Lepton flavor violating Z l + 1 l 2 decay in the general two Higgs Doublet model

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1 Lepton flavor violating Z l l decay in the general two Higgs Doublet model arxiv:hep-ph/668v Sep E. O. Iltan and I. Turan Physics Department, Middle East Technical University Ankara, Turkey Abstract We calculate lepton flavor violating Z l l decay in the framework of the general two Higgs Doublet model. In our calculations we used the constraints for the Yukawa couplings ξ N,τe D and ξ N,τµ D coming from the experimental result of muon electric dipole moment and upper limit of the BRµ eγ. We observe that it is possible to reach the present experimental upper limits for the branching ratios of such Z decays in the model III. address: eiltan@heraklit.physics.metu.edu.tr address: ituran@.metu.edu.tr

2 Introduction Lepton Flavor Violating LFV interactions reached great interest since the related experimental measurements are improved at present. Among them, the lepton flavor changes in Z decays, such as Z eµ, Z eτ and Z µτ are important for the search of neutrinos, their mixing and possible masses, and the physics beyond the Standard model SM. With the Giga-Z option of the Tesla project, the production of Z bosons at resonance is expected to increase [] and this forces to study on such Z decays more precisely. Since the lepton flavor is conserved in the SM, one needs an extended theory to describe the lepton flavor violating Z decays. One of the possibility is the extension of the SM, so called νsm, by taking neutrinos massive and permitting the lepton mixing mechanism []. In this case the lepton sector is analogous to the quark sector. Considering the branching ratio BR BRZ l ± l± = ΓZ l l l l Γ Z, the first predictions for such Z decays are given in [3, 4]. The best experimental limits obtained at LEP [5] are BRZ e ± µ ± <.7 6 [6], BRZ e ± τ ± < [6, 7], BRZ µ ± τ ± <. 5 [6, 8] and with the improved sensitivities at Giga-Z [9] these numbers could be pulled down to BRZ e ± µ ± < 9, BRZ e ± τ ± < f 6.5 8, BRZ µ ± τ ± < f. 8 with f =... In the framework of νsm with light neutrinos the theoretical prediction is extremely small and far from these limits [3, ] BRZ e ± µ ± BRZ e ± τ ± 54, BRZ µ ± τ ± < Another possiblity to increase the corresponding BR is the extension of the νsm with one heavy ordinary Dirac neutrino []. Heavy neutrinos are expected in string-inspired models [] and some GUTs []. In the νsm with one heavy neutrino, it is necessary to include a

3 heavy charged lepton in the theory. In the famework of this model, its is possible to observe their effects from Z decays if the neutrinos with a mass of several hundered GeV exist. Further scenario to increase the BR is the νsm extended with two heavy right-handed singlet Majorana neutrinos []. In this case, it is possible to reach the experimental upper limits in the large neutrino mass region. Lepton flavor Violating Z decays are also studied in the framework of the Zee model [3]. According to this work, among all three lepton flavor violating decay modes of Z, only Z eτ decay has the largest contribution which is less than the present limits. Other two decays are small to be observed in the next linear colliders. In our work, we study Z e ± µ ±, Z e ± τ ± and Z µ ± τ ± decays in the model III version of HDM, which is the minimal extension of the SM. Since there is no CKM type matrix and therefore no charged FC interaction in the leptonic sector according to our assumption, the source of lepton flavor violating Z decays are the neutral Higgs bosons h and A with the Yukawa couplings which allow tree level flavor changing neutral currents FCNC. The choice of complex Yukawa couplings brings the possibility of non-zero electric dipole moments EDM of leptons and this ensures to restrict the Yukawa couplings using the present experimental limits. Further the lepton flavor violating interaction µ eγ is possible in this model and can be used to predict the constraint for the Yukawa couplings see [4]. Calculations are done in one loop level and it is shown that the experimental upper limits for Z decays underconsideration can be reached by playing with the free parameters of the model III respecting the above restrictions. The paper is organized as follows: In Section, we present the explicit expressions for the Branching ratios of Z e µ, Z e τ and Z µ τ in the framework of the model III. Section 3 is devoted to discussion and our conclusions. Z l l decay in the general two Higgs Doublet model. In the SM and the model I and II version of HDM the FCNC at tree level is forbidden. However model III version of HDM permits FCNC interactions at tree level and makes flavor violating interactions possible. With the choice of complex Yukawa couplings, the CP violation can also exist and this leads to appearence of non-zero EDM of fermions. The Yukawa interaction for the leptonic sector in the model III is L Y = ηij l D il φ E jr ξij l D il φ E jr h.c., 4 where i, j are family indices of leptons, L and R denote chiral projections LR = / γ 5, φ i for i =,, are the two scalar doublets, l il and E jr are lepton doublets and singlets respectively.

