PHYSICAL REVIEW D 69, 103004 共2004兲

Propagation of extremely high energy leptons in Earth:
Implications for their detection by the IceCube neutrino telescope
Shigeru Yoshida,* Rie Ishibashi,† and Hiroko Miyamoto
Department of Physics, Faculty of Science, Chiba University, Chiba 263-8522, Japan
共Received 12 December 2003; published 17 May 2004兲
We present the results of numerical calculations on the propagation of extremely high energy 共EHE兲 neutrinos and charged leptons in Earth for trajectories in the whole phase space of nadir angles. Our comprehensive calculation has shown that not only the secondary produced muons but also taus survive without decaying
in the energy range of 10–100 PeV with an intensity approximately three orders of magnitude lower than theneutrino flux regardless of the EHE neutrino production model. They form detectable horizontal or downgoing
events in a 1 km3 underground neutrino telescope such as the IceCube detector. The event rate and the resulting
detectability of EHE signals in comparison with the atmospheric muon background are also evaluated. The
90% C.L. upper limit of EHE neutrino fluxes by a km2 detection area would be placed at E 2 dF/dE⯝3.7⫻10⫺8 GeV/cm2 sec sr for ␯ ␮ and 4.6⫻10⫺8 for ␯ ␶ with energies of 109 GeV in the absence of signals withan energy loss in a detection volume of 10 PeV or greater.
DOI: 10.1103/PhysRevD.69.103004

PACS number共s兲: 98.70.Sa, 95.85.Ry, 98.70.Vc, 98.80.Cq

I. INTRODUCTION

It is well known that there exist extremely high energy
共EHE兲 particles in the Universe with energies up to⬃1020 eV 关1兴. These EHE cosmic rays 共EHECRs兲 may
originate in and/or produce neutrinos by various mechanisms. For example, collisions of EHECRs and cosmic microwave background 共CMB兲 photons photoproduce cosmogenic neutrinos 关2兴, a consequence of the process known
as the Greisen-Zatsepin-Kuzmin 共GZK兲 mechanism 关3兴. The
possible production of EHECRs in the present Universe dueto the annihilation or collapse of topological defects 共TDs兲
such as monopoles and/or cosmic strings 关4兴 could also generate EHE neutrinos with energies even reaching grand unified theory 共GUT兲 scale 关5,6兴. EHE neutrinos provide, therefore, a unique probe to explore the ultrahigh energy
Universe, which is one of the centerpieces of high energy
neutrino astrophysics.
It has been argued that the underground neutrino telescopes being operated and/or planned to be built are capableof detecting such EHE neutrinos 关7兴. In their travel throughEarth to the detection volume in a telescope, EHE neutrinos
collide with nuclei in rock due to the enhancement of thecross section at EHE range and produce secondary leptons
such as muons and taus. The expected mean free path is⬃600( ␳ rock /2.65 g cm⫺3 ) ⫺1 ( ␴ ␯ /10⫺32 cm2 ) ⫺1 km which
is far shorter than the typical path length of the propagationin Earth. Moreover, the decay lifetime is long enough at
EHEs for the produced ␮ and ␶ to survive and possibly reachthe detection volume directly. Successive reactions of interaction and decay are likely to occur in their propagation, andthe propagation processes of EHE particles are rather com-

*Electronic address: syoshida@hepburn.s.chiba-u.ac.jp; URL:
http://www.ppl.phys.chiba-u.jp/†
Now at Ushio Denki, Co.Ltd., 2-6-1 Oote-machi, Chiyoda-ku,
Tokyo 100-0004, Japan.
0556-2821/2004/69共10兲/103004共9兲/$22.50

plex. The accurate understanding of the EHE neutrino andcharged lepton propagation in the earth is, thus, inevitable
for an EHE neutrino search by underground neutrino telescopes.
There has been considerable discussion in the literaturefrom this point of view. In Ref. 关8兴, the transport equations
mainly focusing on ␯ ␶ and ␶ were solved and the resulting
particle fluxes after propagation have been shown for trajectories of several nadir angles in the horizontal directions suchas 85°. It is true that a major fraction of EHE ␶ tracks arecoming from the horizontal directions because Earth isopaque for EHE neutrinos, but a km3 scale neutrino observatory such as IceCube is essentially a 4 ␲ detector withcomparable sensitivities to both muons and taus, and calculation of the EHE particle energy spectra of both muons andtaus over the whole solid angle space including downward
event trajectories would be important to evaluate detectability with reasonable accuracy. Furthermore, they utilized theoften used continuous energy loss 共CEL兲 approximation that
follows only the leading cascade particles. It is a good approximation for taus, but the secondary particle fluxes contributed from the nonleading particles are not negligible formuons at EHEs where their decay does not play a visiblerole. Calculations on the Earth-skimming EHE ␯ ␶ have also
been made in some detail 关9兴. These authors used approximations to neglect the contributions of the leptons generated
from tau interactions and decay in Earth, which would bevalid enough for consideration of Earth-skimming neutrinoinduced air showers. It has been pointed out, however, thatthe secondarily produced ␯ e and ␯ ␮ from tau decay would
also enhance the total neutrino flux 关10兴, which would be abenefit for an underground neutrino observatory. Following
all propagating leptons and taking into account the contributions from particles not only skimming but propagating
deeper in Earth are, therefore, essential for an undergroundbased neutrino observatory.
In this work, we numerically calculate the intensity andenergy distribution of EHE neutrinos and their secondarily
produced ␮ ’s and ␶ ’s during propagation in Earth for the69 103004-1©2004 The American Physical Society

