Fodor, Z

Katz, SD

Ringwald, A

English

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-ph/0105336v1  31 May 2001Possible detection of relic neutrinos and determination of their mass:quantitative analysis⋆Z. Fodor1, S.D. Katz1 and A. Ringwald21Institute for Theoretical Physics, Eötvös University, Pázmány 1, H-1117 Budapest, Hungary2Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22607, Hamburg, Germany(November 5, 2018)ITP-Budapest 569 DESY 01-070We consider the possibility that a large fraction of the ultrahigh energy cosmic rays (UHECRs)are decay products of Z bosons which were produced in the scattering of ultrahigh energy cosmicneutrinos (UHECνs) on cosmological relic neutrinos (Rνs). We compare the observed UHECRspectrum with the one predicted in the above Z-burst scenario and determine the mass of theheaviest Rν as well as the necessary UHECν flux via a maximum likelihood analysis.I. INTRODUCTIONBig-bang cosmology predicts the existence of the cos-mic microwave background radiation (CMBR) and a sim-ilar background of Rνs with an average number densityof 〈nνi〉 ≈ 56 cm−3 per light neutrino species i (mνi < 1MeV). However, the Rνs have not been detected untilnow.A possibility for their detection was discussed sometime ago: the UHECν spectrum should have absorptiondips at energies ≈ Eresνi = M2Z/(2mνi) = 4.2 · 1021 eV (1eV/mνi) due to resonant annihilation with the Rνs intoZ bosons of mass MZ [1–3]. Recently it was realized thatthe same annihilation mechanism might already be visi-ble in the UHECR spectrum [4,5] at energies above thepredicted Greisen-Zatsepin-Kuzmin (GZK) cutoff [6–8]around 4·1019 eV. It was argued that the UHECRs abovethe GZK cutoff are mainly protons from Z decay.This hypothesis was discussed in several papers [9–14].We report here on our recent quantitative investigationof the Z-burst scenario, where we have determined themass of the heaviest Rν as well as the necessary UHECνflux via a maximum likelihood analysis [15].Our comparison of the Z-burst scenario with the ob-served UHECR spectrum was done in four steps. First,we determined the probability of Z production as a func-tion of the distance from Earth. Secondly, we exploitedcollider experiments to derive the energy distribution ofthe produced protons in the lab system. Thirdly, weconsidered the propagation of the protons, i. e. we deter-mined their energy losses due to pion and e+e− produc-tion through scattering on the CMBR and due to theirredshift. The last step was the comparison of the pre-dicted and observed spectra and the extraction of themass of the Rν and the necessary UHECν flux.II. Z-BURST SPECTRUMOur prediction of the contribution of protons from Z-⋆Talk to be presented at the 27th International Cosmic Ray Con-ference, Hamburg, Germany, August 7-15, 2001.bursts to the UHECR spectrum, for degenerate ν masses(mν ≈ mνi), can be summarized asj(E,mν) = I · F−1Z ·∑i∞∫0dEpR0∫0dr∞∫0dǫ (1)Fνi (Eνi , r)nνi(r)σ(ǫ)Q(Ep) (−∂P (r, Ep, E)/∂E) ,where the total time and angle integrated detector areaI and the normalization factor FZ , which is proportionalto the sum of the ν fluxes at centre-of-mass (CM) en-ergy MZ , are determined later by the comparison withthe UHECR data. E is the energy of the protons ar-riving at Earth. Further important ingredients in ourprediction (1) are: the UHECν fluxes Fνi (Eνi , r) at theresonant energy Eνi ≈ Eresν and at distance r to Earth,the number density nνi(r) of the Rνs, the Z productioncross section σ(ǫ) at CM energy ǫ =√2mν Eνi , the en-ergy distribution Q(Ep) of the produced protons withenergy Ep, and the probability P (r, Ep, E) that a protoncreated at a distance r with energy Ep arrives at Earthabove the threshold energy E.The last three building blocks, σ, Q, and P , are verywell determined. At LEP and SLC millions of Z bosonswere produced and their decays analyzed with extremehigh accuracy. We used existing published [16–19] andsome