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Energy spectrum of tau leptons induced by the high energy Earth-skimming neutrinos

Jie-Jun Tseng,1,*Tsung-Wen Yeh,2,†H. Athar,2,3,‡M. A. Huang,4,§Fei-Fain Lee,2,储and Guey-Lin Lin2,¶

1

Institute of Physics, Academia Sinica, Taipei 115, Taiwan

2Institute of Physics, National Chiao-Tung University, Hsinchu 300, Taiwan 3Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan

4Department of Physics, National Taiwan University, Taipei 10617, Taiwan 共Received 28 May 2003; published 26 September 2003兲

We present a semianalytic calculation of the tau-lepton flux emerging from the Earth induced by incident high energy neutrinos interacting inside the Earth for 105⭐E␯/GeV⭐1010. We obtain results for the energy dependence of the tau-lepton flux coming from the Earth-skimming neutrinos, because of the neutrino-nucleon charged-current scattering as well as the resonant␯¯ee⫺scattering. We illustrate our results for several antici-pated high energy astrophysical neutrino sources such as the active galactic nuclei, the gamma-ray bursts, and the Greisen-Zatsepin-Kuzmin neutrino fluxes. The tau-lepton fluxes resulting from rock-skimming and ocean-skimming neutrinos are compared. Such comparisons can render useful information about the spectral indices of incident neutrino fluxes.

DOI: 10.1103/PhysRevD.68.063003 PACS number共s兲: 95.85.Ry, 14.60.Fg, 14.60.Pq, 95.55.Vj

I. INTRODUCTION

The detection of high energy neutrinos (E⬎105 GeV) is crucial to identify the extreme energy sources in the Uni-verse and possibly to unveil the puzzle of cosmic rays with energy above the Greisen-Zatsepin-Kuzmin 共GZK兲 cutoff

关1兴. These proposed scientific aims are well beyond the scope

of conventional high energy gamma-ray astronomy. Because of the expected small flux of the high energy neutrinos, large scale detectors (⭓1 km2) seem to be needed to obtain the first evidence.

There are two different strategies to detect the footprints of high energy neutrinos. The first strategy is implemented by installing detectors in a large volume of ice or water where most of the scatterings between the candidate neutri-nos and nucleons occur essentially inside the detector, whereas the second strategy aims at detecting the air showers caused by the charged leptons produced by the neutrino-nucleon scatterings taking place inside the Earth or in the air, far away from the instrumented volume of the detector. The latter strategy thus includes the possibility of detection of quasihorizontal incident neutrinos, which are also referred to as Earth-skimming neutrinos. These neutrinos are considered to interact below the horizon of an Earth based surface de-tector.

The second strategy has been proposed only recently关2兴. The Pierre Auger observatory group has simulated the antici-pated detection of the air showers from the decays of ␶ lep-tons 关3兴. The tau air shower event rates resulting from the Earth-skimming tau neutrinos for different high energy

neu-trino telescopes are given in关4兴. A Monte Carlo study of the tau air shower event rate was also reported not long ago关5兴. We note that Ref.关4兴 does not consider the tau-lepton energy distribution in the ␯-nucleon scattering, and only the inci-dent tau neutrinos with energies greater than 108 GeV are considered. For Ref.关5兴, we note that only the sum of tau air shower event rates arising from different directions is given. Hence some of the events may be due to tau-leptons or neu-trinos traversing a large distance. As a result, it is not pos-sible to identify the source of the tau-neutrino flux even with the observation of the tau-lepton induced air shower.

In this work, we shall focus on high energy Earth-skimming neutrinos and shall calculate the energy spectrum of their induced tau leptons, taking into account the inelas-ticity of neutrino-nucleon scatterings and the tau-lepton en-ergy loss in detail. Our work differs from Ref. 关5兴 in our emphasis on the Earth-skimming neutrinos. We shall present our results in the form of outgoing tau-lepton spectra for different distances inside the rock, instead of integrating the energy spectra. As will be demonstrated, such spectra are insensitive to the distances traversed by the Earth-skimming

␯␶ and␶. They are essentially determined by the tau-lepton

range. Because of this characteristic feature, our results are useful for setting up simulations with specifically chosen air shower content detection strategy, such as detection of the Cherenkov radiation or the air fluorescence. Our results are also beneficial for the coherent Cherenkov radio emission measurement detectors such as the Radio Ice Cherenkov Ex-periment 共RICE兲 关6兴 and the upcoming Antarctic Impulsive Transient Array 共ANITA兲 关7兴.

We start with our semianalytic description in Sec. II. The transport equations governing the evolutions of neutrino and tau-lepton fluxes will be derived. Using these, we then cal-culate the tau-lepton flux resulting from the resonant¯ee

→W→␯¯scattering. In Sec. III, we summarize our main results, namely, the tau-lepton energy spectra due to neutrino-nucleon scatterings. The implications of our results will be discussed here also. In particular, we shall point out *Email address: gen@phys.sinica.edu.tw

Email address: twyeh@cc.nctu.edu.twEmail address: athar@phys.cts.nthu.edu.tw

§Present address: National United University, Miaoli 360, Taiwan.

