The radiative lepton flavor violating decays in the split fermion scenario in the two Higgs doublet model. 7 0 0 2 E. O. Iltan n ∗ a Physics Department, Middle East Technical University J 4 Ankara, Turkey 2 3 v 3 Abstract 1 0 4 We study the branching ratios of the lepton flavor violating processes µ eγ, τ eγ → → 0 andτ µγ inthesplitfermionscenario,intheframeworkofthetwoHiggsdoubletmodel. 5 → We observe that the branching ratios are relatively more sensitive to the compactification 0 / scale and the Gaussian widths of the leptons in the extra dimensions, for two extra h dimensions and especially for the τ µγ decay. p → - p e h : v i X r a E-mail address: [email protected] ∗ 1 Introduction The radiative lepton flavor violating (LFV) decays exist at least at one loop level and they are rich from the theoretical point of view since the measurable quantities of these decays contain number of free parameters of the model used. In framework of the standard model (SM), these LFV decays are forbidden and, by introducing the neutrino mixing with the non zero neutrino masses, the LFV decays are allowed. However, their branching ratios (BRs) are tiny and much below the experimental limits due to the smallness of neutrino masses. This forces one to go beyond the SM and LFV decays are among the strong candidates for testing the possible new physics effects. There is an extensive experimental and theoretical work done on the radiative LFV decays in the literature. The current limits for the branching ratios (BRs) of µ eγ, τ eγ and → → τ µγ decays are 1.2 10−11 [1], 3.9 10−7 [2] and 1.1 10−6 [3], respectively. The theoretical → × × × analysis of these decays has been performed in various studies [4]-[15]. [4]-[10] were devoted to the analysis in the supersymmetric models. In [11, 12, 13, 14] and [15], they were examined in the framework of the two Higgs doublet model (2HDM) and in a model independent way respectively. WiththeextensionoftheHiggssectorandtheassumptionthattheflavorchanging neutral currents (FCNC’s) at tree level are permitted, the BRs of the radiative decays under consideration can be enhanced theoretically up to the experimental limits. 2HDM with tree level FCNC currents is one of the candidate, and, in this model, the radiative LFV decays are induced by the internal new neutral Higgs bosons h0 and A0. This work is devoted to the LFV processes µ eγ, τ eγ and τ µγ in the 2HDM, → → → with the inclusion of one (two) extra spatial dimension. Here, we choose that the hierarchy of fermion masses is coming from the overlap of the fermion Gaussian profiles in the extra dimensions, so called the split fermion scenario [16, 17]. In this case the fermions are assumed to locate at different points in the extra dimensions with the exponentially small overlaps of their wavefunctions. There are various works done on this scenario in the literature [18]-[35]. The explicit positions of left and right handed components of fermions in the extra dimensions have been predicted in [18]. Using the leptonic W decays and the lepton violating processes, the restrictions on the split fermions in the extra dimensions have been obtained in [19]. The CP violation in the quark sector has been studied in [20] and to find stringent bounds on the size of the compactification scale 1/R, the physics of kaon, neutron and B/D mesons has been analyzed in [21]. [22] is devoted to the rare processes in the split fermion scenario and [23, 24] is related to the shapes and overlaps of the fermion wave functions in the split fermion model. 1 In [25] the electric dipole moments of charged leptons have been predicted, respecting this scenario. In the present work, we study the BRs of the radiative LFV decays by considering that the leptons have Gaussian profiles in the extra dimensions. First, we consider the BRs in the case of a single extra dimension. In the following, we make the same analysis when the number of extra dimensions is two and the charged leptons are restricted to the fifth extra dimension, with non-zero Gaussian profiles. Finally, we estimate the effects of the extra dimensions if the non-zero Gaussian profiles exist in both extra dimensions. We observe that the BRs are relatively more sensitive to the compactification scale and the Gaussian widths of the leptons in the extra dimensions, for two extra dimensions and especially for the τ µγ decay. → The paper is organized as follows: In Section 2, we present the BRs of the radiative LFV decays in the split fermion scenario, in the 2HDM. Section 3 is devoted to discussion and our conclusions. 