Astrophysical Probes of Unification
Asimina Arvanitaki, Savas Dimopoulos, Sergei Dubovsky, Peter W. Graham, Roni Harnik, Surjeet Rajendran
AAstrophysical Probes of Unification
Asimina Arvanitaki,
1, 2
Savas Dimopoulos, Sergei Dubovsky,
3, 4
Peter W. Graham, Roni Harnik, and Surjeet Rajendran
5, 3 Berkeley Center for Theoretical Physics,University of California, Berkeley, CA, 94720 Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720 Department of Physics, Stanford University, Stanford, California 94305 Institute for Nuclear Research of the Russian Academy of Sciences,60th October Anniversary Prospect, 7a, 117312 Moscow, Russia SLAC National Accelerator Laboratory,Stanford University, Menlo Park, California 94025 (Dated: October 26, 2018)
Abstract
Traditional ideas for testing unification involve searching for the decay of the proton and its branchingmodes. We point out that several astrophysical experiments are now reaching sensitivities that allow themto explore supersymmetric unified theories. In these theories the electroweak-mass dark matter particlecan decay, just like the proton, through dimension six operators with lifetime ∼ sec. Interestingly, thistimescale is now being investigated in several experiments including ATIC, PAMELA, HESS, and Fermi.Positive evidence for such decays may be opening our first direct window to physics at the supersymmetricunification scale of M GUT ∼ GeV, as well as the TeV scale. Moreover, in the same supersymmetricunified theories, dimension five operators can lead a weak-scale superparticle to decay with a lifetime of ∼
100 sec. Such decays are recorded by a change in the primordial light element abundances and may wellexplain the present discord between the measured Li abundances and standard big bang nucleosynthesis,opening another window to unification. These theories make concrete predictions for the spectrum andsignatures at the LHC as well as Fermi.
PACS numbers: a r X i v : . [ h e p - ph ] M a y ontents I. Long Lifetimes from the Unification Scale and Astrophysical Signals II. Astrophysical Limits on Decaying Dark Matter
III. Dark Matter Decays by Dimension 6 Operators R -parity Conserving Operators 181. Models with Singlet Parity 182. S -Number Violating Operators 27B. R-parity Breaking Operators 301. Hard R-parity Breaking 322. Soft R-parity Breaking 33 IV. The Primordial Lithium Problems and Dimension 5 decays
V. Models for Lithium and Decaying Dark Matter SO (10) Model 51B. SU (5) × U (1) B − L Model 52C. Supersymmetric Axions 542
I. Astrophysical Signals
VII. LHC Signals
VIII. Conclusions Acknowledgments References I. LONG LIFETIMES FROM THE UNIFICATION SCALE AND ASTROPHYSICALSIGNALS
One of the most interesting lessons of our times is the evidence for a new fundamental scalein nature, the grand unification (GUT) scale near M GUT ∼ GeV, at which, in the presence ofsuperparticles, the gauge forces unify [1, 2]. This is a precise, quantitative (few percent) concor-dance between theory and experiment and one of the compelling indications for physics beyond theStandard Model. Together with neutrino masses [3, 4], it provides independent evidence for newphysics near 10 GeV, significantly below the Planck mass of M pl ≈ GeV. The LHC may con-siderably strengthen the evidence for grand unification if it discovers superparticles. Furthermore,future proton decay experiments may provide direct evidence for physics at M GUT . In this paperwe consider frameworks in which GUT scale physics is probed by cosmological and astrophysicalobservations.In grand unified theories the proton can decay because the global baryon-number symmetry ofthe low energy Standard Model is broken by GUT scale physics. Indeed, only local symmetries can3uarantee that a particle remains exactly stable since global symmetries are generically broken infundamental theories. Just as the proton is long-lived but may ultimately decay, other particles,for example the dark matter, may decay with long lifetimes. If a TeV mass dark matter particledecays via GUT suppressed dimension 6 operators, its lifetime would be τ ∼ π M m = 3 × s (cid:18) TeV m (cid:19) (cid:18) M GUT × GeV (cid:19) (1)Similarly a long-lived particle decaying through dimension 5 GUT suppressed operators has a life-time τ ∼ π M m = 7 s (cid:18) TeV m (cid:19) (cid:18) M GUT × GeV (cid:19) (2)Both of these timescales have potentially observable consequences. The dimension 6 decays causea small fraction of the dark matter to decay today, producing potentially observable high energycosmic rays. The dimension 5 decays happen during Big Bang Nucleosynthesis (BBN) and can leavetheir imprint on the light element abundances. There is, of course, uncertainty in these predictionsfor the lifetimes because the physics at the GUT scale is not known.If the dark matter decays through dimension 6 GUT suppressed operators with a lifetime asin Eqn. 1 it can produce high energy photons, electrons and positrons, antiprotons, or neutrinos.Interestingly, the lifetime of order 10 s leads to fluxes in the range that is being explored by avariety of current experiments such as HESS, MAGIC, VERITAS, WHIPPLE, EGRET, WMAP,HEAT, PAMELA, ATIC, PPB-BETS, SuperK, AMANDA, Frejus, and upcoming experiments suchas the Fermi (GLAST) gamma ray space telescope the Planck satellite, and IceCube, as shown inTable I. This is an intriguing coincidence, presented in section 2, that may allow these experimentsto probe physics at the GUT scale, much as the decay of the proton and a study of its branchingratios would. Possible hints for excesses in some of these experiments may have already started uson such an exciting path.GUT scale physics can also manifest itself in astrophysical observations by leaving its imprinton the abundances of light elements created during BBN. For example neutrons from the decay ofa heavy particle create hot tracks in the surrounding plasma in which additional nucleosynthesisoccurs. In particular, these energetic neutrons impinge on nuclei and energize them, causing acascade of reactions. This most strongly affects the abundances of the rare elements producedduring BBN, especially Li and Li and possibly Be.4 xtragalactic γ -rays Galactic γ ’s antiprotons positrons neutrinosDecay Super-Kchannel EGRET HESS PAMELA PAMELA AMANDA, Frejus qq × s − s − − e + e − × s 2 × s (cid:113) m ψ TeV (K) 10 s 2 × s (cid:16) TeV m ψ (cid:17) × s (cid:0) m ψ TeV (cid:1) µ + µ − × s 2 × s (cid:113) m ψ TeV (K) 10 s 2 × s (cid:16) TeV m ψ (cid:17) × s (cid:0) m ψ TeV (cid:1) τ + τ − s 10 s (cid:113) m ψ TeV (K) 10 s 10 s (cid:16) TeV m ψ (cid:17) × s (cid:0) m ψ TeV (cid:1)
W W × s − × s 4 × s 8 × s (cid:0) m ψ TeV (cid:1) × s ( m ψ = 100 GeV) 2 × s (cid:113) m ψ TeV (K) γγ × s ( m ψ = 800 GeV) 2 × s 8 × s (cid:16) TeV m ψ (cid:17) − × s ( m ψ = 3200 GeV) 5 × s (cid:113) m ψ TeV (NFW) νν × s − s 10 s 10 s (cid:0) m ψ TeV (cid:1)
TABLE I: The lower limit on the lifetime of a dark matter particle with mass in the range 10 GeV (cid:46) m ψ (cid:46)
10 TeV, decaying to the products listed in the left column. The experiment and the observed particle beingused to set the limit are listed in the top row. HESS limits only apply for m ψ >
400 GeV and are shownfor two choices of halo profiles: the Kravtsov (K) and the NFW. PAMELA limits are most accurate in therange 100 GeV (cid:46) m ψ (cid:46) In fact, measurements of both isotopes of Li suggest a discrepancy from the predictions ofstandard BBN (sBBN). The observed Li abundance of LiH ∼ (1 − × − is a factor of severalbelow the sBBN prediction of LiH ≈ (5 . ± . × − [5]. In contrast, observations indicate aprimordial Li abundance over an order of magnitude above the sBBN prediction. The Lithiumabundances are measured in a sample of low-metallicity stars. The Li isotopic ratio in all these starsis similar to that in the lowest metallicity star in the sample: Li Li = 0 . ± .
022 [6]. This implies aprimordial Li abundance in the range LiH ≈ (2 − × − , while sBBN predicts LiH ≈ − [7, 8].The apparent presence of a Spite plateau in the abundances of both Li and Li as a function of stellartemperature and metallicity is an indication that the measured abundances are indeed primordial.5f course, though there is no proven astrophysical solution, either or both of these anomalies couldbe due to astrophysics and not new particle physics. Nevertheless, the Li problems are suggestiveof new physics because a new source of energy deposition during BBN naturally tends to destroy Li and produce Li, a nontrivial qualitative condition that many single astrophysical solutions donot satisfy. Further, energy deposition, for example due either to decays or annihilations, duringBBN most significantly affects the Li abundances, not the other light element abundances, makingthe Li isotopes the most sensitive probes of new physics during BBN. Finally, a long-lived particledecaying with a ∼ Li and produce thecorrect amount of Li without significantly altering the abundances of the other light elements H, He, and He.In this paper we explore astrophysical signals of GUT scale physics in the framework of super-symmetric unified theories (often referred to as SUSY GUTs or supersymmetric standard models(SSM) [2]), as manifested by particle decays via dimension 5 and 6 GUT suppressed operators.In order to preserve the success of gauge coupling unification, we work with the minimal particlecontent of the MSSM with additional singlets or complete SU(5) multiplets. The effects of GUTphysics on a low energy experiment are generally suppressed. However, these effects can accumulateand lead to vastly different physics over long times depending on the details of the higher dimensionoperators generated by GUT physics. For example, particles that would have been stable in theabsence of GUT scale physics can decay with very different lifetimes and decay modes dependingon the particular GUT physics. Conversely, such decays are sensitive diagnostics of the physicsat the GUT scale. The details of these decays depend both upon the physics at the GUT scaleand the low energy MSSM spectrum of the theory. So, astrophysical observations of such decays,in conjunction with independent measurements of the low energy MSSM spectrum at the LHC,would open a window to the GUT scale - just as proton decay would. The role of the water andphotomultipliers that register a proton decay event are now replaced by the universe and eitherthe modified Li abundance, or the excess cosmic rays that may be detected in todays plethora ofexperiments.To characterize the varieties of GUT physics that can give rise to decays of would-be-stableparticles we enumerate the possible dimension 5 and 6 operators, each one of which defines aseparate general class of theories that breaks a selection rule or conservation law that would have6tabilized the particle in question. This approach has the advantage of being far more generalthan a concrete theory and encompasses all that can be known from the limited low-energy physicsexperiments available to us. In other words, all measurable consequences depend on the form of theoperator and not on the detailed microphysics at the unification scale (e.g. the GUT mass particles)that give rise to it.The presence of supersymmetry, together with gauge symmetries and Poincare invariance, andthe simplicity of the near-MSSM particle content greatly reduces the number of possible higherdimension operators of dimension 5 and 6 and allows for the methodic enumeration of the operatorsand the decays they cause. This is what we do in Sections III, IV, and V. We work in an SU(5)framework because any of the SU(5) invariant operators we consider can be embedded into invariantoperators of any larger GUT gauge group so long as it contains SU(5). Of course, such operatorsmay in general contain several SU(5) operators so the detailed conclusions can be affected. As anexample, we study a model that is made particularly simple in an SO(10) GUT in Section V.The dimension 6 operators of Section III have many potentially observable astrophysical signalsat experiments searching for gamma rays, positrons, antiprotons or neutrinos. We separate theminto R-conserving and R-breaking classes and in each case we also build UV models which give riseto these operators. The dimension 5 operators of Section IV have potentially observable effects onBBN, may solve the Lithium problems, and give dramatic out-of-time decays in the LHC detectors.Section V presents frameworks and simple theories that have both dimension 5 and 6 operatorsdestabilizing particles to lifetimes of both 100 sec and 10 sec to explain Li and lead to astrophysicalsignatures today in PAMELA/ATIC and other observatories.In Section VI we similarly look at the consequences for Fermi/Glast and PAMELA/ATIC. InSection VII we outline LHC-observable consequences of some of these scenarios (operators) thatcould lead to their laboratory confirmation.Finally, there are two more classes of theories that fit into the elegant framework of supersym-metric unification. The first of these are SUSY theories that solve the strong CP problem with anaxion. The other is the Split SUSY framework. Both these frameworks provide particles that canhave lifetimes long enough to account for the primordial lithium discrepancies without additionalinputs, and are included in Section IV. 7 I. ASTROPHYSICAL LIMITS ON DECAYING DARK MATTER
In this section we describe the existing astrophysical limits on decaying dark matter, assummarized in Table I. Note that the limits on dark matter decaying into many different fi-nal states (e.g. photons, leptons, quarks, or neutrinos) are similar even though they arise fromdifferent experiments. These different observations are all sensitive to lifetimes in the rangegiven by a dimension 6 decay operator, as in Eqn. (1). This can be understood, at least forthe satellite and balloon experiments, because these all generally have similar acceptances of ∼ (1 m )(1 yr)(1 sr) ≈ × cm s sr. For comparison, the number of incident particlesfrom decaying dark matter is ∼ (cid:82)
10 kpc d rr (0 . GeV m ψ cm )(10 − s − ) ≈ − cm − s − sr − , wherethese could be photons, positrons or antiprotons for example, depending on what is produced in thedecay. This implies such experiments observe ∼ (3 × cm s sr) × (10 − cm − s − sr − ) ≈ ψ with mass m ψ . A. Diffuse Gamma-Ray Background From EGRET
Observations of the diffuse gamma-ray background by EGRET have been used to set limits onparticles decaying either into qq or γγ [14], as reproduced in Table I. We adapted these limits for theother decay modes shown in the Table. We took the age of the universe to be t = 13 . ± .