Email address: huangmh@phys.ntu.edu.tw

Email address: u8727515@cc.nctu.edu.tw

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that the ratio of tau-lepton flux induced by rock-skimming neutrinos to that induced by ocean-skimming neutrinos is sensitive to the spectral index of the incident tau-neutrino flux. In Sec. IV, we discuss some prospects for possible fu-ture observations of the associated radiation from these tau leptons.

II. TAU-LEPTON ENERGY SPECTRUM

Let us begin with the transport equations for tau neutrinos and tau leptons. Considering only the neutrino-nucleon scat-terings, we have ⳵F共E,X兲X ⫽⫺ F共E,X兲 ␭␯共E兲 ⫹nNi

⫽1 3

ymini ymaxi d y 1⫺y ⫻Fi共Ey,Xdi d y 共y,Ey兲 共1兲 and ⳵F共E,X兲X ⫽⫺ F共E,X兲CC共E兲F共E,X兲d共E兲

⫹⳵兵关␣共E兲⫹共E兲E兴FE共E,X兲⫹nN

ymin ymax d y 1⫺yF␯␶共Ey,XdN→␶Y d y 共y,Ey兲, 共2兲

where nNis the number of target nucleons per unit medium mass, and␳ is the mass density of the medium. The␴1,2,3are defined as ␴(␯⫹N→⫹Y), ⌫(␶→␯⫹Y)/cnN, and

␴(␶⫹N→⫹Y), respectively. The quantity X represents the slant depth traversed by the particles, i.e., the amount of medium per unit area traversed by the particle 共and thus in units of g/cm2).␭, d, and␭CCrepresent the␯interaction thickness, the tau-lepton decay length, and the tau-lepton charged-current interaction thickness, respectively, with, say,

⫺1⫽nN␯N and d⫽c␶␶E/m. Ey is equal to E/(1⫺y), where y is the inelasticity of neutrino-nucleon scatterings, such that the initial- and final-state particle energies in the differential cross sections di(y ,Ey)/d y and d

N→␶Y(y ,Ey)/d y are E/(1⫺y) and E, respectively. The limits for y, ymin

i

, and ymax

i

depend on the kinematics of each process. Finally, the energy-loss coefficients␣(E) and(E) are defined by ⫺dE/dX⫽(E)⫹␤(E)E with E being the tau-lepton energy. An equation similar to Eq.共2兲 in the con-text of atmospheric muons was found in Ref.关8兴.

As mentioned before, Eqs. 共1兲 and 共2兲 take into account only neutrino-nucleon scatterings. It is of interest to calculate the tau-lepton fluxes produced by the Glashow resonance

关9,10兴, namely, via ¯ee→W→␯¯, also. The transport equation for¯ethen reads

F␯¯ e共E,X兲X ⫽⫺ F␯¯ e共E,X兲␯¯e共E兲 ⫹nN

ymin ymax dy 1⫺y ⫻F␯¯e共Ey,Xd␯¯ eN→␯¯eY d y 共y,Ey兲. 共3兲 Similarly, the corresponding equation for the tau-lepton flux is given by ⳵F共E,X兲X ⫽⫺ F共E,X兲CC共E兲F共E,X兲d共E兲 ⫹ne

ymin ymax d y 1⫺yF␯¯e共Ey,Xd␯¯ ee→␯¯␶␶⫺ d y 共y,Ey兲, 共4兲

where ne is the number of target electrons per unit medium mass.

Before solving the above coupled transport equations, it is essential to know the energy-loss coefficients ␣(E) and

(E). As pointed out before关11兴, the coefficient␣(E) is due to the energy loss by ionization关12兴, while␤(E) is contrib-uted by the bremsstrahlung 关13兴, the ee⫺ pair production

关14兴, and the photonuclear processes 关11,15兴. It is understood

that the contribution by ␣(E) becomes unimportant for E

⭓105 GeV. The coefficient (E) can be parametrized as

(E)⫽关1.6⫹6(E/109 GeV)0.2兴⫻10⫺7 g⫺1cm2 in standard rock for 105⭐E/GeV⭐1012.

It is of interest to check the tau-lepton range given by our semianalytic approach. To do this, we rewrite Eq. 共2兲 by dropping the neutrino term, i.e.,

F共E,X兲X ⫽⫺ F共E,X兲CC共E兲F共E,X兲d共E兲 ⫹ ⳵关␥共E兲F共E,X兲兴E , 共5兲

with ␥(E)⬅␣(E)⫹␤(E)E. One can easily solve it for F(E,X): F共E,X兲⫽F共E¯,0兲exp

0 X dT

共E¯兲⫺ 1 ␳d共E¯兲 ⫺ 1 ␭CC共E¯兲

, 共6兲 where E¯⬅E¯(X;E) with dE¯/dX⫽(E¯ ) and E¯ (0;E)⫽E. To calculate the tau-lepton range, we substitute F(E,0)⫽␦(E

⫺E0). The survival probability P(E0,X) for a tau lepton with an initial energy E0 at X⫽0 is

P共E0,X兲⫽ ␥共E˜0兲 ␥共E0兲 exp

0 X dT

共E˜0兲⫺ 1 ␳d共E˜0兲 ⫺ 1 ␭CC共E˜ 0兲

, 共7兲

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where 0⬅E˜(X;E0) with dE˜0/dX⫽⫺␥(E˜0) and

E ˜

0(0;E0)⫽E0. The tau-lepton range is simply

R共E0兲⫽

0

dX P共E0,X兲. 共8兲

For E0⫽109 GeV, we find that R␶⫽10.8 km in standard rock (Z⫽11, A⫽22) while R⫽5.0 km in iron. Both values are in good agreement with those obtained by Monte Carlo calculations关11兴. To compare the tau-lepton ranges, we have followed the convention in Ref. 关11兴 by requiring the final tau-lepton energy E˜ (X;E0) to be greater than 50 GeV.

