Model with a gauged lepton flavor SU(2) symmetry

We propose a model having a gauged $SU(2)$ symmetry associated with the second and third generations of leptons, dubbed $SU(2)_{\mu\tau}$, of which $U(1)_{L_\mu-L_\tau}$ is an Abelian subgroup. In addition to the Standard Model fields, we introduce two types of scalar fields. One exotic scalar field is an $SU(2)_{\mu\tau}$ doublet and SM singlet that develops a nonzero vacuum expectation value at presumably multi-TeV scale to completely break the $SU(2)_{\mu\tau}$ symmetry, rendering three massive gauge bosons. At the same time, the other exotic scalar field, carrying electroweak as well as $SU(2)_{\mu\tau}$ charges, is induced to have a nonzero vacuum expectation value as well and breaks mass degeneracy between the muon and tau. We examine how the new particles in the model contribute to the muon anomalous magnetic moment in the parameter space compliant with the Michel decays of tau.


I. INTRODUCTION
A long-standing anomaly in flavor physics that has inspired pursues of physics beyond the Standard Model (SM) is the muon anomalous magnetic moment [1]. A 3.6σ discrepancy between theory calculations within the SM and experimental data [2]: ∆a µ ≡ a exp µ − a SM µ = 288(63)(49) × 10 −11 (1) has been reported 1 . One solution to this problem is to invoke a gauge boson with a sufficiently muon-philic coupling. Such a scenario can be realized in a gauged, anomaly-free U (1) Lµ−Lτ model, as proposed in Refs. [5,6,17] and further examined in Refs. [8,9]. It also features in the ability to explain the neutrino mixing data [11][12][13]. The phenomenology of such a model at the LHC has been studied in Ref. [10]. It is recently analyzed that the Z boson with a mass in the range of O(1 − 100) MeV can explain ∆a µ with a fairly small coupling ∼ 2 × (10 −4 − 10 −3 ), while being consistent with constraints such as the big bang nucleosynthesis bound on the effective number of extra neutrinos ∆N eff ≤ 1 [14] and the neutrino trident production [15]. The remaining parameter space can be soon probed or ruled out by future searches [16].
In this work, we propose a non-Abelian gauged flavor symmetry, SU (2) µτ , between the second and third generations of the SM leptons 2 . After symmetry breaking induced by the vacuum expectation value (VEV) of a scalar SU (2) µτ doublet, the gauged flavor group has three massive gauge bosons, denoted by X 3,± , with X 3 corresponding to the U (1) Lµ−Lτ subgroup and X ± coupled with currents that exchange the muon and tau numbers by one unit. To break the mass degeneracy between µ and τ as otherwise required by the new flavor symmetry, we further introduce a set of scalar fields that transform as a triplet under SU (2) µτ and a doublet under the SM SU (2) L .
An intriguing property of the model is the muon number conservation, respected by the new gauge interactions and Yukawa couplings of the muon and tau with the new scalar bosons. As a consequence, no lepton flavor-violating processes are allowed in the model. 1 Other analyses give somewhat different estimates for the discrepancy, such as ∆a µ = (261 ± 80) × 10 −11 in Ref. [3]. Ref. [4] has some in-depth discussions about ∆a µ in various scenarios that mainly differ in the treatment of leading-order hadronic uncertainties and whether τ data are included. In our numerical analysis, we use the result quoted in Eq. (1). 2 This kind of extensions was briefly mentioned in some earlier works [17,18] Another feature of the model is that no Z-X 3 mixing is allowed at tree level according to the proposed symmetry breaking pattern.
To comply with the current 125-GeV Higgs data, we will consider a nearly alignment limit of this model where the Higgs couplings to the SM particles are SM-like. We compute the contributions of the new gauge bosons and scalar bosons to ∆a µ and the Michel decays of τ → ν τ ν , where = µ, e. To better understand contributions of each set of new particles, we divide the numerical analysis to effects due to purely new gauge bosons, purely lepton flavor-changing scalar bosons, and purely lepton flavor-conserving scalar bosons. We scan for the parameter space that can accommodate ∆a µ while being allowed by the constraint from lepton universality in the Michel decays of τ .
