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2 tic foreground emission, the residual dux was found to could resemble an isotropic extragalactic dux (for a more be roughly isotropic and thus attributed to extragalac- detailed discussion, see [15]). In this expression phaio tic sources. The first analysis of the EGRET data by stands for the dark matter distribution in the Milky Way Sreekumar et of. [13] gave an extragalactic dux with an halo. For our numerical analysis we will adopt a Navarro- energy spectrum described by the power law Frenk-White density prodle [16]

E2-^ = 1.37 x 10 6 f ^ 1 (cm2str s) ^GeV ^°^-r/rc(l + r/rc)2' ^ dE \1 GeV y (2) where r k the dkt^^ to the Galactic center, rc ^ 20 kpc in the energy range 50 MeV-10 GeV. The improved anal ­ is the critical radius and ph ^ 0.33 GeV cm-3. ysis of the galactic foreground by Strong et a! [14], opti­ mized in order to reproduce the galactic emission, shows In Eqs. (3,5) the only undetermined quantity is the a power law behavior between 50 MeV-2 GeV, but a clear energy spectrum of photons produced in the gravitino excess between 2-10 GeV, roughly the same energy range decay, dNY/dE, which depends crucially on the gravitino where one would expect a signal from gravitino decay. mass. If the gravitino is lighter than the W± bosons, it Although it is very tempting to look for explanations for decays mainly into a photon and a neutrino by means of this excess in terms of gravitino decays, in view of all the the photino-neutrino mixing that arises when R-parity is systematic uncertainties involved in the extraction of the violated [11]. Therefore, the spectrum is simply signal from the galactic foreground, we will not attempt to dt our predicted dux to the EGRET data. Nonethe­ (8) less, we will show later the EGRET data superimposed with our predicted dux for comparison. For this case, it was found in [10, 15] that the total The total gamma ray dux received from gravitino dark gamma ray dux received is dominated by the monochro­ matter decay receives two main contributions. The drst matic line coming from the decay of gravitinos in our one stems from the decay of gravitinos at cosmological Milky Way halo, while the redshifted line from the de­ distances, giving rise to a perfectly isotropic extragalactic cay of gravitinos at cosmological distances is somewhat diffuse gamma ray background. Dedning dN^/dE as the fainter. gamma ray spectrum produced in the gravitino decay, On the other hand, if the gravitino is heavier than the the dux received at the Earth with extragalactic origin W± or Z0 bosons, new decay modes are open. In ad­ has the following expression: dition to the decay mode into a photon and a neutrino that follows from the photino-neutrino mixing, Uy V, the 2E2 c r°°d dN7 y-°E2 gravitino can also decay into a W± boson and a charged £2S m3/2 7 Ji d(Ey) y2! + Qa/QmV eg lepton, through the mixing charged wino-charged lepton, (3) Uyf , or mto a Z0 boson and a neutrino, through the mix­ where y = 1 + z, z being the redshift, and ing zino-neutrino, Ug^The decay rates can be straight­ forwardly computed from the interaction Lagrangian of C7 =------3^2/?c 1/9 ~ 10 7 (cm2s str) r3/2 \ 1 10^8 g j a gravitino with a gauge boson and a fermion [17] . The 8nT3/2Ho result for each decay mode reads: (4 Here, G3/2, mid G* are the gravitino, matter and 2™3/2 F(^3/2 ^32^"' cosmological constant density parameters, respectively, M^ Pc k the ^itical density, T3/2 the gravitino lifetime, and f Mw A Ho the present value of the Hubble parameter. r(^3/2 -» W=LfF) - |2™3/2 , In addition to the cosmological contribution, the to­ ' My 1^3/2/ tal gamma ray dux also receives a contribution from the |2 ^3/2 „ Mz A decay of gravitinos in the Milky Way halo. This contri­ F(^3/2 ^ ^-32^" (9) bution reads: M2 ™3/2/

