Probing the supersymmetric type III seesaw: LFV at low-energies and at the LHC
aa r X i v : . [ h e p - ph ] A p r LPT Orsay 11-19CFTP/11-008PCCF RI 1102November 19, 2018
Probing the supersymmetric type III seesaw: LFV atlow-energies and at the LHC
A. Abada a , A. J. R. Figueiredo b , J. C. Rom˜ao b and A. M. Teixeira ca Laboratoire de Physique Th´eorique, CNRS – UMR 8627,Universit´e de Paris-Sud 1, F-91405 Orsay Cedex, France b Centro de F´ısica Te´orica de Part´ıculas, Instituto Superior T´ecnico,Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal c Laboratoire de Physique Corpusculaire, CNRS/IN2P3 – UMR 6533,Campus des C´ezeaux, 24 Av. des Landais, F-63171 Aubi`ere Cedex, France
Abstract
We consider a supersymmetric type III seesaw, where the additional heavy states are embed-ded into complete SU(5) representations to preserve gauge coupling unification. Complyingwith phenomenological and experimental constraints strongly tightens the viable parameterspace of the model. In particular, one expects very characteristic signals of lepton flavourviolation both at low-energies and at the LHC, which offer the possibility of falsifying themodel.
KEYWORDS: Supersymmetry, LHC, slepton mass splittings, lepton flavour violation, seesaw mechanism
Introduction
The seesaw mechanism, in its different realisations, constitutes one of the simplest and yet mostelegant ways to explain neutrino masses and mixings. In the minimal realisations of the seesaw,the Standard Model (SM) can be extended by the addition of fermionic singlets (type I seesaw) [1],scalar triplets (type II) [2] or fermionic triplets (type III) [3]. Although dependent on the sizeof the neutrino Yukawa couplings ( Y ν ), these new states are in general heavy: assuming nat-ural couplings, Y ν ∼
1, their masses can be close to the Grand Unified Theory (GUT) scale, O (10 GeV).If these states are indeed at the origin of neutrino mass generation, it is important to investigatewhich seesaw realisation (or combination thereof) is at work. Indeed, if the mass of the mediatorsis such that production at present colliders is possible (in this case Y ν ∼ − ), then one candevise strategies for their direct searches. On the contrary, if they are very heavy, then theycannot be directly probed, and their indirect signatures in low-energy observables (typically viahigher order corrections) will be extremely suppressed.Other than the mechanism of neutrino mass generation, there are several reasons - theoreticalissues and observational problems - motivating the extension (or embedding) of the SM into alarger framework. Supersymmetry (SUSY) is a well motivated solution for the hierarchy prob-lem that also offers an elegant solution for the non-baryonic dark matter (DM) problem of theUniverse [4–6]. If the Large Hadron Collider (LHC) indeed finds signatures of SUSY, it is thenextremely appealing to consider the embedding of a seesaw mechanism into a supersymmetricframework (the so-called SUSY seesaw).Supersymmetric seesaws lead to a number of possible signatures in the neutral and chargedlepton sectors, both at low and high energies. Among low-energy observables, the most strikingSUSY seesaw impact is perhaps the possibility of having charged lepton flavour violating (LFV)transitions. Indeed, one can have sizable contributions to radiative decays ( ℓ i → ℓ j γ ), three-bodydecays ( ℓ i → ℓ j ) and µ − e transitions in heavy nuclei, well within reach of current and/or futurededicated facilities [7–29]. At high-energy colliders, such as the LHC, several observables mayreflect an underlying SUSY seesaw. Let us begin by noticing that if some components of the seesawmediators are not singlets under the SM gauge group (which is the case in type II and III seesaws),the latter can leave an imprint on the SUSY spectrum, since they can modify the supersymmetric β − functions governing the evolution of the gauge couplings and soft-SUSY breaking parameters.At the LHC, SUSY seesaws can also give rise to several LFV signals: firstly, one can have sizablewidths for LFV decay processes like χ → χ ℓ ± i ℓ ∓ j [27, 30–33]; secondly, one can have observableflavoured slepton mass splittings (MS), ∆ m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ) and ∆ m ˜ ℓ /m ˜ ℓ (˜ µ L , ˜ τ ). These splittingscan be identified since, under certain conditions, one can effectively reconstruct slepton massesvia observables such as the kinematic end-point of the invariant mass distribution of the leptonscoming from the cascade decays χ → ˜ ℓ ± ℓ ∓ → χ ℓ ± ℓ ∓ . If the slepton in the decay chain isreal (on-shell), the di-lepton invariant mass spectrum has a kinematical edge that might then bemeasured with a very high precision (up to 0.1 %) [34–36]. Together with data arising from otherobservables, this information allows to reconstruct the slepton masses [34–38] and hence studythe slepton mass splittings. Finally, one can observe multiple edges in di-lepton invariant massdistributions from χ → χ ℓ ± i ℓ ∓ i , arising from the exchange of a different flavour slepton ˜ ℓ j (inaddition to the left- and right-handed sleptons, ˜ ℓ i L,R ). Under the assumption of a seesaw as theunique underlying source of flavour violation in the leptonic sector (for instance assuming thatSUSY breaking is due to flavour blind interactions), then all the above observables, both at highand low energies, will be strongly correlated.Each seesaw realisation will have a distinct impact on the latter observables. It is thus manda-2ory to conduct an exhaustive study of the many possible experimental signatures, in order totest the seesaw hypothesis, either excluding or substantiating it, and in the latter case, devising astrategy to disentangle among the different seesaw realisations.In a previous