PHYSICAL REVIEW D 69, 103004 共2004兲

YOSHIDA, ISHIBASHI, AND MIYAMOTO

application to a km3 scale neutrino observatory. The resulting fluxes are shown as a function of nadir angle from downward to upward going directions. All the relevant interactions
are taken into account and we follow all particles producedin the reactions whereas the CEL approximation follows only
the leading cascade particles. The initial flux is mainly assumed to be the bulk of the cosmogenic neutrinos, generated
from the decay of pions photoproduced by EHE cosmic ray
protons colliding with the cosmic thermal background photons, since the cosmogenic neutrino model is appropriate as abenchmark as the flux prediction is on the solid theoreticalfoundation. Its implications for detection by the IceCube
neutrino telescope 关11兴, which is currently under construction at Antarctica, are then discussed in some detail.
The paper is outlined as follows. First we briefly review
the interaction and decay channels involved with EHE particle propagation in Earth in Sec. II. The method of our numerical calculations is also briefly explained. In Sec. III weshow the calculated results: the energy distributions and intensities of muons, taus, and neutrinos after their propagation. The energy spectra of these EHE particles are shown forthe cosmogenic neutrino model. Implications on the detection by the IceCube neutrino telescope are discussed in Sec.
IV and the detectability considering the possible backgroundin the experiment is discussed in detail. The sensitivity to
EHE neutrino fluxes by a km3 neutrino observatory is also
shown. We summarize our conclusions and make suggestions for future work in Sec. V.
II. DYNAMICS OF THE PROPAGATION IN EARTH

EHE neutrinos during propagation do not penetrate Earth
but are involved in charged or neutral current interactions
that generate charged leptons and hadronic showers because
their cross sections are expected to be enhanced in the ultrahigh energy regime. Secondarily produced ␮ ’s and ␶ ’s travelin Earth, initiating many radiative reactions to lose their energy. Higher order interactions like ␮ ⫾ pair production 关12兴and charged current disappearance reactions like ␮ N→ ␯ ␮ X
regenerate charged leptons and neutrinos which are subject
to further interactions. Moreover, the ␶ ’s decay channels like
␶ → ␯ ␶ ␮ ␯ ␮ regenerate ␯ ␶ . A primary EHE neutrino particle,
therefore, results in a number of particles with various energies and species which would pass through an instrumented
volume of an underground neutrino telescope. The resulting
energy spectra and their intensities are consequences of thechain processes of interaction and decay. Table I summarizes
the interaction and decay channels as a function of primaryand generated particle species. The main energy loss process
for secondarily produced ␮ ’s and ␶ ’s are e ⫾ pair creation,bremsstrahlung, and photonuclear interactions. The relevant
cross sections are formulated in Ref. 关13兴 for pair creation,
Ref. 关14兴 for bremsstrahlung, and Ref. 关15兴 for photonuclear
interaction. Among these the photonuclear cross section has
the largest theoretical uncertainty because it relies on thedetails of the nuclei structure function, which has to be estimated from extrapolation from the low energy data. In thepresent calculation we use the estimation based on the deepinelastic scattering formalism with the Abramowicz-Levin-

TABLE I. Interactions and decay channels involved in the EHE
particle propagation in Earth. Rows are primary and columns are
generated particles.

␯e
␯e
␯␮
␯␶
␮
␶

NC

␯␮

␯␶

a

e/␥
CC

␮

NC
D/CC
D

hadron

CC
P
P

CC/NC
CC/NC
CC/NC
PN f/CCPN/CC/D

CC
NC

Dc
D

␶

b

D/CC

P d/B e/D
P/B/D

P
P/D

a

Neutral current interaction.
Charged current interaction.
c
Decay.
dPair creation.
eBremsstrahlung.
f
Photonuclear interaction.
b

Levy-Maor 共ALLM兲 parameterization of the structure function 关16兴, which is considered to be the most reliableprediction. We artificially switch off the photonuclear interaction to see its systematic uncertainty in the results later inthis paper. Furthermore, the weak interaction l ⫾ N→ ␯ X
causing muon and tau disappearances, and the heavier lepton
pair production such as ␮ ⫹ ␮ ⫺ 关12兴, are also taken into account in the present calculation, which leads to a visible
contribution to the particle fluxes at EHEs.
An EHE neutrino is subject to charged current 共CC兲 andneutral current 共NC兲 interactions with nucleons. As there isno direct measurement of the relevant interactions in the
EHE range, the predictions for the ␯ N cross sections rely on
incompletely tested assumptions about the behavior of parton
distributions at very small values of the momentum fraction
x. Since we do not have further clues to investigate EHE
neutrino interactions in our hands, we limit our present
analysis to the range of standard particle physics and use thecross section estimated by Ref. 关17兴 using the CTEQ version
5 parton distribution functions 关18兴.
Decay processes are also major channels and competewith the interaction processes depending on energy. The ␮
and ␶ leptonic decay distribution can be analytically calculated from the decay matrix using the approximation that thegenerated lepton mass is negligible compared to that of theparent lepton 关19兴. For z⫽E ␯ l /E l (l⫽ ␮ or ␶ ) it is written as

冉

冊

dn 5
1
4
8
⫽ ⫺3z 2 ⫹ z 3 ⫺ ⫺3z 2 ⫹ z 3 ,
dz 3
3
3
3

共1兲

and for y ␯ ⫽E ␯ e /E ␮ ( ␮ decay兲, E ␯ e, ␮ /E ␶ ( ␶ decay兲,dn⫽2⫺6y 2␯ ⫹4y ␯3 ⫺ 共 ⫺2⫹12y ␯ ⫺18y 2␯ ⫹8y 3␯ 兲 .
dz

共2兲

The hadronic ␶ decay has various modes and its accurate
treatment is rather difficult. Here we use the two-body decay
approximation as in Ref. 关20兴.
The transport equations to describe particle propagation in
Earth are given by

103004-2

PHYSICAL REVIEW D 69, 103004 共2004兲

PROPAGATION OF EXTREMELY HIGH ENERGY . . .

