Muon Anomalous Magnetic Moment and Higgs Potential Stability in the 331 Model from $SU(6)$

We consider a $SU(3)_c \times SU(3)_L \times U(1)_X$ model from a $SU(6)$ Grand Unified Theory (GUT). In order to explain the anomalous magnetic moments of muon and electron, we introduce two new scalar triplets without vacuum expectation values (VEVs) so that the leading contributions to $\Delta a_{\mu}$ and $\Delta a_{e}$ can avoid the suppression from small muon mass. In addition, the Higgs potential stability of this 331 model is studied by giving a set of sufficient conditions to ensure the boundedness from below of the potential.


I. INTRODUCTION
The standard model (SM) has been proved to be a successful theory since the last standard model particle, Higgs, was found at the Large Hadron Collider (LHC). However, there are many problems that SM cannot explain. For instance, the gauge hierarchy problem, dark matter, neutrino masses, gauge coupling unification, etc. There are so many motivations that make people believe the existence of new physics in higher energy beyond SM. In Grand Unified Theory (GUT), it is reasonable to extend the non-Abelian gauge group SU (2) L into a larger non-Abelian gauge group SU (3) L [1][2][3][4][5][6][7][8][9][10][11][12], where the U (1) Y is substituted by U (1) X . The X charges of Higgs sector are critical in generating the breaking of SU (3) L gauge symmetry [13,14]. Thus, it is necessary to find a natural way to obtain the representations of such particles instead of giving the representations by hand. In the 331 model proposed in [2,3,15], all the particle representations under the SU (3) C × SU (3) L × U (1) X gauge group could be obtained from some simple representations of the SU (6) group [15]. The extended Higgs sector brings exotic gauge bosons, in which the Z is constrained to be heavier than ∼ 4.5 TeV [16]. The exotic fermion sector also has rich new physical phenomenology, such as dark matter and neutrino masses and mixing that we have discussed in [3]. In this work, we will study how to explain the muon anomalous magnetic moment, g µ − 2, in this 331 model as well as the Higgs potential stability.
Although there is no doubt that the SM is consistent with most of the experimental data so far, the muon anomalous magnetic moment, a µ = gµ−2 2 , is a long-standing deviation [17][18][19][20][21]. Recently, the first results from the muon g-2 experiment at the Fermi-Lab has been announced. Combining with the previous results by the Brookhaven National Lab (BNL), the measured value deviates from the theoretical prediction of SM around 4.2σ and the discrepancy is given by [22] ∆a µ = a EXP µ − a SM µ = (2.51 ± 0.59) × 10 −9 .
In the rest of the paper, we will first review this 331 model briefly in Section II. The stability of Higgs potential is studied in Section III. In Section IV and Section V, the anomalous magnetic moments of muon and electron are explained, respectively. Finally, we conclude in Section VI.

II. THE MODEL
As the SU (3) C × SU (3) L × U (1) X gauge group of the 331 model proposed in [2,3,15] comes from a SU (6) gauge group, representations of the SU (6) group can be decomposed into representations of the SU (3) C × SU (3) L × U (1) X group to make up of the fermions of this 331 model, where we denote the left-handed (LH) and right-handed (RH) SM fermions respectively as e Li , ν Li , u Li , d Li , and e Ri , u Ri , d Ri , with i = 1, 2, 3 being a generation index. The U (1) X charges of the fermions are given according to that the U (1) X charge operator, denoted aŝ Q X , for the 6 representation of the SU (6) group iŝ Consequently, the U (1) EM charge operator can be expressed aŝ where v t is the vacuum expectation value (VEV) that breaks the SU (3) L × U (1) X gauge symmetry to the SU (2) L × U (1) Y gauge symmetry [42][43][44][45][46], and v u , v d are the VEVs which sequently break the SU (2) L × U (1) Y gauge symmetry to the U (1) EM gauge symmetry. It is known that v 2 d + v 2 u ≈ 246 GeV is required to give masses of the SM fermions and we define The Yukawa terms and Majorana mass terms of this 331 model are where M s , M s and M ss are 3 × 3 matrix. The most general Higgs potential in this 331 model is where V (4) contains all the quartic couplings of the Higgs potential. We can express B and The non-SM gauge bosons in this 331 model are W ± , Z , and V /V * . Details of the neutral and charged scalar mixings can be found in [3]. Eigenstates of the mixing of CP-odd scalars σ i (i = 1, 2, ..5) contain three massless Goldstone bosons a 1 , a 2 , a 3 , and massive (II.17) The mixing of CP-even scalars ρ i (i = 1, 2, ..5) gives four massive eigenstates h j (j = 2, 3, 4, 5) and massless