12 Gyr ≈ . × s and the abundance of dark matter to be Ω DM h ≈ .
11 [15]. For the e + e − , µ + µ − , and νν decay modes the limit comes from assuming that these produce a hard W or Z from final stateradiation ∼ − of the time. The limit on the lifetime is given conservatively as 3 × − times the8imit on the W + W − decay mode, since only one gauge boson is radiated and its energy is slightlybelow m ψ . For the τ + τ − decay mode, the strongest limit comes from considering the hadronicbranching fraction of the τ . The τ decays into leptons, eν e ν τ and µν µ ν τ , 30% of the time. Therest of the decay modes have several hadrons and one ν τ which carries away at most one-half of theenergy [16]. Thus we estimate the hadronic fraction of the energy from the decay as × . ≈ . qq is relatively insensitive to the mass of the decaying particle in our range of interest100 GeV (cid:46) m ψ (cid:46)
10 TeV. We take this to imply that it depends only on the total energy producedin the decay and not as much on the shape of the spectrum, giving a limit on the decay width into τ ’s which is a factor of 0.4 of that into qq . To set a limit on decays into W + W − , the ratio betweenthe photon yield from W + W − and qq is approximated as from [17]. B. Galactic Gamma-Rays From HESS
HESS observations of gamma rays above 200 GeV from the Galactic ridge [18] can also be usedto limit the partial decay rates of dark matter with mass m ψ >
400 GeV. The limit on the flux ofgamma rays comes from this HESS analysis in which the flux from an area near the galactic center( − . ◦ < l < . ◦ and 0 . ◦ < b < . ◦ ) was taken as background and subtracted from the flux in thegalactic center region ( − . ◦ < l < . ◦ and − . ◦ < b < . ◦ ) and the resulting flux reported. Ourlimit on the decay mode ψ → γγ is found by taking a similar difference in the flux from decays andsetting this equal to the observed flux. Because we are considering decays to γγ they give a line inthe gamma-ray spectrum whose intensity need only be compared to the observed flux in one energybin. This is similar to the analysis in [19] and, as a check, their limit on a dark matter annihilationcross section agrees with our quoted limit on the lifetime.The photon flux from decays is given byΦ decay = Γ N γ πm ψ (cid:90) ∆Ω ρ drd Ω (3)where the integral is taken over a line of sight from the earth within a solid angle ∆Ω, r is thedistance from the earth, m ψ is the mass of the dark matter and Γ is its decay rate, and N γ is thenumber of photons from the decay which we will set equal to 2 for our limits. The density of dark9atter is taken as ρ ( s ) = ρ (cid:16) sr s (cid:17) γ (cid:16) (cid:16) sr s (cid:17) α (cid:17) β − γα (4)where s is the radial coordinate from the galactic center. We use the Kravtsov profile [20] with( α, β, γ ) = (2 , , . r s = 10 kpc, and ρ is fixed by ρ (8 . . GeVcm for our limits. Thisgives conservative limits since the flux from the galactic center is much less than in the commonly-used NFW profile [21] with ( α, β, γ ) = (1 , , r s = 20 kpc, and ρ (8 . . GeVcm . Thereis an even more sharply peaked profile, the Moore profile [22], which is defined by Eqn. (4) with( α, β, γ ) = (1 . , , . r s = 28 kpc, and ρ (8 . . GeVcm . There is also the very conservativeBurkert profile [23, 24, 25] ρ ( s ) = ρ (cid:16) sr s (cid:17) (cid:16) s r s (cid:17) (5)where ρ = 0 . GeVcm and r s = 11 . σv , into limits on the decay rate using the flux from annihilationsΦ annihilation = (cid:90) ρ σvN γ πm ψ drd Ω . (6)A conservative limit on decays to γγ is calculated using the Kravtsov profile. If the NFW profileis used instead, the limit is stronger τ > × s (cid:113) m ψ . If we had used the Burkert profile thelimit would have been much weaker τ > × s (cid:113) m ψ . In this case the central profile is soflat that there is essentially no difference between the galactic center signal and the flux from thenearby region used for background subtraction, making the limit from our procedure very weak.However there would then presumably be a much better limit from just comparing the actual fluxat the center (without background subtraction) to what was observed. Thus we believe that thequoted limit in Table I is conservative.The decay widths to e + e − , µ + µ − , and τ + τ − can be limited from the HESS observations. Theselight leptons will bremsstrahlung relatively hard photons with a spectrum that can be estimated as(see for example [26]) d Γ llγ dx ≈ απ (cid:32) − x ) x (cid:33) log (cid:18) m ψ (1 − x ) m l (cid:19) Γ ll (7)10here Γ ll is the decay width of ψ → ll , Γ llγ is the decay width of ψ → llγ , α is the fine structureconstant, x = E γ m ψ , and m l is the mass of lepton l . Ignoring the logarthmic dependence on m ψ weestimate this as giving 10 − photons per decay for the light leptons and − for τ ’s with energyhigh enough to count in the ‘edge’ feature in the final state radiation spectrum. We then scale thelimits from decay to γγ by those factors because the edge is assumed to be visible as the line from γγ . Clearly, a more realistic analysis would include a better determination of the observability ofthe edge feature.We do not place limits on decays to W W or νν because the spectrum of photons producedby final state radiation does not have a large hard component. These can produce many softerphotons but these are better limited by lower energy gamma ray observations such as EGRET andare counted in the first column. Although the qq decay mode may have a large FSR component,this will still give a bound worse than the HESS bound on the γγ mode. Additionally the qq modeproduces π ’s which decay to photons but these are at low energies so HESS cannot place goodlimits on them. So the qq mode is also better limited by the EGRET observations. C. Neutrino Limits From SuperK, AMANDA, and Frejus
To find the limits on dark matter decays from astrophysical neutrino observations we start byfinding the limits on decays directly into two neutrinos using [27]. The given limit on annihilationcross section into νν is almost independent of the dark matter mass m ψ in the range 10 GeV 25 14 τ νν .We do not limit decays to γγ or qq because we expect these to be better limited by directgamma ray and antiproton observations. D. Positrons and Antiprotons from PAMELA We translate recently published limits from PAMELA on the annihilation cross sections of darkmatter into the various final states into limits on the decay rate. This can be done because a darkmatter particle of mass m ψ decaying into one of the given final states (e.g. qq ) yields exactly thesame spectrum of products as two dark matter particles of mass m ψ annihilating into the samefinal state. To translate the limit on annihilation cross section we set Φ decay = Φ annihilation fromEqns (3) and (6) but we must use m ψ instead of m ψ in Eqn. (6) for Φ annihilation . Also, we integrateover the entire sky, ∆Ω = 4 π , but only over a local sphere out to a radius r max = 5 kpc. This is acrude model for the fact that antiprotons and positrons do not propagate simply like gamma raysdo. We can ignore the subtleties of this propagation because we are not computing the actual flux12bserved, just the ratio between the flux from decays and from annihilations. We then just take thesimple model that these particles only arrive at earth from a distance of r max ∼ O (5 kpc). Usingthese assumptions, the lifetime of a decaying dark matter particle with mass m ψ that correspondsto the annihilation rate of a dark matter particle with mass m ψ and cross section σv is: τ = 4 × s (cid:16) m ψ TeV (cid:17) (cid:32) × − 26 cm s σv (cid:33) . (8)It turns out that this is almost independent of halo profile and the size, r max , of the local sphereused to define it. We expect limits from ATIC to be similar to our limits from PAMELA becauseroughly the same signal that fits ATIC will fit PAMELA (see for example Section VI).The limits from positrons in the e + e − , µ + µ − , τ + τ − , and W W channels are translated from theannihilation cross section limits from [28]. Really we use the largest annihilation cross section whichcould explain the observed PAMELA positron excess [9] given propagation uncertainties (model Bof [28]) and translate this into a decay rate. This means that lifetimes around and up to an order ofmagnitude greater than those given in Table I are the best fit lifetimes for explaining the PAMELApositron excess. This is most true for the lepton channels, while decays to W W do not seem to fitthe shape of the positron spectrum very well (see e.g. [17, 28]). Note that these lifetimes are inqualitative agreement with those found in [29]. Note that we do not place a limit on the qq channelbecause this is better limited by the PAMELA antiproton measurement. The limit on νν comesfrom assuming the usual 3 × − factor times the W W limit from one of the neutrinos producing a W from final state radiation. It is possible that a stronger limit could be set by considering soft Wbremsstrahlung from the neutrino, turning the neutrino into a hard positron and the limit wouldthen come from that positron and not from the decay of the W. We do not attempt to estimatethis.The limit from positrons on the γγ decay channel arises when positrons are produced throughan off-shell photon. The relative branching ratio of this decay is well known from π decay [30]Γ ψ → γf + f − Γ ψ → γγ = 4 αQ π (cid:18) ln (cid:18) m ψ m f (cid:19) − (cid:19) (9)where f is a fermion of charge Q , lighter than ψ . Counting the production of muons as well (sincethey always decay to electrons), a conservative estimate for this branching fraction is 4% in ourrange of masses 100 GeV (cid:46) m ψ (cid:46) ∼ m ψ . The scaling of the limit on the e + e − channel with m ψ comes from the scaling of thenumber density of dark matter. Thus the energy of the produced positron is not very relevant inour range of m ψ so the limit on the γγ decay channel is just 4% of the limit on e + e − .The limits from antiprotons on the qq and W W channels come from comparing the antiprotonfluxes computed in [17] with data from PAMELA [31, 32], finding the limit this gives on annihilationcross section, and converting this into a limit on the decay rate using Eqn. (8). These are almostexactly the same limits as would be derived by translating the limits on annihilation cross sectionfrom [33] into limits on decay rate. The limits on the e + e − , µ + µ − , τ + τ − , and νν channels comefrom hard W bremsstrahlung from the leptons producing antiprotons and we take this to be 3 × − of the limit on W W (the same factor as used above). Note that these antiproton limits are mostapplicable in the range 100 GeV (cid:46) m ψ (cid:46) m ψ ).The limit from antiprotons on the γγ decay channel comes when one of the photons is off-shelland produces a qq pair. Using Eqn. (9) and summing the contributions from the four light quarks(using their current masses) with a factor of 3 for number of colors, we estimate a branching ratio B ( ψ → γqq ) ∼ 4% at m ψ = 100 GeV (rising to ∼ 6% at m ψ = 1 TeV). Note that this is in roughagreement with the one-photon rate found in [34]. If we conservatively assume that the quark pairhas energy = m ψ then we find that the limit is 0 . 02 of the limit on the qq decay channel. 1. Explaining the Electron/Positron Excess In the previous section we set limits on the decay rate of a dark matter particle using severalexperiments including PAMELA. The limits from positron observations were less stringent than theywould have been had PAMELA not seen an excess. In this section we consider what is necessaryto explain the PAMELA/ATIC positron excesses. A more detailed analysis of individual models ispresented in Section VI.Due to the large positron signal and hard spectrum detected by PAMELA the best fit to thedata is achieved with a decaying (or annihilating) particle with direct channel to leptons. Further,the lack of a signal in antiprotons disfavors channels with a large hadronic branching fraction suchas qq or W W . Thus the PAMELA excess is fit well by a dark matter particle that decays to e + e − ,14 + µ − , or τ + τ − with the lighter two leptons providing the best fit [17, 28]. As in the previoussection, we translate the cross sections given in [28] using Eqn. (8) to find the range of lifetimeswhich best fit the PAMELA positron data given the propagation uncertainties. We find the bestfit for masses in the range 100 GeV (cid:46) m ψ (cid:46) × s (cid:18) TeV m ψ (cid:19) (cid:46) τ (cid:46) × s (cid:18) TeV m ψ (cid:19) (10)The range comes from the uncertainties in the propagation of positrons in our galaxy and weestimate it by using the maximum and minimum propagation models from [28] (models B and C).It is difficult to fit just the PAMELA positron excess (not even considering antiprotons) with decaysto W W unless the dark matter is light m ψ (cid:46) 300 GeV or very heavy m ψ (cid:38) (cid:46) m ψ (cid:46) . s (cid:46) τ (cid:46) × s and alifetime to decay to qq longer than τ (cid:38) s. III. DARK MATTER DECAYS BY DIMENSION 6 OPERATORS The decays of a particle with dark matter abundance into the standard model are constrainedby many astrophysical observations. These limits (see section II) on the decay lifetimes are in therange 10 to 10 seconds. Current experiments like PAMELA, ATIC, Fermi etc. probe even longerlifetimes. These lifetimes are in the range expected for the decays of a TeV mass particle throughdimension 6 GUT suppressed operators. In this section, we present a general operator analysis ofsuch dimension 6 operators.A dimension 6 operator generated by integrating out a particle of mass M scales as M − . This15trong dependence on M implies a wide range of possible lifetimes from small variations in M .The decay lifetime is also a strong function of the phase space available for the decay, with thelifetime scaling up rapidly with the number of final state particles produced in the decay. Since wewish to explore a wide class of possible operators, we will consider decays with different numbers ofparticles in their final state. Motivated by the decay lifetime ∼ s being probed by experiments,we exploit the strong dependence of the lifetime on the scale M to appropriately lower M to counterthe suppression from multi body phase space factors and yield a lifetime ∼ s. In Table II, wepresent the scale M required to yield this lifetime for scenarios with different numbers of final stateparticles. The scale M varies from the putative scale where the gauge couplings meet ∼ × GeV for a two body decay to the right handed neutrino mass scale ∼ GeV for a five body decay.We note that the scale 10 GeV also emerges as the KK scale in the Horava-Witten scenario. Inthe rest of the paper, we will loosely refer to these scales as M GUT . Number of Final Scale M (GeV)State Particles2 10 × × TABLE II: A rough estimate of the scale M that suppresses the dimension 6 operator mediating the decayof a TeV mass particle in order to get a lifetime ∼ s for decays with various numbers of particles inthe final state. Phase space is accounted for approximately using [36]. Lifetimes scale as M . Specificdecays may have other suppression or enhancement factors as discussed in the text. The observations of PAMELA/ATIC can be explained through the decays of a TeV mass particlewith dark matter abundance if its lifetime ∼ s (see section II). PAMELA, in particular, observesan excess in the lepton channel and constrains the hadronic channels. In our operator analysis, wehighlight operators that can fit the PAMELA/ATIC data. However, we also include operators thatdominantly produce other final states like photons, neutrinos and hadrons in our survey. While theseoperators will not explain the PAMELA data, they provide new signals for upcoming experiments16ike Fermi. For concreteness, we consider SU (5) GUT models. We classify the dimension 6 operatorsinto two categories: R-parity conserving operators and R-breaking ones. In the R-conserving case,we will add singlet superfield(s) to the MSSM and consider decays from the MSSM to the singletsector and vice versa. The singlets may be representatives of a more complicated sector (see sectionV). In the R-breaking part we will consider the decay of the MSSM LSP into standard modelparticles. As a preview of our results, a partial list of operators and their associated final states aresummarized in Table III. Operator in SU(5) Operator in MSSM Final State Lifetime (sec) Mass Scale (GeV)( M GUT ∼ GeV) (lifetime ∼ sec)R-parity conserving S † S † S † SQ † Q, S † SU † U, S † SE † E leptons 10 S † SH † u ( d ) H u ( d ) S † SH † u ( d ) H u ( d ) quarks 10 S † f ¯5 † f f S † QL † U, S † U D † E, S † QD † Q quarks and leptons 5 × S † ¯5 f H † u f S † LH † u E, S † DH † u Q leptons 10 S W α W α S W EM W EM , S W Z W Z γ (line) 10 Hard R violating¯5 f (Σ¯5 f )¯5 f (Σ¯5 f )¯5 f DDDLL quarks and leptons 10 Soft R violating L (cid:51) m SUSY M GUT H u ˜¯5 f m SUSY M GUT H u ˜ (cid:96) quarks 4 × × L (cid:51) m SUSY M GUT ˜ H u ¯5 f m SUSY M GUT ˜ H u (cid:96) leptons 6 × L (cid:51) m SUSY M GUT H d ˜ W ∂/ ¯5 † f m SUSY M GUT H d ˜ W ∂/(cid:96) † γ + ν × TABLE III: A partial list of dimension 6 GUT suppressed decay operators. For each operator, we listits most probable MSSM final state. The lifetime column gives the shortest lifetime that this operatorcan yield when the scale suppressing the operator is ∼ GeV. In the mass scale column, we list thehighest possible scale that can suppress the operator in order for it to yield a lifetime ∼ seconds.Assumptions (see text) about the low energy MSSM spectrum were made in order to derive these results.All the operators are in superfield notation except for the soft R violating operators. . R -parity Conserving Operators The lifetimes of the decays of MSSM particles to the MSSM LSP are far shorter than thedimension 6 GUT suppressed lifetime ∼ s currently probed by experiments. In a R-parityconserving theory, the MSSM cannot lead to decays with such long lifetimes since the MSSM LSP,being the lightest particle that carries R parity, is stable [2]. These decays require the introductionof new, TeV scale multiplets. Limits from dark matter direct detection [37, 38] and heavy elementsearches [39] greatly constrain the possible standard model representations of the new particlespecies. In this paper, we will add a new singlet chiral superfield S in addition to the MSSM[77].The singlet may emerge naturally as one of the light moduli of string theory or it could be arepresentative of another sector of the theory (see section V).Decays between the singlet sector and the MSSM can happen through the dimension 6 GUTsuppressed operators in Table III only if there are no other faster decay modes between the twosectors. Such decay modes might be allowed if there are lower dimensional operators between thesinglet sector and the MSSM. In subsection III A 1, we consider models where lower dimensionaloperators are forbidden by the imposition of a singlet parity under which S has parity -1. Thisparity could be softly broken if the scalar component of the singlet ˜ s develops a TeV scale vev (cid:104) ˜ s (cid:105) . In subsection III A 2, we consider models without singlet parity that require additional modelbuilding to ensure the absence of dangerous lower dimensional operators between the singlets andthe MSSM.For the rest of the paper, we will adopt the following SU (5) conventions: S will refer to a SU (5)singlet, (5 , ¯5) to a fundamental and an antifundamental of SU (5) and (10 , ¯10) to the antisymmetrictensor of SU (5). The subscript f identifies standard model fields, W α denotes standard modelgauge fields and H u and H d are standard model higgs fields. The subscript GUT refers to a fieldwith a GUT scale mass. ˜ l , ˜ e , ˜ ν , ˜ q , ˜ u and ˜ d refer to sleptons and