Lepton flavor violating decay in the two Higgs Doublet model with the inclusion of nonuniversal extra dimensions.
Abstract
We predict the branching ratios of , and decays in the model III version of the two Higgs doublet model, with the inclusion of one and two spatial nonuniversal extra dimensions. We observe that the the branching ratios are not sensitive to a single extra dimension, however, this sensitivity is considerably large for two extra dimensions
1 Introduction
The lepton flavor violating (LFV) interactions are interesting in the sense that they are sensitive the physics beyond the standard model (SM) and they ensure considerable information about the restrictions of the free parameters, appearing in the new models, with the help of the possible accurate measurements. Among LFV interactions, the Z decays with different lepton flavor outputs, such as , and , are rich enough to study and there is an extensive work related to these decays in the literature [1][12]. The GigaZ option of the Tesla project which aims to increase the production of Z bosons at resonance [13] stimulates one to make theoretical works on such Z decays.
In the framework of the SM the lepton flavor is conserved and, for the flavor violation in the lepton sector, there is a need to extend the SM. One of the candidate model is so called SM, which is constructed by taking neutrinos massive and permitting the lepton mixing mechanism [14]. In this model, the theoretical predictions for the branching ratios (BRs) of the LFV Z decays are extremely small in the case of internal light neutrinos [1, 2]
(1) 
They are far from the experimental limits obtained at LEP 1 [15]:
(2) 
and from the improved ones at GigaZ [6]:
(3) 
with . Notice that these numbers are obtained for the decays , namely
(4) 
To enhance the BRs of the corresponding LFV Z decays some other scenarios have been studied. The possible scenarios are the extension of SM with one heavy ordinary Dirac neutrino [2], the extension of SM with two heavy righthanded singlet Majorana neutrinos [2], the Zee model [7], the model III version of the two Higgs doublet model (2HDM), which is the minimal extension of the SM [8], the supersymmetric models [9, 10], topcolor assisted technicolor model [11].
The present work is devoted to predictions of the BRs of , and decays in the model III version of the 2HDM, with the inclusion of one and two spatial extra dimensions. Our motivation is to check whether there is an enhancement in the BRs of these decays due to the extra dimensions. The possible existence of new dimensions reach great interest recently and there is a large amount of work done in the literature [16][32]. The idea of extra dimensions was originated from the study of KaluzaKlein [33] which was related to the unification of electromagnetism and the gravity and the motivation increased with the study on the string theory which was formulated in a spacetime of more than four dimensions. Since the extra dimensions are hidden to the experiments at present (for example see [30]), the most favorable description is that these new dimensions are compactified to the surfaces with small radii, which is a typical size of corresponding extra dimension. This leads to appear new particles, namely KaluzaKlein (KK) modes of the particles in the theory. In the case that all the fields feel the extra dimensions, so called universal extra dimensions (UED), the extra dimensional momentum, and therefore, the KK number at each vertex is conserved. The compactification size has been predicted as large as few hundereds of GeV [17, 18, 19, 20], in the range , using electroweak precision measurements [21], the mixing [22, 23] and the flavor changing process [24] for a single UED.
The assumption that the extra dimensions are at the order of submilimeter distance, for two extra dimensions, the hierarchy problem in the fundamental scales could be solved and the true scale of quantum gravity would be no more the Planck scale but in the order of electroweak (EW) scale [16, 17]. In this case, the gravity is spreading over all the volume including the extra dimensions, however, the matter fields are restricted in four dimensions, called four dimensional (4D) brane, or in 4D surface which has a nonzero thickness in the new dimensions, called fat brane (see for example [25]). This type of extra dimensions, accessible to some fields but not all in the theory, are called nonuniversal extra dimensions (NUED). Contrary to the UED, in the NUED, the KK number at each vertex is not conserved and tree level interaction of KK modes with the ordinary particles can exist. The study in [26] is devoted to the appearence of a very light left handed neutrino in the NUED where only the right handed neutrino is accessible to the extra dimension. In the another work [27], the effect of brane kinetic terms for bulk scalars, fermions and gauge fields in higher dimensional theories, have been studied. In [28] the electric dipole moments of fermions and some LFV decays have been analyzed in the framework of NUED.
Here, we predict the BRs of the LFV Z decays in the model III with the assumption that the extra dimensions are felt by the new Higgs doublet and the gauge sector of the theory. The Z decays under consideration are induced by the internal neutral Higgs bosons and and their KK modes carry all the information about the new dimensions, after the compactification of the single (double) extra dimension on the orbifold ( (. Here, the KK number in the vertices is not conserved, in contrast to the UED case. The nonzero KK modes of neutral Higgs fields have masses () with in one (two) extra dimension. We observe that the BRs of the processes we study enhance almost two orders larger compared to the ones without the extra dimensions, in the case of two NUED, since the neutral Higgs KK modes are considerably crowded.