improved unpublished [20] data to determine theproton momentum distribution Q(Ep). Due to the largestatistics, the uncertainties of our analysis related to Zdecay turned out to be negligible. Similarly, the CMBRis known to a high accuracy. It plays the key role inthe determination of the probability P (r, Ep, E) [21,22],which takes into account the fact that protons of extra-galactic (EG) origin and energies above ≈ 4 ·1019 eV losea large fraction of their energies [6,7] due to pion ande+e− production through scattering on the CMBR anddue to their redshift. P (r, Ep, E), in the form as it hasbeen calculated for a wide range of parameters by two ofthe present authors [22], was an indispensible tool in ourquantitative analysis.Less accurately known in Eq. (1) are the first two ingre-dients, the flux of UHECνs, Fνi(Eνi , r), and the neutrinonumber density nνi(r).1The former was assumed to have the form Fνi(Eνi , r) =Fνi(Eνi , 0) (1 + z)α, where z is the redshift and where αcharacterizes the source evolution (see also [3,10]). Theflux at Earth, Fνi(Eνi , 0), has been determined by thefit to the UHECR data. In our analysis we went up todistances R0 (cf. (1)) corresponding to redshift z = 2(cf. [23]), and uncertainties of the expansion rate [24]were included.The neutrino number density nνi has been treated inthe following way. For distances below 100 Mpc we variedthe shape of the nνi(r) distribution between the homoge-neous case and that of mtot(r), the total mass distribu-tion obtained from peculiar velocity measurements [25].In this way we took into account that the density dis-tribution of Rνs as hot dark matter (DM) follows thetotal mass distribution; however, with less clustering. Itshould be noted that for distances below 100 Mpc thepeculiar velocity measurements [25] suggest relative over-densities of at most a factor 2 ÷ 3, depending on the gridspacing. We did not follow the unnatural assumption ofhaving a relative overdensity of 102 ÷ 104 in our neigh-bourhood, as it was assumed in earlier investigations ofthe Z-burst hypothesis [4,5,9,10,12]. Our quantitative re-sults turned out to be rather insensitive to the variationsof the overdensities within the considered range, whoseeffect is included in our final error bars. For scales largerthan 100 Mpc the Rν density was taken according to thebig-bang cosmology prediction, nνi = 56 · (1 + z)3 cm−3.III. DETERMINATION OF Mν AND UHECνFLUXWe compared the predicted spectrum (1) of pro-tons from Z-bursts with the observed UHECR spectrum(cf. Fig. 1). Our analysis included UHECR data ofAGASA [26,27], Fly’s Eye [28–30], Haverah Park [31,32],and HIRES [33]. Due to normalization difficulties we didnot use the Yakutsk [34] results.The predicted number of UHECR events in a bin wastaken asN(i) =∫ Ei+1EidE[A ·E−β + FZ · j(E,mν)], (2)where Ei is the lower bound of the ith energy bin. Thefirst term is the usual power-law behavior, which de-scribes the data well for smaller energies [26,27]. For thisterm we studied two possibilities. In the first case weassumed that the power part is produced in our galaxy.Thus no GZK effect was included for it (“halo”). In thesecond – in some sense more realistic – case we assumedthat the protons come from uniformly distributed, EGsources and suffer from the GZK cutoff (“EG”). In thiscase the simple power-law-like term is modified, by tak-ing into account the probability P (r, Ep, E), and fallsoff around 4 · 1019 eV (see Fig. 1). The second term ofthe flux in Eq. (2) corresponds to the spectrum of theFIG. 