work [39] we have studied the impact of a type I seesaw, embedded into theconstrained minimal supersymmetric extension of the SM (cMSSM), in what concerns leptonflavour violation both at low-energies and at the LHC. Here we extend the analysis to the typeIII SUSY seesaw. In this case, and in order to accommodate neutrino masses and mixings,one adds (at least two) fermionic SU(2) triplets to the SM particle field content [40], as wellas the corresponding superpartners. If one extends the usual MSSM by just the superfieldsresponsible for neutrino masses and mixings, one would destroy the nice feature of gauge couplingunification. This problem is easily circumvented by embedding the new states in complete SU(5)representations, -plets in the case of a type III seesaw [41]. Note that in addition to the SU(2)triplet, the -plet contains a singlet state which also contributes to neutrino dynamics, so thatin this case one actually has a mixture between type I and type III seesaws.Our study shows that if a type III seesaw is indeed the unique source of neutrino masses andleptonic mixings, and is realised within an otherwise flavour conserving SUSY extension of the SM(specifically the cMSSM), one then expects low-energy LFV observables within future sensitivityreach, as well as interesting slepton phenomena at the LHC. After having identified regions inthe cMSSM parameter space, where the slepton masses could in principle be reconstructed fromthe kinematical edges of di-lepton mass distributions (i.e. χ → χ ℓ ± i ℓ ∓ i can occur, and witha non-negligible number of events), we study the different slepton mass splittings, exploring theimplications for LFV decays. From the comparison of the predictions for the two sets of observables(high- and low-energy) with the current experimental bounds and future sensitivities, one caneither derive information about the otherwise unreachable seesaw parameters, or disfavour thetype III SUSY seesaw as being the unique source of lepton flavour violation.The paper is organised as follows: in Section 2 we define the model, providing a brief overviewon the implementation of a type III seesaw in the cMSSM. In Section 3 we discuss the implicationsof this mechanism for low- and high-energy LFV observables. Our results are presented in Section 4where we study the different high- and low-energy observables in the seesaw case. This willalso allow to draw some conclusions on the viability of a supersymmetric type III seesaw as theunderlying mechanism of LFV. Further discussion is presented in the concluding Section 5. Under the hypothesis that neutrinos are Majorana particles, the smallness of their masses, aswell as their mixings, can be explained via seesaw-like mechanisms due to the exchange of heavystates: fermionic singlets (type I), scalar triplets (type II) or fermionic triplets (type III). Thethree possible seesaw realisations can be easily embedded in the framework of supersymmetricmodels. However, if SUSY’s appealing feature of gauge coupling (and gaugino mass) unification isto be preserved, the new particles present below the Grand Unified scale must belong to completeGUT representations. Under the assumption of an SU(5) gauge group, generating a neutrino massmatrix with at least two non-zero eigenvalues requires the following multiplet content: two copiesof 111 or 242424 (type I and III, respectively) or 151515 + 151515 (type II). The addition of the non-singlet fields(i.e. the 151515- and the 242424-plets) has an important effect on the evolution of several fundamentalparameters, especially on the β -functions for gauge and Yukawa couplings, as well as on the Rank ≥ i (aswell as their superpartners) are added to the MSSM particle content [44]. Each one is embeddedinto a 242424-plet , that decomposes under the SM gauge group, SU(3) × SU(2) × U(1), as242424 = (8 , ,
0) + (3 , , − /
6) + (3 ∗ , , /
6) + (1 , ,
0) + (1 , , b G M + b X M + b X M + c W M + b B M . (2.1)The fermionic components of the last two terms in the above decomposition ( c W M and b B M ) haveexactly the same quantum numbers of a fermionic triplet (Σ) and of a singlet right-handed neutrino( ν R ). It is then clear that if embedded into an SU(5) framework, the realisation of the type IIIseesaw will in general produce a mixture of type I and type III mechanisms.In the unbroken SU(5) phase, the superpotential is given by W SU(5) = √ M i Y ij M j ¯5¯5¯5 H − M i Y ij M j H +555 H M i Y Nij ¯5¯5¯5 M j + 14242424 M i M ij M j , (2.2)with i, j denoting generation (flavour) indices, and where we have not included the terms specifyingthe Higgs sector responsible for the breaking of SU(5). The Majorana mass term in Eq. (2.2) isgauge invariant due to having the triplet superfields in the adjoint SU(2) representation. In thebroken phase, in addition to the usual MSSM terms, the superpotential is given by: W = W MSSM + b H c W M Y M − r b B M Y B ! b L + b H b X M Y X b D c ++ 12 b B M M B b B M + 12 b G M M G b G M + 12 c W M M W c W M + b X M M X b X M . (2.3)After the heavy fields are integrated out, and at lowest order in the expansion in ( v Y B,W /M B,W ) n ( v being the vacuum expectation value of H ), one obtains the light neutrino mass matrix: m ν ≈ − v (cid:18) Y TB M − B Y B + 12 Y TW M − W Y W (cid:19) , (2.4)where we have again omitted flavour indices. From the above formula, it is clear that we areindeed in the presence of a mixed type I and III seesaw, with contributions to m ν arising fromboth the singlets ( ∝ Y B ) and SU(2) triplets ( ∝ Y W ). The model is further specified by W MSSM and by the soft-SUSY breaking Lagrangian. Concerning the latter, we will furthermore assume acMSSM framework, where mSUGRA-inspired universality conditions for the soft-breaking SUSYparameters