The paper is organized as follows: In Section 2, we present the effective vertex and the BRs of LFV Z decays in the model III version of the 2HDM with the inclusion of NUED. Section 3 is devoted to discussion and our conclusions. In appendix section, we give the explicit expressions of the form factors appearing in the effective vertex.
2 decay in the model III with the inclusion of nonuniversal extra dimensions.
The extension of the Higgs sector in the SM brings new contributions to the BRs of the processes and makes it possible to obtain the flavor changing neutral current (FCNC) at tree level, which plays an important role in the existence of flavor violating (FV) interactions. Therefore, the multi Higgs doublet models are worthwhile to study. The 2HDM is one of the candidate for the multi Higgs doublet models. In the model I and II versions of the 2HDM, the FCNC at tree level is forbidden, however, those type of interactions are possible in the model III version of the 2HDM. The lepton flavor violating (LFV) Z decay can be induced at least in the one loop level in the framework of the model III.
The addition of possible NUED, which are experienced by the gauge bosons and the new Higgs particles, brings new contributions to the BRs of the decays under consideration. In the model III, the part of lagrangian which carries the interaction, responsible for the LFV processes in 5 (6) dimension, reads
(5) 
where the couplings are dimensional dimensionful Yukawa couplings which induce the LFV interactions. These couplings can be rescaled to the ones in 4dimension as with lepton family indices ^{1}^{1}1In the following, we replace with where ”N” denotes the word ”neutral”. Here, is the new scalar doublet, R is the compactification radius, and are lepton doublets and singlets, respectively. The scalar and lepton doublets are the functions of and (, ), where () is the coordinate represents the ’th dimension. Here we assume that the Higgs doublet lying in the 4 dimensional brane has a nonzero vacuum expectation value to ensure the ordinary masses of the gauge fields and the fermions, however, the second doublet, which is accessible to the extra dimensions, has no vacuum expectation value, namely, we choose the doublets and and the their vacuum expectation values as
(6) 
and
(7) 
This choice ensures that the mixing between neutral scalar Higgs bosons is switched off and it would be possible to separate the particle spectrum so that the SM particles are collected in the first doublet and the new particles in the second one ^{2}^{2}2Here () is the well known mass eigenstate ().. Here we consider the gauge and invariant Higgs potential in two extra dimensions
(8)  
with constants .
Since, only the new Higgs field is accessible to extra dimensions in the Higgs sector, there appear KK modes of in two spatial extra dimensions after the compactification of the external dimensions on an orbifold ,
(9) 
where , the indices and are positive integers including zero, but both are not zero at the same time. Here, is the 4dimensional Higgs doublet which includes the charged Higgs boson , the neutral CP even (odd) Higgs bosons (). The KK modes of the charged Higgs boson (neutral CP even (odd) Higgs ()) have the mass ( () ), where and . Furthermore, we assume that the compactification radius is the same for both new dimensions. Notice that the expansion for a single extra dimension can be obtained by setting , taking , and dropping the summation over . In addition to the new Higgs doublet, also the gauge fields feel the extra dimensions, however, there is no additional contribution coming from the KK modes of Z boson in the process under consideration since the Z boson does not enter into calculations as an internal line. The KK mode KK mode vertex is the same as the 4dimensional one after the integration over extra dimensions, except a small correction of the coupling due to the gauge field modeKK mode mixing (see section 6 for details).
Now, we would like to present the general effective vertex for the interaction of onshell Zboson with a fermionic current:
(10) 
where is the momentum transfer, , () is vector (axialvector) coupling, () magnetic (electric) transitions of unlike fermions. Here () is the four momentum vector of lepton (antilepton) (see Fig. 2 for the necessary 1loop diagrams due to neutral Higgs particles). Since the LFV Z boson decay exists at least in the loop level, the KK modes of neutral Higgs particles and contribute to the self energy and vertex diagrams as internal lines. The leptons live in the 4D brane and therefore they do not have any KK modes. Notice that, in the case of nonuniversal extra dimension, the KK number needs not to be conserved and there exist KK mode ( KK mode) vertices which can involve two zero modes and one KK mode.
The vector (axialvector) () couplings and the magnetic (electric) transitions () including the contributions coming from a single extra dimension can be obtained as
(11) 
where are the couplings in the 4dimensions and are the ones due to the KK modes of the scalar bosons . The KK mode contributions can be easily obtained by replacing the mass squares in by , with and the compactification radius R. We present the explicit expressions for the couplings in the appendix, by taking into account all the masses of internal leptons and external lepton (antilepton).
If we consider two NUED, the couplings appearing in eq. (11) should be replaced by and the summation would be done over except . Here can be obtained by replacing the mass squares in by , with . Furthermore, the number in front of the summations in eq. (11) would be replaced by .
