We analyze the spectrum of the ultrahigh energy (above \approx 10^{9} GeV)
cosmic rays. With a maximum likelihood analysis we show that the observed
spectrum is consistent with the decay of extragalactic GUT scale particles. The
predicted mass for these superheavy particles is m_X=10^b GeV, where
b=14.6_{-1.7}^{+1.6}.Comment: 4 pages, 3 figures, talk to be presented at the 27th International
  Cosmic Ray Conference, Hamburg, Germany, August 7-15, 200

Fodor, Z.

Katz, S. D.

English

arXiv

Fodor, Z

Katz, SD

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-ph/0105348v1  31 May 2001Ultrahigh energy cosmic rays as a Grand Unification signalZ. Fodor and S.D. KatzInstitute for Theoretical Physics, Eötvös University,Pázmány 1, H-1117 Budapest, HungaryNovember 2, 2018AbstractWe analyze the spectrum of the ultrahigh energy (above≈109 GeV) cosmic rays. With a maximum likelihood analysiswe show that the observed spectrum is consistent with thedecay of extragalactic GUT scale particles. The predictedmass for these superheavy particles ismX = 10b GeV, whereb = 14.6+1.6−1.7.1 IntroductionThe interaction of protons with photons of the cosmic mi-crowave background radiation (CMBR) predicts a sharp dropin the cosmic ray flux above the Greisen-Zatsepin-Kuzmin(GZK) cutoff around5 · 1019 eV (Greisen, 1966; Zatsepinand Kuzmin, 1966). The available data shows no such drop.About 20 events above1020 eV were observed by experi-ments such as AGASA (Takeda et al., 1998), Fly’s Eye (Birdet al., 1993), Haverah Park (Lawrence et al., 1991), Yakutsk(Efimov et al., 1991) and HiRes (Kieda et al., 1999). Fu-ture experiments, particularly Pierre Auger (Boratav, 1996;Guerard, 1999; Bertou et al. 2000), will have a much higherstatistics.Usually it is assumed that at these energies the galacticand extragalactic (EG) magnetic fields do not affect the orbitof the cosmic rays, thus they should point back to their ori-gin within a few degrees. Though there are clustered events(Hayashida et al., 1996; Uchihori et al., 2000) the overalldistribution is practically isotropic (Dubovski and Tinyakov,1998; Berezinsky and Mikhailov, 1999; Medina Tanco andWatson, 1999), which usually ought to be interpreted as asignature for EG origin. Since above the GZK energy theattenuation length of particles is a few tens of megaparsecs(Yoshida and Teshima, 1993 ; Aharonian and Cronin, 1994;Protheroe and Johnson, 1996; Bhattacharjee and Sigl, 2000;Achterberg et al., 1999; T. Stanev et al., 2000) if an ultrahighenergy cosmic ray (UHECR) is observed on Earth it must beproduced in our vicinity. Sources of EG origin should resultin a GZK cutoff, which is in disagreement with experiments.It is generally believed that there is no conventional astro-physical explanation for the observed UHECR spectrum.An interesting idea is that superheavy particles (SP) asdark matter could be the source of UHECRs. In (Kuzmin andRubakov, 1998) EG SPs were studied. A crucial observationwas made (Berezinsky et al., 1997) about the decay of SPsconcentrated in the halo of our galaxy. They used the modi-fied leading logarithmic approximation (MLLA) (Azimov etal., 1985; Fong and Webber 1991) for ordinary QCD and forsupersymmetric QCD (Berezinsky and Kachelrieß, 1998). Agood agreement of the EG spectrum with observations wasnoticed in (Berezinsky et al., 1998). Supersymmetric QCDis treated as the strong regime of the minimal supersymmet-ric standard model (MSSM). To describe the decay spectrummore accurately HERWIG Monte-Carlo (Marchesini et al.,1992) was used in QCD (Birkel and Sarkar, 1998) and dis-cussed in supersymmetric QCD (Rubin 1999, Sarkar 2000),resulting inmX ≈ 1012 GeV and≈ 1013 GeV for the SPmass in SM and in MSSM, respectively.SPs are very efficiently produced by the various mecha-nisms at post inflatory epochs (for a review see Berezinsky2000). Note, that any analysis of SP decay covers a muchbroader class of possible sources. Several non-conventionalUHECR sources produce the same UHECR spectra as de-caying SPs.Here we study the scenario that the UHECRs are com-ing from decaying SPs and we determine the mass of thisXparticlemX by a detailed analysis of the observed UHECRspectrum. We discuss both possibilities that the UHECR pro-tons are produced in the halo of our galaxy and that they areof EG origin and their propagation is affected by CMBR.Here we do not investigate how