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The effects of non-universal extra dimensions on the \tau\to \mu \bar{\nu_i} \nu_i decay

We study \tau\to \mu \bar{\nu_i} \nu_i, i=e,\mu,\tau decays in the two Higgs doublet model, with the inclusion of one and two spatial non-universal extra dimensions. We observe that the branching ratio is sensitive to two extra dimensions in contrary to a single extra dimension.


Introduction
Lepton flavor violating (LFV) interactions are rich to analyze since they exist in the loop level and they are sensitive the physics beyond the standard model (SM). On the other hand they are clean theoretically because they are free from the nonperturbative effects. The experimental work done and the numerical results obtained stimulate the theoretical studies on these decays.
There is an extensive theoretical work done on the LFV decays in the literature. Such interactions are studied in a model independent way in [3], in the seesaw model [4], in the framework of the two Higgs doublet model (2HDM) [5,6], in supersymmetric models [7,8,9,10,11,12,13].
In the present work we study τ → µν i ν i , i = e, µ, τ decay in the model version of the 2HDM with the inclusion of the non-universal extra dimensions. This process exists at least at one loop level in the model III. The lepton flavor violation is driven by the internal scalar bosons h 0 and A 0 and the transition τ → µ is obtained. Furthermore the internal Z boson connects this transition to theνν output (see Fig. 1). Notice that we respect the assumption of the non-existence of Cabibbo-Kobayashi-Maskawa (CKM) type matrix in the leptonic sector and vanishing charged Flavor Changing (FC) interactions.
With the inclusion of additional dimensions there may be an enhancement in the BR of the decay under consideration. The extra dimensions have been studied in the literature extensively [14]- [31]. These new dimensions could not be detected at present and the most favorable description is their compactification on a surface with small radii. From the 4D point of view this compactification results in Kaluza-Klein (KK) modes of the particles with masses regulated by the parameter R, which is a typical size of the extra dimension. If all the fields are accessible to the extra dimensions, the extra dimensional momentum, and therefore, the KK number at each vertex is conserved. Such extra dimensions are called universal extra dimensions (UED) and they are studied in various works [15,16,17,18,20,21,22,23] in the literature. If the extra dimensions are accessible to some fields but not all in the theory, such type of extra dimensions are called non-universal extra dimensions (NUED) (see for example [25,26,30,31]). In this case the KK number at each vertex is not conserved and tree level interaction of KK modes with the ordinary particles can exist.
In our work we take the extra dimensions as non-universal and assume that the first Higgs doublet and the gauge fields are accessible to the extra dimensions, however the leptons, the quarks and the second Higgs doublet, which contains new Higgs particles, live in the 4D brane.
In this case the contributions due to the additional dimensions comes from the internal Z boson KK modes after the compactification of the external dimensions on orbifold S 1 /Z 2 ((S 1 × S 1 )/Z 2 ) for a single (two) extra dimensions. We observe that there is almost two orders larger enhancement on the BR of the process we study in the case of two NUED. This enhancement is obviously due to the abundance of Z boson KK modes.
The paper is organized as follows: In Section 2, we present the theoretical expression for the decay width of the LFV decay τ → µν i ν i , i = e, µ, τ , in the framework of the model III, with the inclusion of one and two NUEDs. Section 3 is devoted to discussion and our conclusions.
In the appendix section, we give explicit expressions of the functions appearing in the general effective vertex for the interaction of off-shell Z-boson with a fermionic current.
where i, j are family indices of leptons, L and R denote chiral projections L(R) = 1/2(1 ∓ γ 5 ), φ i for i = 1, 2, are the two scalar doublets, l iL and E jR are lepton doublets and singlets respec-tively. Here η E 5(6) ij are 5(6)-dimensional dimensionful, ξ E ij are dimensionless Yukawa couplings and η E 5(6) ij can be rescaled to the ones in 4-dimension as η E 5(6) ij = √ 2πR (2πR) η E ij , R is the compactification radius and y (z) is the coordinate represents the 5(6)'th dimension. The coupling ξ E ij , which has complex entries in general, is responsible for the FCNC at tree level 1 . At this stage we take two Higgs doublets φ 1 and φ 2 as respecting that only the first one has a vacuum expectation value but not the second, namely, Therefore the CP even neutral Higgs particles H 1 and H 0 do not mix in the tree level and they are obtained as the mass eigenstates h 0 and H 0 respectively. The neutral Higgs particle H 2 is the well known CP odd A 0 . In this case, the SM particles lie in the first doublet and the new particles in the second one.
The LFV τ → µν i ν i decay is induced by τ → µZ * transition and Z * →ν i ν i process (see Fig. 1). The τ → µZ * transition, which needs the FCNC at tree level, is driven by the internal new neutral Higgs bosons h 0 and A 0 , which are living in the 4D brane. Since the gauge fields are accessible to the extra dimensions, the internal Z boson has KK modes, which have additional contribution to the physical quantities related to the decay studied. The KK modes of gauge fields appear after the compactification of the external dimensions on the orbifold for a single (two) extra dimensions and, for two extra dimensions, the gauge fields can be expanded to the KK modes as: where i = 5, 6, d = 2, the indices n and r are positive integers including zero, but both are not zero at the same time. The KK modes of the gauge field Z have the masses m 2 Z + m 2 n + m 2 r , where m n = n/R and m r = r/R. Notice that the gauge boson mass matrix is diagonal since the Higgs field, which has the non-zero vacuum expectation value, is in the bulk (see [30] for details). Here, we take the compactification radius R as the same for both new dimensions. In the case of a single extra dimension, one should set d = 1, take z = 0, and drop the summation over r in eq. (4). Now, we present the general effective vertex for the interaction of off-shell Z-boson with a fermionic current where k is the momentum transfer, where p (p i , i=1,2,3) is four momentum vector of τ lepton (µ lepton, incoming ν, outgoing ν).