2 The radiative lepton flavor violating decays in the split fermion scenario in the two Higgs doublet model The radiative LFV decays are worthwhile to study since they exist at least in the loop level and they are rich theoretically. The tiny numerical values of the BRs of these decays in the SM forces one to go beyond and the version of the 2HDM, permitting the existence of the FCNCs and the LFV interactions at tree level, is the simplest candidate. The new Yukawa couplings, which are complex in general, play the main role in the calculation of the physical quantities related to these decays. In addition to the extension of the Higgs sector, the inclusion of the spatial extra dimensions brings additional contributions. Here, we take the effects of extra dimensions into account and we follow the idea that the hierarchy of lepton masses are coming from the lepton Gaussian profiles in the extra dimensions. The Yukawa Lagrangian responsible for these interactions in a single extra dimension, respecting the split fermion scenario, reads ¯ = ξE ˆl φ Eˆ +h.c. , (1) LY 5ij iL 2 jR where L and R denote chiral projections L(R) = 1/2(1 γ ), φ is the new scalar doublet and 5 2 ∓ ξE are the flavor violating complex Yukawa couplings in five dimensions. Here, ˆl (Eˆ ), with 5ij iL jR family indices i,j, are the zero mode1 lepton doublets (singlets) with Gaussian profiles in the 1Notice that we take only the zero mode lepton fields in our calculations. 2 extra dimension y and they read ˆl = N e−(y−yiL)2/2σ2 l , iL iL Eˆ = N e−(y−yjR)2/2σ2 E , (2) jR jR with the normalization factor N = 1 . l (E ) are the lepton doublets (singlets) in four π1/4σ1/2 iL jR dimensions and the parameter σ represents the Gaussian width of the leptons, satisfying the property σ << R, where R is the compactification radius. In eq. (2), the parameters y and iL y are the fixed positions of the peaks of left and right handed parts of ith lepton in the fifth iR dimension and they are obtained by taking the observed lepton masses into account [18]. The underlying idea is that the mass hierarchy of leptons are coming from the relative positions of the Gaussian peaks of the wave functions located in the extra dimension [16, 17, 18]. One possible set of locations for the lepton fields in the fifth dimension read (see [18] for details) 11.075 5.9475 P = √2σ 1.0 , P = √2σ 4.9475 . (3) li   ei   0.0 3.1498    −      We choose the Higgs doublets φ and φ as 1 2 1 0 √2χ+ 1 √2H+ φ = + ;φ = , (4) 1 √2 " v +H0 ! iχ0 !# 2 √2 H1 +iH2 ! with the vacuum expectation values, 1 0 < φ >= ;< φ >= 0 , (5) 1 √2 v ! 2 and collect SM (new) particles in the first (second) doublet. Notice that H and H are the 1 2 mass eigenstates h0 and A0 respectively since no mixing occurs between two CP-even neutral bosons H0 and h0 at tree level in our case. The LFV interaction at tree level is carried by the the new Higgs field φ and it is accessible to extra dimension. Following the compactification 2 on the orbifold S1/Z , it reads 2 1 ∞ φ (x,y) = φ(0)(x)+√2 φ(n)(x)cos(ny/R) , (6) 2 √2πR ( 2 2 ) n=1 X (0) (n) where φ (x) (φ (x)) is the Higgs doublet in the four dimensions (the KK modes) including 2 2 the charged Higgs boson H+ (H(n)+), the neutral CP even-odd Higgs bosons h0- A0 (h0(n)- A0(n)). The non-zero nth KK mode of the charged Higgs mass is m2 +m2, and the neutral H± n CP even (odd) Higgs mass is m2 +m2, ( m2 +m2 ), with thqe n’th level KK particle mass h0 n A0 n q q m = n/R. n 3 The radiative decays under consideration exist at least in the one loop level with the help of the intermediate neutral Higgs bosons h0, A0 and their KK modes (see Fig. 1). The ¯ lepton-lepton-S vertex factors Vn in the vertices fˆ S(n)(x) cos(ny/R)fˆ , with LR(RL)ij iL(R) jR(L) S = h0,A0 and the right (left) handed ith flavor lepton fields fˆ in five dimensions (see eq. jR(L) (2)), are obtained by the integration over the fifth dimension and they read n(y +y ) Vn = e−n2σ2/4R2 e−(yiL(R)−yiR(L))2/4σ2 cos[ iL(R) iR(L) ] . (7) LR(RL)ij 2R In the case of n = 0, the factor becomes V0 = e−(yiL(R)−yiR(L))2/4σ2 and we define the LR(RL)ij Yukawa couplings in four dimensions as ξE (ξE†)† = V0 ξE (ξE )† /√2πR . (8) ij ij LR(RL)ij 5ij 5ij (cid:16) (cid:17) (cid:16) (cid:17) Since the LFV processes , µ eγ, τ eγ and τ µγ exist at loop level, there appear → → → the Logarithmic divergences in the calculations and we eliminate them by using the on-shell renormalization scheme 2. Taking only τ lepton for the internal line 3, the decay width Γ, including a single extra dimension, reads as Γ(f f γ) = c ( A 2 + A 2) , (9) 1 2 1 1 2 → | | | | where 1 ∞ A = Q 6m ξ¯E∗ ξ¯E∗ F(z ) F(z )+2 e−n2σ2/2R2 c′ (f ,τ)c (f ,τ) 1 τ48m2 τ N,τf2 N,f1τ h0 − A0 n 1 n 2 τ (cid:16) nX=1 (F(z ) F(z )) +m ξ¯E∗ ξ¯E G(z )+G(z ) × n,h0 − n,A0 f1 N,τf2 N,τf1 h0 A0 ∞ (cid:17) (cid:16) + 2 e−n2σ2/2R2 c (f ,τ)c (f ,τ)(G(z )+G(z )) , n 1 n 2 n,h0 n,A0 ! nX=1 (cid:17) 1 ∞ A = Q 6m ξ¯E ξ¯E F(z ) F(z )+2 e−n2σ2/2R2 c′ (f ,τ)c (f ,τ) 2 τ48m2 τ N,f2τ N,τf1 h0 − A0 n 2 n 1 τ (cid:16) nX=1 (F(z ) F(z )) +m ξ¯E ξ¯E∗ G(z )+G(z ) × n,h0 − n,A0 f1 N,f2τ N,f1τ h0 A0 ∞ (cid:17) (cid:16) + 2 e−n2σ2/2R2 c′ (f ,τ)c′ (f ,τ)(G(z )+G(z )) , (10) n 2 n 1 n,h0 n,A0 ! nX=1 (cid:17) 2Noticethat,inthisscheme,theselfenergydiagramsforon-shellleptonsvanishsincetheycanbe writtenas ¯ (p)=(pˆ m ) (p)(pˆ m ),however,the vertexdiagramsa andbinFig. 1givenon-zerocontribution. In thiscase,th−edifv1ergencesc−anbfe2eliminatedbyintroducingacountertermVC withtherelationVRen =V0+VC, µ µ µ µ Pwhere VRen (V0)Pis the renormalized (bare) vertex and by using the gauge invariance: kµVRen = 0. Here, kµ µ µ µ is the four momentum vector of the outgoing photon. 3We takeintoaccountonlythe internalτ-leptoncontributionsince,werespectthe Sherscenerio[36],results in the couplings ξ¯E (i,j =e,µ), are small comparedto ξ¯E (i=e,µ,τ), due to the possible proportionality N,ij N,τi of them to the masses of leptons under consideration in the vertices. Here, we use the dimensionful coupling ξ¯E with the definition ξE = 4GF ξ¯E where N denotes the word ”neutral”. N,ij N,ij √2 N,ij q 4 for f (f ) = τ;µ(µ or e;e). Here c = G2Fαemm3f1, A (A ) is the left (right) chiral amplitude, 1 2 1 32π4 1 2 z = m2τ, z = m2τ , Q is the charge of τ lepton and the parameters c , c′ read S m2 n,S m2+(n/R)2 τ n n S S n(y +y ) fR τL c (f,τ) = cos[ ] , n 2R n(y +y ) c′ (f,τ) = cos[ fL τR ]. (11) n 2R In eq. (10) the functions F (w), F (w) are given by 1 2 w(3 4w+w2 +2lnw) F(w) = − , ( 1+w)3 − w(2+3w 6w2+w3 +6wlnw) G(w) = − . (12) ( 1+w)4 − Now, we present the amplitudes A and A appearing in the decay width Γ of the radiative 1 2 LFV decays f f γ (see eq. (9)) in the case of two extra dimensions (see Appendix section 1 2 → for details) 1 ∞ A = Q 6m ξ¯E∗ ξ¯E∗ F(z ) F(z )+4 e−(n2+s2)σ2/2R2 c′ (f ,τ) 1 τ48m2τ τ N,τf2 N,f1τ (cid:16) h0 − A0 Xn,s 2(n,s) 1 c (f ,τ)(F(z ) F(z )) +m ξ¯E∗ ξ¯E G(z )+G(z ) × 2(n,s) 2 (n,s),h0 − (n,s),A0 f1 N,τf2 N,τf1 h0 A0 ∞ (cid:17) (cid:16) + 4 e−(n2+s2)σ2/2R2 c (f ,τ)c (f ,τ)(G(z )+G(z )) , 2(n,s) 1 2(n,s) 2 (n,s),h0 (n,s),A0 Xn,s (cid:17)! 1 ∞ A = Q 6m ξ¯E ξ¯E F(z ) F(z )+4 e−(n2+s2)σ2/2R2 c′ (f ,τ) 2 τ48m2τ τ N,f2τ N,τf1(cid:16) h0 − A0 Xn,s 2(n,s) 2 c (f ,τ)(F(z ) F(z )) +m ξ¯E ξ¯E∗ G(z )+G(z ) × 2(n,s) 1 (n,s),h0 − (n,s),A0 f1 N,f2τ N,f1τ h0 A0 ∞ (cid:17) (cid:16) + 4 e−(n2+s2)σ2/2R2 c′ (f ,τ)c′ (f ,τ)(G(z )+G(z )) , (13) 2(n,s) 2 2(n,s) 1 (n,s),h0 (n,s),A0 Xn,s (cid:17)! where the parameter z is defined as, z = m2τ . In eq. (13) the summation (n,s),S (n,s),S m2+n2/R2+s2/R2 S would be done over n,s = 0,1,2..., except n = s = 0. Furthermore the parameters c (f,τ) 2(n,s) and c′ (f,τ) read 2(n,s) n(y +y )+s(z +z ) fR τL fR τL c (f,τ) = cos[ ] , 2(n,s) 2R n(y +y )+s(z +z ) c′ (f,τ) = cos[ fL τR fL τR ]. (14) 2(n,s) 2R 3 Discussion The radiative