Comparison of high-energy galactic and atmospheric
tau neutrino flux
H. Athar
a,b,*, Kingman Cheung
a, Guey-Lin Lin
b, Jie-Jun Tseng
ba
Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan
b
Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan
Abstract
We compare the tau neutrino flux arising from the galaxy and the earth atmosphere for 1036E=GeV 6 1011. The
intrinsic and oscillated tau neutrino fluxes from both sources are calculated. The intrinsic galactic msflux (E P 103GeV)
is calculated by considering the interactions of high-energy cosmic-rays with the matter present in our galaxy, whereas the oscillated galactic msflux is coming from the oscillation of the galactic mlflux. For the intrinsic atmospheric msflux,
we extend the validity of a previous calculation from E 6 106 GeV up to E 6 1011GeV. The oscillated atmospheric m s
flux is, on the other hand, rather suppressed. We find that, for 1036E=GeV 6 5 107, the oscillated m
sflux along the
galactic plane dominates over the maximal intrinsic atmospheric msflux, i.e., the flux along the horizontal direction. We
also briefly mention the presently envisaged prospects for observing these high-energy tau neutrinos. Ó 2002 Elsevier Science B.V. All rights reserved.
PACS: 95.85.Ry; 13.85.Tp; 98.38.)j; 14.60.Pq
1. Introduction
Searching for high-energy tau neutrinos (E P 103GeV) will yield quite useful information about
the highest energy phenomenon occurring in the universe [1]. The same search may also provide evidence for physics beyond the standard model [2]. The latter is suggested by the recent measure-ments of the atmospheric muon neutrino deficit, though yet there is no observation of oscillated
atmospheric tau neutrinos at a significant confi-dence level [3]. Interestingly, it is also only recently that we have the first evidence of existence of tau neutrinos [4].
The high-energy tau neutrinos can be produced in pp and pc interactions taking place in cosmos. These interactions produce unstable hadrons that decay into tau neutrinos. In this paper, we mainly concentrate on pp interactions and will only briefly comment on pc interactions as source interactions for producing high-energy tau neutrinos. There can be several astrophysical sites where the pp in-teractions may occur. Examples of these include the relatively nearby and better known astro-physical sites such as our galaxy and the earth atmosphere, where the basic pp interactions occur as pA interactions. The pp interactions in these
*Corresponding author. Address: Physics Division, National
Center for Theoretical Sciences, Hsinchu 300, Taiwan. E-mail addresses:[email protected] (H. Athar),
[email protected] (K. Cheung), [email protected].
edu.tw (G.-L. Lin), [email protected] (J.-J. Tseng).
0927-6505/02/$ - see front matterÓ 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 6 5 0 5 ( 0 2 ) 0 0 1 8 3 - 4
sites form a rather certain background to the extra-galactic high-energy tau neutrino searches. It is possible that such interactions are the only sources of high-energy tau neutrinos should the search for high-energy tau neutrinos originating from several proposed distant sites such as AGNs, GRBs, as well as groups and clusters of galaxies, turns out to be negative. Therefore, it is rather essential to investigate the high-energy tau neutrino flux ex-pected from our galaxy and the earth atmosphere. Both the galactic and atmospheric tau neutrinos can be categorized into intrinsic and oscillated ones. Here, intrinsic msflux refers to the msproduced
directly by an interaction while oscillated ms refers
to the ms resulted from the ml! ms oscillation.
Presently, there exists no estimate for the intrinsic high-energy ms flux originating from our galaxy in
pp interactions, although the estimates for meand ml
fluxes due to the same interactions are available [5]. In this work, we calculate the intrinsic ms flux
from our galaxy by using the perturbative and non-perturbative QCD approaches to model the pp in-teractions, and taking into account all major tau neutrino production channels up to E 6 1011GeV.
We note that the production of tau neutrinos in the terrestrial context was discussed in Ref. [6], which uses a non-perturbative QCD approach for pp interactions. To calculate the oscillated galactic and atmospheric ms fluxes, we apply the two-flavor
neutrino oscillation analysis [3].
It is essential to compare the above galactic tau neutrino flux with the flux of atmospheric tau neutrinos. The intrinsic atmospheric tau neutrino flux has been calculated for E 6 106 GeV [7]. In
this work, we extend the calculation up to E 6 1011
GeV. Such an extension requires the input of cosmic-ray flux spectrum for an energy range be-yond that considered in Ref. [7]. Furthermore, for a greater neutrino energy, the solutions of cascade equations relevant to the neutrino production be-have differently. For the oscillated ms flux, it is
in-teresting to note that the oscillation length for ml! msfor the energy range 1036E=GeV 6 1011is
much greater than the thickness of the earth at-mosphere. Hence the oscillated atmospheric msflux
in this case is highly suppressed.
