Tau Flavour Violation in Sparticle Decays at the LHC

We consider sparticle decays that violate tau lepton number, motivated by neutrino oscillation data. We work in the context of the constrained minimal supersymmetric extension of the Standard Model (CMSSM), in which the different sleptons have identical masses at the GUT scale, and neutrino Dirac Yukawa couplings mix them. We find that the branching ratio for decay of the heavier neutralino chi_2 to chi + tau mu is enhanced when the LSP mass m_chi = m_stau_1, including the region of CMSSM parameter space where coannihilation keeps the relic chi density within the range preferred by cosmology. Thus chi_2 to chi + tau mu decay may provide a physics opportunity for observing the violation of tau lepton number at the LHC that is complementary to tau to mu + gamma decay. Likewise, chi_2 to chi + e mu decay is also enhanced in the coannihilation region, providing a complement to mu to e + gamma decay.


Introduction
Mixing between different neutrino flavours has now been amply confirmed by experiments on both atmospheric [1] and solar [2,3] neutrinos. The distorted zenith-angle distributions observed by Super-Kamiokande provided a 'smoking gun' for atmospheric-neutrino oscillations, establishing that they are most likely due to near-maximal ν µ − ν τ mixing. More recently, SNO has provided two 'smoking guns' for solar-neutrino oscillations, providing direct evidence for near-maximal ν e → ν µ,τ oscillations [4] through its measurements of the charged-and neutralcurrent scattering rates.
These observations lead one to expect the corresponding charged-lepton numbers to be violated at some level. However, the rates for such processes would be unobservably small if neutrino masses were generated by the seesaw mechanism [5] and there was no lower-energy physics beyond the Standard Model. However, the naturalness of the gauge hierarchy, grand unification of the gauge couplings and the relic density of supersymmetric dark matter all suggest that supersymmetry should appear at an energy scale < ∼ 1 TeV. This suggests that processes violating the different charged lepton numbers might be observable in low-energy experiments. Indeed, charged-lepton-number violating processes could occur at embarrassingly large rates if the soft supersymmetry-breaking masses of the squarks and sleptons were not universal. For this reason, it is often assumed that these masses are equal at the grand-unification scale, as in the constrained minimal supersymmetric extension of the Standard Model (CMSSM).
Even in this case, renormalization of the soft supersymmetry-breaking slepton masses would occur in the minimal supersymmetric version of the seesaw model for neutrino masses, thanks to the Dirac Yukawa couplings of the neutrinos [6]. These are active in the renormalization-group equations at scales between the GUT scale and the heavy singlet-neutrino mass scale, and are not expected to be diagonal in the same basis where the light leptons are flavour-diagonal. This scenario provides the minimal credible amount of charged-lepton-flavour violation: it could be enhanced by GUT interactions and/or non-universal slepton masses at the GUT scale.
Many signatures for charged-lepton-flavour violation have been considered in this scenario [7,8,9,10,11,12], including µ → eγ and related decays, τ → µγ and τ → eγ decays. Any or all of these may be favoured by the (near-)maximal mixing observed amongst the corresponding neutrino species. Other things being equal, one expects these decays to be relatively large when the soft supersymmetry-breaking masses m 1/2 and/or m 0 are relatively small, as has been borne out in specific model-dependent studies. Another possibility that has been considered is the decay χ 2 → χ+e ± µ ∓ [13,14], where χ is the lightest neutralino, assumed here to be the lightest supersymmetric particle (LSP), and χ 2 is the second-lightest neutralino. It has been argued that this decay might have a rate observable at the LHC for certain choices of the CMSSM parameters.
In this paper, we consider the alternative decay χ 2 → χ+τ ± µ ∓ [15]. This has certain theoretical advantages over the decay χ 2 → χ + e ± µ ∓ considered previously, since the feedthrough into the charged-lepton sector may be enhanced by larger Dirac Yukawa couplings and/or lighter singlet-neutrino masses, as compared to the ν µ − ν e sector, if neutrino masses exhibit the expected hierarchical pattern, and ν τ − ν µ mixing is also known to be essentially maximal. On the other hand, the decay χ 2 → χ + τ ± µ ∓ has a less distinctive experimental signature than χ 2 → χ + e ± µ ∓ . Both decays should be explored at the LHC and a possible linear e + e − collider, and which mode offers better prospects may depend on the neutrino-mass model and the experiment.