are imposed at some very high-energy scale, which we take to be M GUT . The MSSMpart of the model is then defined by the usual 4 continuous parameters (the universal gaugino andscalar soft-breaking masses M / and m , the universal trilinear coupling A and the ratio of theHiggs vacuum expectation values, tan β = v /v ) and the sign of the bilinear µ -term in W MSSM ,sign( µ ). Among the representations of lower dimension, only the 242424 does indeed contain a singlet hypercharge field. Y B = Y W = Y X and M B = M W = M G = M X . Although the above parameters do run between the GUT scale and theircorresponding decoupling scales, one has, to a very good approximation, that Y B ≃ Y W and M B ≃ M W as the heavy states decouple. At the seesaw scale (which we define to be ≈ M B ≃ M W ) m ν is approximately given by m ν ≈ − v Y ν T M − N Y ν , (2.5)where we have again used the simplifying notation Y B = Y W = Y N = Y ν , M B = M W = M N .Up to an overall factor (4 / M N , Y ν v = i q M diag N R q m diag ν U MNS † . (2.6)In the above R is a complex orthogonal 3 × U MNS ), and which isparametrized in terms of three complex angles θ i ( i = 1 , , β -functions of the gauge couplings, as well as the runningfor soft gaugino and scalar masses, are strongly affected in type III seesaw models. In fact, RGEeffects are behind the relatively small interval for M N in a type III SUSY seesaw. Assuming thatthe triplet masses are degenerate ( M N i = M ), the interval is bounded from above, M . × GeV, to comply with the atmospheric neutrino mass difference. On the other hand, for tripletmasses below 10 GeV, the running is such that one encounters Landau poles for the gaugecouplings at the GUT scale, while tachyonic sfermions (especially the lighter stau and stop) canalso arise for smaller values of the soft-SUSY breaking parameters.As clear from the above discussion, the new distinctive features of a type III seesaw will likelybe manifest in many phenomena. In what follows we discuss the new contributions of the type IIISUSY seesaw for low-energy lepton flavour violation (e.g. to radiative decays such as µ → eγ ),as well as for LFV at the LHC: in particular, we focus on the study of slepton mass splittingsto probe deviations from the cMSSM and possibly derive information about the SUSY seesawparameters. As for the case of a type I SUSY seesaw, the non-trivial flavour structure of Y ν at the GUT scalewill induce (through the running from M GUT down to the seesaw scale) flavour mixing in theotherwise approximately flavour conserving soft-SUSY breaking terms. In particular, there willbe radiatively induced flavour mixing in the slepton mass matrices, manifest in the LL and LR blocks of the 6 × m L ) ij = −
95 18 π (3 m + A ) ( Y ν † L Y ν ) ij , (∆ A l ) ij = −
95 316 π A Y lij ( Y ν † L Y ν ) ij , (∆ m E ) ij ≃ L kl ≡ log (cid:18) M X M N k (cid:19) δ kl . (3.1)5hen compared to the type I SUSY seesaw, the most important difference corresponds to achange in the overall factor (multiplying the ( Y ν † L Y ν ) ij term). The above sources of flavourmixing will have an impact regarding lepton flavour non-universality and lepton flavour violationin the charged slepton sector, potentially inducing sizable contributions to high- and low-energyLFV observables, as we proceed to discuss.As mentioned in the Introduction, several LFV signals can be observable at the LHC, in strictrelation with the χ → χ ℓ ± ℓ ∓ decay chains. As discussed in [34–38], in scenarios where the χ is sufficiently heavy to decay via a real (on-shell) slepton, the process χ → χ ℓ ± ℓ ∓ is greatlyenhanced while providing a very distinctive signal: same-flavour opposite-charged leptons withmissing energy. The χ → χ ℓ ± ℓ ∓ decay chain thus offers a golden laboratory to study LFV atthe LHC, via the following observables:(i) sizable widths for LFV decay processes like χ → χ ℓ ± i ℓ ∓ j [27, 30–33];(ii) multiple edges in di-lepton invariant mass distributions χ → χ ℓ ± i ℓ ∓ i , arising from theexchange of a different flavour slepton ˜ l j (in addition to the left- and right-handed sleptons, ˜ l i L,R );(iii) flavoured slepton mass splittings.In order to optimise the reconstruction of the leptons’ momentum (which is expected to beeasy, accounting for smearing effects in τ ’s) and, in addition, extract indirect information on themass spectrum of the involved sparticles, the SUSY spectrum must comply with the requirementsof a so-called “standard window”:(a) the spectrum is such that the decay chain χ → ˜ ℓ ℓ → χ ℓ ℓ , with intermediate real sleptons,is allowed;(b) it is possible to have sufficiently hard outgoing leptons: m χ − m ˜ ℓ L , ˜ τ >
10 GeV.In this case, the di-lepton invariant mass spectrum has a kinematical edge that might be measuredwith a very high precision (up to 0.1 %) [34–36]. Together with data arising from other observables,this information allows to reconstruct the slepton masses [34–38], and hence probe slepton massuniversality or test LFV in the slepton sector. In particular, the relative slepton mass splittings,which are defined as ∆ m ˜ ℓ m ˜ ℓ (˜ ℓ i , ˜ ℓ j ) = | m ˜ ℓ i − m ˜ ℓ j | < m ˜ ℓ i,j > , (3.2)can be inferred from the kinematical edges with a sensitivity of O (0 . e L − ˜ µ L and O (1%) for ˜ µ L − ˜ τ .Even in the absence of a seesaw mechanism, it is important to recall that universality betweenthe third and first two slepton generations is broken due to LR mixing and to RGE effectsproportional to the third generation lepton Yukawa coupling. However, in the presence of flavourviolation (as induced by the SUSY seesaw, see Eqs. (3.1)), the mass differences between left-handed selectrons, smuons and staus can be potentially augmented. Similar to the case of a typeI seesaw [39], the relative mass splitting between left-handed sleptons is approximately given by∆ m ˜ ℓ m ˜ ℓ (˜ ℓ i , ˜ ℓ j ) ≈ | (∆ m L ) ij | m ℓ (3.3)where we have neglected LR mixing effects, as well as RGE contributions proportional to thecharged lepton Yukawa coupling. In the R = 1 seesaw limit, where all flavour violation in Y ν stems from the U MNS (see Eq. (2.6)), and assuming that the large flavour violating entries involvingthe second and third generation constitute the dominant source of mixing (and are thus at the6rigin of the slepton mass differences), one can further relate the ˜ e L − ˜ µ L and the ˜ µ L − ˜ τ massdifferences [39]: ∆ m ˜ ℓ m ˜ ℓ (˜ e L , ˜ µ L ) ≈
