Lepton flavor violating processes $l_{i}\longrightarrow l_{j}\nu_{l}\bar{\nu}_{l}$ in topcolor-assisted technicolor models

We study the lepton flavor violating (LFV) processes $l_{i}\longrightarrow l_{j}\nu_{l}\bar{\nu}_{l}$ in the context of the topcolor-assisted technicolor (TC2) models. We find that the branching ratios $B_{r}(\tau\longrightarrow l_{j}\nu_{\tau}\bar{\nu}_{\tau})$ are larger than the branching ratios $B_{r}(\tau\longrightarrow l_{j}\nu_{l}\bar{\nu}_{l})$ in all of the parameter space. Over a wide range of parameter space, we have $B_{r}(\tau\longrightarrow l_{j}\nu_{\tau}\bar{\nu}_{\tau})\sim 10^{-6}$ and $B_{r}(\tau\longrightarrow l_{j}\nu_{l}\bar{\nu}_{l})\sim 10^{-9} (l=\mu$ or $e)$. Taking into account the bounds given by the experimental upper limit $Br^{exp}(\mu\longrightarrow3e)\leq1\times10^{-12}$ on the free parameters of TC2 models, we further give the upper limits of the LFV processes $l_{i}\longrightarrow l_{j}\nu_{l}\bar{\nu}_{l}$. We hope that the results may be useful to partly explain the data of the neutrino oscillations and the future neutrino experimental data might be used to test TC2 models.

It is well known that the individual lepton numbers L e , L µ and L τ are automatically conserved and the tree-level lepton flavor violating (LFV) processes are absent in the standard model (SM). However, the solar neutrino experiments [1], the data on atmospheric neutrinos obtained by the Super-Kamiokande Collaboration [2], and the results from the Kam LAND reactor antineutrino experiments [3] provide very strong evidence for mixing and oscillation of the flavor neutrinos, which imply that the separated lepton number are not conserved. Thus, the SM requires some modification to account for the pattern of neutrino mixing suggested by the data and the LFV processes like l i −→ l j γ and l i −→ l j l k l l are allowed. The observation of these LFV processes would be a clear signature of new physics beyond the SM, which has been widely studied in different scenarios such as Two Higgs Doublet Models, Supersymmetry, Grand Unification [4,5] and topcolor models [6].
On the other hand, neutrino oscillations imply that there are solar ν e −→ ν µ , ν τ transitions and there are atmospheric ν µ −→ ν τ transitions. The standard tau decays τ −→ µν µντ , eν e ν τ and the standard muon decay µ −→ eν eνµ can not explain the experimental fact. However, the LFV processes l i −→ l j ν lνl , where l i = τ or µ, l j = µ or e and l = τ, µ or e, might explain the neutrino oscillation data. With these motivations in mind, we study the LFV processes l i −→ l j ν lνl in the context of topcolor-assisted technicolor (TC2) models [7]. These models predict the existence of the extra U(1) gauge boson Z ′ , which can induce the tree-level FC coupling vertices Z ′ l i l j . The effects of the gauge boson Z ′ on the LFV processes l i −→ l j γ, l i −→ l j l k l l and Z −→ l i l j have been studied in Ref. [6,8]. They have shown that the contributions of Z ′ to these processes are significantly large, which may be detected in the future experiments. In this letter, we show that the Z ′ can generate large contributions to the LFV processes τ −→ µν τντ , eν τντ and µ −→ eν τντ , which may be used to partly explain the data of the neutrino oscillations. Furthermore, considering the constraints of the present experimental bound on the LFV process µ −→ 3e on the free parameters of TC2 models, we give the upper bounds on the branching ratios Br(l i −→ l j ν lνl ), which arise from Z ′ exchange.
For TC2 models, the underlying interactions, topcolor interaction, are non-universal. This is an essential feature of TC2 models, due to the need to single out the top quark for condensate. Therefore, TC2 models predict the existence of the non-universal U (1) gauge boson Z ′ . The new particle treats the third generation fermions differently from those in the first and second generations and can lead to the tree-level FC couplings. The flavor-diagonal couplings of Z ′ to leptons can be written as [7,9]: where g 1 is the ordinary hypercharge gauge coupling constant, θ ′ is the mixing angle with The flavor changing couplings of Z ′ to leptons can be written as: where k ij are the flavor mixing factors. For the sake of simplicity, we consider the case where all three generations of leptons mix with a universal constant k, i.e. k τ µ = k τ e = k µe = k in this letter.
