Broken R-Parity in the Sky and at the LHC

Supersymmetric extensions of the Standard Model with small R-parity and lepton number violating couplings are naturally consistent with primordial nucleosynthesis, thermal leptogenesis and gravitino dark matter. We consider supergravity models with universal boundary conditions at the grand unification scale, and scalar tau-lepton or bino-like neutralino as next-to-lightest superparticle (NLSP). Recent Fermi-LAT data on the isotropic diffuse gamma-ray flux yield a lower bound on the gravitino lifetime. Comparing two-body gravitino and neutralino decays we find a lower bound on a neutralino NLSP decay length, $c \tau_{\chi^0_1} \gsim 30 cm$. Together with gravitino and neutralino masses one obtains a microscopic determination of the Planck mass. For a stau-NLSP there exists no model-independent lower bound on the decay length. Here the strongest bound comes from the requirement that the cosmological baryon asymmetry is not washed out, which yields $c \tau_{\tilde\tau_1} \gsim 4 mm$. However, without fine-tuning of parameters, one finds much larger decay lengths. For typical masses, $m_{3/2} \sim 100 GeV$ and $m_{NLSP} \sim 150 GeV$, the discovery of a photon line with an intensity close to the Fermi-LAT limit would imply a decay length $c\tau_{NLSP}$ of several hundred meters, which can be measured at the LHC.


Introduction
Locally supersymmetric extensions of the Standard Model predict the existence of the gravitino, the gauge fermion of supergravity [1]. For some patterns of supersymmetry breaking, the gravitino is the lightest superparticle (LSP), and therefore a natural dark matter candidate [2]. Heavy unstable gravitinos may cause the 'gravitino problem' [3][4][5] for large reheating temperatures in the early universe. This is the case for thermal leptogenesis [6], where gravitino dark matter has become an attractive alternative [7] to the standard WIMP scenario [8].
Recently, it has been shown that models with small R-parity and lepton number breaking naturally yield a consistent cosmology incorporating primordial nucleosynthesis, leptogenesis and gravitino dark matter [9]. The gravitino is no longer stable, but its decays into Standard Model (SM) particles are doubly suppressed by the Planck mass and the small R-parity breaking parameter. Hence, its lifetime can exceed the age of the Universe by many orders of magnitude, and the gravitino remains a viable dark matter candidate [10].
Gravitino decays lead to characteristic signatures in high-energy cosmic rays, in particular to a diffuse gamma-ray flux [9][10][11][12][13][14][15][16][17]. The recent search of the Fermi-LAT collaboration for monochromatic photon lines [18] and the measurement of the diffuse gamma-ray flux up to photon energies of 100 GeV [19] severely constrain possible signals from decaying dark matter. In this paper we study the implications of this data for the decays of the next-to-lightest superparticle (NLSP) at the LHC, extending the estimates in [9].
We shall restrict our analysis to the simplest class of supergravity models with universal boundary conditions at the Grand Unification (GUT) scale, which lead to neutralino or τ -NLSP. Electroweak precision tests, thermal leptogenesis and gravitino dark matter together allow gravitino and NLSP masses in the range m 3/2 = 10 . . . 500 GeV and m NLSP = 100 . . . 500 GeV [20]. Following [9], the breaking of R-parity is tied to the breaking of lepton number, which leads to a model with bilinear R-parity breaking [21,22]. The soft supersymmetry breaking terms are characteristic for gravity or gaugino mediation.
In order to establish the connection between the gamma-ray flux from gravitino decays and NLSP decays, one needs R-parity breaking matrix elements of neutral, charged and supercurrents. For the considered supergravity models we are able to obtain these matrix elements to good approximation analytically. This makes our results for the NLSP decay lengths rather transparent. As we shall see, the lower bound on the neutralino decay length is a direct consequence of the Fermi-LAT constraints on decaying dark matter. On the other hand, the lower bound on the τ -decay length is determined by the cosmological bounds on R-parity breaking couplings, which follow from the requirement that the baryon asymmetry is not washed out [24,25]. This paper is organized as follows. In Section 2 we discuss the general Lagrangian for R-parity breaking in a basis of scalar SU (2) doublets where all bilinear mixing terms vanish. This leads to new Yukawa and gaugino couplings, some of which are proportional to the up-quark Yukawa couplings. Section 3 deals with the various supersymmetry, R-parity and lepton number breaking terms in the Lagrangian and the relations among them due to a U(1) flavour symmetry of the considered model. The needed R-parity breaking matrix elements of neutral, charged and supercurrent are analytically calculated in Section 4, based on the diagonalization of the mass matrices which is discussed in detail in the appendix. The main results of the paper, the bounds on the NLSP decay lengths and the partial decay widths, are described in Section 5, followed by our conclusions in Section 6.

