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Indirect detection of unstable heavy dark matter

arXiv:hep-ph/9207261v1 27 Jul 1992Uppsala U. PT17-1992June 1992Indirect detection of unstable heavy dark matterPaolo Gondolo1Department of Radiation Sciences, Uppsala University,P.O. Box 535, 75121 Uppsala, SwedenABSTRACTUnstable relics with lifetime longer than the age of the Universe could be thedark matter today.

Electrons, photons and neutrinos are a natural outcome of theirdecay and could be searched for in cosmic rays and in γ-ray and neutrino detectors.I compare the sensitivities of these three types of searches to the mass and lifetimeof a generic unstable particle. I show that if the relics constitute our galactic haloand their branching ratios into electron-positrons, photons and neutrinos are com-parable, neutrino searches would probe the longest lifetimes for masses >∼40 TeV,while electron-positron searches would be better but more uncertain for lighterparticles.

If instead the relics are not clustered in our halo, neutrinos are moresensitive a probe than γ-rays for masses >∼700 GeV. A 1 km2 neutrino telescopeshould be able to explore lifetimes up to ∼1030 s while searching for neutrinosfrom unstable particles above the atmospheric background.1 BITNET address: GONDOLO@SESUF51

1. IntroductionPopular elementary particle candidates for cold dark matter (e.g.

neutrinos,cosmions and sneutrinos) have been ruled out or severely constrained by the SLCand LEP measurements of the Z decay width combined with the non-observation ofa signal in direct and indirect dark matter searches, in low-background germaniumdetectors and proton-decay experiments respectively (see e.g. ref.

[1]).It is so natural that alternative possibilities for non-baryonic cold dark matterare being (re)explored. Here I consider a class of unstable dark matter candidatesthat are heavy and long-lived, decaying on cosmological time scales.

Many particlesof this sort have been already proposed before the above-mentioned experimentalresults for various reasons.Massive long-lived particles with a substantial relic density may arise in techni-color models, which are an interesting alternative to the standard Higgs mechanismfor spontaneous SU(2)×U(1) symmetry breaking. It was pointed out in ref.

[2] thatthe lightest technibaryon, which likely has mass m ∼TeV and lifetime 1027−32 scould account for the missing mass if there is a technibaryon-antitechnibaryonasymmetry comparable to the baryon-antibaryon asymmetry. Non-perturbativefermion-number violating processes in the Standard Model could generate such anasymmetry in a natural way [3].Motivated by a preliminary report of an anomalous high energy (> 10 GeV)positron component in cosmic rays which is unlikely to be generated by spallationprocesses, new massive long-lived particles have been proposed as dark mattercandidates.

It has been suggested that the excess positrons come from the decayof ∼30 GeV right-handed neutrinos with lifetime ∼1025 s [4] or from 1-3 TeVGUT particles with lifetime ∼1024 s [5].The ‘lightest supersymmetric particle’ can be unstable and long-lived in su-persymmetric models with very weak R-parity breaking. A decaying gravitino ofmass ∼100 GeV [6] and a neutralino-neutrino mixed state of mass 10−50 GeV [7],2

both with any lifetime longer than ∼1017 s, have been examined in a cosmologicalcontext.As a final example, a simple solution via confinement [8] to the problem offractional charges in models derived from the superstring [9] results in severalinteger-charged composite particles, named ‘cryptons’ [10], some of which couldbe long-lived with lifetimes as long as 1023 s and masses ∼1012 GeV and could, inprinciple, constitute the dark matter [11]. Their expected relic density is howeververy uncertain since it depends on the amount of entropy released in the decaysof short-lived cryptons (and other particles) after the long-lived cryptons go out ofchemical equilibrium.It is therefore interesting to examine long-lived heavy particles, with masses>∼GeV and lifetimes >∼1017 s, as dark matter candidates.