4 Here φ and φ are chosen as φ = [ v H χ iχ ] ; φ = H H ih, 5 and the vacuum expectation values are < φ >= v ; < φ >=. 6 This choice ensures that the SM particles are collected in the first doublet and the ones beyond in the second doublet. The part which produce FCNC at tree level is L Y,FC = ξ D ij l il φ E jr h.c.. 7 Here the Yukawa matrices ξ D ij have in general complex entries. Note that in the following we replace ξ D with ξ D N where N denotes the word neutral and define ξ D N which satisfies the equation ξn D = 4 G F ξd N. In our analysis, we take H and H see eq. 5 as the mass eigenstates h and A respectively since no mixing between CP-even neutral Higgs bosons h and the SM one, H, occurs at tree level. The general effective vertex for the interaction of on-shell Z-boson with a fermionic current is given by Γ µ = γ µ f V f A γ 5 i m W f M f E γ 5 σ µ ν q ν 8 where q is the momentum transfer, q = p p, f V f A is vector axial-vector coupling, f M f E magnetic electric transitions of unlike fermions. Here p p is the four momentum vector of lepton anti-lepton. The necessary -loop diagrams due to neutral Higgs particles are given in Fig.. Taking into account all the masses of internal leptons and external lepton anti-lepton, the explicit expressions for f V, f A, f M and f E read as { g f V = dx c 64 π cos θ W m m V m l m l l l m i η i m l x η V Lself, h i ln m µ i η i m l x ηi V m i η i m l c A m l m l x η V i ln Lself, A m i ηi m l x ηi A ln Lself, h µ, A µ m µ i η i m l x ηi V ln Lself ln Lself, h m µ i ηi m l x ηi A 3 ln Lself, h µ

5 } m i ηi m l x ηa i ln Lself, A m µ i ηi m l x ηa i ln Lself, A µ g { x dx dy m 64 π i cos θ c A ηi A c V ηi V W A h x y m i c A m l m l ηi c h V m l m A l η i h A c A ηi A c V ηi V q xy m l m l x y ln Lver h A h A µ µ η A i xm l y m l m i ηi m l m l x y ηa i xm l y m l m i η i A h h A ηa i ln Lver A h } h A, µ µ { f A = g 64 π cos θ W dx m l m i ηi m l x ηi A m i η i m l c A m l m l m i η i m l x ηi V m i η i m l g 64 π cos θ W m i x y c V η A i c A η V i m l m l m l ln Lself, A c V m l x η A i ln Lself, h m l ln Lself, A µ m µ i ηi m l x ηi A m µ i ηi m l x ηi A ln Lself, h µ Lself, A ln m µ i η i m l x η V i ln Lself, h x x y ηv i ln Lver A h µ µ f M = g m W 64 π cos θ W { x η V i m µ i η i m l x ηi V y m l m i η i A h ln Lself, A µ dx dy m i c V ηi A c A ηi V A h c V m l m l ηi c A m l m l η i h q xy m l m l x y h η V i xm l ηv i xm l h A dx }, x dy { m i c A x y η i c V η i x y x y c V ηi V c A ηi A xm l h } ln Lself, h µ A ln Lver h A y m l h A y m l A µ µ m i η i 4