12 ∆ m ˜ ℓ m ˜ ℓ (˜ µ L , ˜ τ ) . (3.4)As discussed in [39], in the framework of a type I seesaw, the slepton mass differences can besufficiently large as to be within the reach of LHC sensitivity.Before proceeding, let us briefly notice that, depending on the amount of flavour violation, onecan be led to regimes where two non-degenerate mass eigenstates have almost identical flavourcontent (maximal flavour mixing). To correctly interpret a mass splitting between sleptons withquasi-degenerate flavour content, one has to introduce an “effective” mass m (eff) i ≡ X X =˜ τ , ˜ µ L , ˜ e L m ˜ l X (cid:16) | R ˜ lXi L | + | R ˜ lXi R | (cid:17) , (3.5)where R ˜ l is the matrix that diagonalizes the 6 × (cid:18) ∆ mm (cid:19) (eff) (˜ ℓ i , ˜ ℓ j ) ≡ | m (eff) i − m (eff) j | m (eff) i + m (eff) j . (3.6)The seesaw-generated flavour violating entries of Eqs. (3.1) will also give rise to low-energyLFV phenomena, such as radiative ℓ i → ℓ j γ decays, which are induced by 1-loop diagrams viathe exchange of gauginos and sleptons. These can be described by the effective Lagrangian [7], L eff = e m ℓ i ℓ i σ µν F µν ( A ijL P L + A ijR P R ) ℓ j + h.c. , (3.7)where P L,R = (1 ∓ γ ) are the usual chirality projectors and the couplings A L and A R arise fromloops involving left- and right-handed sleptons, respectively. Using Eq. (3.7), the branching ratio ℓ i → ℓ j γ is given byBR( ℓ i → ℓ j γ ) = 48 π αG F (cid:16) | A ijL | + | A ijR | (cid:17) BR( ℓ i → ℓ j ν i ¯ ν j ) . (3.8)where G F is the Fermi constant and α is the electromagnetic coupling constant. In our numericalcalculation we use the exact expressions for A L and A R . However, for an easier understandingof the numerical results, we note that the relations between these couplings and the slepton soft-breaking masses are approximately given by | A ijL | ∼ | (∆ m L ) ij | tan βm ≃ (cid:12)(cid:12)(cid:12)(cid:12)
95 tan β π (3 m + A ) m ( Y ν † L Y ν ) ij (cid:12)(cid:12)(cid:12)(cid:12) , A ijR ∼ (∆ m E ) ij tan βm ≃ , (3.9)where m SUSY denotes a generic (average) SUSY mass and where we have used Eqs. (3.1).It is important to notice here that when compared to other seesaws, and for the same cMSSMparameters, the sparticle spectrum is lighter. Together with the larger Yukawa couplings (a con-sequence of the larger seesaw scale), the type III seesaw typically leads to larger LFV observablesthan in either type I or II [44]. The exact formulae for the branching ratios of the radiative LFV decays, as used in our numerical computation,can be found, for example, in [46]. µ − e conversions in heavy nuclei, as they offer chal-lenging experimental prospects: the possibility of improving experimental sensitivities to valuesas low as ∼ − renders this observable an extremely powerful probe of LFV in the muon-electron sector. In the limit of photon-penguin dominance, the conversion rate CR( µ − e , N) andBR( µ → eγ ) are strongly correlated, since both observables are sensitive to the same leptonicmixing parameters [28].In the following section we numerically analyse the above discussed points. For the numerical computation, we have used the public code
SPheno (v3.beta.51) [47] to carryout the numerical integration of the RGEs. The RGEs of the SU(5) type III SUSY seesaw werecalculated at 2-loop level in [44], using the public code
SARAH [48].
SPheno further computesthe sparticle and Higgs spectrum, as well as the various low-energy LFV observables. The darkmatter relic density is evaluated through a link to micrOMEGAs v2.2 [49].Regarding low-energy neutrino data, current (best-fit) analyses favour the following intervalsfor the mixing angles [50] θ = (34 . ± . ◦ , θ = (42 . +4 . − . ) ◦ , θ = (5 . +3 . − . ) ◦ ( ≤ . ◦ ) , (4.1)while for the mass-squared differences one has∆ m = (7 . ± . × − eV , ∆ m = (cid:26) ( − . ± . × − eV (+2 . ± . × − eV , (4.2)where the two ranges for ∆ m correspond to inverted and normal neutrino spectrum. In Table 1we summarise the current bounds and the future sensitivities of dedicated experimental facilities,for the low-energy LFV observables considered in our numerical discussion.LFV process Present bound Future sensitivityBR( µ → eγ ) 1 . × − [51] 10 − [52]BR( τ → µγ ) 4 . × − [53] 10 − [54]CR( µ − e , Ti) 4 . × − [51] O (10 − ) ( O (10 − )) [55] ( [56])Table 1: Present bounds and future sensitivities for several LFV observables.In the first part of the analysis we assume a degenerate spectrum for the three families oftriplet fermions. Moreover, we consider the conservative limit in which flavour violation solelyarises from the U MNS leptonic mixing matrix, i.e. R = 1 in Eq. (2.6). Leading to the resultsdisplayed in this section, we have taken into account all available LEP and Tevatron bounds onthe Higgs boson and SUSY spectrum [51, 57, 58], as well as the most recent results on negativeSUSY searches from the LHC collaborations [59, 60]. In general, the limit R = 1 translates into a “conservative” limit for flavour violation: apart from possiblecancellations, and for a fixed seesaw scale, this limit typically provides a lower bound for the amount of generatedLFV.