It is to be noted that we obtain Rby using the continuous tau-lepton energy-loss approach, rather than the stochastic approach adopted in Ref. 关11兴. In the muon case, the con-tinuous approach to the muon energy loss is known to over-estimate the muon range 关16兴. Such an overestimate is not significant in the tau-lepton case, because of the decay term in Eq. 共7兲. In fact, tau-lepton decay term dictates the tau range in the rock until E⭓107 GeV. Even for E

⬎107 GeV, the tau-lepton range is still not entirely deter-mined by the tau-lepton energy loss. Hence different treat-ments on the tau-lepton energy loss do not lead to large differences in the tau-lepton range, in contrast to the case for the muon range. Our results for the tau-lepton range up to 1012 GeV are plotted in Fig. 1. This is an extension of the result in Ref. 关11兴, where the tau-lepton range is calculated only up to 109 GeV. Our extension is seen explicitly in the addition of a charged-current scattering term on the right-hand side 共RHS兲 of Eq. 共5兲. This term is necessary because 1/␭CC becomes comparable to 1/␳d in rock for E

⭓1010GeV; whereas one does not need to include the con-tribution by the tau-lepton neutral-current scattering, since such a contribution cannot compete with the last term in Eq.

共5兲 until E⭓1016GeV 关11兴. We remark that our extended results for Rare subject to the uncertainties of the neutrino-nucleon scattering cross section at high energies. We use the CTEQ6 parton distribution functions 关17兴 in this work, and at the high energy 共the small x region, namely, for x

⬍10⫺6), we fit these parton distribution functions into the

form proportional to x⫺1.3as a guide.

Having checked the tau-lepton range, we now proceed to calculate the tau-lepton flux. It is instructive to begin with the simple case: the ¯ee⫺ resonant scattering. It is well known that 关9,10兴 ␴共¯ee→W→␯¯兲⫽GF 2 mW4 3␲ • s 共s⫺mW 22⫹m W 2 W 2 , 共9兲

with s⫽2meE¯e and 1/␴•d/dz⫽3(1⫺z)2, where z

⫽E/E␯¯e. We shall focus on only those␯

¯e’s for which E ␯¯e

satisfies the resonance condition, i.e., E␯¯

e⬇ER⬅mW

2

/2me. It is clear from Eq. 共4兲 that F(E,X) depends only on F␯¯

e(ER,X), because of the narrow peak nature of

¯ee scat-tering cross section. One also expects that F(E,X) is sig-nificant only for E around the resonance energy ER. In this energy region, one may neglect the first term on the RHS of Eq. 共4兲 in comparison with the second term. In the narrow width approximation, the last term in Eq. 共4兲 can be recast into 1

3(1⫺E/ER)2(␲⌫W/LRmW)F␯¯e(ER,X), whereW is the width of the W boson while LR is the interaction thick-ness of the resonant¯ee→W⫺ scattering共see Appendix A for details兲. The tau-lepton flux can be readily obtained once F␯¯

e(ER,X) is given. We observe that the regeneration term

in Eq.共3兲 共second term on the RHS兲 can be neglected as it is necessarily off the W boson peak. Hence, we easily obtain F␯¯

e(ER,X)⫽exp(⫺X/LR)F␯¯e(ER,0). Substituting this

expres-sion into Eq.共4兲, we obtain F共E,X兲 F␯¯ e共ER,0兲 ⫽3.3⫻10⫺4

E ER

1⫺ E ER

2 ⫻exp

X LR

共10兲

in the limit XⰇ␳d. The prefactor 3.3⫻10⫺4is obtained by assuming a standard-rock medium. In water it becomes 1.4

⫻10⫺4. It is to be noted that E⬍E

R in the above equation. We shall see later that the contribution to F(E,X) by the W resonance is negligible compared to that by the␯-N scatter-ing.

Let us now turn to the case of tau-lepton production by

␯␶-N charged-current scattering. The tau-lepton flux can be

calculated from Eqs.共1兲 and 共2兲 once the incoming␯flux is given. The ␯ flux can be obtained by the following ansatz

关18兴:

F

共E,X兲⫽F␯␶共E,0兲exp

X

共E,X兲

, 共11兲

where ⌳(E,X)⫽␭(E)/关1⫺Z(E,X)兴, with the factor Z(E,X) arising from the regeneration effect of the flux. On the other hand, the tau-lepton flux is given by

5 6 7 8 9 10 11 12 10-3 10-2 10-1 100 101 102 103

Tau lepton decay length Tau lepton range in water Tau lepton range in rock

T a u lepton range (km) log(E / GeV)

FIG. 1. The tau-lepton range in rock and in water using Eq.共8兲 and the tau-lepton decay length din km as a function of tau-lepton energy in GeV.

