Muon-Electron Conversion in a Family Gauge Boson Model

We study the $\mu$-$e$ conversion in muonic atoms via an exchange of family gauge boson (FGB) $A_{2}^{\ 1}$ in a $U(3)$ FGB model. Within the class of FGB model, we consider three types of family-number assignments for quarks. We evaluate the $\mu$-$e$ conversion rate for various target nuclei, and find that next generation $\mu$-$e$ conversion search experiments can cover entire energy scale of the model for all of types of the quark family-number assignments. We show that the conversion rate in the model is so sensitive to up- and down-quark mixing matrices, $U^{u}$ and $U^{d}$, where the CKM matrix is given by $V_\text{CKM} = U^{u\dagger} U^d$. Precise measurements of conversion rates for various target nuclei can identify not only the types of quark family-number assignments, but also each quark mixing matrix individually.


discuss the feasibility of it.
This work is organized as follows. First we briefly review the FGB model. We illustrate three types of quark family-number assignments. Then we introduce four types of quark mixing to describe the interaction between FGBs and quarks. Next, in Sec. 3, we formulate the µ-e conversion rate in the FGB model. In Sec. 4, we give numerical results, and show that the FGB model can be confirmed or ruled out at µ-e conversion search experiments in near future. We discuss feasibility for discriminations among three types of quark family-number assignments and four types of quark mixing matrices. Sec. 5 is devoted to summarize this work.

(2.2)
In the absence of family-number dependent factor log(m 2 e i ), the running masses m e i (µ) also satisfy the relation (2.1). In order to understand this puzzle, Sumino has proposed a U (3) FGB model so that a factor log(m 2 e i ) from the QED correction is canceled by the FGB loop contribution log(M 2 ii ) [1]. Here, the masses of FGBs A j i , M ij , are given by M 2 ij = k(m n e i + m n e j ), (2.3) where k is a constant with dimension of (mass) 2−n . The cancellation mechanism holds for any n, because log M n ii = n log M ii . The original model has studied the n = 1 case [1]. The cancellation requires the following relation between the family gauge coupling g F and QED coupling e, Here θ w is the Weinberg angle. Note that the cancellation mechanism holds only at the one loop level. Sumino has speculated the scale of U (3) family symmetry is an order of 10 3 TeV [1,2]. In the FGB model, the family symmetry is broken by a scalar Φ with (3, 3) of U (3) × O(3). The family-numbers of quarks and leptons, which are triplets of U (3), are changed only by exchanging ΦΦ, not the single Φ. Thus, the FGB contribution to pseudo-scalar meson oscillations is highly suppressed. The FGB mass matrix is diagonal in the flavor basis in which Table 1: Three extended FGB models. q 0 stands for eigenstates of the U (3) family gauge symmetry. Note that this lower bound on M 12 is derived from P 0 -P 0 mixing measurements [4], not from µ-e conversion search experiments. the charged lepton mass matrix is diagonal, because those masses are generated by the common scalar Φ. Therefore, family-number violation does not occur in the charged lepton sector. In the original model, charged leptons (e Li , e Ri ) are assigned to (3, 3 * ) of U (3) family symmetry, which makes the sign of FGB loop correction to be opposite to the QED correction for the cancellation. So the original model is not anomaly free. In order to avoid this anomaly problem, Yamashita and one of the authors (YK) have proposed an extended FGB model [3]: two scalars Ψ and Φ are introduced, which are (3, 3 * ) of U (3) × U (3) ′ . Charged lepton masses are generated via the VEV of Φ only. FGB masses are achieved via the VEVs of Φ and Ψ. Relations of these VEVs are Ψ ∝ Φ −1 and Ψ ≫ Φ . These relations lead the FGB spectrum (2.3) with negative n, in contrast to the original FGB model in which a VEV of single scalar field generates both of masses of charged leptons and FGBs. We can therefore realize the cancellation with a normal assignment (e Li , e Ri ) = (3, 3) of U (3) family symmetry, because of log M n ii = n log M ii < 0 with the negative n. In this paper, we call the extended FGB model Model A, and call the original model Model B. The characteristics of these models are summarized in Table 1. In order to relax the severe constraints from the observed P 0 -P 0 mixings, we consider that the lightest FGB interacts with only the third generation quarks. We define the family-number as (e 1 , e 2 , e 3 ) = (e − , µ − , τ − ). In Table 1, we list "optimistic" lower limit on M 12 which is not conflict with all of observed P 0 -P 0 mixings [4].

