Spectroscopy of Family Gauge Bosons

Spectroscopy of family gauge bosons is investigated based on a U(3) family gauge boson model proposed by Sumino. In his model, the family gauge bosons are in mass eigenstates in a diagonal basis of the charged lepton mass matrix. Therefore, the family numbers are defined by $(e_1,e_2, e_3)=(e, \mu, \tau)$, while the assignment for quark sector are free. For possible family-number assignments $(q_1, q_2, q_3)$, under a constraint from $K^0$-$\bar{K}^0$ mixing, we investigate possibilities of new physics, e.g. production of the lightest family gauge boson at the LHC, $\mu^- N \rightarrow e^- N$, rare $K$ and $B$ decays, and so on.


Introduction
The most exciting subject in particle physics is to understand the origin of "flavor". It seems to be very attractive to understand "families" ("generations") in quarks and leptons from concept of a symmetry [1]. Since the observed masses of quarks and leptons are in range of 10 −3 − 10 2 GeV, we may suppose a possibility that the lightest family gauge boson can be observed by terrestrial experiments, e.g. at the LHC.
However, when we try to consider such a visible family gauge boson model, we always meet with constraints from the observed pseudo-scalar-anti-pseudo-scalar meson mixings P 0 -P 0 (P = K, D, B, B s ). The constraints are too tight to allow family gauge bosons with lower masses. It is usually taken that a scale of the symmetry braking is considerably high (e.g. an order of, at least, 10 4 TeV). However, there is a family gauge boson model [2] in which such severe constraints from the P 0 -P 0 mixings can be considerably loosen. In the model, the family gauge symmetry is U(3), so that a number of the family gauge bosons are nine (not eight), and quarks and leptons interacts with the family gauge bosons A j i is given by where (u 0 i , d 0 i ) are eigenstates of the family symmetry U(3) and those are define by (u 0 i , d 0 i ) = (U u ij u j , U d ij d j ). (The expression (1.1) is based on an extended version [2] of the Sumino model [3]. See in the next section.) Note that in the limit of no quark mixing, the family number is exactly conserved, so that the whole P 0 -P 0 mixings are forbidden. (A brief review is given in the next section.) Another remarkable point in the Sumino model is that the family gauge coupling constant g F and ratios among the family gauge boson masses M ij are not free, and when once a model is settled, g F and M ij /M kl are fixed. Therefore, the model can give a clear answer to observations. The family number in the Sumino model [3] is defined by the charged lepton sector e i = (e, µ, τ ) and the gauge boson masses are given proportionally to the charged lepton masses. On the hand, family number in the quark sector may be d 0 i = (d 0 , s 0 , b 0 ), but it may be an inverted assignment d 0 i = (b 0 , s 0 , d 0 ), and also a twisted assignment d 0 i = (b 0 , d 0 , s 0 ). (Of course, we consider the same assignments for u 0 i because of SU(2) L symmetry.) There are six possible assignments of (u 0 i , d 0 i ) correspondingly to e i = (e, µ, τ ). (Hereafter, for convenient, we will denote q 0 i as q i simply.) In the present paper, based on the Sumino model [3] (and also an extended Sumino model [2]), we investigate visible effects of the family gauge bosons, i.e. the deviations from the e-µ-τ universality, rare K and B decays, µ-e conversion, direct production of the lightest family gauge boson, and so on. We will conclude that the case with a twisted assignment d 0 i = (b 0 , d 0 , s 0 ) can give rich phenomenology to us.

