (µiEE)2,

(mie)

2 = v2(yLeµ

iEE − yLEµiEe)2

(µiEe)2 + (µiEE)2

. (4)

The contributions to the light charged lepton masses inthe perturbative limit are depicted in Fig. 1. In the limit

2 Nonzero µEE or µEe values in Eq. (1) are also essential, sincein the µEE , µEe → 0 limit the charged lepton masses becomeuniversal again.

of large µEE or µEe, the heavy lepton effects do not de-couple; lepton masses generally continue to be split bythe heavy-Ei effects even for arbitrarily heavy Ei.

Since we assume that 〈Φ8〉 is diagonal, mixing with theheavy states does not cause any flavor-violating effectsin the charged lepton sector. However, the mixing of thelight states with the heavy ones is constrained by elec-troweak precision data (EWPD). Diagonalizing the massmatrix in Eq. (2) yields mass eigenstates (eiL,R, E

iL,R)′,

given by(eiL,REiL,R

)=

(cθiL,R−sθiL,R

sθiL,RcθiL,R

)(eiL,REiL,R

)′. (5)

To second order in v/µ, the mixing angles are given by

tan 2θiL =2v (yLeµ

iEe + yLEµ

iEE)

(µiEe)2 + (µiEE)2

,

tan 2θiR = 2µiEEµ

iEe + v2yLEyLe

(µiEE)2 − (µiEe)2

. (6)

In the limit where µiEE or µiEe � v, the angle θiL issuppressed, but θiR can be relatively large. EWPD con-straints require that θR(τ, µ, e) . (0.03, 0.02, 0.02) [61].3

To sufficiently suppress θiR, there must be a hierarchyµiEe � µiEE or µiEE � µiEe for each value of i.

To give a concrete example, consider the case µiEe .10−2 µiEE; the derivation for the opposite possibility iscompletely analogous. In this limit, the masses of theheavy charged leptons are simply

miE ' µiEE , (7)

while the mass of the SM charged leptons becomes

mie∼= v

(yLe − yLE

µiEeµiEE

). (8)

Taking for simplicity yLE = 1 and yLe = 0 (yLe .10−6yLE is the sufficient condition), the ratios

µiEeµiEE

' −mie

v∼ (10−2, 10−3, 10−6) (9)

are determined by the measured charged lepton masses.Using Eq. (3) and taking into account the fact that Φ8

is traceless, one sees that achieving the hierarchies inEq. (9) requires some cancellations between terms in thenumerator or denominator. For example, take 〈Φ8〉 =

3 The precise upper limits are model-dependent; we have chosenthe strongest limits corresponding to models that give dominantright-handed mixing.

4

diag(−µEE, µEE, 0). Then Eq. (9) is realized by choosing

µEe =me

vµEE ' 3× 10−6 µEE ,

yEE =mτ +mµ

mτ −mµ' 1.1 ,

yEe =2

v

mτmµ

mτ −mµ' 1.3× 10−3 . (10)

The required order-of-magnitude enhancement of mτ rel-ative to mµ is achieved by a ∼ 10% cancellation in thedenominator of Eq. (9) for i = τ , while the smallness ofme is due to µEe � µEE and 〈Φ8〉ee = 0.

Having fixed the charged lepton masses, one is still freeto choose µEE and therefore the absolute mass scale ofthe charged lepton partners. If they are sufficiently light,they may be directly accessible at the Large Hadron Col-lider (LHC). They can be pair-produced by the Drell-Yanprocess, and their main decays are Ei → Li + h via theyLE interaction. Ref. [62] showed that electroweak-singletvectorlike leptons (VLL’s) are very difficult to observe atthe LHC; indeed, existing constraints [63, 64] have fo-cused on the search for doublet VLL’s. VLL models sim-ilar to ours were studied in Ref. [65], where current LHClimits were found to be as low as mi

E & 300 GeV. Thisleaves a large range of allowed VLL masses for future dis-covery, consistent with our requirement 〈Φ8〉 & 1 TeV,that will be derived in Section 3.

In the following sections, we adopt the parameter val-ues described above as our principal benchmark model(BM1). However, other possibilities for fixing the con-stants in the Lagrangian (1) exist. There are eight freeparameters: two dimensionful scales µEE and µEe, fourYukawa couplings yEE, yEe, yLE, and yLe, and two in-dependent entries of 〈Φ8〉. Reproducing the observedcharged lepton masses imposes three constraints throughEq. (4). In addition, the mixings in Eq. (6) must be suf-ficiently small. In order to explore the parameter spaceand find other viable models, we employed a MarkovChain Monte Carlo method. Two further examples ofparameters that satisfy all the phenomenological require-ments but differ qualitatively from BM1 are given in Ta-ble II.

In the second example (BM2), also expressed in termsof the unconstrained parameter µEE, the heavy leptonmasses are all approximately mi

E∼= µEE. Satisfying the

EWPD bounds on mixing leads to the largest Yukawacoupling, yLE = 3.6, being close to unitarity limits. Inthis example, since µEE � 〈Φ8〉, the heavy leptons are farabove the TeV scale and therefore not directly accessiblein current collider experiments.