From Eq. (1) and Eq. (2), one can see that the LFV processes l i −→ l j ν lνl can be generated via gauge boson Z ′ exchange at tree-level. The relevant Feynman diagrams are depicted in Fig.1. The partial widths can be written as: Where C 2 W = cos 2 θ W , θ W is the Weinberg angle, M Z ′ is the mass of the non-universal U(1) gauge boson Z ′ predicted by TC2 models. In above equations, we have assumed m µ ≈ 0, m e ≈ 0 for the processes τ −→ l j ν lνl and m e ≈ 0 for the processes µ −→ eν lνl .
The widths of the processes l i −→ l j ν µνµ are equal to those of the processes l i −→ l j ν eνe . This is because the gauge boson Z ′ only treats the fermions in the third generation differently from those in the first and second generations and treats the fermions in the first generation same as those in the second generation.
To obtain numerical results, we take the SM parameters as C 2 W = 0.7685, α e = 1 128.8 , m τ = 1.78GeV , m µ = 0.106GeV [10]. It has been shown that vacuum tilting and the constraints from Z-pole physics and U(1) triviality require k 1 ≤ 1 [11]. The limits on the Z ′ mass M Z ′ can be obtained via studying its effects on various experimental observables [9]. For example, Ref. [12] has been shown that to fit the electroweak measurement data, the Z ′ mass M Z ′ must be larger than 1T eV . As numerical estimation, we take the M Z ′ and k 1 as free parameters.
The branching ratios Br 1 and Br 2 are ploted in Fig.2 and Fig.3 as functions of M Z ′ for k = λ = 0.22 (λ is the Wolfenstein parameter [13]) and three values of the parameter k 1 : k 1 = 0.2(solid line), 0.5(dotted line), 0.8(dashed line). One can see that the value of Br 1 is larger than that of Br 2 in all of the parameter space of TC2 models. This is because the extra U(1) gauge boson Z ′ couple preferentially to the third generation fermions. The value of the branching ratio Br 1 increases from 3.09 × 10 −8 to 7.91 × 10 −6 as M Z ′ decreasing from 4T eV to 1T eV for k 1 = 0.5 and the value of branching ratio Br 2 increases from 1.26 × 10 −11 to 3.23 × 10 −9 . For k 1 = 1, M Z ′ = 1T eV , the branching ratio The extra U(1) gauge boson Z ′ can also contribute to the LFV process µ −→ 3e. The relevant decay width arisen from the Z ′ exchange can be written as: The current experimental upper limit is Br exp (µ −→ 3e) ≤ 1 × 10 −12 [14]. Therefore, the present experimental bound on the LFV process µ −→ 3e can give severe constraints on the free parameters of TC2 models. Then the branching ratios Br 1 , Br 2 , Br 3 and Br 4 can be written as: Br exp (τ −→ eν eντ )Br exp (µ −→ 3e), Br exp (τ −→ eν eντ )Br exp (µ −→ 3e), Br exp (µ −→ 3e), Observably Extra gauge bosons Z ′ are the best motivated extensions of the SM. If discovered they would represent irrefutable proof of new physics, most likely that the SM gauge groups must be extended [15]. If these extensions are associated with flavor symmetry breaking, the gauge interactions will not be flavor-universal [12], which predict the existence of non-universal gauge bosons Z ′ . After the mass diagonalization from the flavor eigenbasis into the mass eigenbasis, the non-universal gauge interactions result in the tree-level FC couplings. Thus, the Z ′ may have significant contributions to some FCNC processes. In this letter, we study the contributions of the non-universal gauge bosons Z ′ predicted by TC2 models to the LFV processes l i −→ l j ν lνl . We find that the branching ratios B r (τ −→ l j ν τντ ) are larger than the branching ratios B r (τ −→ l j ν lνl )(l = µ or e) in all of the parameter space. Over a wide range of parameter space, we have B r (τ −→ l j ν τντ ) ∼ 10 −6 and B r (τ −→ l j ν lνl ) ∼ 10 −9 . For k 1 = 1, M Z ′ = 1T eV and k = 1/ √ 2, the value of the branching ratio B r (τ −→ l j ν τντ ) can reach 1.63 × 10 −4 . Considering the bounds given by the experimental upper limit Br exp (µ −→ 3e) ≤ 1 × 10 −12 on the free parameters of TC2 models, we further give the upper limits of the LFV processes l i −→ l j ν lνl . The results are Br(τ −→ l j ν τντ ) ≤ 3.42 × 10 −6 , Br(τ −→ l j ν lνl ) ≤ 3.49 × 10 −10 , Br(µ −→ eν τντ ) ≤ 1.96 × 10 −9 and Br(µ −→ eν lνl ) ≤ 2 × 10 −13 (l = µ or e). We hope that the results may be useful to partly explain the data neutrino oscillations. The future neutrino experiment data might be used to test TC2 models.