A model with flavor-dependent gauged $U(1)_{B-L_1}\times U(1)_{B-L_2-L_3}$ symmetry

We propose a new model with flavor-dependent gauged $U(1)_{B-L_1}\times U(1)_{B-L_{2}-L_{3}}$ symmetry in addition to the flavor-blind one in the standard model. The model contains three right-handed neutrinos to cancel gauge anomalies and several Higgses to construct the measured fermion masses. We show the generic feature of the model and explore its phenomenology. In particular, we discuss the current bounds on the extra gauge bosons from the K and B meson mixings as well as the LEP and LHC data and focus on their contributions to the lepton flavor violating processes of $\ell_{i+1}\to \ell_i\gamma$ (i=1,2).


I. INTRODUCTION
It is known that two vector U(1) gauge bosons often appear in grand unified theories (GUTs) such as SO(10) gauged symmetry [1] when it spontaneously breaks down, in which a flavor-blind gauged U(1) B−L can be naturally induced along with right-handed neutrinos. On the other hand, the flavor-dependent U(1) gauged symmetries are one of the promising scenarios to explain several anomalies beyond the standard model (SM), such as semi-leptonic decays involving b → sℓl, muon anomalous magnetic moment, and so on [2]. This paper is organized as follows. In Sec. II, we first construct our model by showing its field contents and their charge assignments and then give the concrete renormalizable Lagrangian with scalar and vector gauge boson sectors. After that, we discuss the phenomenologies, including the interaction terms, the bounds from the K and B meson mixings, the LEP [4] and LHC [5] experiments, and lepton flavor violations (LFVs). In Sec.
III, we perform the numerical analysis. In Sec. IV, we extend our model to explain several anomalies indicated from the current experiments. Finally, we conclude in Sec. V with some discussions.

II. MODEL SETUP AND PHENOMENOLOGY
First of all, we impose two additional U(1) B−L 1 × U(1) B−L 2 −L 3 gauge symmetries by including three right-handed neutral fermions N R 1,2,3 with the subscripts representing the family indices. The field contents of fermions (scalar bosons) are given in Ta-  Fermions cause the two additional charge assignments are orthogonal each other. In Table II, H 1 is expected to be the SM Higgs, while H 2 is another isospin doublet scalar boson, which plays a role in providing the mixings of the 1-2 and 1-3 components in the CKM matrix, as we will see below. Under these symmetries, the renormalizable Lagrangian for the quark and lepton sectors and scalar potential are given by respectively, whereH ≡ (iσ 2 )H * with σ 2 being the second Pauli matrix, and i runs over 2 to 3.

Scalar sector:
The scalar fields are parameterized as with all four CP-odd bosons z 1,2,ϕ 1 ,ϕ 2 massless, in which three of them are absorbed by vector gauged bosons Z SM , Z ′ and Z ′′ , respectively, where Z SM ≡ (g 2 The feature of the singly charged bosons is same as the typical two-Higgs doublet model. Consequently, the mass-squared, mixing and eigenvalue-squared matrices are found as respectively, where the above massless eigenstate is absorbed by the SM gauge boson W ± , and c β (s β ) = cos β(sin β) with tan β ≡ v 1 /v 2 . As for the CP-even sector in the basis of 1 We remark that the dangerous physical Goldstone boson from H 2 can be evaded by introducing an isospin singlet boson ϕ 3 of (-1/3,1/3) under U (1) B−L1 × U (1) B−L2−L3 , resulting in additional terms (H † 1 H 2 )ϕ 3 and ϕ * 1 ϕ 2 ϕ 3 3 /Λ that give the non-vanishing CP-odd mass. Here, Λ is the cut-off scale, expected to be O(100) TeV at most. Then, the CP-odd Higgs mass with O(100) GeV is found. Even though ϕ 3 affects the vector gauge boson masses, we neglect the contribution hereafter, by assuming v ′ ϕ3 << v ′ ϕ1,2 . Note here that ϕ 3 does not contribute to the fermion masses.
[h 1 , h 2 , ϕ R 1 , ϕ R 2 ] t , we get a four-by-four mass matrix squared M 2 R , which can be diagonalized by the mixing matrix Here, we identify H 1 ≡ h SM . Fermion sector: The SM Dirac fermions are diagonalized by bi-unitary mixing matrices as D u,d,e = (U u,d,e ) L m u (U † u,d,e ) R , and the active neutrinos are derived by an unitary mixing matrix as D ν = U * ν m ν U † ν , while the observed mixing matrices can be defined by V CKM ≡ U † uL U dL , and V M SN ≡ U † ν U eL , respectively [3]. However, we impose U uL = 1 for simplicity. Hence, we reduce the formula to be V CKM ≡ U dL . In the lepton sector, we classify the case of Here, the neutrino mass matrix m ν is induced via the canonical seesaw mechanism in Eqs. (2) and (3). where Here, we can identify the mass of Z 1 as the SM one, since we expect v 2 << v 1 < v ′ 1,2 in order to reproduce the SM fermion masses and the LEP measurement of m Z 1 ∼ m Z SM . This approximation is in good agreement with the current experimental data as the mass difference between m Z SM and m Z 1 should be less than O(10 −3 ) GeV.
The other part can be reduced to be which is diagonalized by the two-by-two mixing matrix Note here that we have to satisfy the following condition: that arises from the vector boson masses to be positive real.
Here, we evaluate the typical scale of v 2 that should be suppressed by the deviation of where δm Z should satisfy δm Z 2.1 × 10 −3 GeV from the electroweak precision test. As a result, we find e.g., v 2 19.5 GeV for v ′ 1,2 ∼ 10 5 GeV and g ′ 1,2 ∼ 10 −3 . Interacting Lagrangian: The interactions in the kinetic term between the neutral vector bosons and quarks in terms of the mass eigenstates are given by where we have used the central values for the CKM elements in V CKM [6]. While the interactions between the neutral vector bosons and charged-leptons depend on the parame- and respectively, by taking the best fitted results in ref. [6] for V M N S .