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F共E,X兲⫽

0 X dTG共E¯,T兲exp

T X dT

共E¯兲⫺ 1 ␳d共E¯兲 ⫺ 1 ␭CC共E¯兲

共12兲

with E¯⬅E¯(X⫺T;E), and G共E,X兲⫽nN

ymin ymax d y 1⫺yF共Ey,XdN→␶Y d y 共y,Ey兲. 共13兲

It is easy to see that the factor Z(E,X) enters into the ex-pression for F(E,X) through the function G(E,X). Simi-larly, Z(E,X) also depends on F(E,X). It is possible to solve for Z(E,X) and F(E,X) simultaneously by the itera-tion method关18兴. The details are given in Appendix B.

III. RESULTS AND DISCUSSION

In the following, we show the tau-lepton fluxes resulting from three kinds of diffuse astrophysical neutrino fluxes: the active galactic nuclei 共AGN兲 关19兴, the gamma-ray burst

共GRB兲 关20兴, and the GZK 关21兴 neutrino fluxes. In these

rep-resentative models, F

␶ arises because of neutrino flavor

mixing 关22兴. The p␥ interactions are the source of intrinsic F

, and F␯␶⫽1/2•F␯␮ because of共two兲 neutrino flavor

os-cillations during propagation. Our convention for F

␶is that F

⫽dN␯␶/d(log10E) in units of cm⫺2s⫺1sr⫺1. The same convention is used for the outgoing tau-lepton fluxes. For completeness, let us remark here that the recent upper bound on diffuse astrophysical F

共not F␯␶) from the Antarctic

Muon and Neutrino Detector Array 共AMANDA兲 B10 is of the order of ⬃8.4⫻10⫺7cm⫺2s⫺1sr⫺1GeV for 6⫻103

⭐E␯/GeV⭐106 关23兴. This 90% classical confidence upper

bound is mainly for upward going␯with E⫺2energy spec-trum and includes the systematic uncertainties. As far as the AMANDA B10 upper bound on F

␮ is concerned, all three

of our representative neutrino flux models are clearly com-patible with this upper bound within its energy range.

In Fig. 2, we show the outgoing tau-lepton energy spectra resulting from the propagation of incident AGN neutrinos inside rock (␳⫽2.65 g/cm3) for X/␳⫽10 km, 100 km, and 500 km, respectively. It is interesting to see that the tau-lepton energy spectra remain almost unchanged for the above three different slant depth/matter density ratio values. This feature can be understood by two simple facts. First of all, the neutrino-nucleon charged-current interaction length, which is related to the interaction thickness by␭CC⫽␳lCC, is given by lCC⫽2⫻104 km关(1 g/cm3/␳)兴关E␯/(106 GeV)兴⫺0.363. Secondly, the tau leptons, which eventually exit the Earth, ought to be produced within a tau-lepton range distance to the exit point. For a tau-lepton produced far away from the exit point, it loses energy and decays before reaching the exit point. Hence the tau-lepton flux is primarily determined by the ratio of tau-lepton range to the

charged-current neutrino-nucleon interaction length. The total slant depth X which the tau-neutrino 共tau-lepton兲 traverses inside the Earth is then unimportant, unless X is large enough that the tau-neutrino flux attenuates significantly before the tau neutrino is converted into the tau lepton. We note that the typical energy for the AGN neutrinos, in which this flux peaks, is between 105 and 108 GeV. The corresponding neutrino-nucleon neutral-current interaction length then ranges from 42 000 km down to 3400 km, given lNC

⫽2.35•lCC. Hence, even for X/␳ as large as 500 km, the attenuation of the tau-neutrino flux is negligible. This ex-plains the insensitivity of tau-lepton flux with respect to our chosen X/␳ values for the AGN case. The situation is rather similar for the tau-lepton flux resulting from the GRB tau neutrinos共see Fig. 3兲. On the other hand, a slight suppression is found for the GZK case at E⬎109 GeV as one increases X/␳ from 10 km to 500 km共see Fig. 4兲. This is because the typical GZK tau-neutrino flux peaks in the energy range be-tween 107 and 1010GeV, which corresponds to attenuation lengths ranging from 7800 km down to 640 km. One notices

5 6 7 8 9 10 10-23 10-21 10-19 10-17 10-15 10-13 AGNντ 10 km 100 km 500 km F τ (c m -2 s -1 sr -1 ) log(E / GeV)

FIG. 2. The tau-lepton energy spectrum induced by the AGN neutrinos in rock for three different X/␳ ratio values 共see text for more details兲. The incident tau-neutrino flux is shown by the thin solid line. 5 6 7 8 9 10 10-23 10-21 10-19 10-17 10-15 10-13 GRBν τ 10 km 100 km 500 km F τ (c m -2 s -1 sr -1 ) log(E / GeV)

FIG. 3. The tau-lepton energy spectrum induced by the GRB neutrinos in rock for three different X/␳ ratio values 共see text for more details兲. The incident tau-neutrino flux is shown by the thin solid line.

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that 640 km is rather close to the distance 500 km which we choose for X/␳. Hence a slight suppression in the tau-lepton flux occurs for X/␳⫽500 km.