squarks. ˜ H u and ˜ H d refer tohiggsinos and ˜ W refers to a wino. 1. Models with Singlet Parity The R parity conserving operators in Table III that also conserve singlet parity are18 † S ¯5 † f ¯5 f M GUT , S † S † f f M GUT , S † SH † u ( d ) H u ( d ) M GUT and S W α W α M GUT (11)The decay topologies of these operators are determined by the low energy spectrum of the theory.SUSY breaking will split m s and m ˜ s , the singlet fermion and scalar masses respectively. It isconceivable that SUSY breaking soft masses may make m s negative leading to a TeV scale vev (cid:104) ˜ s (cid:105) for ˜ s . In this case, additional interactions in the singlet sector will be required to stabilize the vev atthe TeV scale. A decay mode with a singlet vev will typically dominate over modes without a vevsince the former have fewer particles in their final state leading to smaller phase space suppression(see discussion in sub section III A 1 a).Upon SUSY breaking, there are two distinct decay topologies:1. A component of the singlet is heavier than the MSSM LSP and this component can then decayto it. The relic abundance of the singlet can be generated if the singlet is a part of a morecomplicated sector, for example, through the decays of heavier standard model multiplets (seesection V).2. Alternatively, the MSSM LSP is heavier than the singlets. In this case, the MSSM LSP willdecay to the singlets.We now divide our discussion further based on the final state particles produced in the decay.Motivated by PAMELA/ATIC and Fermi, we will be particularly interested in operators thatproduce leptons and photons. a. Leptonic decays The R and singlet parity conserving operators in Table III that contain leptonic finalstates are S † S ¯5 † f ¯5 f M GUT and S † S † f f M GUT (12)19 inglet fermion lepton slepton< s >~ FIG. 1: A singlet fermion decaying to a lepton, slepton pair with a singlet scalar vev (cid:104) ˜ s (cid:105) insertion. These operators can be generated by integrating out a GUT scale U (1) B − L gauge boson underwhich both the MSSM and the S fields are charged. At low energies the first of these operators willlead to couplings of the form ˜ sM GUT ˜ l ∗ s † ∂/l and ˜ sM GUT ˜ d ∗ s † ∂/d (13)while the second operator will lead to similar operators involving u , q and e . Here we havesuppressed all flavor indices.We first consider the case when the singlet scalar develops a TeV scale vev (cid:104) ˜ s (cid:105) . In this case,the operators in (13) mediate the decay of the singlets to the MSSM LSP or vice versa. With a vevinsertion, the operators in (13) yield: (cid:104) ˜ s (cid:105) M GUT ˜ l ∗ s † ∂/l and (cid:104) ˜ s (cid:105) M GUT ˜ d ∗ s † ∂/d (14)When the singlet fermion is heavier than the MSSM LSP, these interactions mediate its decayinto MSSM states. In particular, when the singlet fermion is heavier than a slepton, it can decayto a slepton, lepton pair. The component operators of (14) produce the two body final state (seefigure 1), s → l ± ˜ l ∓ (15)20 − e + s ˜ e × h ˜ s i FIG. 2: MSSM neutralino decaying to a singlet LSP and an e + e − pair. The decay is dominantly intoleptons because sleptons are typically lighter than squarks. with a lifetime of τ s → l ± ˜ l ∓ ∼ × (cid:18) m (cid:19) (cid:18) (cid:104) ˜ s (cid:105) (cid:19) (cid:18) M GUT GeV (cid:19) sec (16)per lepton generation, where ∆ m = m s − m ˜ l .When the MSSM LSP is heavier than the singlet fermion, it will decay to the singlet fermionand a lepton, anti lepton pair through the operators of equation (12). These decays (see Figure 2)are of the form LSP → l + + l − + s (17)The lifetime for this three body decay is τ LSP → sl + l − ∼ (cid:18) m χ (cid:19) (cid:18) (cid:104) ˜ s (cid:105) (cid:19) (cid:18) M GUT GeV (cid:19) (cid:18) R l . (cid:19) sec (18)where R l is the ratio of the LSP mass to the slepton mass and we assumed that m χ (cid:29) m s ; if this isnot the case m χ should be replaced by the available energy in the decay ∆ m = m χ − m s . Similarly,when the singlet fermion is heavier than the MSSM LSP but lighter than the slepton, the singletfermion will decay to the MSSM LSP through a 3 body decay mediated by an off-shell slepton. Notethat the decay lifetime ∼ s when the mass scale suppressing the decay is M GUT ∼ GeV.In a U (1) B − L UV completion of these operators, this scale is the vev of the broken B − L gaugesymmetry. The B − L symmetry must be broken slightly below the GUT scale ( i.e. the putative21 ~ ~< s >slepton slepton FIG. 3: A singlet scalar ˜ s decaying to a slepton pair with a singlet scalar vev (cid:104) ˜ s (cid:105) insertion. scale where the gauge couplings meet) in order for the three body decays mediated by this gaugesector to have lifetimes of interest to this paper.The decays discussed above can also produce quarks in their final states. The decay rate isa strong function of the phase space available for the decay and is hence a strong function of thesquark and slepton masses. As discussed in section VII B, the hadronic branching fraction of thesedecays can be suppressed if the squarks are slightly heavier than the sleptons. Since squarks aregenerically heavier than sleptons due to RG running, the suppressed hadronic branching fractionobserved by PAMELA is a generic feature of these operators. An interesting possibility emergeswhen the spectrum allows for the decay of the singlet fermion to an on-shell slepton, lepton pair.This decay produces a primary source of monoenergetic hot leptons. However, the subsequent decayof the slepton to the MSSM LSP will also produce a lepton whose energy is cut off by the sleptonand LSP mass difference. With two sources of injection, this decay could explain the secondary”bump” seen by ATIC in addition to the primary bump (see sections VII B and VI).The scalar singlet ˜ s can decay when the singlet gets a vev. In the presence of such a vev, ˜ s candecay to a pair of scalars ( i.e. sleptons and squarks) or fermions ( i.e. leptons and quarks). Thedecays of ˜ s to a pair of sleptons is also mediated by the operators in (12) which in components yieldterms of the form (cid:104) ˜ s (cid:105) M GUT ˜ s ∗ ∂ ˜ l ∗ ∂ ˜ l and (cid:104) ˜ s (cid:105) M GUT ˜ s ∗ ∂ ˜ d ∗ ∂ ˜ d (19)as well as terms involving ˜ q , ˜ u , and ˜ e . 22 s will now decay directly to two sleptons (see figure 3) or two squarks, with a lifetime τ ˜ s → ˜ l ± ˜ l ∓ ∼ × (cid:18) m ˜ s (cid:19) (cid:18) (cid:104) ˜ s (cid:105) (cid:19) (cid:18) M GUT GeV (cid:19) sec (20)per generation (and per representation). The hadronic branching fraction of this decay is genericallysuppressed since squarks are expected to be heavier than sleptons (see section VII B). ˜ s can alsodecay to a pair of fermions. These decays are mediated by the operators (cid:104) ˜ s (cid:105) M GUT ˜ s ∗ l † ∂/l and (cid:104) ˜ s (cid:105) M GUT ˜ s ∗ d † ∂/d (21)which can also be extracted from (12). However, the decays of a scalar to a pair of fermions (ofmass m f ) are helicity suppressed by (cid:16) m f m ˜ s (cid:17) [78]. These decays will predominantly produce the mostmassive fermion pair that is allowed by phase space.Owing to this helicity suppression, the hadronic branching fraction from the decays of ˜ s canbe smaller than 0.1 and accommodate the PAMELA anti-proton constraint if one of the followingconditions are satisfied by the scalar singlet mass m ˜ s , the slepton mass m ˜ l and the top quark mass m t . • m ˜ s (cid:29) m ˜ l , m ˜ s (cid:29) m t : In this case, the spectrum allows both sleptons and the top quark to beproduced on shell. Hadrons are produced in this process from the decays of the top quark.The branching fraction for top production is (cid:16) m t m ˜ s (cid:17) which is smaller than 0.1 if m ˜ s (cid:39) m t . • m ˜ s > m ˜ l and m ˜ s < m t : In this case, sleptons are produced on shell. Hadrons can beproduced in this process either through direct production of the b quark or through off-shelltops. The branching fractions of these hadronic production channels are smaller than 0.1since the direct production of b is suppressed by (cid:16) m b m ˜ s (cid:17) and the decays mediated by off-shelltops are suppressed by additional phase space factors.When the scalar singlet does not get a vev, S parity is conserved. Decays between the singletsand the MSSM must either involve the decay of the heavier component of the singlet to its lighterpartner and the MSSM or the decay of the MSSM to the two singlet components. We first considerthe case when one of the singlet components is heavier than the MSSM LSP. Without loss ofgenerality, we assume this component to be the scalar singlet ˜ s . ˜ s can decay to its fermionic partner23nd a lepton-slepton pair through a 3 body decay mediated by the operators in (13) if the sleptonis light enough to permit the decay. The lifetime for this decay mode is τ ˜ s → s ˜ l ± l ∓ ∼ (cid:18) m (cid:19) (cid:18) M GUT GeV (cid:19) sec , (22)The 3 body decay of ˜ s to the singlet fermion and slepton-lepton pair may be kinematicallyforbidden if the slepton is heavy. ˜ s then decays to the singlet fermion and the MSSM through afour body decay: ˜ s → l + + l − + s + LSP . (23)The lifetime in this case is τ ˜ s → LSP sl ± l ∓ ∼ (cid:18) m (cid:19) (cid:18) M GUT × GeV (cid:19) (cid:18) R l . (cid:19) sec , (24)If the MSSM neutralino is heavier than the singlets, then its decays through the operators in(13) are also four body decays similar to the decay discussed above. The lifetime from this decayis ∼ s when M GUT ∼ × GeV, which is roughly the scale of the right-handed neutrino ina see-saw scenario. In fact, if this decay is mediated by a U (1) B − L gauge boson, the scale M GUT that suppresses this decay is the vev that breaks the U (1) B − L gauge symmetry which is roughlythe mass of the right handed neutrino.The decays discussed in this section involve decays between the singlet sector and the MSSMLSP. In order for these dimension 6 GUT suppressed operators to be involved in the decays betweenthese sectors, it is essential that there are no other faster decay modes available in the model. Onesuch mode can be provided by a light gravitino. If the gravitino is the MSSM LSP, then thesuperparticles of the MSSM will rapidly decay to the gravitino with lifetimes ∼ TeV F . If onecomponent of the singlet is heavier than the gravitino, then that component will decay to itssuperpartner and the gravitino with a lifetime ∼ ∆ m F where ∆ m is the phase space available forthis decay. Since we are interested in the decays of TeV mass particles, this scenario is relevant onlywhen the gravitino mass (cid:16) FM pl (cid:17) is less than a TeV i.e. F (cid:47) (10 GeV) . When F (cid:47) (10 GeV) ,the above decays occur with lifetimes ∼ s which are far too rapid.Another possibility is for the gravitino to be the MSSM LSP and be heavier than the singlets. Inthis case, the gravitino will decay to the singlet sector with a decay rate ∼ (cid:16) F M pl (cid:17) yielding a lifetime24 s (cid:18) ( GeV ) F (cid:19) which is also far too rapid. The dimension 6 GUT suppressed operatorsdiscussed in this section lead to astrophysically interesting decays only when the gravitino is notthe MSSM LSP and has a mass larger than ∼ TeV. This forces the primordial SUSY breaking scale F (cid:39) (10 GeV) , making the gravitino heavier and forcing it off-shell in the decays mediated byit. Integrating out the gravitino, operators of the form (cid:16) m s F (cid:17) (cid:16) M pl F (cid:17) (cid:16) ˜ s ∗ s ˜¯5 ∗ f ¯5 f (cid:17) are generated. Themasses m ˜ s in this operator are the singlet and soft SUSY breaking masses ∼ (cid:39) s for F (cid:39) (10 GeV) and will not compete withdimension 6 GUT operators discussed in this section.The constraints on the gravitino mass can be evaded if there are more singlets in the theory.For example, if the dark sector contains flavor, standard model particles may be emitted during“flavor-changing” decays in the dark sector. Consider for example an SU (6) extension of the GUTgroup on an orbifold. The chiral matter fields and their decomposition to SU (5) representationsare ¯6 i = ¯5 i + S i i = 10 i + ˆ5 i . (25)where the index i represents flavor. We break the SU (6) symmetry by projecting out the lightmodes of the 5-plet ˆ5 by orbifold boundary conditions at the GUT scale. In the absence of the5-plet, the singlet superfields S do not have yukawa couplings that connect them to the MSSMfields, eliminating direct decay modes between the singlets and the MSSM.Once the heavy off-diagonal SU (6) gauge multiplets are integrated out, operators of the form1 M GUT S † i S j L † j L i or 1 M GUT S † i S j D † j D i (26)are generated. These operators may lead to dark-flavor changing decays of s i → s j + l i + l j or s i → s j + d i + d j if there are mass splittings amongst the singlets. These splittings can arise due toexplicit SU (6) breaking terms (which may be present on the brane which breaks this symmetry).Soft SUSY breaking will also contribute to mass splittings in the singlet scalar sector. Thesesplittings can cause the decay of a scalar singlet ˜ s i to a singlet fermion s j and a lepton l i , slepton˜ l j pair. In this case, the lepton and slepton emitted in this process will belong to different families.The hadronic branching fraction of these operators relative to the leptonic channel depends stronglyon the masses of the various broken SU (6) gauge bosons. This branching fraction is suppressed if25he SU (6) gauge boson that connects the singlets and the leptons is lighter than the boson thatconnects the singlets and the quarks.The decays mediated by these operators are immune to the effects of the gravitino since thesedecays explicitly require off-diagonal gauge bosons. A relic abundance of the singlets can again begenerated through non-thermal processes as discussed in section V. b. Decays to Higgses The R and singlet parity conserving operators in Table III that contain higgs final states are S † SH † u ¯ H u M GUT S † SH † d H d M GUT . (27)These operators are very similar to the operators discussed in subsection III A 1 a. They can also begenerated by integrating out a GUT scale U (1) B − L gauge sector and the topologies of the decaysmediated by these operators are also similar to the decay topologies of the operators discussed inthat subsection. However, since these operators involve final state higgses, they will always yieldan O (1) hadronic branching fraction.In any particular UV completion, the leptonic operators discussed in subsection III A 1 a and thehiggs operators presented in this section may be simultaneously present. The hadronic branchingfraction of the decays between the singlets and the MSSM is a strong function of the phase spaceavailable for the various decay modes. If the sleptons are lighter than the squarks and the higgsinos,the decays will predominantly proceed via the leptonic channels and hence these UV completionswill also be compatible with the constraints on the hadronic channel imposed by PAMELA. c. Decays to Gauge Bosons The only operator in Table III that contains gauge boson final states is W α W α S M GUT (28)This operator may be generated by integrating out a heavy axion-like or dilaton field that coupleslinearly to both W and to S . For example, at the GUT scale we may write a superpotential W = M GUT GUT GUT + M GUT ¯ X GUT X GUT + X GUT GUT GUT + X GUT S . (29)where M GUT is a GUT scale mass. Integrating out the heavy 10 GUT s will lead to a one loop couplingof the form X GUT W α W α /M GUT (see subsection IV C). Using this effective operator and integrating26ut X GUT leads to the operator α πM S W α W α (30)The decay topology of this operator is similar to the topologies already discussed in subsectionIII A 1 a but leads to new final states. For example, when the scalar singlet ˜ s develops a TeV scalevev, this operator can lead to decays between the MSSM LSP and the singlet fermion that resultin the direct production of a monochromatic photon. The lifetime for this decay is τ s → γ + LSP ∼ × (cid:18) (cid:104) ˜ s (cid:105) (cid:19) (cid:18) ∆ m (cid:19) (cid:18) × GeV M GUT (cid:19) sec (31)with ∆ m = m s − m LSP .This signal is noteworthy since Fermi will have an enhanced sensitivity to a photon line up to aTeV. Direct decays to monochromatic photons is also a qualitatively different feature permitted fordecaying dark matter. The production of monochromatic photons from dark matter annihilationsis loop suppressed and hence annihilations always lead to bigger signals in other standard modelchannels before yielding signals in the photon channel. However, direct decays of dark matter tomonochromatic photons can happen independently of its decays to other channels.The operator discussed in this section will also induce decays involving Z s or decays to achargino and a W boson. The relative rates compared to the photon decay will be set by thedecay topology, the bino vs. wino composition of dark matter, as well as the relative size of the U (1) Y vs. SU (2) L couplings. Hadronic decays to a gluino and a gluon are also possible if they arekinematically allowed. S -Number Violating Operators A different class of operators are Kahler terms that break the global S number, for example S † f H † u ¯5 f S † f H † d f S † f ¯5 † f f (32)The first two operators involve an R-even S and allow the lightest neutralino of the MSSM to decayto the fermion component of S . If the neutralino has a significant Higgsino component the decayproceeds just through the single dimension 6 vertex and so is three-body. In this case the first27perator produces both leptons and hadrons and the second produces only hadrons. In this case,it is possible to make the first operator produce mostly leptons if SU(5) breaking effects in the UVcompletion make the coefficient of the S † H † u LE component dominate over that of the S † H † u QD . Ifthe neutralino is mostly gaugino the decay proceeds through an off-shell Higgsino, squark, or sleptonand is four-body. In this case, the first operator could produce both hadrons and leptons. Here, thelepton-only channel will dominate if a slepton is lighter than all the Higgsinos and squarks. Again,the second produces only hadrons, as in Table III.The third operator in Eqn. (32) has an R-odd S , so the neutralino will decay to the scalarcomponent of S. This generally produces both leptons and hadrons N → ˜ s + l ± + 2 jets N → ˜ s + ν + 2 jets or N → ˜ s + 3 jets (33)If the right handed sleptons are the lightest sleptons the channel with charged leptons can dominate.The decays from the MSSM to S mediated by the operators in Eqn. (32) are either three orfour body and their rates are as shown in Table III, probing scales as shown in Table II.Of course, these operators also allow the S to decay to MSSM particles, if S has a primordialabundance. A primordial abundance of S could be generated, for example, by decays at around1000 s by dimension 5 operators, as happens in the model in Section V B. The second and thirdoperators in Eqn. (32) will always produce hadrons in the decay of S , but the first operator couldproduce mostly leptons if, for example, the lepton component dominates due to SU(5) breakingeffects as described above. In a theory with this operator S † f H † u ¯5 f , if the LSP of the MSSM hasa large Higgsino component then the decay of the fermion S will be three-body to a Higgsino andtwo leptons. In such a scenario, the scalar ˜ s could have two-body decays to lepton or slepton pairs.Also possible are three-body decays to a Higgs and two leptons or to a Higgsino, a slepton, and alepton with the slepton then decaying to the LSP and a lepton. Even if the ˜ s is lighter than theLSP it will still decay just to a Higgs and two leptons or just to two leptons if the channel wherethe Higgs goes to its VEV dominates. The details of the decay depend on the mass spectrum of theMSSM particles. Such a decay could yield an interesting electron/positron spectrum. Cosmic rayobservations could then give important evidence for the mass spectrum of the MSSM, as discussedin Section VI.One example of a UV theory which can generate the operator S † f H † d f is shown in Fig. 4.28 GUT GUT H d f f GUT GUT S FIG. 4: A way to generate the operator S † f H † d f . We have added two new fields at the GUT scale, 10 GUT and 24 GUT and their conjugate fields, 10 GUT and 24 c GUT , in order to give them vector-like GUT masses. The superpotential is taken to be W = S GUT GUT + 10 f GUT GUT + H d GUT GUT (34)This preserves a ‘heavy parity’ under which the GUT scale fields 10 GUT and 24 GUT and theirconjugates are odd and everything else is even. This ensures no mixing happens between thenew GUT scale fields and the MSSM fields. Further this preserves a PQ symmetry with charges Q ( H u ) = Q ( H d ) = Q (24 GUT ) = 2, Q (10 f ) = Q (5 f ) = Q (10 GUT ) = − Q ( S ) = − 4. Thesesymmetries and R-parity forbid all dangerous operators of dimension lower than 6 that would causea faster decay, except for SH u H d which is a superpotential term and so will not be generated if itdoes not exist at tree level.Similar box diagram ways exist to generate the other S -number violating operators. A combi-nation of PQ symmetry and R parity forbids dangerous operators of dimension 5 or lower for thethree operators in Eqn. (32) except for SH u H d in the case of the first two operators and S f H u for the third operator. These are in the superpotential and so will not be generated if not there attree level. 