LFV decays f f γ exist at least in the one loop level in the 2HDM where 1 2 → the tree level FCNC interactions are permitted. The Yukawa couplings ξ¯E , i,j = e,µ,τ, N,ij 5 which are the free parameters of the model, play an essential role on the physical parameters of these decays. Since we follow the idea that the hierarchy of lepton masses are due to the lepton Gaussian profiles in the extra dimensions, there appear exponential suppression factors, originated from the different locations of various flavors and their left and right handed parts of lepton fields, in the Yukawa part of the lagrangian, after the integration of the extra dimension(s) (see the eq. (7) (eq. (17)) for n = 0 (n,s = 0)). We take the Yukawa couplings in four dimensions as the combination of these new factors and the higher dimensional one (see eq. (8) and (18)) and consider that the couplings ξ¯E , i,j = e,µ are smaller compared to N,ij ξ¯E i = e,µ,τ since latter ones contain heavy flavor. Furthermore, we assume that, in four N,τi dimensions, the couplings ξ¯E is symmetric with respect to the indices i and j. N,ij Our analysis is devoted to the prediction of the effects of the extra dimensions on the LFV radiative decays. Here we choose the appropriate numerical values for the Yukawa couplings, by respecting the current experimental measurements of these decays (see Introduction section) and the muon anomalous magnetic moment (see [37] and references therein). Notice that, for the Yukawa coupling ξ¯E , we use the numerical value which is greater than the upper limit N,ττ of ξ¯E . We also study the effects of the parameter ρ = σ/R, where σ is the Gaussian width N,τµ of the fermions (see [18] for details). For the compactification scale 1/R, there are numerous constraints for a single extra dimension in the split fermion scenario. The direct limits from searching for KK gauge bosons imply 1/R > 800 GeV. The precision electro weak bounds on higher dimensional operators generated by KK exchange place a far more stringent limit 1/R > 3.0 TeV [38]. In [22], the lower bounds for the scale 1/R have been obtained as 1/R > 1.0 TeV from B φK , 1/R > 500 GeV from B ψK and 1/R > 800 GeV from S S → → the upper limit of the BR, BR(B µ+µ−) < 2.6 10−6. We make our analysis by choosing s → × an appropriate range for the compactification scale 1/R, by respecting these limits in the case of a single extra dimension. For two extra dimensions we used the same broad range for 1/R. Inthemodelweusetherearevariousfreeparameters, thenewYukawa couplings, themasses of new Higgs bosons and the ones coming from the split fermion scenario, namely, the com- pactification scale and the possible locations of fermions in the extra dimensions. It is obvious that the predictions are sensitive to those parameters and it is not easy to decide whether the enhancement comes from the broad region of one parameter set belonging to the 2HDM part, or the one belonging to the split fermion scenario. However, with the possible forthcoming experimental results of the processes which are more sensitive to the new parameters coming from the new Higgs doublet, the more stringent restrictions can be obtained. This would lead 6 to more accurate analysis of the effects due to the split fermion scenario. In our case, we try to estimate the sensitivity of the BRs to the split fermion scenario by fixing the other param- eters and we expect that the more accurate discussion of this scenario can be reached with forthcoming experiments which reduces the sensitivities at several orders. In the present work, we take split leptons in a single and two extra dimensions and use a possible set of locations to calculate the extra dimension contributions. We make the analysis in one and two extra dimensions. In the case of a single extra dimension (two extra dimensions) we use the estimated location of the leptons given in eq. (3) (eq. (19)) to calculate the lepton- lepton-Higgs scalar KK mode vertices. For two extra dimensions, first, we take that the leptons arerestricted to the fifthextra dimension, with non-zero Gaussian profiles. This