PHYSICAL REVIEW D 69, 033004 共2004兲

Tracing very high energy neutrinos from cosmological distances in ice
J. Jones,1 I. Mocioiu,1,3 M. H. Reno,2 and I. Sarcevic1,3
1

2

Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA
3
KITP, UCSB, Santa Barbara, California 93106, USA
共Received 28 August 2003; published 25 February 2004兲

Astrophysical sources of ultrahigh energy neutrinos yield tau neutrino fluxes due to neutrino oscillations. We
study in detail the contribution of tau neutrinos with energies above 106 GeV relative to the contribution of theother flavors. We consider several different initial neutrino fluxes and include tau neutrino regeneration intransit through the Earth and the energy loss of charged leptons. We discuss the signals of tau neutrinos indetectors such as IceCube, RICE, and ANITA.
DOI: 10.1103/PhysRevD.69.033004

PACS number共s兲: 13.15.⫹g, 96.40.Tv

I. INTRODUCTION

trinos produce hadronic showers. In addition, tau decays
contribute to both electromagnetic and hadronic showers that
could be detected by IceCube, RICE, or ANITA.
In this paper, we investigate the effect of ␯ ␶ regenerationfrom tau decays and tau energy loss for neutrinos with energies above 106 GeV, with a particular interest in the higher
energy range relevant for RICE and ANITA. Attenuation
shadows most of the upward-going solid angle for neutrinos,
so we concentrate on incident neutrinos which are nearlyhorizontal or slightly upward going. Considering the correct
propagation of tau neutrinos is important for a number ofreasons. The effective volume for the detection of ␯ ␶ is larger
than for the other flavors due to the regeneration effects. The
initial interaction happens far outside the detector, but theneutrinos 共taus兲 produced in the neutrino interaction–tau decay cascade interact 共decay兲 inside the detector. This can leadto enhanced event rates for particular energies and trajectories. The regenerated neutrinos and taus contribute, however,at lower energies than the initial ones, so a detailed quantitative analysis is necessary to understand where the regeneration effects are significant. In addition to the possible enhancement in the rates, a good discrimination between all thedifferent neutrino flavors is important. The flavor composition of the detected neutrino fluxes could provide a better
understanding of the sources of such neutrinos and neutrino
properties. The separate identification of each flavor could be
important for this purpose 关11兴.
We illustrate our results with a variety of fluxes. We concentrate our analysis on the flux of Greisen-Zatsepin-Kuzmin
共GZK兲 neutrinos 关12兴. These are produced in the decay ofpions from the interaction of cosmic ray protons with thebackground microwave photons, so they are a ‘‘guaranteed’’
source. In addition, they are expected to dominate the neutrino flux at energies for which the radio Cherenkov detection methods are most sensitive if the Z burst model discussed below is not an explanation of the highest energy
cosmic rays. Detection of GZK neutrinos could be essential
for the understanding of very high energy cosmic ray protons. The normalization of the GZK flux is somewhat uncertain. Our results are based on a conservative value, which
considers ‘‘standard’’ cosmological evolution 关12兴. For‘‘strong’’ source evolution the same general behavior of theflux is valid, but the flux is a factor of 4 higher, leading toVery high energy neutrinos can be unique probes of both
astrophysics and particle physics. They point back to theirsources, escape from the most extreme environments, andhave energies much higher than those available in colliders.
The SuperKamiokande experimental data on atmospheric
neutrinos 关1兴 show evidence of nearly maximal ␯ ␮ ↔ ␯ ␶ mixing. As a result, astrophysical sources of ␯ ␮ become sourcesof ␯ ␮ and ␯ ␶ in equal proportions after oscillations over astronomical distances 关2兴. We do not differentiate between
neutrinos and antineutrinos, since they cannot be distinguished experimentally. The neutral current cross sectionsfor ␯ ␮ and ␯ ␶ are identical, and above ⬃1 TeV the charged
current cross sections are identical as well. Nevertheless, signals from ␯ ␶ have the potential to contribute differently fromthe ␯ ␮ ’s because their propagation through matter is different. Tau neutrinos produce tau leptons via charged current
interactions in the Earth. Having a short lifetime, the tau
leptons decay, producing ␯ ␶ , which then interact and produce ␶ , resulting in a cascade that produces ␯ ␶ and ␶ withenergies lower than the original flux 关3,4兴. The leptonic tau
decays also produce secondary ␯ ␮ and ␯ e neutrinos 关23兴. All
neutrinos from this cascade can then interact in the detector.
The decays of taus in the detector also contribute to the signal 关5,6兴. For muons, the electromagnetic energy loss
coupled with the long muon lifetime make regeneration of␯ ␮ from muon decay negligible, and high energy ␯ ␮ ’s getlarge attenuation as they propagate through the Earth.
Signals of neutrino interactions in the rock below the ice
or in the ice depend on the energy and flavor of the neutrino.
Muon neutrino charged current 共CC兲 conversions to muons
are noted by the Cherenkov signal of upward going muons ina detector such as IceCube 关7兴. High energy electromagnetic
showers from ␯ e →e CC interactions produce Cherenkov
radiation which is coherent for radio wavelengths 关8兴. The
Radio Ice Cherenkov Experiment 共RICE兲 has put limits on
incident isotropic electron neutrino fluxes which produce
downward-going electromagnetic showers 关9兴. The Antarctic
Impulsive Transient Antenna 共ANITA兲 also uses the ice as aneutrino converter 关10兴. These balloon missions will monitor
the ice sheet for refracted radio frequency signals with aneffective telescope area of 1⫻106 km2 . All flavors of neu0556-2821/2004/69共3兲/033004共12兲/$22.50

