Arteaga-Velazquez, J C

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29th International Cosmic Ray Conference Pune (2005) 9, 155158Influence of the regeneration by neutral currents in the observable fluxof upward-going muons induced by astrophysical neutrinosJ. C. Arteaga-Vela´zquez and A. ZepedaDepartment of Physics, Cinvestav, Apartado Postal 14-740, 07000, Mexico, DF, Mexico.Presenter: A. Zepeda (zepeda@fis.cinvestav.mx), mex-zepeda-A-abs3-he22-posterThe telescopes of astrophysical high-energy neutrinos based on the Cherenkov technique are very differentfrom the conventional telescopes in that the first ones look down, to the interior of the Earth. These Cherenkovdetectors are designed to observe the upward-going muons produced by muon neutrinos that traverse throughour planet, when they interact weakly with matter around the detector. In order to interpret the data fromupward-going muons it is important to understand how the neutrino flux is modified when it travels throughthe Earth. Besides absorption, the neutrino flux can be altered by other mechanisms, for example, regenerationby means of neutral currents. In this work we explore the effect of the regeneration through neutral currentsin the propagation of neutrinos and its influence on the observable flux of upward-going muons for differentastrophysical and exotic models of high-energy neutrinos.1. IntroductionThe detection of extraterrestrial high-energy neutrinos will open a new window to the farthest and most ener-getic phenomena in the universe. So far, these elusive particles have escaped from observation but it is thoughtthat with the next generation of 1 km Cherenkov detectors, like Icecube, the first detection of very high-energyneutrinos from outer space could be achieved [1]. The neutrino telescopes based on the Cherenkov techniquelook in the direction of the Earth and use our planet as a filter for neutrinos and as an extension of the detector[2]. They are designed to observe the upward-going muons produced by muon neutrinos in charged currentinteractions with the rock under the detector [3].In its transit through the Earth, the neutrino flux is modified mainly by absorption due to interactions,whereis a nucleon from the medium. However, there are secondary mechanisms, such as regeneration bymeans of neutral currents, that alter the shape of the neutrino flux and hence the rate of detectable upward-goingmuons. Therefore, those processes must be studied in order to understand properly the data from Cherenkovdetectors. In this article, we will be concerned with the role of the regeneration by neutral currents in thepropagation of muon neutrinos through the Earth and with its contribution to the detectable flux of upward-going muons from different sources.2. Regeneration by means of neutral currentsAt high energies, neutrinos which undergo charged current (CC) interactions are removed from the initial flux,but those which interact through neutral currents (NC) only degrade their energy, that is, present regeneration[4]. This phenomenon is possible because at high energies the final lepton tends to conserve the direction ofthe initial neutrino [5], and because in each collision the neutrino, in average, loses a small fraction  ofits initial energy (For 	 GeV, ﬁﬀﬂ 0.3  0.2 [6]).Neglecting  ﬃ interactions, the transport equation that applies to a muon neutrino flux that traverse throughthe Earth (in absence of  oscillations and muon decay) is [4] !#"%$&('*) '+!("%$ﬁ&#'*)-,/.1032#465$7)98:<;>=@?#AB;B!