Finally, the BR for can be written in terms of the couplings , , and as
(12) 
where and is the total decay width of Z boson. In our numerical analysis we consider the BR due to the production of sum of charged states, namely
(13) 
3 Discussion
The LFV Z decays , and strongly depend on the Yukawa couplings ^{3}^{3}3The dimensionfull Yukawa couplings are defined as . in the model III version of 2HDM and these couplings are free parameters which should be restricted by using the present and forthcoming experiments. At first, we assume that the couplings which contain at least one index are dominant similar to the ChengSher scenario [34] and, therefore, we consider only the internal lepton case among others. Furthermore, we assume that the Yukawa couplings are symmetric with respect to the indices and . As a result, we need the numerical values for the couplings , and .
The upper limit of is predicted as (see [35] and references therein) by using the experimental uncertainty, , in the measurement of the muon anomalous magnetic moment and assuming that the new physics effects can not exceed this uncertainty. Using this upper limit and the experimental upper bound of BR of decay, BR , the coupling can be restricted in the range, (see [36]). For the Yukawa coupling , we have no explicit restriction region and we use the numerical values which are greater than . Furthermore, the addition of the extra dimensions bring new parameter, namely the compactification radius which arises from the compactification of the a single (double) extra dimension on the orbifold (().
In the present work, we study the prediction of the NUED on the BR of the LFV processes , in the framework of the type III 2HDM. We see that the contribution coming from two extra dimensions are considerably large compared to the one coming from a single extra dimension, due to the crowd of neutral scalar Higgs boson KK modes.
Throughout our calculations we use the input values given in Table (1).
Parameter  Value 

(GeV)  
(GeV)  
(GeV)  
(GeV)  
Fig. 3 is devoted to dependence of the BR for , and . The soliddashedsmall dashed lines represent the BR without extra dimensionincluding a single extra dimension for including two extra dimensions for . It is observed that the BR is not sensitive to the extra dimension effects for a single extra dimension. However, for two NUED, there is a considerable enhancement, almost two orders, in the BR compared to the one without extra dimensions, even for the small values of the coupling . This is due to the crowd of neutral Higgs boson KK modes. This enhancement can be observed also in Fig. 4 where the BR is plotted with respect to the compactification scale for , , and . In this figure the soliddashedsmall dashed lines represent the BR without extra dimensionincluding a single extra dimension including two extra dimensions. It is seen that in the case of two extra dimensions the BR reaches almost twice of the one without extra dimensions, for the values of the compactification scale, . This enhancement becomes negligible for the larger values of the compactification scales, . The possible enhancement due to the effect of two NUED on the theoretical value of the BR of the corresponding Z decay is worthwhile to study.
In Fig. 5, we present dependence of the for , and . The soliddashedsmall dashed lines represent the BR without extra dimensionincluding a single extra dimension for including two extra dimensions for . Similar to the previous process, the BR is not sensitive to the extra dimension effects for a single extra dimension and this sensitivity increases for two NUED. The enhancement of the BR of two NUED is more than two orders larger compared to the one without extra dimensions. Fig. 6 is devoted to the compactification scale dependence of the BR for , , and . In this figure the soliddashedsmall dashed lines represent the BR without extra dimensionincluding a single extra dimension including two extra dimensions. The enhancement in the BR for the intermediate values of the compactification scale, namely , is more than one order. Similar to the previous decay, this enhancement becomes small for the larger values of the compactification scales, .
Finally, Fig. 7 (8) is devoted to the (the compactification scale ) dependence of the BR of the decay for (, ), and . In Fig. 7 the soliddashedsmall dashed lines represent the BR without extra dimensionincluding a single extra dimension for including two extra dimensions for . In the case of two extra dimensions, even for small Yukawa couplings, it is possible to reach the experimental upper limit of the BR of the corresponding decay, since the enhancement in the BR is two order larger compared to the case without extra dimensions. In Fig. 8, the soliddashedsmall dashed lines represent the BR without extra dimensionincluding a single extra dimensionincluding two extra dimensions. It is observed that, in the case of two extra dimensions, the BR reaches almost twice of the one without extra dimensions, even for intermediate values of the compactification scale, . For the larger values of the compactification scales, , there is no enhancement in the BR of the present decay.
At this stage we would like to present our results briefly.

With the inclusion of a single NUED, the enhancement in the BR of the LFV Z decays is small for the intermediate values of the compactification scale .

In the case of two NUED, even for the small values of the Yukawa couplings, it is possible to reach the experimental upper limits of the BRs of the LFV Z decays, since the enhancement in the BR is two order larger compared to the case without extra dimensions for the intermediate values of the compactification scale . This enhancement is due to the crowd of the KK modes and it is an interesting result which may ensure an important information to test the existence of the NUED, and if it exists, to decide its number and to predict the lower limit of the compactification scale, with the help of more accurate experimental results.
As a summary, the effect of two NUED on the BRs of LFV Z decays is strong and the more accurate future experimental results of these decays will be useful to test the possible signals coming from the extra dimensions.
4 Acknowledgement
This work has been supported by the Turkish Academy of Sciences in the framework of the Young Scientist Award Program. (EOITUBAGEBIP/200118)