Gravitino Decay in High Scale Supersymmetry with R-parity Violation
Emilian Dudas, Tony Gherghetta, Kunio Kaneta, Yann Mambrini, Keith A. Olive
CCPHT-RR034.052018UMN–TH–3718/18, FTPI–MINN–18/08LPT-Orsay-18-62
Gravitino Decay in High Scale Supersymmetrywith R-parity Violation
Emilian Dudas a , Tony Gherghetta b , Kunio Kaneta b,c ,Yann Mambrini d , and Keith A. Olive b,ca CPhT, Ecole Polytechnique, 91128 Palaiseau Cedex, France b School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA c William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN55455, USA d Laboratoire de Physique Th´eorique Universit´e Paris-Sud, F-91405 Orsay, France
Abstract
We consider the effects of R-parity violation due to the inclusion of a bilinear µ LH u superpotential term in high scale supersymmetric models with an EeV scale gravitinoas dark matter. Although the typical phenomenological limits on this coupling (e.g.due to lepton number violation and the preservation of the baryon asymmetry) arerelaxed when the supersymmetric mass spectrum is assumed to be heavy (in excess ofthe inflationary scale of 3 ˆ GeV), the requirement that the gravitino be sufficientlylong-lived so as to account for the observed dark matter density, leads to a relativelystrong bound on µ À
20 GeV. The dominant decay channels for the longitudinalcomponent of the gravitino are
Zν, W ˘ l ¯ , and hν . To avoid an excess neutrino signalin IceCube, our limit on µ is then strengthened to µ À
50 keV. When the boundis saturated, we find that there is a potentially detectable flux of mono-chromaticneutrinos with EeV energies. a r X i v : . [ h e p - ph ] M a y Introduction
Naturalness and potential solutions to low energy phenomenological quandaries such as thediscrepancy between the theoretical and experimental determinations of the anomalous mag-netic moment of the muon [1, 2] pointed to low energy supersymmetry. Indeed, statisticalanalyses of a multitude of low energy observables predicted [3, 4] a supersymmetric spectrumwell within reach of the LHC. However, to date, there has been no experimental confirma-tion of low energy supersymmetry [5]. Supersymmetry may still lie within the reach of theLHC, and discovery may occur in upcoming runs. Nevertheless it is also possible that su-persymmetry lies beyond the LHC reach, and in that case, it is unclear whether the scale ofsupersymmetry, r m is just beyond its reach, r m „
10 TeV, or far beyond its reach, r m ą GeV, for example.If supersymmetry plays a role in nature below the Planck scale, it may still be brokenat some high energy scale [6]. If that scale is above the inflationary scale, „ ˆ GeV,supersymmetric particles, with one exception, may not have participated in the reheatingprocess and were never part of the thermal background in the early Universe. The exceptionmay be the gravitino with an approximately EeV mass [7]. In this case, we still have a viablesupersymmetric dark matter candidate, namely the gravitino which is produced from thethermal bath during reheating [8, 7, 9]. Interestingly, in the context of an SO(10) GUT, suchhigh scale supersymmetric models are still able to account for gauge coupling unification,radiative electroweak symmetry breaking and the stability of the Higgs vacuum [10]. Howeverif the gravitino is stable, as would be the case if R -parity is conserved, there are very fewdetectable signatures of the model. R -parity is typically imposed in supersymmetric models to insure the stability of theproton [11] by eliminating all baryon and lepton number violating operators. Of course,a consequence of R -parity conservation is that the lightest supersymmetric particle (LSP)is stable and becomes a dark matter candidate [12]. Limits on R -parity violating (RPV)couplings can be derived by requiring baryon and lepton number violating interactions toremain out-of-equilibrium in the early universe to preserve the baryon asymmetry [13]. How-ever in high scale supersymmetry, these limits are relaxed as supersymmetric partners werenever in the thermal bath and did not mediate interactions which could wash out the baryonasymmetry. Therefore, it is possible that some amount of RPV is acceptable. If present,RPV violating operators would render the lightest supersymmetric particle, the gravitino in1his case, unstable. If long-lived, the decay products may provide a signature for the EeVgravitino.In this paper we consider a minimal addition to the minimal supersymmetric standardmodel (MSSM). Namely, we include a single RPV interaction, generated by the LH u bilinearterm in the superpotential. This term is sufficient to allow for the decay of the LSP gravitino,and demanding that it remains long-lived to account for the dark matter, will enable us toset a limit on the “ µ ”-term