We find that the branching ratio for χ 2 → χ + τ ± µ ∓ decay is enhanced when m χ 2 > mτ 1 > m χ , whereτ 1 is the lighter stau slepton. This occurs in a wedge of the (m 1/2 , m 0 ) parameter plane in the CMSSM that is complementary to that explored by τ → µγ. The region of CMSSM parameter space where this enhancement occurs includes the region where χ −l coannihilation suppresses the relic density Ω χ , keeping it within the range 0.1 < Ω χ h 2 < 0.3 preferred by astrophysics and cosmology, even if m 1/2 is comparatively large. The interest of this coannihilation region has been accentuated by the latest experimental constraints on the CMSSM, such as m h and b → sγ decay, which disfavour low values of m 1/2 . We show that the branching ratio for χ 2 → χ + τ ± µ ∓ decay may be a large fraction of that for the flavourconserving decay χ 2 → χ + µ ± µ ∓ . An analogous enhancement is expected for the flavourviolating decay χ 2 → χ + e ± µ ∓ considered by other authors [13,14], although the absolute branching ratio is expected to be smaller. Nevertheless, this decay may provide another way of probing lepton flavour violation in the coannihilation region.

Calculational Framework
We assume the minimal supersymmetric extension of the seesaw mechanism for generating neutrino masses, in which there are three heavy singlet-neutrino states N i , and the leptonic sector of the superpotential is: where Y ν is the neutrino Dirac Yukawa coupling matrix, M ij is the Majorana mass matrix for the N i , the L j and H I are lepton and Higgs doublets, and the E c i are singlet chargedlepton supermultiplets. The superpotential of the effective low-energy theory, obtained after the decoupling of heavy neutrinos is [16] In the basis where the charged leptons and the heavy neutrino mass matrices are diagonal, where the v = 174 GeV and tan β = v 2 /v 1 .
As mentioned above, we work in the context of the CMSSM, where the soft supersymmetrybreaking masses of the charged and neutral sleptons are assumed to be universal at the GUT scale, with a common value m 0 . In the leading-logarithmic approximation, the non-universal renormalization of the soft supersymmetry-breaking scalar masses is by an amount We note that, in this approach, non-universality in the soft supersymmetry-breaking left-slepton masses is much larger than that in right-slepton masses when the trilinear soft supersymmetrybreaking parameter A 0 = 0, as we assume here 1 . The pattern of charged-lepton-flavour violation induced by renormalization depends on the details of (Y ν ) ij .
In plausible mixing textures, the renormalization of the soft supersymmetry-breaking parameters at low energies can be understood approximately in terms of the dominant non-universality in the third-generation left-slepton mass: where a typical value of the non-universality factor is x ∼ 0.9. Correspondingly, we assume there is an off-diagonalτ L −μ L mixing term in the soft mass-squared matrix: where φ is the mixing angle between the second and third generation in the charged-lepton Yukawa matrix. For the type of non-universalities introduced in (5), this angle can be quite large without entering in conflict with the current bounds for τ → µγ, though in this case large mixing in the 2-3 sector must be combined with a small mixing angle between the first and second generation, due to the very restrictive bound in the µ → eγ decay [17]. This mixing leads to lepton-flavour violation ∼ sin 2 (2φ), as long as sin(2φ) is not too large 2 .
We give below numerical results for sample choices of the parameters (x, φ) that may be representative of the possibilities in specific models. We also show how the results vary as (x, φ) are varied.
In the following, we consider mixing between the left-handed τ -and µ-flavoured sleptons, but τ −ẽ mixing might also be present, or even favoured in some models. In such cases, the results would be rather similar to those we present, simply with µ replaced by e in the L τ -violating decay modes studied.
We consider the following flavour-violating and -conserving χ 2 decays: 1 In the case A 0 = 0, this parameter would also be renormalized analogously to m 2 L (4). 2 We have checked that this parametrization is generally a very good approximation for τ → µγ decay, as well as for χ 2 → χτ ± µ ∓ .
The first two decays are the only ones in which flavour violation may be expected, and it would of course be unobservable in χ 2 → χνν decay. The intermediate sleptons are produced on-shell if they are lighter than the χ 2 , while the Z and the h are always on-shell for the range of parameters that are of interest to us. Slepton exchanges and h decays may give significantly different rates for the various flavor-conserving decays χ 2 → χℓ + i ℓ − i , suppressing the cases ℓ = µ, e relative to the case ℓ = τ , an effect we see in subsequent plots.
Our calculations are similar to [14], except that we also include the Yukawa interactions, which are relevant for decays into τ leptons at large tan β. Furthermore, we include finite-width effects in our calculations of Z 0 and slepton exchanges. The neutralino and slepton widths, which arise mainly from two-body decays, were calculated using the ISAJET package [18], and a check with calcHEP [19] found good agreement. For the decays χ 2 → χ + µ ± µ ∓ , we found good agreement between our code and calcHEP, once we incorporated the VEGAS adaptive Monte Carlo programme for the momentum integrals in three-body decays. The results from VEGAS differ by several orders of magnitude from those obtained using ISAJET for χ 2 → χ + µ ± µ ∓ decay close to theτ resonances.