One may argue that the interaction of the high-energy cosmic-rays with the ubiquitous cosmic
microwave background (CMB) photons present in cosmos (pc rather than pp interactions) could also be an important source for high-energy astro-physical tau neutrinos. The center-of-mass energy (pffiffis) needed to produce a s lepton and a ms is at
least 1.8 GeV. In a collision between a proton with an energy Ep and a CMB photon with an
energy EcCMB, the center-of-mass energy squared of
the system satisfies m2
p< s <4EpEcCMBþ m 2 p. Since
the peak of the CMB photon flux spectrum with a temperature 2.7 K is at about 2:3 104 eV, it
requires a very energetic proton with EpJ2:5
1012GeV in order to produce a sm
spair. Thus, the
contribution of the intrinsic tau neutrino flux from the interaction between the cosmic proton and the CMB photon is negligible, unless we are con-sidering extremely high-energy protons, beyond the presently observed highest energy cosmic-rays [8]. To compute the oscillated ms flux in this case,
we consider the non-tau neutrino flux generated by the pc interaction via the D resonance, com-monly referred to as the Greisen–Zatsepin–Kuz-min (GZK) cutoff interaction, which assumes that the proton travels a cosmological distance [9]. A recent calculation of the intrinsic non-tau GZK neutrino flux indicates that this flux peaks typically at E 109GeV [10], beyond the reach of presently
operating high-energy neutrino telescopes such as AMANDA and Baikal experiments [1]. This flux decreases for E < 109 GeV. In fact, it falls below
the intrinsic non-tau galactic-plane neutrino flux for E 6 5 107 GeV. The neutrino flavor
oscil-lations of the intrinsic non-tau GZK neutrinos into tau neutrinos result in a msflux comparable to
the original non-tau neutrino flux [11]. Therefore, in the absence (or smallness) of tau neutrino flux from other possible extra-galactic astrophysical sites, the only source of high-energy tau neutrinos besides the atmospheric background is from our (plane of) galaxy, typically for 1036E=GeV 6
5 107. This is an energy range to be explored by
the above high-energy neutrino telescopes in the near future.
The organization of the paper is as follows. In Section 2, we discuss the calculation of intrinsic high-energy tau neutrino flux from our galaxy, including the description of the flux formula, the galaxy model, and the various tau neutrino
pro-duction channels taken into account in the calcu-lation. Although this flux will be shown small, we shall go through some details of the calculation since they are also useful for the calculation in the next section. In Section 3, we present our result on the intrinsic atmospheric ms flux and compare it
with the galactic one. In Section 4, we discuss the effects of neutrino flavor mixing, which are used to construct the oscillated ms fluxes from the galaxy
and the earth atmosphere respectively. The total galactic ms flux (the sum of intrinsic and oscillated
fluxes) is compared to its atmospheric counterpart, and the dominant energy range for the former flux is identified. We also mention the currently envi-saged prospects for identifying the high-energy tau neutrinos. We summarize in Section 5.
2. The intrinsic galactic tau neutrino flux
2.1. The tau-neutrino flux formula and the galaxy model
We use the following formula for computing the ms flux: dNms dE ¼ Z 1 E dEp/pðEpÞf ðEpÞ dnpp!msþY dE : ð1Þ
In the above equation, E is the tau neutrino energy and the cosmic-ray flux spectrum, /pðEpÞ, is given
by [12] /pðEpÞ ¼
1:7 ðEp=GeVÞ2:7 for Ep< E0;
174 ðEp=GeVÞ3 for EpP E0;
ð2Þ where E0¼ 5 106 GeV and /pðEpÞ is in units of
cm2s1sr1GeV1. We assume directional
iso-tropy in /pðEpÞ for the above energy range. A
more recent measurement of cosmic-ray flux spec-trum between 2 105and 106GeV agrees with the
/pðEpÞ given by Eq. (2) within a factor of 2 in
this energy range [13]. The function fðEpÞ is equal
to R=kppðEpÞ, where kppðEpÞ ¼ ðrinclpp npÞ1 is the pp
interaction length and R is a representative dis-tance in the galaxy along the galactic plane. The target particles are taken to be protons with a constant number density of 1 cm3 and R is
taken to be 10 kpc, where 1 pc ’ 3 1018 cm.
The rincl
pp is the total inelastic pp cross section. Since
the high-energy protons traverse a distance R much shorter than the proton interaction length, the proton flux spectrum /pðEpÞ is assumed to be
constant over the distance R. Furthermore, we calculate the intrinsic tau neutrino flux along the galactic plane only to obtain the maximal expected tau neutrino flux. The matter density decreases exponentially in the direction orthogonal to the galactic plane, therefore the amount of intrinsic tau neutrino flux decreases by approximately two orders of magnitude for the energy range of our interest. For further details, see Ingelman and Thunman in Ref. [5]. The function fðEpÞ in Eq. (1)
basically gives the number of inelastic pp collisions in the distance R, while the distribution dn=dE is the differential cross section normalized by rincl
pp , i.e., dnpp!msþY dE ¼ 1 rincl pp drpp!msþY dE : ð3Þ
The above distribution gives the fraction of in-elastic pp interactions that goes into msÕs. We can
now simplify Eq. (1) into dNms dE ¼ Rnp Z 1 E dEp/pðEpÞ drpp!msþY dE : ð4Þ
It is clear that the task of computing dNms=dE relies
on the evaluation of the differential cross section dr=dE in pp interactions. In this work, we include all major production channels of tau neutrinos, namely, via the Ds meson, b-hadron, ttt, W , and
Z . In general, the symbol m
s shall be used to
ac-count for the contribution of ms and mms unless
otherwise mentioned. We note that the heavy in-termediate states, such as the Dsmeson, b-hadrons
and other heavier states, decay (into ms) before
they interact with other particles in the galactic plane. This is due to the rather small matter den-sity of the medium and the large distance between the proton source and the earth. Before we pro-ceed, let us remark that the intrinsic tau neutrino production by the galactic pc interactions is sup-pressed relative to that in the galactic pp inter-action, because nc np for the energy range of
2.2. Tau neutrino production 2.2.1. Via Dsmesons
The lightest meson that can decay into a s–ms
pair is the Ds meson. It was pointed out that the
production and the decay of Dsmeson is the major
production channel for tau neutrinos in the AGN [15]. We expect the same will be true for galactic tau neutrinos. The Dsmeson decays into a charged
s lepton and a ms. The charged s lepton
subse-quently decays into the second ms plus other
par-ticles, which can be one prong or three prong. The kinematics of the s lepton decay is treated by the Monte Carlo technique. For simplicity, we assume that the s lepton decays into a ms and a particle
Y with the mass mY satisfying 0:1 GeV < mY <
ms 0:1 GeV. Consequently, the second msis much
more energetic than the first one because the Ds
mass is only slightly larger than ms. We take the
branching ratio BðDs! sþmsÞ as ’0.074 (0.04)
[16].