69 033004-1©2004 The American Physical Society

PHYSICAL REVIEW D 69, 033004 共2004兲

JONES et al.

FIG. 1. 共Color online兲 Initial neutrino fluxes.

higher rates. Other studies 关13兴 obtain fluxes different by up
to one order of magnitude. Since the normalization we use is
the smallest, our results are conservative and rates could actually be higher.
In order to discuss how the results depend on the shape of
the initial flux, we also show two generic distributions 1/E
and 1/E 2 . The 1/E flux describes well the neutrinos from Z
burst models 关14,15兴. Z burst models could explain the highest energy cosmic rays by extremely high energy neutrinos
scattering on nearby massive relic neutrinos. We also show
results for the neutrino fluxes from active galactic nuclei
共AGN兲 for the model in Ref. 关16兴. This flux has an approximate 1/E 2 behavior for E⫽106 –109 GeV, so toward thelower end of that energy range our 1/E 2 results apply. All of
the fluxes used in our analysis are shown in Fig. 1, except forthe 1/E 2 distribution, which would correspond to just a horizontal line.
In the following section we describe our methods of computing the fluxes of neutrinos and charged leptons after
propagation through the Earth. In Sec. III we discuss theeffect of regeneration and lepton energy loss on the propagation of the initial neutrino fluxes. In Sec. IV we present our
results for the distributions of electromagnetic and hadronic
showers from tau decays and neutrino interactions relevantto the different types of experiments. Other recent papers
关5,17兴 have focused on IceCube detection or on air showers.
Related work was recently presented at ICRC 2003 关18兴.
Here, we consider IceCube, as well as RICE and ANITA,
which also use the ice as detector and have higher energy
sensitivities. In the case of ANITA, one has dramaticallylonger trajectories through the ice to consider.

FIG. 2. Neutrino interaction length 共solid line兲 and tau effective
decay length neglecting energy loss 共dashed line兲 and including
electromagnetic energy loss in water 共solid line兲.

essentially equal because the cross sections are sea quark
dominated. The neutral current contribution to the total crosssection is about 1/2 of the charged current contribution.
To compare the interaction lengths with physical distances, we note that D⫽2R 丣 cos ␪␳⯝6⫻108 cm waterequivalent 共w.e.兲 for ␪ ⫽80° and D⯝2⫻107 cm w.e. for ␪⫽89° where R 丣 ⫽6.37⫻108 cm is the radius of the Earth.
Neutrino attenuation is clearly an important effect, even fornearly horizontal incident neutrinos. In the figures below, we
mainly show results for a nadir angle of 85° where attenuation and regeneration effects in the propagation of ␯ ␶ ’s are ineffect without dramatically reducing the flux. We also compare fluxes with those from incident 80° and 89° nadir
angles.
The effective decay length of the tau shows that, for energies above about 108 GeV, the tau is more likely to interact electromagnetically than to decay 关20兴. For an initial tau
energy of 1012 GeV, the average energy just before it decays
is a few⫻108 GeV, depending on the density of the material
the tau is passing through. Its effective decay length is oforder 50 km in water. We use a density of ␳ ⫽0.9 g/cm3 forice.
For tau neutrinos, we take into account the attenuation bycharged current interactions, the shift in energy due to neutral current interactions, and the regeneration from tau decay.
For tau leptons we consider their production in charged current ␯ ␶ interactions, and their decay, as well as electromagnetic energy loss. The tau neutrino and tau fluxes satisfy thefollowing transport equations:

II. NEUTRINO AND CHARGED LEPTON PROPAGATION

Neutrino attenuation and regeneration are governed by interaction lengths and decay lengths. In Fig. 2, we show theneutrino interaction length 共in water equivalent distance兲, aswell as the tau decay length and the effective decay length
when one includes the tau energy loss in water 关19,20兴.
The upper curve for the neutrino interaction length isequally applicable to antineutrinos, since at high energies theneutrino and antineutrino cross sections with nucleons are
033004-2

⳵ F ␯ ␶ 共 E,X 兲
⳵X⫽⫺N A ␴ tot 共 E 兲 F ␯ ␶ 共 E,X 兲
⫹N A

冕

冕

dE y

⫹

⬁

E

dE y F ␯ ␶ 共 E y ,X 兲⬁Ed ␴ NC共 E y ,E 兲dEF ␶ 共 E,X 兲 dn
共 E y ,E 兲 ,␭ ␶dec dE

共1兲

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TRACING VERY HIGH ENERGY NEUTRINOS FROM . . .

teristic of each of the secondary neutrinos, and we include
them here, even though they are very small.

⳵ F ␶ 共 E,X 兲
F ␶ 共 E,X 兲⫽⫺ dec⳵X␭ ␶ 共 E,X, ␪ 兲
⫹N A

冕

⫺

dE y F ␯ ␶ 共 E y ,X 兲d ␴ CC共 E y ,E 兲 , 共2兲dEdE ␶⫽␣⫹␤E␶ .
dX

共3兲

⬁

E

Here F ␯ ␶ (E,X)⫽dN ␯ ␶ /dE and F ␶ (E,X)⫽dN ␶ /dE are thedifferential energy spectra of tau neutrinos and taus, respectively, for lepton energy E, at a column depth X in the medium defined by冕␳ ⬘ ⬘LX⫽

0

共 L 兲 dL .

共4兲We use the average density ␳ (L) of the medium along theneutrino path, as given by the preliminary earth model 关21兴.
We note that Antarctica is covered by a sheet of ice with afew kilometers thickness, so that some of the neutrino trajectories can go mostly or even entirely through ice rather thanrock. For the neutrino interaction cross sections we use
CTEQ6 parton distribution functions 关22兴 and their extrapolation in the regions of interest to us 关19兴. The decay length of
the tau is ␭ ␶dec (E,X, ␪ )⫽ ␥ c ␶ ␶ ␳ . We use the decay distributions dn/dE written explicitly in Ref. 关4兴, and we approximate the energy distribution of the neutrino interaction process by d ␴ /dy(E,y)⯝ ␴ (E) ␦ (y⫺ 具 y 典 ) with y⫽(E⫺E ⬘ )/E
for incident neutrino energy E and outgoing lepton energy E ⬘and 具 y 典 ⫽0.2.
Equation 共3兲 describes the approximate energy loss of thetau. The parameter ␣ is determined by the ionization energy
loss and is negligible at high energy. The parameter ␤ is due
to radiative energy loss through bremsstrahlung, pair production, and photonuclear scattering. The photonuclear scattering becomes dominant at very high energies. As a first approximation we use ␤ ⫽0.8⫻10⫺6 cm2 /g for the ␶ energy
loss. There is a negligible change (ⱗ a few percent兲 if one
uses a more realistic ␤ (E)⫽ 关 0.16⫹0.069 log10(E ␶ /GeV) 兴⫻10⫺6 cm2 /g 关20兴, except at very high energies (E ␶
⬎1010 GeV), where the flux is extremely low. We use anenergy loss parameter ␤ ⫽7⫻10⫺6 cm2 /g for the muons.
It should be added here that for ¯␯ e the additional scattering on electrons should be considered because of the W boson resonance at 6.3⫻106 GeV. However, this resonance isextremely narrow 共narrower than the energy resolution of theexperiments兲 and it contributes a negligible amount to total
rates 关6,19兴.
In 关23兴 it was pointed out that the secondary fluxes of ¯␯ eand ¯␯ ␮ from the tau decays could also have a significant
contribution for a flux of monoenergetic neutrinos. In 关24兴 it
was shown, however, that secondary neutrinos have a negligible contribution for large nadir angles 共i.e., for ␪ ⬎60°) for
generic initial neutrino fluxes 1/E and 1/E 2 . The fluxes ofsecondary neutrinos are described by a transport equation
similar to Eq. 共1兲, with the decay distribution dn/dE charac-

