$L_{\mu}-L_{\tau}$ effects to quarks and leptons from flavor unification

In the family grand unification models (fGUTs), we propose that gauge U(1)'s beyond the minimal GUT gauge group are family gauge symmetries. For the symmetry $L_\mu-L_\tau$, i.e. $Q_{2}-Q_{3}$ in our case, to be useful for the LHC anomaly, we discuss an SU(9) fGUT and also present an example in Georgi's SU(11) fGUT.


I. INTRODUCTION
The most interesting problem remaining in the Standard Model (SM) is the flavor or family problem. The family problem forces on the symmetry of all the massless chiral fields surviving below the grand unification (GUT) scale M GUT or the Planck scale M P . Chiral fields with quantum numbers consistent with the observed weak and electromagnetic phenomena were a crucial achievement of the SM. With the electromagnetic and charged currents (CCs), the leptons need representations which are a doublet or bigger. A left-handed (L-handed) lepton doublet (ν e , e) alone is not free of gauge anomalies because the observed electromagnetic charges are not ± 1 2 . The anomalies from the fractional electromagnetic charges of the u and d quarks add up to make the total anomaly from the first family vanish [1,2]. It is very difficult to obtain another kind of chiral model free of gauge anomalies. If another chiral model is found consistently with some observed fact, that model should include some truth in it. The same gauge structures of the first family, {ν e , e, u, d}, repeats two more times in the µ family and τ family.
A correct treatment of flavors is necessary not only in the familiar field of particle theory and high-energy physics but also in astronomy and especially cosmology. Big Bang Nucleosynthesis calculations can place upper limits on the number of active neutrinos. The mixings of six flavors of quark allow a CP violating phase which is successful in agreeing precisely with data on CP violation in K and B decays, and yet the correct derivation of baryogenesis which itsef needs CP violation and the tiny ratio η = (∆n B /n γ ) 9 × 10 −11 remains challenging, especially whether the CP violation known in quark flavor mixing can suffice to explain the matter-antimatter asymmetry of the universe. These are merely two examples of cosmolological applications of flavor theory.
Recent phenomenological studies on the flavor problem centered around (i) "Why are there three families?", and (ii) "What are the symmetries giving the observed Cabibbo-Kobayashi-Maskawa (CKM) and the Pontecorvo-Maki-Nakagawa-Skata (PMNS) matrices?". The first part of the family problem was formulated in simple gauge groups 40 years ago by Georgi [3]. Some interesting models appeared along this line in [4,5]. The second part is discussed recently in [6][7][8][9] in relation to the CKM and PMNS matrices. Until recently, there has not appeared any significant deviation from the CKM and PMNS matrices. If some deviation were to be observed, then it might predict beyond the SM (BSM) physics, probably in the fourth family, sterile neutrinos, or in the important scalar interactions as presented in this paper.
The obvious family dependences are in the masses and the CKM and PMNS matrices. Given these, the next level is to check lepton family universality in the decays of mesons. With hundreds of millions of B decays already found at the LHC, it is possible to check the universality of a ratio of the type where H represents a hadron. In the last few years, observation of an anomaly in R H with a 2.5 σ level significance [10,11] (see also [12]) attracted a great deal of attention [13][14][15][16][17][18][19]. In the leptonic sector, the lepton family dependence was used to make the BNL (g − 2) µ observation to draw near to the SM prediction. This needed an additional arXiv:2004.08234v2 [hep-ph] 29 Apr 2020 interaction for the muon family. If it is a gauge U(1) interaction, cancellation of anomalies necessitates to have contributions from other family members, for example in the form of the quantum number such as L µ − L τ .
Within fGUT models, L µ − L τ symmetry can affect the prediction of R H phenomenology since the quantum number L µ − L τ applies also to the quark members in the same family.
Consider the three family indices, I = 6, 7, 8, for U(3) representations, which are equivalently used as I → {e ≡ 1, µ ≡ 2, τ ≡ 3}. The upper (X IJ··· ) and lower indices (X IJ··· ) are distinguished. The Levi-Civita symbols, IJK and IJK for the U(3) group, will be used to raise and lower the indices. Thus, X (0,1,1) = X (1,0,0) , etc. Note also that complex conjugation corresponds to taking the hermitian conjugate, X I * = X −I . Now, we consider the following completely antisymmetric representations which split into The example for SU (9) given in [5] thus becomes, after removing vectorlike representations, The remaining three lepton (doublet) families, out of the second line from the bottom of Eq. (3), will be 3 ψ α(0,0,0) which do not carry L µ − L τ . To avoid confusion, we will write quantum numbers of L µ − L τ always as subscripts within square brackets.
Georgi's fGUT model is for N = 11 [3], Defining the first three slots for we obtain which leads to, removing vectorlike representations, We want to define family numbers such that electron carries electron number, muon carries muon number and tau carries tau number. Let S i be SU(5)-singlet scalars.