In the third example (BM3) and similarly to BM1, theVLL are of order 〈Φ8〉, making them potentially accessi-ble at the LHC. In this case (as in BM1), all Yukawa cou-plings are well below perturbative unitarity bounds. Aqualitatively different feature of BM3 is that µτEE � µτEe,while in the others µiEE � µiEe for all flavors. Moreover,the VVL-charged lepton mixing angles fall well below theexperimental limits in this model.

BM1 BM2 BM3

Input parameters

φµ µEE 0.014µEE −3.0µEE

φe 0 0.016µEE −3.2µEE

yLE 1 3.6 −0.011

Derived parameters

µEe 3× 10−6 µEE 0.023µEE 1.55µEE

yEE 1.1 0.32 0.16

yEe 1.3× 10−3 −0.055 −0.48

yLe . 10−6 0.56 −8.5× 10−4

Phenomenological quantities

miE (1, 1, 1)µEE (1, 1, 1)µEE (4.5, 1.5, 1.5)µEE

θτR 1× 10−2 0.021 −1× 10−3

θµR 6× 10−4 5.6× 10−4 6× 10−3

θeR 3× 10−6 0.024 6× 10−3

TABLE II: Input parameters, derived parameters and quan-tities relevant for phenomenology for three benchmark mod-els that can reproduce the observed charged lepton proper-ties. The mixing angles θiR between right-handed chargedleptons and heavy vectorlike leptons of mass mi

E are consis-tent with EWPD constraints [61]. The traceless octet VEVis parametrized as 〈Φ8〉 = diag(−φµ − φe, φµ, φe). The di-mensionful parameter µEE, which controls the scale of theoctet VEV and the VLL masses, is left as freely adjustableparameter.

2.2. Neutrino masses

The right-handed neutrinos Ni get a Majorana massmatrix from the sextet VEVs, MN = yN〈Φ6〉, while theDirac neutrino mass matrix is proportional to the unitmatrix, with coefficient mD = yLN v. The light neutrinomass matrix is therefore mν = −m2

DM−1N from the see-

saw mechanism, and we can solve for the sextet VEVs interms of the PMNS mixing matrix UPMNS and the lightneutrino mass eigenvalues D = diag(m1,m2,m3):

〈Φ6〉 = −m2D

yNUPMNSD

−1 UT

PMNS . (11)

Recall that we have interchanged the first and third rowsof UPMNS in our ordering convention for the lepton flavors.This allows us to refer to the Lµ−τ gauge boson as the oneassociated with the Gell-Mann matrix λ3. A quantitativeanalysis of the right-handed neutrino mass and mixingspectrum is deferred to Sec. 4.

5

2.3. Gauge boson masses

The mass-squared matrix of gauge bosons is given by

M2ab

g′2= 1

4Φ†3{λa, λb}Φ3 − 12 tr([Φ8, λ

a][Φ8, λb])

+ 12 tr(

[Φ†6λa + λa∗Φ†6][Φ6λ

b∗ + λbΦ6]), (12)

where the VEVs of the scalar fields are understood. Inthe approximation that 〈Φ6〉 = 0, while 〈Φ3〉 ∼ 〈Φ8〉,all gauge bosons get masses except for A3, which cor-responds to the Lµ−τ gauge boson in our numberingscheme. Thus A3 gets its mass solely from 〈Φ6〉. Theoverall magnitude of 〈Φ6〉 is determined by the small-est light neutrino mass mν1 ; hence one prediction of themodel is that mν1 cannot be arbitrarily small.

The various contributions of the scalar VEVs to themass eigenvalues M2

i /g′2 are summarized in Table III.

Before turning on the Φ6 VEV, the gauge boson massmatrix is diagonal, and the eigenvalues can be labelledwith the number of the corresponding generator.

We first consider the contribution of the Φ8 VEV. Fol-lowing the discussion in the previous section, in bench-mark scenario BM1 it takes the form

〈Φ8〉 ≡ φ8 diag(−1, 1, 0) , (13)

with φ8 = µEE.4 The resulting contributions to the dif-ferent gauge boson masses are shown in the third columnof Table III. Purely diagonal 〈Φ8〉 VEVs leave the A3 andA8 gauge bosons massless. To generate mass for A8, weintroduce the VEV

〈Φ3〉 = φ3 (0, 0, 1)T , (14)

which further lifts the A4,5,6,7 masses, leaving only A3

massless. The numerical values of the contributions toM2i /g′2 are shown in the fourth column of Table III. As

we will see in the next section, phenomenological con-straints on the leptophilic gauge bosons lead to the re-quirements φ8 & 1 TeV and φ3 & 10 TeV.