Bilinear R-Parity Breaking
Supersymmetric extensions of the Standard Model with bilinear R-parity breaking contain mass mixing terms between lepton and Higgs fields in the superpotential 1 , as well as the scalar potential induced by supersymmetry breaking, These mixing terms, together with the R-parity conserving superpotential the scalar mass terms where higher order terms in the R-parity breaking parameters have been neglected.
It is convenient to discuss the predictions of the model in a basis of SU(2) doublets where the mass mixings µ i , B i and m 2 id in Eqs. (1) and (2) are traded for R-parity breaking Yukawa coulings. This can easily be achieved by field redefinitions. First one rotates the superfields H d and l i , Then the bilinear term (1) vanishes for the new fields, i.e., µ ′ i = 0, and one obtains instead the cubic R-parity violating terms where The new R-parity breaking mass mixings are given by The corrections for R-parity conserving mass terms are negligable.
In a second step one can perform a non-supersymmetric rotation among all scalar SU(2) doublets, where ε is the usual SU(2) matrix, ε = iτ 2 . Choosing the H uli andl † H d mixing terms vanish in the new basis of doublets. According to (6) also the scalar lepton VEVs ν i vanish in this basis.
It is straightforward to work out the R-parity violating Yukawa couplings which are induced by the rotation (11). We are particularly interested in the terms containing one light superparticle, i.e, a scalar lepton, bino or wino. The corresponding couplings read, after dropping prime and double-prime superscripts on all fields 3 , where the Yukawa couplings are given by Since the field transformations are non-supersymmetric, the couplings λ ijk andλ ijk are no longer equal as in Eq. (9). Furthermore, a new coupling of right-handed up-quarks, λ ′ ijk , has been generated. After electroweak symmetry breaking one obtains new mass mixings between higgsinos, gauginos and leptons, where we have defined Given the Yukawa couplings h u ij , h d ij and h e ij , the Lagrangian (14) predicts 108 Rparity breaking Yukawa couplings in terms of 9 independent parameters which may be chosen as These parameters determine lepton-gaugino mass mixings, lepton-slepton and quarkslepton Yukawa couplings, and therefore the low-energy phenomenology. The values of these parameters depend on the pattern of supersymmetry breaking and the flavour structure of the supersymmetric standard model. 3 Our notation for gauge fields, field strengths and left-handed gauginos reads: B µ , B µν , b etc.

3 Spontaneous R-parity breaking
Let us now compute the parameters ǫ i , ǫ ′ i and ǫ ′′ i in a specific example where the spontaneous breaking of R-parity is related to the spontaneous breaking of B-L, the difference of baryon and lepton number [9].
We consider a supersymmetric extension of the standard model with symmetry group In addition to three quark lepton generations and the Higgs fields H u and H d the model contains three right-handed neutrinos ν c i , two non-Abelian singlets N c and N, which transform as ν c and its complex conjugate, respectively, and three gauge singlets X, Φ and Z. The part of the superpotential responsible for neutrino masses has the usual form where M P = 2.4 × 10 18 GeV is the Planck mass. The expectation value of H u generates Dirac neutrino masses, whereas the expectation value of the singlet Higgs field N generates the Majorana mass matrix of the right-handed neutrinos ν c i . The superpotential responsible for B-L breaking is chosen as where unknown Yukawa couplings have been set equal to one. Φ plays the role of a spectator field, which will finally be replaced by its expectation value, Similarly, Z is a spectator field which breaks supersymmetry and U(1) R , Z = F Z θθ. The superpotential in Eqs. (22) and (23) is the most general one consistent with the R-charges listed in Table 1, up to nonrenormalizable terms which are irrelevant for our discussion.