At present, it seems ap-propriate to study such weakly unstable massive particles (WUMPs)∗in a model-independent way.In general, a WUMP x is characterized by its mass mx, its lifetime τx and itsbranching fractions by into different decay channels y. I focus here on WUMPswhich constitute the dark matter today and I consider two classes of WUMP dis-tributions: a uniform mass distribution throughout the Universe with critical valueρc = 10.5 keV cm−3h2 (unclustered WUMPs) and an inhomogeneous distributionwith WUMPs clustered in our galactic halo with mass density ρ⊙= 0.3 GeV cm−3in the solar neighborhood (clustered WUMPs). Other values of the relic WUMPdensity have been considered in refs.

[11,12]. As usual, h is the Hubble constantin units of 100 km s−1 Mpc−1.Of the WUMP decay products, I consider here neutrinos, electrons and pho-tons, which may give origin to diffuse extraterrestrial fluxes.In the following,I examine the range of WUMP masses and lifetimes that could be explored bybackground- and flux-limited detectors searching for such indirect signals fromunstable dark matter particles.∗I adopt the terminology of ref.

[4].3

2. Diffuse photon, electron-positron and neutrino fluxesBranching ratios and energies of the decay products, and even the nature ofthe decay products, depend of course on the particular model for the decayingdark matter.For the sake of definiteness, I imagine that WUMPs undergo 3-body decays, in which one (or more) decay product(s) is a photon, an electron, apositron or a neutrino, and acquires a typical energy E0 = mx/3.

Furthermore,when comparing sensitivities to signals of different origin, I take all branching ratiosby to be equal, in particular I use by = 1 when stating limits on the lifetime τx.All results can be trivially rescaled to the values appropriate to a specific model.More complicated generation scenarios, e.g. secondary products or jets, can alsobe easily incorporated by including the decay product multiplicity and possiblecontinuous energy spectra in an energy dependent by.The maximum WUMP lifetime explorable by background-limited detectors isobtain as a function of the WUMP mass by imposing that the electron-positron,photon and neutrino fluxes from WUMP decays be smaller than the respectivebackgrounds.

I separately discuss now the three cases of decay photons, electron-positrons and neutrinos. I will use t0 = 1010 yr and h = 1.2.1 PhotonsThe photon flux expected from uniformly distributed decaying dark matter isa superposition of the photon spectra generated in decays occuring at differenttimes.

The resulting present day flux Iγ(E) can be written as an integral over theredshift 1 + z = E′/E of the photon spectrum per decaying particle Sγ(E):Iγ(E) = 38πρct0mxτxE1/2∞ZEdE′E′3/2Sγ(E′). (1)(This equation is obtained for a matter-dominated universe and for τx ≫t0.

)4

For WUMPs clustered in the galactic halo no integration over redshift is nec-essary and the decay photon flux is given byIγ(E) = 14π1mxτxZρh(x)|x −x⊙|2d3x Sγ(E),(2)where x⊙denotes the position of the solar system and ρh(x) is the WUMP distri-bution in the halo. If the dark matter distribution is taken to beρh(x) =2ρ⊙1 + (r/a)2,(3)the integral in eq.

(2) evaluates to 31.0aρ⊙and for ρ⊙= 0.3 GeV cm−3 and a⊙=8 kpc, it is ≃2.3ρct0.Two cases arise for the source function Sγ(E) according to the energy of theprimary decay photon E0.If it is energetic enough to produce e+e−pairs incollisions against the microwave photons, it can trigger electromagnetic cascades.Otherwise it is able to propagate to us without cascading. In the latter case, thesource spectrum can be approximated by a lineSγ(E) ≃bγ δ(E −E0).

(4)The cascading case is more complicated.The electromagnetic cascade de-velopes until the energies of the photons have fallen below the pair productionthreshold Emax. For a blackbody target this is given by [13]Emax ≃m2e20.4T1 + 12 lnη7×10−102+ 12 lnEmaxme2,(5)where me is the electron mass, T is the blackbody temperature and η is the baryonto photon ratio, which for simplicity I take to be just 7 × 10−10.

At the presentepoch, Emax(t0) ≃3.4 × 1012 eV and cascades can be generated if the WUMP5

mass mx>∼mcrit ≃1013 eV. The spectrum of the ‘breakout’ photons below thepair-production threshold falls as ∼E−1.5 until ∼0.04Emax and then steepens to∼E−1.8 before being cutoffat Emax [14].