1. The available UHECR data with their error barsand the best fits from Z-bursts [15]. Note that there are noevents above 3× 1020 eV (shown by an arrow). Bins with noevents show the 1σ upper bounds on the flux. Therefore theexperimental value of the integrated flux is in the “hatched”region with 68% confidence level (“hatching” is a set of in-dividual error bars; though most of them are too large tobe depicted in full). The dotted line shows the best fit forthe “halo”-case. The bump around 4 · 1019 eV is due to theZ-burst protons, whereas the almost horizontal contribution isthe first, power-law-like term of Eq. (2). The solid line showsthe “extragalactic”-case. The first bump at 4 · 1019 eV repre-sents protons produced at high energies and accumulated justabove the GZK cutoff due to their energy losses. The bumpat 3 · 1021 eV is a remnant of the Z-burst energy. The dashedline shows the contribution of the power-law-like spectrumwith the GZK effect included. The predicted fall-off for thisterm around 4 · 1019 eV can be observed.Z-bursts, Eq. (1). A and FZ are normalization factors.Note that the following implicit assumptions have beenmade through the form of Eq. (2): i) We have assumed(and later checked) that the UHE photons from Z-burstscan be neglected. ii) We have assumed that there areno significant additional primary UHE proton fluxes be-yond the extrapolation of the above power-law. This con-straint will be relaxed in a future publication [35].The expectation value for the number of events in abin is given by Eq. (2). To determine the most probablevalue for mν we used the maximum likelihood methodand minimized [36] the χ2(β,A, FZ ,mν),χ2 =26.0∑i=18.52 [N(i)−No(i) +No(i) ln (No(i)/N(i))] , (3)where No(i) is the total number of observed events in theith bin. As usual, we divided each logarithmic unit intoten bins. Since the Z-burst scenario results in a quitesmall flux for lower energies, the “ankle” is used as alower end for the UHECR spectrum: log(Emin/eV) =18.5. Our results are insensitive to the definition of the2upper end (the flux is extremely small there) for whichwe choose log(Emax/eV) = 26. The uncertainties of themeasured energies are about 30% which is one bin. Usinga Monte-Carlo method we included this uncertainty inthe final error estimates.In our fitting procedure we had four parameters:β,A, FZ and mν . The minimum of the χ2(β,A, FZ ,mν)function is χ2min at mν min which is the most proba-ble value for the mass, whereas χ2(β′, A′, F ′Z ,mν) ≡χ2o(mν) = χ2min + 1 gives the 1σ (68%) confidence in-terval for mν . Here β′, A′, F ′Z are defined in such a waythat the χ2(β,A, FZ ,mν) function is minimized in β,Aand FZ at fixed mν .Qualitatively, our analysis can be understood in thefollowing way. In the Z-burst scenario a small Rν massneeds large Eresν in order to produce a Z. Large Eresν re-sults in a large Lorentz boost, thus large Ep. In thisway the shape of the detected energy (E) spectrum de-termines the mass of the Rν. The sum of the necessaryUHECν fluxes was then determined from the obtainednormalization FZ .Our best fits to the observed data can be seen in Fig. 1,for evolution parameter α = 1. We found a neutrino massof 2.34+1.29(3.74)−0.84(1.66) eV for the “halo”- and 0.26+0.20(0.50)−0.14(0.22) eVfor the “EG”-case, respectively. The first numbers arethe 1σ, the numbers in the brackets are the 2σ errors.This gives an absolute lower bound on the mass of theheaviest ν of 0.06 eV at the 95% CL. The fits are rathergood; for 21 non-vanishing bins and 4 fitted parametersthey can be as low as χ2 = 15.1. We determined mν fora wide range of cosmological source evolution (α = 0÷3)and Hubble parameter (H0 = 0.64 ÷ 0.78 km/sec/Mpc)and observed only a moderate dependence on them. Theresults remain within the above error bars. We performeda Monte-Carlo analysis studying higher statistics. In thenear future, Auger [37,38] will provide a ten times higherstatistics, which reduces the error bars in the neutrinomass to ≈ one third of their present values.The necessary UHECν flux at Eresν has been obtainedfrom our fit values of the normalization FZ . We havesummarized them in Fig. 2, together with some exist-ing upper limits and projected sensitivities of present,near future and future observational projects. It is ap-parent that the flux