Sumino mechanism
Priori to our investigation, let us give a brief review of the Sumino model and its extended version.
The necessity of the family gauge bosons was first pointed out by Sumino [3]. Sumino has paid why the charged lepton mass relation [4] is well satisfied by the pole masses (not by the running masses). The running masses m ei (µ) are given by [5] m ei (µ) = m ei If the factor log(m 2 ei /µ 2 ) in Eq.(2.2) is absent, then the running masses m ei (µ) are also satisfy the formula (2.1). Sumino has required that contribution of family gauge bosons to the charged lepton mass m ei (µ) cancels the factor log(m 2 ei /µ 2 ) due to photon. Therefore, masses of the gauge bosons have to be given by (n = 1 in the original Sumino model [3]), with the cancellation condition Here, the family gauge boson coupling constant g F is defined by The unfamiliar current form in Eq.(2.5) is due to an assignment (f L , f R ) = (3, 3 * ) of U(3) family symmetry (f = u, d, ν, e). The assignment is inevitably required in order to obtain the minus sign for the cancellation between log m 2 ei and log M 2 ii . However, the assignment (f L , f R ) = (3, 3 * ) in the original Sumino model causes the following problems: (i) The model cannot be anomaly free. (ii) Effective current-current interactions with ∆N f am = 2 (N f am is a family number) appear inevitably.
In order to evade these problems, an extended version of the Sumino model (K-Y model) [2] has been proposed by Yamashita and the author: 3), so that the model is anomaly free. (ii) In order to obtain the minus sign of cancellation, the family gauge boson masses are given by an inverted mass hierarchy , (2.6) (n = 1 in the K-Y model). The cancellation condition is given by Here, the coupling constant g F is defined by (For convenience of comparison with the Sumino model, the coupling constant g F in the original K-Y model [2] has been changed into g F / √ 2.) Note that, differently from the original Sumino model, the cancellation in the K-Y model is satisfied only approximately. The factor ζ in Eq.(2.7) is a fine tuning factor which gives K(µ) ≃ 2/3 almost independently of µ, and it is numerically given by ζ = 1.752 in the case of n = 1.
In the present investigation, it is essential that the family gauge boson interactions are given by Eq.(1.1). The interaction (1.1) has been derived from the following scenario: The family symmetry breaking is not caused by scalars 3 and/or 6 of U(3), but it is caused by a scalar ( The gauge boson masses M ij are dominantly generated by VEV of scalar Ψ α i which is (3, 3 * ) of U(3)×U(3) ′ , and whose VEV is given by ⟨Ψ α i ⟩ = δ α i v i . Then, we obtain family gauge boson masses where "+ · · · " denotes contributions from other scalars which are negligibly small, so that the family gauge boson masses M ij ≡ M (A j i ) satisfy relations (2.10) (Note that the, in the Sumino model, family gauge bosons acquire their masses dominantly by a scalar Φ = (3, 3 * ) of U(3)×U(3) ′ which also gives masses of the charged leptons, while, in the K-Y model, the gauge bosons dominantly acquire their masses by the scalar Ψ which is different scalar from Φ, and where a relation |⟨Ψ⟩| 2 ≫ |⟨Φ⟩| 2 has been assumed. Exactly speaking, the relation (2.10) is satisfied only approximately in both models.) In the present paper, we investigate the following two Cases A and B which satisfy the Sumino cancellation mechanism: Case A: The inverted gauge boson mass hierarchy (K-Y model like) The mass ratios are given by Eq.
In the original Sumino model, the currents with an unwelcome form as shown in Eq.(2.5) appear inevitably. We want less contribution of the family gauge bosons to the P 0 -P 0 mixing. Therefore, in the present investigation in Case B, we slightly change the original Sumino model into a modified model where leptons , so that the quark sector is anomaly free. In Case B, the gauge boson interactions are given by instead of Eq.(2.5). However, the lepton currents with the unwelcome form still appear. (We will provide additional heavy leptons in order to remove anomaly in the lepton sector.)

