Lepton Flavour Violating Decays of Supersymmetric Higgs Bosons

We compute the lepton flavour violating couplings of Higgs bosons in the Minimal Supersymmetric Standard Model, and show that they can induce the decays (h,H,A) -->mu tau at non-negligible rates, for large tan(beta) and sizeable smuon-stau mixing. We also discuss the prospects for detecting such decays at LHC and other colliders, as well as the correlation with other flavour violating processes, such as tau -->mu gamma and tau -->3 mu.


Introduction
The recent important indications of neutrino oscillations [1] reveal that flavour violation also occurs in the lepton sector and further motivate the search for alternative signals of lepton flavour violation (LFV). The Minimal Supersymmetric extension of the Standard Model (MSSM) is a natural framework where several such signals could be significant, provided the mass matrices of the leptons and of the sleptons are not aligned. Well known examples are the LFV radiative decays of charged leptons, µ → eγ, τ → µγ, τ → eγ. In this Letter we would like to explore another class of such processes, namely the LFV decays of the neutral Higgs bosons (h 0 , H 0 , A 0 ). An important feature of these decays is that the corresponding amplitudes do not vanish in the limit of very heavy superpartners, since the leading contributions are induced by dimension-four effective operators, at variance with the case of radiative decays.
Related investigations on flavour violating Higgs couplings in the MSSM framework have mainly focused on processes with virtual Higgs exchange (see e.g. [2,3,4,5]) and regard either quark or lepton flavour violation. The decays of physical Higgs bosons into fermion pairs have been explored in the case of quark flavour violation in the MSSM [6,7], whereas in the case of lepton flavour violation existing studies [8,9] have mainly used phenomenological parametrizations of the LFV couplings 1 .
This Letter is organized as follows. In Section 2 we present the effective LFV Higgs couplings in the MSSM framework, focusing on the second and third lepton generations. We explicitly compute the one-loop contributions to those couplings and the branching ratios of the decays (h 0 , H 0 , A 0 ) → µτ . New results on flavour conserving Higgs couplings are also presented. In Section 3 we give a numerical discussion on the LFV Higgs couplings and branching ratios, and also discuss the prospects at future colliders. Finally in Section 4 we comment on the correlation of the LFV Higgs decays with other LFV processes, such as τ → µγ and τ → 3µ, and summarize our results.

Higgs-muon-tau effective interactions
The MSSM contains two Higgs doublets H 1 and H 2 , with opposite hypercharges. Downtype fermions, which only couple to H 1 at the tree level, also couple to H 2 after the inclusion of radiative corrections [11]. In particular, for the charged leptons of second and third generations the tree-level couplings read as where H 0 1 is the neutral component of H 1 and Y µ , Y τ are the Yukawa coupling constants 2 . Also the leading effective interactions with H 2 , which arise once superpartners are integrated out, are described by dimension-four operators. These can be either flavour conserving (FC): or flavour violating (FV): where ∆ µ , ∆ ′ µ , ∆ τ and ∆ L , ∆ R are dimensionless functions of the MSSM mass parameters, to be described below. In eqs. (2) and (3) we have only retained the dominant terms, proportional to Y τ , besides the first term in ∆L F C proportional to Y µ . In the following we are mostly interested in the effects induced by the terms in (3). In the mass-eigenstate basis for both leptons and Higgs bosons, the FV couplings read as: where tan β = H 0 2 / H 0 1 , α is the mixing angle in the CP-even Higgs sector [ and A 0 is the physical CP-odd Higgs field. The expression in (4) holds up to O(∆ τ tan β) corrections, which arise from eq. (2) and can be O(10%) for large tan β. For our purposes it is not compelling to include and resum such higher-order (tan β-enhanced) terms.