FIG. 1. The energy distribution of EHE leptons after propagation in Earth with nadir angle of
89.5°. ␮ ’s and ␶ ’s are secondarily
produced. The left panel shows
the distributions of leptons with ␮
flavor while the right panel shows
the case of ␶ flavor. The input
spectrum is 1010 GeV monochromatic with ␯ ␮ and ␯ ␶ having equal
intensity of 1 in these arbitrary
units.

dJ ␯
dX⫽⫺N A ␴ ␯ N,CC⫹NC J ␯ ⫹
⫹N A
⫹N A

dJ l
dX

冕 ⬘␴
冕 ⬘␴
d

dE ␯

dE ␯

d

dE l

⫹N A

⫹

ml
c ␳ ␶ dl E l

冕 ⬘␴
冕 ⬘␴

ml
c ␳ ␶ dlddE ␯

␯ N,CC

dE l

d

dE l

lN,CC

dE ␯

⫽⫺N A ␴ lN J l ⫺⫹N A

␯ N,NC

lN

dE l

冕 ⬘⬘
dE l

ml
c ␳ ␶ dl冕dE l

1 dn dlE l dE ␯

J l共 E l 兲J ␯ 共 E ␯⬘ 兲
J l 共 E ⬘l 兲 ,共3兲JlJ ␯ 共 E ␯⬘ 兲

J l 共 E ⬘l 兲

1 dn dlE l dE l

J l 共 E l⬘ 兲 ,共4兲

where J l ⫽dN l /dE l and J ␯ ⫽dN ␯ /dE ␯ are the differential
fluxes of charged leptons and neutrinos, respectively, N A is
Avogadro’s number, ␳ is the local density of the medium
共rock/ice兲 in the propagation path, ␴ is the relevant interaction cross section, dn dl /dE is the energy distribution of the
decay products, which is derived from the decay rate per unit
energy and given by Eqs. 共1兲 and 共2兲, c is the speed of light,
and m l and ␶ dl are the mass and the decay lifetime of the
lepton l, respectively. The density profile of the rock, ␳ (L),is given by the preliminary Earth model 关21兴. A column density X is defined by X⫽ 兰 L0 ␳ (L ⬘ )dL ⬘ .
Equation 共3兲 describes neutrino propagation. The firstterm is a loss due to the neutrino interaction, the second
represents a contribution due to the decay, and the rest of theterms account for generation of neutrinos by the neutrino andcharged lepton interactions. The fourth term represents neutrino appearance by CC interactions such as ␮ N→ ␯ ␮ X.
Equation 共4兲 describes charged lepton propagation and has
similar terms to those of Eq. 共3兲, but also a term to represent
loss due to lepton decay.
We calculated these equations numerically by building thematrices describing the particle propagation over infinitesi-

mal distances as described in Refs. 关22,23兴. The energy differential cross sections are derived from the ones for the
inelasticity parameter y⫽1⫺E ⬘ /E, i.e., d ␴ /dy. Let us show
two examples to show the behavior of the EHE particle
propagation in Earth. Figure 1 shows the energy distributionof EHE leptons after propagation in Earth, entering with nadir angle of 89.5°. The corresponding propagation distancein Earth is ⬃110 km. The primary input spectrum is amonochromatic energy distribution of 1010 GeV of ␯ ␮ and␯ ␶ with equal intensities. Sizable amounts of the secondarily
produced ␮ ’s and ␶ ’s are found. As the ␮ bulk from ␯ ␶ ismainly generated from ␶ decay, which occurs less frequently
in the high energy region, their intensity decreases withhigher energy. For the same reason, the secondary ␶ energy
distribution is harder than that of ␮ ’s. Note that ␶ originatedin primary ␯ ␮ denoted as ␯ ␮ → ␶ in the right panel in the
figure are produced in heavy lepton pair creation ␮→ ␮␶ ⫹ ␶ ⫺ .
The intensities of ‘‘prompt’’ muons and taus, whose energies are approximately the same as those of the primary neutrinos, are four to five orders of magnitude lower than the
primary neutrino flux as indicated in the figures, but the low
energy bulk of the secondary muons and taus which havesuffered energy loss during their propagation makes a significant contribution to the flux for a given neutrino energy
spectrum. It should also be remarked that the muons generated from secondarily produced tau decay, denoted as ␯ ␶→ ␮ , constitute a major fraction of the intensity below
108 GeV. We see in the next section that they form a sizable
flux for the EHE neutrino model producing a hard energy
spectrum like the cosmogenic neutrinos generated by the
GZK mechanism.
When particles are propagating more vertically upward,i.e., their propagation distance is longer, all the prompt component disappears and no particles essentially survive in the
EHE range, because of the significant energy losses. A typical case is shown in Fig. 2 for nadir angle of 70°. One cansee that most of the secondary muons and neutrinos are absorbed and they remain only in the low energy range.
The energy distributions and intensities of EHE particles
propagating in Earth are, consequently, strongly dependent
on the zenith 共or nadir兲 angle of the trajectory, and also on
the initial neutrino energy spectrum. One must solve thetransport equation in the entire phase space in the zenith
angle in order to make accurate estimations of the fluxes we
see in an underground neutrino telescope for a given neutrino
initial flux.