6 x y c V η V i c A η A i xm l η A i xm l x y y m l A h f E = g m { x W dx dy 64 π cos θ W where and m i c A x y η i c V ηi x y y m l m i c A x y ηi c V η i x y A h A m i η i h A x y c V η A i c A η V i xm l h x y c V η A i c A η V i xm l x y η V i m l x m l y A h A h }, y m l y m l m i c A x y η i c V ηi x y h A L self, h = m h x m i m l L self, A = L self, h m h m A, L self, h = L self, h m l L self, A = L self, A m l m l, m l, m i η i x x, h = m h x y m i x y q xy, h A = m A x m i x y m h q x y, A A h = Lver h m h m A, A h h A A }, 9 = Lver h A m h m A, ηi V ηi A η i ηi = ξ D N,l iξ D N,il ξ D N,il ξ D N,l i, = ξ D N,l i ξd N,il ξ D N,il ξ D N,l i, = ξ D N,il ξ D N,il ξ D N,l i ξd N,l i, = ξ D N,il ξ D N,il ξ D N,l i ξd N,l i. The parameters c V and c A are c A = and c 4 V = 4 sin θ W. In eq. the flavor changing couplings ξ N,l D j i represent the effective interaction between the internal lepton i, i = e, µ, τ and outgoing incoming j = j = one. Here we take ξ N,l D j i complex in general and use the parametrization ξn,il D j = ξn,il D j e iθ ij, 5

7 where i, l j denote the lepton flavors and θ ij are CP violating parameters which are the sources of the lepton EDM. Now, using the couplings f V, f A, f M and f E the BR for Z l l can be written as BRZ l l = 48 π m Z Γ Z { f V f A cos θ W f M f E } 3 where α W = g and Γ 4 π Z is the total decay width of Z boson. Note that, in general, the production of sum of charged states is considered with the corresponding BR BRZ l ± l ± = ΓZ l l l l Γ Z, 4 and in our numerical analysis we use this branching ratio. 3 Discussion In the model III, there are number of free parameters such as the masses of charged and neutral Higgs bosons, complex Yukawa couplings. The Yukawa couplings in the lepton sector are ξ N,ij D, i, j = e, µ, τ and they should be restricted by present and forthcoming experiments. The first assumption is that ξ D N,ij, i, j = e, µ, are small compared to ξ D N,τ i i = e, µ, τ since the strength of these couplings are related with the masses of leptons denoted by the indices of them, similar to the Cheng-Sher scenario [5]. Second we assume that ξ D N,ij is symmetric with respect to the indices i and j. In our work we study on the decays Z e ± µ ±, Z e ± τ ± and Z µ ± τ ±. In the case of Z e ± µ ± decay we need the couplings ξ D N,µi and ξ D N,ei with i = e, µ, τ. Here we use the first asssumption and neglect the contributions of ξ N,ij D where i, j = µ, e, by taking only the internal τ lepton into account. Now, we should restrict ξ D N,iτ, i = e, µ. For ξ D N,µτ coming from the experimental limits of µ lepton EDM [6], see [4] for details. we use the constraint.3 9 e cm < d µ < 7. 9 e cm 5 For the restriction of the coupling ξ N,µτ D, the deviation of the anomalous magnetic moment AMM of muon over its SM prediction [7] due to the the recent experimental result of muon AMM by g- Collaboration [8], can also be used. However, AMM of muon is posssible in the model III even for vanishing complex Yukawa couplings. In the LFV Z l l decay, the part which depends on the couplings η A i and η i see eq. is non-vanishing when the complex Yukawa couplings are permitted in the model. Here, we choose EDM of muon since the 6