The effective couplings (4) contribute to LFV low-energy processes, such as the decay τ → 3µ and other ones, through Higgs boson exchange [3,4,5]. We will comment later on τ → 3µ. Here we are interested in a more direct implication of those LFV couplings, i.e. the decays Φ 0 → µ ± τ ∓ where Φ 0 = h 0 , H 0 , A 0 . It is straightforward to compute the branching ratios BR(Φ 0 → µ + τ − ) = BR(Φ 0 → µ − τ + ), and it is convenient to relate them to those of the flavour conserving decays Φ 0 → τ + τ − : where the C Φ coefficients are: Since non-negligible effects can only arise in the large tan β limit, in eq. (5) we have approximated 1/ cos 2 β ≃ tan 2 β. We now present explicit expressions for the quantities ∆ L and ∆ R , i.e. the coefficients of the dimension-four operators in (3). The relevant one-loop diagrams, which involve the exchange of sleptons, gauginos and Higgsinos, are shown in Fig. 1. The diagrammatic computation is consistently performed in the gauge symmetry limit, at zero external momentum 3 . In the superfield basis in which the charged lepton mass matrix is diagonal, the mass matrices of the left-handed and right-handed sleptons read: We are interested in scenarios with large LFV, either inM 2 Analogous relations hold for the right-handed sleptons: The mixing parameters satisfy the following relations: and for ∆ R : The function I, which has mass dimension −2, is the standard three-point one-loop integral: Our results for the LFV diagrams in Fig. 1 can be compared with similar ones presented in [3,5]. However, one notices some differences in those works: i) there LFV effects were treated at linear order, through the mass insertion approximation; ii) only LFV in the lefthanded sleptons was considered, since LFV was related to the seesaw generation of neutrino masses; iii) the relative signs between theB diagram and gaugino-Higgsino diagrams differ from ours. This sign is crucial to correctly determine the interference effects, as we will see below. Notice that such a sign discrepancy does not depend on the fact that we use a different sign convention for the µ parameter.
For the sake of completeness we also present the expressions of the FC parameters ∆ µ , ∆ ′ µ , ∆ τ , which are relevant for establishing the relations between the lepton masses (m µ , m τ ) and the corresponding Yukawa couplings. Such quantities are induced by diagrams analogous to those in Fig. 1 but with the same flavour in the external fermion lines (either muon or tau flavour): These formulas are quite general as they include possible LFV in the slepton mass matrices. By setting c L = c R = 1, s L = s R = 0 one can easily recover for ∆ µ and ∆ τ the corresponding cases 4 without LFV, whereas ∆ ′ µ vanishes as this term requires both (LFV) L and (LFV) R . Incidentally, notice that ∆ ′ µ in eq. (2) is multiplied by Y τ . Thereby, if (LFV) L and (LFV) R are both large, the relation between the muon mass and Yukawa coupling could receive large (tan β and Y τ /Y µ enhanced) corrections 5 , and the ratios BR(Φ 0 → µ + µ − )/BR(Φ 0 → τ + τ − ) could differ significantly from the tree level expectation (m µ /m τ ) 2 . However, having simultaneously large (LFV) L and (LFV) R does not seem very natural if the smallness of m µ /m τ is related to an underlying supersymmetric flavour symmetry.

Numerical results and implications at colliders
Now we give some numerical examples to appreciate the size of the effects we are discussing. For definiteness, we discuss separately the case of large (LFV) L , with negligible (LFV) R , and the complementary case of large (LFV) R , with negligible (LFV) L . Let us redefine in (7)m 2 Lτ τ ≡m 2 L andm 2 Rτ τ ≡m 2 R . As a representative case of large (LFV) L , we choosẽ m 2 Lµµ =m 2 L andm 2 Lµτ = 0.8 ·m 2 L , whilem 2 Rµτ ∼ 0. Analogously, for the case of large (LFV) R we choosem 2 Rµµ =m 2 R andm 2 Rµτ = 0.8 ·m 2 R , whilem 2 Lµτ ∼ 0. We show the quantity |50∆ L | 2 as a function of |µ|/m L in Fig. 2 and |50∆ R | 2 as a function of |µ|/m R in Fig. 3, for fixed values of other mass ratios. We have inserted a factor 50 to make it easier the numerical estimate of eq. (5) for the reference case of tan β = 50. The curves depicted exhibit a 4 In this limit of vanishing LFV, different expressions for ∆ τ can be found in the literature [12,13,3], and some discrepancies exist among them. Our result for ∆ τ is consistent with that in [13], taking into account that we use an opposite sign convention for the µ parameter and include left-right slepton mixing at linear order. To our knowledge no explicit expression for ∆ µ or ∆ ′ µ appears in the literature. In principle ∆ µ can be distinct from ∆ τ . 5 In this limit of large (LFV) L and (LFV) R , analogous enhancement effects also appear in the muon magnetic and electric dipole operators, see e.g. [14]. For similar enhancement effects in the relation between quark masses and Yukawa couplings, see e.g. [7].  common behaviour 6 with respect to the ratio |µ|/m L or |µ|/m R : for each curve there is a deep minimum which separates the right-side region, where the pureB 0 diagram dominates as that mass ratio increases (diagram (a) for (LFV) L and diagram (e) for (LFV) R in Fig. 1), from the left-side one in which the Higgsino-gaugino diagrams dominate. The deep wells for either |∆ L | 2 or |∆ R | 2 are due to the destructive interference of the above mentioned diagrams. Notice that the interference would be constructive if the sign of M 1 were opposite to that of M 2 .