The layout of this paper is as follows. In Section II, we introduce the model, discussing the extended SU (2) µτ gauge sector, scalar sector, and lepton sector. We show in Section III how various new particles can contribute to the muon g − 2 and explain the current discrepancy between experimental data and the SM expectation. Section IV discusses the constraint on the model from the Michel decays of τ → ν τ ν . An extension of the model to explain neutrino mass and a discussion of collider phenomenology are given in Section V. Section VI concludes with a summary of our findings in this work. We collect detailed formulas of scalar field mass matrices in Appendix A, new gauge interactions of µ and τ in Appendix B, and new Yukawa interactions of µ and τ in Appendix C.

II. MODEL OF GAUGED SU (2) µτ
In addition to the SM gauge group SU (3) C × SU (2) L × U (1) Y , the model considered in this work has a gauged SU (2) µτ symmetry, which is the lepton flavor symmetry within the second-and third-generation leptons. To break the SU (2) µτ symmetry, we introduce a scalar field S, which is a singlet under the SM gauge group and a doublet under the gauge flavor symmetry. Moreover, to break the degeneracy in the mass of the second-and third-generation leptons, the model also has another new scalar field Φ that transforms as an SU (2) L doublet and an SU (2) µτ triplet. The new scalar fields are all colorless. Quantum numbers of the lepton and scalar fields under the SU (2) µτ , SU (2) L and U (1) Y groups are listed in Table. I. Also explicitly shown are the SU (2) µτ indices α, β ∈ {1, 2}. In this paper, we reserve the first few Greek letters for the flavor indices of SU (2) µτ and the others for the Lorentz indices. We note in passing that with the above-mentioned particle content, the model is free of gauge anomalies. 3 The most general scalar potential of Φ 0 , Φ and S is given by where λ ΦS , λ 5 , κ, λ µ , λ µ1 and λ µ2 are generally complex parameters. Hereafter, we take all of them to be real for simplicity. The SU (2) L indices are shown explicitly by i, j. The flavor adjoint Higgs doublet Φ is parameterized as 3 The Witten anomaly is known to occur if an SU (2) model contains only an odd number of doublets charged under it. In our SU (2) µτ model, this anomaly disappears when we introduce the flavor-doublet right-handed neutrinos, which are required to generate active neutrino mass by the seesaw mechanism, to be discussed in Section V.
where a = 1, 2, 3, and the SU (2) µτ flavor space is denoted by the square brackets. We assume that the SM singlet field S develops a VEV at an energy higher than the electroweak scale.
Without loss of generality, we parameterize the S field as where S − ≡ (S + ) , S z 0 will be identified as a would-be Nambu-Goldstone (NG) boson corresponding to the longitudinal mode of the new flavor gauge boson X λ 3 , and the superscripts h, z here denote respectively the real and imaginary components of the field. The VEV of S breaks the SU (2) µτ symmetry completely. The flavor-singlet Higgs doublet field Φ 0 is assumed to develop a VEV at the electroweak scale, and can be parameterized as The flavor adjoint Higgs doublet Φ is required to have VEV's as well in order to generate mass splitting between µ and τ . In general, the VEV's of Φ can be induced by those of S and Φ 0 . In particular, the VEV's of Φ ± will contribute to the off-diagonal elements of the charged lepton mass matrix and make the vacuum structure more complicated. For simplicity, we consider that only Φ 3 is induced to have a VEV, φ 0 3 = v 3 / √ 2, as the easiest way to generate the mass splitting. In this case, we have v 2 = v 2 0 + v 2 3 = (246 GeV) 2 and tan β ≡ v 0 /v 3 . To realize this simple setup, we set λ ΦS = λ µ1 = λ µ2 = 0 in the following discussions. 4 We then parametrize the flavored Higgs and scalar doublet fields as where we take the phase convention that φ − ∓ ≡ (φ + ± ) * . Note that in the limit where S does not develop a VEV, the flavor-diagonal VEV of Φ induced by Φ 0 would also break SU (2) µτ to U (1) Lµ−Lτ . 5 The scalar bosons in this model are thus classified according to their (i) muon numbers, (ii) CP parities, and (iii) electric charges. 4 These terms can also be removed by imposing a Z 3 charge "ω" (or a new U (1) charge) for the S field. 5 The global "muon number" symmetry L µ is realized and rearranged since L µ S = 0 and [ L µ , Φ ] = 0, The mass eigenstates are defined in terms of the following field rotations: where ω ± and z are the would-be NG bosons corresponding to the longitudinal modes of the Z λ and W ± λ bosons, and with the notation (c i , s i ) ≡ (cos θ i , sin θ i ) for i = 1, 2, 3. Moreover, the flavor off-diagonal Higgs bosons are given by where tan β ≡ 2v 3 /v S . Details of the mass matrices are given in Appendix A.