We now turn to the production of Ds mesons.
Here, we employ two approaches to calculate the production of Dsmesons: (i) the perturbative QCD
(PQCD) and (ii) the quark–gluon string model (QGSM). In the PQCD approach, we use the leading-order result for pp! ccc:
rðpp ! cccÞ ¼X
ij
Z Z
dx1dx2fi=pðx1Þ
fj=pðx2Þ^rrðij ! cccÞ; ð5Þ
where fi=pðxÞ are the parton distribution functions
(we use the CTEQv5 [17]), while the parton sub-processes are qqq; gg! ccc. We use a K factor, K¼ 2, to account for the NLO corrections [18]. The matrix elements for these subprocesses can be found in Appendix A. The c or ccthen undergoes fragmentation into the Dsmeson, which we model
by the Peterson fragmentation function [19] with 0:029 [20] (see Appendix A). The probability fc!Ds of a charm quark fragmenting into a Ds is
0.19 [20] (we have added the fc!Ds and fc!D s).
Typically, a fraction z (z < 1) of the charm-quark energy is transferred to the Ds meson so that the
energy spectrum of Ds is softer than that of the
charm quark.
The QGSM approach is non-perturbative and is based on the string fragmentation. It contains a number of parameters determined by experiments [21]. The production cross section of the Dsmeson
is given by the sum of n-Pomeron terms drDsðs; xÞ dx 1 x2þ x2 ? X1 n¼1 rpp nðsÞ / Ds n ðs; xÞ; ð6Þ where x¼ 2pk= ffiffis p and x? ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm2 Dsþ p 2 ?Þ=s q . The functions rpp nðsÞ and / Ds
n ðs; xÞ are given in
Appen-dix B.
A comparison of these two approaches for Ds
meson is shown in Fig. 1. The dashed line in the figure is the spectrum of the injected proton flux given by Eq. (2). The msspectra calculated by these
two approaches agree well with each other for E 6106 GeV. Beyond this energy, the QGSM
ap-proach gives a relatively harder spectrum. In fact, this behavior was already seen in the dr=dx dis-tribution. Nevertheless, in the region where the two approaches differ, the tau neutrino flux is already small. To our knowledge, the heaviest meson production that the QGSM has been ap-plied to is the Ds meson production. The current
highest energy collider experiment for Ds meson
production is at the FERMILAB TEVATRON
Fig. 1. A comparison between the PQCD and QGSM ap-proach to the energy spectrum of the intrinsic galactic msflux
coming from the Ds meson. The thick dashed curve is the
with pffiffis¼ 1:8 103 GeV, which corresponds to
an Ep 1:7 106 GeV, as s 2mpEp in our
set-ting. Note that up to this pffiffis, the agreement between the two approaches is quite good, ac-cording to Fig. 1. We have used a factorization scale Q2¼ ^ss=4 and the one-loop running strong
coupling constant aswith the value asðQ2¼ MZ2Þ ¼
0:118, and the mc ¼ 1:35 GeV.
An important quantity in the neutrino flux calculation is the average fraction of the injected proton energy being transferred to the tau neu-trino, i.e., the ratio r E=Ep. The average value of
r is given either by the mean hri ¼ R r dr dE dr R dr dE dr ; ð7Þ
or by the value of r at which the distribution dr=dE attains the peak. We have found that both averages of r are very close to each other. Thehri ranges from 5 103 to 5 107 for E
p from 103
to 1011 GeV for the production channel pp!
ccc! Dsþ Y ! msþ Y , using the PQCD approach.
The higher the injected proton energy, the smaller is the fraction of the incident Ep that goes into
hadrons.
2.2.2. Via bbb, ttt, W and Z
The production of bbb and ttt in pp interactions can be calculated quite reliably by the PQCD ap-proach, similar to the calculation of ccc. The rele-vant matrix elements, including the ones for W
and Z , are listed in Appendix A.
The results are shown in Fig. 2. Let us remark that after taking into account the sources of un-certainties such as values of mc, mb, mt, R, np, etc.,
we estimate that our calculation of the intrinsic galactic ms flux is reliable within an order of
mag-nitude. A few observations can be drawn from the Fig. 2. (i) The production via Dsmesons dominates
for E 6 109 GeV, followed by b-hadrons, W , Z ,
Fig. 2. Intrinsic galactic tau neutrino flux calculated via various intermediate states and channels: Ds, b-hadron, W , Z , and ttt. The
and ttt respectively. (ii) For E P 109 GeV all these
production channels become comparable. (iii) The intrinsic tau neutrino flux is about 10–12 orders of magnitude smaller than the injected proton flux [22].