500 1000 1500 2000 2500 3000 100 200 300 400 500 600 700 800 900 M / [ G e V ] m [GeV] B R ( µ → e γ ) = . × - -12 -10 -9 -8 B R ( χ → χ µ µ ) = . % -13 -12 -11 -10 -9
0 50 100 150 200 B R ( µ → e γ ) m [GeV] M - G e V - = Ω h ± σ± σ± σ Figure 1: On the left, m − M / plane (in GeV), with A = 0, tan β = 10, for a seesaw scale M ∼ × GeV and θ = 0 . ◦ . The shaded region on the left is excluded due to the presenceof a charged LSP, while the yellow (red) region is excluded in view of m h bounds ( m h andLHC bounds). Several regions do not fulfil the “standard window” requirements: solid regionscorrespond to having m χ < m ˜ ℓ L + 10 GeV (cyan) and m χ < m ˜ τ + 10 GeV (blue). The dashedblue region corresponds to m χ < m ˜ ℓ L , ˜ τ while blue crosses correspond to m χ < m ˜ τ + m τ .The centre white region denotes the parameter space obeying the “standard window” constraint.Green lines denote isocurves for BR( µ → eγ ), while the dashed-dotted lines correspond to differentvalues of BR( χ → χ µµ ) as indicated in the plot. A small black region in the lower left cornercorresponds to a WMAP7 compatible χ relic density. On the right panel, BR( µ → eγ ) as afunction of m (in GeV), for A = 0, tan β = 5, θ = 0 . ◦ and several values of M : 1 . × GeV . M . × GeV (from lower to upper curves). Horizontal lines correspond to thecurrent bound and future sensitivity. The yellow gridded region is excluded due to violation of m h bounds. The colour code denotes compatibility with the WMAP7 bounds on Ω h .Concerning the WMAP7 bound for the observed dark matter relic density [6],0 . . Ω h . . , (4.3)we do not systematically impose it as a viability requirement in our analysis. Nevertheless, we dorequire the lightest neutralino to be the lightest SUSY particle (LSP). We will return to this issueat a later stage.Let us then begin our discussion by investigating how the requirements of a “standard window”,as well as compatibility with experimental bounds, constrain the type III SUSY seesaw parameterspace in the case of degenerate fermion triplets (i.e., M N i = M ).On the left-hand side of Fig. 1, we display the m − M / parameter space for a type IIISUSY seesaw, taking A = 0, tan β = 10, and a seesaw scale M ∼ × GeV, settingalso θ = 0 . ◦ . The excluded (shaded) areas correspond to a charged LSP, to the violation ofcollider constraints on the Higgs and sparticle spectrum, and to kinematically disfavoured regimes(kinematically closed χ → ˜ ℓℓ channel, excessively soft outgoing leptons, etc.). The requirementsof a “standard window” (see section 3) are fulfilled on the central white region. For this choice ofSUSY seesaw parameters, a large part of the latter viable region is excluded since it is associated9ith an excessively large µ → eγ branching ratio, as can be verified from the isocurves for theBR( µ → eγ ). Additional isocurves (dashed-dotted lines) denote BR( χ → χ µµ ). In the regioncomplying with the “standard window” requirements, the latter range from 5% to 7%; for theLHC operating at √ s = 7 TeV, hardly any events would be observable, while for √ s = 14 TeV,one could expect some 10 to 1000 events (for an integrated luminosity of 100 fb − ). This impliesthat these χ decay chains could indeed be studied at the higher luminosity and higher energyphase of LHC.Concerning dark matter, it is important to notice that, although the requirements imposedon the χ → ˜ ℓℓ decay usually lead to a region where the correct dark matter relic density couldin principle be obtained from co-annihilations of the LSP with the next-to-LSP (NLSP), findingpoints for which Ω h is indeed in agreement with WMAP7 data proves to be challenging. Forthe particular SUSY seesaw configuration investigated in Fig. 1, we verify that the regions whereone finds the correct dark matter relic density are already excluded due to having an excessivelylarge BR( µ → eγ ). Although viable DM scenarios in the type III SUSY seesaw are indeed veryconstrained [44], regions can be found where either by a different choice of seesaw parameters (e.g.setting δ , the Dirac phase in U MNS , δ = π ) or for smaller tan β values, a viable Ω h can be obtained,but still associated with a considerable fine tuning of the parameters. This is illustrated on theright-hand side plot of Fig. 1 for tan β = 5, where we display BR( µ → eγ ) as a function of m forseveral (7) choices of the seesaw scale, 1 . × GeV . M . × GeV. When compatibilitywith the WMAP7 3 σ interval for Ω h is indeed possible, M / has been varied (correspondingto the coloured solid regions as well as the gridded ones - which are already excluded by colliderconstraints); else, we display the value of M / that minimises the deviation from the WMAP73 σ interval (black curves). Typically, the correct relic density is obtained for nearly degenerateLSP and NLSP.Contrary to the type I seesaw, where the requirements of observing the χ → χ ℓ ℓ chain didnot significantly alter the expected low-energy SUSY spectrum, important changes are expectedin the type III seesaw, especially due to the (strong) running of the gaugino masses. Moreover,and as discussed previously, the allowed interval for the triplet masses ( M ) is also severelyconstrained. To illustrate the impact of a “standard window” on the spectrum, we display inFig. 2 the (geometrically) averaged squark masses as a function of the triplet mass, for differentvalues of m . We consider two regimes of tan β , tan β = 10, 40. For each point a scan over M / isconducted to determine its lowest possible value complying with the requirement of a “standardwindow”. We also differentiate between the ranges allowed with and without applying the currentbound on BR( µ → eγ ). Regarding mixings in the neutrino sector, we again work in the limit R = 1 and set θ = 0 . ◦ .As can be seen from Fig. 2, and as hinted on section 2, the allowed interval for the seesaw scale(here represented by M ) ranges from 10 GeV to just below 10 GeV, corresponding to theresults of [44]. It is worth emphasising that there are regions where, in addition to complying withall accelerator and neutrino data, the type III seesaw still leads to scenarios of LFV in agreementwith low-energy data (the most stringent constraint arising from the µ → eγ decay). This divergesfrom the findings of [44], where only very light SUSY spectra were considered. Regimes of heaviersparticles (large M / and m ) are preferred, confirming that these scenarios would be more likelyto be observed at the LHC for √ s = 14 TeV. It is important to remark that, even for a regime ofsmall m , we are always led to a very heavy SUSY spectrum (here