associated with this bilinear. We will compare this limit withthe one imposed from the preservation of the baryon asymmetry in both weak scale andhigh scale supersymmetry models. Furthermore, as we will show, while there is a γν decaymode, the dominant decay channel actually proceeds through the longitudinal mode of thegravitino to Zν , W ˘ l ¯ , and hν . Thus this model predicts a monochromatic source of „ EeVneutrinos.The paper is organized as follows. In the next section we discuss the expected abundanceof the heavy gravitino produced during reheating. We also make some preliminary remarksconcerning the expected effects of including the LH u RPV term. In section 3, we introducethe LH u term and discuss its role in the neutralino and chargino mixing matrices and itsrole as a source for neutrino masses. Constraints arising from other relevant operators arealso discussed. In section 4, we compute the lifetime and branching ratios of the gravitinoand in section 5, we discuss the observational consequences of its decay. Our conclusions aresummarized in section 6. Generically, in weak scale supersymmetry models with a gravitino LSP, the gravitino massis typically O p q GeV. Higher masses lead to an overabundance of gravitinos, independentof the reheat temperature due to the decays of the next-to-lightest supersymmetric particle,often a neutralino, to the gravitino. It is difficult to obtain neutralinos with masses in excessof a few TeV, with relic densities still compatible with CMB observations [14]. By combiningthe limit on the relic density with limits from big bang nucleosynthesis, one can derive anupper limit of roughly 4 TeV on the gravitino mass [7]. This limit is evaded in high scalesupersymmetry models, when no superpartners other than the gravitino are produced duringreheating and a new window of gravitino masses opens up above O p . q EeV [7].In high scale supersymmetry models with the gravitino as the only superpartner lighter2han the inflaton, gravitinos can be pair produced during reheating [8, 7]. The gravitinoproduction rate density was derived in [8] R “ n x σv y » . ˆ T M P m { , (1)where M P “ . ˆ GeV is the reduced Planck mass, and n is the number densityof incoming states. The gravitino abundance can be determined by comparing the rateΓ „ R { n „ T { M P m { to the Hubble expansion rate so that n { { n γ „ Γ { H „ T { M P m { .More precisely, we find,Ω { h » . ˆ . m { ˙ ˆ T RH . ˆ GeV ˙ . (2)In the absence of direct inflaton decays, we see that a reheating temperature, T RH , of roughly10 GeV is required. This was shown to be quite reasonable in a more detailed modelwhich combined inflation with supersymmetry breaking [9]. In that model, the dominantmechanism for reheating involved inflaton decays to Standard Model Higgs pairs.Without R-parity violation, the gravitino remains stable and experimental signatures arelimited. Instead R-parity violation allows the possibility for gravitino decays and perhaps anindirect signature for gravitino dark matter. Here, we concentrate on the effects of addingan LH u term to the superpotential leading to decays such as νγ , νZ , νh , and lW . We nextoutline the channels we expect to dominate in gravitino decay. Our argument here will belargely heuristic and a more detailed derivation follows in section 4.To estimate the decay width, one can consider the coupling of the gravitino, ψ µ to amassive gauge field. For simplicity, we consider the abelian Higgs model with a U p q gaugegroup. The coupling is generated through the gravitational interaction L int “ ´ i ? M P D µ φ : ¯ ψ ν γ µ γ ν χ L ` h.c. , (3)between the gravitino, ψ ν , the Higgs field, φ and Weyl fermion, χ L (which plays the role ofthe Higgsino). The Lagrangian can be written as function of the Goldstino component, ψ ofthe gravitino, and the Higgs field components ψ ν „ B ν ψm { or iγ ν ψ ; φ “ ? p v ` h q e ´ i θv , (4)where v is the Higgs vacuum expectation value, h is the radial component (Higgs boson)and θ the corresponding Nambu-Goldstone boson. A more detailed calculation (see the3ppendix) shows that the dominant contribution arises from γ ν ψ , leading to the interaction L int „ M P B µ θ ¯ ψγ µ χ L ` h.c. . (5)In the massless χ L limit, the amplitude squared then becomes | M | „ m { M P . (6)Anticipating that the LH u term will induce a mixing, parameterized by (cid:15) , between χ L (orthe Higgsino) and the neutrino (to be discussed in detail below), we can write χ L „ (cid:15) ν . Thedominant decay channel is then ψ µ Ñ νZ { h , with a widthΓ { „ | M | s m { „ (cid:15) m { M P . (7)From the above argument, we can also anticipate that the Goldstino decay to νγ will besuppressed since the photon does not have a longitudinal component. In the detailed cal-culation the