("C$ED&('*)-, .5F65$D&) G EDﬁH(1)156 J.C. Arteaga-Vel·azquez et al10-1910-1810-1710-1610-1510-1410-1310-1210-1110-1010-910-810-7103 104 105 106 107 108 109 1010E ν dΦν µ+ν¯ µ/dE ν (cm-2  s-1  sr-1 )Eν (GeV)Flux E-1Limit WBAtmosphericAGNGZKTD101101102103104103 104 105 106 107 108 109 1010Ψν+ν¯(Eν)Eν (GeV)θ = 0o40o80o0Limit WB 0 0.1 0.2 0.3 0.4103 104 105 106 107 108 109 1010Z ν+ν¯(Eν, θ)Eν (GeV)θ = 0o40o80o(c)(b)(a)Figure 1. (a) Astrophysical and exotic uxes of high-energy muon neutrinos. See text for description. In (b) and (c) wecan see the regeneration and the IKJ factor, respectively, for the Waxman-Bahcall bound as a function of the energy LJ forthree different nadir angles.A similar equation applies for muon antineutrinos. In the above equation, M is the nadir angle of observation,'9$ M ) ; the column depth of matter found by a neutrino along the M direction after traversing a given distanceinside the Earth;-,, the Avogadro’s number;.032#465, the total  cross section; .5FN5$D&7)(O  , thedifferential one for the corresponding neutral current process, and!#"%$&('*) represents the flux of muonneutrinos at '9$ M ) . The first term on the right-hand side of relation (1) is associated with absorption and thesecond one, with regeneration. The effect of absorption and regeneration in!"%$	&#'*) can be factorized usingthe following expression [7]:!("%$ﬁ&#'9$M)#)!"$)#ﬃ7P3QRTSVUXWVY*Z\[^]B(_R;BU#`a$&('9$M)()b& (2)where!("$) is the initial flux that arrives to the Earth surface and c032#465$7)dOe,K.032#465$7) , the to-tal  interaction length of a muon neutrino with energy   . The effect of the regeneration is put inside`a$&('*) , which has become known as the regeneration factor and satisfies `a$  &#'*)f  . The factor `a$  &#'*)is ruled by the following integro-differential equation [7]: `a$G&#'*) 'e,:;@=>?#AB;B ED .5FN5$D&	) !("$D)!("$7)ﬃ7P5/g%h iZ[^]B(_R;>jBUkPiZ[^]BV_R;BUmlnQ7`a$ED&#'*)6& (3)with the initial condition `a$  &  ) o . We solved numerically this equation for apq6ros GeV anddifferent initial neutrino fluxes in order to find!#"%$&#'*) in each case.Here, we considered the models displayed in graph 1a. In that figure, three different spectra of the form P3tare shown, with spectral index uvw [8], x (Waxman-Bahcall limit [9]) and 3.6 (angle-averaged atmosphericneutrinos [6, 10], named Atmospheric in the figure). We also present the Mannheim’s model for the AGNdiffuse flux of muon neutrinos [11], that from Engel et al. for the spectra of GZK muon neutrinos [12] and aSigl’s model associated with topological defects (TD) [13].On the other hand, the rate of detectable upward-going muons above a given threshold, ypzt, in a Cherenkovdetector of effective area {| km } was estimated through the relation [6]~$pzt)x9{: ﬂ6M:<;>=>?(AB;=9" !#"%$&#'*$M)#)Ł-,@.F@F65$7)1$&pzt)ﬀH(4)Influence of the regeneration by neutral currents . . . 157 0 0.2 0.4 0.6 0.8 1 1.2103 104 105 106 107 108 109 1010Φν µ+ν¯ µ(Eν, θ)/Φ0 νµ+ν¯ µ(Eν)Eν (GeV)θ = 0o40o80oInitialLimit WB Abs.Abs. + Reg.10-210-1100101102103104105103 104 105 106 107 108Nµ- + µ+ [year-1  in 2pi sr]Eν (GeV)Flux E-1Limit WBAtmosphericAGNGZKTD(a) (b)Figure 2. (a) Final ux of neutrinos, after traversing the Earth, compared with its corresponding initial ux for various nadirangles. The results with ŁJ>J equal to and different from one are shown in the graph. In the rst case only absorption isinvolved (Abs.) and, in the second one, regeneration is included (Abs. + Reg.). (b) Annual rate of detectable upward-goingmuons and antimuons in a 1 km  Cherenkov detector from several astrophysical and exotic sources of high-energy muonneutrinos.In this expression,  1$  &  pzt)ﬀ represents the average range of a muon in standard rock and '\C$ M ) , thetotal column depth of matter along the M direction inside the Earth. The cross section was calculated atleading order in the framework of the Standard Model using the quark-parton model and consideringasan isoscalar nucleon. We also employed the CTEQ6-L1 [14] and Double Logarithmic Approximation [15] todescribe the parton distributions of the nucleon. The interior of the Earth was described with the PreliminaryEarth Model, which we took from [6]. Finally, in order to evaluate the average range of a muon in standardrock we used MUM v1.4 [16] and studied the propagation of  muons in this medium.3. Results and discussionsIn order to illustrate our results, we present in figure 1b the regeneration factor for the Waxman-Bahcall boundas a function of M and 	 . The regeneration factor, in