determination depends much moreon the evolution uncertainties than the mass determi-nation. The necessary ν flux appears to be well belowpresent upper limits and is within the expected sensitiv-ity of AMANDA, Auger, and OWL.IV. COMPARISON WITH △M2νFROM νOSCILLATIONSOne of the most attractive patterns for ν masses issimilar to the one of the charged leptons or quarks: themasses are hierarchical, thus the mass difference betweenthe families is approximately the mass of the heavier par-FIG. 2. Differential neutrino fluxes required by theZ-burst hypothesis for the “halo” and the “extragalactic”case [15]. The evolution parameter α takes values 0,1,2,3 fromtop to bottom for both cases, respectively. The horizontal er-rors indicate the 1σ uncertainty of the mass determinationand the vertical errors include also the uncertainty of theHubble expansion rate. Also shown are upper limits fromFly’s Eye [39] and the Goldstone lunar ultrahigh energy neu-trino experiment GLUE [40], as well as projected sensitivitiesof AMANDA [41], Auger [10,42] and OWL [10,43].ticle. Using the mass difference of the atmospheric ν os-cillation for the heaviest mass [24], one obtains valuesbetween 0.03 and 0.09 eV. It is an intriguing feature ofour result that the smaller one of the predicted masses iscompatible on the ≈ 1.3σ level with this scenario.Another popular possibility is to have 4 neutrino types.Two of them – electron and sterile neutrinos – are sepa-rated by the solar ν oscillation solution, the other two –muon and tau – by the atmospheric ν oscillation solution,whereas the mass difference between the two groups is ofthe order of 1 eV. We studied this possibility, too. On ourmass scales and resolution the electron and sterile neu-trinos are practically degenerate with mass m1 and themuon and tau neutrinos are also degenerate with massm2. The best fit and the 1 σ region in the m1−m2 planeis shown in Fig. 3 for the “EG”-case. The dependenceof this result on the cosmological evolution and on theUHECν spectrum will be discussed elsewhere [35]. Sincethis two-mass scenario has much less constraints the al-lowed region for the masses is larger than in the one-massscenario.V. DIFFERENCES WITH RESPECT TOYOSHIDA ET AL. (1998)Numerical simulations of Z-burst cascades for mν ∼ 1eV, taking into account all known EG propagation ef-fects, were performed by Yoshida et al. [10]. Based on3FIG. 3. The best fit and the 1σ (68% confidence level)region in a scenario with two non-degenerate ν masses [15].case studies, relative overdensities of Rνs ranging from20÷ 103 on a scale of 5 Mpc were argued to be necessaryin order to get a successful description of the UHECRevents and rate above the GZK cutoff without violat-ing lower energy photon flux limits and without invokinginconceivable UHECν fluxes. For such large overdensi-ties, most of the UHECRs from Z-bursts originate nearbyand their attenuation to the Earth can be neglected. Inour case, with realistic overdensities ≤ 2 ÷ 3 on scales≤ 100 Mpc, most of the UHECRs from Z-bursts origi-nate from cosmological distances. Therefore, despite ofthe fact that by construction the overall rate of UHECRsfrom Z-bursts observed at Earth is the same in both in-vestigations, the predicted spectra are quite different. Nolarge overdensity is needed to reproduce the data. Notethat the EG scenario is dominated not by the nearby Z-burst but by the pile-up of Z-burst protons due to theGZK effect (cf. Fig. 1).VI. CONCLUSIONSWe reported on a comparison of the predicted spec-trum of the Z-burst hypothesis with the observed UHE-CR spectrum [15]. The mass of the heaviest Rν turnedout to be mν = 2.34+1.29−0.84 eV for halo and 0.26+0.20−0.14 eV forEG scenarios. The second mass, with a lower bound of0.06 eV on the 95% CL, is compatible with a hierarchicalν mass scenario with the largest mass suggested by theatmospheric ν oscillation. The above ν masses are in therange which can be explored