2 E2 dNy where (5) _ halo TU3/2 ^ dE ' f (x) ‘-k (10) where E^ ^ defined as

E-y = %------/ Aialo(f)d/. (6) The fragmentation of the W± ^d the Z0 gauge bosons 8nT3/2 ./log will eventually produce photons, mainly from the decay of neutral pions. We have simulated the fragmentation The integration extends over the line of sight, so E^ of the gauge bosons with the event generator PYTHIA has an angular dependence on the direction of observa­ 6.4 [18] and calculated the spectra of photons in the tion, yielding a slightly anisotropic gamma ray flux that W ± mid Z0 ^^nds, whidi we de note by dNW/dE and 3 dN"Z/dE, respectively. The total spectrum is therefore TABLE I: Branching ratios for gravitino decay in different given by: R-parity violating channels for different gravitino masses. ^ BR(V3/2 - 7Wj + 772.3/2 BR(^3/2 BR(^3/2 -7 tVf) BR(Vs/2 ^ z°y) 10 GeV 1 0 0 dNW dNZ BR(^/2 ^ + BR(^/2 ^ - (11) 85 GeV 0.66 0.34 0 100 GeV 0.16 0.76 0.08 The branching ratios in the different decay channels 150 GeV 0.05 0.71 0.24 are determined by the size of the R-parity breaking mix­ 250 GeV 0.03 0.69 0.28 ing parameters, U^v, Uzv mid U^, and by the kine- matical function f(x) dedned in Eq. (10). The mixing parameters stem from the diagonalization of the 7 x 7 neutralino-neutrino and 5 x 5 chargino-charged lepton which under the assumption of gaugino mass universality mass matrices, whose explicit form can be found in the at the Grand Unified Scale yields |U'fl| — 1.09|Ugv |. vast existing literature on R-parity violation [19]. The Hence, under this assumption, the three relevant mixing parameters me in the ratio precise expression for the mixing parameters in terms of the R-parity breaking couplings in the Lagrangian is |Uev| : |Uzv| : |Uf,| — 1 : 3.2: 3.5 , (17) fairly cumbersome and will not be reproduced here. How­ ever, to derive the branching ratios, only the ratio among and thus the branching ratios for the different decay them is relevant, and not their overall value. modes only depend on the gravitino mass (see Table I). To derive the relation between U^^ mid Uzv, we first Once the spectrum of photons from gravitino decay note that the photino does not couple directly to the neu­ has been determined, Eq. (11), it is straightforward to trino (since neutrinos do not couple to photons). Never­ compute the gamma ray dux received at the Earth from theless, an effective photino-neutrino mixing is generated our local halo and from cosmological distances, by us­ through the mixing photino-zino and the mixing zino- ing Eqs. (3,5). Assuming universality of gaugino masses neutrino. The result reads: at high energies, the photon dux received from gravitino decay depends essentially on the gravitino mass, which M ez |Uev1 - IU (12) determines the shape of the energy spectrum, and the Zv I gravitino lifetime, which determines its overall normal ­ ization. Therefore, the relation between U^^ mid Uzv follows from In Fig. 1 we show the different contributions to the the 2x2 gaugino sub-block of the neutralino mass matrix, gamma ray dux for m3/2 _ 150 GeV and T3/2 — 2 x 1026 s. that in the (—iy, —iE) basis reads To compare our results with the EGRET data [14], also shown in the dgure, we have averaged the halo signal n _( M1 CW + M2sW (M2 — Ml)sWcwX qx 2x2 ((M2 — Mi)swcw MisW + M2 CW J over the whole sky excluding a band of ±10° mound the Galactic disk, and we have used an energy resolu­ Here, M^ ^d M^ ^e the U(1)y mid SU(2)^ gaugino tion of 15%, as quoted by the EGRET collaboration in masses, and cw (sw) denotes the cosine (sine) of the this energy range. The energy resolution of the detec­ weak mixing angle. Therefore, tor is particularly important to determine the width and the height of the monochromatic line stemming from the (M2 — M\)swcw |Uev1 - |UZv 1 , (14) two body decay ^3/2 ^ ^v. The three contributions MlC^y + M^S^y me dominated by the halo component, the extragalactic that depends only on the gaugino masses at the elec- component being smaller by a factor of 2-3. Finally, to troweak scale. Assuming gaugino mass universality at compute the total dux received, we have adopted an en­ the Grand Unified Scale, Mx _ 2 x 1016 GeV, we ob­ ergy spectrum for the back^ound described by the power law = 4 x 10-T (c#v)"° ^ (cm2str s)-^GeV. tain at low energies M2/M1 — 1.9, whi^ yieIds |Uev | — 0.31IUZ, |. The predicted energy spectrum shows two qualitatively The mixing parameter Uf on the other hand, is re­ different features. At energies between 1-10 GeV, we ex­ lated to Ugv by SU(2)^ gauge invmiance. The relation pect a continuous spectrum of photons coming from the approximately reads: fragmentation of the gauge bosons. As a result, the pre­ dicted spectrum shows a departure from the power law M |Uf i\ — zz in this energy range that might be part of the appar ­ ^v1 ' (15) Mf ent excess inferred from the EGRET data by Strong et a! [14]. The upcoming satellite-based gamma ray ex­ where M"f _ M2 is the wino mass at the electroweak scale. Using Eq. (13), we finally obtain periments GLAST and AMS-02 will measure the energy spectrum with unprecedented accuracy, providing very Ml S^y + M^Cyy valuable information for the scenario of decaying grav­ |Uf i\ — |Uev I , (16) itino dark matter. M2 4

