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Higgs-Mediated e ->mu transitions in II Higgs doublet Model and Supersymmetry

We study the phenomenology of the e-mu lepton flavour violation (LFV) in a general two Higgs Doublet Model (2HDM) including the supersymmetric case. We compute the decay rate expressions of mu ->e gamma, mu ->eee, and mu ->e conversion in nuclei at two loop level. In particular, it is shown that mu ->e gamma is generally the most sensitive channel to probe Higgs-mediated LFV. The correlations among the decay rates of the above processes are also discussed.


Introduction
The Supersymmetric (SUSY) extension of the Standard Model (SM) is one of the most promising candidate for physics beyond the SM. Besides direct searches for SUSY particles, it is also important to analyze implications of such a theory in the low-energy phenomena, through virtual effects of SUSY particles. Lepton flavor violating (LFV) processes are excellent candidates to explore such virtual SUSY effects from low energy experiments [1]. In fact, in a SM framework with massive neutrinos, all the LFV transitions are expected at a very suppressed level, very far from any future and reasonable experimental resolutions. On the other hand, the observation of neutrino oscillation have established the existence of lepton family number violation. This mixing is expected to be manifested also in the charged lepton sector through the observation of rare decay processes such as µ → eγ, τ → µγ etc. Any experimental signals of such a process would clearly indicate the presence of a non standard mechanism. In a supersymmetric (SUSY) framework, new direct sources of flavor violation appear, provided the presence of off-diagonal soft terms in the slepton mass matrices and in the trilinear couplings [2]. In practice, flavor violation would originate from any misalignment between fermion and sfermion mass eigenstates. LFV processes arise at one loop level through the exchange of neutralinos (charginos) and charged sleptons (sneutrinos). The amount of the LFV is regulated by a Super-GIM mechanism that can be much less severe than in the non supersymmetric case [3,4,5] 1 . Another potential source of LFV in models such as the minimal supersymmetric standard model (MSSM) is the Higgs sector. In fact, extensions of the Standard Model containing more than one Higgs doublet generally allow flavor-violating couplings of the neutral Higgs bosons with fermions. Such couplings, if unsuppressed, will lead to large flavor-changing neutral currents in direct opposition to experiments. The MSSM avoids these dangerous couplings at the tree level segregating the quark and Higgs fields so that one Higgs (H u ) can couple only to up-type quarks while the other (H d ) couples only to d-type. Within unbroken supersymmetry this division is completely natural, in fact, it is required by the holomorphy of the superpotential. However, after supersymmetry is broken, couplings of the form QU c H d and QD c H u are generated at one loop [8]. As shown in ref. [9,10], the presence of these loop-induced non holomorphic couplings also leads to the appearance of flavor-changing couplings of the neutral Higgs bosons that are particularly relevant at large values of tan β. As a natural consequence, a variety of flavor-changing processes such as B 0 → µ + µ − [10],B 0 − B 0 [11], K → πνν [13] etc. is generated 2 . Higgs-mediated FCNC can have sizable effects also in the lepton sector [14]: given a source of nonholomorphic couplings, and LFV among the sleptons, Higgs-mediated LFV is unavoidable. These effects have been widely discussed in the recent literature both in a generic 2HDM [15,16,17] and in supersymmetry [17,18,19,20,21] frameworks. In particular, it has been shown that a tree level Higgs-exchange leads to τ → l j l k l k [14], τ → l j η [18], B 0 → l j τ [20] and µ → e conversion in Nuclei [21]. Recently, it was pointed out that Higgs-mediated LFV effects can also generate violations of lepton universality at the 1% level in the R = Γ(K → eν)/Γ(K → µν) ratio [22]. Moreover, Higgs-mediated FCNC can have a sizable impact also in loopinduced processes, such as τ → l j γ [17]. In this letter, we investigate the effects of Higgs mediated LFV in the e − µ transitions both in a generic two Higgs Doublet Model (2HDM) and in Supersymmetry. We evaluate analytical expressions and correlations for the rates of µ → eγ, µ → eee and µ → e conversion in nuclei at two loop level, establishing which the most promising channels to detect LFV signals are. 1 As recently shown in ref. [6], some of these effects are common to many extensions of the SM, even to non-susy scenarios, and can be described in a general way in terms of an effective field theory. Moreover, in the context of general SU (2) L ×U (1) Y seesaw scenarios, large LFV effects can be induced by the exchange of left-handed and/or right-handed neutral singlets [7]. 2 For a recent and detailed analysis of the B physics phenomenology at large tan β within the Minimal Flavor Violating framework, see [12].