by future laboratory exper-iments like the β decay endpoint spectrum and the ν lessββ decay [13,44]. They compare also favourably with thetau(?) neutrino mass range 0.04÷ 4.4 eV found recentlyfrom a detailed analysis of the latest CMBR measure-ments [45]. We analysed a possible two-mass scenarioand gave the corresponding confidence level region. Thenecessary UHECν flux was found to be consistent withpresent upper limits and detectable in the near future.ACKNOWLEDGMENTSWe thank S. Barwick, O. Biebel, S. Bludman, W.Buchmüller, K. Mannheim, H. Meyer, W. Ochs, and K.Petrovay for useful discussions. We thank the OPAL col-laboration for their unpublished results on hadronic Zdecays. This work was partially supported by HungarianScience Foundation grants No. OTKA-T34980/T29803/-T22929/M28413/OM-MU-708.[1] Weiler, T., Phys. Rev. Lett. 49, 234, 1982.[2] Roulet, E., Phys. Rev. D 47, 5247, 1993.[3] Yoshida, S. et al., Astrophys. J. 479, 547, 1997.[4] Fargion, D., Mele, B. and Salis, A., Astrophys. J. 517,725, 1999.[5] Weiler, T.J., Astropart. Phys. 11, 303, 1999; 12, 379,2000 (Err.).[6] Greisen, K., Phys. Rev. Lett. 16, 748, 1966.[7] Zatsepin, G.T. and Kuzmin, V.A., Pisma Zh. Eksp. Teor.Fiz. 4, 114, 1966.[8] Bhattacharjee, P. and Sigl, G., Phys. Rept. 327, 109,2000.[9] Waxman, E., astro-ph/9804023[10] Yoshida, S., Sigl, G., and Lee, S., Phys. Rev. Lett. 81,5505, 1998.[11] Gelmini, G. and Kusenko, A., Phys. Rev. Lett. 82, 5202,1999.[12] Blanco-Pillado, J.J., Vázquez, R.A., and Zas, E., Phys.Rev. D 61, 123003, 2000.[13] Päs, H. and Weiler, T.J., hep-ph/0101091[14] Fargion, D. et al., astro-ph/0102426[15] Fodor, Z., Katz, S.D., and Ringwald, A., hep-ph/0105064[16] Akers, R. et al., Z. Phys. C 63, 181, 1994.[17] Abreu, P. et al., Nucl. Phys. B 444, 3, 1995.[18] Buskulic, D. et al., Z. Phys. C 66, 355, 1995.[19] Abe, K. et al., hep-ex/9908033[20] Abe, K. et al., OPAL PN299, unpublished.[21] Bahcall, J.N. and Waxman, E., Astrophys. J. 542, 543,2000.[22] Fodor, Z. and Katz, S.D., Phys. Rev. D 63, 023002, 2001.[23] Waxman, E., Astrophys. J. 452, L1, 1995.[24] Groom, D.E. et al., Eur. Phys. J. C 15, 1, 2000.[25] Dekel, A. et al., Astrophys. J. 522, 1, 1999.[26] Takeda, M. et al., Phys. Rev. Lett. 81, 1163, 1998.[27] Takeda, M. et al., astro-ph/9902239[28] Bird, D.J. et al., Phys. Rev. Lett. 71, 3401, 1993.[29] Bird, D.J. et al., Astrophys. J. 424, 491, 1994.[30] Bird, D.J. et al., Astrophys. J. 441, 144, 1995.[31] Lawrence, M.A., Reid, R.J.O., and Watson, A.A., J.Phys. G 17, 773, 1991.4[32] Ave, M. et al., Phys. Rev. Lett. 85, 2244, 2000.[33] Kieda, D. et al., in Proceedings of the 26th InternationalCosmic Ray Conference, Salt Lake, 1999.[34] Efimov, N.N. et al., in Proc. of the Astrophysical Aspectsof the Most Energetic Cosmic Rays (World Scientific, Sin-gapore, 1991).[35] Fodor, Z., Katz, S.D., and Ringwald, A., in preparation.[36] Fodor, Z. and Katz, S.D. Phys. Rev. Lett. 86, 3224, 2001.[37] Guerard, C.K., Nucl. Phys. Proc. Suppl. 75A, 380, 1999.[38] Bertou, X., Boratav, M., and Letessier-Selvon, A., Int. J.Mod. Phys. A15, 2181, 2000.[39] Baltrusaitis, R.M. et al., Phys. Rev. D 31, 2192, 1985.[40] Gorham, P.W. et al., astro-ph/0102435[41] Barwick, S., www.ps.uci.edu/˜amanda[42] Capelle, K.S. et al., Astropart. Phys. 8, 321, 1998.[43] Ormes, J.F. et al., in Proc. of the 25th International Cos-mic Ray Conference, Potchefstroom, 1997.[44] Farzan, Y., Peres, O.L.G., and Smirnov, A.Yu., hep-ph/0105105[45] Wang, X., Tegmark, M., and Zaldarriaga, M., astro-ph/01050915

Possible detection of relic neutrinos and determination of their mass

https://edit.elte.hu/xmlui/bitstream/handle/10831/49441/Possible%20detection%20of%20relic%20neutrinos%20and%20determination%20of%20their%20mass.pdf?sequence=1&amp;isAllowed=y

Possible detection of relic neutrinos and determination of their mass

Abstract

Similar works

Full text

Available Versions

ELTE Digital Institutional Repository (EDIT)