We denote the SU (2) µτ gauge fields by while the global "tau lepton number" L τ is not conserved in the scalar sector because L τ S = 0, where The eigenvalues are calculated as Therefore, the subscripts ±, 0 (3) denote the muon number ±1, 0 (0), respectively, in the broken phase.
Again the subscripts ± associated with the new gauge bosons do not represent their electric charges. In fact, all the new gauge bosons are electrically neutral. The gauge interactions of matter fields are encoded in their kinetic terms involving the covariant derivative where D λ is the covariant derivative in the SM, and g X denotes the associated gauge coupling strength. The masses of new gauge bosons are generated by the VEV's of Φ and S as This shows that X ± is slightly heavier than X 3 . But as long as v 3 v S , as assumed here, we can safely neglect this small difference and consider the new gauge bosons virtually degenerate in mass ∼ M X . No mass mixing occurs between X 3 and the SM Z boson because only Φ couples to both SU (2) L and SU (2) µτ gauge bosons and the VEV of Φ breaks SU (2) µτ to U (1) Lµ−Lτ associated with X 3 . The latter point is also reflected in the fact that the mass of X 3 does not receive the contribution of v 3 , as shown in Eq. (14). Therefore, the model does not have Z-X 3 mixing at tree level. The new gauge interactions for µ and τ are given There are two kinds of SU (2) µτ -invariant Yukawa interactions for µ and τ : The explicit form of the Yukawa interactions for µ and τ in the scalar mass eigenbasis is given in Appendix C. It is observed that in the limit of sin(β − α) 1 and large tan β, where α ≡ α 3 , the SM-like Higgs couplings to µ and τ are SM-like. We note in passing that the Φ α β and S α fields do not directly couple with quarks due to their nontrivial SU (2) µτ quantum charges. For definiteness, we will make the assumption of α 1 = α 2 = 0 for the mixing angles. Therefore, in this nearly alignment limit, the couplings of the SM-like Higgs boson h with the quarks and electron are SM-like, while the extra CP-even Higgs boson H are suppressed by a factor of cos(β − α) − sin(β − α)/ tan β in the large tan β case.
In summary, we are considering in this model the alignment limit of sin(β − α 3 ) 1 and large tan β in order to be consistent with the current Higgs data, and taking the mixing angles α 1 = α 2 = 0 for simplicity. For concreteness, we assume mass degeneracy for heavy scalar bosons unless we explicitly vary some of them. Deviations from the above are not further investigated as the muon g − 2 can already be accommodated within this parameter space. We note that electroweak precision tests and other constraints, particularly those that could arise from significant nonzero couplings between the scalars and quarks, could be relevant in the general case.