LFV in the Higgs Sector
SM extensions containing more than one Higgs doublet generally allow flavor-violating couplings of the neutral Higgs bosons with fermions. Such couplings, if unsuppressed, will lead to large flavor-changing neutral currents in direct opposition to experiments. The possible solution to this problem involves an assumption about the Yukawa structure of the model. A discrete symmetry can be invoked to allow a given fermion type to couple to a single Higgs doublet, and in such case FCNC's are absent at tree level. In particular, when a single Higgs field gives masses to both types of fermions the resulting model is referred as 2HDM-I. On the other hand, when each type of fermion couples to a different Higgs doublet the model is said 2HDM-II. In the following, we will assume a scenario where the type-II 2HDM structure is not protected by any symmetry and is broken by loop effects (this occurs, for instance, in the MSSM). Let us consider the Yukawa interactions for charged leptons, including the radiatively induced LFV terms: where H 1 and H 2 are the scalar doublets, l Ri are lepton singlet for right handed fermions, L k denote the lepton doublets and Y l k are the Yukawa couplings.
In the mass-eigenstate basis for both leptons and Higgs bosons, the effective flavor-violating interactions are described by the four dimension operators: where α is the mixing angle between the CP-even Higgs bosons h 0 and H 0 , A 0 is the physical CP-odd boson, H ± are the physical charged Higgs-bosons and t β is the ratio of the vacuum expectation value for the two Higgs (where we adopt the notation, c x , s x = cos x, sin x and t x = tan x). Irrespective to the mechanism of the high energy theories generating the LFV, we treat the ∆ ij L,R terms in a model independent way. In order to constrain the ∆ ij L,R parameters, we impose that their contributions to LFV processes do not exceed the experimental bounds [17]. On the other hand, there are several models with a specific ansatz about the flavor-changing couplings. For instance, the famous multi-Higgs-doublet models proposed by Cheng and Sher [23] predict that the LFV couplings of all the neutral Higgs bosons with the fermions have the form Hf i f j ∼ √ m i m j .
In Supersymmetry, the ∆ ij terms are induced at one loop level by the exchange of gauginos and sleptons, provided a source of slepton mixing. In the so called MI approximation, the expressions of ∆ ij L,R are given by respectively, where µ is the the Higgs mixing parameter, M 1,2 are the gaugino masses and m 2 L(R) stands for the left-left (right-right) slepton mass matrix entry. The LFV mass insertions (MIs), i.e. δ 3ℓ XX = (m 2 ℓ ) 3ℓ XX /m 2 X (X = L, R), are the off-diagonal flavor changing entries of the slepton mass matrix. The loop function I ′ (x, y, z) is such that I ′ (x, y, z) = dI(x, y, z)/dz, where I(x, y, z) refers to the standard three point one-loop integral which has mass dimension -2 The above expressions, i.e. the Eqs.3,4, depend only on the ratio of the susy mass scales and they do not decouple for large m SU SY . As first shown in Ref. [19], both ∆ ij R and ∆ ij L couplings suffer from strong cancellations in certain regions of the parameter space due to destructive interferences among various contributions. For instance, from Eq.4 it is clear that, in the ∆ ij R case, such cancellations happen if µ = m L .