result (7) will be generalized to the non-Abelian, supersymmetric two Higgsdoublet case. In section 5, we will derive limits on (cid:15) from existing experimental constraints,requiring in addition, that sufficiently many gravitinos are present today to supply the darkmatter. The simplest model including RPV only introduces a bilinear RPV operator: W “ W MSSM ` W RPV , (8) W MSSM “ µH u H d ` y e LH d e c ` y u QH u u c ` y d QH d d c , (9) W RPV “ µ LH u . (10)In general the RPV mass parameter µ depends on the lepton flavor, but here we omit theflavor dependence for simplicity (for more detailed discussion, see, e.g., [15]). Note that wehave suppressed all generation indices in both (9) and (10). Since lepton number is no longerconserved, L and H d cannot be distinguished in this setup, and thus there is a field basis As will be shown in the Appendix, the piece ψ ν „ B ν ψ { m { leads to | M | „ m { m A { M P where m A isthe gauge boson mass, which is highly suppressed when m A ! m { . L and H d fields. For instance, when we take L Ñ c ξ L ` s ξ H d and H d Ñ c ξ H d ´ s ξ L with s ξ “ sin ξ , c ξ “ cos ξ and tan ξ “ µ { µ , we can eliminate thebilinear RPV term. Instead, we obtain trilinear RPV terms, such as y e s ξ LLe c and y d s ξ QLd c .Though the observables do not depend on our choice of basis, we need to clarify which basiswe use. We will work in a basis where we define the linear combination of the four fields, L and H d , which picks up a vacuum expectation value to be the Higgs and write Eq. (10)without any additional trilinear terms. In either case, while lepton number is violated,baryon number is still conserved, so this model is free from proton decay constraints. In thefollowing calculation, we will take the basis that explicitly keeps only the bilinear term givenin W RPV . The inclusion of the RPV bilinear term induces a mixing between the charged leptons andthe charged Higgsinos. In the relevant fermionic part of the Lagrangian, the mass matrix forthe charged fermions in the form of L mass “ ´p ˜ W ` , r H ` u , l c q M C p ˜ W ´ , r H ´ d , l q T ` h.c. is givenby M C “ ¨˚˚˝ M gv d gv u µ µ y e v d ˛‹‹‚ . (11)Without loss of generality, we can take a lepton field basis such that y e becomes diagonal.For the neutral fermions, the mass matrix in the field basis p ˜ B, ˜ W , r H u , r H d , ν q is given by M N “ ¨˚˚˚˚˚˚˚˝ M g v u ? ´ g v d ? M ´ gv u ? gv d ? g v u ? ´ gv u ? ´ µ ´ µ ´ g v d ? gv d ? ´ µ ´ µ ˛‹‹‹‹‹‹‹‚ ” ˜ ˆ M ˆ m ˆ m T ˆ µ ¸ , (12)where we have definedˆ M “ ˜ M M ¸ , ˆ m “ ˜ g v u ? ´ g v d ? ´ gv u ? gv d ? ¸ , ˆ µ “ ¨˚˚˝ ´ µ ´ µ ´ µ ´ µ ˛‹‹‚ . (13)5ow it is clear that the neutrino acquires a mass due to a non-vanishing µ , which is givenby m ν » (cid:15) c β ˆ c W M ` s W M ˙ M Z , (14)where c β “ cos β with tan β “ v u { v d , tan θ W “ g { g , s W “ sin θ W , c W “ cos θ W , (cid:15) “ s ξ “ µ { ¯ µ ” µ { a µ ` µ « µ { µ when µ ! µ as will assume later. Note that this mass istoo small to account for the physical neutrino masses. To derive Eq. (14) we diagonalizedthe mass matrix perturbatively as follows: suppose a unitary matrix U diagonalizes M N as U T M N U “ M diag N . We may take U “ ˜ ˆ V ¸ exp ˜ θ ´ θ T ¸ » ˜ ˆ V ¸ ˜ ´ θθ T θ ´ θ T ´ θ T θ ¸ , (15)where θ and V are 2 ˆ ˆ V satisfies V T ˆ µV “ ˆ µ diag ,which allows V to be written as a function of µ and µ , given by V “ ? ¨˚˚˝ ´ c ξ c ξ ´? s ξ s ξ s ξ ? c ξ ˛‹‹‚ . (16)The matrix θ can be obtained by solving the conditions r U T M N U s ij “ i ‰ j . Inthis parametrization the solution is θ “ ? ˜ M Z M ` ¯ µ s W p c β c ξ ´ s β q M Z M ´ ¯ µ s W p c β c ξ ` s β q ´ ? M Z M s W c β s ξ ´ M Z M ` ¯ µ c W p c β c ξ ´ s β q ´ M Z M ´ ¯ µ c W p c β c ξ ` s β q ? M Z M c W c β s ξ ¸ , (17)where s β “ sin β . Then, by ignoring the mixing with gauginos, namely, at the leadingorder in the perturbative diagonalization, the mass eigenstate χ ” p ˜ h, r H, ν q is relatedto the gauge eigenstate N ” p r H u , r H d , ˆ ν q as χ “ V T N , so, for instance, ν is given by ν “ p´ s ξ r H d ` c ξ ˆ ν q{? r H u component. Similarly, it is clear that r H u doesnot have a ν component to order θ , thus the non-zero contribution in U r H u ν comes from theperturbation in θ T θ “ O p M Z { r m q . In contrast, the r H d has a term ´ s ξ ν which is the leadingcontribution in U r H d ν . Therefore, at leading order, U r H u ν is suppressed by a factor of p M Z { r m q compared to U r H d ν „ s ξ p“ (cid:15) q . We have taken the neutrino component as a massless eigenstate.
In the following calculation we neglect O p θ q terms in solving r U T M N U s ij “ θ . Indeed, this isa good approximation as long as r m is sufficiently large compared to the weak scale. „ ˆ GeV), the constraint on the neutrino masses, ř i m ν i ă .