103004-3

PHYSICAL REVIEW D 69, 103004 共2004兲

YOSHIDA, ISHIBASHI, AND MIYAMOTO

FIG. 2. Same as Fig. 1, but for nadir angle of 70°. Shown are
the distributions of leptons with ␮ flavor when the input ␯ ␮ and ␯ ␶is monochromatic energy of 1010 GeV.
III. THE COSMOGENIC NEUTRINO FLUX AT
UNDERGROUND DEPTH

In this section, we discuss the case when the initial fluxesof ␯ ␮ and ␯ ␶ are given by the GZK mechanism, EHE neutrino production by collisions of EHECRs with CMB photons in extragalactic space, as this model has been thought tobe the most conventional mechanism to generate EHE neutrinos without new physics and/or speculative assumptions.
The biggest uncertainty in the intensity of the cosmogenic
fluxes is related to the cosmic ray source distributions. Assuming homogeneously distributed astrophysical sources,
however, variations of the magnitude of the neutrino flux
above 109 GeV are restricted approximately within a factorof 10 关24兴. Although assuming an extremely hard cosmic ray
injection spectrum like ⬃E ⫺1 or very strong source evolution allows larger fluxes which can still be consistent with
the EHECRs and the EGRET ␥ -ray observations 关25兴, here
we limit the present calculations to the conventional case thathomogeneously distributed astrophysical sources are responsible for the observed EHECR flux below 1020 eV.
We solve Eqs. 共3兲 and 共4兲 to evaluate the particle fluxes at
an underground depth where a kilometer-scale neutrino observatory is expected to be located. The IceCube neutrino
telescope is constructed at 1400 m depth below the ice surface and we take this number as a representative depth. It has
been found that changing this depth within a factor of 2
would not affect the overall EHE particle intensity in a significant manner and the conclusion remains the same. Neutrino oscillation with full mixing is assumed and the ␯ ␮ initial flux is identical to that of ␯ ␶ . For the parametersconstrained by the SuperK experiment 关26兴, the oscillation
probability in the earth in the EHE range is negligible, however, and we do not account for the oscillation in the present
calculation on propagation.
Figure 3 shows the fluxes with nadir angle of 85° and
70°. The initial primary cosmogenic neutrino fluxes aretaken from Ref. 关22兴. Taus notably dominate muons because
their heavy mass makes them penetrate Earth and because
decay is less important than interactions for the relevant energy range. The case of nadir angle of 70° exhibits the strong
attenuation, however, due to the fact that the mean free pathsof all the relevant interactions including the weak interac-

FIG. 3. Fluxes of EHE particles at the IceCube depth for the
scenario of neutrino production by the GZK mechanism. Two casesof nadir angle are shown in the figure.

tions of neutrinos are far shorter than the propagation distance. This implies that most of the upgoing events in aneutrino observatory come from horizontal directions.
The ␯ ␶ flux becomes dominating over that of ␯ ␮ in the
low energy range where the ␶ decay is significantly more
important than interactions. This enhancement is caused bythe ␯ ␶ → ␶ → ␯ ␶ regeneration process. Note that the small
bump in the ␯ e spectrum is not a propagation effect but generated primarily by EHECR neutron decay in space 关22,24兴.
The intensity strongly depends on the nadir angle. Figure
4 shows the dependence of the secondary muon and tau
fluxes on the zenith angles. Strong attenuation by Earth canbe seen but the fluxes are more or less stable in the region ofthe ‘‘downward’’ events where cos ␪⭓0. Particles in thisrange are propagating in ice ( ␳ ⫽0.917 g/cm3 ) to enter into
the detection volume. We numerically solved the transport
equation in the ice medium to derive the downward fluxes.
The downward fluxes constitute a major fraction of events inan underground neutrino observatory. The detection issues
are discussed in Sec. IV.

FIG. 4. Dependence of the muon and tau fluxes originating in
the cosmogenic neutrinos on the cosine of zenith angle. The integrated flux above 10 PeV is plotted on a linear scale. The atmospheric muon fluxes are also shown by the solid curve for a conservative estimation with low energy extrapolation and by thedashed curve for a CORSIKA-based estimation. The details of theatmospheric fluxes are discussed in Sec. IV.

103004-4

PHYSICAL REVIEW D 69, 103004 共2004兲

PROPAGATION OF EXTREMELY HIGH ENERGY . . .

FIG. 5. Energy spectra of ␯ e , ␯ ␮ , ␯ ␶ , ␮ , ␶ originating in the cosmogenic neutrinos at the IceCube depth. The intensities are integrated oversolid angle and shown for the upward region (cos ␪⭐0, left panel兲 and the downward region (cos ␪⭓0, right panel兲. The two dashed lines
represent the atmospheric muon intensities. The upper line shows a conservative estimation based on simple extrapolation from thecalculation at 5 TeV, while the lower line is derived by Monte Carlo simulation with the CORSIKA package.