B. Phenomenology
Since Z ′ 1,2 interact with the SM fermions in a non-universal manner as discussed before, the constraints are unlikely to be the same as those in the typical U(1) B−L models. Here, we will examine the bounds on the extra gauge bosons from the K and B meson mixings as well as the LEP data, and discuss the lepton flavor violating processes of ℓ i+1 → ℓ i γ (i=1 and 2). for the mass splittings are given by [7]  2. Bounds on Z ′ 1,2 from LEP and LHC From Eqs. (20) and (21), we obtain the effective Lagrangians as As a results, the bounds for Z ′ 1,2 from the measurements of e + e − → ff at LEP [4] and qq → eē(µμ) at LHC [5] are found to be and where f = e, µ, τ, d and u. It is worthwhile mentioning that these neutral gauge boson searches will be carried out by experiments such as International Linear Collider (ILC) [8], and more stringent constraints should be obtained in the near future.

Lepton flavour violating processes
For V M N S ≈ U † ν , one does not need to consider the lepton flavor violations from the Z ′ 1,2 mediations, because the charged-leptons are diagonal from the beginning. On the other hand, if V M N S ≈ U eL , the lepton flavor violating processes due to Z ′ 1,2 can be induced. In this case, we get that where G F and α em are is the Fermi and fine structure constants, respectively, while C µe ≈ 1, C τ e ≈ 0.1784 and C τ µ ≈ 0.1736. The current experimental limits are given by [9,10]: These constraints are imposed in the numerical analysis below. 2 2 One can consider the anomalous magnetic moment because of evading the stringent constraint of the trident production via the Z ′ boson (flavor eigenstate) [12]. In our case, its value is of the order 10 −14 , which is much smaller than the experimental value.

III. NUMERICAL ANALYSIS
In our numerical analysis, we explore the allowed gauge parameters of g ′ 1,2 and m Z ′ 1,2 by taking s θ = 0 and s θ = 1/ √ 2. We scan the parameter regions as follows:  fig. 2 are similar to those in fig. 1. While g ′ 2 is restricted to be g ′ 2 ≤ 0.2, the allowed regions for g ′ 1 -g ′ 2 and m Z ′ 1 -m Z ′ 2 are more degenerate than the case of V M N S ≈ U † ν for s θ = 1/ √ 2. For the lepton flavor violating processes at the bottom in fig. 2, we see that BR(µ → eγ) reaches the current experimental bound in Eq. (33), which is clearly testable in the near future for both cases of s θ . However, BR(τ → µγ) is much lower than the limit in Eq. (33). Note here that BR(τ → eγ) ≈ BR(τ → µγ).
We remark that the current bounds on the masses of the extra gauge bosons are around 3 TeV by the LHC experiments [11] 3 , consistent with all the cases in our analyses with g ′ 1 = g ′ 2 = g Z ≈ 0.72. Finally, we also mention that the muon anomalous magnetic moment cannot be explained in our present model due to the constraint of the trident production via Z ′ [12]. Moreover, the new contributions to the semi-leptonic decays of b → sℓ + ℓ − from the Z ′ 1,2 mediations are negligible small, so that our model sheds no light to solve the recent anomalies in B → K ( * ) µ + µ − unlike those with the extra Z ′ in the literature [2]. Thus, we minimally extend our model to explain these issues in the next section.