right intensity would support the gravitino dark mat ­ ter decay hypothesis. While scenarios with neutralino Total flux dark matter also predict a continuous spectrum and a monochromatic line coming from the annihilation chan ­ nels x0x0 ^ YY, ZY [20], these channels only arise at the Background quantum level, and thus the intensity of the monochro­ matic line is greatly suppressed compared to the contin­ uum. One should note, however, that the presence of an intense gamma line is not unique to the scenario with decaying gravitino dark matter and is also expected, for example, from the annihilation of inert Higgs dark mat ­ ter [21]. To summarize, in this letter we have computed the E [GeV] gamma ray dux from gravitino dark matter decay in sce­ narios with R-parity violation. These scenarios are very FIG. 1: Contributions to the total gamma ray dux for m3/2 = appealing theoretically, as they naturally lead to a his­ 150 GeV and T3/2 ^ 2 x 1026 s compared to the EGRET data. tory of the Universe consistent with thermal leptogenesis In dotted lines we show the photon dux from the fragmenta ­ tion of the Z boson, in dashed lines from the fragmentation and primordial nucleosynthesis. The predicted dux es­ of the W boson, and in dot-dashed lines from the two body sentially depends on two parameters: the gravitino mass, decay ^3/2 ^ YV. The background is shown as a long dashed which determines the shape of the energy spectrum, and line, while the total dux received is shown as a thick solid line. the gravitino lifetime, which determines its overall nor­ malization. If the gravitino is lighter than the W± and Z0 gauge bosons, the predicted energy spectrum is essen­ tially monochromatic. On the other hand, if it is heavier, In addition to the continuous component, the energy the energy spectrum consists of a continuous component spectrum shows a relatively intense monochromatic line and a relatively intense gamma ray line. This gamma ray at higher energies arising from the decay channel ^ dux might have already been observed by EGRET. Fu­ Yv. This line could be observed not only by GLAST or ture experiments, such as GLAST, AMS-02 or Cherenkov AMS-02, but also by ground-based Cherenkov telescopes telescopes, will provide unique opportunities to test the such as MAGIC (with an energy threshold of 70 GeV) or decaying gravitino dark matter scenario. VERITAS (50 GeV). The intense gamma line is very characteristic of this Acknowledgements; We are grateful to W. Buchmiiller, scenario, and the observation of this feature with the G. Bertone, L. Covi and L. Fieri for useful discussions.

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