can they be of halo or EGorigin, we just analyze their effect on the observed spectruminstead. We assume that the SP decays into two quarks (otherdecay modes would increasemX in our conclusion). Af-ter hadronization these quarks yield protons. The result ischaracterized by the fragmentation function (FF)D(x,Q2)which gives the number of produced protons with momen-tum fractionx at energy scaleQ. For the proton’s FF presentaccelerator energies can be used (Binnenwies et al., 1995;Kniehl et al., 2000). We evolve the FFs in ordinary (Gri-bov and Lipatov 1972; Lipatov 1975; Altarelli and Parisi,1977; Dokshitzer, 1977) and in supersymmetric (Jones andLlewellyn Smith 1983) QCD to the energies of the SPs. Thisresult can be combined with the prediction of the MLLAtechnique , which gives the initial spectrum of UHECRs atthe energymX . Altogether we study four different models:halo-SM, halo-MSSM, EG-SM and EG-MSSM.1flavor Q(GeV) N α βu = 2d 1.41 0.402 -0.860 2.80s 1.41 4.08 -0.0974 4.99c 2.9 0.111 -1.54 2.21b 9.46 40.1 0.742 12.4t 350 1.11 -2.05 11.4g 1.41 0.740 -0.770 7.69q̃i, g̃ 1000 0.82 -2.15 10.8Table 1: The fragmentation functions of the different partonsusing the parametrizationD(x) = Nxα(1− x)β at differentenergy scales (second column).2 Decay and fragmentation of heavyparticlesThe UHECRs are most likely to be dominated by protons(Dawson et al., 1998); in our analysis we use them exclu-sively.The FF of the proton can be determined from present ex-periments (Binnenwies et al., 1995; Kniehl et al., 2000). TheFFs atQ0 energy scale areDi(x,Q20), wherei represents thedifferent partons (quark, squark or gluon, gluino). The FFscan not be determined in perturbative QCD; however, theirevolution inQ2 is governed by the DGLAP equations (Gri-bov and Lipatov 1972; Lipatov 1975; Altarelli and Parisi,1977; Dokshitzer, 1977):∂Di(x,Q2)∂ lnQ2=αs(Q2)2π∑j∫ 1xdzzPji(z, αs(Q2))Dj(xz,Q2), (1)wherePji(z) is the splitting function. We solve the DGLAPequations numerically with the conventional QCD (SM case)splitting functions and with the supersymmetric (MSSM ca-se) ones (Jones and Llewellyn Smith 1983). For the top andthe MSSM partons we used the FFs of ref. (Rubin 1999).While solving the DGLAP equations each parton is includedat its own threshold energy.At small values ofx, multiple soft gluon emission can bedescribed by the MLLA. It describes the observed hadropro-duction quite accurately in the smallx region (see eg. Abreuet al., 1999; Abbiendi et al., 2000). For large values ofx theMLLA should not be used. We smoothly connect the solu-tion for the FF obtained by the DGLAP equations and theMLLA result at a givenxc value. Our final result onmX israther insensitive to the choice ofxc, the uncertainty is in-cluded in our error estimate. We also determined the FF ofthe pion. Fig. 1 shows the FF for the proton and pion atQ = 1016 GeV in SM and MSSM.Figure 1: The FFs averaged over the quark flavors atQ =1016 GeV for proton/pion in SM (solid/dotted line) and inMSSM (dashed/dashed-dotted line) in the relevantx region.To show both the small and largex behavior we change fromlogarithmic scale to linear atx = 0.01.3 Comparison of the predicted and theobserved spectraUHECR protons produced in the halo of our galaxy can prop-agate unaffected and the production spectrum should be com-pared with the observations.Particles of EG origin and energies above≈ 5 · 1019 eVloose a large fraction of their energies due to interactionswith CMBR (Greisen, 1966; Zatsepin and Kuzmin, 1966).This effect can be quantitatively described by the functionP (r, E,Ec), the probability that a proton created at a dis-tancer with energyE arrives at Earth above the thresholdenergyEc (Bahcall and Waxman 2000). This function hasbeen calculated for a wide range of parameters in (Fodor andKatz, 2001a), which we use in the present calculation. Theoriginal UHECR spectrum is changed at least by two differ-ent ways: (a) there should be a steepening due to the GZKeffect; (b) particles loosing their energy are accumulatedjustbefore the cutoff and produce a bump. We study the ob-served spectrum by assuming a uniform source distributionfor UHECRs.Our analysis includes the published and the unpublished(from the www pages of the experiments) UHECR data ofAGASA, Fly’s Eye, Haverah Park and HiRes. Since the de-cay of SPs results in a non-negligible flux for lower energieslog(Emin/eV) = 