With the normalizationRdEESγ(E) =E0, the source spectrum per decaying WUMP with mx>∼3Emax isSγ(E) ≃34bγE0E−1/2max E−3/2,for 0 ≤E ≤0.04Emax,310bγE0E−0.2max E−1.8,for 0.04Emax ≤E ≤Emax,0,for E ≥Emax. (6)The source functions (4) and (6) can then be inserted in eqs.

(1) and (2) andthe resulting photon fluxes can be compared with the diffuse background γ flux,which at E > 3 MeV I approximate as (see ref. [15])Ibkgdγ<∼2 × 10−33 MeVE2cm−2 s−1 sr−1 MeV.

(7)I assume a detector with 10% energy resolution to smear the delta function occuringin the decay neutrino flux from non-cascading clustered WUMPs.The maximum WUMP lifetimes accessible to γ-ray astronomy turn then outto beτx>∼(1.73 × 1025bγ s,for mx < mcrit,1.46 × 1024bγ s,for mx > mcrit,(8)for unclustered WUMPs andτx>∼(2.6 × 1026bγ s,for mx < mcrit,6.0 × 1026bγ s,for mx > mcrit,(9)for clustered WUMPs. These lifetimes (with bγ = 1) are shown in Fig.

1 as thesolid lines labelled γ. The different directions of the step at mcrit ≃1013 eV aredue to the rapid falling of the background flux (7) with energy combined with thefact that the ratio between the WUMP-generated and the background fluxes ismaximum at E = Emax(t0) in the clustered case and at E ≃0.18Emax(t0) in theunclustered one.6

2.2 Electrons and positronsIf the decaying dark matter is clustered in the galactic halo, the galactic mag-netic field is able to contain the decay electrons and positrons in the galaxy. Thecontainment time is quite uncertain, typically thought to be of the order of 1016 sfor 1 GeV electrons, and possibly varying with energy.

This is a source of uncer-tainty in the determination of the electron-positron fluxes from clustered WUMPs,and renders the corresponding bounds on the lifetimes less reliable than in thephoton and neutrino cases.For these reason, it is therefore sufficient to describe the high energy e± densityne(E) using a leaky box model [16], with the e± sources uniformly distributed overthe halo,ne(E)τcont(E) + ddE [f(E)ne(E)] = Qe(E). (10)Here f(E) = −βE2, with β ∼3 × 10−17 GeV−1 s−1, is the rate of energy loss ofelectrons and positrons due to synchrotron radiation and inverse-Compton scat-tering offmicrowave photons, Qe(E) is the e± source function, which I take tobeQe(E) = beρ⊙mxτxδ(E −E0),(11)and τcont(E) is the containment time in the halo.

This can be estimated usinga diffusion model [16] in which τcont(E) ∼R2halo/D(E), taking for the halo sizeRhalo ∼10kpc and for the diffusion coefficient D(E) ∼1029E1/3GeV cm2 s−1 [16].The containment time results τcont(E) ∼1016 s E−1/3GeV . The electron-positron fluxis then obtained by solving eq.

(10):Ie±(E) = be14πρ⊙mxτxβE2 exp"EcE02/3−EcE2/3#,(12)with Ec =23βτcont(1 GeV)−3/2 ∼10 GeV.7

The combined cosmic-ray electron and positron flux has been measured up toabout 2 TeV [17]. I adopt here the following parametrization for it (cfr.

ref. [18]):Ibkgde±(E) ≃(4.4 × 10−7 E20 GeV−2.7 cm−2 s−1 sr−1 GeV−1,for E < 20 GeV,4.4 × 10−7 E20 GeV−3.5 cm−2 s−1 sr−1 GeV−1,for E > 20 GeV.

(13)Comparing the decay and background fluxes (12) and (13), the maximumWUMP lifetime accessible to background-limited electron-positron searches is ob-tain asτx>∼(3.8 × 1027bem−0.310s,for mx < 60 GeV,9.5 × 1026bem1/210 s,for mx > 60 GeV. (14)This is shown in Fig.