29nother possible UV model to generate these operators is to expand the GUT gauge groupbeyond SU(5) and integrate out the heavy (GUT scale) gauge bosons. For example, the operator S † f ¯5 † f f may be generated by integrating out an SO (10) gauge boson at the GUT scale. Ifthis is the case, the field S is a right handed neutrino and some model building would be requiredto assure the decay of the neutralino does not happen by dimension five operators mediated byYukawa couplings. This may happen if the lightest right handed sneutrino has no Yukawa coupling,suggesting one of the neutrinos would be completely massless. Generating one of the other operatorsinvolving a Higgs would require an even larger group than SO(10). We do not consider such modelsfurther. B. R-parity Breaking Operators The minimal extension of the SSM allowing the dark matter to decay without introducingany new light particles arises if R-parity is broken. R-parity is a symmetry imposed to forbidrenormalizable superpotential operators, U DD, QDL, LLE, and H u L, (35)that would otherwise cause very rapid proton decay. As a byproduct, it stabilizes the LightestSupersymmetric Particle (LSP) which, if neutral, makes an excellent dark matter candidate. Con-sequently, dark matter decay may indicate that R-parity is broken.In this section, we will connect the smallness of R-breaking to the hierarchy between the weakand the GUT scales. If R-parity violating effects are mediated through GUT scale particles, theireffects can be suppressed by the GUT scale. But the SUSY non-renormalization theorem is notenough to protect the theory from dimension 4 or dimension 5 R-breaking operators, which wouldlead to too rapid LSP decay; for example, if R-parity is broken and there is no additional symmetryreplacing it, kinetic mixings, such as H † d L , are allowed and are not suppressed by the high scale.To illustrate this point consider a dimension 6 operator H u H d ¯5 f ¯5 f f . In the presence of theMSSM Yukawa interactions, there is no symmetry that forbids the dimension 5 operator 10 f H † u H d ,and prevents it from being generated in a UV completed theory, as can be seen from the existenceof diagram 5. Note, that in this example there is no problem at the effective field theory level30ecause the diagram in Fig. 5 cannot generate 10 f H † u H d . If one starts with an effective field theoryinvolving only dimension 6 operators, the low energy loops will never generate dimension 5 termsbecause Lorentz symmetry leads to cancellation of the linear divergencies. So the diagram in Fig.5 by itself gives rise only to dimension 6 operators like D α f H † u H d , but not to dimension 5 terms.Still the existence of such a diagram is a signal that it will be challenging to UV complete such atheory without generating the 10 f H † u H d term as well. H † u H d f ¯5 f ¯5 f f H d FIG. 5: D α f H † u H d generated by H u H d ¯5 f ¯5 f f and the MSSM Yukawas In fact, there are also examples of operators for which the problem arises directly at the effectivefield theory level. For example, consider the superpotential dimension 6 operator, M GUT H u LW α W α .By closing the gaugino legs (see Fig. 6) it gives rise to the R-parity breaking H u L term. This is asuperpotential term, so one may think it is not generated. Indeed, this loop is zero in the SUSYlimit. However, in the presence of a SUSY breaking gaugino mass m / the loop is non-zero andquadratically divergent, so it gives rise to the term m / H u L . The presence of such a quadraticdivergence does not contradict the lore that the quadratic divergencies are cancelled in softly brokenSUSY, because at the end of the day we obtained a mass term of order the SUSY breaking scale m / .These problems lead us to two possible ways of consistently implementing R-parity breakingimplying dimension 6 dark matter decays. The first possibility is to replace R-parity by anotherdiscrete symmetry that forbids the dimension 4 terms in (35) and also dimension 5 operators thatgive rise to the dark matter decays, but allows dimension 6 decays. An alternative proposal is thatthe R-parity is violated by a tiny amount, that is not put in by hand, but related to the SUSY31 u Lm / FIG. 6: Gaugino loop generating LH u from M GUT H u LW α W α after SUSY breaking breaking scale. Let us illustrate each of these options in more detail. 1. Hard R-parity Breaking R-parity is not the only discrete symmetry that forbids the dangerous lepton and baryon numberviolating operators (35). Alternative discrete symmetries may arise from broken gauge symmetriesand insure the longevity of the proton. Heavy fields may have couplings that preserve these sym-metries but not R-parity [40, 41]. As a result, these symmetries allow for the LSP to decay at thenon-renormalizable level. One such operator, DDDLLM GUT (36)arises when there is GUT scale antisymmetric representation of SU(5), while the fundamentaltheory obeys a Z symmetry (see Table III B 1). This operator is generated by the combination ofthe couplings W ⊃ GUT ¯5 f ¯5 f + ¯10 GUT ¯10 GUT ¯5 f + Σ10 GUT ¯10 GUT , (37)that violates R-parity. Σ is the adjoint that breaks SU(5) down to the SM gauge group. It isessential that it splits the colored and the electroweak parts of 10 GUT , otherwise R-parity violationwould come through (¯5 f ) which is zero. 32 article Z charge10 f e i π ¯5 f H u e i π H d e i π GUT Z discrete that substitutesR-parity The Z symmetry also insures that there are no kinetic mixings between light and heavy fieldsthat introduce rapid DM decay. The LSP decays to 5 SM fermions through a sparticle loop(see Fig.7) χ → (cid:96) (38)The rate of decay is of order,Γ ψ ∼ − m ψ M GUT ∼ (3 × sec ) − (cid:0) m ψ (cid:1) (cid:0) m GUT GeV (cid:1) , (39)where we took into account the five body phase space and loop suppression.This example illustrates two generic features of LSP decays in the presence of Z N symmetriesreplacing R-parity. Typically the allowed operators have a large number of legs, such that the decayrates are significantly suppressed by final state phase space. Such operators also tend to involvequarks, resulting in order one hadronic branching fractions. 2. Soft R-parity Breaking Even if R-parity is violated only through GUT fields, in the absence of a symmetry, kineticmixings can still generate order one R-parity violating effects. This problem could be avoided ifR parity is broken only through SUSY breaking effects involving GUT fields. These effects will becommunicated to the MSSM through the GUT fields resulting in suppressions ∼ (cid:16) m SUSY M GUT (cid:17) . Take,33 d ˜ d ˜ d " "d d ˜ d FIG. 7: LSP decay in a theory with a Z symmetry substituting R-parity for example, two pairs of heavy 5 ⊕ ¯5 and two heavy singlets, S and S . The interactions betweenthe GUT and MSSM fields are: W ⊃ S ¯5 GUT H u + S GUT ¯5 f + M GUT S + M GUT S + M GUT ¯5 GUT GUT + M GUT ¯5 GUT GUT . (40)The important point is that the R-parities of these heavy fields are not fully defined with thissuperpotential– S , GUT have equal R-parity and S , GUT have opposite R-parity. To breakR-parity one needs two soft terms for heavy fields, for instance: m SUSY ˜ S ˜ S and m SUSY ˜¯5 GUT ˜5 GUT (41)Consequently, the R-parity breaking coefficients in the MSSM sector are proportional to the productof the two soft masses and are always suppressed by at least M GUT . Indeed, the loop of heavyfields generates the R-breaking Bµ -term m SUSY M GUT h u ˜¯5 f , which is effectively dimension 6.Even though this scenario works at the spurion level, it is hard to implement in a full theory ofSUSY breaking. First, we need to sequester the source of R-parity violation from the MSSM fields34ut not from the GUT fields. In a toy model with one extra dimension, the MSSM and R-breakingfields are located on different branes, while the GUT fields are free to propagate in the bulk. ThenR-parity breaking is communicated to the MSSM fields only through loops of GUT particles. Carehas to be taken in order to suppress the effects of gravity that propagates everywhere and maycommunicate unsuppressed R-violation to the MSSM. This is ensured if the soft terms for theheavy fields are generated by a gauge mediation mechanism. However, as discussed in sub sectionIII A 1, the MSSM LSP can decay through these dimension 6 GUT suppressed operators only ifthe LSP is not the gravitino. This requirement forces the F -term responsible for the MSSM softmasses to be much larger than the one responsible for R-breaking. Consequently, we need two verydifferent scales of SUSY breaking, one for the MSSM sector and another for the GUT sector. It isnot clear if such a SUSY breaking mechanism can be successfully embedded into a UV completion. IV. THE PRIMORDIAL LITHIUM PROBLEMS AND DIMENSION 5 DECAYS Recent observations [42] of the Li/H and Li/H ratio in metal-poor halo stars suggest a discrep-ancy between the standard big bang nucleosynthesis (BBN) and observationally inferred primordiallight element abundances. As pointed out in [43, 44], the decay of a particle χ with a lifetime τ ∼ − χ h of χ and the hadronicbranching fraction Br ( H ) are such that Ω χ h Br ( H ) ∼ − . The required density of χ at thetime of its decay is similar to the expected relic density of a particle of mass M χ ∼ 100 GeV withelectroweak interactions. The decay of such a particle can explain the Li and Li abundances ifits lifetime τ ∼ − ∼ . ∼ 500 GeV [46] making it difficult to test this scenario at the LHC. Other previous work hassuggested that late decays can affect nucleosynthesis [47, 48].In this paper, we point out that supersymmetric GUT theories provide a natural home toanother wide class of models that could explain the Lithium anomalies. Dimension 5 operatorssuppressed by the GUT scale are a generic feature of supersymmetric GUT theories. If χ decaysthrough such an operator, then its decay rate Γ ∼ (cid:16) M χ M GUT (cid:17) is naturally ∼ (100 s) − .In the following, we perform a general operator analysis of the possible dimension 5 GUTsuppressed operators that can solve the primordial Lithium abundance problem. For concreteness,we consider SU (5) grand unification. A solution to the primordial Lithium problem involvingdimension 5 GUT suppressed operators requires the introduction of a new TeV scale particle species χ . We restrict ourselves to models that involve the addition of complete, vector-like SU (5) multipletsin order to retain the success of gauge coupling unification in the MSSM. We will only consideroperators that preserve R-parity, automatically ensuring the existence of a stable dark mattercandidate. The operators are classified on the basis of the SU (5) representation of χ . In each case,we discuss the relic abundance and hadronic branching fraction of the decaying species that solvesthe primordial Lithium problem and the corresponding LHC signatures. A. Operator Classification We illustrate the possible dimension 5 GUT suppressed operators in Table V. The operators areclassified on the basis of the SU (5) representation of χ . The R-parity of χ is indicated by subscripts e (R-even) and o (R-odd) and chosen to make the operators in Table V R-invariant. We restrictourselves to the cases when χ is a singlet, a fundamental (5 , ¯5) or an antisymmetric tensor (10 , ¯10)of the SU (5) group. As illustrated in the second and third column of Table V, it is straightforwardto construct a number of dimension five operators inducing the decay of χ or LSP decay, dependingwhich one is lighter, in all these cases. The hadronic branching fraction for the corresponding decaysis typically quite significant, as the operators involve either Higgs fields or quarks.Note that fundamental or antisymmetric representations of SU (5) are not good Dark Mattercandidates; a stable fundamental would imply dirac DM, which is excluded by direct detection36earches [37, 38], while an SU (5) antisymmetric does not have a neutral component. Only when χ is a singlet is the LSP phenomenologically allowed to be heavier than χ and decay to it. In thiscase, χ naturally inherits the LSP thermal relic abundance. If this singlet is heavier than the LSP,one needs to introduce TeV scale interactions that would give χ a thermal calculable abundance,or rely on non-thermal production mechanisms such as tuning of the reheat temperature, in orderto yield the required χ abundance. Due to this difference between χ ’s that do and do not carrySM charges, we have only included the quadratic in χ operator in the of a singlet χ . This operatorallows for the LSP decay into χ ’s, but doesn’t make the lightest component of χ unstable.A small subtlety in all the cases is that we can also construct relevant and marginal gaugeinvariant operators involving χ and the MSSM fields. If present, they would mediate the decay of χ without GUT scale suppression. These dangerous operators are collected in the last column of TableV. In particular, the kinetic mixing terms such as ¯ χ † o ¯5 f (here ¯ χ o is an R-odd antifundamental GUTmultiplet; the effect of this term is equivalent to the mixing between ¯ χ o and ¯5 f in the MSSM Yukawainteractions) are not protected by SUSY non-renormalization theorems and will be inevitably gen-erated unless forbidden by some additional symmetries. However, it is relatively straightforward toimpose Peccei–Quinn symmetries that either forbid these operators or make them GUT suppressedas well.Let us illustrate how this works in several concrete examples. If χ is an R-even singlet, the onlydangerous operator is χ e H u H d . We can forbid it by imposing a discrete Z symmetry, χ → − χ .With this symmetry, the only allowed dimension five operator is χ e H u H d . As a result, χ is the darkmatter, and the lithium problem is solved by the decay of the LSP into pairs of χ particles.Other superpotential dimension 5 terms for an R-even singlet are of the form χW i , where W i areYukawa terms present in the MSSM Lagrangian. In this case, we need to use a symmetry other than Z under which χ is neutral. The Higgs fields carry charges Q u and Q d such that Q u + Q d (cid:54) = 0, andthe charges of the MSSM matter fields are determined by requiring that the MSSM Yukawa termsdo not violate the symmetry. Then the MSSM µ term is forbidden by this symmetry, however, itcan be generated if the symmetry is spontaneously broken by the TeV vev of the field S with charge( − Q u − Q d ) coupled to Higgses through the term SH u H d . The new massless Goldstone bosonwill not appear if the actual symmetry of the action is just a discrete subgroup of this continuoussymmetry, so that one can make all components of S massive. The lowest dimension operator37nvolving χ , S and MSSM Higgses is SχH u H d . When S develops a vev it gives rise to the marginaloperator χH u H d mediating χ decay, whose coefficient is naturally suppressed by the ratio of thesoft PQ breaking scale, µ , to the scale where the dimension five operator is generated, M GUT .It is straightforward to generalize these arguments to other cases as well. For instance, forthe fundamental χ to avoid the kinetic mixings with Higgses or matter fields one can impose adiscrete symmetry which is a subgroup of the continuous PQ symmetry with the following chargeassignments for the MSSM fields Q f = 1, Q H d = − Q H u = 2, Q ¯5 f = − χ isequal to Q χ = 8 the dimension 5 operator χ e H u ¯5 f ¯5 f is allowed, while all other operators mediating χ decay are forbidden. In fact, one can check that this is the only possible charge assignment thatavoids kinetic mixing and allows χ to decay without requiring soft breaking of the PQ symmetry.There are more possibilities if the Higgs charges are not opposite so that the PQ symmetry is softlybroken by the µ term. Finally, let us use this example to illustrate that there is no problem to gobeyond the effective theory analysis and construct renormalizable models generating the requiredoperators at the GUT scale. Namely, let us consider the following renormalizable superpotential W = 10 GUT ¯5 f ¯5 f + ¯10 GUT H u χ e + M GUT ¯10 GUT GUT . Here (10 GUT , ¯10 GUT ) is a pair of R-even GUT scale fields with the PQ charge Q = 6. By integratingout the heavy fields one obtains the dimension five operator χ e H u ¯5 f ¯5 f at low energies.In addition to decays involving the chiral supermultiplets of the MSSM, singlet χ s can alsohave decays involving the gauge supermultiplets W α through operators of the form χW α W α . Theseoperators can be generated through an axion-like mechanism where χ is the goldstone boson ofsome global symmetry broken at the GUT scale. B. Relic Abundance The energy density Ω χ h Br ( H ) that must be injected into hadrons is plotted against the lifetime τ of the decaying particle in Figure 8. The Li problem can be solved when the lifetime τ ∼ SU (5) Rep. Superpotential Terms Kahler terms Soft PQ breakingSinglet χ e f f H u , χ e f ¯5 f H d , χ e † f f , χ e H † u H u , (cid:16) µM GUT (cid:17) χ e H u H d χ e,o H u H d , χ o f ¯5 f ¯5 f , χ o ¯5 † f H d (cid:16) µM GUT (cid:17) χ o H u ¯5 f χ e W α W α (5 , ¯5) χ e H u ¯5 f ¯5 f , ¯ χ e H u H u H d , ¯ χ † e f f , (cid:16) µM GUT (cid:17) (cid:16) χ † e H u , ¯ χ † e H d , ¯ χ † o ¯5 f (cid:17) χ o f f f , χ o ¯5 f H u H d χ o † f H u µ (cid:16) µM GUT (cid:17) χ o ¯5 f (10 , ¯10) χ e f f H d , ¯ χ e f ¯5 f H u , ¯ χ † e f ¯5 † f , ¯ χ † e ¯5 f ¯5 f (cid:16) µM GUT (cid:17) (cid:16) χ † o f , χ e ¯5 f ¯5 f (cid:17) ¯ χ o ¯5 f ¯5 f ¯5 f , ¯ χ e ¯5 f ¯5 f H d ¯ χ o H u ¯5 † f µ (cid:16) µM GUT (cid:17) ¯ χ o f TABLE V: The possible dimension 5 GUT suppressed operators classified on the basis of their generationin the superpotential or through soft breaking of PQ symmetry or through kinetic mixing in the Kahlerpotential. The subscript f denotes standard model families, W α are gauge fields and H u , H d are the Higgsfields of the MSSM. The R-parity of χ is denoted by its subscripts e and o for even and odd paritiesrespectively. 100 s (cid:16) . χ h Br ( H ) (cid:17) . The operators described in section IV A can cause decays between the MSSMand the χ sector with these lifetimes. 