18.5 is used as a lower end for the UHECRspectrum. Our results are insensitive to the definition of theupper end (the flux is extremely small there) for which we2Figure 2: The available UHECR data with their error barsand the best fit from a decaying SP. Note that there are noevents above3×1020 eV (shown by an arrow). Neverthelessthe experiments are sensitive even in this region. Zero eventdoes not mean zero flux, but a well defined upper bound forthe flux (given by the Poisson distribution). Therefore theexperimental value of the integrated flux is in the ”hatched”region with 68% confidence level. (”hatching” is a set ofindividual error bars; though most of them are too large to bedepicted in full) Clearly, the error bars are large enough tobeconsistent with the SP decay.chooselog(Emax/eV) = 26. As it is usual we divided eachlogarithmic unit into ten bins. Using a Monte-Carlo methodwe included this uncertainty in the final error estimates. Thepredicted number of events in a bin is given byN(i) =∫ Ei+1Ei[A · E−3.16 +B · j(E,mX)], (2)whereEi is the lower bound of the ith energy bin. The firstterm describes the data below1019 eV according to (Takedaet al., 1998), where the SP decay gives negligible contribu-tion. The second one corresponds to the spectrum of the de-caying SPs. A and B are normalization factors.The expectation value for the number of events in a binis given by eqn. (2) and it is Poisson distributed. To deter-mine the most probablemX value we used the maximum-likelihood method by minimalizing theχ2(A,B,mX) forPoisson distributed data (Groom et al., 2000)χ2 =26.0∑i=18.52 [N(i)−No(i) +No(i) ln (No(i)/N(i))] ,(3)whereNo(i) is the total number of observed events in theith bin. In our fitting procedure we have three parameters:Figure 3: The most probable values for the mass of thedecaying ultra heavy dark matter with their error bars and thetotalχ2 values. Note that 21 bins contain nonzero number ofevents and eqn.(2) has 3 free parameters.A,B andmX . Fig. 2 shows the measured UHECR spectrumand the best fit in the EG-MSSM scenario. The first bumpof the fit represents particles produced at high energies andaccumulated just above the GZK cutoff due to their energylosses. The bump at higher energy is a remnant ofmX . Inthe halo models there is no GZK bump, so the relatively largex part of the FF moves to the bump around5 × 1019 GeVresulting in a much smallermX than in the EG case. The ex-perimental data is far more accurately described by the GZKeffect (dominant feature of the EG fit) than by the FF itself(dominant for halo scenarios).4 ResultsTo determine the most probable value for the mass of the SPwe studied 4 scenarios. Fig. 3 contains theχ2min values andthe most probable masses with their errors for these scenar-ios.The UHECR data favors the EG-MSSM scenario. Thegoodnesses of the fits for the halo models are far worse. TheSM and MSSM cases do not differ significantly. The mostimportant message is that the masses of the best fits (EGcases) are compatible within the error bars with the MSSMgauge coupling unification GUT scale (Amaldi at al., 1991).The SP decay will also produce a huge number of pionswhich will decay into photons. Our spectrum contains 94%of pions and 6% of protons. Thisπ/p ratio is in agree-ment with (Bhattacharjee and Sigl, 2000; Bhattacharjee andSigl, 2000) which showed that for different classes of mod-3els mX <∼ 1016 GeV , which is the upper boundary of ourconfidence intervals, the generated gamma spectrum is stillconsistent with the observational constraints. We performedthe whole analysis including the pion producedγ-s in eqn.(3). The results agree with our results of Fig. 3 within error-bars.In the near future the UHECR statistics will probably beincreased by an order of magnitude (Boratav, 1996; Guerard,1999; Bertou et al. 2000). Performing our analysis for such astatistics the uncertainty ofmX was found to be reduced bytwo orders of magnitude.Since the decay time should be at least the age of the uni-verse it might happen that such SPs overclose the universe.Due to the large mass of the SPs a single decay results ina large number of UHECRs, thus a relatively small numberof SPs can describe the observations. 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Ultrahigh energy cosmic rays as a Grand Unification signal

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