8 restriction is sensitive to complex phases, existing also in the Z l l decay. The coupling ξ D N,eτ is restricted using the experimental upper limit of the BR of the process µ eγ and the above constraint for ξ D N,µτ, since µ eγ decay can be used to fix the Yukawa combination ξ D N,µτ ξ D N,eτ see [4]. This ensures us to determine the upper and lower limits of the coupling ξ D N,eτ. The maximum value of the BRZ µ e is calculated by taking the combination ξ D N,µτ ξ D N,eτ, which respects the upper bound of µ eγ decay. For the minimum value of BRZ µ e, we use the combination ξ D N,µτ ξ D N,eτ if each coupling is at its minimum value. In fact, this minimum value is artificial. Note that, ξ D N,eτ can be restricted and the minimum value of BRZ µe can be obtained by using the experimental result of the EDM of electron [9]. However, we expect that the experimental result of the EDM of electron is not more reliable than the one of the process µ eγ. This analysis shows that ξ D N,µτ, ξ D N,eτ is at the order of the magnitude of 3 4 GeV. Note that these couplings are chosen complex to be able to describe the EDM which is posssible in the case of CP violating interactions. Throughout our calculations we use the input values given in Table. Parameter Value m µ.6 GeV m τ.78 GeV m W 8.6 GeV m Z 9.9 GeV G F GeV Γ Z.49 GeV sin θ W.35 Table : The values of the input parameters used in the numerical calculations. Fig. 3 represents sinθ τe dependence of the maximum minimum value of the BR Z µ ± e ± for sin θ τµ =.5, m h = 7 GeV and m A = 8 GeV. Here the maximum and minimum values are predicted by taking upper and lower limits of µ lepton EDM into account. The maximum minimum value of the BR is 7 3 for small values of sinθ τe and its sensivity to this parameter is weak. BR decreases at the order of the magnitude 5% for sinθ τe.5 and it becomes more sensitive to sinθ τe. sinθ τµ dependence of the maximum minimum value of the BR Z µ ± e ± for sinθ τe =.5, m h = 7 GeV and m A = 8 GeV almost the same as sinθ τe dependence of the BR under consideration. In Fig. 4 we present m A dependence of the minimum value of the BR Z µ ± e ± for sinθ τe =.5, sinθ τµ =.5 and m h = 7 GeV. The BR is strongly sensitive to m A and 7

9 decreases with increasing values of m A. The same dependence appears for the maximum value of the BR. In the Z τ ± e ± decay, the couplings ξ D N,τi and ξ D N,ei with i = e, µ, τ play the main role, in the model III. Again the first assumption permits us to neglect the contributions of the couplings ξ D N,ij where i, j = µ, e, by taking only the internal τ lepton into account. For the coupling ξ D N,eτ the same restriction is used, however for ξ D N,ττ we do not use any constraint. Fig. 5 6 shows sinθ τe and sinθ τµ dependence of the maximum minimum value of the BR Z τ ± e ± for sinθ τµ =.5 and sinθ τe =.5 respectively. Here the coupling ξ D N,ττ is taken as ξ D N,ττ = 3 GeV. The maximum minimum value of the BR for this process is at the order of the magnitude of. BR increases with increasing values sinθ τµ, however this behaviour appears in reverse for sinθ τe dependence. The sensitivity of BR Z τ ± e ± to both CP violating parameters, sinθ τe and sinθ τµ, is strong. We present the coupling ξ D N,ττ dependence of BR Z τ ± e ± in Figs. 7 and 8. These figures shows that the BR enormously increases with increasing values of the coupling ξ D N,ττ. Note that we take the coupling ξ D N,ττ real in our calculations. Finally, we study the Z µ ± τ ± decay by taking τ lepton as an internal one similar to previous analysis. Here the experimental upper limit for BRZ µ ± τ ± can be reached for the small values of ξ N,ττ D. Further, the theoretical calculations are consistent with this upper limit for the case where the mass differences of neutral Higgs bosons h and A are large, even for large values of the coupling ξ D N,ττ. As a summary, we study the BR s of the decays Z e ± µ ±, Z e ± τ ± and Z µ ± τ ± and observe that it is possible to reach the present experimental upper limits in the model III, playing with the model parameters in the restriction region. This result is important since the theoretical work in the SM shows that the branching rates are less than 54 and compared to this number large rates are expected with massive and mixing neutrinos. In our analysis, we predict that the BR for the Z e ± µ ± decay can reach to the values at the order of the magnitude. BR for the process Z e ± τ ± depends strongly on the Yukawa coupling ξ D N,ττ and for its large values, 3 4 GeV, it can be in the range 9. The process Z µ ± τ ± can have larger BR compared to the previous ones, since the Yukawa couplings entering in the expressions are ξ D N,µτ and ξ D N,ττ. Furthermore, BR s of the processes under consideration are sensitive to the CP-violating parameters since the source of the parts which depend on couplings η A i and η i III. are the non-vanishing complex Yukawa couplings, in the model 8