The energy spectra integrated over zenith angle are shownin Fig. 5. Secondary muons and taus form a potentially detectable bulk with intensity ⬃ three orders of magnitude
lower than the neutrino fluxes. The main energy range is 10
PeV to 10 EeV (⫽1010 GeV). Regardless of the neutrino
production model, the intensity of ␮ and ␶ relative to ␯ ␮ and␯ ␶ remains approximately unchanged. It should be remarked
that the intensity of the downward going muons and taus islarger than the upward one by an order of magnitude. As also
seen in Fig. 3, the tau flux dominates over the muons above
108 GeV. Enhancement of ␯ ␶ intensity by regeneration also
appears in the upward going trajectories.
The uncertainty in the muon and tau flux estimations
mainly arises from the fact that we do not know the photonuclear cross section accurately in the EHE range. For example, using the updated approach to deduce the photonuclear cross section including the soft part of thephotonuclear interaction would lead to ⬃30% enhancementof the total tau energy loss in the EHE range 关27兴. To beconservative, in Fig. 6 we show a comparison of the fluxes
with and without photonuclear reactions. Switching off the

FIG. 6. Dependence of the muon and tau upward fluxes onphotonuclear interactions. The integrated fluxes over nadir angles of
0° to 90° are shown.

photonuclear interactions results in a factor of 2 variance in
the intensity, which would represent the error range of thesecondary tau flux estimations in a conservative manner.
IV. DETECTION BY A KILOMETER-SCALE NEUTRINOOBSERVATORYThe event rate for a neutrino observatory can be estimatedby integrating the energy spectra shown in Fig. 5 above athreshold energy multiplied by the effective area of the detector, which is 1 km2 in the case of IceCube. The downward
events are major contributions and it is necessary to consider
the atmospheric muon background, however. The atmospheric muon flux estimation in the relevant energy range is
not straightforward because there is no measurement available and numerical calculation is also time consuming as one
must fully simulate EHE air shower cascades. Here we use
two methods to estimate the flux. One is to extrapolate thecalculation at 5 TeV 关28兴 which has been confirmed to beconsistent with the measurement. Because the cosmic rayenergy spectrum follows E ⫺2.7 in the TeV region while highenergy cosmic ray spectra above 10 PeV are steeper, following E ⫺3 , this extrapolation would overestimate the flux, but
it gives a conservative evaluation. The other method is to run
the CORSIKA air shower simulation 关29兴 with the energy
spectrum of the observed E ⫺3 dependence under the assumption that the whole mass composition is protons, andcount the number of high energy muons reaching the ground.
Then we solve the transport equations for the derived muon
fluxes at the surface. The results obtained for the background
intensity in downward events are shown in the right panel ofFig. 5 by two dashed lines. One can see that the muon background spectrum is quite steep. Setting a higher thresholdenergy, therefore, would allow elimination of the backgroundcontamination. The flux dependence on the zenith angle isshown in Fig. 4 when the threshold energy is 10 PeV. It isclearly seen that the muon background attenuates faster than
the neutrino-induced EHE muons and taus, and there is awindow where the signals dominate the muon background.
Table II summarizes the intensity with threshold energy of 10

103004-5

PHYSICAL REVIEW D 69, 103004 共2004兲

YOSHIDA, ISHIBASHI, AND MIYAMOTOTABLE II. Integral flux intensities for several EHE neutrino models.
I ␯ ␮ , ␶ (E⭓10 PeV) a(cm⫺2 sec⫺1 2 ␲ ⫺1 )I ␮ (E⭓10 PeV)(cm⫺2 sec⫺1 )I ␶ (E⭓10 PeV)(cm⫺2 sec⫺1 )I ␮ (E loss ⭓10 PeV)(cm⫺2 sec⫺1 )I ␶ (E loss ⭓10 PeV)(cm⫺2 sec⫺1 )5.97⫻10⫺16
5.97⫻10⫺16
9.92⫻10⫺15——

5.90⫻10⫺19
3.91⫻10⫺20
5.48⫻10⫺18
2.06⫻10⫺18
7.25⫻10⫺19

5.97⫻10⫺19
6.63⫻10⫺20
5.11⫻10⫺18——

4.75⫻10⫺19
2.57⫻10⫺20
3.75⫻10⫺18
1.74⫻10⫺19
5.34⫻10⫺20

3.28⫻10⫺19
2.64⫻10⫺20
2.94⫻10⫺18——

GZK b downward
GZK upward
TD c downward
Atmospheric ␮Atmospheric ␮ daIntensity at surface before propagating in Earth.
Cosmogenic neutrinos with (m,Z max )⫽(4.0,4.0) in Ref. 关22兴.
cTopological defect scenario using supersymmetry-based fragmentation function in Ref. 关6兴.
d
Estimation based on the CORSIKA simulation.
b

PeV for the various EHE neutrino models together withthose of the atmospheric muon background.
In fact, what neutrino detectors can measure in a directmanner is not the energy of muon or tau tracks but the energy loss in the detection volume. The relation between energy and energy loss is approximately ⫺dE/dX⬃ ␤ E. Here
␤ is the average inelasticity given by

␤⫽

冕

y max

y min

dy ⬘ y ⬘

d␴
dy ⬘

共5兲

.