Model A
According to the extended FGB model, Model A is characterized by the following inverted mass hierarchy of FGB mass [3], (n is a positive integer). Interaction Lagrangian of quarks and leptons with the FGBs is given by Here q 0 i = U q ij q j is an interaction eigenstate of the U (3) symmetry, where q j and U q ij represent mass eigenstate and quark mixing matrix, respectively. The interactions are a type of pure vector, so that the model is anomaly free. The gauge coupling g F in Model A is given as [3] where α em (m µ ) = 1/137, and ζ = 1.752 is a fine tuning factor which is obtained from phenomenological study.

Model B
Model B is characterized by the following relation of FGB mass,

n is a positive integer). Interaction Lagrangian of quarks and leptons with the FGBs is given by
Here, note that the leptonic currents have an unfamiliar form, . Since this assignment in the quark sector leads unwelcome large K 0 -K 0 mixing, we use pure vector current form as far as quark currents are concerned. The gauge coupling g F is given by In order to avoid the severe constraints from the observed P 0 -P 0 mixing, the lightest FGB A 1 1 couples only to the third generation quarks, so that we have the following two scenarios for the family-number assignment [4]: (2.11)

Typical cases of quark mixing
In the FGB model, µ-e conversion branching ratio B(µ − N → e − N ) is sensitive to the quark mixing matrices, U u and U d . Each explicit form is not determined yet, though the combination is measured as V CKM = (U u ) † U d . We calculate B(µ − N → e − N ) by using some typical mixing matrices from the practical point of view.
The family numbers do not always correspond to the generation numbers in Model B. In order to avoid confusing, hereafter, we denote U u , U d and V CKM in the generation basis and, e.g., we denote (U d ) 12 as (U d ) ds .
As the first case (Case I), we consider following mixing, Case I is the most likely case. Since we know m t /m u ≫ m b /m d , we consider that the CKM mixing almost comes from down-quark mixing U d . Besides, we know an empirical well-satisfied relation V us ≃ m d /m s without m u /m t [17]. In fact, Case I is practically well satisfied in most of mass matrix models. We adopt the standard expression for the explicit form of V CKM ,  (2.18) The mixings in (2.18) have been derived in a mass matrix model [19] which is notable one: a unified description of the quark-and lepton-mixing matrices and mass ratios has been described by using only the observed charged lepton masses as family-number dependent parameters. It is worth investigating the potential of the µ-e conversion to determine the quark mixing. To do this, we consider Case IV with following parametrization: 3 µ-e conversion in the FGB model We formulate the reaction rate of µ-e conversion in muonic atoms via A 1 2 exchange based on Ref. [20]. Note that in the FGB model other muon lepton family violating (LFV) reactions Table 2: C X,α L(u) and C X,α L(d) for each model and for each quark mixing matrix. V qq ′ andŨ q stand for the CKM matrix and the mixing matrices derived in Ref. [19], respectively.
in muonic atom [21], and so on) arise at higher order. These reaction rates are suppressed by higher order couplings, gauge invariance, and so on. Hence we do not study these reactions here.
The µ-e conversion via A 1 2 exchange is described by the effective interaction Lagrangian 1 , Here X and α denote the model, X ∋ {A, B 1 , B 2 }, and the type of quark mixing matrices, α ∋ {I, II, III, IV}, respectively. The coefficients C X,α L(q) and C X,α R(q) are derived from interaction Lagrangian in each model discussed in previous section. We list C X,α L(q) in the generation basis in Table 2. C X,α R(q) is related with C X,α L(q) as follows, 3) The branching ratio of µ-e conversion is defined by B(µ − N → e − N ) = ω conv /ω capt , where ω conv and ω capt represent the reaction rates of µ-e conversion and of the muon capture process, respectively. The reaction rate ω conv is calculated by the overlap integral of wave functions of the initial muon, the final electron, and the initial and final nucleus. In the FGB model, ω conv is (3.4) 1 We omit the contribution via the kinetic mixing of A 1 2 and Z boson. The contribution is suppressed by the loop factor and quark mixings, and is sub-dominant relative to direct ones of A 1 2 .  Here m µ is the muon mass. The overlap integral of wave functions of muon, electron, and protons (neutrons) gives V (p) (V (n) ) (explicit formulae and details of the calculation are explained in Ref. [20]). We list V (p) and V (n) for relevant nuclei of SINDRUM-II (Au), DeeMe (C and Si), COMET (Al and Ti), Mu2e (Al and Ti), and PRISM (Al and Ti) in Table 3. We also list them for U nucleus. The µ-e conversion search with the U target can assist to confirm the FGB model and to determine the quark mixings.