Discussion
The τ → µν i ν i is induced by the LFV τ → µ transition which depends on the various Yukawa couplingsξ E N,ij 2 i = µ, τ ; j = e, µ, τ and they need to be restricted by using the present and forthcoming experiments. In our work, we take only the τ lepton as an internal lepton and, therefore, we choose the couplingsξ E N,τ τ andξ E N,τ µ as non-zero. Here we expect that the couplings which contain at least one τ index are dominant similar to the Cheng-Sher scenario [32]. The upper limit of the couplingξ E N,τ µ has been estimated as 30 GeV (see [33] and references therein) by assuming that the new physics effects can not exceed experimental uncertainty 10 −9 in the measurement of the muon anomalous magnetic moment. In our numerical calculation we chooseξ E N,τ µ = 1 GeV by respecting this upper limit. Since there is no restriction for the Yukawa couplingξ E N,τ τ , the numerical values we use are greater thanξ E N,τ µ . This work is devoted to the a single and two NUED effects on the BR of the LFV processes τ → µν i ν i , in the type III 2HDM. In the case of two extra dimensions, we observe that the contribution of KK modes enhances the BR considerably, due to the crowd of Z boson KK  Now we make the same analysis for two NUED. In this case the crowd of Z boson KK modes cause to enhance the BR of the decay analyzed and these effects can be observable.
Notice that there is a possible divergence problem due to the abundance of KK modes in the summation of the KK mode contributions, however, the ratio there is an enhancement in the BR.

Appendix
The explicit expressions for the functions f 1 , f 2 , f 3 and f 4 appearing in eq. (5) read where and The parameters c V and c A are c A = − 1 4 and c V = 1 4 − sin 2 θ W . In eq. (9) the flavor changing couplings ξ E N,ji represent the effective interaction between the internal lepton i, (i = e, µ, τ ) and outgoing (incoming) j = l 1 (j = l 2 ) one. Here we take only the τ lepton in the internal line and we neglect all the Yukawa couplings except ξ E N,τ τ and ξ E N,τ µ in the loop contributions (see Discussion section). The Yukawa couplings ξ E N,ji are complex in general, however, in the present work, we take them real.