For the e ⫾ pair creation of muons in ice, ␤ ⯝1.3
⫻10⫺6 cm2 /g. Therefore the average energy loss fraction
due to the pair creation is⌬E⫾⯝ ␤ e ⌬X
E
⫽0.12

冉

␤e

⫾

1.3⫻10⫺6冊冉

␳ ice
0.92 g cm⫺2

冊冉 冊

⌬L
,
1 km

共6兲

indicating that 10% of the muon primary energy is depositedin a detection volume. Because radiative interactions likebremsstrahlung have a stochastic nature, ⌬E fluctuates significantly on an event by event basis, however. We carriedout a Monte Carlo simulation to see the fluctuation. The
simulation code uses the same cross section and decay tables
but calculates the energy of a particle after an infinitesimal
propagation length ⌬X with the Monte Carlo method insteadof solving the transport equations. Figure 7 shows the distribution of the energy loss of muons in running over 1 km in
ice. The energy loss distribution due to the pair creation may
be narrow enough for the CEL approximation; this is not thecase for the distributions due to the other interactions, however. It is not appropriate to approximate the entire distribution by a ␦ function, which implies that the energy loss
rather than the energy would be better to describe the event
characteristics.
As a more realistic criterion, we introduce the threshold of
the energy loss in ice instead of the energy itself. Figure 8
shows the GZK integrated flux dependences on the zenith
angle in the case of the 10 PeV threshold of the energy loss.
One can see in comparison to Fig. 4 that the GZK fluxes arelarger than or comparable to the muon background intensity
in all the zenith directions in this energy-loss-based criterion.
This indicates that it is probable that an EHE neutrino searchusing downward events can be made under an almost
background-free environment.
It should be noted that the tau flux is lower than the muon
flux with this criterion. This is because the heavier mass ofthe tau suppresses the energy loss compared to that of muons
with the same energy. This situation is illustrated in Fig. 9
where the tau fluxes are plotted as functions of energy andenergy loss in ice during 1 km propagation. The intensity
above 107 GeV is reduced because of the energy loss suppression. Higher energy loss takes place in the form of hadronic cascades initiated by the photonuclear interaction.
Table II lists the intensity of muons and taus above 10 PeVof the energy loss for the fluxes of the cosmogenic 关22兴 andtop-down 关6兴 models. The event rate under this criterion isfound to be 0.27( ␮ ⫹ ␶ )/km2 yr for cosmogenic neutrino
fluxes with moderate source evolution. Note that the downward event rate is 0.25/km2 yr and dominates in the overallrate. The event rates for the various neutrino production
models are summarized in Table III.
The IceCube sensitivity to EHE neutrinos can be evaluated by the event rate per energy decade dN/d log E. For agiven energy of primary neutrinos, the secondary muon andtau fluxes are calculated by the transport equations Eqs. 共3兲
and 共4兲 as a function of zenith angles. The probability that

FIG. 7. Distribution of energy loss in propagation of muons over
1 km in ice. The primary energy of muons is 1010 GeV. Contributions from each interaction are shown separately.

103004-6

PHYSICAL REVIEW D 69, 103004 共2004兲

PROPAGATION OF EXTREMELY HIGH ENERGY . . .

FIG. 8. Dependence of the muon and tau fluxes originating in
the cosmogenic neutrinos on the cosine of zenith angle. The integral
flux above 10 PeV of the energy loss is plotted on a linear scale.
The atmospheric muon fluxes are also shown by the solid curve forthe conservative estimation with the low energy extrapolation andby the dashed curve for the CORSIKA-based estimation.

energy loss with the threshold value or greater occurs is estimated by a Monte Carlo simulation and convoluted withflux integration over energy and zenith angle to give the rate.
Figure 10 shows the resulting sensitivity of the IceCube detector with 1 km2 detection area. The various model predictions are also shown for comparison. The 90% C.L. upper
limit, i.e., 2.3 event/energy decade/10 yr is plotted for a 10
PeV and 1 PeV threshold of the energy loss, respectively.
The ␯ ␮ sensitivity is better than that for ␯ ␶ below the
108 GeV region because muon energy loss in a detection
volume is larger than that of taus with the same energy, but
tau decay, which results in a large energy deposit in the detection volume, makes the dominant contribution in ␯ ␶ sensitivity in this relatively low energy range, forming a slightbump structure in the sensitivity curve. The 90% C.L. upper
limit of EHE neutrino fluxes in a km2 detection area would
be placed at E 2 dF/dE⯝3.7⫻10⫺8 GeV/cm2 sec sr for ␯ ␮
and 4.6⫻10⫺8 for ␯ ␶ with energies of 109 GeV in the absence of signals with energy loss in a detection volume of 10
PeV or greater.
This bound would not exclude the cosmogenic neutrino
production model but strongly constrains the cosmic ray injection spectrum in the model. Cosmic ray nucleon injection
spectra harder than E ⫺1.5 would violate the bound 关25兴. On
the other hand, as long as the injection spectrum is softer
than E ⫺2 , which is very likely in the case of astrophysicalcosmic ray sources, the IceCube bound would constrain thesource evolution less as seen in Fig. 10, where we plotted an
extreme scenario of cosmological evolution (1⫹z) 5 where z
is the redshift 关25兴. Stronger evolution possibilities than thiscase are inconsistent with the diffuse background ␥ -ray observation by EGRET 关30兴, since the GZK mechanism also
initiated electromagnetic cascades 关22,23,25兴 via photoproduced ␲ 0 decay and e ⫾ pair creation by EHECR collisions
with CMB photons, forming the photon flux below 100 GeV
which is constrained by the observation.
The topological defect scenario, on the other hand, would
be severely constrained by the absence of EHE event detection by IceCube. The expected event rate is⬃2 events/yr km2 as one can calculate from Table II. The

FIG. 9. The tau fluxes at IceCube depth originating in the cosmogenic neutrinos as a function of energy 共the dashed curve兲, totalenergy loss in ice 共the solid curve兲, energy loss in the form ofelectromagnetic cascades 共the dash-dotted curve兲, and in the formof hadronic cascades 共the dotted curve兲.