In the SUSY seesaw model, the MIs of the slepton mass matrix appear in the left-handed sleptons through the neutrino Yukawa interactions. The superpotential of the lepton sector is given by W where N c is the superfields corresponding to the right-handed neutrinos. The neutrino mass matrix is obtained by integrating out the heavy right-handed neutrinos as where v is the vacuum expectation value (VEV) of the Higgs field (v = 246 GeV). The correct size of the neutrino masses is obtained for right-handed neutrinos as heavy as 10 14 GeV for f ν ∼ O(1). The Yukawa coupling Y ν violates the lepton flavor conservation and this violation is communicated to the slepton mass matrix at low-energy. The renormalization group equation (RGE) running effect induces the following off-diagonal components in the left-handed slepton mass matrix where the SUSY breaking parameters m 0 and a stand for the scalar mass and the trilinear scalar coupling at the GUT scale, respectively.
Given our ignorance about the mixings in Y ν , we consider two extremal benchmark cases as discussed, within the SO(10) framework, in [5]. As a minimal mixing case we take the one in which the neutrino and the up-quark Yukawa unify at the high scale, so that the mixing is given by the CKM matrix; this case is named 'CKM-case'. As a maximal mixing scenario we take the one in which the observed neutrino mixing is coming entirely from the neutrino Yukawa matrix, so that Y ν = U P M N S · Y diag u , where U P M N S is the Pontecorvo-Maki-Nakagawa-Sakata matrix; in this case the unknown U e3 PMNS matrix element turns out to be crucial in evaluating the size of LFV effects. The maximal case is named 'PMNS-case'. As regards the δ 21 LL MI one obtains [5] where we set a = 0. So, in the CKM-case, it turns out that δ 21 LL ≃ 3·10 −5 while in the PMNS-case, taking U e3 = 0.07 at about half of the current CHOOZ bound, we get δ 21 LL ≃ 10 −2 .
3 e − µ transitions in the non-decoupling limit In this section, we will analyze e − µ transitions through the study of µ → eγ, µ → eee and µ → e conversion in nuclei in the non-decoupling limit of a 2HDM, where s β−α = 0 and t β is large. In particular, we derive analytical expressions and correlations for the examined branching ratios in order to establish which the most promising channels to detect Higgs mediated LFV are. µ → eγ is generated by a dipole operator arising, at least, from one loop Higgs exchange. However, higgs mediated dipole transitions imply three chirality flips: two in the Yukawa vertices and one in the lepton propagator. This strong suppression can be overcome at higher order level. Going to two loop level, one has to pay the typical g 2 /16π 2 price but one can replace light fermion masses from yukawa vertices with heavy fermion (boson) masses circulating in the second loop [15,24]. In this case, the virtual higgs boson couples only once to the lepton line, inducing the needed chirality flip. As a result, two loop amplitudes provide the major effects and we find that Br(µ → eγ) is given by where N τ,b = 1, 3, q f is the electric charge of the fermion f and a W = m 2 W /m 2 h . The term proportional to F (a W ) arises from two loop effects induced by Barr-Zee type diagrams [24] with a W boson exchange. The loop function F (z) is given by with the Barr-Zee loop integrals given by: For z ≪ 1 it turns out that: The first term of Eq.9 refers to one loop contributions while the last term arises from two loop effects induced by fermionic loops. In the computation, we retained only the h 0 γγ effective vertex neglecting the (1 − 4 sin 2 W ) suppressed contributions arising from the h 0 Zγ vertex. To get a feeling on the relative size among different contributions, we note that two loop fermionic (bosonic) amplitudes are enhanced by an m 2 f /m 2 µ (m 2 W cot β/m 2 µ ) factor with respect to the one loop amplitude. In fact, one gets heavy fermionic (bosonic) masses both from the fermionic (bosonic) propagators and from the Hf f ∼ m f t β (HW W ∼ m W ) couplings. Two loop effects generated by the top quark are generally subdominant. In fact, bearing in mind that any Model II 2HDM predicts that Htt ∼ m t /t β and noting that the top amplitude isn't enhanced by large logarithm factors one finds naively that where ω Al capt ≃ 0.7054 · 10 6 sec −1 . In fact, in contrast to µ → 3e, that is suppressed by the electron