Quark family arrangements and P 0 -P 0 mixing
Effective quark current-current interactions with ∆N f am = 2 are given by For example, in a case of K 0 -K 0 mixing, λ i are given by These λ i with k ̸ = l satisfy a unitary triangle condition We define the effective coupling constant G ef f in the current-current interaction as Note that all family gauge bosons contribute to the P 0 -P 0 mixing as seen in Eq.(3.1).
In order to demonstrate numerical results, we tentatively assume s mixings, so that it is good news for the present purpose. However, the case brings a more severe constraint on the gauge boson masses from the observed value of D 0 -D 0 mixing.
The assumption U d ≃ V CKM leads to values of λ i , Therefore, in the limit of λ 3 ≃ 0 and λ 1 ≃ −λ 2 , we obtain approximate relation Thus, the K 0 -K 0 mixing put a severe constraint on the lower bound of the family gauge boson mass M 11 for M 11 2)] ∼ 340 TeV, where we have used a vacuum-insertion approximation (with no QCD correction) and If we give the parameters a and b in Eq.(2.12) [or (2.14)], we can estimate G ef f without approximation (3.7). In the next section, we will calculate constraints for M ij /(g F / √ 2) directly from Eq.(3.5) and by using V CKM with CP violation phase. Here, note that the CKM matrix V CKM is defined in the generation basis u i = (u, c, t) and d i = (d, s, b). Therefore, the notations M ij in Eqs.(3.1) are different from those defined by the diagonal bases of the charged lepton mass matrix M e . In this paper, we investigate various assignments of q i = (q 1 , q 2 , q 3 ). As far as quark sector is concerned, the use of generation basis d i = (d, s, b) is convenient. Therefore, hereafter, for example, for Case B 1 with family number d i = (b, s, d) (the case is defined in the next section), we denote M 11 , M 12 , M 22 , · · · as M bb , M bs , M ss , · · · , respectively, in order to distinguish those from M ij defined in the family numbers. (For convenience, we use down-quark names as the quark family numbers.) The physics is highly dependent on the quark family assignments. The details are discussed in the next section.

Which quark-family assignment is favorable ?
We find that K 0 -K 0 mixing puts the most severe constraints on the family gauge boson masses M i compared with other P 0 -P 0 mixings. As seen in Eqs.(3.6) and (3.7), because of |λ b | 2 ≪ |λ s | 2 ≃ |λ d | 2 , the observed K 0 -K 0 mixing put a constraint on M dd or M ss , but it does not put a constraint on M bb . Therefore, for our purpose of visible family gauge bosons, we should regard the third generation quark (t, b) as (t, b) = (u 3 , d 3 ) in Case A with the inverted gauge boson mass hierarchy, and (t, b) as (t, b) = (u 1 , d 1 ) in Case B with the normal gauge boson mass hierarchy. As a result, we have the following four candidates of the quark family assignments: , s). Cases A 1 with n = 1 and B 1 with n = 1 correspond to the K-Y model and the Sumino model, respectively.
In Table 1  As seen in Table 1, Case A 1 and Case A 2 lead to large values ofM ij , so that these cases are not interesting to us. Case A with n ≥ 3 can haveM 33 smaller than a few TeV, but the case givesM 11 ∼ 10 6 TeV. Phenomenology in Case A 1 with n = 1 has already been investigated in Refs. [2,9]. Phenomenology for Case B with d i = (d, s, b) has investigated in Ref. [10]. The results for visible effects of the family gauge bosons was negative.
We consider that Case B with n = 2 is phenomenologically most attractive, because the lightest family gauge boson A 1 1 has mass of an order of a few TeV which is visible at the LHC (remember M 11 = (g F / √ 2)M ii ). Besides, even the heaviest gauge boson has, at most, a mass

Case
Family gauge boson masses of an order of 10 4 TeV.

Phenomenology of the family gauge bosons in Cases B 1 and B 2
In this section, let us investigate phenomenology of the family gauge bosons in Cases B 1 and B 2 with n = 2. From a point of view of model-building, too, the case n = 2 is not so unlikely, because we can consider a VEV relation In this case, from Eq.(2.4), the gauge coupling constant g F / √ 2 is given by where, for convenience, we have used [6] α(m τ ) = 1/133.471.