3. The intrinsic atmospheric tau neutrino flux The earth atmosphere is an interesting extra-terrestrial site where the basic pp interaction occurs in the form of a pA collision with A the nuclei present in the earth atmosphere. Incidentally, it is the only known nearby extra-terrestrial site from where the intrinsic neutrinos are observed as a result of high-energy cosmic-ray interactions.
We have calculated the downward and hori-zontal intrinsic atmospheric ms flux for the energy
range 1036E=GeV 6 1011. We have used the
non-perturbative QCD approach mentioned in the last section to model the production of Ds mesons in
pA interactions. We have used the /pðEpÞ given by
Eq. (2) and the Z-moment description for the calculation of intrinsic tau neutrino flux, which is appropriate for a varying target density medium [23]. Since the tau neutrino flux is determined by the flux of Dsmeson, we briefly discuss the cascade
equation for the Dsflux. In general, we have [23]
d/Ds dX ¼ /Ds kDs /Ds qdDs þ ZDsDs /Ds kDs þ ZpDs /p kp ; ð8Þ where X is the slant depth, i.e., the amount of at-mosphere (in g/cm2) traversed by the D
s meson
(X ¼ 0 at the top of the atmosphere), /pðE; X Þ and
/DsðE; X Þ are the fluxes of protons and Dsmesons
respectively. The kDs and dDs are the interaction
thickness (in g/cm2) and the decay length of the D s
meson respectively. Finally, the Z-moments ZpDs
and ZDsDs describe the effectiveness of generating
Ds meson from the higher-energy protons and Ds
mesons respectively. We have
ZpDsðEÞ ¼ Z 1 E dE0/pðE0;0Þ /pðE; 0Þ kpðEÞ kpðE0Þ
dnpA!DsþYðE; E
0Þ dE ; ZDsDsðEÞ ¼ Z 1 E dE0/DsðE 0;0Þ /DsðE; 0Þ kDsðEÞ kDsðE0Þ
dnDsA!DsþYðE; E
0Þ
dE ;
ð9Þ
where dnpA!DsþY=dE and dnDsA!DsþY=dE are
de-fined according to Eq. (3). We note that Eq. (8) should be solved together with the cascade equa-tion governing the propagaequa-tion of high-energy cosmic-ray protons. In fact, the proton flux equa-tion can be easily solved such that
/pðE; X Þ exp X Kp /pðE; 0Þ; ð10Þ where Kp kp=ð1 ZppÞ is the proton attenuation
length with kp the proton interaction thickness (in
g/cm2) and the Z pp given by ZppðEÞ ¼ Z 1 E dE0/pðE 0;0Þ /pðE; 0Þ kpðEÞ kpðE0Þ dnpA!pþYðE; E 0Þ dE : ð11Þ
An analytic solution of Eq. (8) can be obtained for either the low or the high energy limit. Such limits are characterized by whether the Dsdecays before
it interacts with the medium or vice versa. The critical energy separating the two limits is ap-proximately 8:5 107 GeV. In the low energy
limit, we disregard the first and third terms in the RHS of Eq. (8). On the other hand, one can drop the second term in the high energy limit. With /Ds determined, the ms flux can be calculated by
con-sidering the decay Ds! mss and the subsequent
decay s! msþ Y as treated in Section 2. We first
obtain two msfluxes, valid for low and high energy
limits respectively, in terms of Z-moments and then interpolate the two fluxes. A complete numerical solution without the interpolation is given in a separate work [24].
In Fig. 3, we show our result for the intrinsic atmospheric ms flux along the horizontal direction.
For comparison, the results by Pasquali and Reno (PR) [7], valid for 1036E=GeV 6 106, are also
shown. The ms flux along all the other direction is
small. For example,1 the downward ms flux is
about eight times smaller than the horizontal one
1The upward going high-energy m
s with a energy E P 104 GeV is degraded in energy to about E 6 103GeV after crossing the earth.
for E P 108 GeV. We remark that the major
un-certainty for determining the above ms flux is the
Z-moment ZpDs. In Ref. [7], the authors
calcu-late ZpDsusing two different approaches, which then
give rise to different results for the msflux. The first
approach is based upon next-to-leading order (NLO) perturbative QCD [25], while the second approach rescales the ZpD0 given by the PYTHIA
[26] calculation of Thunman, Ingelman, and Gon-dolo (TIG) [27]. In the inset of Fig. 3, we also show our calculated ZpDs in comparison with the one
given by rescaling TGIÕs result for ZpD0. We do
not show the ZpDs obtained by NLO
perturbat-ive QCD since it is not explicitly gperturbat-iven in Ref. [7].
4. Effects of oscillations and prospects for observa-tions
In the context of two neutrino flavors, mland ms,
the total ms flux, dNmtots =dðlog10EÞ, is given by [28]
dNtot
ms =dðlog10EÞ ¼ P dNml=dðlog10EÞ
þ ð1 P Þ dNms=dðlog10EÞ:
ð12Þ Here P P ðml! msÞ ¼ sin22h sin2ðl=loscÞ. The
neutrino flavor oscillation length for ml! ms is
losc ðE=dm2Þ. For 1036E=GeV 6 1011 and with
dm2 103 eV2, we obtain 1086losc=pc 6 1. We assume maximal flavor mixing between ml and ms.