mass through the H(A)ēe ∼ m e coupling, µN → eN is not suppressed by the light constituent quark m u and m d but only by the nucleon masses, because the Higgs-boson coupling to the nucleon is shown to be characterized by the nucleon mass using the conformal anomaly relation [25]. In particular, the most important contribution turns out to come from the exchange of the scalar Higgs boson h and H which couples to the strange quark [26] 3 . Moreover, from a previous analysis [17], we know that µ → eγ * (chirality conserving) monopole amplitudes are generally subdominant compared to (chirality flipping) dipole effects. In addition, the enhancement mechanism induced by Barr-Zee type diagrams is effective only for chirality flipping operators so, in the following, we will disregard chirality conserving one loop effects. Let us derive now the approximate relations among µAl → eAl, µ → eγ and µ → eee branching ratios where the approximate expression for F (a W ) is given by Eq.13. In the above equations we retained only dominant two loop effects arising from W exchange. The exact behavior for the examined processes is reported in fig.1 where we can see that µ → eγ gets the largest branching ratio except for a region around m H ∼ 700Gev where strong cancellations among two loop effects sink its size. For a detailed discussion about the origin of these cancellations and their connection with non-decoupling properties of two loop W amplitude, see ref. [15]. On the other hand, µ → eγ amplitude can receive large one loop contributions by a double LFV source, namely by (∆ 21 ) ef f. = ∆ 23 ∆ 31 and 3 As discussed in [21], the coherent µ − e conversion process, where the initial and final nuclei are in the ground state, is expected to be enhanced by a factor of O(Z) (where Z is the atomic number) compared to incoherent transition processes. Since the initial and final states are the same, the elements N |pp|N and N |nn|N are nothing but the proton and the neutron densities in a nucleus in the nonrelativistic limit of nucleons. In this limit, the other matrix elements N |pγ 5 p|N and N |nγ 5 n|N vanish. Therefore, in the coherent µ − e conversion process, the dominant contributions come from the exchange of h and H, not A. therefore, the resulting Br(µ → eγ) is: Now, in contrast to one loop contributions with a single LFV coupling (see the first term of Eq.9), it is always possible to pick up m τ instead of m µ both at the LFV Yukawa vertices and at the fermion propagator. However, if the LFV couplings are generated radiatively (as it happens for instance in a Susy framework), the above enhancement is modulated by the loop suppression. In practice, the dominance of one loop effects (with two LFV couplings) over two loop effects (with one LFV coupling) depends on the specific model we are treating, namely on the size of ∆ ij terms. Assuming that the contributions with a double source of LFV (see Eq.16) dominate over those with a single LFV source (see Eq.9), the following ratios are expected: The µ → eγ * dominance in Br(µAl → eAl) and Br(µ → eee) is the reason of the above correlations. On the other hand, the same correlations are expected, for instance, in a Susy framework with gaugino mediated LFV and then, the predictions of Eq.17 prevent us from distinguishing between the two scenarios. Possible deviations from Eq.17 can arise only through tree level Higgs exchange effects to µAl → eAl.

e − µ transitions in the decoupling limit
The decoupling limit of a 2HDM is a particularly appealing scenario in that it is achieved by Supersymmetry. In this context, the higgs bosons masses are nearly degenerate m A ≃ m H ≃ m H ± being the mass splitting of order O(m 2 Z /m A ) and, in addition, it turns out that c β−α = 0 and m Z /m A → 0. In particular, the couplings of the light Higgs boson h are nearly equal to those of the SM Higgs boson. In a Supersymmetric framework, besides the higgs mediated LFV transitions, we have also LFV effects mediated by the gauginos through loops of neutralinos (charginos)-charged sleptons (sneutrinos). On the other hand, the above contributions have different decoupling properties regulated by the mass of the heaviest scalar mass (m H ) or by the heaviest mass in the slepton gaugino loops (m SU SY ). However, in both cases, the effective operator for l i → l j γ is m l i m 2 H,SU SYl i σ µν l j F µν .