The Φ6 VEV contributes to all of the masses andcauses mixing between the gauge bosons. We assumethat it is much smaller than the other two VEVs so thatthe mixing, as well as the shifts to the heavy masses, canbe neglected, or computed perturbatively. Therefore the〈Φ6〉3,3 element directly determines M3, and thereby thenew physics contribution to (g − 2)µ.

Using Eq. (11), the light gauge boson mass can bewritten as

M3 ∼ g′m2D

yN mν1

≡ g′φ6 , (15)

4 More generally, φ8/µEE could differ from 1, and the relations(10) would be modified by factors of φ8/µEE . We allow for thismore general form in the following.

Gauge

boson

Flavor

structureΦ8 Φ3 Φ6

A1,2

0 ∗ 0

∗ 0 0

0 0 0

τ

µ

e

4φ28 7 3

A3

1 0 0

0 −1 0

0 0 0

7 7 3

A4,5

0 0 ∗

0 0 0

∗ 0 0

φ28

12φ23 3

A6,7

0 0 0

0 0 ∗

0 ∗ 0

φ28

12φ23 3

A81√3

1 0 0

0 1 0

0 0 −2

7 23φ23 3

TABLE III: Flavor structure of the couplings of the 8 lep-tophilic gauge bosons in the SU(3)` model, together with thecontributions to M2

i /g′2, their squared masses divided by g2,

arising from the various scalar VEVs (see Eq. 13). The contri-butions from Φ6 are small compared to those of Φ3 and Φ8 byconstruction, and they lead to mixing between the tabulatedLagrangian states.

where the exact proportionality factor depends on theabsolute scale of neutrino masses and varies between∼ 0.5 and 1.5 within the range of interest. It will beshown in the next section that for values of g′ and M3

that can explain the muon anomalous magnetic moment,φ6 = O(20 − 200) GeV. If yN ∼ 1, this is also thescale of the sterile neutrino masses, whose mass matrixis mN = yN〈Φ6〉. In the following sections, we discussthe phenomenological consequences of the new particlespredicted by the model.

3. PHENOMENOLOGY OF THE LEPTOPHILICGAUGE BOSONS

The phenomenology of gauge bosons associated withlepton number gauge symmetries has been widely stud-ied; see e.g. [37, 66] and references therein. In the SU(3)`model presented here, neglecting the small mixing be-tween gauge bosons, there are six purely flavor-violatingcurrents, mediated by A1,2 for µ ↔ τ , A4,5 for e ↔ τ ,and A6,7 for e↔ µ. The flavor-conserving A3 couples toµ−τ , and A8 to the 2e−µ−τ current. These leptophilicgauge bosons have implications for a number of differentobservables.

3.1. Muon anomalous magnetic moment

By construction, A3 is much lighter than the rest of thevector bosons, due to the symmetry-breaking pattern ofthe model. This enables it to explain the current 4.2σ dis-crepancy between experimental measurements [8, 9] andSM predictions [7] of the anomalous magnetic moment of

6

100 101 102 103

Lµ − Lτ gauge boson mass M3 [MeV]

10−4

10−3

10−2

SU

(3) `

gaug

eco

uplin

gg′

(g − 2)µ

CCFR

BaBar

∆Neff > 0.5

FIG. 2: Region of the gauge coupling versus mass param-eter space for which a massive dark photon coupled to theLµ − Lτ current can explain the observed value of the muong − 2 factor. The blue-dotted region shows the 2σ preferredvalues obtained from the recently reported measurement [9]and SM calculations [7]. The other shaded areas correspondto constraints from Babar [68] (orange), CCFR [69] (green),and cosmology [70] (purple).

the muon.5 A gauge bosonA′ coupling to the muonic vec-tor current with strength g′ contributes to aµ = (g−2)µ/2at the one-loop level [12] with

∆aµ =g′2

4π2

∫ 1

0

dxm2µ x (1− x)2

m2µ(1− x)2 +m2

A′ x. (16)

This has the right sign to address the experimentalanomaly and is phenomenologically viable as long as A′

couples negligibly to electrons, which is the case for A3.Such a Lµ − Lτ gauge boson was searched for in thee+e− → 4µ channel at BaBar [68] and through neutrinotrident production at CCFR [69]. Taking the ensuingconstraints into account, the A3 gauge boson can explainthe (g − 2)µ discrepancy as long as its mass lies in the10 . mA3

. 200 MeV range and the gauge coupling is3× 10−4 . g′ . 10−3 [29, 30] as shown in Fig. 2.

3.2. Di-lepton searches at LEP

The LEP experiments searched for e+e− → `+`−

events beyond the Standard Model [5], parametrizing thenew physics (NP) contributions by contact interactionsof the form

4π

Λ2Je,α

[12J

αe + Jαµ + Jατ

], (17)

5 A recent lattice evaluation of (g − 2)SM [67] may help alleviatethis tension, but it is in disagreement with the other SM predic-tions [7].