151 eV (95% CL) [16] or similar limits [17], can be easily evaded. For instance, if we take M „ M „ ˆ GeV with tan β “ O p q , the neutrino mass constraint on the RPVparameter is no stronger than (cid:15) À Another way of encoding gravitino couplings to matter is by using the equivalence theorem[18] and using the Goldstino couplings, which are present in particular in the soft terms in thelow-energy effective supersymmetric Lagrangian. Some of them correspond to non-universalcouplings of the Goldstino to matter [19], which are not related to the usual low-energytheorems. Let us denote in what follows the supersymmetry breaking spurion superfield by X “ x ` ? θψ ` θ F , (18)where ψ is the Goldstino, x its scalar partner and F “ ? m { M P is the supersymmetrybreaking scale. Then operators containing soft terms and Goldstino couplings to StandardModel particles describe also through the equivalence theorem, the gravitino couplings toStandard Model particles. However, since the equivalence theorem is valid only for momentawell above the gravitino mass, these Goldstino couplings can only be used for high-energyprocesses and not for gravitino decays, which still have to be computed from the originalgravitino/supercurrent interactions.The relevant non-vanishing operators for our discussion are: ‚ The soft term associated with µ : B µ F ż d θ XLH u Ñ B µ ˜ lh u . (19)This operator generates mixing between a slepton and a Higgs, and can be compared withthe mixing between leptons and Higgsinos. This operator would not dominate the gravitinodecay rate so long as B µ { r m ă (cid:15) . If we write B µ “ B µ , this puts a constraint on B ă r m having assumed that µ „ r m . In principle there is another coupling proportional to µ betweenthe gravitino, leptons and the scalars associated with H u . However, for on-shell gravitinos(as they must be in gravitino decay), γ µ ψ µ “
0, causing this vertex to vanish. ‚ The gravitino coupling related to the soft term associated with µ : B µ F ż d θ XH u H d Ñ Bµ h u h d , (20)7lso vanishes for an on-shell gravitino, as does an additional operator proportional to µ . ‚ The dimension-four operator: c M P ż d θ p LH u qp H u H d q Ñ c µv M P ˜ l h , (21)where the operator is assumed to be generated at the Planck scale. This operator induces amixing between the sleptons and the Higgs. Assuming µ „ m ˜ l „ r m “ ´ M P , one obtainsthe estimate c µv r m M P À µ µ , (22)where µ is assumed to generate a Higgsino-neutrino mixing. Due to the suppression fromthe electroweak vacuum expectation value and assuming µ „ r m there is no meaningfulconstraint on c . ‚ A Giudice-Masiero-like contribution to µ and B µ is possible if the following term isadded to the K¨ahler potential: c GM p LH u ` h.c. q Ă K (23)leading to the shift µ Ñ µ ` c GM m { and a shift in B µ “ B µ Ñ B µ ` c GM m { . Insection 5, we will derive a limit on µ of order µ À ´ GeV for m { „ EeV, and this canbe translated into a limit on c GM ă µ { m { À ´ . The shift in B µ gives a weaker limit(again from Higgs slepton mixing), c GM ă µ r m { m { À ´ .In the rest of the paper we will assume that the main contribution to gravitino decayscomes from the bilinear µ term and therefore that the effect of all other operators like theones above satisfy the constraints which render them sub-dominant.Finally, a possible origin for a small µ term is to assume minimal flavor violation, whichcan generate RPV terms with coefficients that are proportional to Yukawa couplings [20].Even though the holomorphic spurions do not carry lepton number, a bilinear LH u termcan be generated after supersymmetry breaking. A large suppression can then be obtainedif the neutrino Yukawa coupling y ν !
1. A complete study of this possible origin is beyondthe scope of this work.
Before concluding this section, we note that in weak scale supersymmetric models, it ispossible to derive a relatively strong limit on µ [13]. The presence of an LH u mixing term,8ill induce one-to-two processes involving a Higgsino, lepton, and a gauge boson. Thethermally averaged rate at a temperature, T for these lepton number violating interactionsis given by Γ Ñ “ g θ T π ζ p q » . g µ m f T , (24)where g is a gauge coupling, and θ » µ { m f is the mixing angle induced by µ for afermion with mass m f . Comparing the interaction rate (24) with the Hubble rate, H » a π N { T { M P , where N is the number of relativistic degrees of freedom at T , gives usthe condition µ ă ? N TM P m f . (25)By insisting that any lepton number violating rate involving µ remains out-of-equilibriumwhile sphaleron interactions are in equilibrium, i.e., between the weak scale and „ GeV(where the latter is determined by comparing the sphaleron rate „ α W T to the Hubble rate),the limit (25) is strongest for m f „ T , where T is of order the weak scale. For weak scalesupersymmetry, the fermion can be either a lepton or Higgsino, N “ { T „ µ ă ˆ ´ GeV . (26)For weak scale supersymmetry this limit translates to (cid:15) À ´ . This is stronger than thelimit from neutrino masses in weak scale supersymmetry models [15, 21].In the case of high scale supersymmetry, while the Higgsino cannot be part of the ther-mal bath, it can still mediate lepton number violating interactions, but the limit on µ issignificantly weaker. For example, the process HH Ø LL will involve two insertions and issuppressed by the supersymmetry breaking scale. The rate can be estimated asΓ Ñ » ´ g µ µ r m T , (27)where ˜ m „ µ is the gaugino mass. Setting Γ Ñ ă H gives us µ À ? N µ r m T M P , (28)This limit should now be applied at the highest temperatures at which sphalerons are inequilibrium ( T „ GeV), with N “ {