In the case of (LFV) L we can see that values of |50∆ L | 2 larger than ∼ 5×10 −4 are achieved both in the left and right ranges in Fig. 2. The example withm R =m L /3 (dashed line) provides larger values in the range |µ|/m L > ∼ 3 since the pureB 0 diagram is further enhanced by the smallerm R . In the case of (LFV) R , values of |50∆ R | 2 larger than ∼ 5 × 10 −4 can be obtained for large values of |µ|/m R (see Fig. 3). An enhancement appears form L =m R /3 (dashed line), in analogy to the (LFV) L example mentioned above. On the other hand, in the left-side region the values of |50∆ R | 2 are smaller with respect to the analogous ones of |50∆ L | 2 . Indeed, in this range |∆ R | 2 is dominated by theH-B diagram (proportional to g ′2 ), while |∆ L | 2 is dominated by theH-W diagrams (proportional to g 2 ).
We now make contact with the physical observable, i.e. the BR(Φ 0 → µ + τ − ) in (5), and discuss the phenomenological implications. We recall that the Higgs boson masses and the angle α in the coefficients C Φ are also affected, through radiative corrections, by a set of MSSM parameters not involved in the determination of ∆ L , ∆ R , such as the mass parameters of the squark-gluino sector (see e.g. [15] and references therein). The latter parameters indirectly affect also the BR(Φ 0 → τ + τ − ) through radiative corrections to BR(Φ 0 → bb) (see e.g. [16,13]). We do not make a definite choice of those parameters and only outline some general features of BR(Φ 0 → µ + τ − ) at large tan β and the prospects for these decay channels at the Large Hadron Collider (LHC) and other colliders 7 . It is convenient to schematically separate the three Higgs bosons into two groups. The CP-odd and one of the CP-even Higgs bosons have about the same mass, non-standard (enhanced) couplings with down-type fermions and suppressed couplings with up-type fermions and electroweak gauge bosons. These bosons, which are mainly contained in H 0 GeV. The other CP-even Higgs has a mass ∼ m ⋆ and Standard Model-like couplings with up-type fermions and electroweak gauge bosons. It is mainly contained in H 0 2 and corresponds to h 0 (H 0 ) for m A > ∼ m ⋆ (m A < ∼ m ⋆ ). Let us discuss the different Higgs bosons, assuming for definiteness tan β ∼ 50, |50∆| 2 ∼ 10 −3 (∆ = ∆ L or ∆ R ) and an integrated luminosity of 100 fb −1 at LHC.
2. If Φ 0 denotes the other (more 'Standard Model-like') Higgs boson, the factor C Φ · BR(Φ 0 → τ + τ − ) strongly depends on m A , while the production cross section at LHC, which is dominated by top-loop mediated gluon fusion, is σ ∼ 30 pb. For m A ∼ 100 GeV we may have C Φ · BR(Φ 0 → τ + τ − ) ∼ 10 −1 and BR(Φ 0 → µ + τ − ) ∼ 10 −4 , which would imply ∼ 300 µ + τ − events. The number of events is generically smaller for large m A since C Φ scales as 1/m 4 A , consistently with the expected decoupling of LFV effects for such a Higgs boson. However, an enhancement can occur under certain conditions. In particular, for a range of m A values the (radiatively corrected) offdiagonal element of the Higgs boson mass matrix could be over-suppressed. In this case the Φ 0 bb c , Φ 0 τ τ c couplings would also be suppressed and as a result the number of µ + τ − events could be even O(10 3 ).