For intrinsic neutrinos produced along the galactic-plane, we take dNml=dðlog10EÞ given by
Ingelman and Thunman in Ref. [5] by extra-polating it up to E 6 1011GeV, whereas for dN
ms=
dðlog10EÞ, we use our results obtained in Section 2.
For galactic-plane neutrinos, we note that losc l,
where l 5 kpc is the typical average distance the intrinsic high-energy muon neutrinos traverse after being produced in our galaxy. Eq. (12) then im-plies that, on the average, half of the muon neu-trino flux will be oscillated into tau neuneu-trino flux, reducing its intrinsic level to one half.
Fig. 3. Intrinsic horizontal msflux via production and decay of the Dsmeson in the earth atmosphere. For 1036E=GeV 6 106, the
results by PR are also shown. In the inset, we compare our calculated ZpDswith the one given by rescaling TIGÕs ZpD0. The total intrinsic galactic-plane tau neutrino flux is also shown.
For downward going neutrinos produced in the earth atmosphere, we take l’ 20 km as an ex-ample. We use the (prompt) dNml=dðlog10EÞ given
in [29], whereas for dNms=dðlog10EÞ, we use our
results obtained in Section 3. For horizontal and upward going atmospheric neutrinos, l 103–104
km. Here, losc l, and so the intrinsic
atmos-pheric tau neutrino flux dominates over the oscil-lated one for E P 103GeV, essentially irrespective
of the incident direction. We present these results in Fig. 4, along with the GZK oscillated tau neu-trino flux briefly mentioned in Section 1. For the GZK neutrinos, l P Mpc and dNml=dðlog10EÞ is
taken from Ref. [10]. From the figure, we note that the galactic-plane oscillated ms flux dominates over
the intrinsic atmospheric ms flux for E 6 5 107
GeV, whereas the GZK oscillated tau neutrino flux dominates for E P 5 107 GeV.
A prospective search for high-energy tau neutri-nos can be done by appropriately utilizing the char-acteristic s lepton range in deep inelastic (charged current) tau–neutrino–nucleon interactions, in ad-dition to the attempt of observing the associated showers. For E close to 6 106 GeV, the
(anti-electron) neutrino–electron resonant scattering is also available to the search for (secondary) high-energy tau neutrinos [30]. The main advantages of
using the latter channel are that the neutrino flavor in the initial state is least affected by neutrino flavor oscillations and that this cross section is free from theoretical uncertainties [31]. The appropriate uti-lization of the characteristic s lepton range in both interaction channels not only helps to identify the incident neutrino flavor but also helps to bracket the incident neutrino energy as well, at least in principle.
For downward going or near horizontal high-energy tau neutrinos, the deep inelastic neutrino– nucleon scattering, occurring near or inside the detector, produces two (hadronic) showers [32]. The first shower is due to a charged-current neu-trino–nucleon deep inelastic scattering, whereas the second shower is due to the (hadronic) decay of the associated s lepton produced in the first shower. It might be possible for the proposed large neutrino telescopes such as IceCube to con-strain the two showers simultaneously for 1066E=
GeV 6 107, depending on the achievable shower
separation capabilities [33] (see, also, [1]). Here, the two showers develop mainly in ice. Using the same shower separation criteria as given in [33], we note that the(100 m)3 proposed neutrino
detec-tor, commonly called the megaton detector [34], may constrain the two showers separated by P10 m, typically for 5 1056E=GeV 6 106. The two
nearly horizontal showers may also possibly be contained in a large surface area detector array such as Pierre Auger, typically for 5 1086E=
GeV 6 109 [35]. In contrast to previous situations,
here the two showers develop mainly in air. Sev-eral different suggestions have recently been made to measure only one shower, which is due to the s lepton decay, typically for 108< E=GeV < 1010,
while the first shower is considered to be mainly absorbed in the earth [36]. The upward going high-energy tau neutrinos with E P 104 GeV, on the
other hand, may avoid earth shadowing to a cer-tain extent because of the characteristic s lepton range, unlike the upward going electron and muon neutrinos, and may appear as a rather small pile up of ml (l¼ e, l, and s) with E 103 GeV [37,
38]. However, the empirical determination of in-cident tau neutrino energy seems rather challeng-ing here.
Fig. 4. Galactic-plane, horizontal atmospheric and GZK tau neutrino fluxes under the assumption of neutrino flavor oscil-lations.
The above studies indicate that for a rather large range of tau neutrino energy, a prospective search may be carried out. The event rate in each experimental configuration is directly proportional to the incident ms flux and the effective area of the
detector concerned. Presently, no direct empirical upper bounds (or observations) for high-energy tau neutrinos exist.
To have an idea of the event rate, let us consider the downward going high-energy tau neutrinos originating from the galactic-plane due to neutrino flavor oscillations. The flux of such neutrinos has been given in Fig. 4. Following Ref. [33], we note that these galactic tau neutrinos give a represen-tative event rate of 6 1 per year per steradian for two separable and contained showers with E 106
GeV in a km3 volume neutrino telescope such as
the proposed IceCube.