Numerical result
We are now in a position to show numerical results. Table 4 shows the lower bound on the FGB mass M 12 by the µ-e conversion search at SINDRUM-II, B(µ − Au → e − Au) < 7 × 10 −13 [22]. Current most stringent limits of M 12 are obtained from observed P 0 -P 0 oscillations (Table 1), not from the µ-e conversion search. Next we show the feasibility of FGB search in µ-e conversion search experiments. Fig. 1 shows B(µ − Al → e − Al) as a function of M 12 . In light of the cancellation, the FGB masses are supposed to be up to ∼ 10 4 TeV [1, 2] (see Sec. 2). As is shown in Fig. 1, next generation experiments cover most of this mass region, and the discovery of µ-e conversion via A 1 2 exchange is expected in near future. To put it the other way around null results of µ-e conversion search can rule out the FGB model.
After the discovery of µ-e conversion, we need to check whether the observed event is a signal of A 1 2 or not. Table 6 lists the ratio of branching ratios, B(µ − N → e − N )/B(µ − Al → e − Al).  The µ-e conversion events will be confirmed as the signal of A 1 2 through precise measurements of the ratios. Also, a type of quark mixing matrix can be identified by the precise measurements. The µ-e conversion search by using large nucleus target is important. Indeed, although it is hard to distinguish the case I and III(A) from the ratios B(Ti)/B(Al), B(C)/B(Al), and B(Si)/B(Al), it can be possible for large nucleus, i.e., B(Au)/B(Al), and B(U)/B(Al). It is probably impossible to distinguish the case III(A) and III(B 1 ) from the ratios. To do this, we need additional observables via the FGB exchange, e.g., LFV kaon decays, LFV collider signals, and so on. Some of experiments are running or will launch in near future to search for these signals [24,23]. Therefore it is important to simulate what correlations are expected and how sensitivity is required for the purpose. It is however beyond the scope of this paper and we leave them in future work [25].
One may wonder why, in Table 6, B(N )/B(Al) is insensitive to Model in Case I and II. This is understood as follows. The branching ratios can be decomposed into Model independent and dependent part as For any target nuclei, the Model dependent part g X B(µ − Al → e − Al). Particularly, in Model A, since the ratios B(N )/B(Al) also depends on θ, the quark mixing can be accurately determined by accumulating a large number of µ-e conversion events. Fig. 2 emphasizes an importance of the µ-e conversion searches with various target nuclei. In Model A, even if the signal of µ − Al → e − Al will never be found, a number of events can be observed at experiments with other target nucleus. On the other hand, in Models B 1 and B 2 , the ratios B(N )/B(Al) are independent of θ, and are equal to those of Case I. This is because that the branching ratios can be decomposed into θ dependent part and independent part as follows

Concluding remarks
We have investigated the µ-e conversion via an exchange of family gauge boson A 1 2 in a U (3) FGB model. In the model there are various types of FGB spectrum and of family-number assignments. We have considered three well-motivated models: a model with inverted familynumber assignment (Model A), and models with twisted ones (Model B 1 and B 2 ). We also have a degree of freedom of choice of quark mixing U u and U d . We have introduced four types of mixing: a most likely mixing, U u ≃ 1 and U d ≃ V CKM (Case I), an opposite type of Case I, U u ≃ V † CKM and U d ≃ 1 (Case II), a phenomenologically derived mixing (2.18), U u ≃Ũ u and U d ≃Ũ d (Case III), and a parametrized mixing, U u = R 3 and U d = R T 3 V CKM (Case IV). We have calculated the branching ratio of µ-e conversion process, B(µ − N → e − N ), in Models A, B 1 and B 2 for each type of quark mixing. We have shown that next generation µ-e conversion search experiments will cover entire energy scale of the FGB model, and could confirm or rule out the FGB model. Muon-number violating decays except for the µ-e conversion is extremely suppressed in the FGB model. Thus we have emphasized the importance of precise measurements of the ratios B(N )/B(Al), which is necessary to confirm the FGB model. Searches for LFV decays of mesons should assist the confirmation. This interesting possibility is left for future work [25].
In the FGB model it is, in principle, possible to individually determine quark mixing matrix U u and U d , in contrast within the SM. However, since V (p) ≃ V (n) in the most nuclei, it is hard to observe the difference between U u and U d . We hope that further precise search for the µ-e conversion with heavy nuclei, e.g., Au and/or U which V (p) and V (n) are sizably different.