III. BOUNDEDNESS FROM BELOW
As there are three scalar triplets in this 331 model, it is necessary to find the criteria for the parameters to ensure a stable scalar potential. The relevant issues can be found in [47][48][49][50][51][52][53][54]. In this Section, we will derive a set of conditions to ensure the BFB of the potential in this 331 model.
According to the four solutions given above, we obtain that It is direct to get the expressions of M BFB i which we do not give here. The BFB is realized as long as that co-positivity constraints on the four M BFB i (i = 1, 2, 3, 4) are satisfied at the same time.
It is noted that the conditions, derived in this Section, are sufficient but not necessary to ensure the BFB of the potential in our 331 model since the left side in Eq. III.13 could be larger than the right side.

IV. g µ -2
In this 331 model, new contributions to the muon anomalous moment a µ arise from loop diagrams involved the BSM fermions, scalars, and gauge bosons. According to [16], |v t | is required to be larger than 10 TeV to satisfy the constraint M Z > 4.5 TeV, which also makes M V ≈ M W > 3.2 TeV. So the contributions to a µ involving the non-SM gauge bosons are negligible due to the heavy masses of these gauge bosons. We also neglect the contributions given by the non-SM charged scalars and exotic neutrinos since in most viable parameter space they are very heavy. So, we only focus on the contributions induced by the neutral scalars.
The contributions to a µ involving neutral scalars come from the following parts of the Lagrangian of this 331 model, Supposing that y e ij and y L ij are proportional to δ ij for simplicity, we define where h i (a j ) stands for the eigenstate of the CP-even (CP-odd) scalar mixing, V h (V a ) is the corresponding matrix for the mixing, and Note that h 1 and a j (j = 1, 2, 3) are not included in the summations in Eq. IV.4 because they are Goldstone bosons. The f LR (x) increases with x and satisfies It is know that y µ is related to the mass of µ by According to Eq. II.16, Eq. II.17, and Eq. II.18, we obtain that Considering Eq. IV.6 to Eq. IV.10, we can estimate that where |v t | > 10 TeV has been used. The contribution involving muon and the non-SM scalar is shown in the upper right panel of FIG. 1, which can be expressed as where h 2 (the Higgs boson) is not included because the relevant contribution is not BSM, and , the magnitude of ∆a µ2 can be estimated as which will be about ∼ 0.4 × 10 −13 if we choose that tan θ = 6 and m h i /a j = 600 GeV. We conclude that it is impossible to account for the muon anomalous moment with the scalars now available. To account for the muon anomalous moment in this 331 model, we introduce another two scalar triplets, T and T d , which are in the same representation with T (T d ) but have no VEVs Very similar to [55], the following terms can be added in the Lagrangian of this 331 model It should be emphasized that we do not add all the gauge invariant terms involving T and T d for simplicity. New mixing of neutral scalars arises. The mass matrix and the mixing matrix satisfy where +A v u / √ 2 in the mass matrix in company with U 1 is for the mixing of ρ 7 and ρ 8 (also for the mixing of σ 6 and σ 9 ), and −A v u / √ 2 in the mass matrix together with U 2 is for the mixing of ρ 6 and ρ 9 (also for the mixing of σ 7 and σ 8 ). It is direct to obtain the physical masses and mixing, which are Supposing that λ e ij and λ L ij are proportional to δ ij for simplicity, we define λ e 22 = λ µ , (IV.21) Similarly, the leading contributions involving µ and neutral scalars from T and T d are the chirally-enhanced ones shown in the lower left panel of FIG. 1, which can be expressed as is used. The contribution involving muon and neutral scalars from T d is shown in the lower right panel of FIG. 1, which can be expressed as is used. Based on the above discussions, we only need to consider contributions ∆a µ3 and ∆a µ4 , and thus have ∆a µ ≈ ∆a µ3 + ∆a µ4 . (IV.25) To study the dependences of ∆a µ on λ µ , λ µ , and m µ , we set other parameters to that m 4 = 300 GeV, m 5 = 600 GeV, A = 10 GeV, and tan θ = 6. In FIG. 2, points in the orange (blue) region on the m µ − λ µ plane give values of ∆a µ within 2σ (1σ) deviation from the measured value.