We are now ready to discuss new contributions to the muon anomalous magnetic moment in this model. From the interactions listed in Appendices B and C, we expect three kinds of potentially large contributions from: (i) new gauge bosons X λ 3 and X λ ± , (ii) flavor-changing scalar bosons h ± and H ± , and (iii) flavor-conserving scalar bosons H and A. To evaluate the contributions from the scalar bosons, we impose the relations

Effects of new gauge bosons X λ
3 and X λ ± : Contributions from the new gauge bosons are calculated as where ε B ≡ M 2 /M 2 B and ρ ≡ M τ /M µ 16.82. The X 3 contribution is identical to the one in the U (1) Lµ−Lτ model, and is approximately given by ∆a µ (X 3 ) ≈ (g 2 X /48π 2 )ε µ X for ε µ X 1. The sign of this new contribution is favored by the observed muon g − 2 anomaly.
The X ± contributions are (6ρ − 4) 96.9 times larger than that of X 3 due to the chirality flipping effect in the same limit. 6 Thanks to the muon number conservation, no lepton flavor violation is generated by the new gauge bosons. Fig. 1 shows the regions in the (M X , g X ) plane that can accommodate the muon g − 2 at the 2σ level (i.e., 12.8 < ∆a NP µ × 10 10 < 44.8) with the contributions of the X bosons. The orange (blue) region corresponds to the contribution from X 3 (X ± ) alone. Since ∆a µ (X ± ) ∆a µ (X 3 ), there is almost no change in the preferred parameter region from the blue region when contributions from all the X bosons are taken into account. 6 In Ref. [19], phenomenological consequences of a new gauge boson with some properties similar to our X ± have been discussed. However, the muon number is not conserved in their interactions.
For ε B 1 and v v S , the sum of the two contributions is approximated by where The correction has an overall enhancement for large tan β. The third term in the curly brackets also receives a chirality flipping enhancement similar to the X ± contribution. Since this is proportional to ρ, the mass ratio of m τ to m µ , it dominates in a wide region of parameter space. Moreover, the sign of the contribution can be made positive by choosing an appropriate mixing angle α . The flavor-changing scalar interactions also respect the muon number conservation, so that lepton flavor violation is forbidden in this model. 7 Effects of the flavor-changing Higgs boson in the type-III two-Higgs-doublet model have been studied in Ref. [20]. The scalar contributions discussed here is induced by Φ ± that does not develop a VEV.    The left, middle, right plots are the cases with α = π/4, 0.26π, π/2, respectively.
In Fig. 2, the contributions to the muon g − 2 from the flavor-changing h ± and H ± bosons are evaluated, with all colored regions satisfying the muon g − 2 data at 2σ level.
For definiteness, we have taken β = 0 (corresponding to the large v S limit) in numerical evaluations. The left, middle, right plots are drawn for the cases of α = π/4, 0.26π, π/2, respectively. Comparing the left and middle plots, we observe that the correction has a more sensitive dependence on α around π/4. In the upper row, the blue, orange, and green regions correspond respectively to the choices of tan β = 10, 30, 100. For α π/4, the mass of h ± is preferred by data to fall in the regime of a few tens to a few hundreds of GeV and has less dependence on the mass ratio R when it is smaller than 1. For α = π/2 in contrast, M h ± is of O(0.1 − 1) TeV unless R > 1. The peculiar behavior of the allowed regions around R = 1 in this case originates from the sign change of the last term in the curly brackets of Eq. (20). This effect is less prominent in the other two plots because c 2α ≈ 0.
In the lower row of Fig. 2, the blue, orange, and green regions correspond respectively to the choices of M h ± /M H ± = 1, 0.9, 0.1. In the limit R → 1, where α becomes undefined, the contributions with dependence on α vanish. In the large tan β regime of interest to us, the contribution to the muon g − 2 simply scales as tan 2 β.

Effects of flavor-conserving scalar bosons H and
It is interesting to note that each of the integrals in the above two expressions is divergent In both large and small tan β limits, 1/t 2β and 1/s 2 β become large. In the former case of interest to us, they interfere destructively. If the contribution from the chirality flipping term in Eq. (20) is small, the effects of flavor-conserving scalar bosons can be comparable to those of flavor-changing scalar bosons. However, we will show in the next section that such parameter space is ruled out by the Michel decays.
In Fig. 3, we draw regions that can accommodate the muon g − 2 anomaly at 2σ level.