0.1 1 10 100Octet VEV scale φ8 [TeV]

0.1

1

10

100

Tri

plet

VE

Vsc

aleφ

3[T

eV]

CAA

A(τ→µ)A(τ→e)

A(τ→µ)A(µ→e)

A(τ→e)A(µ→e)

LEP

FIG. 3: 95% CL contours on the model parameters φ8 andφ3 from LEP dilepton analyses and lepton flavor universalityconstraints as labelled in the figure. The blue-dotted regionshows the 2σ preferred values to address the CKM unitaritydeficit, also known as the Cabibbo Angle Anomaly (CAA).The dotted orange line shows the edge of the preferred regionfor the slight ∼ 2σ excess in the LFU ratio (21).

where Jµi is the vector current for charged leptons offlavor i. Different limits on Λ were derived depending onthe combinations of final states observed. However, thederived limits do not directly apply to states such as A8

that couple with different strengths to different flavors.However, Ref. [5] provides the observed differential dis-

tributions for the e+e− → `` processes studied by LEP,enabling us to derive limits on nonabelian leptophilicgauge bosons such as those in the present model. Fordimuon and ditau final states we use the data for thetotal cross-section σT and forward-backward asymmetryAfb as a function of

√s, while for the e+e− final state

we use the full averaged differential cross section givenin [5]. Employing a χ2 test statistic on the SM versusSM + NP models leads to the following bounds on theindividual gauge boson masses at 95% CL:

M4,5

g′> 4.6 TeV,

M6,7

g′> 5.2 TeV,

M8

g′> 8.9 TeV. (18)

We further construct the combined χ2 for all the afore-mentioned channels, including the effects of all relevantgauge bosons parametrized by the two model parametersφ3 and φ8. Doing so leads to the 95% CL constraints onthe VEVs corresponding to the yellow contours in Fig. 3.As shown in Eq. (18), the strongest limits are on A8, itbeing the only state coupling diagonally to the electroncurrent. Since the mass of A8 comes solely from 〈Φ3〉,this constrains φ3 independently of the value of φ8, whichexplains the shape of the exclusion contour in Fig. 3.

7

3.3. Lepton flavor universality violation

Vector bosons with flavor off-diagonal couplings can in-duce lepton decays of the form `i → `j νν, which interferewith the SM amplitudes and spoil the flavor-universalnature of these weak decays. Following the prescriptionof [37], these effects can be measured using the amplituderatios

R(`i → `j) =A(`i → `j νν)

A(`i → `j νν)SM= 1 + 2

g′2

g22

m2W

m2A′, (19)

where g2 denotes the SU(2)L gauge coupling and we haveexpanded to first order in g′2, i.e., keeping only SM-NPinterference terms. These can be confronted with exper-imental measurements of the ratios of partial widths inpurely leptonic decays [31],6

A(τ → µνν)

A(τ → eνν)= 1.0018± 0.0014, (20)

A(τ → µνν)

A(µ→ eνν)= 1.0029± 0.0014, (21)

A(τ → eνν)

A(µ→ eνν)= 1.0010± 0.0014, (22)

with the correlations given in Ref. [31]. In our model,the R-ratios are completely determined by the Φ3 andΦ8 VEVs, which can be parametrized using φ3 and φ8 asdefined in Eqs. (13) and (14). Applying the experimen-tal constraints in Eqs. (20)-(22) at 95% CL leads to theexclusion limits shown in Fig. 3.

3.4. Cabibbo angle (or CKM) anomaly

In addition to LFU violation, a modification of the µ→eνν decay rate changes the inferred value of the Fermiconstant, which affects the determination of Vud from βdecays. The latter measurements currently exhibit a ∼3σ [32–35] tension with the SM predictions, which couldbe eased by a NP contribution at the level of [37]

R(µ→ e) = 1.00075± 0.00025 . (23)

This experimental anomaly was originally formulated asa breakdown of unitarity in the first row of the CKMmatrix, and is thus commonly referred to as the Cabibboangle anomaly (CAA). Since the sum |Vui|2 appears to beless than unity, one interpretation is that weak decays ofnuclei are suppressed relative to those of leptons. The ad-ditional contributions of A6,7 gauge interactions to muondecays can explain the anomaly if the effective Fermi con-stant Gµ inferred from muon decays is increased by a

6 The τ − µ universality ratio can also be measured in semi-hadronic decays, but with a larger uncertainty that does notsignificantly strengthen the constraints.

factor of (1 + δµ), with δµ = 7 × 10−4 [32]. It can beachieved in the present model with

M6,7

g′∼= 13 TeV (24)

due to the additional neutral current contribution to µ→eνµνe. This is consistent with the LEP and LFU bounds,as can be seen in Fig. 3, where the 2σ preferred regionto explain the CAA anomaly corresponds to the blue-dotted contour. If a NP origin of the anomaly were to beconfirmed, it could be explained by the nonabelian modelwith VEVs φ3 ∼ 10− 20 TeV and φ8 ∼ 1− 15 TeV.7