10 In future, with the reliable experimental result of upper limits of the BR s of above processes it would be possible to test models beyond the SM and free parameters of these models 4 Acknowledgement The work of E.O.I was supported by Turkish Academy of Sciences TÜBA/GEBIP. References [] R. Hawkings and K. Mönig, Eur. Phys. J. direct C [] B. Pontecorvo, Zh. Eksp. Teor. Fiz , Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys , B. Pontecorvo, Sov. Phys. JETP [3] T. Riemann and G. Mann, Nondiagonal Z decay: Z eµ, in Proc. of the Int. Conf. Neutrino 8, 4-9 June 98, Balatonfüred, Hungary A. Frenkel and E. Jenik, eds., vol. II, pp. 58-, Budapest, 98, scanned copy at riemann, G. Mann and T. Riemann, Annalen Phys [4] V. Ganapathi, T. Weiler, E. Laermann, I. Schmitt, and P. Zerwas, Phys. Rev. D , M. Clements, C. Footman, A. Kronfeld, S. Narasimhan, and D. Photiadis, Phys. Rev. D [5] Particle Data Group Collaboration, C. Caso et al., Eur. Phys. J. C [6] OPAL Collaboration, R. Akers et al., Z. Phys. C [7] L3 Collaboration, O. Adriani et al., Phys. Lett. B [8] DELPHI Collaboration, P. Abreu et al., Z. Phys. C [9] G. Wilson, Neutrino oscillations: are lepton-flavor violating Z decays observable with the CDR detector? and Update on experimental aspects of lepton-flavour violation, talks at DESY-ECFA LC Workshops held at Frascati, Nov 998 and at Oxford, March 999, transparencies obtainable at and at DESY OXFORD/scans/5 wilson.pdf. 9

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12 h, A j Z i k a b Z i i j h, A k c d Figure : One loop diagrams contribute to Z k j decay due to the neutral Higgs bosons h and A in the HDM. i represents the internal, j k outgoing incoming lepton, dashed lines the vector field Z, h and A fields.

13 BR sin Figure : The maximum value of BR Z µ ± e ± as a function of sinθ τe for sinθ τµ =.5, m h = 7 GeV and m A = 8 GeV.

14 3 BR sin Figure 3: The same as Fig. but for the minimum value of BR Z µ ± e ±.. 3 BR ma GeV Figure 4: The minimum value of BR Z µ ± e ± as a function of m A for sinθ τµ =.5, sinθ τe =.5 and m h = 7 GeV. 3

15 BR BR sin Figure 5: The maximum value of BR Z τ ± e ± as a function of sinθ for ξ D N,ττ = 3 GeV, m h = 7 GeV and m A = 8 GeV. Here solid line represents the dependence with respect to sinθ τµ for sinθ τe =.5 and dashed line to sinθ τe for sinθ τµ = sin Figure 6: The same as Fig. 5 but for the minimum value of BR Z τ ± e ±. 4

16 3.5 9 BR D N; GeV Figure 7: The maximum value of BR Z τ ± e ± as a function of ξ D N,ττ for sinθ τµ =.5, sinθ τe =.5, m h = 7 GeV and m A = 8 GeV. 5 4 BR D N; GeV Figure 8: The same as Fig. 7 but for the minimum value of BR Z τ ± e ±. 5