4. Thus µ ă ˆ ´ ˆ µ r m { GeV { ˙ GeV . (29)9he limit on (cid:15) then becomes (cid:15) ă ˆ ´ p ˜ m { GeV q { and for r m „ GeV, we have only (cid:15) À We turn now to a more detailed derivation the gravitino decay into a gauge/Higgs bosonand lepton through the RPV bilinear term. In the supergravity Lagrangian, the relevantinteraction of gravitino ψ µ to a gauge multiplet p A µ , λ q and a chiral multiplet p φ, χ L q is givenby L “ ´ i M P ¯ λγ µ r γ ν , γ ρ s ψ µ F νρ ` „ ´ i ? M P D µ φ : ¯ ψ ν γ µ γ ν χ L ` h.c. . (30)Calculations of the gravitino decay width have been previously performed in several works[22, 23, 24, 25, 26, 27, 28, 29] .A promising signal for observing gravitino decay through the LH u term would be amonochromatic photon-neutrino pair [22, 24, 25, 30, 27]. In this decay channel, the bino ˜ B and the neutral wino ˜ W are related to the neutrino mass eigenstate ν by the mixing matrix U ˜ Bν “ U « θ and U ˜ W ν “ U « θ , respectively, and thus the decay width is given byΓ p ψ µ Ñ γν q » m { πM P | c W U ˜ Bν ` s W U ˜ W ν | » m { πM P M Z ˇˇˇˇ µ ¯ µ M ´ M M M s W c W c β ˇˇˇˇ , (31)where the neutrino mass has been neglected, and the mixing between the bino/neutral winoand neutrino are given by U ˜ Bν » θ » ´ (cid:15) M Z M s W c β , U ˜ W ν » θ » (cid:15) M Z M c W c β . (32)For high scale supersymmetry, we see that this channel carries a significant suppression oforder p M Z { r m q where M „ M „ r m .Similarly, we can compute the partial rate for gravitino decays into Z and ν , whose decay Note that our notation for (cid:15) , which parametrizes the RPV effect, differs from the notation used in someof the literature, and introduces an overall factor of c β that appears in the decay widths. p ψ µ Ñ Zν q » m { πM P β Z “ | c W U ˜ W ν ´ s W U ˜ Bν | F Z ` M Z m { Re rp c W U ˜ W ν ´ s W U ˜ Bν qp s β U ˚ r H u ν ` c β U ˚ r H d ν qs J Z ` | s β U r H u ν ` c β U r H d ν | H Z , (33)where β X “ ´ M X m { , (34) F X “ ` M X m { ` M X m { , (35) J X “ ` M X m { , (36) H X “ ` M X m { ` M X m { . (37)As stated in Section 3.1, the mixing angle between r H u and ν comes from θ T θ , and is propor-tional to max[ M Z { r m , M Z {p ¯ µ r m q ] which is negligible in our case . Recall that the mixingbetween r H d and ν is given by U r H d ν » ´ (cid:15) . While each term in the decay width is propor-tional to (cid:15) , for M Z { r m ! U r H d ν and is the only term which does not lead to a suppression which is at least M Z { r m or M Z {p r mm { q [21]. The source of this term is the gravitino decay into the longitudinalcomponent of Z leading to a relative enhancement over the terms involving the transversecomponents. Thus for M Z { r m !
1, we haveΓ p ψ µ Ñ Zν q » (cid:15) c β m { πM P . (38)As can be seen from Eq.(11), there is mixing between ˜ W ´ and l , opening the decaychannel ψ µ Ñ W ` l ´ with decay widthΓ p ψ µ Ñ W ` l ´ q » m { πM P β W „ | U ˜ W l | F W ` M W m { Re r c β U ˜ W l U ˚ r Hl s J W ` | c β U r Hl | H W , (39) Note that θ given in Eq. (17) is the solution obtained by neglecting O p θ q , and thus it cannot be usedto compute U Ă H u ν . U ˜ W l » (cid:15) ? M W M c β , U r Hl » ´ (cid:15) ´ (cid:15) M W M ¯ µ s β c β . (40)As in the decay channel discussed above, the final term in (39) carries only the suppressionproportional to (cid:15) without the additional high scale supersymmetry suppression of M W { r m or M W {p r mm { q , and thus for M W { r m !
1, we haveΓ p ψ µ Ñ W ` l ´ q » (cid:15) c β m { πM P . (41)Finally, the longitudinal component of the gravitino also decays into hν where h is thelightest Higgs boson. The decay width of this channel is given byΓ p ψ µ Ñ hν q » m { πM P β h | s β U r H u ν ` c β U r H d ν | , (42)where again the last term dominates bearing only the suppression proportional to (cid:15) .Figure 1 (top) shows the branching ratios of the two-body gravitino decays. While wetake M “ M { “ µ “ GeV in the figure, the result is largely independent of thosescales as long as r m " O p q GeV. Since M Z { r m ! p ψ µ Ñ γν q is muchsmaller than Γ p ψ µ Ñ W l q , and thus the branching ratio of the ψ µ Ñ W l channel dominatessoon after m { becomes larger than „ M W . For m { Á ψ µ Ñ Zν { W l { hν converge to their asymptotic values with the relationship2Γ p ψ µ Ñ Zν q “ Γ p ψ µ Ñ W l q “ p ψ µ Ñ hν q , as expected by the equivalence theorem.Thus, the decay channels ψ µ Ñ Zν { W l { hν are all much larger than the γν channel for r m " O p q GeV, due to the enhancement of the decay into the Higgs/Nambu-Goldstoneboson (longitudinal components of the gauge bosons) which can be traced to the fact thatthe Higgsino-lepton mixings are larger than the gaugino-neutrino mixing. In the large m { limit, each decay width is given by ÿ i Γ p ψ µ Ñ Zν i q » (cid:15) c β m { πM P , (43) ÿ i Γ p ψ µ Ñ W l i q » (cid:15) c β m { πM P , (44) ÿ i Γ p ψ µ Ñ hν i q » (cid:15) c β m { πM P , (45)12 m [ GeV ] b r an c h i ng r a t i o γν WlZ ν h ν