IV. AN EXTENSION
We now extend our model by introducing two extra vector-like fermions: Q ′ L/R = (3, 2, 1/6, 1/2, −2/3) and L ′ L/R = (1, 2, −1/2, 1/2, −2) along with a neutral inert complex scalar S = (1, 1, 0 resulting in the following additional Lagrangian: where i = 2, 3. Here, we have assumed the mass eigenstates for the above down-quark and charged-lepton sectors in the SM, and f 3 << f 2 . As a result, the b → seē excess is negligible that is consistent with the current experimental data, while τ → µγ at one-loop level is also suppressed to avoid the current experimental bound. Note that S is a complex boson that is assured by the charge assignment under U(1) B−L 1 × U(1) B−L 2 −L 3 , and its mass is denoted by m S .
B → K * μ µ anomaly: The effective Hamiltonian for the b → sµ + µ − transition is induced via the box diagram [15], given by where with V tb ∼ 0.9991 and V ts ∼ −0.0403 being the CKM matrix elements [6]. Here, we take C 9 = −C 10 , which is one of the promising relations to explain the anomaly [16], and the experimental result is given by where the best fit value is −0.68.
Neutral meson mixing: The neutral meson mixing gives the bounds on g i and M Q ′ at the low energy, where our valid process is the B s −B s mixing in our case. Similar to the B → K * µμ anomaly, the formula is derived by [7]: where the above parameters are found to be f Bs = 0.200 GeV [7], m Bs = 5.367 GeV [6].
Dark matter candidate: We suppose that S is a DM candidate. First, we assume that any annihilation modes coming from the Higgs potential are negligibly small. This is a reasonable assumption, because we can avoid the strong constraint coming from the spin independent scattering cross section reported by several direct DM detection experiments, such as LUX [17]. Second, we do not consider the modes through Z ′ 1,2 coming from the kinetic term, since this is enough suppressed by the masses of m Z ′ 1,2 . We comment here that there are two resonant solutions at around the points of m Z ′ 1 = 2m S and m Z ′ 2 = 2m S . Subsequently, the dominant contribution to the thermal relic density comes from f and g, and the cross section is approximately given by [18] in the limit of massless final-state leptons and m 2 Q i,j /M 2 Q ′ << 1. Here, the approximate formula is obtained by expanding the cross section in powers of the relative velocity; v rel : σv rel ≈ a eff + b eff v 2 rel , where a eff = 0. The resulting relic density is found to be where the present relic density is 0.1199 ± 0.0108 [19], g * (x f ≈ 25) ≈ 100 counts the degrees of freedom for relativistic particles, and M PL ≈ 1.22 × 10 19 GeV is the Planck mass.

Numerical analyses:
We now perform the numerical analysis to satisfy the anomalies of the muon g − 2, B → K * μ µ, the constraints of the correct relic density, and the neutral meson mixing, as discussed above. We randomly select the input parameters as follows: where 1.2m S is used to avoid the coannihilation processes among Q ′ , L ′ and S, for simplicity.
We show the allowed regions in fig. 3 constraints or their discoveries will be found at ILC with its sensitivity of the cut-off scale being around 50-100 TeV, which are stronger than the LEP ones.
In addition, the possible effects on the flavor violating processes have been explored.
Particularly, we have shown that the branching ratio of µ → eγ for the case of V M N S ≈ U eL can be large, which is testable by the future experiment.
Finally, we have discussed the possibility to explain the muon g − 2, B → K ( * )μ µ, and dark matter candidate, by introducing vector-like fermions Q ′ , L ′ and an inert complex boson S with appropriate charge assignments under SU(3) C × SU(2) L × U(1) Y × U(1) B−L 1 × U(1) B−L 2 −L 3 . We have also shown the allowed regions to satisfy all the anomalies and constraints, and found 0.5 f 2 √ 4π for both of blue and red points, and 10 m S 170(90) GeV in red(blue) points. It is worthwhile to mention that Z boson decay modes of Z → f ifj at one-loop level could restrict our parameter spaces, where f i represent all the SM fermions. It is expected that the sensitivities of these modes further increase at future experiments, such as CEPC [20], by several orders of magnitude.