general, increases with the angle M and, therefore, withthe total column depth '  $ M ) . Moreover, it exhibits a maximum at high energies, in particular, near pq6r. Thevalue of `a$  &('*) depends on the specific model, too. In the case of the power law spectra, it grows up as thespectral index u decreases, behavior that was already pointed out by Nicolaidis et al. [4].From graph 1b, it can be observed that the value of the regeneration factor is bigger for large column depths andvery high energies. But, before any conclusion about the importance of the regeneration in the flux of neutrinosthrough the Earth can be drawn it must be remembered that the contribution of the regeneration to the final fluxcompete with the effect of attenuation. For all the models that were analysed, the attenuation always dominatesover the regeneration effects, as can be seen, for instance, in figure 2a, since!#"%$&('*$M)#)VO!("$7) , except for the GZK model where!#"%$&#'*$M)#)VO!"$7)vf at low energies. Then, it results thatthe effective attenuation length  $ G &('*) defined through!"3$G&('*)!("$G)6>'9$M)VO$G&('*)k in[17], in order to describe the combined influence of absorption plus regeneration in the neutrino flux, becomesnegative and takes an infinity value when!#"%$&('7)!("$) .The regeneration factor can be also be expressed in terms of the  factor introduced by Naumov and Perronein reference [17] in the following way: `  $ G & M )  exp  ﬂ $ 	 & M )'CO c032#465$	)k . This ﬂ $ G & M ) factor hasinformation about the competition between regeneration and absorption in the neutrino flux when traveling158 J.C. Arteaga-Vel·azquez et alacross the Earth. In graph 1c, the  factor of the Waxman-Bahcall limit is shown for three different nadirangles. We can observe that the value of the  factor is less than one. This implies that absorption is moreimportant than regeneration. In addition, we can see that the  factor increases when the observation anglebecomes closer to the horizontal direction. These characteristics of the  factor were also observed for theother models, but the first one for the GZK flux. In this case, at low energies it results that  f  , whichmeans that the regeneration effects are more important than those from attenuation.Now, the rates of detectable events with absorption and regeneration are shown in figure 2b as a functionof the threshold energy at the detector. According to our evaluations, the contribution from regeneration tothe ~ﬁ  ¡$  pzt) flux is less than 16 %. The biggest contributions are observed in the case of the Pspectrum, the GZK model and the Sigl’s flux from topological defects. We also see that, from all the P3tspectra, the enhancement of the~   ¡$ epzt) rates due to regeneration effects is smaller for those fluxeswith higher spectral indexes, n.4. ConclusionsThe regeneration by means of neutral currents recycles neutrinos which undergo a neutral current  inte-raction during their propagation through the Earth. This effect is flux dependent. An increase on the amountof matter traversed by the neutrinos results in a bigger regeneration factor, although, it is not enough to com-pensate the effect of the attenuation. Only at low energies the first one could balance or overcome the decreaseof absorption specifically for fluxes which behave like the GZK spectra. The final enhancement of the numberof detectable upward-going muons due to regeneration is modest, less than 16 % for the analysed models, butit should be incorporated for a detailed description of the problem.5. Acknowledgements. We acknowledge partial support from CONACyT.References[1] F. Halzen, electronic archive: astro-ph/0301143.[2] T. K. Gaisser, F. Halzen y T. Stanev, Phys. Rep. 258, 173 (1995).[3] J. Ahrens et al., Phys. Rev. Lett. 90, 251101 (2003); Phys. Rev. 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Influence of the regeneration by neutral currents in the observable flux of upward-going muons induced by astrophysical neutrinos

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