We have compared our AGN-type tau-lepton flux with that obtained by Monte Carlo simulations, adopting a sto-chastic approach for the tau-lepton energy loss关24兴. The two tau-lepton fluxes agree within⬃10%. This is expected since the tau-lepton ranges obtained by the above two approaches agree well, as pointed out before. It is easily seen from Figs. 2– 4 that the AGN case has the largest tau-lepton flux be-tween 106 and 108 GeV. Since the resonant¯e-e⫺scattering cross section peaks at E⫽6.3⫻106 GeV, it is of interest to compare the integrated tau-lepton flux resulting from this scattering to the one arising from neutrino-nucleon scatter-ing. For the former case, we integrate the tau-lepton energy spectrum from 106 GeV to 6.3⫻106 GeV, and obtain ⌽R

⫽0.08 km⫺2sr⫺1yr⫺1. For neutrino-nucleon scattering, we

find that⌽CC⫽2.2 km⫺2sr⫺1yr⫺1 by integrating the corre-sponding tau-lepton energy spectrum from 106 GeV to 107 GeV. The detailed results for ⌽CC are summarized in Table I. The entries in the table entitled ‘‘Full’’ are obtained using the F obtained in this work, whereas the approxi-mated values entitled ‘‘Approx’’ are obtained by following the description given in Ref.关4兴, which uses a constant␤and a constant inelasticity coefficient for ␯N scattering. We re-mark that the authors of Ref. 关4兴 have taken E to be greater

than 108 GeV. Hence the integrated fluxes in the column ‘‘Approx’’ with energies less than 108 GeV are taken as ex-trapolations. Thus, one should compare the two integrated fluxes only for E⬎108 GeV. One can see that the two inte-grated fluxes seem to agree for E⬎108 GeV. In addition to the integrated fluxes for E⬎108 GeV, we also obtain inte-grated tau-lepton fluxes for 106⭐E/GeV⭐108. It is easily seen that, in this energy range, the integrated tau-lepton flux from Earth-skimming AGN neutrinos is relatively signifi-cant.

It is possible that the tau-neutrino skims through a part of the ocean in addition to the Earth before exiting the interac-tion region关25兴. Hence, it is desirable to compare the result-ing tau-lepton fluxes as the tau neutrinos skim through media with different densities, while the slant depths of the media, are held fixed as an example. As stated before, the tau-lepton flux is essentially determined by the probability of ␯N charged-current interaction happening within a tau-lepton range. Furthermore, from Fig. 1, it is clear that the tau-lepton range equals the tau-lepton decay length for E less than 107 GeV. One therefore expects Frock(E,X)/Fwater(E,X)

⫽␳rock/waterfor E

␶⬍107 GeV. This is clearly seen to be the

case from Fig. 5 and Fig. 6, as we compare Frock with Fwater(E,X) for X⫽2.65⫻106 g/cm2 and X⫽2.65

⫻107 g/cm2, respectively. For E

␶⬎107 GeV, the tau-lepton

range has additional dependencies on the mass density and the atomic number of the medium. Hence the ratio

5 6 7 8 9 10 10-25 10-23 10-21 10-19 10-17 GZKντ 10 km 100 km 500 km F τ (c m -2 s -1 sr -1 ) log(E / GeV)

FIG. 4. The tau-lepton energy spectrum induced by the GZK neutrinos in rock for three different X/␳ ratio values 共see text for more details兲. The incident tau-neutrino flux is shown by the thin solid line.

TABLE I. Comparison of the integrated tau-lepton flux (km⫺2yr⫺1sr⫺1) in different energy bins for the AGN, the GRB, and the GZK neutrinos without and with approximation 共see text for details兲. The distance traversed is taken to be 10 km in rock here. For 109 ⭐E/GeV⭐1010

, the incident AGN neutrino flux is too small so that its induced tau-lepton flux is not shown.

Energy interval

AGN GRB GZK

Full Approx Full Approx Full Approx

106⭐E/GeV⭐107 2.23 2.12 9.63⫻10⫺3 1.05⫻10⫺2 7.38⫻10⫺5 2.08⫻10⫺5 107⭐E/GeV⭐108 4.89 5.12 7.12⫻10⫺3 6.82⫻10⫺3 1.14⫻10⫺2 1.90⫻10⫺2 108⭐E/GeV⭐109 1.95⫻10⫺1 1.52⫻10⫺1 5.39⫻10⫺4 4.63⫻10⫺4 8.17⫻10⫺2 8.47⫻10⫺2 109⭐E/GeV⭐1010 1.13⫻10⫺5 1.24⫻10⫺5 3.31⫻10⫺2 3.52⫻10⫺2 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 AGN GRB GZK ρwa te r / ρrock F τ roc k (1 0 k m)/F τ wa te r (26. 5 k m ) log(E / GeV)

FIG. 5. The ratio of Fin rock and water induced by the AGN, the GRB, and the GZK neutrinos for X⫽2.65⫻106g/cm2.