In the upper two panels of FIG. 2 where λ µ = λ µ , it is shown that a µ4 /a µ3 increases with m µ and ∆a µ4 dominates (a µ4 /a µ3 > 1) when m µ is larger than 1 TeV. When m µ is larger than 5.9 TeV, ∆a µ4 contributes more than 99% of ∆a µ and both the 1σ and 2σ regions have no dependence on m µ since ∆a µ4 is independent of m µ . In the m µ >5.9 TeV region, λ µ = λ µ within the 1σ and 2σ regions needs to be in the ranges of (1.92, 2.43) and (1.60, 2.65), respectively. ∆a µ3 dominates (a µ4 /a µ3 < 1) when m µ is less than 1 TeV and the allowed λ µ = λ µ within the 1σ and 2σ regions increases with m µ .
In the lower panel of FIG. 2 where λ µ = −λ µ , ∆a µ4 is positive while ∆a µ3 is negative. Both the 1σ and 2σ regions need to ensure that m µ > 1 TeV to make ∆a µ > 0. In the 1σ region, λ µ = −λ µ decreases with m µ since the negative ∆a µ3 approaches zero when m µ increases. Similarly, ∆a µ4 contributes almost the whole ∆a µ (−∆a µ3 /∆a µ4 < 0.01) when m µ is larger than 5.9 TeV and the 1σ and 2σ regions are approximately independent of m µ .
In principle, we can add extra terms like −λ e ij f i Xf c j T − λ L ij f i Xf c j T d to account for the ∆a µ without introducing T and T d . However, the VEV of T (T d ) will lead to mixing between µ L and µ L (µ R and µ R ), which can bring about serious fine tuning problem. So we introduce T and T d with no VEVs to account for the ∆a µ in this Section. Much like the discussions for g µ -2, the non-SM particles in this 331 model (with T and T d included) also induce new contributions to the electron anomalous moment a e . Similarly, we define There is no doubt that the leading contributions to a e can be expressed as ∆a e ≈ ∆a e3 + ∆a e4 , (V.5) In the left panel of FIG 2 where λ e = −λ e , ∆a e3 is negative while ∆a e4 is positive. The 1σ and 2σ regions, which need to give negative value of ∆a e , must be at the left side of the line where −∆a e4 /∆a e3 = 1. So the allowed m e within the 1σ (2σ) region has a largest value, which is 4.1 TeV (5.6 TeV) when λ e = −λ e = 3.
Because ∆a e3 and ∆a e4 are both positive in the right panel of FIG 2 where λ e = λ e , the 1σ and 2σ regions disappear. Since ∆a e4 = 1.1 × 10 −13 when λ e = 3, the region where λ e < 3 is always within the 3σ region as long as ∆a e4 dominates (|∆a e4 /∆a e3 | > 1).

VI. CONCLUSION
We derive a set of sufficient conditions to ensure the boundedness from below of the potential in the 331 model proposed in [3] which has three scalar triplets. Since the quartic couplings V (4) are more complex than those in [54], inequalities have to be used during the derivations which makes that the conditions obtained are sufficient but not necessary to ensure the BFB of the potential.
We focus on the BSM contributions to ∆a µ and ∆a e involving neutral scalars and charged fermions. The analysis shows that the contributions induced by the neutral scalars from T and T d are too small to account for the muon and electron anomalous moments unless there are some serious fine tuning problems. So another two triplets T and T d with non VEVs are introduced to provide new couplings to fermions. With the neutral scalars from T and T d , the leading contributions to ∆a µ (∆a e ) involve both the BSM µ (e ) and µ (e) from the SM. The chirally-enhanced contributions dominate the contributions from µ (e ), whose magnitudes decrease with m µ (m e ), while the contributions from µ (e) are independent of m µ (m e ). The former dominates over the latter when m µ < 1 TeV (m e < 7.5 TeV) if the other parameters are set as the Section IV.