5. Discussion and conclusions
We have calculated the ms flux due to pp
inter-actions in our galaxy. This flux consists of intrinsic tau neutrino flux and that arising from the osci-llations of muon neutrinos. We note that the lat-ter flux is dominant over the former by four to five orders of magnitude for the considered neu-trino energy range. From Fig. 4, one can see that the main background for the search of high-energy extra-galactic tau neutrino is due to the muon neutrinos produced in the galactic-plane, which then oscillate into tau neutrinos. Such a flux dom-inates for 1036E=GeV 6 5 107. Therefore it is
clear that searching for extra-galactic tau neutri-nos orthogonal to the galactic-plane is more pro-spective.
In the calculation of galactic tau neutrino flux, we have used a simplified model of matter dis-tribution along our galactic plane to obtain the maximal intrinsic tau neutrino flux. We have ex-plicitly calculated the contribution of heavier states such as bbb, ttt as well as W and Z in
ad-dition to the more conventional Dschannel to the
intrinsic tau neutrino flux. We have estimated the average fraction of the incident cosmic-ray energy that goes into tau neutrinos and found it to be
less than 1%. The contributions from bbb, ttt, W
and Z channels are comparable to D
s for E P
109 GeV. For D
s channel, we have used both
perturbative and non-perturbative QCD ap-proaches.
We have extended a previous calculation of intrinsic atmospheric ms flux from E 6 106GeV up
to E 6 1011 GeV. Here, we used the
non-pertur-bative QCD approach to calculate the production of Ds mesons in pA interactions. In comparison
with the intrinsic galactic-plane ms flux, it is large.
However, since the distance between the detector and the neutrino source in the galactic plane is sufficiently large, the neutrino flavor oscillations of non-tau neutrinos into tau neutrinos makes the eventual tau neutrino flux along the galactic plane greater than the atmospheric tau neutrino flux for 1036E=GeV 6 5 107. However, the intrinsic
at-mospheric ms flux dominates over the oscillated
galactic ms flux in the direction orthogonal to the
galactic plane. We have also briefly mentioned the presently envisaged prospects for observations. In summary, we have completed the compilation of all definite sources of tau neutrino flux, i.e., those from our galaxy and from the earth atmosphere. Such a compilation is needed before one conducts the search for tau neutrinos from extra-galactic sources.
Acknowledgements
H.A. and K.C. are supported in part by the Physics Division of National Center for Theo-retical Sciences under a grant from the National Science Council of Taiwan. G.L.L. and J.J.T. are supported by the National Science Council of R.O.C. under the grant number NSC90-2112-M009-023.
Appendix A. Formulas for PQCD
In this appendix, we list the matrix element squared for the subprocesses of pp! Q QQ, where Q¼ c; b; t, used in the PQCD calculation.
where g2 s ¼ 4pas, b¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð4m2 Q=^ssÞ q and ^ss, ^tt, ^uu are the usual Mandelstem variables with ^tt m2
Q¼
ð^ss=2Þð1 b cos h Þ.
The subprocesses for high-energy tau neutrino production via W and Z are qqq0! W ! sm
s
and qqq! Z ! m
smms. The spin- and color-averaged
amplitude squared for these subprocesses are given by X Mðudd ! W ! sþm sÞ ¼g 4 12 1 ð^ss m2 WÞ 2 þ C2 Wm 2 W ^ttð^tt m2 sÞ; X Mðduu ! W ! smm sÞ ¼g 4 12 1 ð^ss m2 WÞ 2 þ C2 Wm 2 W ^ u uð^uu m2 sÞ; X Mðqqq ! Z ! msmmsÞ ¼ g 4 3 cos4h w 1 ð^ss m2 ZÞ 2 þ C2 Zm 2 Z gm Lg q L 2 ^ u u2 h þ gm Lg q R 2^ tt2i; ðA:2Þ
where gfL ¼ T3f Qfsin2hw and hw is the weak
mixing angle, T3f is the third component of the
weak isospin and Qf is the electric charge in units
of proton charge of the fermion f.
The Peterson fragmentation function is given by
DQ!hQðzÞ ¼ N
zð1 zÞ2
½ð1 zÞ2þ z2; ðA:3Þ where N is the normalization constant and is given in the text.
Appendix B. Formulas for QGSM In Eq. (6), the functions rpp
nðsÞ and / Ds n ðs; xÞ are given as follows: /Ds n ðs; xÞ ¼ a Ds 0 F Ds qqðxþ; nÞFqDsðx; nÞ h þ FDs q ðxþ; nÞFqqDsðx; nÞ þ 2ðn 1ÞFDs seaðxþ; nÞFseaDsðx; nÞ i ; ðB:1Þ where FDs q ðx; nÞ ¼ 2 3 Z 1 x dx1fpuðx1; nÞGDus x x1 þ1 3 Z 1 x dx1fpdðx1; nÞGDds x x1 ; ðB:2Þ FDs qqðx; nÞ ¼ 2 3 Z 1 x dx1fpudðx1; nÞGDuds x x1 þ1 3 Z 1 x dx1fpuuðx1; nÞGDuus x x1 ; ðB:3Þ d^rr dcosh qqq ! Q QQ¼ g 4 sb 72p^ss3 m 2 Q ^tt2þ m2 Q ^uu 2 þ 2^ssm2 Q ; d^rr dcosh ðgg ! Q QQÞ ¼ g4 sb 768p^ss 4 ð^tt m2 QÞ 2 ( m4 Q 3m 2 Q^tt m 2 Quu^þ ^uu^tt þ 4 ð^uu m2 QÞ 2 m4 Q 3m 2 Quu^ m 2 Q^ttþ ^uu^tt þ m 2 Q ð^uu m2 QÞð^tt m2QÞ 2m2Q þ ^ttþ ^uuþ 181 ^ ss2 m 4 Q m2 Q ^tt þ ^uuþ ^tt^uu þ 9 ^tt m2 Q 1 ^ ss m4 Q 2m 2 Q^ttþ ^uu^tt þ 9 ^ u u m2 Q 1 ^ ss m4 Q 2m 2 Quu^þ ^uu^tt ) ; ðA:1Þ
FDs seaðx; nÞ ¼ 1 4þ 2dsþ 2dc Z 1 x dx1fpuseaðx1; nÞ GDus x x1 þ GDs u u x x1 þ Z 1 x dx1fpdseaðx1; nÞ GDds x x1 þ GDs d d x x1 þ ds Z 1 x dx1fpsseaðx1; nÞ GDss x x1 þ GDs ss x x1 þ dc Z 1 x dx1fpcseaðx1; nÞ GDcs x x1 þ GDs cc x x1 : ðB:4Þ In the above, fi
pðx; nÞÕs are the distribution
tions describing the n-Pomeron distribution func-tions of quarks or diquarks (i¼ u; d; uu . . .) with a fraction of energy x from the proton, and Gh
iðzÞ 0
s are the fragmentation functions of the quark or diquark chain into a hadron h which carries a fraction z of its energy.