Direct production of the lightest gauge boson
From the value given in Table 1  It should be noted that the gauge boson A 1 1 can interact only with the third generation quarks (t, b), although it does with the first generation leptons (ν e , e) for leptons. Therefore, the gauge boson A 1 1 will be produced by gluon fusion (Fig.1) as at the LHC. (In future, we will also observe A 1 1 production in the ILC as e + + e − → A 1 Figure 1: A 1 1 production at the LHC. We have decay modes of A 1 1 into t +t, b +b, e − + e + and ν e +ν e with branching fractions as follows: Note that the branching ratio Br(A 1 1 → ν eνe ) = 1/15 = 6.7% is one in the case of Majorana neutrinos. If neutrinos are Dirac neutrinos, the branching ratios is given Br(A 1 1 → ν eνe ) = 2/16 = 12.5%. The large difference between both is due to the large leptonic branching ratio in the family gauge boson decays. Therefore, in future, when data of the direct production of A 1 1 are accumulated, we will be able to conclude whether neutrinos are Dirac or Majorana by observing whether Br(A 1 1 → ν eνe ) is 6.7% or 12.5%. The search for A 1 1 production at the LHC is done by a similar way of the Z ′ search (for a review, see, for example, [11]). Although there has been an experimental report on Z ′ search [12], the result cannot be applicable for A 1 1 search, because A 1 1 cannot interact with the first generation quarks, so that the cross section is considerably small compared with Z ′ production. The cross section of A 1 1 in the original Sumino model has been discussed in Ref. [10], but the case was a different family gauge boson A 1 1 which can interact with the first generation quarks. Since the purpose of the present paper is to give an overview of the family gauge bosons with visible energy scale, estimate of the production rate σ(pp → A 1 1 ) will be given elsewhere. If the real mass M 11 is smaller than 500 GeV, we may expect an observation at the ILC in future, too.

Contribution of family gauge bosons to the rare decay K + → π + νν
Let us estimate contributions of family gauge bosons to the rare decay K + → π + νν, because only a finite value of the branching ratio has been reported [6] at present: It is usually taken that this value is consistent with the standard model prediction [13] Br(K + → π + νν) SM = (0.80 ± 0.11) × 10 −10 .
We are interested in whether Case B is consistent or not with the present experimental result (5.5). In the present model, all family gauge bosons can, in principle, contribute to each rare decay mode. For example, in Cases B 1 and B 2 , a transition K → π is mediated by the gauge bosons A d s ≡ A 3 2 and A d s ≡ A 2 3 , respectively. However, as seen in Table 1, the mass of M 23 is of the order of 10 3 TeV, so that the effect is invisible. Remember that family-number violating transitions are possible in the quark sector. Since the effective mass value ofM 11 ≡M bb is too small, the contribution of A 1 1 is dominated compared with other gauge boson exchanges even considering the existence of the suppression factor |U d * bd U d bs | (the value is 0.0155 in the approximation U d ≃ V CKM ). Then, the branching ratio due to the family gauge boson exchange A 1 1 are estimated as follows: K + → π + e − µ + as follows: (We have neglected the lepton masses.) Here, the factor ξ denotes mixing effects in quarks, and in this case, ξ is given by For rare B and K decays, we can estimate their branching ratios by a similar way to Eq.(5.7). We investigate only the decay modes via the family gauge boson A 1 2 , because other gauge bosons are considerably heavy, so that such gauge boson effects are obviously invisible. Note that since the family number in the quark sector is assigned unconventionally, for example, the gauge boson A 1 2 causes the decay B → K + e + + µ − with mixing factor ξ = |U bb U ss |/|V us | for Case B 1 with A 2 1 ≡ A s b , and B → π + e + + µ − with mixing factor ξ = |U bb U dd |/|V us | for The branching ratios of decay modes via A 1 2 are, for example, as follows: where we have assumed U d ≃ V CKM . These results are invisible for a time, because the present experimental lower limits [6] are Br(B + → K + µ − e + ) < 9.1 × 10 −8 and Br(B + → π + µ ∓ e ± ) < 1.7 × 10 −7 . The family gauge boson A 1 2 can also contribute to rare K decays. However, the predicted branching ratios are of orders of 10 −15 − 10 −17 because small values of quark mixing factors, so that the effects invisible.