is the case that the enhancement in the BRs of the present decays is relatively large. The reason beyond the enhancement is the well known KK mode abundance of Higgs fields. Finally, we assume that the leptons have non-zero Gaussian profiles also in the sixth dimension and using a possible set of locations in the fifth and sixth extra dimensions (see eq. (19)), we calculated BRs of the decays under consideration. In this case the additional the exponential factor appearing in the second summation further suppresses the BRs. In Fig. 2 (3 ; 4), we plot the BR of the decay µ eγ (τ eγ ; τ µγ) with respect to the → → → compactificationscale1/Rforρ = 0.01,m = 100GeV,m = 200GeV andtherealcouplings h0 A0 ξ¯E = 10GeV, ξ¯E = 0.001GeV (ξ¯E = 100GeV, ξ¯E = 1GeV ; ξ¯E = 100GeV, N,τµ N,τe N,ττ N,τe N,ττ ξ¯E = 10GeV)4. Here the solid (dashed, small dashed, dotted) line represents the BR without N,τµ extra dimension (with a single extra dimension, with two extra dimensions where the leptons have non-zero Gaussian profiles in the fifth extra dimension, with two extra dimensions where the leptons have non-zero Gaussian profiles in both extra dimensions). It is observed that BR is weakly sensitive to the parameter 1/R for the 1/R > 500GeV for a single extra dimension. The enhancement of the BR is relatively larger for two extra dimensions due to the Higgs scalar KK mode abundances and, in this case, the sensitivity is weak for 1/R > 1.0TeV. However, these contributions do not increase extremely due to the suppression exponential factor appearing in the summations. Furthermore, the numerical values of BRs are slightly greater in the case that the leptons have non-zero Gaussian profiles only in the fifth extra dimension. Now, we would like to present the amount of the enhancements in the BRs of the decays we study, by taking the existing bounds of the compactification scale 1/R into account and to 4For τ eγ we take the numerical value of the coupling ξ¯E , ξ¯E = 1GeV. Here we try to reach the → N,τe N,τe new experimental result of the BR of this decay (see [2]). With the more sensitive future measurements of the BRs of these decays these couplings would be fixed more accurately. 7 discuss the possibility of observations of these additional contributions. For µ eγ (τ eγ ; → → τ µγ) decay, the enhancement in the BR for a single extra dimension is at the order of 0.7% → (1.0% ; 1.3%) for 1/R 800GeV, compared to the case where there is no extra dimension. ∼ For the greater value of the scale 1/R, 1/R 3.0TeV, the enhancement reads 0.01% (0.03% ; ∼ 0.05%). These numbers show that the enhancement in the BR of the decay τ µγ is larger → compared to the BRs of the others and this decay is probably the better candidate among the present LFV decays to detect the effects of the extra dimensions for a single extra dimension. Notice that, for 1/R 3.0TeV, the enhancements in the BRs of these decays are weak and ∼ difficult to observe. For two extra dimensions, where the leptons have non-zero Gaussian profiles in the fifth extra dimension, the enhancement in the BR of the decay µ eγ (τ eγ ; τ µγ) is at → → → the order of the magnitude of 3.6% (9.5% ; 10.6%) for 1/R 800GeV, compared to the case ∼ that there is no extra dimension. For 1/R 3.0TeV, the enhancement reads 0.05% (0.5% ; ∼ 0.6%) 5. This shows that the additional contributions cause to increase the BR at the order of 10% for τ eγ and τ µγ decays, in the case of 1/R 800GeV. Therefore, for ∼ → → ∼ two extra dimensions, τ eγ and τ µγ decays can ensure valuable information on the → → effects of extra dimensions in the split fermion scenario. Notice that, the lower bound of the compactification scale probably will be different in the case of two extra dimensions, compared to the one existing for a single extra dimension. However, in our calculations, we used the same broad range of the scale 1/R for a single and two extra dimensions. Fig. 5 (6 ; 7) is devoted to the parameter ρ dependence of the BR of the decay µ eγ → (τ eγ ; τ µγ) for 1/R = 500GeV, m = 100GeV, m = 200GeV and the real couplings h0 A0 → → ξ¯E = 10GeV, ξ¯E = 0.001GeV (ξ¯E = 100GeV, ξ¯E = 1GeV ; ξ¯E = 100GeV, N,τµ N,τe N,ττ N,τe N,ττ ξ¯E = 10GeV). Here the solid (dashed, small dashed, dotted) line represents the BR without N,τµ extra dimension (with a single extra dimension, with two extra dimensions where the leptons have non-zero Gaussian profiles in the fifth extra dimension, with two extra dimensions where the leptons have non-zero Gaussian profiles in both extra dimensions). The BR of the decay µ eγ (τ eγ ; τ µγ) changes 0.8% (1.0% ; 0.8%) in the given interval of the parameter → → → ρ and for 1/R = 500GeV, in the case of a single extra dimension. This shows that the BRs of the present decays are weakly sensitive to ρ, which is the main parameter controlling the Gaussian widths and the possible locations of the fermions in the extra dimensions. For two 5For two extra dimensions where the leptons have non-zero Gaussian profiles in both extra dimensions, the enhancement in the BR is at the orderof the magnitude of 3.5%(8.6% ; 9.7%) for 1/R 800GeV, for µ eγ ∼ → (τ eγ ; τ µγ) decay, compared to the case that there is no extra dimension. For 1/R 3.0TeV, the → → ∼ enhancement reads 0.05% (0.4% ; 0.5%). 8 extra dimensions, the sensitivity of the BRs to the parameter ρ increases. For the decay µ eγ → (τ eγ ; τ µγ) the BR changes 3.6% (19.0% ; 15.0%) in the given interval of the parameter → → ρ and for 1/R = 500GeV, in the case of two extra dimensions where the leptons have non-zero Gaussian profiles in the fifth extra dimension. For two extra dimensions where the leptons have non-zero Gaussian profiles in both extra dimensions, the BR of the decay µ eγ (τ eγ ; → → τ µγ) changes 5.1% (24.0% ; 18.0%) in the given interval of the parameter ρ. It is shown → that the sensitivities of the BRs of studied LFV decays are relatively greater for two extra dimensions and this sensitivity increases for the case where the leptons have non-zero Gaussian profiles in both extra dimensions. In addition to this, the BRs of tau decays are sensitive to the parameter ρ and this sensitivity increases for ρ < 0.05, in the case of two extra dimensions. At this stage we would like to discus the possibilities of detecting the additional contri- butions due to the extra dimensions with the present and possible forthcoming experimental measurements. The experimental work for the lepton flavor violating decays has been done since the discoveries of heavy leptons. A new experiment, to search for the lepton flavor vi- olating decay µ eγ [39] at PSI has been described and the aim of the experiment was to → reach to a sensitivity of BR = 10−14, improved by three order of magnitudes with respect to previous searches. At present the experiment (PSI-R-99-05 Experiment) to search the µ eγ → decay is still running in the MEG [40]. For the τ µγ decay, recently, an upper limit of → BR = 9.0(6.8)10−9 at 90% CL has been obtained [41] ([42]) and this result is an improvement almost by oneorder ofmagnitudes with respect toprevious one. The futuremeasurement ofthe radiative µ eγ decay with the sensitivity of BR = 10−14, hopefully, would make it possible → to detect possible the additional contributions, especially, coming from two extra dimensions, even for the scale 1/R 3.0TeV. On the other hand, for the decay τ µγ, the enhancement ∼ → in the BR in the case of two extra dimensions is at the order of the magnitude of 0.6% ∼ for 1/R 3.0TeV, and the one order improved experimental value of the BR would ensure a ∼ possible detection of the extra dimension effects even for large values of the compactification scale 1/R. As a summary, the BR is weakly sensitive to the parameter 1/R for 1/R > 500GeV for a single extra dimension, however, this sensitivity increases for two extra dimensions. The expo- nential suppression factor appearing in the summations reduces the extreme enhancement due to the Higgs KK mode abundance. Furthermore, the BR is weakly sensitive to the parameter ρ especially for a single extra dimension case. In the two extra dimensions, this sensitivity is slightly larger compared to the one in the single extra dimension. With the help of the forth- 9