3.5. Gauge boson mixing

The Φ6 VEV generates mixing between all the gaugeboson Lagrangian states in the model, with mixing an-gles of order sin ξij ∼ (φ6/φ3,8)2 . 10−4 for the VEVs ofinterest here. To see that these are phenomenologicallyharmless, we consider the most sensitive observable prob-ing transitions enabled by them, µ→ eγ, whose branch-ing ratio is constrained to be Br(µ → eγ) ≤ 4.2× 10−13

[71–73]. This process is generated at one loop viaA1,2 −A4,5 mixing (internal τ lepton) or A6,7 −A8 mix-ing (internal electron and muon). Following Ref. [37], wederive the bounds

Mi,j

g′≥ 3.7 TeV

√sin ξij10−4

,

Mk,8

g′≥ 48 GeV

√sin ξk810−4

, (25)

where i = 1, 2, j = 4, 5, and k = 6, 8. The first limit isstronger than the second one by a factor of

√mτ/me ∼

60 due to a cancellation of the µ-exchange contributionin the latter. In any case, neither of the two competewith the previously discussed LEP or LFU bounds.

4. PHENOMENOLOGY OF THE HEAVYNEUTRAL LEPTONS

The spectrum of heavy neutral leptons Ni, and theirmixing with the light neutrinos νi, is largely dictated bythe measured νi masses and mixings, once the lightestmass mν,l is specified. In particular, there is a strictrelation between pairs of mνi and MNi eigenvalues,

mνiMNi = m2D, i = 1, 2, 3 , (26)

7 The LFU ratio (21) also deviates by 2σ from the SM prediction,which would single out an even more restricted range φ8 ∼ 1 −5 TeV for the Φ8 VEV as indicated by the orange dotted line inFig. 3.

8

Normal Ordering (NO) Inverted Ordering (IO)

m2ν

M2N

ν3

ν2ν1

ν1

ν3

ν2

N1

N2

N3

N3

N2

N1

Δm2atm

Δm2atm

νe νμ ντ

∝Δm2atm

∝Δm2atm

νe νμ ντ

Uei Uμi Uτi

Uei Uμi Uτi

Activ

e ne

utrin

osH

eavy

Neu

tral

Lep

tons

Δm2sol

∝Δm2sol

∝Δm2sol

Δm2sol

FIG. 4: Schematic representation of the mass and mixing hier-archies of the active neutrinos and the heavy neutral leptonsin the gauged SU(3)` model, for both normal and invertedorderings. The masses of the HNLs are predicted to be in-versely proportional to the light neutrino ones, so that theyare distributed following the reflected pattern represented inthe figure. The mixings of each HNL mass eigenstate with thelight neutrino flavors, represented by colors, exactly match theflavor admixture of the corresponding active neutrino, whichis dictated by the PMNS matrix.

where mD is the universal neutrino Dirac mass. Becauseof SU(3)` symmetry, mD is just a number, not a matrix.This implies that the logarithmic range of Ni masses isthe same as that of the light neutrinos, as is representedschematically in Fig. 4. The mixing angles between Niand the νi are diagonal in the basis of mass eigenstates,and are approximately

Ui =mD

MNi

=

√mνi

MNi

. (27)

Thus, heavier HNLs are more weakly mixed and the fla-vor mixing pattern of each one exactly matches that ofthe corresponding active neutrino, as is illustrated inFig. 4. We now proceed to quantify this picture moreprecisely.

4.1. HNL masses

There are three parameters that determine the HNLmass and mixing spectrum, together with the Φ6 contri-bution to gauge boson masses: the Yukawa couplings yNand yLN and the mass of the lightest active neutrino mν,l.This state is labeled as ν1 in the normal mass hierarchy(NO), and ν3 in the inverted one (IO), as shown in Fig.4. The choice of hierarchy as well as the relative signs ofactive neutrino masses affects the HNL mixings.

0.000 0.005 0.010 0.015 0.020 0.025 0.030

Lightest neutrino mass mν,l [eV]

0.0

0.2

0.4

0.6

0.8

1.0

Rat

ioof

HN

Lm

asse

s

NO, MN2/MN1

NO, MN3/MN1

IO, MN1/MN3

IO, MN2/MN3

FIG. 5: Ratios of the absolute value of the lighter two HNLmasses to the heaviest one, as a function of the mass of thelightest active neutrino. For normal ordering, there is a singleHNL that is substantially lighter than the other two, whilefor inverted ordering (IO) there are two nearly degenerateHNLs that are lighter than the third one. For IO, the curvesterminate when the cosmological limit on

∑i |mν,i| becomes

saturated.

From Eq. (11), the sextet VEV can be written as

〈Φ6〉 = −y2LN

yN

v2

mν,lUPMNS D

−1 UTPMNS , (28)

where D is the diagonal active neutrino mass matrixrescaled by 1/mν,l and UPMNS is the PMNS matrix withcolumns arranged according to our flavor ordering scheme(τ, µ, e). The Φ6 VEV determines the ratio M3/g

′ forthe µ − τ gauge boson via Eq. (12), and thus its abilityto explain the (g − 2)µ anomaly. As was noted in theprevious section,

M3

g′∈ (18 GeV, 200 GeV) (29)

is the preferred range at the 2σ level, corresponding tog′ ∈ (3.7, 13)×10−3. The precise value ofM3 predicted inthis way has a weak dependence on the relative sign of theactive neutrino masses, which amounts to a factor of ∼ 2for mν,l = 0.03 eV and disappears for mν,l � 0.03 eV.