In principle, the m SU SY and m H masses can be unrelated so, we can always proceed by considering only the Higgs mediated effects (assuming a relatively light m H and an heavy m SU SY ) or only the gaugino mediated contributions (if m H is heavy). So, taking into account the only Higgs-mediated effects, we get the following branching ratio for µ → eγ 4 : are squarks mixing angles and the loop function h(z) is given by: The asymptotic form of h(z), which may be useful for an easy understanding of the results, is given by: The first two terms of Eq.18 refer to two loop effects induced by fermionic and W loops, respectively, while the last term appears only in a supersymmetric framework and it is relative to squark loops [27]. In the previous section, we have seen that W effects dominate over the fermionic ones. Moreover, being the H and A masses almost degenerate in the decoupling limit, the H and A contributions partially cancel themselves in the fermionic amplitude because of their opposite signs. This is in contrast to the W amplitude that turns out 4 In a SUSY framework, the couplings between the scalar and the fermions are given by −i( √ 2G F ) 1/2 tan βHξ f m f f f where the parameters ξ f are equal to one at tree level but they can get large corrections from higher order effects. For instance, ξ b gets contributions from gluino-squark loops (proportional to α s t β ) that enhance or suppress significantly the tree level value of ξ b [8]. In the ξ τ case the leading one loop effects induced by chargino-sneutrino contributions (proportional to α w t β ) do not affect ξ τ so significantly. For simplicity's sake, we disregard the above factors in the following.
to be sensitive only to H effects. As regards the squark loop effects, it is very easy to realize that they are negligible compared to W effects. In fact, it is well known that Higgs mediated LFV can play a relevant or even a dominant role compared to gaugino mediated LFV provided that slepton masses are not below the T eV scale while maintaining the Higgs masses at the electroweak scale (and assuming large t β values). In this context, it is natural to assume squark masses at least of the same order as the slepton masses (at the T eV scale) so that xf H ≫ 1. So, even for maximum squark mixings, namely for sin 2θt ,b ≃ 1, and large A b and µ terms, two loop squark effects remain much below the W effects, as it is straightforward to check by Eqs. 18, 19,20. In the decoupling limit, µ → eee and µAl → eAl are still dominated by the two loop µ → eγ * amplitude and by a three level Higgs exchange, respectively. Finally one gets the following relations: As we can note, the above predictions are exactly the same ones we found in the non-decoupling limit and the numerical results are reported in fig.1. However, this property is no longer true for processes associated to e − τ and µ − τ transitions [17]. Let us now consider the one loop contributions to µ − e transitions arising from (∆ 21 ) ef f. = ∆ 23 ∆ 31 . The corresponding Br(µ → eγ) is: where δm = m A − m H . The proportional term to ∆ 23 R ∆ 31 L ∼ δ 23 RR δ 31 LL in eq.16 is enhanced by an m τ /m µ factor compared to the proportional term to ∆ 23 LL . On the other hand, this enhancement is not effective in a Susy framework. In fact, the upper bounds on δ 23 RR δ 31 LL imposed by the gaugino mediated effects to Br(µ → eγ) are stronger than those relative to δ 23 LL δ 31 LL of the same m τ /m µ factor [28], as we will discuss. Differently from the non-decoupling limit case, now one loop effects with two LFV couplings are suppressed by the mass splitting δm/m A . In a SUSY framework, if δm/m A ≃ 10%, ∆ 21 ∼ 10 −3 δ 21 and δ 21 ∼ δ 23 δ 31 we get Br µ→eγ 1−loop , roughly two-three order of magnitude below the Br µ→eγ 2−loop obtained from two loop effects with a single LFV coupling. However, in a generic model II 2HDM, one loop effects can still provide the major effects depending on the size of the ∆ ij terms. In the following, we are interested to make a comparison between Higgs and gaugino mediated LFV effects. To this purpose let us report the branching ratio of l i → l j γ induced by the one loop exchange of neutralinos, charginos and sleptons In the decoupling (non-decoupling) limit m h refers to the heaviest (lightest) Higgs boson mass. Right: Branching ratios of µ → eγ, µ → eee and µAl → eAl in the gaugino mediated LFV case vs a common SUSY mass m SU SY . In both of figures we set t β = 50 and δ 21 LL = 10 −2 (that corresponds, in a generic 2HDM, to ∆ 21 L ≃ 5 · 10 −6 ).