III. NEUTRINO AND TAU FLUXES

From Eqs. 共1兲–共3兲 for ␶ ’s, and from suitably modified
equations for muons, we evaluate the charged lepton fluxes
at the end of the trajectory of a neutrino incident with nadir
angle ␪ .
For most astrophysical sources, the neutrinos are produced in pion decays, which determine the flavor ratio␯ e : ␯ ␮ : ␯ ␶ to be 1:2:0. After propagation over very long
distances, neutrino oscillations change this ratio to 1:1:1
because of the maximal ␯ ␮ ↔ ␯ ␶ mixing. For the GZK flux,the ␯ e and ␯ ␮ incident fluxes are different because of theadditional contributions from ¯␯ e from neutron decays and ␯ efrom ␮ ⫹ decays 关12兴. Because of this, the flavor ratio at the
Earth is affected by the full three-flavor mixing and is different from 1:1:1. Given fluxes at the source F ␯0 , F ␯0 , ande␮F ␯0 , the fluxes at the Earth become␶F ␯ e ⫽F ␯0 ⫺ 41 sin2 2 ␪ 12共 2F ␯0 ⫺F ␯0 ⫺F ␯0 兲 ,
e
e
␮
␶

共5兲

F ␯ ␮ ⫽F ␯ ␶ ⫽ 21 共 F ␯0 ⫹F ␯0 兲␮

␶

⫹ 18 sin2 2 ␪ 12共 2F ␯0 ⫺F ␯0 ⫺F ␯0 兲 ,
e

␮

␶

共6兲

where ␪ 12 is the mixing angle relevant for solar neutrino
oscillations. We have assumed that ␪ 23 , the mixing angle
relevant for atmospheric neutrino oscillations, is maximaland ␪ 13 is very small, as shown by reactor experiments, aswell as atmospheric and solar data. We use the initial GZK
flux evaluated by Engel, Seckel, and Stanev 关12兴 for standard
evolution and we get the fluxes at the Earth from Eqs. 共5兲
and 共6兲. The fluxes of ␯ ␮ and ␯ ␶ are still equal, due to the
maximal ␯ ␮ ↔ ␯ ␶ mixing. The main effect of the three-flavor
oscillations is to transform some of the low energy ␯ e ’s into␯ ␮ and ␯ ␶ . This could be useful since IceCube has very goodsensitivity for detecting tracks, which are enhanced in thiscase.
To illustrate the effect of the charged lepton lifetime andelectromagnetic energy loss in propagation, we begin withmuon and tau distributions. Figure 3 shows the tau and muon
distributions for the GZK neutrinos at the end of the trajectory through the Earth with a nadir angle of 85°, with andwithout taking into account the energy loss of the charged
lepton. As noted above, the incident ␯ ␮ and ␯ ␶ fluxes areequal. It is clear from here that the energy loss is extremely
important and strongly limits the fluxes at very high energy.
Including energy loss, the difference between the ␶ and ␮
fluxes above 107 GeV comes from the ␯ ␶ pileup from interaction and regeneration, as well as to the difference in decay
length and energy loss for taus and muons. Most of thecharged leptons are produced in the last step of neutrino
propagation. The muon flux below 107 GeV dominates thetau flux because of the much longer decay length of themuon.

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JONES et al.

FIG. 3. 共Color online兲 Charged lepton distributions for GZK
neutrinos, at a nadir angle of 85°.

Figure 4 shows the distribution of ␯ ␶ , ␯ ␮ , and ␯ e fluxes
after propagation through the Earth for the same initial flux
and angle. For ␯ e and ␯ ␮ the transport equations are effectively decoupled from those of the charged leptons and theirpropagation is given by Eq. 共1兲 in which the last term is notpresent. On the same figure we show the flux of secondary␯ ␮ neutrinos, which is the same as that of the secondary ␯ e .
It can be seen that their contribution is negligible, with maximum corrections of the order of a few percent around
107 GeV and much smaller for most energies. At larger nadir
angles, the secondary neutrino contribution will be an evensmaller fraction of the ␯ e and ␯ ␮ fluxes. At smaller nadir
angles, the secondary flux will be a larger percentage of theoverall neutrino flux, but the flux will become more strongly
attenuated.
Figure 5 shows the same ␯ ␶ distribution as in the previous
figure, together with the distributions obtained if the energy
loss of tau leptons or the shift in energy due to neutral current is not included in the propagation. Electromagnetic energy loss is extremely important at energies above 108 GeV,
and this effect causes strong suppression of the neutrino
fluxes at very high energy. If the tau did not lose energy in its
propagation, the regeneration effect would be much bigger at

FIG. 4. 共Color online兲 Neutrino distributions for GZK neutrinos,
at a nadir angle of 85°.