represented by a geometricalaverage of the squark masses). Complying with all the above requirements implies that even for m as low as 50 GeV, one must have < m ˜ q > min ∼ m = 0). By itself, this result is important in the sense that should any light SUSY spectrum10 < m q ~ > [ T e V ] M [GeV] m [TeV] µ → e γ ) = 1.2 × -11 µ → e γ ) = 1.2 × -11 µ → e γ ) = 1.2 × -11 < m q ~ > [ T e V ] M [GeV] m [ T e V ] µ → e γ ) = 1.2 × -11 µ → e γ ) = 1.2 × -11 µ → e γ ) = 1.2 × -11 Figure 2: Average squark mass range (in TeV) as a function of the triplet mass (in GeV), fordifferent values of m : 50 GeV (blue/cyan), 500 GeV (black/grey) and 1 TeV (red/pink), thecolour code further denoting imposing/not imposing the bound on BR( µ → eγ ). Gridded regionscorrespond to cases where one has a charged LSP. The brown region is excluded due to violationof LHC or m h bounds. The left (right) figure corresponds to tan β = 10 (40). In both cases, θ = 0 . ◦ , A = {− , , } TeV, with M / set to the lowest possible value complying with therequirement of a “standard window”.be discovered at the LHC in association with the χ → ˜ ℓℓ decay chain, this would strongly suggestthat a type III seesaw is not at work. It is also important to notice that the steep increase of < m ˜ q > for lower values of M is a direct consequence of having imposed the requirement ofa “standard window”. In particular the strong running of M would imply that for lower M the mass of the sleptons would be much larger than that of the neutralinos, thus preventing thecascade decay χ → ˜ ℓℓ .Increasing the value of tan β has an effect on the SUSY contributions to the LFV observables(which grow with tan β , see Eq. (3.9)), implying that larger values of the SUSY spectrum (andhence of M / ) are required to comply with the experimental constraints. Furthermore, theaugmentation of the LR mixing in the stau sector implies that having a neutral LSP becomesincreasingly difficult. For tan β = 40, as depicted on the right-hand side of Fig. 2, the allowedregions are extremely reduced: only a thin blue band (corresponding to m = 1 TeV) survivesall constraints. To further clarify and illustrate the above discussion regarding the dependenceof the sparticle spectrum on the seesaw scale (under the requirements of a “standard window”and compatibility with experimental bounds), we present on Fig. 3 the electroweak gaugino andslepton masses as a function of the triplet mass ( M ), also explicitly denoting the value of A ineach case. Being essentially driven by M / , the running of their values is similar to that of the(averaged) squark masses.Finally, let us notice that variations of the still unknown Chooz angle, θ , have a comparativelysmall impact: they only contribute to some of the LFV observables and compatibility with theexperimental bound is easily recovered through a minor augmentation of M / , which in turnleads to a heavier sparticle spectrum (for fixed values of m ).We now focus our discussion on the slepton mass differences, as potentially measurable at theLHC. We recall that the expected (conservative) sensitivities for the slepton mass splittings are11 M a ss e s [ T e V ] M [GeV] m = 200 GeVm χ m χ m τ ~ M a ss e s [ T e V ] M [GeV] m = 200 GeVm e~ L m µ ~ L m τ ~ Figure 3: Gaugino and slepton masses (in TeV) as a function of the triplet mass, M (in GeV),for m = 200 GeV. On the left, m χ , and m ˜ τ ; on the right m ˜ e L , m ˜ µ L and m ˜ τ . We have takentan β = 10 and set θ = 0 . ◦ . M / is set to the lowest possible value complying with therequirement of a “standard window” and with the bound on BR( µ → eγ ). In both cases, thedifferent lines correspond to distinct values of A : -1 TeV (full), 0 (dashed), and 1 TeV (dotted).An interrupted line signals the onset of a charged LSP region.of O (0 . m ˜ ℓ /m ˜ ℓ (˜ e, ˜ µ ) and O (1%) for ∆ m ˜ ℓ /m ˜ ℓ (˜ µ, ˜ τ ). In Figs. 4 we display the sleptonmass splittings (effective mass difference in the case of ˜ e L − ˜ µ L ) as a function of the seesaw scale,for the same parameter scan as in Figs. 2. One verifies that ∆ m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ) can be as largeas 3% and ∆ m ˜ ℓ /m ˜ ℓ (˜ µ L , ˜ τ ) ∼ m regimes (where the largest amount of flavour violation, still compatible with experimental boundsand with the requirements of a “standard window”, occurs). For larger values of tan β , one couldhave slightly larger ˜ µ L − ˜ τ mass splittings (mostly in association with larger LR mixings in thestau sector), but the viable regions in the parameter space are much smaller, as mentioned before.In all cases one always has ∆ m ˜ ℓ /m ˜ ℓ (˜ µ L , ˜ τ ) . m ˜ ℓ /m ˜ ℓ (˜ µ R , ˜ e R ) . . ∼
3% for ˜ µ L − ˜ e L and ∼
5% for ˜ µ L − ˜ τ ) and that the reconstructed value of m is found to belarge (around 1 TeV) then, as seen from Figs. 4, this would suggest that the seesaw scale wouldbe M ∼ GeV (for the limiting case R = 1).In Fig. 5 we present the comparison of the ˜ e L − ˜ µ L and ˜ µ L − ˜ τ mass differences, as well astheir ratio, as a function of the seesaw scale. Similar to what occurs for a type I SUSY seesaw,and as discussed in Section 3, the mass differences are strongly correlated (being driven by the(∆ m L ) entry in the slepton mass matrix). With the exception of the regions correspondingto smaller values of M , the relation ∆ m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ) ≈ O (1 / m ˜ ℓ /m ˜ ℓ (˜ µ L , ˜ τ ) (Eq. (3.4))typically holds to a very good approximation (with corrections due to fact that flavour conservingradiative corrections driven by the tau Yukawa coupling now play a non-negligible rˆole). For lower12 ∆ m l ~ / m l ~ ( e ~ L , µ ~ L ) [ % ] M [GeV] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.510 ∆ m l ~ / m l ~ ( µ ~ L , τ ~ ) [ % ] M [GeV] Figure 4: On the left, ˜ e L − ˜ µ L mass difference (normalised to an average slepton mass) as a functionof the triplet mass, M (in GeV). On the right, ˜ µ L − ˜ τ effective mass difference (normalised tothe corresponding average slepton mass) also as a function of the seesaw scale. In both caseswe take tan β = 10, θ = 0 . ◦ , and consider different values of m : 50 GeV (red), 500 GeV(black), and 1 TeV (blue). Gridded regions correspond to a charged LSP. For each point onevaries A = {− , , } TeV, while M / is set to the lowest possible value complying with therequirement of a “standard window” and with the bound on BR( µ → eγ ). ∆ m l ~ / m l ~ ( µ ~ L , τ ~ ) [ % ] ∆ m l~ / m l~ (e~ L , µ ~ L ) [%]m = 200 GeVA = -1 TeVA = 0A = 1 TeV 0.35 0.4 0.45 0.5 0.55 0.6 0.6510 ∆ m l ~ / m l ~ ( e ~ L , µ ~ L ) { ∆ m l ~ / m l ~ ( µ ~ L , τ ~ ) } - M [GeV] Figure 5: On the left, ∆ m ˜ ℓ /m ˜ ℓ (˜ µ L , ˜ τ ) as a function of ∆ m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ), for m = 200 GeV,tan β = 10, θ = 0 . ◦ and taking A = {− , , } TeV (green, blue and black lines, respectively).An interrupted (dashed) line signals the onset of a charged LSP regime towards larger values ofthe mass splittings. On the right, ratio of slepton mass differences, ∆ m ˜ ℓ (˜ e L , ˜ µ L ) / ∆ m ˜ ℓ (˜ µ L , ˜ τ )(normalised to the corresponding average slepton mass), as a function of the triplet mass (in GeV),for different values of m , with tan β = 10, A = {− , , } TeV and θ = 0 . ◦ . Scan and colourcode as in Fig. 4.values of the seesaw scale, where the requirement of a “standard window” (i.e. χ → ˜ ℓℓ decay, withhard outgoing leptons) forces a rapid increase of M / , a small deviation to this strict correlation13 -14 -13 -12 -11 -10 -9 -8 -7 -3 -2 -1 -3 -2 -1 -16 -15 -14 -13 -12 -11 -10 -3 -2 -1 B R ( µ → e γ ) CR ( µ - e , T i ) ∆ m l~ / m l~ (e~ L , µ ~ L ) A = -1 TeV A = 0 A = 1 TeV -13 -12 -11 -10 -9 -8 -7 -6 -3 -2 -1 -3 -2 -1 -3 -2 -1 B R ( τ → µ γ ) ∆ m l~ / m l~ (e~ L , µ ~ L )A = -1 TeV A = 0 A = 1 TeV Figure 6: BR( µ → eγ ) and BR( τ → µγ ) as a function of the ˜ e L − ˜ µ L slepton mass difference(normalised to an average slepton mass), corresponding to the left- and right-hand side panels. Inboth cases we have set m = 100 GeV, tan β = 10 and θ = 0 . ◦ , and considered different valuesof the triplet scale M and of M / . Each sub-panel corresponds to a distinct choice of A . Cyanregions correspond to fulfilling the requirements of a “standard window”. The bounds on m h areviolated in the yellow regions, LHC bounds on SUSY spectrum are violated in orange regions,while red regions are excluded due to both. Further excluded regions are due to failing to meet thekinematical constraints (blue), having a charged LSP (magenta) or violating another LFV bound(grey). Inset into each plot are “horizontal” isolines for M / (ranging from 1.5 TeV to 6 TeV,from top to bottom) and “vertical” isolines for M : from left to right, 10 GeV to 9 × GeV.The secondary y-axis on the left-hand panel illustrates the corresponding values of CR( µ − e , Ti).Horizontal lines denote the current experimental bounds (full) and future sensitivities (dashed).is observed. This can also be seen in the left-hand side of Fig. 5, zooming into the lower end ofthe lines. We have verified that this behaviour occurs irrespective of the value of θ and for alltan β regimes (provided that the regions are phenomenologically and experimentally viable).The correlation of low- and high-energy LFV observables is explored in Fig. 6, where wepresent BR( µ → eγ ) and BR( τ → µγ ) as a function of the ˜ e L − ˜ µ L slepton mass difference,taking m = 100 GeV, and considering different values of the triplet scale, M . We also provideadditional information about the CR( µ − e , Ti). As seen from both panels of Fig. 6, only a smallregion of the scanned parameter space complies with the requirements of a “standard window”while being in agreement with the several experimental and phenomenological constraints. Similarto what occurs for a type I SUSY seesaw, larger, negative values of A translate into largermass splittings. The maximal amount of flavour violation, both regarding radiative decays andslepton mass splittings, is obtained for: (i) a seesaw scale as large as possible (without violatingperturbativity arguments, specifically on Y ν ), as can be understood from Eqs. (2.6, 3.1); (ii)lower values of M / (leading to a lighter SUSY spectrum, see Eq. (3.9)). Regarding the τ → µγ decays, as can be seen from the right panel of Fig. 6, the regions in parameter space associatedwith BR( τ → µγ ) within the sensitivity of SuperB are in fact excluded by the present boundson µ → eγ decays. Although we do not present the corresponding results, a similar study with m = 1 TeV leads to scenarios of somewhat larger mass splittings, and smaller branching ratios forthe radiative decays (due to the much heavier spectrum). It is nevertheless interesting to remark14 -15 -14 -13 -12 -11 -10 -9 -2 -1 -2 -1 -17 -16 -15 -14 -13 -12 -2 -1 B R ( µ → e γ ) CR ( µ - e , T i ) ∆ m l~ / m l~ (e~ L , µ ~ L ) A = -1 TeV A = 0 A = 1 TeV -13 -12 -11 -10 -9 -8 -7 -6 -2 -1 -2 -1 -2 -1 B R ( τ → µ γ ) ∆ m l~ / m l~ (e~ L , µ ~ L )A = -1 TeV A = 0 A = 1 TeV Figure 7: Non-degenerate triplet masses: BR( µ → eγ ) and BR( τ → µγ ) as a function of the˜ e L − ˜ µ L slepton mass difference (normalised to an average slepton mass), corresponding to theleft- and right-hand side panels. Same scan as leading to Fig. 6, except that now M N , are fixed,with varying M N : M N = 10 GeV . M N . × GeV = M N . Same line and colour codeas in Fig. 6, the only exception being that the inset “vertical” isolines for M N decrease from leftto right.that in this regime of very large m , one can have maximal mixings in the lightest slepton - nowa composition of ˜ τ L , ˜ τ R and ˜ µ L - possibly leading to scenarios of very large mass splittings (albeitfor a tiny fraction of the parameter space).Assuming that a type III seesaw is indeed the only source of LFV, and given the extremelyconstrained parameter space, one finds that in the conservative case of R = 1, the correspondingslepton mass splittings will always lie around the % level, and are thus within the expectedsensitivity of the LHC. Furthermore, these mass splittings correspond to values of BR( µ → eγ )well within the expected sensitivity of MEG (or even already ruled out by current searches).Moreover, the regions lying below MEG sensitivity have an associated CR( µ − e , Ti) within thereach of future experiments (PRISM/PRIME).We now consider more general scenarios of non-degenerate spectrum for the heavy triplets. Inorder to investigate this regime, we fix the heaviest (lightest) triplet mass to the upper (lower)limits of the M interval previously obtained, and allow the next-to-lightest triplet mass to varybetween the latter limits. For such a non-degenerate triplet spectrum, we display in Fig. 7 ananalogous study to that of Fig. 6 (same choice of the SUSY parameters, still working in thelimiting case of R = 1). As can be observed, the area complying with all requirements (cyanband) is now comparatively larger. The ˜ e L − ˜ µ L slepton mass differences are also enhancedwhen compared to the degenerate case: for all three regimes of A = − , , . ∆ m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ) . m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ) ∼ µ → eγ ). Concerning the amount of LFV