The expectation value of Φ leads to the breaking of B − L, where the first equality is a consequence of the U(1) B−L D-term. This generates a Majorana mass matrix M for the right-handed neutrinos with three large eigenvalues Integrating out the heavy Majorana neutrinos one obtains the familiar dimension-5 seesaw operator which yields the light neutrino masses.
Since the field Φ carries R-charge −1, the VEV Φ breaks R-parity, which is conserved by the VEV Z . Thus, the breaking of B − L is tied to the breaking of R-parity, which is then transmitted to the low-energy degrees of freedom via higher-dimensional operators in the superpotential and the Kähler potential. Bilinear R-parity breaking, as discussed in the previous section, is obtained from a correction to the Kähler potential, Replacing the spectator fields Z and Φ, as well as N c and N by their expectation values, one obtains the correction to the superpotential where m 3/2 = F Z /( √ 3M P ) is the gravitino mass. Note that Θ can be increased or decreased by including appropriate Yukawa couplings in Eqs. (22) and (23). The corresponding corrections to the scalar potential are given by The corresponding R-parity conserving terms are generated by [26] K which yields Higher dimensional operators yield further R-parity violating couplings between scalars and fermions. However, the cubic couplings allowed by the symmetries of our model are suppressed by one power of M P compared to ordinary Yukawa couplings and cubic soft supersymmetry breaking terms. Note that the coefficients of the nonrenormalizable operators are free parameters, which are only fixed in specific models of supersymmetry breaking. In particular, one may have µ 2 , m 2 i > m 2 3/2 and hence a gravitino LSP. All parameters are defined at the GUT scale and have to be evolved to the electroweak scale by the renormalization group equations.
The phenomenological viability of the model depends on the size of R-parity breaking mass mixings and therefore on the scale v B−L of R-parity breaking as well as the parameters a i . . . c ′ i in Eq. (25). Any model of flavour physics, which predicts Yukawa couplings, will generically also predict the parameters a i . . . c ′ i . As a typical example, we use a model [27] for quark and lepton mass hierarchies based on a Froggatt-Nielsen U(1) flavour symmetry, which is consistent with thermal leptogenesis and all contraints from flavour changing processes [28].
The mass hierarchy is generated by the expectation value of a singlet field φ with charge Q φ = −1 via nonrenormalizable interactions with a scale Λ = φ /η > Λ GU T , η ≃ 0.06. The η-dependence of Yukawa couplings and bilinear mixing terms for multiplets ψ i with charges Q i is given by The charges Q i for quarks, leptons, Higgs fields and singlets are listed in Table 2. The neutrino mass scale m ν ≃ 0.01 eV implies for the heaviest right-handed neutrinos M 2 ∼ M 3 ∼ 10 12 GeV. The corresponding scales for B − L breaking and R-parity breaking are For the small R-parity breaking considered in this paper the neutrino masses are dominated by the conventional seesaw contribution [9].
The R-parity breaking parameters µ i , B i and m 2 id strongly depend on the mechanism of supersymmetry breaking. In the example considered in this section all mass parameters are O(m 3/2 ), which corresponds to gravity or gaugino mediation. From Eqs. (26), (27) and (31) one reads off ψ i 10 3 10 2 10 1 5 * . Correspondingly, one obtains for ǫ-parameters (cf. (12), (13)) with a, b, c = O(1). Our phenomenological analysis in Section 5.2 will be based on this parametrization of bilinear R-parity breaking.
Depending on the mechanism of supersymmetry breaking, the R-parity breaking soft terms may vanish at the GUT scale [22], Non-zero values of these parameters at the electroweak scale are then induced by radiative corrections. The renormalization group equations for the bilinear R-parity breaking mass terms read (cf. [22], t = ln Λ): In bilinear R-parity breaking, the R-parity violating Yukawa couplings vanish at the GUT scale. One-loop radiative corrections then yield for the soft terms at the electroweak scale (cf. Eqs. (36),(37); ǫ i = µ i /µ): This illustrates that the bilinear R-parity breaking terms µ 2 i , B i and m 2 id are not necessarily of the same order of magnitude at the electroweak scale.