expected EHE neutrino flux in the Z burst model 关31兴, thescenario that the collisions of EHE neutrinos with the cosmological background neutrinos explain the EHECR fluxes
without the GZK cutoff, is well below the IceCube bound if
the injection neutrino spectrum is E ⫺1 as described in Ref.
关32兴.
Although less significant, there are ␮ and ␶ events produced by neutrinos inside the detector instrumented volume.
In this case the charged leptons produced propagate in only apart of the observation volume. We carried out the same
Monte Carlo simulation deriving the results of Fig. 7 but inwhich ␯ ␮ and ␯ ␶ were initially entering into the ice volume.
The probability that neutrinos interact inside the 1 km3 volume and that the produced muon or tau loses energy greaterthan 10 PeV was estimated and convoluted with the neutrino
intensity at the IceCube depth. The detection sensitivities by
this channel are shown as thick dashed curves in Fig. 10. In
the EHE regime above ⬃108 GeV, the intensity of internallyproduced muon and tau events is too small to contribute tothe overall sensitivity because the neutrino target volume islimited by the size of the detector, i.e., 1 km3 . Below
108 GeV, on the other hand, including this channel improves
the sensitivity in a sizable manner because the energy lossesof muons and taus during their propagation over long distances are more likely to transfer them out of the energy
range above the 10 PeV threshold, which leads to reductionof the effective neutrino target volume for producing EHE
TABLE III. The event rates for several EHE neutrino models.
The notation for the model name is the same as in Table II.
N ␮ (E loss ⭓10 PeV) N ␶ (E loss ⭓10 PeV)(km⫺2 yr⫺1 )(km⫺2 yr⫺1 )GZK downward
GZK upward
TD downward
Atmospheric ␮Atmospheric ␮ 共CORSIKA兲

103004-7

0.15
0.0081
1.18
0.055
0.016

0.10
0.00830.93——

PHYSICAL REVIEW D 69, 103004 共2004兲

YOSHIDA, ISHIBASHI, AND MIYAMOTO

FIG. 10. The IceCube sensitivities to the EHE neutrino fluxes. 90% C.L. limits with a 1 km2 detection area with 10 yr observation are
drawn. The left panel shows the case of ␯ ␮ and the right panel shows the ␯ ␶ case. Labels refer to GZK 共关22兴 for the lower curve, 关25兴 forthe upper curve兲, TD 关6兴, and Z burst 关32兴. The dashed curves show the sensitivities determined by events of neutrinos interacting inside thedetector volume.

muons and taus outside the detector volume. There is little
gain in EHE neutrino searches, however, because the proposed EHE neutrino models have their main energy range
above 108 GeV.
V. SUMMARY AND OUTLOOK

are heavy nuclei 关34兴 and may not constitute a background in
the EHE neutrino search. Even if the prompt muon intensity
is sizable, their energy spectrum will still be much steeper
than the expected spectrum in the proposed EHE cosmic
neutrino models, however, and one can easily distinguish thesignal detections from the prompt muon background events
if the neutrino observatory has reasonable resolution for theenergy loss of muon and tau tracks. The detector resolution
issues require detailed detector Monte Carlo simulations forfurther investigations. The AMANDA experience in relatively low energy muon reconstruction would lead to energy
resolution of ⌬ log E⯝0.3 关35兴. The development of a detector Monte Carlo simulation is in progress and its application
to the present results will be important future work toward
the search for EHE neutrinos by the IceCube observatory.

We calculated the propagation of the EHE neutrinos andcharged leptons in Earth to derive their intensities and theirdependence on nadir angle. The secondarily produced muonsand taus form detectable fluxes at the IceCube depth, with anintensity three orders of magnitude lower than the neutrino
fluxes. A realistic criterion, requiring energy deposit greaterthan 10 PeV in 1 km3 volume of ice, leads to⬃0.27 events/yr for the cosmogenic neutrinos in the case ofmoderate source evolution. The topological defect scenario
would be severely constrained.
The atmospheric muon background is likely to be negligible even for downgoing events. The background rate is⬃0.05 event/km2 yr. It should be noted that we ignored thepossible contributions of prompt muons from the charm decay in EHE cosmic ray air showers 关33兴, however. The atmospheric muon intensity can be increased by an order ofmagnitude but a large uncertainty remains due to highly uncertain cross sections in charm production. The mass composition of cosmic rays in the relevant energy range would
also be a deciding factor of the prompt muon flux intensity:
It can be reduced by an order of magnitude if the cosmic rays

We wish to acknowledge the IceCube Collaboration for
useful discussions and suggestions. We thank Dmitry Chirkin, Thomas Gaisser, and Esteban Roulet for helpful discussions on atmospheric muon issues. We also thank John Beacom and Igor Sokalski for their valuable comments. Special
thanks go to Mary Hall Reno for providing the cross section
table evaluated by the CTEQ version 5 parton distribution.
This work was supported in part by Grants-in-Aid 共Grants
No. 15403004 and No. 15740135兲 in Scientific Researchfrom the MEXT 共Ministry of Education, Culture, Sports, Science, and Technology兲 in Japan.

关1兴 For a review see, e.g., M. Nagano and A.A. Watson, Rev. Mod.
Phys. 72, 689 共2000兲; J.W. Cronin, ibid. 71, S165 共1999兲; S.
Yoshida and H. Dai, J. Phys. G 24, 905 共1998兲.
关2兴 V.S. Beresinsky and G.T. Zatsepin, Phys. Lett. 28B, 423
共1969兲.
关3兴 K. Greisen, Phys. Rev. Lett. 16, 748 共1966兲; G.T. Zatsepin and
V.A. Kuzmin, Pis’ma Zh. Eksp. Teor. Fiz. 4, 114 共1966兲 关JETP
Lett. 4, 78 共1966兲兴.