1 for be = 1 as the light line labelled e±. The dashed portioncorresponds to the extrapolation of the background e± flux (13) above the highestmeasured energies.2.3 NeutrinosThe diffuse neutrino flux from decays of unclustered WUMPs is obtained byintegration over redshift of a monochromatic decay spectrum with energy E0.

Forlifetimes τx ≫t0, it is given byIν(E) = bν38πρcmxE0t0τx EE01/2. (15)This equation applies separately to each neutrino and antineutrino flavor.As a background to the decay neutrino flux I consider the νµ + ¯νµ atmosphericneutrinos produced in collisions of cosmic rays with nuclei of the upper atmosphere.8

Their spectrum in the vertical direction has been estimated in ref. [19] asIbkgdνµ+¯νµ<∼4.4 E−3.69GeV + 2.4 × 10−5 E−2.65GeVcm−2 s−1 sr−1 GeV−1,(16)for E < 2.3 × 106 GeV, andIbkgdνµ+¯νµ<∼4.4 E−3.69GeV + 3.9 × 10−3 E−3GeVcm−2 s−1 sr−1 GeV−1,(17)for E > 2.3 × 106 GeV.Comparison of the decay neutrino flux (15) with the atmospheric backgroundat E = E0 gives the maximum explorable lifetime for unclustered WUMPsτx>∼1.4 × 1022 sbνm1.69101 + 2 × 10−5m1.0410,for mx < 6.9 × 1015 eV,1.4 × 1022 sbνm1.69101 + 2 × 10−3m0.6910,for m10 > 0.6.9 × 1015 eV.

(18)This lifetime is shown in Fig. 1 as the heavy solid line labelled ν, drawn for bν =bνµ + b¯νµ = 2.For WUMPs clustered in our galactic halo, no integration over redshift needsto be performed and the decay neutrino flux is given byIν(E) = bν14π1m2xτxZρh(x)|x −x⊙|2d3x δ(E −E0).

(19)Assuming a detector with 10% energy resolution to smear the delta function theaccessible lifetimes are a factor 15 larger than those for unclustered WUMPs. Theselifetimes are shown in Fig.

1, again for bν = 2, as the light solid line labelled ν.Present neutrino detectors are already able to exclude a sizeable region of theWUMP mass-lifetime plane. The dotted line shows the lower bound on τx obtainedin ref.

[12] using IMB and Fly’s Eye published data. Analogous calculations showthat a future 1 km2 neutrino telescope would be able to explore the region limitedby the dotted line labelled 106 at the flux level of 1 muon per year.9

3. ConclusionsIf the dark matter WUMPs are uniformly distributed in the universe, a background-limited neutrino detector would be able to explore longer lifetimes than those ac-cessible to photon searches for any WUMP masses larger than ∼700 GeV.If the WUMPs constitute the galactic halo, the longest lifetimes would beprobed by electron-positron searches for mx<∼10 TeV.

This occurs because of theeffective magnetic confinement of electrons and positrons in the galaxy. Unfortu-nately, the containment time is quite poorly known and the bounds so obtainedwould be subject to uncertainty.

Searches for decay neutrinos seem again the mostpowerful for WUMP masses above ∼700 GeV.In both cases, a 1 km2 neutrino telescope might be able to reach lifetimes oforder 1030 s for masses mx>∼100 TeV. Such a detector would be quite a good probeof long-lived dark matter particles.AcknowledgementsI would like to thank Subir Sarkar for valuable discussions.10

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FIGURE CAPTIONSFigure 1. Maximum accessible WUMP lifetime τx versus WUMP mass mx in electron-positron, photon and neutrino background-limited detectors.Light andheavy lines correspond to clustered and unclustered WUMP distributionsrespectively.The dotted lines indicate the present lower limits on τx im-posed by the IMB and Fly’s Eye neutrino data and the region explorable bya neutrino telescope sensitive to 1 muon km−2 yr−1 (curve 106).12

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