Neutral, charged and supercurrents
In Section 2 we have discussed the R-parity breaking Yukawa couplings in our model. For a phenomenological analysis we also need the couplings of the gauge fields, i.e., photon, W-bosons and gravitino, to charged and neutral matter, 9 The corresponding currents read The gravitino and the gauginos are now Majorana fermions, where the superscript c denotes charge conjugation. In Eqs. (41) -(44) we have only listed contributions to the currents which will be relevant in our phenomenological analysis.
The R-parity breaking described in the previous section leads to mass mixings between the neutralinos b, w 3 , h 0 u , h 0 d with the neutrinos ν i , and the charginos w + , h + u , w − , h − d with the charged leptons e c i , e i , respectively. The 7 × 7 neutralino mass matrix reads in the gauge eigenbasis where we have neglected neutrino masses. Correspondingly, the 5 × 5 chargino mass matrix which connects the states (w − , h − d , e i ) and (w + , h + u , e c i ) is given by Note that all gaugino and higgsino mixings with neutrinos and charged leptons are parametrized by the three parameters ζ i .
In the following section we shall need the couplings of gravitino, W -and Z-bosons to neutralino and chargino mass eigenstates. Since ζ i ≪ 1, diagonalization of the mass matrices to first order in ζ i is obviously sufficient. We shall also consider supergravity models where the supersymmetry breaking parameters satisfy the inequalities (cf. Fig. 1) The gaugino-higgsino mixings are O(m Z /µ), and therefore suppressed, and χ 0 1 , the lightest neutralino, is bino-like.
The mass matrices M N and M C are diagonalized by unitary and bi-unitary transformations, respectively, where These unitary transformations relate the neutral and charged gauge eigenstates to the mass eigenstates (χ 0 a , ν ′ i ) (a = 1, . . . , 4) and (χ − α , e ′ i ), (χ + α , e ′c i ) (α = 1, 2), respectively. Inserting these transformations in Eqs. (42) -(44) and dropping prime superscripts, one obtains neutral, charged and supercurrents in the mass eigenstate basis: where we have defined the photino matrix elements In the appendix the unitary transformations between gauge and mass eigenstates and the resulting matrix elements of neutral and charged currents are given to next-to-leading order in m Z /µ. As we shall see, that expansion converges remarkably well.
In the next section we shall need the couplings of the lightest neutralino χ 0 1 to charged leptons and neutrinos, and the coupling of the gravitino to photon and neutrino. From the formulae in appendix A one easily obtains 4 (s 2β = 2s β c β ) Note that the charged and neutral current couplings agree up to the isospin factor at leading order in m 2 1i LO /2. The mass of the lightest neutralino is given by We have numerically checked that varying M 1 between 120 and 500 GeV, the relative corrections in Eqs. (54) -(57) are less than 10%.

Fermi-LAT and the LHC
We are now ready to evaluate the implications of recent Fermi-LAT data [18,19] and cosmological constraints [24,25] for signatures of decaying dark matter at the LHC. We shall first discuss monochromatic gamma-rays produced by gravitino decays and then analyze the implications for a neutralino and a τ -NLSP, respectively.
In order to keep our analysis transparent we shall not study the most general parameter space of softly broken supersymmetry, but only consider two typical boundary conditions for the supersymmetry breaking parameters of the MSSM at the grand unification scale, (A) m 0 = m 1/2 , a 0 = 0, tan β = 10 , which yields the right-handed stau as NLSP. In both cases, the trilinear scalar coupling a 0 is put to zero for simplicity. Choosing tan β = 10 as a representative value of the Higgs vacuum expectation values, only the gaugino mass parameter m 1/2 remains as independent variable; the mass parameters µ and B are determined by requiring radiative electroweak symmetry breaking with the chosen ratio tan β. For both boundary conditions (58) and (59), the gaugino masses at the electroweak scale satisfy the familiar relations For the chosen supergravity models, consistency with electroweak precision tests, gravitino dark matter (GDM) and thermal leptogenesis leads to the following allowed mass ranges of gravitino and lightest neutralino [20], where we have used m χ 0 1 ≃ M 1 (cf. (57)). Note that the masses M 1 and m 3/2 cannot be chosen independently. The GDM constraint implies that for a given gravitino mass the maximal bino mass is M max 1 ≃ 270 GeV(m 3/2 /100 GeV) 1/2 [20].