The mass matrix for the HNLs is in turn determinedby the sextet VEV as

MN = yN 〈Φ6〉

= −y2LNv2

mν,lUPMNS D

−1 UTPMNS . (30)

Using the top line in Eq. (30), the (g − 2)µ-preferredvalues for 〈Φ6〉 result in the prediction that the scale ofHNL masses lies in the O(1 − 100)GeV range for O(1)values of yN .

Eq. (30) implies that the PMNS matrix also representsthe unitary transformation that diagonalizes MN . As

9

10−1 100 101

Lightest HNL mass MN3[GeV]

10−12

10−10

10−8

10−6H

NL

-neu

trin

om

ixin

gsU

2Terrestrialsearches-µ

SU(3)` model

BBN

U2e3

U2:U2µ3

U2:U2τ3

U2= 0.05 : 1 : 0.95

Normal Ordering

10−1 100 101

Lightest HNL masses MN1'MN2

[GeV]

10−12

10−10

10−8

10−6

HN

L-n

eutr

ino

mix

ingsU

2

Terrestrialsearches-e

SU(3)` model

BBN

U2e1

U2:U2µ1

U2:U2τ1

U2= 1 : 0.20 : 0.23

U2e2

U2:U2µ2

U2:U2τ2

U2= 0.45 : 0.50 : 0.52

Inverted Ordering

FIG. 6: Dark blue: predictions from the gauged SU(3)` lepton flavor model for the mass and mixing of light HNLs. The mixingsare normalized to the muon (left) or electron (right) one as indicated in the insets. We show bounds (grey) from terrestrialsearches as compiled in [74] as well as the BBN limit (green) computed following [75, 76]. For the normal mass hierarchy (left),there is a single light HNL with suppressed e mixing and we therefore show terrestrial searches based on the µ mixing. Forinverted hierarchy (right), there are two quasi-degenerate light HNLs with fairly universal mixings and the leading terrestrialconstraints therefore come from e mixing.

a consequence, each HNL mass eigenstate is strictly in-versely proportional to the corresponding active neutrinomass,

MNi = −y2LNv2

mνi

. (31)

This means that in the normal ordering case, there isone light and two heavy HNLs, while for inverted order-ing there are two light and one heavy HNLs. This isschematically represented in Fig. 4, while the quantita-tive mass ratios are shown in Fig 5 as a function of mν,l.The largest considered value of mν,l = 0.03 (0.015) eVin the NO (IO) case, roughly saturates the cosmologicalconstraint on the sum of the absolute value of neutrinomasses8

∑ |mνi | ≤ 0.12 eV [78].

4.2. HNL mixings

Transforming to the HNL mass eigenbasis (denoted byprimes), it is straightforward to show that the mixingbetween the light neutrino flavors and the HNLs is givenby

να ∼= ν′α + UαiN′i

Ni ∼= −Uαi ν′α +N ′i , (32)

8 A recent evaluation [77] finds a tighter limit∑|mνi | ≤ 0.09 eV,

which would compromise the viability of the IO scenario butwould not significantly affect the NO one.

where

Uαi = UPMNS

αi

mνi

yLN v. (33)

The mixing structure is completely determined by thePMNS matrix: the mixing pattern of the HNL masseigenstate i is proportional to the flavor composition ofthe corresponding active neutrino mass eigenstate (seeFig. 4 for a graphical representation). The approximatescale of the mixing is U = (mν/MN )1/2 ∼ 10−6 − 10−5

for HNLs in the 1− 100 GeV mass range.While the HNL masses and mixings predicted by the

SU(3)` model are out of reach for present and upcom-ing terrestrial experiments, they have significant impli-cations for cosmology. In particular, HNLs decaying tooclose to the time of Big Bang Nucleosynthesis (BBN)can affect the predictions for light element yields [79, 80].The strongest current constraints come from the hadronicdecays of the HNLs, which inject light mesons into theplasma that can act to imbalance the proton-to-neutronratio at BBN [75].

Following the prescription of Ref. [76] together withthe width calculations of Ref. [81], we obtain the con-straints shown in Fig. 6 for normal and inverted masshierarchies. For NO, there is a single light HNL whosemixings match those of ν3 (see Fig. 4). As a consequence,its e mixing is suppressed by a factor of ∼ 5 comparedto the µ and τ mixings. Then the BBN bound givesMN3

& 2.4 GeV or U2µ3 . 1 × 10−11, which translates

into yLN & 6.4×10−8. IO has two light HNL with quasi-degenerate masses and mixings dictated by the flavorcomposition of ν1 and ν2 (see Fig. 4). The BBN boundarising from the combined effect of both HNLs leads toMN1

∼MN2& 0.95 GeV and U2

e1 . 4× 10−11, implyingthat yLN & 4.0× 10−8.