100 200 500 1000 20000.20.51 m [ GeV ] r Figure 1: Branching ratios (top) and the deviation r (47), from the asymptotic value forΓ tot (bottom) with M “ M { “ µ “ r m “ GeV.where the charge conjugate of the final state and the number of neutrinos are incorporated .Thus the total decay width is given byΓ tot » (cid:15) c β m { πM P , (46)which is indeed a good approximation for m { Á We have assumed that µ is flavor universal. M “ M { “ µ “ GeV, which is parametrized by r “ Γ tot { ˜ (cid:15) c β m { πM P ¸ . (47)Thus, in the large m { limit, the gravitino lifetime is given by τ { » ˆ . ˆ ´ (cid:15)c β ˙ ˆ m { ˙ s . (48)In the next section, we derive a constraint on (cid:15) , by ensuring that a) we have sufficient darkmatter and b) that the decay products do not exceed observational backgrounds. Cosmological constraints on models with high scale supersymmetry are severe. Indeed,the only way to produce the gravitino in the early Universe if the supersymmetry break-ing scale lies above the reheating temperature , T RH , is through the exchange of highlyvirtual sparticles with Planck-suppressed couplings, such as t-channel processes of the type G G Ñ ˜ G Ñ ψ µ ψ µ , with G, ˜ G representing the gluon and gluino, respectively [8]. Becausethe production rate is doubly Planck-suppressed, the abundance of dark matter producedfrom the bath is very limited (proportional to T RH [8] as in Eq. (2)), requiring a massivegravitino to compensate its low density. Moreover, it was shown in [7, 9] that consideringreheating processes involving inflaton decay imposes a lower bound on T RH Á ˆ GeVimplying from Eq.(2) a lower bound on the gravitino mass m { Á . µ „ r m " µ ñ (cid:15) “ µ a µ ` µ » µ µ » µ r m . (49) To be more precise, above the maximum temperature of the thermal bath T max which is different from T RH if one considers non-instantaneous reheating [31].
14e can then rewrite Eq.(48): τ { » ˆ r m GeV ˙ ˆ .
44 keV µ c β ˙ ˆ m { ˙ s . (50)One of the interesting features in this framework is that the scale of the gravitino massrequired to obtain the experimentally determined relic abundance from Eq. (2) is aroundthe PeV-EeV scale (and higher). The decay of a particle with this mass would provide asmoking gun signature: a monochromatic neutrino from its decay into Zν or hν (Eq. (50))which could be observed by IceCube [33] or ANITA [34].Combining the relic density constraint Eq. (2) with Eq. (50), we can eliminate the grav-itino mass and write µ c β “
14 keV ˆ Ω { h . ˙ { ˆ s τ { ˙ { ˆ r m GeV ˙ ˆ . ˆ GeV T RH ˙ { . (51)We see that while the high scale supersymmetry framework does not yield a strong constraintfrom lepton number violation ( µ À µ » r m » GeV from Eq. (29)) just requiring thelifetime to exceed the current age of the Universe ( τ U » . ˆ s), would give the limit µ À
20 GeV, for c β » .
1. However, as we will see below, observational constraints willactually require a lifetime in excess of 10 s, which further restricts µ ă
140 keV, for c β » .
1, as given in Eq. (51).These limits can be contrasted with those derived in weak-scale supersymmetric models,where µ ă
20 keV from the preservation of the baryon asymmetry as given in Eq.(26). Inthe weak scale supersymmetry scenario, gravitinos are singly produced from the thermalbath and the relic abundance can be expressed as [31, 35]Ω { h » . ˆ
100 GeV m { ˙ ˆ T RH . ˆ GeV ˙ ˆ M {
10 TeV ˙ , (52)where M { is a typical gaugino mass and we have assumed m { ! M { . Repeating thesteps outlined above, we can again relate µ to the gravitino lifetime, µ c β » . ˆ
10 TeV r m ˙ ˆ Ω { h . ˙ { ˆ s τ { ˙ { ˆ . ˆ GeV T RH ˙ { , (53) We have utilized non-instantaneous reheating in solving the complete set of Boltzmann equations [31]with T max “ ˆ T RH T RH needed to obtain the correct gravitino relicdensity in both limits.As one can see, in both cases (high-scale supersymmetry and weak-scale supersymmetry)the constraints imposed on the RPV couplings from the lifetime of the gravitino (whenassumed to be a dark matter candidate) are comparable or stronger than the limits imposedby the lepton number violating constraints in Eq.(26) for reheating temperatures compatiblewith inflationary scenario.Due to a possible signature in neutrino telescopes such as IceCube or ANITA from theobservation of ultra high energy (monochromatic) neutrinos emerging in the Zν or hν finalstates of gravitino decay, we next show that it is possible to test or set new constraints onthe parameter µ once the telescope or satellite limits are combined with PLANCK data. We next go beyond setting the relation in Eq.(51) which sets a limit on µ for a fixedgravitino lifetime, and use the experimental limits from IceCube as a function of the gravitinomass and/or inflationary reheat temperature. Indeed, unstable gravitinos decaying intomonochromatic neutrinos are severely constrained by searches from the Galactic center orthe Galactic halo. The IceCube collaboration has set a lower bound on the lifetime of heavydark matter candidates [36, 37, 38] (and [33, 39] for older analyses). We can also expectgamma ray fluxes produced by Z -decay, and although it was shown in [40] that the gamma-ray bounds are comparable to the ones derived from neutrino fluxes, the branching fractionto gamma-rays in the model discussed here is suppressed by p M Z { r m q which is negligible.The level of interest in ultra-high energy neutrinos has been raised by the PeV eventsmeasured in the last few years by the IceCube collaboration. IceCube recently released thecombination of two of their results in [36, 38]. The first analysis used 6 years of muon-neutrinodata from the northern hemisphere, while the second analysis uses 2 years of cascade datafrom the full sky . We combined both analyses ( Zν and hν channels) with PLANCK [32]constraints to obtain limits on µ as function of the gravitino mass and reheating temperature.IceCube is sensitive to energies above Á GeV. For energies of the order of the electroweakscale, we applied the limit from the Fermi satellite observation of the galactic halo [42], and
See also [41] for an alternate recent analysis ). We present ourresults in Fig. 2. Using Eq. (50), we can set a limit on µ as a function of m { over the massrange considered by IceCube. Bearing in mind, that in high-scale supersymmetric models,we must have m { ą . µ À c β “ .