Inserting the matrix element (56) one obtains the gravitino lifetime where α is the electromagnetic fine-structure constant, and we have defined The corrections to the leading order expression in (63) are less than 10%. Using Eq. (60) and M P = 2.4 × 10 18 GeV, one obtains (65) Recent Fermi-LAT data yield for dark matter decaying into 2 photons the lower bound on the lifetime τ DM (γγ) > ∼ 1 × 10 29 s, which holds for photon energies in the range 30 GeV < E γ < 200 GeV [18]. For gravitino decays into photon and neutrino this implies Since according to the GDM constraint the largest allowed bino mass scales like M max 1 ∝ m 1/2 3/2 , the largest lifetime (65), and therefore the most conservative bound on ζ, is obtained for the smallest value of m 3/2 . For small gravitino masses, a rough lower bound on the lifetime can be obtained from the isotropic diffuse gamma-ray flux. The recent Fermi-LAT data give E 2 dJ/dE| 5 GeV ≃ 3 × 10 −7 GeV (cm 2 s str) −1 [19]. From the analysis in [12] one then obtains τ 3/2 > ∼ 10 28 s. 6 Together with Eq. (65) one then obtains the approximate upper bound on the R-parity breaking parameter On the other hand, the observation of a photon line corresponding to a gravitino lifetime close to the present bound would determine the parameter ζ as 7 ζ obs = 10 −9 5 × 10 28 s τ 3/2 (γν) Note the strong dependence of ζ obs on the gravitino mass. In (68) we have normalized these masses to central values suggested by thermal leptogenesis, electroweak precision tests and gravitino dark matter [20].

Neutralino NLSP
A neutralino NLSP heavier than 100 GeV dominantly decays into charged lepton and W-boson or neutrino and Z-boson [30] (cf. Fig. 3). The partial decay widths are given by Here V (χ,e) 1i LO and V (χ,ν) 1i LO are the charged and neutral current matrix elements at leading order, which are given in Eqs. (55) and (54), respectively, and is a phase space factor which becomes important for neutralino masses close to the lower bound for m χ 0 1 of 100 GeV (cf. Fig. 4). The total neutralino NLSP width is the sum Using the matrix elements (54) and (55), one obtains the branching ratios Furthermore, the flavour structure of our model implies Using the matrix elements (54), (55) and (56) for neutral, charged and supercurrent, respectively, one can express the neutralino lifetime directly in terms of the gravitino lifetime, With the mass relations (57) and (60) one then obtains for the minimal neutralino decay length In Eqs. (75) and (76) the corrections to the leading order expressions are less than 10%. We emphasize again the strong dependence of this lower bound on the neutralino and gravitino masses. For instance, for a gravitino mass of 100 GeV and the Fermi-LAT bound τ 3/2 > ∼ 5 × 10 28 s, which applies for gravitino masses in the range 60 GeV < m 3/2 < 400 GeV, one obtains cτ χ 0 1 ≃ 2 km for a neutralino mass of 150 GeV. It is very interesting that such neutralino lifetimes are detectable at the LHC [31].
We conclude that, given the current bounds on the gravitino lifetime, a neutralino NLSP may still decay into gauge boson and lepton inside the detector, yielding a spectacular signature. However, for most of the parameter space a neutralino NLSP decays outside the detector, leading to events indistinguishable from ordinary neutralino dark matter.