The above discussion suggests that LHC may offer good chances to detect the decays Φ 0 → µτ , especially in the case of non-standard Higgs bosons. This indication should be supported by a detailed study of the background (which is beyond the scope of this work), for instance by generalizing the analyses in [9]. At Tevatron the sensitivity is lower than at LHC because both the expected luminosity and the Higgs production cross sections are smaller. The number of events would be smaller by a factor 10 2 − 10 3 . A few events may be expected also at e + e − or µ + µ − future colliders, assuming integrated luminosities of 500 fb −1 and 1 fb −1 , respectively. At a µ + µ − collider an enhancement may occur for the non-standard Higgs bosons if radiative corrections strongly suppress Y b , since in this case both the resonant production cross section [σ ∼ (4π/m 2 A )BR(Φ 0 → µ + µ − )] and the LFV branching ratios BR(Φ 0 → µ + τ − ) would be enhanced. As a result, for light m A , hundreds of µ + τ − events could occur.

Final remarks and conclusions
A few comments are in order about possible correlations between the decays Φ 0 → µτ and other LFV processes. We have seen that non-negligible rates for Φ 0 → µτ can only be obtained for large tan β and large LFV. In this limit also the decay rate for τ → µγ, which is dominated by diagrams analogous to those of Fig. 1 with an extra photon attached [20], is enhanced and could exceed the experimental limit. However, we recall that the rate of τ → µγ decreases as the superparticle masses increase, whereas the rate of Φ 0 → µτ does not, since the latter is induced by dimension-four effective operators and only depends on mass ratios. Therefore to obtain an adequate suppression of τ → µγ the superparticle spectrum has to be raised towards the TeV region, although some slepton may be lighter. For instance, in the case 1) of (LFV) L shown in Fig. 2 (M 1 = M 2 =m R =m L ), for |µ|/m L ∼ 1 we obtain |50∆ L | 2 ∼ 6 × 10 −4 . In this particular example the present bound BR(τ → µγ) < 6 × 10 −7 [19] constrainsm L > ∼ 1.4 TeV for tan β = 50, which implies min(m L 2 ,m L 3 ) > ∼ 0.6 TeV, max(m L 2 ,m L 3 ) > ∼ 1.9 TeV and M 1 , M 2 ,m R , |µ| > ∼ 1.4 TeV. The decays Φ 0 → µτ are also correlated to the decay τ → 3µ. We recall that the latter receives tan β-enhanced contributions of two types: from dipole LFV operators via photon exchange [20] and from the scalar LFV operators (4) via Higgs exchange [3,5]. The dipole contribution is directly related to the τ → µγ decay rate and is consequently bounded, i.e. BR(τ → 3µ) γ * ∼ 2.3 × 10 −3 BR(τ → µγ) < ∼ 1.4 × 10 −9 . As for the Higgs-mediated contribution, we obtain the following estimate: Therefore, this contribution can exceed the dipole induced one 8 and be not far from the present bound, BR(τ → 3µ) < 3.8 × 10 −7 [21]. Notice that the parameter region in which this occurs is also the most favorable one for the observation of the Φ 0 → µτ decays, so an interesting correlation emerges. Throughout our work we have focused on the second and third generations, implicitly assuming that large slepton mixing only appears in that sector. In a scenario in which staus are mainly mixed with selectrons rather than with smuons, our discussion and numerical estimates concerning Φ 0 → µτ decays can be directly translated to Φ 0 → eτ decays, with obvious substitutions. The case of large smuon-selectron mixing is somewhat different. Although the strong constraints from µ → eγ can be satisfied by taking sufficiently heavy superparticles, the decays Φ 0 → µe are generically suppressed by the presence of Y µ . The latter decays could be Y τ -enhanced if both (LFV) L and (LFV) R were present, and staus were mixed with both smuons and selectrons.
In summary, we have studied the LFV couplings of Higgs bosons in a general MSSM framework, allowing for generic LFV entries in the slepton mass matrices, but without invoking any specific mechanism to generate them. We have computed the branching ratios of Φ 0 → µτ decays, which depend on ratios of MSSM mass parameters, and increase for increasing tan β and LFV. Although cancellations can occur in some regions of parameter space, O(10 −4 ) values are achievable, and they are compatible with the bounds on τ → µγ for a superparticle spectrum in the TeV range. If the Higgs spectrum is relatively light (m A < ∼ 300 GeV), our results indicate that future colliders (in particular LHC) may be able to detect the decays Φ 0 → µτ , especially in the case of non-standard Higgs bosons. Moreover, the detection of these decays is closely correlated with that of τ → 3µ, which may be observed in the near future.