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Frock(E,X)/Fwater(E,X) starts deviating from ␳rock/␳water. It is worthwhile to mention that the tau-lepton flux ratios for the AGN and GRB cases behave rather similarly. On the other hand, the ratio in the GZK case has a clear peak in the range 107.5⬍E/GeV⬍108.5. Such a peak is even more appar-ent for the slant depth X⫽2.65⫻107 g/cm2. The appearance of this peak has to do with the relatively flat behavior of the incident GZK neutrino spectrum, while the position of this peak is related to the energy dependencies of the tau-lepton range and the neutrino-nucleon scattering cross sections. We have confirmed our observations by computing the flux ra-tios with simple power-law incident tau-neutrino fluxes. The above peak in the tau-lepton flux ratio implies the suppres-sion of tau-lepton events from ocean-skimming neutrinos compared to those from rock-skimming neutrinos. As stated earlier, the suppression of ocean-skimming neutrinos is re-lated to the spectral index of the incident neutrino flux. It is therefore useful to perform a detailed simulation for it 关26兴. Such a detailed study is needed because the slant depths traversed by the above two kinds of neutrinos are generally different.

IV. PROSPECTS FOR POSSIBLE FUTURE OBSERVATIONS

To observe the above tau leptons, the acceptance of a detector must be of the order of⬃km2sr. For AGN neutri-nos, the tau-lepton energy spectrum peaks at around 107–108 GeV, which is below the threshold of a fluores-cence detector, such as the High Resolution Fly’s Eye

共HiRes兲 关27兴. Also, these tau leptons come in near

horizon-tally. At present, it seems very difficult to construct a ground array in the vertical direction. A Cherenkov telescope seems to be a feasible solution. In this context, the NuTel Collabo-ration is developing Cherenkov telescopes to detect the Earth-skimming high energy neutrinos 关25兴. However, be-cause of the small opening angle of the Cherenkov light cone and only a 10% duty cycle共optical observations are limited to moonless and cloudless nights only兲, such a detector must cover a very large area and field of view. A potential site for NuTel is at Hawaii Big Island, where two large volcanos, namely, Mauna Loa and Mauna Kea, could be favorable

can-didates for high energy neutrinos to interact with. For a de-tector situated on top of Mount Hualalai and to look at both Mauna Kea and Mauna Loa, the required angular field of view is⬃8°⫻120°. Furthermore, this telescope should have an acceptance area larger than 2 km2sr so as to detect more than one event per year.

Concerning the GZK neutrinos, we note that the recent observation of ultrahigh energy cosmic rays by HiRes seem to be consistent with the GZK cutoff. Therefore a future observation of GZK tau neutrinos will provide a firm support to GZK cutoff. In particular, the slight pileup of tau leptons between 108 GeV and 109 GeV, induced by the Earth-skimming high energy GZK neutrinos, should be a candidate signature for GZK neutrinos. The integrated tau-lepton flux in this energy range is approximately 0.08 km⫺2sr⫺1yr⫺1. To detect one event per year from this flux, the acceptance of a detector must be larger than 120 km2sr for a fluorescence detector 共assuming a duty cycle of 10%兲. Although HiRes can reach 1000 km2sr at energy greater than 3⫻109 GeV, it would be a technical challenge to lower the threshold to 108 GeV. Using a system similar to HiRes, the Dual Imag-ing Cherenkov Experiment 共DICE兲 was able to detect Cher-enkov light from extensive air showers at energy as low as 105 GeV 关28兴. However, the field of view of DICE is also quite small, and thus several Cherenkov telescopes would be needed. An alternative method is a hybrid detection of both Cherenkov and fluorescence photons关29兴. That is, a detector similar to HiRes, which looks at both land and sea and de-tects both Cherenkov and fluorescence photons, may observe the associated signal of GZK neutrinos.

In summary, we have given a semianalytic treatment of the problem of simultaneous propagation of high energy tau neutrinos and tau leptons inside the Earth. Our treatment explicitly takes into account the inelasticity of neutrino-nucleon scatterings as well as the tau-lepton energy loss. We specifically considered the Earth-skimming situation and provided detailed results for the energy dependencies of emerging tau-lepton fluxes resulting from a few anticipated astrophysical neutrino fluxes. The effect of matter density on the tau-lepton flux is also studied. Such an effect is found to be related to the spectrum index of the incident neutrino flux. Our treatment thus provides a basis for a more complete and realistic assessment of high energy neutrino flux measure-ments in the large neutrino telescopes under construction or being planned.

ACKNOWLEDGMENTS

We thank N. La Barbera for communicating his Monte Carlo–based results to us. H.A. thanks the Physics Division of NCTS for support. M.A.H. is supported by Taiwan’s Min-istry of Education under ‘‘Research Excellence Project on Cosmology and Particle Astrophysics: Sub-project II’’ with the grant number 92-N-FA01-1-4-2. F.F.L., G.L.L., J.J.T., and T.W.Y. are supported by the National Science Council of Taiwan under the grant numbers NSC91-2112-M009-019 and NSC91-2112-M-001-024. 5 6 7 8 9 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 AGN GRB GZK ρwa te r /ρ ro c k Fτ ro ck (100 k m )/ Fτ wa te r (265 k m ) log(E / GeV)

FIG. 6. The ratio of Fin rock and water induced by the AGN, the GRB, and the GZK neutrinos for X⫽2.65⫻107g/cm2.