10

10−2 10−1 100 101

yN

10−8

10−7

10−6y L

N Uni

tari

ty

BBN

(g − 2)µmν1

= 0.03 eV

mν1= 0.0

1 eV

mν1= 0.0

01 eV

mν1= 0.0

001 eV

Normal Ordering

10−2 10−1 100 101

yN

10−8

10−7

10−6

y LN U

nita

rity

BBN

(g − 2)µ mν3= 0.0

15 eV

mν3= 0.0

1 eV

mν3= 0.0

01 eV

mν3= 0.0

001 eV

Inverted Ordering

FIG. 7: Regions of the yN -yLN parameter space for which the gauged SU(3)` model can address the (g − 2)µ anomaly whilesatisfying all other phenomenological constraints, for various values of the lightest neutrino mass as labeled in the figure. Theblue-dotted regions show where the A3 gauge boson mass and coupling g′ are consistent with the 2σ preferred values shownin Fig 2. The green region is the BBN exclusion from Fig. 6 and the orange one is forbidden by perturbative unitarity of theHNL Yukawa coupling, yN ≤

√4π. Left (right) panel corresponds to the NO (IO) active neutrino mass hierarchy, respectively.

The BBN constraint on yLN , together with the boundyN ≤

√4π arising from perturbative unitarity in NN →

NN scattering, rule out regions of the yLN versus yNparameter space, including some relevant for explainingthe (g−2)µ anomaly. They are shown in Fig. 7 for severalchoices of the lightest active neutrino mass and for thetwo possible neutrino mass orderings. Here we assumethat mν3 has opposite sign to mν1 and mν2 , which givesthe model the greatest latitude for consistency. Otherchoices of relative signs differ by factors . 2 for the extentof (g − 2)µ preferred regions.

Several conclusions can be drawn from Fig. 7. Theminimum value for mν,l that is compatible with the ex-perimental measurement of (g − 2)µ is mν,l ' 10−4 eV.Consequently, the maximum possible mass for the heav-iest HNL is MN1 ' 1.2 TeV for NO and MN3 ' 1 TeVfor IO. At the other extreme, values of yLN as large as10−6 are possible if the mass of the lightest neutrinosaturates the cosmological upper bound. This revealsa novel connection between the scale of neutrino massesand the anomalous magnetic moment of the muon withinthe SU(3)` paradigm. It constitutes a unique predictionof the proposal, which could exclude it definitively inlight of more precise experimental data.

5. CONCLUSIONS

In this work, we have proposed a framework for gaug-ing the vectorial SU(3) family symmetry of lepton flavorin a minimal way. The construction can consistently re-produce the observed patterns of charged lepton massesand neutrino masses and mixings. The new particle con-tent of the model, in addition to the eight leptophilicgauge bosons, consists of a right-handed neutrino triplet

and a triplet of vectorlike isosinglet charged lepton part-ners. In addition, three scalar multiplets in the fun-damental, symmetric two-index, and adjoint representa-tions of SU(3)` provide the symmetry breaking requiredto reproduce the observed lepton masses and mixings,and allowed gauge boson masses. The charges and inter-actions are collected in Table I and Eq. (1), respectively.

One of the new leptophilic gauge bosons, the one as-sociated with the Lµ − Lτ current, is taken to lie in the10 − 200 MeV range, being parametrically lighter thanthe others. This allows it to address the current discrep-ancy between experimental measurements and SM pre-dictions of the anomalous magnetic moment of the muon.Moreover, the Cabibbo angle anomaly can be explainedby the pair of vector bosons that couple off-diagonallyto eµ, with masses in the 1− 10 GeV range, while com-plying with all other LEP and lepton-flavor universalityconstraints.

Interestingly, the relative lightness of the Lµ−Lτ gaugeboson is shown to be linked to a low (∼ 1-100 GeV) massscale for the right-handed neutrinos that participate inthe seesaw mechanism. As shown schematically in Fig. 4,the masses of the HNL eigenstates are inversely propor-tional to the light neutrino ones, and their mixings withthe active flavors match the flavor composition of the cor-responding light neutrino mass eigenstates. These twoconnections lead us to predict that the lightest activeneutrino cannot be arbitrarily light, but must rather havea mass larger than ∼ 0.1 meV.