关4兴 P. Bhattacharjee, C.T. Hill, and D.N. Schramm, Phys. Rev.
Lett. 69, 567 共1992兲.
关5兴 G. Sigl, S. Lee, D.N. Schramm, and P.S. Coppi, Phys. Lett. B
392, 129 共1997兲.
关6兴 G. Sigl, S. Lee, P. Bhattacharjee, and S. Yoshida, Phys. Rev. D
59, 043504 共1999兲.
关7兴 J. Alvarez-Muñiz and F. Halzen, Phys. Rev. D 63, 037302
共2001兲.

ACKNOWLEDGMENTS

103004-8

PHYSICAL REVIEW D 69, 103004 共2004兲

PROPAGATION OF EXTREMELY HIGH ENERGY . . .
关8兴 J. Jones, I. Mocioiu, M.H. Reno, and I. Sarceiv, Phys. Rev. D
共to be published兲, hep-ph/0308042.
关9兴 J.L. Feng, P. Fisher, F. Wilczek, and T.M. Yu, Phys. Rev. Lett.
88, 161102 共2003兲; K. Giesel, J.-H. Jureit, and E. Reya, Astropart. Phys. 20, 335 共2003兲.
关10兴 J.F. Beacom, P. Crotty, and E.W. Kolb, Phys. Rev. D 66,
021302 共2002兲.
关11兴 S. Yoshida, in Proceedings of the 28th ICRC, H.E.2.3, 1369,
2003; http://icecube.wisc.edu/
关12兴 S. Kelner, R.P. Kokoulin, and A.A. Petrukhin, Phys. At. Nucl.
63, 1603 共2000兲.
关13兴 R.P. Kokoulin and A.A. Petrukhin, in Proceedings of the 12th
ICRC 共Hobert兲, Vol. 6, p. A2436, 1971; S.R. Kelner, R.P. Kokoulin, and A.A. Petrukhin, Phys. At. Nucl. 62, 1894 共1999兲.
关14兴 Yu.M. Andreev, L.B. Bezrukov, and E.V. Bugaev, Phys. At.
Nucl. 57, 2066 共1994兲; S.R. Kelner, R.P. Kokoulin, and A.A.
Petrukhin, ibid. 60, 576 共1997兲; I.A. Sokalski, E.V. Bugaev,
and S.I. Klimushin, Phys. Rev. D 64, 074015 共2001兲.
关15兴 S. IyerDutta, M.H. Reno, I. Sarcevic, and D. Seckel, Phys.
Rev. D 63, 094020 共2001兲.
关16兴 H. Abramowicz and A. Levy, hep-ph/9712415.
关17兴 R. Gandhi, C. Quigg, M.H. Reno, and I. Sarcevic, Astropart.
Phys. 5, 81 共1996兲; Phys. Rev. D 58, 093009 共1998兲.
关18兴 CTEQ Collaboration, H. Lai et al., Phys. Rev. D 55, 1280
共1997兲.
关19兴 T.K. Gaisser, Cosmic Rays and Particle Physics 共Cambridge
University Press, Cambridge, England, 1990兲.

关20兴 S.I. Dutta, M.H. Reno, and I. Sarcevic, Phys. Rev. D 62,
123001 共2000兲.
关21兴 A.M. Dziewonsky and D.L. Anderson, Phys. Earth Planet. Inter.
25,297
共1981兲;S.V.
Panasyuk,
http://
cfauvcs5.harvard.edu/lana/rem/index.html
关22兴 S. Yoshida and M. Teshima, Prog. Theor. Phys. 89, 833 共1993兲.
关23兴 R.J. Protheroe and P.A. Johnson, Astropart. Phys. 4, 253
共1996兲.
关24兴 R. Engel, D. Seckel, and T. Stanev, Phys. Rev. D 64, 093010
共2001兲.
关25兴 O.E. Kalashev, V.A. Kuzmin, D.V. Semikoz, and G. Sigl, Phys.
Rev. D 66, 063004 共2002兲.
关26兴 SuperKamiokande Collaboration, Y. Fukuda et al., Phys. Rev.
Lett. 85, 3999 共2000兲.
关27兴 I. Sokalski 共private communication兲.
关28兴 V. Agrawal, T.K. Gaisser, P. Lipari, and T. Stanev, Phys. Rev.
D 53, 1314 共1996兲.
关29兴 D. Heck et al., Report No. FZKA 6019, Forschungszentrum
Karlsruhe, 1998.
关30兴 P. Sreekumar et al., Astrophys. J. 494, 523 共1998兲.
关31兴 D. Fargion, B. Mele, and A. Salis, Astrophys. Lett. 517, 725
共1999兲; T.J. Weiler, Astropart. Phys. 11, 303 共1999兲.
关32兴 S. Yoshida, G. Sigl, and S. Lee, Phys. Rev. Lett. 81, 5505
共1998兲.
关33兴 See, e.g., T.S. Sinegovskaya and S.I. Sinegovsky, Phys. Rev. D
63, 096004 共2001兲, and references therein.
关34兴 J. Candia and E. Roulet, J. Cosmol. Astropart. Phys. 09, 005
共2003兲.
关35兴 IceCube Collaboration, J. Ahrens et al., astro-ph/0305196.

103004-9