The left plot shows a result similar to the analogous plot in Fig. 2 for α 0.26π. As noted above, ∆a µ gets an enhancement in either tan β → 0 or tan β → ∞ limit, thus the turning behavior in the allowed regions shown in the right plot.

IV. MICHEL DECAYS OF τ
The Michel decays of the tau lepton can be modified by both X ± and H ± bosons at tree level. We consider the ratio as the parameter showing violation in µ/e universality. The measured value R exp τ →µ/τ →e = 0.979 ± 0.004 is compared to the SM prediction of 0.9726 with a comparatively negligible uncertainty [2].
For definiteness, we consider two limiting scenarios: M H ± M X /g X and M H ± M X /g X . In the former case, the τ → µν µ ν τ decay gets an additional contributing amplitude mediated by the X ± gauge boson while the τ → eν e ν τ decay does not. With the definition we find where As numerically ρf (ρ −2 ) g(ρ −2 ), we observe constructive interference between the W and The experimental constraint of η X < 1.011 (1.015) at 1σ (2σ) level is shown in light yellow (light green) in Fig. 4. As shown in the plot, the difference between the 1σ and 2σ bounds is very small. It is seen that the entire parameter region favored by the muon g − 2 anomaly is excluded. Therefore, it is impossible to explain the muon g − 2 anomaly while being consistent with the lepton universality in the Michel decays of τ with the X gauge bosons alone.
In the limit of M H ± M X /g X , we add only the contribution of H ± to the SM amplitude and obtain where we take the M e → 0 limit. Note that the interference term is further suppressed by 1/ρ. Thus, the bound from the τ Michel decays is expected to be weak.
The constraint of η H ± < 1.011 (1.015) at 1σ (2σ) level along with the ∆a µ favored regions are drawn in Fig. 5 for the cases of α = π/4 (left plot), 0.26π (middle plot), and π/2 (right plot). For definiteness, here we assume the mass relations 8 Only the contributions of flavor-changing scalar bosons to muon g − 2 are taken into account because of the additional chirality enhancement and logarithmic enhancement seen in Eq. (20). We have also shown the direct search constraint on H ± , which would be expected to be the same as that from the slepton search from the electroweak production in the massless neutralino limit. We show here the left-handed selectron and smuon search lower mass bound of 300 GeV in Ref. [22] and the latest slepton mass lower bound of 500 GeV (this bound would be an overestimate for our purpose since both left-and righthanded sleptons of all three flavors are summed over) [23]. Such a direct search bound can 8 The parameter choice, M 2 H ∼ M 2 A ∼ M H ± , is naturally realized in the decoupling limit, where the new mass scale in the Higgs sector controls their masses. This choice is also favored by the electroweak precision data, where their contributions to S and T parameters vanish in the degenerate mass limit [21]. be relaxed if the H ± → HW ± , AW ± decay modes are allowed when sufficient mass splitting exists among the particles.
We note in passing that if M X /g X and M are comparable and therefore all X ± , h ± , and H ± can contribute to the muon g − 2 and the Michel decays of τ leptons, then it is possible to find some more allowed parameter space in the (M X , g X ) plane and the (M, tan β) plane.
To see this cooperative effect clearly, we make plots in the (M, M X ) plane for fixed (g X , α ) in Fig. 6 for several values of tan β. In these plots, all the contributions from the new gauge bosons and scalar bosons are included in the calculations. The choice of g X = 10 −2 and α = π/2 is taken as an example. A change in the value of g X would only result in a different range of M X . We set the mixing angle α = π/2 in oder to show the maximal contribution to the muon g − 2. By varying α , one would obtain a smaller contribution. From Fig. 6, we see that the contribution from the flavor-changing scalar is essential, thanks to the chirality flip and logarithmic enhancements. The logarithmic enhancement reflects in the preference of a larger mass hierarchy between H ± and h ± . A comparison among the three plots indicates that M X needs to be greater than a few tens of GeV and that more parameter space is allowed for larger tan β.