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APPENDIX A: THE CONTRIBUTION FROM RESONANT

␯¯EEÀSCATTERING

The transport equations for¯eand the tau lepton are given by Eqs. 共3兲 and 共4兲. For convenience, let us write 1⫺y⫽z. The last term in Eq. 共4兲 can be simplified using

d␯¯ ee→␯¯␶␶⫺ dz 共z,E/z兲⫽ mW4GF2 ␲ s共1⫺z兲2 共s⫺mW 22⫹m W 2 W 2 共A1兲

and the narrow width approximation 1 ␲ mWW 共s⫺mW 22⫹m W 2 W 2 ⬇␦共s⫺mW 2兲. 共A2兲 We arrive at ⳵F共E,X兲X ⫽⫺ F共E,X兲d共E兲 ⫹ 1 3

1⫺ E ER

2 ⫻

␲⌫W LRmW

F␯¯ e共ER,X兲, 共A3兲 where ER⫽mW 2/2m

e is the¯e energy such that the W boson is produced on shell in the ¯ee⫺ scattering. LR

⬅1/ne␯¯ee→W⫺is the interaction thickness for such a scat-tering. To solve for F(E,X), we need to input F␯¯

e(ER,X).

Obviously, the¯eflux at the resonant-scattering energy ER is mainly attenuated by the resonant scattering itself. Hence F␯¯

e(ER,X)⫽exp(⫺X/LR)F␯¯e(ER,0). Substituting this result

into Eq.共A3兲, we obtain F共E,X兲⫽1 3

1⫺ E ER

2

␲⌫W LRmW

F␯¯ e共ER,0兲 ⫻exp

dX共E兲

0 X dZ ⫻exp

冋冉

1 d共E兲⫺ 1 LR

Z

. 共A4兲

The integration over Z can be easily performed. In practice, it is obvious that XⰇ␳d(E). In this limit, we have

F共E,X兲⫽␲ 3

1⫺ E ER

2

W mW

冊冉

d共E兲 LR

F␯¯ e共ER,0兲 ⫻exp

X LR

. 共A5兲

Let us consider standard rock as the medium for¯ee⫺ scat-tering; we then have ␳/LR⫽ne␳␴␯¯ee→W⫺. Given

␯¯ee→W⫺⫽4.8⫻10

⫺31cm2at the W boson mass peak, and

ne␳⫽2.65⫻6.0/2⫻1023/cm3 in standard rock, we obtain

/LR⫽(26 km)⫺1. Furthermore, we can write d(E)

⫽49 km⫻(E/106 GeV). We then obtain the following ratio:

F共E,X兲 F␯¯ e共ER,0兲 ⫽3.3⫻10⫺4

E ER

1⫺ E ER

2 ⫻exp

X LR

. 共A6兲

This is the result given by Eq.共10兲 in the main text. APPENDIX B: THE ITERATION METHOD

FOR OBTAINING Z␯„E,X… AND F␶„E,X… The evolution for F

␶is given by Eq.共1兲. With the ansatz

F

共E,X兲⫽F␯␶共E,0兲exp

X

⌳␯共E,X兲

, 共B1兲

we obtain the following equation for Z(E,X):

XZ共E,X兲⫽

0 X dX

0 1 dy 1⫺y

F ␶ (0)共E yF

(0)共E兲 exp关⫺X

D共E,Ey,X

兲兴⌽␯ NC共y,E兲F共Ey,X

F ␶ (0)共E兲

␭␯共E兲d共E兲

exp

X

⌳␯共E,X

d共y,E兲⫹F共Ey,X

F ␶ (0)共E兲

␭␯共E兲 ␭␶共E兲

exp

X

⌳␯共E,X

CC共y,E兲

, 共B2兲

(8)

where F

(0)(E)⬅F

(E,0), while ⌽␯NC , ⌽␶CC, and ⌽␶d are

respectively given by ⌽NC共y,E兲⫽

T nT共d␴␯T→Y/d y兲共y,Ey

T nTT tot 共E兲 , 共B3兲 ⌽CC 共y,E兲⫽

T nT共d␶T→␯Y/d y兲共y,Ey

T nT␶Ttot共E兲 , 共B4兲 ⌽␶d共y,E兲⫽ 1 ⌫␶共E兲 d␶→␯Y d y 共y,Ey兲, 共B5兲

with nTthe number of targets per unit mass of the medium, and D共E,Ey,X兲⫽ 1 ⌳␯共Ey,X兲 ⫺ 1 ⌳␯共E,X兲. 共B6兲

For simplicity in the notation, we take the lower and upper limits for the y integration to be 0 and 1, respectively. In reality, the limits depend on the actual kinematics of each process. One may impose these limits in the functions⌽

NC,

⌽␶CC, and⌽␶d.

To perform the iteration, we begin by setting Z␯(0)⫽0. In this approximation, we have

F

␶(0)共E,X兲⫽F共E,0兲exp

X

␭␯共E,X兲

. 共B7兲

Substituting F

␶(0)(E,X) into Eq.共12兲, we obtain the lowest order␯flux, F␶(0)(E,X). The first iteration for Z, denoted by Z␯(1) is calculable from Eq. 共B2兲 by substituting F

␶(0)(E,X), F␶(0)(E,X), and Z␯(0) into the RHS of this equation. From Z␯(1), we can then calculate F

␶(1)(E,X) and

F␶(1)(E,X), which corresponds to the results presented in this paper. We have checked the convergence of the iteration procedure and have found negligible differences between Z␯(2) and Z␯(1) and their associated␯and␶ fluxes.