where the A L,R amplitudes are given by As we can see from Eq. (23), the A L amplitude includes both U(1) (the terms proportional to tan 2 θ W ) and SU (2) given by: A particularly interesting feature of the above amplitude is the m τ /m µ enhancement with respect to the usual Bino-like mediated processes. This is due to the implementation of the chirality flip in the internal sfermion line through δ LR 33 ∼ m τ µ tan β and not by δ LR 22 ∼ m µ µ tan β, as usual. were retained because enhanced by a (m τ /m µ ) 2 factor compared to the corresponding effects proportional to ∆ RR,LL
The processes µ → eee and µ-e conversion in Nuclei get contributions not only from penguin-type diagrams (both with photon or Z-boson exchange) but also from box-type diagrams. In fact, the dipole µ → eγ * contribution in these processes is also given by Eqs. 23,24 and therefore is enhanced by a tan β factor. On the other hand the other contributions, monopole or boxes, are not proportional to tan β. So the dipole contributions usually dominate specially in the large tan β regime and one can find the simple theoretical relations In order to make the comparison between Higgs and Chargino mediated LFV effects as simple as possible, let us consider the simple case where all the susy particles are degenerate. In this case, it turns out that In Fig.1 we report the branching ratios of the examined processes as a function of the heaviest Higgs boson mass m H (in the Higgs LFV mediated case) or of the common susy mass m SU SY (in the gaugino LFV mediated case). We set t β = 50 and we consider the PMNS scenario as discussed in Section 2 so that (δ 21 LL ) P M N S ≃ 10 −2 . Subleading contributions proportional to (δ 23 LL(RR) δ 31 RR(LL) ) P M N S (see Eqs.22,25) were neglected since, in the PMNS scenario, it turns out that (δ 23 LL(RR) δ 31 RR(LL) ) P M N S /(δ 21 LL ) P M N S ≃ 10 −3 [5]. As we can see from Fig.1, Higgs mediated effects start being competitive with the gaugino mediated ones when m SU SY is roughly one order of magnitude larger then the Higgs mass m H . Moreover, we stress that, both in the gaugino and in the Higgs mediated cases, µ → eγ gets the largest effects. In particular, within the PMNS scenario, it turns out that Higgs mediated Br(µ → eγ) ∼ 10 −11 when m H ∼ 200GeV and t β = 50, that is just closed to the present experimental resolution.

Conclusions
In this letter we have studied Higgs-mediated LFV e − µ transitions in 2HDM and Supersymmetry frameworks. The sources of LFV were parametrized in a model independent way in order to be as general as possible. In particular, we have analyzed µ → eγ, µ → eee and e − µ conversion in nuclei finding that µ → eγ is generally the most sensitive channel to probe Higgs-mediated LFV. Analytical expressions for the rates of the above processes and their correlations have been established up to two loop level. Particular emphasis was given to the correlations among the processes as an important signature of the theory. In fact, while it is rather difficult to predict the absolute branching ratio value for any given process (depending on the amount of LFV sources and on the mass spectrum), possible correlations with other processes seem to be a more powerful tool to disentangle different scenarios. In this respect, experimental improvements in all the examined e − µ transitions would be very welcome. On the other hand, we have shown that the Higgs-mediated contributions to LFV processes can be within the present or upcoming experimental resolutions and provide an important chance to detect new physics beyond the Standard Model.