µ
So far, phenomenological merits of Cases B 1 and B 2 has been almost equal. In this subsection, we would like to emphasize that Most sensitive test in the near future for Cases B 1 and B 2 is to observe the so-called µ-e conversion. (For a review of the µ-e conversion and more detailed calculations, for example, see Ref. [14] and Ref. [15], respectively.) The present experimental limit is, for instance, for Au, [17] The reaction µ − N → e − N is caused by an exchange of the family gauge boson A 1 2 . It means the exchange of . At present, we do not know values of |U q ij | (q = u, d). Therefore, it is not practical, at this stage, to estimate a µ-e conversion rate strictly. Instead, we roughly estimate a µ-e conversion rate in the quark level as follows: where q = u, d, and (r 2 12 ) is defined by Eq.(5.8). (Although the estimated value R q has different physical meaning from the value R(Au), we consider that the order of the value R q can provide one with useful information.) In Eq.(5.13), ξ is a quark mixing factor similar to Eq.(5.9), and the value of ξ is given by In this approximation, we may regard the ratios R q as R u ≪ R d , so that we can neglect contribution to nucleon from R u compared with that from R d . Then, we can roughly estimate values of R q where we have usedM 12 = 260 TeV from Table 1.
In the near future, the COMET experiment [16] will reach a single-event sensitivity of 2.6×10 −17 . Therefore, the value R q ∼ 10 −16 in Case B 2 become within reach of our observation, but the value R q ∼ 1.32 × 10 −17 in Case B 1 is critical for its observation.
Since the decay µ − → e − + γ is highly suppressed in the present scenario, if we observe µ − N → e − N without observation of µ − → e − +γ, then it will strongly support our family gauge boson scenario. (The decay µ − → e − + γ can occur through a quark-loop diagram. However, such a diagram is highly suppressed.)

Deviations from the e-µ-τ universality
Previously, we pointed out [9] a possibility of a deviation from the e-µ universality in tau decays τ → µνν/eνν by assumingM 23 ≪M 31 . However, in the present model, we cannot observe such a deviation because the mass spectrum in the present model givesM 23 ≃M 31 , and besides, we have a large valueM 23 ∼ 10 3 TeV in Case B.
On the other hand, we have a possibility of sizable deviations from the e-µ-τ universality in the Υ decays Υ → τ + τ − /µ + µ − /e + e − , because the value ofM 11 ≡M bb is considerably small in Case B. We have matrix elements for the decays Υ → τ + τ − /µ + µ − /e + e − , as follows: At present, we have not observed such a deviation [6]. However, the value (5.16) will become visible in future experiments.

Concluding remarks
We have investigated possibility of visible family gauge boson effects for six family assignments in the quark sector (d 1 , d 2 , d 3 ) = (d, s, b), (d 1 , d 2 , d 3 ) = (b, d, b), and so on, under the Sumino cancellation condition. In the Sumino model, the family number is defined by the diagonal basis of the charged lepton mass matrix M e = diag(m e , m µ , m τ ). The P 0 -P 0 mixings (P = K, D, B, B s ) are caused only through quark mixings U u ̸ = 1 and U d ̸ = 1. We have found that the most interesting case is Case B 2 , (d 1 , d 2 , d 3 ) = (b, d, s). In Case B 2 , a direct production of A 1 i at the LHC, µ-e conversion µ − N → e − N , and a deviation from e-µ-τ universality in the Υ decay will be observed in future experiments. Also, Case B 1 , (d 1 , d 2 , d 3 ) = (b, s, d), is attractive, although the case is somewhat hard to observe in µ − N → e − N compared with Case B 2 .
In Case B, the leptons take a Sumino-like structure (so that Sumino's cancellation mechanism is satisfied), while quarks takes a twisted family-number assignment. At present, there is no theoretical ground for such family-number assignments. In order to make the twisted familynumber assignment (d 1 , d 2 , d 3 ) = (b, d, s) more reliable, we, at least, have to build a unified mass matrix model of quarks and leptons under such the twisted family-number assignment. It is a task in future.
We hope that many physicists turn their attention to a possibility of visible family gauge bosons and of a twisted family-number assignment versus generation-numbers.