V. NEUTRINO AND COLLIDER PHENOMENOLOGY
The neutrino mass in the model can be generated through the usual Type-I seesaw mechanism. The relevant terms for neutrino mass are given by where Φ 0 ≡ iσ 2 Φ * 0 , N R and N α R are the right-handed SU (2) µτ singlet and doublet neutrinos, respectively. Here we introduce another scalar field Σ β γ which is a SM singlet and SU (2) µτ triplet that develops a VEV, v Σ , as induced by that of S. 9 The Dirac mass and the Majorana mass matrices are given respectively by The structures of these mass matrices are exactly the same as those discussed in Ref. [24].
Note that the "muon number" of N R can be defined through the λ eµ coupling. However, the muon number is softly broken by the M ee term. With such an extension in the neutrino sector, the muon number is no longer a conserved charge.
Here we make some comments about collider searches of the new gauge bosons and scalar bosons in the model. Since the X 3,± gauge bosons couple to µ, τ and neutrinos, they can be produced at the CERN LHC in association with Drell-Yan µ, τ or ν pairs from finalstate bremsstrahlung, followed by leptonic decays and leading to multi-lepton final states (without/with missing energy) [10]. The relation of the lepton invariant mass M µµ ∼ M τ τ ∼ M X would be a first step to confirm the diagonal subgroup of SU (2) µτ . In addition, our model predicts a peculiar decay channel X ± → µτ with M µτ ∼ M X . This signal is the smoking-gun signature of the proposed model. 10 The beam dump experiments [16,26,27] can also probe the X 3 boson through the γ-X 3 mixing at loop level, but not for X ± . The lepton flavor-conserving and -changing scalar bosons can also be searched for through the 9 Here the diagonal VEV of Σ is assumed to be induced in a way similar to that of Φ, as can be arranged with a similar setup in the scalar potential. 10 A similar process for flavon has been discussed in the literature [25]. multi-lepton final states, with the obvious changes in coupling and mass. In addition, since some of these scalar bosons are charged under the SU (2) L , they can be pair produced directly on shell if the collider energy is sufficiently high. The signals from the flavor-conserving scalar bosons have shared a common feature with the Type-X two-Higgs-doublet model [28][29][30][31], the muon-specific two-Higgs-doublet model [32], and the A 4 symmetric neutrino mass models [33][34][35].

VI. SUMMARY AND DISCUSSION
In this work, we propose a model that extends the Standard Model (SM) gauge group with the SU (2) µτ symmetry that transforms between the second and third generations of the SM leptons. To break such a symmetry and generate mass splitting between the µ and τ leptons, we introduce two exotic scalar fields in addition to the SM Higgs doublet. Since the new gauge bosons and the flavor-violating scalar bosons mediate the τ → µν τ ν µ decay but not the τ → eν τ ν e decay, we also discuss the constraint from the measured lepton universality ratio. We find that if only the new gauge bosons are considered, the lepton universality constraint completely rules out the parameter region favored by the muon g − 2.
If only the flavor-changing scalar bosons are included, on the other hand, a significantly large parameter space is still allowed by both data. Current search limits of the charged Higgs boson rule out part of the region. The mass matrices for the scalar fields associated with the SU (2) µτ symmetry are decomposed as where the electric charges, CP-parity, and lepton flavor property of each set of scalar fields are summarized in Table II. The mass-squared matrices in Eq. (A1) are given by where M 2 is defined as (M 2 /v 2 )v 3 v 0 ≡ λ µ v 2 S /4, and λ 345 ≡ λ 3 + λ 4 + λ 5 . The parameters µ 2 S , µ 2 0 and m 2 Φ are eliminated by vacuum conditions. The unitary matrix V is given by where 1/t 2β = (1 − t 2 β )/2t β , 1/s 2β = (1 + t 2 β )/2t β , and the Yukawa coupling constants are determined by the charged lepton masses as We note that the requirement of perturbativity for the Yukawa coupling |y| < 4π leads to a lower bound on v 3 : The flavor off-diagonal Yukawa interactions, on the other hand, are given by