1) . For larger values of µ , the gravitino lifetime is too short, yielding aneutrino signal in excess of that observed by IceCube [36]. Note that we have assumed asupersymmetry breaking scale of 10 GeV, and our limit on µ scales linearly with r m .In the bottom panel of Fig. 2, we show the corresponding limit on µ as a function ofthe inflationary reheat temperature which combines Eq. (51) with the limit from IceCube.The vertical line at T RH “ ˆ GeV corresponds to the lower bound on the reheatingtemperature if one considers inflationary-inspired models of reheating. We begin the scanat T RH ą . ˆ GeV, corresponding to m { ą M Z extracted from Eq. (2) to allow theopening of the Zν channel. Once again in order to avoid the overdensity of the UniverseEq. (2), we require a massive gravitino and hence a reheating temperature above „ GeV.On the other hand, if we are not tied to inflationary models, there remains the possibilityfor µ ą O p q GeV if T RH À GeV.
The Antarctic Impulsive Transient Antenna (ANITA) was designed to look for Ultra HighEnergy (UHE) neutrinos produced by the decay of cosmic ray products. The experimentmeasures radio pulses produced by the interaction of neutrinos in the ice (the Askaryan effect[46]) and the balloon transporting the detector has flown three times since 2015. Recently,ANITA detected a „ . ˘ . . o below the horizon [47]. Moreintriguingly, an even more recent flight has observed a similar 0 . ` . ´ . EeV event at an angleof 35 o below the horizon [48]. The measurements are consistent with the decay of an upgoing τ generated by the interaction of an UHE ν τ inside the Earth. However, it is difficult tointerpret this event as an UHE ν τ generated in cosmic ray fluxes because the Earth is quiteopaque to such energetic „ EeV neutrinos. Indeed, a 1 EeV neutrino has an interactionlength of only 1600 kilometers water–equivalent, corresponding to an attenuation coefficient
During the completion of our work, we noticed that the MAGIC telescope released new limits on the ν τ cosmic flux [45], but these limits are currently less stringent than the ones obtained by IceCube. µ c β from the hν ` Zν channel taking into accountthe relic abundance constraints from PLANCK [32] as function of the gravitino mass (top)and as function of the reheating temperature (bottom).of „ ˆ ´ for 27 . o incidence angle [47]. 18ifferent explanations have been proposed, including invoking dark matter decay into asterile neutrino [49] transforming into an active one while passing through the Earth or aheavy 480 PeV right handed neutrino decaying into a Higgs and left-handed neutrino [50].Both interpretations avoid the attenuation problem by the fact that sterile neutrinos havea much longer mean free path in water [49]. In the case of the right handed neutrino, theauthors of [50] claimed that the capture rate of the right handed neutrino is sufficientlystrong to justify a high density of dark matter in the Earth. The probability that a darkmatter particle decays not so far from the ice surface is then not negligible, and can be ofthe order of one decay per year as seems to be observed by ANITA.The EeV energy measured by ANITA is particularly intriguing as this is the mass rangepredicted for the gravitino in the high-scale supersymmetry models we are considering. Itseems natural, therefore, to ask whether or not an EeV gravitino could be responsible forthe events observed by ANITA. Unfortunately, the capture rate of a gravitino by the Earthis Planck suppressed and is ridiculously low. The only possible dark matter decays whichcan give rise to this signal are from the local dark matter density. Using a local dark matterdensity of 0.3 GeVcm ´ , the radius of the Earth of 6371 kilometers, a simple computationgives, for a lifetime of τ { “ . ˆ seconds (the IceCube limit) and a gravitino mass of0.1 EeV, the number of decaying gravitino per year N decay3 { » ˆ . m { ˆ V earth τ { » . .Although not completely ruled out, the observation of two events in 3 years seems to be intension with our estimate. While much of the high energy physics community would be overjoyed with the detection ofweak scale supersymmetry at the LHC, we have no guarantee that the sparticle spectrum lieswithin the reach of the LHC. With the possible exception of the fine-tuning associated withthe hierarchy problem, nearly all of the motivating factors pointing to supersymmetry can beaccounted for in either non-supersymmetric or high-scale supersymmetric GUT models. Inthe latter we have argued that the gravitino is a dark matter candidate if its mass, m { ą . A more precise computation should be done using not the entire Earth, but only a slice correspondingto the mean free path of a 0.1 EeV neutrino, but is beyond the scope of the paper in the view of our result. R -parity violation. Here, we considered the simplest case of the effects of an µ LH u bilinearterm in the superpotential. While the limits from the preservation of the baryon asymmetryare greatly relaxed in high-scale supersymmetric models, the limits on this lepton numberviolating operator are strong. We have used the limits on the high-energy neutrino flux fromIceCube to constrain µ as a function of the gravitino mass and reheat temperature afterinflation. For m { ą . µ ă
50 keV for c β “ . O p q EeV neutrinos at IceCube and other neutrino experiments such as ANITA.While it may be unlikely that the two high energy neutrino events observed by ANITA arerelated to gravitino dark matter, this conclusion may need to be revisited if no other eventsare observed in the next 140 (or so) years.