1. Electroweak Relics Let us first consider how the Lithium problems can be solved by the colorless componentsof fundamental or antisymmetric representations of SU (5). Standard model gauge interactionsgenerate a thermal abundance of the electroweak multiplets in (5 , ¯5) and (10 , ¯10). We focus on thestandard model operators that are extracted from the SU (5) invariant operators in Table V andcontain the electroweak multiplets (the lepton doublet L in (5 , ¯5) and the right handed positron E in (10 , ¯10)) from the χ . These operators can be classified into three categories: operators thatinvolve quarks or only contain higgses, operators that are purely leptonic and operators that involveleptons and higgs doublets. This classification is presented in Table VI.Operators in Table V that contain higgs triplets when the χ s are electroweak multiplets have notbeen included in Table VI since the higgs triplets are at the GUT scale and cannot cause a dimension39 -1 -2 -3 -4 -5 -6 τ (sec) Ω X h B h Y p >0.258 D/H>4x10 -5 Bailly, Jedamzik, Moultaka 2008 FIG. 8: (Color online) This figure from [44] plots the energy density that must be injected into hadronsversus the decay lifetime in order to solve the primordial lithium problems. Decays in the red region solvethe Li problem and decays in the green region solve the Li problem. χ . For example, with χ a (10 , ¯10), the only gauge invariant operator thatcan be extracted from ¯ χ ¯5 f ¯5 f H d in Table V is ¯ E χ DDH Td where H Td is the color triplet higgs. Thisoperator cannot cause a dimension 5 decay of the E χ and is not listed in Table VI. There are alsooperators in Table V from which standard model operators that belong to more than one category inTable VI can be extracted. For example, the operator ¯ χ f ¯5 f H u generates an operator containingquarks, ¯ E χ Q f D f H d , and an operator containing leptons and higgses, ¯ E χ E f L f H d . We include thisoperator in both categories in Table VI. A UV completion of this operator involves integrating outGUT scale SU (5) multiplets. O (1) SU (5) breaking effects at the GUT scale (like doublet-tripletsplitting) can result in one of the operators(say, ¯ E χ Q f D f H d ) being suppressed relative to the other( ¯ E χ E f L f H d ).The relic energy density Ω χ h of these electroweak multiplets is ∼ . (cid:16) M χ TeV (cid:17) . The decaysmediated by the operators in Table VI that involve quarks or only contain higgses have O (1)40 lepton LSPlepton FIG. 9: The decay of a slepton to a lepton and the LSP. slepton lepton W, ZLSP FIG. 10: The decay of a slepton to a lepton, gauge boson and the LSP through an off-shell chargino. hadronic branching fractions. These operators can solve the Li problem if the χ lifetime is ∼ 100 s (cid:16) TeV M χ (cid:17) (figure 8). However, these decays cannot solve the Li problem. A solution to the Liproblem requires a hadronic energy density injection Ω χ h Br ( H ) (cid:47) − around a 1000 s. Colliderbounds on charged particles imply that M χ > 100 GeV. Consequently, the relic energy density Ω χ h is greater than 10 − . Due to the O (1) hadronic branching fraction, the hadronic energy densityinjected is also greater than 10 − . This injection is too large and over produces Li (figure 8).The MSSM products of the decays mediated by the purely leptonic operators in Table VI mayor may not contain sleptons. For example, in the decay of the fermionic component χ ino of the χ , the operator ¯ χ † ¯5 f ¯5 f yields a lepton, slepton pair while the operator ¯ χ † ¯5 f does not produce anysleptons. The hadronic branching fraction of decays that involve sleptons depends upon the MSSMspectrum. An O (1) hadronic branching fraction will be produced from the decay of the sleptonsif the SUSY spectrum contains charginos or other neutralinos between the slepton and the LSP.41 iggs sleptonlepton FIG. 11: χ decay to a slepton, lepton and higgs. These decays can solve the Li problem but not the Li problem. When this is not the case, theslepton predominantly decays to its leptonic partner and the LSP (figure 9). The slepton decay candirectly produce hadrons through off-shell charginos (figure 10). But, these decays are suppressedby phase space factors and additional gauge couplings. The branching fraction for these processesis ∼ − . The leptons produced in these decays could be τ s. However, even though the τ has an O (1) branching fraction into hadrons, the hadrons produced in this process are pions. A solutionto the primordial lithium problem requires the injection of neutrons[43, 44] and hadronic energyinjected in the form of pions is ineffective in achieving this goal. The dominant hadronic branchingfraction in these leptonic decays is provided by final state radiation of Z and W bosons off theproduced lepton doublets. These bosons decay to hadrons with an O (1) branching fraction. Usingthe branching fraction for final state radiation of Z and W bosons from [52], we estimate that therelic abundance and hadronic branching fraction from the decays of a 600 GeV - 1 TeV χ satisfythe constraint Ω χ h Br ( H ) ∼ − . These decays can solve the Li and Li problems if the lifetime ∼ O (1) hadronicbranching fraction since the higgs decays predominantly to b quarks. However, the higgs operatorsin these fields can be replaced with the higgs vev (cid:104) h (cid:105) , resulting in an effective purely leptonic decaymode. These leptonic decay modes produce hadrons through final state radiation of Z and W bosonswith a branching fraction ∼ − as discussed above. The hadronic branching fraction of operators42 h >sleptonlepton FIG. 12: χ decay to a slepton and lepton with a higgs vev (cid:104) h (cid:105) insertion. with both leptons and higgses is the ratio of the decay rate to processes involving the higgs and therate to processes where the higgs is replaced by (cid:104) h (cid:105) . For example, the operator χH u ¯5 f ¯5 f can causethe χ to decay to a higgs, slepton and lepton (figure 11) with a rate Γ h ∼ (cid:16) M χ π M GUT (cid:17) . When thehiggs is replaced by (cid:104) h (cid:105) , this operator causes the χ to decay to a lepton, slepton pair (figure 12)with a rate Γ l ∼ (cid:16) (cid:104) h (cid:105) πM GUT (cid:17) M χ . The decay to a higgs directly produces hadrons and the branchingfraction for this decay mode is Γ h Γ l = (cid:0) π (cid:1) (cid:16) M χ (cid:104) h (cid:105) (cid:17) ∼ − (cid:16) M χ 500 GeV (cid:17) . With M χ ∼ 500 GeV, thishadronic branching fraction is sufficiently small to allow this decay to solve both the Li and Liproblems. However, we cannot replace the higgs field by (cid:104) h (cid:105) in every such operator in Table VIthat has leptons and higgses. After electroweak symmetry breaking, the masses of the electricallycharged l + χ and neutral l χ components of the lepton doublet in (5 , ¯5) are split. The charged fermion l f + χ is heavier than the neutral fermion l f χ by ∼ αM Z [53, 54] while the masses of the correspondingscalar components are additionally split by ∼ cos (2 β ) M W [55]. When the fermion componentsare lighter than the scalars, all the components of l χ rapidly decay to l f χ . In this case, the lithiumproblems can be solved with (5 χ , ¯5 χ )s only through the decays of l f χ . With l f χ , the operator χ † f H u does not lead to any SU (3) × U (1) EM invariant operators when H u is replaced by (cid:104) h (cid:105) . The onlydecay mode for the l f χ that is permitted by this operator is a decay to the right handed positronand a charged higgs, resulting in an O (1) hadronic branching fraction. As discussed earlier, thedecays mediated by this operator cannot solve the Li problem but can address the Li problem.The relic abundance of a singlet χ is model dependent. A thermal abundance of χ can be43enerated if the χ is coupled to new, low energy gauge interactions like a U (1) B − L . It couldalso be produced through the decays of new TeV scale standard model multiplets or through atuning of the reheat temperature of the Universe. The hadronic branching fraction of the singlet χ operators in Table V is also model dependent. Operators like χ f f H u have an O (1) hadronicbranching fraction since χQ f U f H u , the only standard model operator that can be extracted from it,contains quark fields. However, an operator like χ f ¯5 f H d yields both χQ f D f H d and χE f L f H d . AUV completion of this operator involves integrating out GUT scale SU (5) multiplets. O (1) SU (5)breaking effects at the GUT scale (like doublet-triplet splitting) can result in the operator χQ f D f H d being suppressed relative to χE f L f H d . Due to these tunable model dependences, the decays ofsinglet χ s can solve both Li and Li problems as long as their relic abundance Ω χ h (cid:39) − .Yet another possibility available in the singlet case is that the MSSM LSP is heavier than theR-odd component of χ . Then the LSP will decay to χ and the χ abundance will be close to thedark matter abundance today. In fact, if no other stable particles are added then χ itself will be adark matter particle. This scenario shares many similarities with the scenario where the gravitinois the lightest R-odd particle, so that the LSP can decay. If there is no additional mechanismfor generating χ (such as the coupling to new low energy gauge interactions like U (1) B − L ) thedecaying MSSM LSP should have rather high relic abundance, Ω LSP h (cid:38) . 1, depending on themass ratio between the LSP and χ . This makes it somewhat challenging to solve Li problem. Thisis achievable though if the LSP is a slepton coupled to the singlet through purely leptonic operators.It is worth stressing that a generic property of all our models is the presence of several long-living particles with somewhat different lifetimes and masses. There are two sources for proliferationof different long-living species. The first is related to R-parity. Indeed, as the χ -parity χ → − χ is broken only by dimension five operators, the lightest particle in the χ multiplet is always long-lived. However, if SUSY breaking mass splitting between the lightest R-odd and R-even particlesis smaller than the mass of the MSSM LSP, both of these particles are metastable and will decayonly through χ parity violating dimension five operators. Their presence doesn’t change much inour discussion. Another reason for the existence of several long-lived particles is that we are addingnew fields in the complete GUT multiplets that contain also colored components. Let’s now discusstheir story. 44 SU (5) Rep. Quark and Higgs operators Purely Leptonic Leptonic operatorsOperators with higgs doubletsSinglet χ f ¯5 f H d (= χQ f D f H d ) , χ f ¯5 f H d (= χE f L f H d ) , (cid:16) µM GUT (cid:17) χH u ¯5 f , χ f f H u , χ H u H d , χ f ¯5 f ¯5 f , χ f ¯5 f ¯5 f , χ † f f χ ¯5 † f H d χ † f f , χW α W α , χH † u H u (5 , ¯5) ¯ χ † f f (= ¯ l † χ Q f U f ), ¯ χH u H u H d , (cid:16) µM GUT (cid:17) ¯ χ † ¯5 f , χ ¯5 f H u H d , χ † f H u ,χ f f f µ (cid:16) µM GUT (cid:17) χ ¯5 f (cid:16) µM GUT (cid:17) (cid:0) χ † H u , ¯ χ † H d (cid:1) (10 , ¯10) ¯ χ f ¯5 f H u (cid:0) = ¯ E χ Q f D f H d (cid:1) , ¯ χ † ¯5 f ¯5 f , µ (cid:16) µM GUT (cid:17) ¯ χ f ¯ χ f ¯5 f H u (cid:0) = ¯ E χ E f L f H d (cid:1) , χ f f H d , ¯ χ ¯5 f ¯5 f ¯5 f , ¯ χ † f ¯5 † f (cid:16) µM GUT (cid:17) (cid:0) χ † f , χ ¯5 f ¯5 f (cid:1) , ¯ χH u ¯5 † f TABLE VI: The classifications of dimension 5 operators based on the standard model operators that can beextracted from them. These operators are generated using the electroweak multiplets in χ . Operators thatrequire the higgs triplet fields are ignored. The subscript f denotes standard model families and H u , H d arethe Higgs fields of the MSSM. The R-parity of χ is chosen in order to make these operators R-invariant. 2. Colored Relics For the fundamental χ we have at least one pair of long-lived quarks χ d, ¯ d , where χ d has quantumnumbers of the right-handed MSSM d -quarks (and, as before, if the mass splitting between R-evenand R-odds χ -quarks is small enough, we have another pair of long-lived colored particles). Forantisymmetric χ we have long-lived vector-like χ -quarks both with quantum numbers of the right-handed MSSM u -quark χ u, ¯ u and left-handed u and d quarks, χ Q, ¯ Q .The evolution history in the early Universe is significantly more involved for colored particles[58] and the corresponding relic abundances are border-line to be uncalculable. The point is thatin general heavy long-lived colored particles experience two epochs of annihilation as the Universeexpands. First, they suffer from the conventional perturbative annihilations at high temperaturesbefore the QCD phase transition. The resulting relic abundance of the colored particles is at thelevel Ω χ h ∼ − (cid:16) M χ T eV (cid:17) . This is a very interesting number—as follows from Figure 8, if correct itwould imply that a long-lived colored particle with a mass in the subTeV range solves both Lithium45roblems.However, colored particles experience a second stage of annihilations after the QCD phasetransition, that can significantly reduce the abundance. Indeed, after the QCD phase transitionthey hadronize—get dressed by a soft QCD cloud of the size of order ∼ Λ − QCD . It is plausiblethat when two slowly moving hadrons involving heavy colored particles collide, a bound statecontaining two heavy particles forms with geometric cross-section ∼ 30 mbarn. This conclusion issomewhat counterintuitive—naively, one may think that the soft QCD cloud cannot prevent twoheavy particles from simply passing by each other without forming a bound object. The argument,however, is that the reaction goes into the excited level of the two χ system of the size of order ∼ Λ − QCD . At low enough temperatures the angular momentum of such a state is close to the typicalangular momentum of two colliding hadrons, L i ∼ ( mχT ) / Λ − QCD so that one may satisfy theangular momentum conservation law by emission of a few pions. Assuming that the reaction tosuch an excited level is exothermic the geometrical cross-section appears to be a reasonable estimate.After the excited state with two χ ’s forms it decays to the ground level and χ ’s annihilate. As aresult the relic abundance can be reduced to the values below Ω χ h ∼ − for a TeV mass particles,where they don’t affect the Lithium abundance.Definitely, many of the details of this story are rather uncertain at the quantitative level andthis conclusion has to be taken with a grain of salt. We discuss some of the involved uncertainties inmore detail in section IV D. Interestingly, in the models we are discussing here, one may avoid goinginto the detailed discussion of this complicated process and be rather confident that the residualabundance of the colored particles is close to that given by the perturbative calculation. The reasonis that in order for the above mechanism to operate the two χ particles in the bound state should beable to annihilate with each other. This is the case for some candidate long-lived colored particles,such as gluino in the split SUSY scenario or stop NLSP decaying into gravitino, but not alwaystrue.For instance, for antisymmetric χ annihilation is possible in some of the bound states (e.g., χ u χ ¯ u ), but not in the others (e.g., χ u χ ¯ Q ). Once formed, these bound states go to the Coulombicground state, which is compact and doesn’t get converted into other mesons any longer. Throughweak decays such a meson decays to the energetically preffered neutral ground state, that survivesuntil individual χ particles decay. In fact, it is likely that an original χ -hadron is a baryon, given46hat the reaction converting χ -meson and ordinary proton into χ -baryon and pion is exothermic.This doesn’t change the story much; after two transitions one obtains in this case baryons containingthree χ ’s, such as χ u χ Q χ Q . Again, such a baryon will decay to the stable state and will survive tillindividual χ particles decay. Similarly, for fundamental χ an order one fraction of them ends upbeing in the compact baryonic state χ d χ d χ d which is safe with respect to annihilations.We don’t attempt here to analyze the above processes at the precision level, but this discus-sion implies that the resulting abundance of colored χ ’s while being somewhat reduced from itsperturbative value is still high enough to solve Li problem, or even both Lithium problems. C. Supersymmetric Axion We already mentioned dimension 5 decays involving the gravitino as one of the solution of theLithium problems, that does not involve new particles beyond those present in the MSSM. Thereis another well motivated particle in the MSSM that may have similar effects – the axino (see, [34]for a detailed discussion of axino properties and cosmology).The axion is a well-motivated new pseudo-scalar particle. It solves the strong CP-problem,and may constitute a fraction, or all, of the dark matter. Axion-like particles are also genericin string models. In supersymmetric models, the axion is a part of a chiral supermultiplet S , soit comes together with the scalar (saxino) and the fermionic (axino, ˜ a ) superpartners. Being a(pseudo)Goldstone boson, the axion supermultiplet couples to the MSSM fields suppressed by thePQ breaking scale f a . The leading interactions are with the gauge fields L = (cid:90) d θ Sf a (cid:88) i =1 C i α i π W i ) + h.c. , (42)where C i are model-dependent coefficients of order one. These interactions are often generated atthe one-loop level, for instance, by integrating out vector-like fields acquiring the mass from the vevof S due to interactions like S ¯ QQ (KSVZ model [59, 60]). As a result, as compared to the generalmodel-independent analysis above these dimension 5 operators contain extra one-loop suppressionfactors. Consequently, to be relevant for the Lithium problem the high-energy scale f a enteringhere has to be somewhat lower than the GUT scale, f a ∼ GeV. Still, this scale is intriguinglyclose to the GUT scale. 