τ -Lepton NLSP
Contrary to the neutralino NLSP decay, the R-parity violating decays of a τ 1 -NLSP strongly depend on the flavour structure and the supersymmetry breaking parameters. The relative strength of the various decay modes becomes most transparent in the field basis where all bilinear R-parity breaking terms vanish, as discussed in Section 2. Since the R-parity breaking Yukawa couplings are proportional to the ordinary Yukawa couplings, decays into fermions of the second and third generation dominate. The leading partial decay widths of left-and right-handed τ -leptons are (cf. (14)) In the flavour model discussed in Section 3, the order of magnitude of the various decay widths is determined by the power of the hierarchy parameter η (η 2 ≃ 1/300), The lightest mass eigenstate τ 1 is a linear combination of τ L and τ R , From the above equations one obtains the τ 1 -decay width The total width is dominated by the contributions τ R → τ L ν, µ L ν and τ L →t R b L , respectively, and it can be directly expressed in terms of the τ -lepton and top-quark masses, where we have assumed This corresponds to the parameter choice a = b = c = 1 in Eq. (34). Note that τ 1decay width and branching ratios have a considerable uncertainty since these parameters depend on the unspecified mechanism of supersymmetry breaking. From Eqs. (18), (26) and η ≃ 0.06, one obtains for the R-parity breaking parameter which is consistent with the present upper bound (67) within the theoretical uncertainties.
The dependence of the mixing angle θ τ on m τ 1 is shown in Fig. 5 for the boundary condition (59). For masses below the top-bottom threshold only leptonic τ 1 -decays are possible. When the decay into top-bottom pairs becomes kinematically allowed, sin 2 θ τ is small. However, the suppression by a small mixing angle is compensated by the larger Yukawa coupling compared to the leptonic decay mode. This is a direct consequence of the couplingsλ ′ which were not taken into account in previous analyses.
Due to the competition between mixing angle suppression and hierarchical Yukawa couplings, the top-bottom threshold is clearly visible in the τ 1 -decay length as well as   Branching Ratio¯t the branching ratios into leptons and heavy quarks. This is illustrated in Figs. 6 and 7, respectively, where these observables are plotted as functions of m τ 1 . Representative values of the τ 1 -decay lengths below and above the top-bottom threshold are (90) Choosing for ǫ the representative value (68) from gravitino decay, ǫ = ζ obs = 10 −9 , one obtains cτ τ 1 = 4 km(1 km) for m τ 1 = 150 GeV(250 GeV). It is remarkable that such lifetimes can be measured at the LHC [31,32].
Is it possible to avoid the severe constraint from gravitino decays on the τ 1 -decay length? In principle, both observables are independent, and the unknown constants in the definition of ǫ, ǫ ′ and ǫ ′′ can be adjusted such that ζ = 0. However, this corresponds to a strong fine-tuning, unrelated to an underlying symmetry. To illustrate this, consider the case where the soft R-parity breaking parameters vanish at the GUT scale, B i = m 2 id = 0, which was discussed in Section 3. In bilinear R-parity breaking, also the R-parity violating Yukawa couplings vanish at the GUT scale. With the one-loop radiative corrections at the electroweak scale (cf. (39); ǫ i = µ i /µ), and M 1,2 ∼ µ, one reads off from Eqs. (10), (12) and (13) Hence, all R-parity breaking parameters are naturally of the same order, unless the finetuning also includes radiative corrections between the GUT scale and the electroweak scale.
Even if one accepts the fine-tuning ζ = 0, one still has to satisfy the cosmological bounds on R-parity violating couplings, which yield ǫ i = µ i /µ < ∼ 10 −6 [25]. In the flavour model discussed in Section 3 this corresponds to the choice a = 20 in Eq. (33). For the smaller τ 1 -mass, which is preferred by electroweak precision tests, one then obtains the lower bound on the decay length However, let us emphasize again that current constraints from Fermi-LAT on the diffuse gamma-ray spectrum indicate decay lengths several orders of magnitude larger. 20

Planck Mass Measurement
It has been pointed out in [9] that, in principle, one can determine the Planck mass from decay properties of a τ -NLSP together with the observation of a photon line in the diffuse gamma-ray flux, which is produced by gravitino decays. This is similar to the proposed microscopic determination of the Planck mass based on decays of very long lived τ -NLSP's in the case of a stable gravitino [33].