The value of Z depends on the spectrum index of the neutrino flux, since it effectively gives the regeneration effect in the neutrino-nucleon scattering. In general, a flatter neu-trino spectrum implies a larger Z. Z is, however, not sen-sitive to the slant depth X. In the case of GRB neutrinos, where the flux decreases as E⫺2 for E⬍107 GeV and de-creases as E⫺3 for energies greater than that, we obtain ZGRB⬇0.2. For the AGN neutrino, ZAGNchanges from 0.96 to 0.35 as Eruns from 105 GeV to 106 GeV. In this energy range, the neutrino flux decreases more slowly than E⫺0.5. For E greater than 108 GeV, ZAGNdrops below 0.2 as the neutrino flux spectrum begins a steep fall. The values for ZGZKalso follow a similar pattern.

关1兴 S.W. Barwick, lectures presented at 28th SLAC Summer

Insti-tute on Particle Physics: Neutrinos from the Lab, the Sun, and the Cosmos共SSI 2000兲, Stanford, California, USA, 2000; H. Athar, hep-ph/0209130. For a recent review article, see H. Athar, hep-ph/0212387.

关2兴 G. Domokos and S. Kovesi-Domokos, hep-ph/9801362;

hep-ph/9805221; see also D. Fargion, Astrophys. J. 570, 909

共2002兲.

关3兴 X. Bertou, P. Billoir, O. Deligny, C. Lachaud, and A.

Letessier-Selvon, Astropart. Phys. 17, 183共2002兲.

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88, 161102共2002兲.

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see also, M. Chiba et al., in Radio Detection of High Energy Particles, edited by D. Saltzberg and P. Gorham, AIP Conf. Proc. No. 579共AIP, Melville, NY, 2001兲, p. 204.

关7兴 http://www.ps.uci.edu/⬃anita/

关8兴 L.V. Volkova, G.T. Zatsepin, and L.A. Kuzmichev, Yad. Fiz.

29, 1252共1979兲 关Sov. J. Nucl. Phys. 29, 645 共1979兲兴.

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关10兴 For a recent discussion, see H. Athar and G.-L. Lin, Astropart.

Phys. 19, 569共2003兲, and references cited therein.

关11兴 S. Iyer Dutta, M.H. Reno, I. Sarcevic, and D. Seckel, Phys.

Rev. D 63, 094020共2001兲.

关12兴 B. Rossi, High Energy Particles 共Prentice-Hall, Englewood

Cliffs, NJ, 1952兲.

关13兴 A.A. Petrukhin and V.V. Shestakov, Can. J. Phys. 46, S377 共1968兲.

关14兴 R.P. Kokoulin and A.A. Petrukhin, in Proceedings of the XII

International Conference on Cosmic Rays, Hobart, Tasmania, Australia, 1971, Vol. 6.

关15兴 L.B. Bezrukov and E.V. Bugaev, Yad. Fiz. 33, 1195 共1981兲 关Sov. J. Nucl. Phys. 33, 635 共1981兲兴. For a recent discussion,

see E.V. Bugaev and Y.V. Shlepin, Phys. Rev. D 67, 034027

共2003兲.

关16兴 P. Lipari and T. Stanev, Phys. Rev. D 44, 3543 共1991兲. 关17兴 J. Pumplin, D.R. Stump, J. Huston, H.L. Lai, P. Nadolsky, and

W.K. Tung, J. High Energy Phys. 07, 012共2002兲.

关18兴 V.A. Naumov and L. Perrone, Astropart. Phys. 10, 239 共1999兲. 关19兴 A. Neronov, D. Semikoz, F. Aharonian, and O. Kalashev, Phys.

Rev. Lett. 89, 051101 共2002兲; O.E. Kalashev, V.A. Kuzmin, D.V. Semikoz, and G. Sigl, Phys. Rev. D 66, 063004共2002兲.

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Phys. Rev. D 59, 023002共1999兲.

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Athar, K. Cheung, G.-L. Lin, and J.-J. Tseng, Astropart. Phys. 18, 581共2003兲.

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关24兴 N. La Barbera 共private communication兲.

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Workshop on Astroparticle Physics, Taiwan, 2001, edited by H. Athar, G.-L. Lin, and K.-W. Ng共World Scientific, Singapore, 2002兲, pp. 105–116, astro-ph/0204145. See also http:// hep1.phys.ntu.edu.tw/nutel/

关26兴 M.A. Huang, G.-L. Lin, and J.-J. Tseng 共in progress兲.

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Pro-ceedings of the 28th International Cosmic Ray Conference, Tsukuba, Japan, 2003共in preparation兲.

數據

FIG. 1. The tau-lepton range in rock and in water using Eq. 共8兲 and the tau-lepton decay length d ␶ in km as a function of tau-lepton energy in GeV.
FIG. 2. The tau-lepton energy spectrum induced by the AGN neutrinos in rock for three different X/ ␳ ratio values 共see text for more details 兲
FIG. 5. The ratio of F ␶ in rock and water induced by the AGN, the GRB, and the GZK neutrinos for X ⫽2.65⫻10 6 g/cm 2 .
FIG. 6. The ratio of F ␶ in rock and water induced by the AGN, the GRB, and the GZK neutrinos for X ⫽2.65⫻10 7 g/cm 2 .

參考文獻

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