47he relevance of axino for the Lithium problems crucially depends on its mass. We will focusour attention on the gravity mediated SUSY breaking, so that there is no light gravitino. Then thesaxion receives a mass of the same order as other soft masses, ∼ F susy /M P l . On the other hand,the mass of axino is a hard SUSY breaking term and its value is highly model-dependent. If it isgenerated at the loop level, it is suppressed by at least one extra loop factor, and varies between ∼ GeV down to the keV range. It may also be generated at the tree level after SUSY is broken, ifSUSY breaking triggers some singlet fields to develop vev’s giving rise to the axino mass. In thisway the axino acquires mass of the same order as other soft masses.In all other respects, the axino is a particular example of adding an MSSM singlet. If the axinois lighter than the LSP, and the MSSM LSP is bino-like, it will decay to axino and photon. Thecorresponding lifetime is τ ∼ sec (cid:18) f a GeV (cid:19) (cid:18) m χa (cid:19) , where ∆ m χa is the axino-LSP mass difference. The hadronic branching fraction in this case is dueto decays with virtual photon producing quarks and is at the few percent level, see Eqn. (9). Ifthe resulting axino is the only cold dark matter component now these decays may solve the Liproblem, but not the Li problem. If the dominant component of the cold dark matter is the axion,and the LSP is light, so that its thermal abundance is low, the LSP decay to the axino may solveboth Lithium problems. D. Split SUSY In the Split SUSY framework [56], the SUSY breaking scale is not the TeV but an intermediatescale up to 10 GeV. All the scalars are at that scale except one Higgs that is tuned to be light.The gauginos and Higgsinos are protected by chiral symmetries and they are at the TeV scale,giving thermal dark matter and allowing for the gauge couplings to still unify at the GUT scale.The gluino of split SUSY can only decay through an off-shell squark (see Fig. 13) and its lifetime48s set by the SUSY breaking scale:Γ ˜ g ∼ π m gluino m SUSY ∼ (100 sec) − (cid:16) m gluino (cid:17) (cid:18) × GeV m SUSY (cid:19) (43)When the SUSY breaking scale is 10 − GeV, and the gluino lifetime is 100 − ˜ g h ∼ − − − . ˜ g q ˜ q ψq † FIG. 13: Gluino decay through an off-shell squark Even though the gluino lifetime is well determined by the squark masses, its abundance isdifficult to know because it involves strong dynamics [57, 58]. At a temperature of ∼ m gluino thegluino will thermally freeze-out. Its cross-section is calculable, since QCD is still perturbative atthose times, and its abundance is roughly given by Ω h ∼ − (cid:0) m gluino (cid:1) . After the QCD phasetransition, the gluino gets dressed into colored singlet states, R-hadrons, which have a radius of ∼ Λ − QCD . The question is if these extended states can bind to form gluinonium states, ˜ g − ˜ g , with ageometric cross-section, π Λ − QCD ∼ 30 mbarn, that will lead to a second round of gluino annihilationand suppress its abundance by several orders of magnitude.Whether this second round of annihilations happens depends on the details of R-hadron spec-troscopy.For example, even if the lightest state is an R- π , the reaction:R- π + baryon → R-baryon + π (44)49ay well be exothermic due to the lightness of π and, if it has a geometric cross-section, most ofthe gluina will end up in R-baryon states. Gluino annihilation now depends on the reaction:R-baryon + R-baryon → ˜ g -˜ g + 2 baryons . (45)If this reaction is exothermic, it may suppress the gluino abundance by many orders of magnitude. Ifthis reaction is endothermic for high angular momentum gluinonium states, it may leave a significantnumber of gluinos in R-hadrons and there is no significant reduction in the gluino abundance. Theseuncertainties render the final gluino abundance incalculable.However, the case where there is no second round of annihilations after the QCD phase transitionprovides an interesting scenario for the LHC. The gluino mass range that gives the measured Liabundance is 300 − ∼ Li and Liproblems (see Fig. 8) and it will be abundantly produced at the LHC. The potential discovery of agluino with a lifetime 100 − Li abundances and sBBN. V. MODELS FOR LITHIUM AND DECAYING DARK MATTER The primordial lithium abundance discrepancies and the observations of PAMELA/ATIC canbe explained by the decays of a TeV mass particle through dimension 5 and 6 GUT suppresseddecays (see sections III and IV). The dimension 6 operators S † m S m † f f M , S † m S m ¯5 † f ¯5 f M , S † m S m H † u ( d ) H u ( d ) M and S m W α W α M (46)in section III allow decays between a singlet sector S m and the MSSM with lifetimes long enoughto explain the signals of PAMELA/ATIC. A large class of dimension 5 operators were discussed insection IV. In particular, operators of the form10 m S m ¯5 f ¯5 f M GUT and S m W α W α M GUT (47)50llow for the population of the singlet S m sector through dimension 5 decays from the MSSM or anew TeV scale SU (5) vector-like sector, for example 10 m . In this section, we will show that bothof these decays can be naturally embedded into SUSY GUT models. These models can naturallysolve both the primordial lithium abundance problem and the observations of PAMELA/ATIC. Wealso consider models yielding mono-energetic photons that give qualitatively new signals for Fermi. A. SO (10) Model In SO (10), the MSSM superpotential is: W MSSM = λ f f f h + µ h h (48)where 16 f and 10 h are family and higgs multiplets. We add TeV scale multiplets (16 m , ¯16 m ) and aGUT scale 10 GUT along with the following interactions: W (cid:48) = λ m f GUT + m m ¯16 m + M GUT GUT GUT (49)These interaction terms allow for R-parity assignments -1 for 10 GUT and +1 for 16 m and pre-serves a m parity under which 16 m and 10 GUT have odd parity. Integrating out the 10 GUT field andthe SO (10) gauge bosons, we generate the dimension 5 and 6 operators: (cid:90) d θ (cid:18) m m f f M GUT (cid:19) , (cid:90) d θ (cid:18) π (cid:19) (cid:32) m m † h M GUT (cid:33) and (cid:90) d θ (cid:32) † m m † f f M B − L (cid:33) (50)where M B − L is the vev that breaks the SO (10) U (1) B − L gauge symmetry. The R and m parityassignments forbid dangerous, lower dimensional Kahler operators like 10 † GUT h and 16 † m f . Thedimension 5 operators in (50) connect the components of the 16 m multiplet that are charged underthe standard model to the singlet component S m of the 16 m and the MSSM. However, the onlyoperator in (50) that allows for two singlet fields S m to be extracted from 16 m is the dimension 6operator (cid:18) † m m † f f M B − L (cid:19) which yields (cid:18) S † m S m † f f M B − L (cid:19) . The phenomenology of this model is identicalto that of the SU (5) × U (1) B − L model discussed below. The decays of the standard model multipletsin 16 m to the singlets S m and the MSSM fields at ∼ ∼ s can reproducethe observations of PAMELA/ATIC. B. SU (5) × U (1) B − L Model We consider a SU (5) × U (1) B − L model, with U (1) B − L broken at the scale M B − L near the GUTscale. The MSSM is represented in this model by the superpotential: W MSSM = λ uf f f H u + λ df f ¯5 f H d + µH u H d (51)where 10 f and ¯5 f are the standard model generations, λ uf and λ df are the yukawa matrices and H u and H d are the higgs fields. We add a GUT scale (10 GUT , ¯10 GUT ), a TeV scale (10 m , ¯10 m ) and asinglet S m to this theory with the following additional terms W (cid:48) in the superpotential: W (cid:48) = λ m S m ¯10 GUT + λ GUT ¯5 f ¯5 f + M GUT GUT ¯10 GUT + m m ¯10 m + m s S m S m (52)The mass terms m and m s are at the TeV scale and M GUT is at the GUT scale. These inter-actions allow R parity assignments +1 for 10 m , S m and 10 GUT . The superpotential also conservesa m parity under which S m and (10 m , ¯10 m ) have parities -1. Soft SUSY breaking will contributeto the scalar masses and lead to mass splittings between the fermion and scalar components. Inparticular, the singlet fermion mass m fs will be different from the singlet scalar mass m ˜ s . Integratingout the GUT scale field 10 GUT and the broken U (1) B − L gauge sector, we get the dimension 5 and6 operators: (cid:90) d θ (cid:18) m S m ¯5 f ¯5 f M GUT (cid:19) , (cid:90) d θ (cid:18) S † m S m Y † YM B − L (cid:19) and (cid:90) d θ (cid:18) π (cid:19) (cid:18) S † m S m Y † YM (cid:19) (53)Here the Y represent the other chiral multiplets 10 m , 10 f , ¯5 f , H u and H d in the model. The gaugesymmetries of the standard model, supersymmetry and R and m parities ensure that the operatorsin (53) are the lowest dimension operators that connect particles carrying m parity ( i.e. S m and10 m ) and the MSSM.Consider the phenomenology of this theory when the mass m of the 10 m particles are greaterthan the singlet masses m s and m ˜ s . The 10 m are produced with a thermal abundance due to52 0m Sm f55f FIG. 14: 10 m decaying to S m and sleptons. standard model gauge interactions. They will decay to the singlets S m and the MSSM particlesthrough the dimension 5 operator in (53) with lifetime τ ∼ (cid:0) m (cid:1) (cid:0) M H GeV (cid:1) where ∆ m is the mass difference between the 10 m and the singlet (see figure 14). Following the discussion insection IV, the decays of the electroweak and colored multiplets in the 10 m can solve the primordial Li and Li abundance problems. The decays of the 10 m generates a relic abundance of the singlets s m and ˜ s m . Since R and m parities are conserved in this model, decays between the singlets andthe MSSM have to involve the dimension 6 operators in (53) when the 10 m fields are heavier thanthe singlets. These operators are identical to the R-parity conserving operators discussed in sectionIII. Following the discussion in that section, the decays mediated by these operators can explainthe observations of PAMELA/ATIC.A decaying particle can explain the observations of ATIC if its mass is ∼ . ∼ s m is the decaying particle with mass m ˜ s ∼ . s m with a mass m s and the MSSM LSP with mass M LSP are light. A relic abundance of ˜ s m is generatedin the early universe from the decays of the thermally produced 10 m . The relic energy density ofthe 10 m is ∼ . (cid:0) m (cid:1) . 10 m decays to both s m and ˜ s m . However, since s m is lighter than ˜ s m ,the branching fraction for decays to s m is higher due to the larger phase space available for thedecay. When m s (cid:28) m ˜ s , the branching fraction for decays to ˜ s m is ∼ (cid:0) − m ˜ s m (cid:1) . The relic energydensity Ω ˜ s m h of ˜ s m is ∼ . (cid:0) m (cid:1) (cid:0) m ˜ s m (cid:1) (cid:0) − m ˜ s m (cid:1) . For m ∼ . m ˜ s ∼ . ˜ s m h ∼ − . The decays of ˜ s m can explain the observations of PAMELA/ATIC if the lifetime53 a γ γ FIG. 15: Axino decay to the lighter axino and two photons. ∼ s. This lifetime can be obtained from the dimension 6 GUT suppressed operators in (53) ifthe U (1) B − L symmetry is broken slightly below the GUT scale with a vev M B − L ∼ × GeV.The other possible decay topology is for the MSSM LSP to decay to the singlets, with the LSPmass M LSP ∼ . m abundance to decay to the MSSMand the singlets. The 10 m can decay to the MSSM and the singlets s m through the dimension 5operator in (53) if the spectrum permits the decay. Since the 10 m has R parity +1, its fermioniccomponents 10 m can decay only if their mass is greater than the LSP mass ∼ . m can decay to the standard model and the singlets as long as the scalars are heavier than thesinglets. However, if the scalars ˜10 m and the singlets are too light, the MSSM LSP will decay to˜10 m and the singlet fermion through the dimension 5 operator in (53). This decay occurs with alifetime ∼ m and singlet masses are larger than the LSP mass. C. Supersymmetric Axions The interesting example of the setup that provides both dimension 5 and dimension 6 mediateddecays is a mild generalization of the axino model for Lithium. Namely, let’s assume that thereare two axion-like particles corresponding to different PQ symmetries, so that they are coupled to54SSM through different combinations of operators as in (42) (in other words, coefficients C i aredifferent in the two cases).In this case there are two axinos and after the LSP decays the dark matter will be a mixtureof the two. The heavier of the axinos is unstable, but its decays involve insertion of two dimensionfive operators, so effectively it has dimension 6 suppression. Typically, the fastest decay channel is˜ a → ˜ a + 2 γ (see Fig. 15), giving τ ∼ sec (cid:18) f a GeV (cid:19) (cid:16) m χ TeV (cid:17) (cid:18) TeV∆ m (cid:19) , where m is the axino mass difference. The decays involving other gauge bosons are suppressedby the higher mass of the intermediate gaugino in the diagram in Fig. 15. As discussed in sectionIV C, axino masses are highly model dependent and may be as high as of order m SUSY . If that isthe case for one of the axinos, its decays will produce monoenergetic photons potentially observableby Fermi.Note that at the one loop level the two axions (and axinos) are mixed by the gauge loops (seeFig. 16). This mixing is generated at the high scale, so there is no reason for it to be small. However,this does not change the conclusion that the decay of one axino to the other is effectively dimension6. Indeed, after one diagonalizes axinos kinetic terms and mass matrices, at the dimension 5 levelthe resulting eigenstates have only couplings of the form (42). An intuitive way to understand thisis to note that axion is a (pseudo)Goldstone boson, so each axion (and axino) carries a factor of f − a with it. Consequently processes involving two of them have dimension 6 suppression.In principle, one may generalize this story to the broader class of models explaining Lithiumproblems by dimension 5 decays. Namely, one may consider two singlet fields (not necessarilyaxinos), that do not couple directly to each other and decay through the dimension 5 operatorslisted in the Table V. However, generically, there is a danger that the dimension 5 operator involvingtwo singlets can be generated and as a result the transition of one singlet to another will be rapid.One can avoid this problem for some choices of operators by imposing additional symmetries, andfor other choices of operators one may check that there are UV divergent diagrams already at thelevel of effective theory that makes this proposal invalid. The advantage of the axino scenario, isthat the absence of the dangerous dimension 5 operators is built in automatically.55 a FIG. 16: Mixing between two axinos induced by the gauge loop. VI. ASTROPHYSICAL SIGNALS In this section we consider the astrophysical signals of dark matter decaying through dimension6 operators at current and upcoming experiments including PAMELA, ATIC, and Fermi. The mainastrophysical signals of long-lived particles decaying at around 1000 s from dimension 5 operators arechanges to the light element abundances as produced during BBN. As well as solving the Lithiumproblems, such a decay could make a prediction for the abundance of Be [44]. These decaysmay also generate a significant component of the current dark matter abundance of the Universe(see section V), resulting in the production of naturally warm dark matter that may contribute toobserved erasure of small structure [70]. A. Electrons and Positrons and Signals of SUSY The PAMELA satellite has reported [9] a rise in the observed positron ratio above the expectedbackground starting at about 10 GeV and continuing to 100 GeV where the reported data ends. TheATIC balloon [10] experiment has observed an interesting feature in the combined flux of electronsand positrons which begins to deviate from a simple power law around 100 GeV, peaks at around650 GeV, and descends steeply thereafter. The HESS experiment [62] has added new data above700 GeV, clearly observing the steep decline of the spectrum. It is attractive to explain both thePAMELA and the ATIC signals as coming from a new population of electrons and positrons whichare the products of dimension six decay of dark matter suppressed by the GUT scale. As we havenoted in section II D 1, such a GUT suppressed decay agrees with the ATIC and PAMELA rates,both of order 10 seconds. In this section we will show that the spectral shape observed by both56xperiments can be explained as well. We will find that decays of dark matter allow for a muchbetter fit to ATIC data, as compared with models considered thus far, when dark matter is allowedto decay to both standard model particles and superpartners.Beyond the rough agreement in rates, additional important information may be gained bystudying the spectral shape of the ATIC feature as well as the PAMELA rise. These shapes willbecome even clearer as upcoming experiments will add new data. The PAMELA experiment isexpected to release positron data up to ∼ 270 GeV as well as measure the combined electronpositron flux up to 2 TeV. The Fermi satellite, though optimized for gamma rays, has a largeacceptance and is also capable of measuring the combined flux. HESS is also capable of measuringthe combined flux and may be able to add to their already published by measuring the flux atenergies down to ∼ 200 GeV, checking the ATIC bump. As has been shown for annihilations [63],assuming a new smooth component of positrons that would explain the PAMELA rise accompaniedby an equal spectrum of new electrons and extrapolating to higher energies, gives rough qualitativeagreement with the height and slope of the ATIC feature. Here we would like to stress that analysisof the precise shape of electron-plus-positron flux, particularly as data improves, may allow forsurprising discoveries. This may already be demonstrated by examining ATIC data closely andconsidering models that fit its shape.In what follows, we will take the published ATIC data with only statistical errors as an example,and assume for now that there are no large systematics. Thus, we will discuss in detail how thespectrum may be fit, including its various features. Such an assumption may well be wrong. ATICmay have many systematic effects in the data, rendering small features meaningless. We will takeit seriously merely to give an example of all the information that may be extracted from a fullspectrum, including small features on top of the overall bump. From this we learn an interestingnew qualitative feature, that dark matter decays can generically produce several features in thespectrum and not merely a smooth rise and fall around the dark matter mass. However, we do notbelieve that the ATIC data are conclusive yet. Future data with better systematics, including fromthe Fermi satellite, are necessary before we can definitively conclude anything from the details ofthe electron spectrum.The ATIC spectrum, shown in figure 18, may in fact contain two features – a soft feature around100-300 GeV and a hard feature at 300-800 GeV. The hard feature may extend to yet higher energies57f HESS data is considered. The collaboration itself and the following literature has generallyhighlighted the hard and more pronounced feature, however it should be noted that the statisticalerrors in the 100-300 GeV range are quite small, and when compared with a power law backgroundextrapolated from low energies, the soft feature may well be more significant statistically. The softfeature also has a peculiar shape; a sharp rise just below 100 GeV, followed by a nearly flat, orslightly descending spectrum up to 300 GeV. The shape of the hard feature is less clear, and maybe interpreted either as ending sharply at 800 GeV or ending more smoothly around 2 TeV if HESSdata is added. It is tempting to fit the two ATIC features by two different products of the samedecay which would give these shapes.If dark matter is a singlet which is heavier than the MSSM LSP, the products of its decay maybe superpartners. If this is the case, the decay of dark matter may potentially allow us to discoversupersymmetry and measure its spectrum. Consider for example the simple dimension six operatorof equation (12), which allows the singlet fermion to decay to an electron and a selectron with alifetime given by τ s → ˜ ee ∼ × (cid:18) m (cid:19) (cid:18) (cid:104) ˜ s (cid:105) (cid:19) (cid:18) M GUT GeV (cid:19) sec . (54)This is a two body decay and both particles are mono-energetic. In particular the electron (orpositron) energy will be E e = m s − m e m s (55)The selectron, being a scalar will then decay isotropically in its rest frame. Assuming the decay isdirectly to an electron and a neutralino LSP the isotropic decay will give a flat energy distributionin the “lab” frame between two edges E + = m s ( m e − m LSP )2 m e E − = m e − m LSP m s (56)The combined spectrum of the electron-positron pair emitted in the decay is shown schematicallyin figure 17. Note that if one were to measure the three energies E e , E − and E + , the mass of thedark matter, the selectron and the LSP may be solved for without ambiguity. We further noticethat the flat spectrum that the selectron produces is reminiscent of the plateau above 100 GeV inthe ATIC flux. The hard monochromatic electron may produce the hard ATIC feature.58 e E + E − electron selectron d N / d E Energy FIG. 17: A schematic plot of a spectrum of the electron-positron pair emitted when a singlet dark matterparticle decays to an electron and a selectron. Such a spectrum provides an explanation of the two featuresin the ATIC flux. In the top left panel of figure 18 we show that such a decay can indeed fit the ATIC dataremarkably well. We have used GALPROP (v5.01) [64] to generate electron and positron back-ground and also to inject and propagate the signal, modifying the dark matter annihilation packageto one for decays. In order to reproduce the ATIC