A. Notation for Lµ − Lτ
Since ψ α contains lepton doublets, it is better to define lepton family number by ψ AB → ψ α I . So, the lepton family number is defined by the subscript I: electron doublet from ψ α 6 , muon doublet from ψ α 7 , and tau doublet from ψ α 8 .
Here, m, M 1 and M 2 can be superheavy but m 3 must be smaller than or at the electroweak scale.
The LHS and RHS contributions of Fig. 1 sandwiched betweenū(p ) and u(p) are LHS ∝ m 2 m 3 eh [22] h [23] and similarly for the RHS. Here we assumed m 2 is the H α[0] mass and treated m 3 as the mass of ψ αβ[+3] ⊕ ψ α [−3] , and Thus, from Fig. 1 the anomalous magnetic moment is estimated as [22] h [23] where · · · is The calculation through the Feynman parametrization is sketched in the Appendix.
The BNL value of (g − 2) µ minus the SM prediction is [21], Thus, there is some region of the parameter space pulling the (g − 2) µ of the SM value to the BNL value with m = O(10 6 ) GeV, with m 3 at the electroweak scale with heavy m, M 1 , and M 2 .
If we consider the symmetry U(1) µ−τ , the calculation is the same.
C. Models for RK,K * For the fGUT interaction discussed in Subsec. II B, let us consider what can be its effect to R H with the symmetry U(1) µ−τ . From Eq. (10), the L µ − L τ quantum numbers of three families in the GG model are (20) and the BSM Higgs field H can be one from (ψ α [0] ⊕ψ α[0] )'s. To have the coupling (11), H must be neutral in L µ −L τ . Thus, the H couplings takes the form, From (21), we note that b L can decay to s R , In the next section, we will use the following interaction where h [IJ] couplings are set to real values by absorbing their phases to quark or lepton fields.
The first report on the family dependence is the Run1 result of LHCb, R K = 0.745 +0.090 −0.074 ± 0.036 in the q 2 interval of q 2 = 1.1 − 6 GeV 2 [10], which gives a 2.6 σ level anomaly, away from the SM prediction. The recent result from LHCb is R K = 0.846 +0.060+0.016 −0.054−0.014 [11], which is a 2.5 σ anomaly. However, the recent Belle report is consistent with the SM but with larger error bars, R K = 0.98 +0. 27 −0.23 ± 0.06 [12]. We will use the Run1 result of LHCb since it covers a wide range of q 2 , There exists a claim that new physics by scalar mediation cannot explain the R K phenomenology [23] (see also [24]). But, their assumptions deriving this conclusion do not include our scenario. Firstly, they assumed only the SM gauge group while our symmetry below the cutoff scale Λ is the SM gauge group times U(1) µ−τ . Second, the constraint, for example Eq. (15) of [23], is from purely leptonic data. But, our interaction Eq. (24) is not related to the SM interaction and hence the parameters in Eq. (24) are restricted only by the (g − 2) µ phenomenology, whose allowed region will be given together with the R K bound.
Incorporating the new operator C bsµµμ µsP R(L) b, R K is given by where, (36) In our model, the contribution of C bsµµ relative to the one in the SM is given by h [23] h [22]  h [23] h [22] m 2 0 2.5 × 10 −9 GeV −2 . Determining h [23] h [22] m 2 0 in the way, we can compare the (g − 2) µ shift by Fig. 1 with the measured value at the BNL. We note that there are more unknown parameters for the expression on the muon (g − 2) µ : m 3 , M 1 , M 2 for a given h [23] h [22] m 2 0 .
To glimpse a behavior for (g − 2) µ , we choose M 2 1 /m 2 0 to be 10 and look for the allowed region of h [23] h [22]  Imposing the experimental result, Eq (19), we can get the allowed region of parameters space ( , m 3 ), which is shown in Fig. 2. In Fig. 2, the blue and orange regions correspond to the alowed regions of parameter space obtained by taking R K to be the 1 σ upper limit of R K , and the 1 σ lower limit of R K .

IV. CONCLUSION
In a family grand unification model, we related the BNL anomaly on the muon anomalous magnetic moment and the LHC anomaly on R K via the symmetry L µ − L τ . As a fGUT grand unification example, we used Georgi's SU (11).