The list of the fi
pðx; nÞ are given by fpuðx; nÞ ¼ Cð1 þ n 2aNÞ Cð1 aRÞCðaR 2aN þ nÞ xaRð1 xÞaR2aNþðn1Þ; ¼ fusea p ; fd pðx; nÞ ¼ Cð2 þ n 2aNÞ Cð1 aRÞCðaR 2aN þ n þ 1Þ xaRð1 xÞaR2aNþn; ¼ fdsea p ; fuu p ðx; nÞ ¼ Cð2 þ n 2aNÞ CðaRþ nÞCðaR 2aNþ 1Þ xaR2aNþ1ð1 xÞaRþðn1Þ; fud p ðx; nÞ ¼ Cð1 þ n 2aNÞ CðaRþ nÞCðaR 2aNþ 2Þ xaR2aNð1 xÞaRþðn1Þ; fssea p ðx; nÞ ¼ Cð1 þ n þ 2aR 2aN 2a/Þ Cð1 a/ÞCð2aR 2aN þ n a/Þ xa/ð1 xÞ2aR2aNþðn1Þa/; fcsea p ðx; nÞ ¼ Cð1 þ n þ 2aR 2aN 2awÞ Cð1 awÞCð2aR 2aN þ n awÞ xawð1 xÞ2aR2aNþðn1Þaw;
where C is the usual Gamma function. The list of the GDs i ðzÞ are given by GDs u;uu;d;ddðzÞ ¼ ð1 zÞ kawþ2aRa/; GDs uu;udðzÞ ¼ ð1 zÞ
kawþaR2aNa/þ2;
GDþs s ðzÞ ¼ ð1 zÞ kawþ2ð1a/Þ; ¼ GDs ss ðzÞ; GDs s ðzÞ ¼ ð1 zÞ kaw ð1 þ a 1z2Þ; ¼ GDþs ss ðzÞ; GDs c;ccðzÞ ¼ z1awð1 zÞ ka/:
In the above, the input parameters are as follows: aR¼ 0:5; aN ¼ 0:5; a/ ¼ 0; aw¼ 2:18;
k¼ 0:5; ds¼ 0:25; aD0s¼ 0:0007; a1¼ 5;
dc¼ 0ð0:01Þ; if charm sea contribution is
turned off ðonÞ: The function rpp
nðsÞ is given by the following
formulas: rpp nðnÞ ¼ rp nz 1 expð zÞ Xn1 k¼0 zk k! ! ; ðB:5Þ where n¼ ln s 1 ðGeVÞ2 ! ; z¼ 2Ccp R2þ a0 pn expðnDÞ; rp¼ 8pcpexpðnDÞ:
The best fit parameters are as follows: (i) for pffiffis6103GeV
cp¼ 3:64 ðGeVÞ 2
; R2¼ 3:56 ðGeVÞ2; a0p¼ 0:25 ðGeVÞ2; C¼ 1:5; D¼ 0:07: (ii) forpffiffisP103 GeV
cp¼ 1:77 ðGeVÞ 2
; R2¼ 3:18 ðGeVÞ2; a0p¼ 0:25 ðGeVÞ2; C¼ 1:5; D¼ 0:139:
References
[1] For a recent review article, see, for instance, F. Halzen, arXiv:astro-ph/0111059 and references therein.
[2] Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562.
[3] Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Rev. Lett. 85 (2000) 3999.
[4] DONUT Collaboration, K. Kodama et al., Phys. Lett. B 504 (2001) 218;
DONUT Collaboration, J. Sielaff, arXiv:hep-ex/0105042. [5] F.W. Stecker, Astrophys. J. 228 (1979) 919;
G. Domokos, B. Elliott, S. Kovesi-Domokos, J. Phys. G 19 (1993) 899;
V.S. Berezinsky, T.K. Gaisser, F. Halzen, T. Stanev, Astropart. Phys. 1 (1993) 281;
G. Ingelman, M. Thunman, arXiv:hep-ph/9604286. [6] A. De Rujula, E. Fernandez, J.J. Gomez-Cadenas, Nucl.
Phys. B 405 (1993) 80;
M.C. Gonzalez-Garcia, J.J. Gomez-Cadenas, Phys. Rev. D 55 (1997) 1297.