shape, including its sharp rises in flux, weused a propagation model with a relatively thin diffusion region, “model B” of [28] ( L = 1 kpc)which was found to agree with cosmic ray data. In addition, to the propagation model the ATICflux depends on the electron background spectrum which is still highly uncertain [65] (a slope of ∼ − . ± . − . E e , E − , E + ) = (700 , , m s ∼ m ˜ e ∼ m LSP ∼ . (57)It is remarkable that such a good fit to both ATIC features may be achieved by assuming the same59 æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à ATIC 08HESS 08 s ® e Ž e æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à ATIC 08HESS 08 s ® e Ž e 50 100 500 1000 5000102050100200500 Energy H GeV L E d N (cid:144) d E H G e V m - s - s r - L æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ PAMELA 08 s ® e Ž e 10 20 50 100 2000.1000.0500.2000.0300.1500.070 Energy H GeV L P o s it r on F r ac ti on FIG. 18: (Color online) Left: The combined electron-positron flux for a heavy dark matter candidatedecaying to an electron and a selectron. The selectron subsequently decays to a neutralino and a positrongiving a two features in the observed flux. The dotted curves are the various components of the flux -electron, selectron and background. ATIC (red circles) and HESS (blue squares) data are shown. Thedarker grey band is the HESS systematic error and the lighter grey is the area covered by this error afterincluding the uncertainty in the energy scale shift in both directions. Right: The positron fraction for thesame decay and PAMELA data. The dotted line is the expected background. rate of injection for both the hard electron and the selectron, strengthening the case for them tocome from the same decay. Assuming the singlet s makes up all of dark matter we can deduce itslifetime from the rate that fits th ATIC spectrum to be 1 . × seconds. Using equation (54),this decay is probing a scale of 0 . × , remarkably close to the GUT scale.The heavy selectron is required so that the selectron will be mildly boosted producing thefeature at relatively low energies. The superpartner spectrum is quite sensitive to the values of( E e , E − , E + ) and thus to the shape of the combined electron-positron flux. The uncertainties onthese masses are thus large, being affected by propagation uncertainties as well as the statisticaland systematic errors of ATIC, HESS and future data. It will be interesting to perform a more60ystematic analysis as data improves.It is also interesting to consider other ways to get a similar spectrum. For example, if darkmatter is a scalar singlet, it may be decaying both to an e + e − pair and to a ˜ e + ˜ e − pair. This wouldbe given for example by the operator S † f H † u ¯5 f from Section III A 2 where the Higgs gets a vev.This operator does not have a helicity suppression because the scalar ˜ s decays to two left-handedspinors. In order to reproduce the hard ATIC feature at 700 GeV and the soft feature between 100and 200 GeV, the spectrum is lighter m ˜ s ∼ m ˜ e ∼ 660 GeV m LSP ∼ 500 GeV . (58)However, if the operators that lead to decays to electrons and to selectrons are suppressed by thesame high scale, as would be expected in a supersymmetric theory, one would expect that the decayto electrons would be more rapid due to phase space. The ATIC spectrum on the other handrequires the two rates to be equal, which would imply the two rates must be tuned independentlysomehow.On the top right panel of figure 18 we show the positron fraction this decay, s → ˜ e ± e ∓ , produces.Qualitatively the positron fraction agrees with that seen by PAMELA, a monotonic rise. However,the positron fraction in this case stays flat until roughly 40 GeV before climbing rather abruptlywhereas the PAMELA data goes up more smoothly. Though the fit is not perfect, the quantitativedisagreement is not very significant. At yet higher energies, beyond the range of current data, thepositron fraction undergoes a wiggle, transitioning from the soft to the hard components of thespectrum. This feature is an interesting prediction for future PAMELA data or other experimentssuch as AMS2.The mild disagreement between the PAMELA data and the spectrum that fits ATIC exemplifiesthat there is a mild tension between the detailed shape of the spectra observed by the two experi-ments. This tension may be phrased model independently by noticing that given a smooth powerlaw for the electron background, say the dotted background line on the right panel of figure 18, thedifference between it and the ATIC spectrum may be interpreted as signal, half of which is electronsand half of which is positrons. This new component of positrons can be combined with the positronbackground and the total flux to give a positron fraction which has the peculiar shape seen in theright panel of figure 18. The mild tension is thus between the two data sets, regardless of whether61 æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à ATIC 08HESS 08 s ® e + e - æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à ATIC 08HESS 08 s ® e + e - 50 100 500 1000 5000102050100200500 Energy H GeV L E d N (cid:144) d E H G e V m - s - s r - L (cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) s (cid:174) e (cid:43) (cid:32) e (cid:45) PAMELA 0810 20 50 100 2000.1000.0500.2000.0300.1500.070 Energy (cid:72) GeV (cid:76) P o s it r on F r ac ti on FIG. 19: (Color online) Left: The combined electron-positron flux for a heavy dark matter candidatedecaying to an electron-positron pair. The dotted curves are the various components of the flux - signaland background. ATIC (red circles) and HESS (blue squares) data are shown. The darker grey band is theHESS systematic error and the lighter grey is the area covered by this error after including the uncertaintyin the energy scale shift in both directions. Right: The positron fraction for the same decay and PAMELAdata. The dotted line is the expected background. In this decay channel the fit to ATIC data is inferiorto that in the previous case, but the fit to PAMELA data is slightly improved the signal originates from annihilation or decays.An alternative approach is to begin with a smoother signal that reproduces the PAMELA shapevery well, but to give up on the various ATIC features and treat ATIC as a single smoother bump.This is shown in figure 19 in which both the combined flux and the positron fraction are shown fora dark matter with a mass of 1.4 TeV that is decaying to an electron-positron pair. The lifetime ofdark matter in this case is 3 × seconds which may result by a dimension six operator suppressedby a scale of 1 . × GeV. This spectrum does not require sharp features and a more standardpropagation model was used, (the diffusive convective model of [66]).Though the hard ATIC peak by itself fits a sharp electron positron peak at around 700 GeV,62hen combined with recent HESS data the hard feature may be interpreted as a broader bump,extending to a couple of TeV. The ambiguity in interpretation arises from the uncertainty of whetherHESS is seeing the background beyond the ATIC bump, or the decline of the bump itself to alower and steeper background. Though the HESS statistical errors are small, these should beadded to a larger systematic error (shown as a grey band in the figures), and to an overall energyscale uncertainty (whose effect is roughly shown in the figure as a wider and lighter grey band).Nonetheless, the HESS decline motivates fitting the harder ATIC feature with a smoother spectrum.This may be done without losing the good fit to the soft feature given by a mono-energetic selectron.In figure 20 we show the combined flux and positron fraction from a heavy dark matter singletdecaying to a mono-energetic muon, with an energy of 2 TeV, and a slow slepton, that produces abox-like spectrum as in figure 17. The shape of the hard ATIC peak including the HESS declineis qualitatively reproduced, within the propagation uncertainties. This particular spectrum maybe achieved if a very heavy dark matter candidate (of order 8 TeV) is decaying to a heavy slepton(at 5.6 TeV) which decays to a slightly lighter neutralino (5.5 TeV). Though this spectrum is veryheavy, is remarkable that a good fit to both features may be gotten from a single decay. B. Gamma-Rays Gamma rays can provide the best evidence that dark matter is the explanation for the observedelectron/positron excesses. Gamma rays produced in our galaxy at energies of hundreds of GeVare not bent or scattered and so provide clean directional and spectral information. Such highenergy gamma-rays have been searched for with Atmospheric Cherenkov Telescopes (ACTs) suchas HESS and will be searched for in the upcoming Fermi telescope (GLAST). Gamma-rays fromdark matter decays or annihilations will appear as a diffuse background with greater intensity indirections closer to the galactic center. There is also an expected astrophysical diffuse backgroundand so to distinguish the dark matter signal may require spectral information as well. The classicsignals of dark matter are lines, edges, or bumps in the gamma-ray spectrum.Dark matter annihilations do not generically produce gamma rays as a primary annihilationmode. By contrast, dark matter decays may directly produce photons as a primary decay mode,as discussed in Section III. Operators such as S W α W α or m SUSY M GUT H d W ∂/ ¯5 † f cause decays of ψ → γγ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à ATIC 08HESS 08 s ® e Ž Μ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à ATIC 08HESS 08 s ® e Ž Μ 50 100 500 1000 5000102050100200500 Energy H GeV L E d N (cid:144) d E H G e V m - s - s r - L æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ PAMELA 08 s ® e Ž Μ 10 20 50 100 2000.1000.0500.2000.0300.1500.070 Energy H GeV L P o s it r on F r ac ti on FIG. 20: (Color online) Left: The combined electron-positron flux for a heavy dark matter candidatedecaying to a hard muon (2 TeV) and a selectron. The selectron subsequently decays to a neutralino anda positron giving a two features in the observed flux. The dotted curves are the various components of theflux - muon, selectron and background. The more rounded spectrum of the hard muon is in qualitativeagreement with HESS data. ATIC (red circles) and HESS (blue squares) data are shown. The darker greyband is the HESS systematic error and the lighter grey is the area covered by this error after including theuncertainty in the energy scale shift in both directions. Right: The positron fraction for the same decayand PAMELA data. The dotted line is the expected background. or ψ → γν which produce monoenergetic photons and neutrinos. This will appear as a line in thegamma-ray spectrum, easily distinguishable from backgrounds if the rate is fast enough. This wouldalso be a line in the diffuse neutrino spectrum, perhaps detectable at upcoming experiments such asIceCube, but it would likely be seen first in gamma rays. The exact reach of upcoming experimentsdepends on the energy of the line as well as the halo profile of the dark matter. However, it is clearfrom the conservative limits in Table I that HESS observations are already in the range to detectsuch decays. Fermi and future ACT observations will significantly extend this reach, probing GUTscale suppressed dimension 6 operators. 64 00 400 500 600 700 800 900 1000E H GeV L - dN dE H GeV - L - FIG. 21: The spectrum of final state radiation from a m ψ = 1600 GeV dark matter particle decaying inthe galaxy with lifetime τ = 10 s as would be seen at Fermi with a field of view = 1 sr near the galacticcenter. The black curve is for the decay channel ψ → e + e − , the dashed is for decays to µ + µ − . Gray is theexpected background. Even if the primary decay mode does not include photons, they will necessarily be produced byfinal state radiation (FSR), i.e. internal bremsstrahlung, from charged particles that are directlyproduced. In particular, if dark matter decay explains the electron/positron excesses then thedecay must produce a large number of high energy electrons and positrons. These decays will thenalso produce high energy gamma-rays through FSR with a spectrum given in Eqn. (7). Suchdecays could come, for example, from the operators from Section III. Figure 21 shows the spectrumof final state radiation expected from decays of dark matter in the galaxy to e + e − and µ + µ − aswould be seen by Fermi. We take the effective area = 8000 cm , the field of view = 1 sr annulusaround the galactic center but at least than 10 ◦ away from it, and the observing time = 3 yr. Thisspectrum exhibits an ‘edge’ at half the dark matter mass which could allow these gamma-rays to65e distinguished spectroscopically from the diffuse astrophysical background.The third general mechanism which produces photons from dark matter decays (or annihila-tions) is hadronic decay (e.g. to qq ) producing π ’s which then decay to photons. These generallyyield a soft spectrum, similar to the expected diffuse astrophysical backgrounds. Thus, we will notconsider these signals, as a definitive detection could be challenging. However it is worth notingthat these signals can be quite large because they are not suppressed by the factor of ∼ − thatsuppresses final state radiation.Fig. 22 shows an estimate for the sensitivity of the Fermi telescope to dark matter that decayseither to γγ or e + e − . The sensitivity to decays to e + e − comes observing the edge in the gamma-rayspectrum from final state radiation, as in Fig. 21. Note though, the plot shows the reach of Fermiin the lifetime of the primary decay ψ → e + e − and not the FSR decay ψ → e + e − γ . Roughly wesee that the reach for e + e − is a factor of 10 worse than for γγ , which is the expected probabilityto emit final state radiation. The sensitivity to the gamma ray line signal ( ψ → γγ ) was found bydemanding that the signal (which is entirely within one energy bin) be larger than 3 times the squareroot of the expected diffuse astrophysical background in that bin. The signals were calculated fromEqns. (3) and (7). The expected background was estimated as [26, 67, 68] d Φ dEd Ω = 3 . × − (cid:18) 100 GeV E (cid:19) . cm − s − sr − GeV − . (59)The energy resolution of Fermi was estimated from [69] as the given function below 300 GeV and30% between 300 GeV and 1 TeV. The sensitivity to the edge feature was estimated (similarly to[26]) by requiring that the number of signal photons within 50% of the energy of the edge be largerthan 5 times the square root of the number of background photons within that same energy band.This sensitivity increases as the size of the energy bin used is increased and does not depend onthe actual energy resolution of the detector (except for the assumption that it is smaller than theenergy bin used). This is clearly only a crude approximation of the actual statistical techniqueswhich would be used to search for an edge.We exclude a 10 ◦ half-angle cone around the galactic center from our analysis because thediffuse gamma ray background is expected to be much lower off the galactic plane. The annulus weconsider, though it overlaps the galactic plane, is a good approximation to the dark matter signalfrom off the galactic plane. Ignoring the galactic center reduces the signal. To be conservative66 500 1000 1500 2000m Ψ H GeV L Τ H s L Ψ ® ΓΓΨ ® e + e - Ψ H GeV L Τ H s L FIG. 22: A guess for the sensitivity of Fermi. The solid lines are the expected reach of the Fermi telescopein the lifetime to decay into the modes ψ → γγ and ψ → e + e − as a function of the mass of the decayingdark matter. The signal from the second mode comes from internal bremsstrahlung gamma-rays fromthe electrons (but the plotted reach is for the primary decay mode into just e + e − ). The sensitivities areconservatively estimated using the Burkert (isothermal) profile, though the NFW profile gives essentiallythe same result. The grey band is the region of lifetimes to decay to e + e − that would explain the PAMELApositron excess from Eqn. (10). The ATIC excess would be explained in this same band for masses near m ψ ≈ we use Eqn. (59) for the background everywhere, though it is probably an overestimate for thebackground off the galactic plane. Fermi may also be able to resolve point sources of gamma rays,further reducing the astrophysical diffuse background.From Fig. 22 we see that the most of the range of lifetimes which could explain the PAMELAand ATIC excesses as a decay to e + e − should be observable at Fermi from the final state radiation67 Π Π Π Π Angle10 - - - - Flux H cm - s - sr - L - - - - FIG. 23: (Color online) The flux of galactic gamma rays versus the angle from the galactic center fordark matter decay to 2 photons with m ψ = 1 TeV and τ = 4 × s versus dark matter annihilation to 2photons with m ψ = 500 GeV and σv = 3 × − 26 cm s . Note that the spike at 0 angle is cut off by a finiteangular resolution taken to be 3 × − sr. Solid lines are the fluxes from decays, dashed lines are fromannihilations. Listed in order from top to bottom on the left edge of the plot: Moore profile (red), NFWprofile (blue), Kravtsov profile (black), Burkert or isothermal profile (green). produced. The same conclusion would hold for decays to µ + µ − . However, decays to three bodyfinal states would soften the produced electron and positron spectrum, thus softening the spectrumof FSR gamma-rays. This signal could be more difficult to observe with Fermi, though the injectedelectrons and positrons must always be rather high energy in order to explain ATIC so there is alimit to how soft the gamma-ray spectrum could be for any model which explains ATIC.It may be possible to distinguish dark matter decays from annihilations, so long as the dark68atter-produced component of the total diffuse gamma ray background can be distinguished fromthe spectrum. The intensity of dark matter produced gamma rays can then be measured as afunction of the observing angle. Because the decay rate scales as n while the annihilation ratescales as n , the dependence of the gamma ray intensity on the angle from the galactic center isdifferent for decays and annihilations for a given halo profile. Of course, uncertainty in the haloprofile could make it difficult to distinguish between the two. In Fig. 23 we have plotted the galacticgamma ray flux as a function of the angle from the galactic center at which observations are made(these halo profiles are spherically symmetric around the galactic center so only the angle fromthe center matters). The intensity is shown for both decays and annihilations to 2 photons from avariety of halo profiles. We have chosen to scale the rates to the standard annihilation cross section( σv = 3 × − 26 cm s ) and the decay rate that corresponds to this from Eqn. (8) ( τ = 4 × s).The shape of these curves is independent of the overall normalization. The decay curves in Fig. 23exhibit a universal behavior for large angles (cid:38) π , independent of the halo profile. This stems fromthe fact that the integral of n is essentially just the total amount of dark matter and is relativelyinsensitive to the distribution. Note that this universal shape of the decay curve is significantlydifferent from the shape of the annihilation curves. With enough statistics, this difference wouldbe readily apparent. The annihilation curve from the Burkert profile is most similar to the decaycurve and so would be the most difficult to distinguish.A telescope such as Fermi is ideally suited to the task of measuring the angular distribution ofthe gamma-rays because of its large, O (1 sr), field of view and coverage of the entire sky. Further,astrophysical backgrounds are significantly lessened away from the galactic ridge, giving an advan-tage to Fermi over the existing HESS observations which are mostly dominated by