Acknowledgments
We thank W. Buchmuller for discussions. This work was supported by the France-US PICSno. 06482 and PICS MicroDark. Y.M. acknowledges partial support from the EuropeanUnion Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie:RISE InvisiblesPlus (grant agreement No 690575), the ITN Elusives (grant agreement No674896), and the ERC advanced grants Higgs@LHC. E.D. acknowledges partial support fromthe ANR Black-dS-String. The work of T.G., K.K., and K.A.O. was supported in part bythe DOE grant DE–SC0011842 at the University of Minnesota.20 ppendix A Goldstino contribution in the decay widths
In the decay widths given in the text, we have summed over all gravitino spin states, whilethere are two distinctive contributions, namely, spin ˘ { ˘ { ˘ { ψ µ into spin 1 and 1 { (cid:15) µ and ψ , respectively.Incorporating Clebsch-Gordan coefficients, we have ψ ˘ { µ “ (cid:15) ˘ µ ψ ˘ , ψ ˘ { µ “ c (cid:15) ˘ µ ψ ¯ ` c (cid:15) µ ψ ˘ , (54)where (cid:15) ˘ , µ and ψ ˘ denote the spin ˘ (cid:15) µ and ˘ { ψ .For the ψ µ Ñ γν decay channel, the corresponding interaction in the Lagrangian inmomentum space may be written as ´ i M P ¯ λ p k q γ µ r γ ν , γ ρ s ψ µ p q q F νρ p p q „ i m { M P c
23 ¯ λ p k q{ q r{ p, { A p p qs ψ p q q , (55)where we have used ψ µ p q q „ b q µ m { ψ p q q . Due to the RPV coupling, the gaugino (inthe gauge eigenstate) can be written as λ „ (mixing angle) ˆ ν where ν is the neutrino masseigenstate. Also by using q “ p ` k and the Dirac equation for ν , we obtain ¯ λ p k q{ q „ ¯ ν p k qp{ p ` { k q “ ¯ ν p k qp{ p ` m ν q . Moreover, we have p “ p ¨ A “ { p r{ p, { A s “
0. Therefore, only the amplitude proportional to the neutrino mass can appear forthe decay of the Goldstino mode in this channel.On the other hand, this is not the case for the decays involving a massive gauge boson(or the Higgs boson). For the massive gauge boson case, there appears a large enhancementfor the decay into a fermion and longitudinal mode. In the same manner, we may write therelevant interaction as1 ? M P gA µ p p q φ ˚ ¯ ψ ν p q q γ µ γ ν χ L p k q ` h.c. » g x φ y? m { M P ¯ ψ p q q{ (cid:15) r p p q{ (cid:15) s p q q χ L p k q ` h.c., (56)where we have assumed φ and χ L are the (up or down) Higgs and Higgsino fields, respectively,and the polarization tensors of a gauge field A µ and gravitino are represented by (cid:15) r p p q and (cid:15) s p q q with r, s labelling the polarization states. Each squared amplitude denoted by | M p r, s q| then becomes | M p˘ , ˘q| , | M p˘ , q| , | M p , q| „ ˆ m A M P ˙ m { , | M p , ˘q| „ m { M P , (57) It can be verified by a direct calculation that the contributions of the other polarization states vanish. g x φ y „ m A with m A the gauge boson mass, and we have taken the massless limit for χ L .Thus, it turns out that the Goldstino mode, especially ψ ˘ { µ „ b (cid:15) ˘ µ ψ ¯ , gives the dominantcontribution in the decay into a gauge boson and neutrino pair, and by incorporating themixing between the neutrino and Higgsino, we obtainΓ p ψ µ Ñ Z L ν q „ m { M P | U r Hν | „ m { M P (cid:15) , (58)where Z L denotes the longitudinal mode of the Z -boson, and r H « r H d which has a large mix-ing with the neutrino, as discussed in Section 3.1. Note that this enhancement also appearsin the decay channel ψ µ Ñ W l . For the ψ µ Ñ hν channel, the squared amplitude behaves as m h m { { M P and m { { M P for the spin state ψ µ „ b (cid:15) µ ψ ˘ and b (cid:15) ˘ µ ψ ¯ , respectively, andthus, the latter is the dominant contribution and the resultant decay width becomes similarin size to the ψ µ Ñ Zν, W l channels.
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