From our analysis of NLSP decays in this section it is clear that neutralino NLSP decays are particularly well suited for a measurement of the Planck mass, which does not require any additional assumptions. Eq. (75) implies (G F = √ 2/(4v 2 )), As expected, for gravitino and neutralino masses of the same order of magnitude, the ratio of the two-body lifetimes is determined by the ratio of the electroweak scale and the Planck mass, Quantitatively, using the relation (60) for the gaugino masses, one finally obtains (v = 174 GeV), M P =3.6 × 10 18 GeV m 3/2 m χ 0 1 3/2 τ 3/2 (γν) It is remarkable that the observation of a photon line in the diffuse gamma-ray flux, together with a measurement of the neutralino lifetime at the LHC, can provide a microscopic determination of the Planck mass.

Summary and conclusions
We have studied a supersymmetric extension of the Standard Model with small R-parity breaking related to spontaneous B − L breaking, which is consistent with primordial nucleosynthesis, thermal leptogenesis and gravitino dark matter. We have considered supergravity models with universal boundary conditions at the GUT scale, which lead to scalar tau or bino-like neutralino as NLSP. Supersymmetry breaking terms have been introduced by means of higher-dimensional operators. The size of the soft terms corresponds to gravity or gaugino mediation.
We have analyzed our model, which represents a special case of bilinear R-parity breaking, in a basis of scalar SU(2) doublets, where all bilinear terms vanish. In this basis one has R-parity violating Yukawa and gaugino couplings. They are given in terms of ordinary Yukawa couplings and 9 R-parity breaking parameters ǫ i , ǫ ′ i and ǫ ′′ i , i = 1, ..., 3, which are constrained by the flavour symmetry of the model. The R-parity violating couplings include terms proportional to the up-quark Yukawa couplings, which were not taken into accound in previous analyses.
The main goal of this paper are the quantitative connection between gravitino decays and NLSP decays, and the corresponding implications of recent Fermi-LAT data on the isotropic diffuse gamma-ray flux for superparticle decays at the LHC. To establish this connection one needs the relevant R-parity breaking matrix elements of neutral, charged and supercurrents. For the considered supergravity models these matrix elements can be obtained analytically to good approximation, since the diagonalization of the neutralino-neutrino and chargino-lepton mass matrices in powers of m Z /µ converges well, as demonstrated in the appendix. The analytic expressions for the decay rates make the implications of the Fermi-LAT data for NLSP decays very transparent.
Our main quantitative results are the branching ratios for NLSP decays and the lower bounds on their decay lengths. For a neutralino NLSP with m χ 0 1 = 150 GeV, the Fermi-LAT data yield the lower bound cτ χ 0 1 > ∼ 30 cm. This bound does not depend on details of the superparticle mass spectrum or the flavour structure of the model. It directly follows from the comparison of two-particle gravitino and neutralino decays. On the contrary, there exists no model independent lower bound on the τ 1 -decay length. The natural relation between gravitino and τ -decay widths can be avoided by fine-tuning. In this case the cosmological constraint that the baryon asymmetry is not washed out leads to the lower bound cτ τ 1 > ∼ 4 mm.
Without fine-tuning parameters the diffuse gamma-ray flux produced by gravitino decays constrains the lifetime of a neutralino as well as a τ -NLSP. For typical masses, m 3/2 ∼ 100 GeV and m NLSP ∼ 150 GeV, the discovery of a photon line with an intensity close to the present Fermi-LAT limit would imply a decay length cτ NLSP of several hundered meters. This is a definite prediction of a class of supergravity models. It is very interesting that such lifetimes can be measured at the LHC [31,32].
Finally, it is intriguing that the observation of a photon line in the diffuse gammaray flux, together with a measurement of the neutralino lifetime at the LHC, can yield a microscopic determination of the Planck mass, a crucial test of local supersymmetry.

A.1 Mass matrix diagonalization
The mass matrices M N and M C in the gauge eigenbasis were explicitly given in Eqs. (46) and (47), respectively, For non-vanishing R-parity breaking parameters ζ i , i = 1, . . . , 3, they induce a mixing between gauginos, Higgsinos and leptons,