backgrounds.Fermi has an acceptance of ∼ × cm s sr. Thus we can see from the scale in Fig. 23 thatat least with a relatively strong decay mode into photons, Fermi could have enough statistics todifferentiate the decay signal from an annihilation signal.If the WMAP haze is due to dark matter decays producing electrons and positrons in the galacticcenter it may be possible to detect inverse Compton scattered photons with Fermi. A study hasbeen done for annihilating dark matter [71]. It may be interesting to check both the possibility ofexplaining the original WMAP haze and of observing the corresponding inverse Compton gammarays at Fermi, for decaying dark matter as well.69 II. LHC SIGNALS In this section, we study the LHC signals of the dimension 5 and 6 GUT suppressed operatorsconsidered in this paper. The dimension 5 operators lead to particle decays with lifetimes ∼ − ∼ s and these decays will not be visible at the LHC. However, the requirement that theseoperators explain the observations of PAMELA/ATIC leads to constraints on the low energy SUSYspectrum and these constraints can be probed at the LHC. In the following subsections, we analyzethe signals of these two kinds of operators. A. Signals of Dimension 5 Operators The dimension 5 operators introduced in section IV mediate decays between the MSSM and anew TeV scale χ . The χ is in a vector-like representation of SU (5). In the following subsections,we analyze the cases when χ is a (10 , ¯10), a (5 , ¯5) or a singlet of SU (5) and summarize our resultsin Table VII. 1. Decouplet Relic The most striking collider signals arise in the case when χ is a (10 , ¯10). In this case, all theparticles in the multiplet are either colored or carry electric charge. When these particles areproduced at a collider, they will leave charged tracks as they barrel out of the detector. Thissignal should be easily visible over backgrounds at the LHC [46]. A significant fraction of theseparticles will stop in the calorimeters and their late time ( ∼ − ∼ Li problemcan be solved through the decays of a electroweak multiplet only if its mass M E χ (cid:47) χ could also be light enough to be produced at the LHC.The production of these colored multiplets allows for more dramatic signatures at the LHC. Letus first consider the case when the fermionic components of the colored SU (2) doublets ( U, D ) in(10 , ¯10) are lighter than the scalar components, so that the scalar components, if produced, decayrapidly to the fermionic components. The masses of the components of the colored SU (2) doubletfermions (cid:0) U f , D f (cid:1) in (10 , ¯10) are split by ∼ αM Z ∼ 350 MeV due to electroweak symmetry breaking[53, 54]. When the heavier component of the doublet is produced at the LHC, it will decay to thelighter one with a lifetime ∼ G F (350 MeV) ∼ × − s = 1 . ∼ 350 MeV resulting in the production of soft pions, muons and electrons.The lighter component will barrel through the detector where it will leave a charged track and mayeventually stop and decay after 100 − SU (2) colored multiplet.It is also possible that the scalar components of the colored doublets are lighter than theirfermionic partners. In this case, the fermionic components will decay rapidly to the scalar com-ponents. The contribution from electroweak symmetry breaking to the mass splitting between thecomponents of the colored SU (2) scalar doublets is ∼ cos (2 β ) M W [55]. The decays of the heavierscalar component to the lighter component will be rapid, producing the lighter component and hardjets or leptons. The lighter component will again barrel through the detector producing a chargedtrack and a late decay. This will also be a striking signal at the LHC.Since the χ are always pair produced and decay at similar times, it may be possible to determinetheir lifetime by correlated measurements of double decay events in the calorimeter [61]. If all theparticles in the multiplet are light enough to be produced at the LHC, a measurement of their massand lifetime will be a direct probe of the GUT scale. The decay rate Γ of a particle of mass M through a dimension 5 operator scales as Γ ∼ M . A measurement of the masses and decay ratesof the colored and electrically charged multiplets can then be used to determine if Γ and M scaleas expected for a dimension 5 decay operator. Upon establishing this, a measurement of the scaleΛ mediating the decay can be inferred from the relation Γ ∼ M Λ .71 . Fiveplet Relic The colored particles in (5 , ¯5) will give rise to charged tracks and late time decays similar to thedecays of the colored multiplets in (10 , ¯10) as discussed earlier in subsection VII A 1. Electroweaksymmetry breaking causes mass splittings between the electrically charged component l + χ and theneutral l χ . The collider phenomenology of the lepton doublet l χ depends upon the spectrum of thetheory. We first consider the case when the scalar components ˜ l χ are lighter than their fermionicpartners l fχ , so that the l fχ , if produced, decay rapidly to the ˜ l χ . Upon electroweak symmetrybreaking, the masses of the scalar components ˜ l + χ and ˜ l χ are split by ∼ cos (2 β ) M W [55]. Dependingupon the value of cos (2 β ), the ˜ l + χ can be lighter than ˜ l χ . In this case, the ˜ l + χ will produce chargedtracks as it traverses through the detector. Some of the ˜ l + χ s will stop in the detector and result inlate time decays. The ˜ l + χ will rapidly decay to the ˜ l χ when the ˜ l χ is lighter than the ˜ l + χ . This decayproduces hard jets and leptons along with missing energy from the ˜ l χ that leave the detector anddecay outside it.We now consider the case when the fermionic components l fχ are lighter than their scalar partners˜ l χ . In this case, the scalars ˜ l χ , if produced, will rapidly decay to their fermionic partners l fχ .Electroweak symmetry breaking makes the electrically charged fermionic component l f + χ heavierthan the neutral component l f χ by ∼ αM Z ∼ 350 MeV [53, 54]. The l f + χ decays to the l f χ with alifetime ∼ G F (350 MeV) ∼ × − s = 1 . ∼ 350 MeV. The softness of these particlesmakes it difficult to trigger on them unless the production of these particles is accompanied byother hard signals like initial or final state radiation of photons [54]. One possible source for thishard signal is the decay of the scalars ˜ l χ . If the ˜ l χ are produced, they will rapidly decay to theirfermionic counterparts. The LSP will also be produced in this process resulting in the activation ofmissing energy triggers.Another possible source for this hard signal are the colored particles in the multiplet. In themodel discussed in sub section V B, the colored particles in the SO (10) multiplet 16 χ can decayto the lepton doublets in the multiplet through dimension 5 operators. In this case, the late timedecays of the colored multiplet can produce l f + χ . This decay will be accompanied by a release ofhadronic energy at the location of the colored particle and a charged track that follows the trajectory72f the produced l f + χ . This track ends in a displaced vertex when the l f + χ decays to the l f χ . The latetime decay followed by the charged track can be used to trigger on this event. The discovery ofsoft particles at the location of the displaced vertex, while difficult experimentally, will unveil thepresence of the lepton doublet. 3. Singlet Relic The LHC signals for decays involving operators with singlet χ s is dependent on the MSSMspectrum. If the MSSM LSP is heavier than χ , then the MSSM LSP will decay to χ at ∼ χ . In this scenario, since the MSSM LSP is no longer the dark matter, theMSSM LSP could be charged. The SUSY particles produced at the LHC will rapidly decay to theMSSM LSP and if this MSSM LSP is charged or colored, it will give rise to charged tracks andlate decays in the detector. The signals of SUSY in this scenario are significantly altered sincethe decays of SUSY particles are no longer associated with missing energy signals as the chargedMSSM LSP can be detected. However, if the MSSM LSP is neutral, it will decay to the χ outsidethe detector and the LHC signals of this scenario will be identical to that of conventional MSSMmodels.The cosmological lithium problem could also be solved by decays of singlet χ s to the MSSM ifthe χ is heavier than the MSSM LSP. In this case, an abundance of χ must be generated through nonstandard model processes. This abundance can be generated thermally through new interactionslike a low energy U (1) B − L or through the decays of heavier, thermally produced standard modelmultiplets to χ . In this scenario, these new particles could be discovered at the LHC, for examplethrough Z (cid:48) gauge boson searches. It is also possible that the initial abundance of χ was generatedthrough a tuning of the reheat temperature of the Universe. This scenario is devoid of new signalsat the LHC. 73 SU (5) Rep. Spectrum Prominent Signals and Features(10 , ¯10) M χ > M LSP Charged tracks from colored and electrically charged particles.Measurement of mass, lifetime from stopped colored and charged particles.Potentially visible displaced vertex from colored doublet decays.Infer dimensionality and scale of decay from mass, lifetime measurements.(5 , ¯5) M χ > M LSP Charged tracks from colored particles.Measurement of mass, lifetime from stopped colored particles.Charged track from scalar lepton doublet.Potentially visible displaced vertex from lepton doublet decays.Singlet M LSP > M χ Possibility of charged or colored LSP giving rise to charged tracks.Charged LSP detection enables better measurement of SUSY spectrum. M χ > M LSP Discover new particles ( e.g. TeV scale U (1) B − L ) for thermally produced χ .TABLE VII: Prominent signals of the dimension 5 decay operators considered in section IV. The signalsare classified on the basis of the SU (5) representation of the new particle χ and its mass M χ . When M χ > M LSP , the primordial lithium problem is solved by the decays of χ to the MSSM and vice-versawhen M LSP > M χ . Bounds on dark matter direct detection [37] force χ to decay to the MSSM LSP when χ ( e.g (5 , ¯5), (10 , ¯10)) has dirac couplings to the Z . B. Signals of Dimension 6 Operators The dimension 6 GUT suppressed operators discussed in this paper mediate decays betweensinglet S s and the MSSM when the operators conserve R-parity or cause decays of the LSP tothe standard model when the operators violate R-parity (see section III). The decay rate for theseprocesses is Γ ∼ − s − . The largest production channel for these particles at the LHC wouldbe through the decays of colored particles. For example, the LSP would be produced in the decaysof gluinos. The production cross-section for a 200 GeV gluino at the LHC is ∼ fb[61]. Witha integrated luminosity of 100 fb − , the LHC will produce ∼ LSPs through the decays of thegluino. This number is far too small to allow for the observation of the decay. Similarly, the number74f singlet S s that can be produced either through a new low energy gauge symmetry ( e.g. U (1) B − L )or through the decays of other particles that carry standard model quantum numbers is also toosmall to allow for the observation of the decay of the S .The most promising signatures of these dimension 6 GUT suppressed operators are the astro-physical signatures of their decay. In section III, we show that these dimension 6 GUT suppresseddecays can explain the observations of PAMELA/ATIC. PAMELA has observed an excess in thepositron channel while it does not see an excess in the anti proton channel. The hadronic branchingfraction of these decays must be smaller than 10 percent in order to account for the PAMELA signal(see section III). The need to suppress the hadronic branching fraction of these decays can be usedto arrive at constraints on the superpartner spectrum.For concreteness, we consider the model in section III A where the scalar component ˜ s of the S gets a TeV scale vev (cid:104) ˜ s (cid:105) . This leads to the operators (cid:104) ˜ s (cid:105) s ˜ llM GUT and (cid:104) ˜ s (cid:105) s ˜ qqM GUT where s denotes the fermioniccomponent of S , (cid:16) ˜ l, l (cid:17) represent slepton and lepton fields and (˜ q, q ) represent squark and quarkfields. This operator will mediate the decay of s to the MSSM LSP if the mass M s of s is greaterthan the LSP mass M LSP . The hadronic branching fraction of this decay depends upon the masssplittings ∆ M s ˜ q and ∆ M s ˜ l between s and the squarks and sleptons respectively. The followingoptions emerge: • M s > M ˜ l and M s > M ˜ q : This decay produces on shell sleptons and squarks. The hadronicbranching fraction is ∼ (cid:16) ∆ M s ˜ q ∆ M s ˜ l (cid:17) and this fraction is smaller than 10 percent if ∆ M s ˜ q < . 45 ∆ M s ˜ l . • M s < M ˜ l and M s < M ˜ q : This decay produces electrons and quarks through off-shell sleptonsand squarks respectively. The hadronic branching fraction of this decay mode is (cid:16) M ˜ l M ˜ q (cid:17) where M ˜ l and M ˜ q are slepton and squark masses respectively. This branching fraction is smallerthan 10 percent if M ˜ l < . M ˜ q . • M s > M ˜ l and M s < M ˜ q : This decay produces sleptons on shell. The dominant hadronicbranching fraction of this decay mode arises from final state radiation of W and Z bosonsyielding a branching fraction ∼ − for M s > − since it is suppressedby additional phase space factors and couplings.75he cases where the slepton is light enough to be produced on shell ( e.g. M s > M ˜ l ) allow forthe possibility of another correlated measurement at the LHC and astrophysical experiments likePAMELA/ATIC. The decay of the s to a on-shell lepton and slepton will generate a primary sourceof hot electrons from the leptons produced in this primary decay. These electrons will producea ”bump” in the spectrum that will be cut off at roughly ∼ M s . The sleptons produced in thisprocess are also unstable and will rapidly decay to a lepton and the LSP. This process produces asecondary lepton production channel that will also lead to a spectral feature cut off at M ˜ l − M LSP .This astrophysical measurement can be correlated with measurements of this mass difference at theLHC and these independent measurements can serve as a cross-check for this scenario.The other possible decay in this model is the decay of the LSP to the s which will happen if M LSP > M s . In this case, leptons and quarks are produced through off-shell sleptons and squarksrespectively (see figure 2). The hadronic branching fraction of this decay is given by (cid:16) M ˜ l M ˜ q (cid:17) . Thisbranching fraction is smaller than 10 percent if M ˜ l < . M ˜ q . We note that the model discussedin this section naturally allows for small hadronic branching fractions. The hadronic branchingfraction is determined by slepton and squark masses and since squarks are generically expectedto be heavier than sleptons due to RG running, the hadronic branching fraction of these modelsis generically small. These decays have a hadronic branching fraction that is at least ∼ − for M s ∼ Z and W bosons. A measurement of anti-protonsat PAMELA that indicates a hadronic branching fraction smaller than ∼ − will rule out thismodel. A hadronic branching fraction that is larger than 10 − will however constrain the squarkand slepton masses as discussed earlier in this section. VIII. CONCLUSIONS In broad classes of theories, global symmetries that appear in nature, such as baryon number,are accidents of the low energy theory and are violated by GUT scale physics. If the symmetrystabilizing a particle, such as the proton, is broken by short-distance physics then it will decay witha long lifetime. In a sense, a particle is continually probing the physics that allows it to decay.Thus, though the effects of GUT scale physics are suppressed at low energies, this suppressionis compensated for by the long time scales involved. Intriguingly, short-distance physics can be76nveiled by experiments sensitive to long lifetimes.Astrophysics and cosmology provide natural probes of long lifetimes. The dark matter particleappears stable, but dimension 6 GUT suppressed operators can cause it to decay with a long lifetime ∼ s. This can lead to observable signals in a variety of current experiments including PAMELA,ATIC, HESS, and Fermi. Such observations can also probe the TeV scale physics associated withdark matter annihilations. However, annihilations cannot proceed through GUT scale particlessince such a cross section would be far too small. Thus, the observation of annihilations can probeTeV physics but does not probe GUT physics directly.This leads to interesting differences between the expected signals from dark matter decaysand annihilations [53, 72]. Decaying dark matter can directly produce photons with an O (1)branching ratio, as seen in Section III. It can also produce light leptons without helicity or p-wavesuppression and give signals in the range necessary to explain the PAMELA/ATIC excess withoutboost factors [35, 73, 74, 75, 76]. Additionally, in our framework the hadronic branching fractionscan be naturally suppressed since sleptons tend to be lighter than squarks, perhaps explainingthe lack of an antiproton signal at PAMELA. We have also found that the signals produced bydecays may show many interesting features beyond just a bump in the spectrum of electrons andpositrons. For example, the spectrum from decays may accommodate the secondary feature visiblein the ATIC data at energies between 100 - 300 GeV. The SUSY spectrum can allow for the directproduction of a slepton, lepton pair from the decaying dark matter. In this case, the subsequentdecay of the slepton provides a secondary hot lepton that can give rise to the observed secondaryATIC feature. It is also common in our framework to have two dark matter particles, scalar andfermion superpartners, that both decay at late times producing interesting features in the observedspectra of electrons or photons.Such models will be tested at several upcoming experiments. The spectrum of electrons andpositrons will be measured to increasing accuracy by Fermi, HESS, and PAMELA, testing theobserved excesses. These measurements could potentially reveal not just the nature of dark matter,but also the mass spectrum of supersymmetry and GUT scale physics. Excitingly, Fermi andground-based telescopes such as HESS or MAGIC will be providing measurements of the gamma-ray spectrum in the near future. These measurements have the potential to test many of ourmodels, and in particular may be able to observe final state radiation from models that explain the77lectron/positron excess. Further, they could distinguish decaying and annihilating dark matterscenarios by the different angular dependence of the gamma ray signals.The supersymmetric GUT theories in which the dark matter decays through dimension 6 oper-ators often have TeV mass particles that decay via dimension 5 operators with a lifetime of ∼ ∼ Acknowledgments We would like to thank Nima Arkani-Hamed, Douglas Finkbeiner, Raphael Flauger, StefanFunk, Lawrence Hall, David Jackson, Karsten Jedamzik, Graham Kribs, John March-Russell, IgorMoskalenko, Peter Michelson, Hitoshi Murayama, Michele Papucci, Stuart Raby, Graham Ross,Martin Schmaltz, Philip Schuster, Natalia Toro, Jay Wacker, Robert Wagoner, and Neal Weiner forvaluable discussions. 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In fact, extrapolatinga power law to high energies and taking the steep decline seen by HESS into account, may imply thatthe background is on the steeper end of the uncertainty.his region may be somewhat contaminated by signal, loosening the constraint. In fact, extrapolatinga power law to high energies and taking the steep decline seen by HESS into account, may imply thatthe background is on the steeper end of the uncertainty.