For the decay χ 2 → χ + τ ± τ ∓ the channels mediated by Higgses are important in the areas where mχ 2 − mχ < mτ 1 . The widths have been obtained using calcHEP, after adding to the package the one-loop QCD corrected Higgs widths from HDECAY [20]. For flavour-violating decays, our calculation agrees with calcHEP, once we modify the MSSM Lagrangian included in this package to allow flavour-mixing amongτ 1 ,τ 2 andμ L .
To quite a good approximation, R(τ µ/µµ) scales by the square of the factor (1 − x) sin(2φ) shown in (6). This makes it relatively easy to reinterpret our illustrative results in the context of any specific flavour texture model that makes definite predictions for x and φ.
The darker (green) shaded regions in the bottom right corners of the panels in Fig. 3 are excluded because there the LSP is the lighter stau:τ 1 . Such a charged LSP would be in conflict with basic astrophysics. The lighter (grey) shaded regions are those in which the cosmological relic density of the neutralino LSP χ is in the range preferred by cosmology: 0.1 < Ω χ h 2 < 0.3 as calculated using MICROMEGAS [23], and in agreement with our previous calculations [24,25,26], where h ∼ 0.7 is the current Hubble expansion rate in units of 100 km/s/Mpc. In each panel, tan β = 40, µ > 0, for x = 0.9 and φ = π/6. The regions disallowed at low m 1/2 and m 0 by the E821 measurement of a µ at the 2-σ level [28] are dark (brown) shaded, the dark (green) shaded regions at large m 1/2 and low m 0 are excluded because the LSP is the chargedτ 1 , the light grey shaded regions are those with 0.1 < Ω χ h 2 < 0.3 that are preferred by cosmology (calculated using MICROMEGAS [23]), and the medium (blue) shaded regions are excluded by b → sγ [25] and the dotted line is m h = 114.1 GeV (calculated using FeynHiggs [27]).
there is a region at small m 1/2 that is disfavoured by laboratory experiments. The regions at small m 1/2 excluded by the b → sγ decay rate are medium (blue) shaded, the regions disfavoured by g µ − 2 at small m 1/2 and m 0 are darker (brown) shaded, and the (dotted) line is where m h = 114.1 GeV as calculated using FeynHiggs [27]. Together, these constraints favour the coannihilation strip where mτ 1 ∼ m χ in all the panels, and the channels at large m 1/2 and m 0 in panel (d) where direct-channel χχ → A, H annihilation is relatively rapid.
In cases (a, b, c), the only region of the (m 1/2 , m 0 ) plane that survives these constraints is the strip parallel to the boundary of the disallowed region, where m χ /mτ 1 ∼ 1.1 − 1.2, and coannihilation keeps Ω χ h 2 within the range allowed by astrophysics and cosmology. This is precisely the region where R(τ µ/µµ) is maximized, and hence the chances of observing the decay may be maximized. We do note, however, that R(τ µ/µµ) has a tendency to fall as m 1/2 increases along this strip, which is apparent in panels (c) and (d). We further note in panel (d) that R(τ µ/µµ) > ∼ 10 −2 also on the right side of the rapid χχ → A, H annihilation channel, but may be significantly lower on the left side of this channel.

Conclusions
We have demonstrated in this paper that the decay χ 2 → χτ ± µ ∓ provides an opportunity to look for τ flavour violation at the LHC that is largely complementary to the search for τ → µγ. Essentially all the above analysis would apply also if the slepton mixing texture favours χ 2 → χτ ± e ∓ and τ → eγ over χ 2 → χτ ± µ ∓ and τ → µγ: it is even possible that both χ 2 → χτ ± µ ∓ /e ∓ decays may be observable at the LHC.
We have phrased this analysis as model-independently as possible. Specific models will predict values for the mixing parameters x and φ, and the scaling of our results with these parameters is quite simple. We would expect the relevant mixing parameters to be much smaller in the case of χ 2 → χµ ± e ∓ decay, but the corresponding R(µe/µµ) would be enhanced in a similar region of the CMSSM parameter space.
We note that the µ ∓ produced in χ 2 → χτ ± µ ∓ decay are likely to have significant transverse momentum, and any event in which the χ 2 is produced is likely to have considerable missing transverse energy and jet activity associated with the decays of other sparticles. Therefore, we do not expect such events to be suppressed badly at the trigger level at the LHC, though it might be more difficult to see χ 2 → χτ ± e ∓ decays. However, a detailed simulation goes beyond the scope of this paper. There should be even less problem seeing χ 2 → χ + τ ± µ/e ∓ decays at a linear e + e − collider. We therefore urge more detailed simulations of this decay mode for this machine, as well as for the LHC.