inducing the µ → eγ transitions, one finds that, similar to what occurs in the degenerate case, the largest BRs areassociated with M N close to its maximal allowed value (i.e. ∼ M N , leading to degenerateheavy and next-to-heavy triplets) and minimal values of M / . While the latter leads to a lighterspectrum, the former allows to enhance the ( Y ν † LY ν ) contributions proportional to M N , which15re not suppressed by the smallness of θ . However, it is important to notice that the same doesnot occur regarding BR( τ → µγ ), which is maximal for both minimal values of M / and M N (nowdegenerate with the lightest triplet). For fixed values of M N , , while the flavour violating entriesresponsible for µ → eγ transitions and other decays involving the first lepton family (i.e. (∆ m L ) and (∆ m L ) ) increase with increasing M N , (∆ m L ) - which induces BR( τ → µγ ) - remainsapproximately constant: in fact it actually decreases by a small factor, since the contributionsproportional to M N have the opposite sign of those associated to M N .Finally, and to conclude our numerical study, we have considered deviations from the R = 1limit, i.e., allowing for additional mixings in the seesaw mediators. Non-vanishing angles θ i lead tolarger Y νij , with implications for LFV observables: as expected (and aside from eventual accidentalcancelations), there is a large enhancement of the contributions to low-energy LFV observables,as well as an increase of the mass splittings. More concretely, this would displace the cyan regionsin Figs. 6 and 7 towards larger values of BR( µ → eγ ) - potentially excluded by current bounds- and towards slightly larger ˜ e L − ˜ µ L mass differences. Notice that when compared to the type ISUSY seesaw, the effects of R = 1 are somewhat less important, since due to the much narrowerinterval of the seesaw scale (which is also heavier), perturbativity of Y ν effectively constraints thevalues of θ i . Concerning the impact of these variations on the SUSY spectrum (RGE induced),we have verified that deviations from R = 1 have no effect on the gaugino and squark spectra.To summarise, let us re-emphasise that should the χ → χ ℓℓ decay chain be reconstructed atthe LHC, a type III SUSY seesaw will be manifest in both low- and high-energy LFV observables,which will lie within the sensitivity of present/future experimental facilities. In other words,finding regions in the type III SUSY seesaw parameter space where the χ → χ ℓℓ is present,without observing neither µ → eγ transitions at MEG, nor ∆ m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ) at the LHC, is almostimpossible. Although it is a very appealing hypothesis to explain the origin of neutrino masses and mixings,the seesaw mechanism is in general very hard to probe directly. When embedded into a largerframework (as for instance SUSY models), where new states are active between the seesaw scaleand the electroweak one, the seesaw mechanism can give rise to many distinct signatures, de-pending on the nature of the mediators: scalar or fermionic (gauge singlets or triplets). In thisstudy we considered a supersymmetric type III seesaw where, in order to preserve gauge couplingunification, the additional states are embedded into complete SU(5) representations. The manyexperimental constraints (LEP, LHC, low-energy experiments) strongly reduce the available pa-rameter space of the model, so that one expects very characteristic signals (SUSY spectrum andcharged LFV, both at low-energies and at the LHC), which offer the possibility of falsifying themodel. Using the correlation between the different LFV observables (inherent from the assump-tion that the seesaw provides the only source of flavour violation in the model), we have focusedour analysis on the interplay between low-energy radiative decays (e.g. µ → eγ ) and potentialLFV signatures appearing in association with the χ → ˜ ℓℓ cascade decays at the LHC, such asflavoured slepton mass splittings, ∆ m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ).Firstly, requiring that the spectrum allows for the reconstruction of slepton masses from the χ cascade decay chains (and assuming a χ LSP), the type III SUSY seesaw leads to scenarios wherea heavy SUSY spectrum (e.g. m ˜ q ∼ χ has the correctrelic density. Such scenarios typically arise in association with the low m regime. Concerningdark matter, it is important to recall that other candidates might be present and have a relicdensity in agreement with WMAP bounds, as could be the case for gravitinos. However, this issueclearly lies beyond the scope of the present work.Assuming that a type III seesaw is indeed the only source of LFV, and given the extremelyconstrained parameter space, one finds that the corresponding slepton mass splittings will alwayslie around the % level, and are thus within the expected sensitivity of the LHC. A hierarchicalfermionic triplet spectrum further boosts the expected mass splittings: one is led to a regimewhere, even in the conservative limit of R = 1, one has 1% . ∆ m ˜ ℓ /m ˜ ℓ (˜ e L , ˜ µ L ) . µ → eγ ) well within the expected sensitivity of MEG (or, in very limiting cases,within PRISM/PRIME sensitivity for CR( µ − e , Ti)).In the more general case of an increased mixing involving the triplet sector (i.e. R = 1), thereis an enhancement of the contributions to low-energy LFV observables, as well as a small increasein the slepton mass splittings, without further impact on the remaining SUSY spectrum.Unlike what occurs for a type I SUSY seesaw, the very constrained range for the type IIIseesaw scale strongly tightens the predictions for LFV: the expected flavoured mass splittings areindeed well within the sensitivity range of the LHC, while at the same time low-energy scale LFVmust unavoidably lie within the present and future sensitivity of either MEG or PRISM/PRIME(observation of a τ → µγ signal at SuperB will be much more challenging). If supersymmetryis discovered at the LHC, and a type III seesaw is at the origin of flavour mixing in the leptonsector, then this model can be easily falsified in the near future. This work has been done partly under the ANR project CPV-LFV-LHC NT09-508531. Thework of A. J. R. F. has been supported by
Funda¸c˜ao para a Ciˆencia e a Tecnologia through thefellowship SFRH/BD/64666/2009. A. J. R. F. and J. C. R. also acknowledge the financial supportfrom the EU Network grant UNILHC PITN-GA-2009-237920 and from
Funda¸c˜ao para a Ciˆencia ea Tecnologia grants CFTP-FCT UNIT 777, CERN/FP/83503/2008 and PTDC/FIS/102120/2008.
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