Once fit to the observed lepton and neutrino prop-erties (including (g − 2)µ), the gauged SU(3)` scenariois highly predictive and makes potentially testable fore-casts for existing and upcoming astrophysical and par-ticle physics experiments. Focusing on the HNL sector,there are three main avenues that complement each other

11

in testing the parameter space in Fig. 7: (i) more preciseexperimental measurements and SM predictions for themuon anomalous magnetic moment; (ii) refined measure-ments and theoretical calculations of the light-elementyields and the number of relativistic degrees of freedomat Big Bang Nucleosynthesis, and (iii) improved deter-minations of the absolute scale of neutrino masses fromcosmology or terrestrial experiments like KATRIN [82].In the charged lepton sector, electroweak precision con-straints on lepton mixing with heavy vectorlike partners,needed for realistic lepton mass generation, constitute apowerful test of the model. These lepton partners mightbe discoverable at the High-Luminosity LHC or otherfuture colliders. In contrast, although they are energet-ically accessible, the new scalar fields in the model areweakly coupled to the SM states, making their possibledetection difficult.

Although it is not mandatory for the model, it is possi-ble to introduce an additional fermionic vectorlike tripletΨi as a dark matter candidate. As in the U(1)Lµ−Lτmodel of Refs. [29, 83], the observed relic density couldbe achieved through resonantly enhanced Ψµ/τ Ψµ/τ →µ+µ−+νµ,τ νµ,τ annihilations. The obstacle in the SU(3)`model is that the Ψe component would not be sufficientlysuppressed unless its mass also happens to allow for reso-

nantly enhanced (co)annihilations through the exchangeof the heavier gauge bosons. This would require addi-tional model building, which we do not pursue here.

A remaining challenge for future study is to constructa suitable scalar potential leading to the required Φ3,Φ6, and Φ8 VEVs. Although the preliminary study inAppendix A makes it plausible that the desired symmetrybreaking pattern can be achieved, further investigation isrequired to prove it.

It is in principle possible to construct a similar modelfor quark masses and mixing based on a spontaneouslybroken SU(3)B symmetry. This extension, combinedwith the present proposal including mixing between theSU(3)` and SU(3)B gauge bosons, might explain the ac-cumulating evidence for lepton-universality violation inb → s`+`− decays [36]. Work in this direction is under-way.

Acknowledgments. We thank B. Grinstein, A. Criv-ellin, W. Altmannshofer, and F. Kahlhoefer for helpfulcorrespondence. This work was supported by NSERC(Natural Sciences and Engineering Research Council,Canada). G.A. is supported by the McGill Space Insti-tute through a McGill Trottier Chair Astrophysics Post-doctoral Fellowship.

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Appendix A: Scalar interactions

Obtaining the desired pattern of symmetry breaking ina self-consistent way from a scalar potential is a challeng-ing task which is beyond the scope of the present work.That said, in this appendix we take a preliminary stepin this direction by writing the renormalizable couplingsin the scalar sector that are consistent with the gaugesymmetries.

The Higgs-like symmetry breaking terms in the scalarpotential for the three SU(3)` multiplets have the form

V 3 λ3(Φ†3Φ3 − v23)2 + λ6(tr Φ†6Φ6 − v26)2

+ λ8(tr Φ28 − v28)2 . (A1)

14

Beyond these, the dimensionless interaction terms are

V 3 λ3,8Φ†3Φ8Φ8Φ3 + λ3,6Φ†3Φ†6Φ6Φ3

+ λ6,6 tr(Φ†6Φ6Φ†6Φ6) + λ8,8 tr(Φ48)

+ λ6,8 tr(Φ28Φ†6Φ6) + λ′6,8 tr(Φ8Φ†6ΦT

8Φ6)

+ λ′′3,8 Φ†3Φ3 tr(Φ28) + λ′′3,6 Φ†3Φ3 tr(Φ†6Φ6)

+ λ′′6,8 tr(Φ28) tr(Φ†6Φ6) , (A2)

while the dimension-3 couplings correspond to

V 3 µ3,6 ΦT

3Φ6Φ3 + H.c.

+ µ3,8 Φ†3Φ8Φ3 + µ8 tr(Φ38)

+ µ6,8 tr(Φ6Φ8Φ†6) + µ′6,8 tr(Φ6Φ†6ΦT

8 ) . (A3)

As a first step, we can set Φ6 = 0, Φ3 = (0, 0, a)T , andΦ8 = bλ3+cλ8 to determine whether the desired values ofa, b, c can give rise to a stationary point in the potentialwithin this restricted slice of the field space. There are9 relevant parameters in the potential, three equations

of stationarity for given values of the VEVs, and threestability conditions, so this seems generically possible.

The more difficult step may be to arrive at the desiredform of 〈Φ6〉. In a linear analysis, due to the source termµ3,6Φ3,iΦ3,j , one would expect the Φ6 VEV to be domi-nantly aligned in the (3, 3) direction. This is in contrastwith the need for relatively democratic entries in order torecover the observed mixing pattern of active neutrinosas dictated by the PMNS matrix. One should then sup-press the µ3,6 coupling and obtain the desired solutionat the nonlinear level of the minimization equation. Thiswould yield the schematic form

AΦ6 +BΦ36 = ε , (A4)

where ε is the suppressed source term. There are 14real potential parameters involving Φ6, compared to 6extremization and 6 stability conditions; therefore it maybe possible to obtain the desired pattern of Φ6 VEVs. Wedefer a more thorough analysis to future study.