[7] L. Pasquali, M.H. Reno, Phys. Rev. D 59 (1999) 093003. [8] M. Nagano, A.A. Watson, Rev. Mod. Phys. 72 (2000) 689. [9] K. Greisen, Phys. Rev. Lett. 16 (1966) 748;
G.T. Zatsepin, V.A. Kuzmin, JETP Lett. 4 (1966) 78 [Pisma Zh. Eksp. Teor. Fiz. 4 (1966) 114].
[10] R. Engel, D. Seckel, T. Stanev, Phys. Rev. D 64 (2001) 093010.
[11] H. Athar, arXiv:hep-ph/0008121.
[12] JACEE Collaboration, T.H. Burnett et al., Astrophys. J. Lett. 349 (1990) L25.
[13] Tibet AS(gamma) Collaboration, M. Amenomori et al., Phys. Rev. D 62 (2000) 112002.
[14] For an estimate of galactic nc, see, Berezinsky et al., in
Ref. [5].
[15] H. Athar, Nucl. Phys. Proc. Suppl. 76 (1999) 419. [16] Particle Data Group Collaboration, D.E. Groom et al.,
Eur. Phys. J. C 15 (2000) 1.
[17] CTEQ Collaboration, H.L. Lai et al., Eur. Phys. J. C 12 (2000) 375.
[18] See, for instance P. Nason, S. Dawson, R.K. Ellis, Nucl. Phys. B 327 (1989) 49 [Erratum: Nucl. Phys. B 335 (1989) 260].
[19] C. Peterson, D. Schlatter, I. Schmitt, P.M. Zerwas, Phys. Rev. D 27 (1983) 105.
[20] ALEPH Collaboration, R. Barate et al., Eur. Phys. J. C 16 (2000) 597.
[21] A.B. Kaidalov, Phys. Lett. B 116 (1982) 459;
A.B. Kaidalov, O.I. Piskunova, Sov. J. Nucl. Phys. 43 (1986) 994 [Yad. Fiz. 43 (1986) 1545];
G.G. Arakelian, P.E. Volkovitsky, Z. Phys. A 353 (1995) 87;
G.H. Arakelian, Phys. Atom. Nucl. 61 (1998) 1570 [Yad. Fiz. 61 (1998) 1682];
G.H. Arakelian, S.S. Eremian, Phys. Atom. Nucl. 62 (1999) 1724 [Yad. Fiz. 62 (1999) 1851];
See also, M.C. Gonzalez-Garcia, J.J. Gomez-Cadenas, in Ref. [6].
[22] For a more detailed discussion, see K. Cheung, in: H. Athar, G.-L. Lin, K.-W. Ng (Eds.), Proceedings of the First NCTS Workshop on Astroparticle Physics, 2001, Kenting (Taiwan), World Scientific, Singapore, in press. [23] T.K. Gaisser, Cosmic Rays and Particle Physics,
Cam-bridge University Press, New York, 1990. [24] J.-J. Tseng, Ph.D. Thesis, NCTU, submitted.
[25] S. Frixione, M.L. Mangano, P. Nason, G. Ridolfi, Nucl. Phys. B 431 (1994) 453.
[26] T. Sjostrand, Comput. Phys. Commun. 82 (1994) 74. [27] M. Thunman, G. Ingelman, P. Gondolo, Astropart. Phys.
5 (1996) 309.
[28] See, for instance H. Athar, M. Jezabek, O. Yasuda, Phys. Rev. D 62 (2000) 103007, and references therein. [29] L.V. Volkova, G.T. Zatsepin, Phys. Atom. Nucl. 64 (2001)
266 [Yad. Fiz. 64 (2001) 313]. [30] D. Fargion, arXiv:astro-ph/9704205. [31] H. Athar, G.-L. Lin, arXiv:hep-ph/0108204;
H. Athar, G.-L. Lin, arXiv:hep-ph/0201026.
[32] J.G. Learned, S. Pakvasa, Astropart. Phys. 3 (1995) 267. [33] H. Athar, Astropart. Phys. 14 (2000) 217;
H. Athar, G. Parente, E. Zas, Phys. Rev. D 62 (2000) 093010.
[34] See, for instance, VLBL Study Group H2B-1 Collabora-tion, H. Chen et al., arXiv:hep-ph/0104266.
[35] H. Athar, arXiv:hep-ph/0004083. [36] D. Fargion, arXiv:astro-ph/0002453;
X. Bertou, P. Billoir, O. Deligny, C. Lachaud, A. Letessier-Selvon, arXiv:astro-ph/0104452;
J.L. Feng, P. Fisher, F. Wilczek, T.M. Yu, arXiv:hep-ph/ 0105067;
A. Kusenko, T. Weiler, arXiv:hep-ph/0106071.
[37] F. Halzen, D. Saltzberg, Phys. Rev. Lett. 81 (1998) 4305.
[38] F. Becattini, S. Bottai, Astropart. Phys. 15 (2001) 323; S.I. Dutta, M.H. Reno, I. Sarcevic, arXiv:hep-ph/0110245 and references therein;
J.F. Beacom, P. Crotty, E.W. Kolb, arXiv:astro-ph/ 0111482.