FIG. 5. 共Color online兲 Tau neutrino distributions for the GZKflux, at a nadir angle of 85°.

high energy. The ␯ ␶ → ␶ → ␯ ␶ regeneration is a significant effect for energies between 106 and 108 GeV, as can be seen
more clearly in Fig. 7 below. The neutral current interactionsof the neutrinos are also very important, as can be seen in
Fig. 5 by comparing the flux of ␯ ␶ after correct propagation
with a simple attenuation with exp关⫺D/L ␯int (E) 兴 for column
depth D and interaction length L evaluated using the charged
current cross section. The difference can be as large as twoorders of magnitude at very high energy.
Figure 6 shows the ␶ and ␯ ␶ distributions for the same
initial neutrino flux, but for different nadir angles. As theneutrinos pass through more matter, the regeneration effects
become relatively more important, but the overall fluxes get
significantly attenuated. At a nadir angle of 80° the Earth isalready opaque to neutrinos with energies above 1010 GeV.
Figure 7 shows the ratio of the tau neutrino flux after
propagation to the incident tau neutrino flux, for 89°, 85°,
and 80°. This ratio illustrates a combination of the regeneration of ␯ ␶ due to tau decay and the attenuation of all neutrino
fluxes. For 89°, where both the total distance and the density
are smaller, the attenuation is less dramatic, and the flux canbe significant even at high energy. The regeneration in thiscase can add about 25% corrections at energies between 107

FIG. 6. 共Color online兲 The ␶ and ␯ ␶ distributions for GZK neutrinos for different nadir angles.

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TRACING VERY HIGH ENERGY NEUTRINOS FROM . . .

tances, as will be the case for ANITA.
There are two contributions to electromagnetic showers
from tau decays. One is from the decay of the taus produced
outside the detector that decay electromagnetically inside thedetector, giving ‘‘lollipop’’ events in IceCube and electromagnetic showers in RICE. The distribution of these showers
is given as a function of the energy of the electron producedin the decays by

冕

em,1
F sh,
␶ 共 Ee兲⫽Ee⫻FIG. 7. 共Color online兲 Ratio ␯ ␶f inal / ␯ initialfor GZK neutrinos, at␶nadir angles of 80°, 85°, and 89°.

and 108 GeV. For 85° the relative enhancement is around
80% and peaked at slightly lower energies, while at 80° it isalmost a factor of 3 at low energy. At 80°, however, the flux
is very strongly attenuated for energies above a few⫻107 GeV.
It is already clear from Fig. 7 that the total rates will bedominated by the nearly horizontal trajectories that go
through a small amount of matter. The largest pileups occur
when the trajectory of the neutrino passes through a larger
column depth. For the higher energies relevant to RICE and
ANITA, this does not translate to higher fluxes of ␯ ␶ ’s. Attenuation is the main effect at those energies. Rates can get
significant enhancements at low energies where the regeneration from tau decays adds an important contribution even for
longer trajectories.

E max

dE ␶

d␳

0

dec

dxF ␶ 共 E ␶ 兲 e ⫺x/␭ ␶(E ␶ e ⫺ ␤ x )dn ␶ →e
共 E ␶ e ⫺ ␤ x ,E e 兲 ,
dE e

1
⫺␤x
␭ dec
兲
␶ 共 E ␶e

共7兲

where F(E)⬅dN/dE is the flux of taus just before the final
column depth d in ice and ␳ ⫽0.9 g/cm3 is the density of ice.
For negligible energy loss 共which is a very good approximation below 109 GeV in 1 km of ice兲 this becomes
em,1
F sh,
␶ 共 Ee兲⫽

冕

E max

dec

dE ␶ F ␶ 共 E ␶ 兲共 1⫺e ⫺d ␳ /␭ ␶

Ee⫻(E ␶ )

兲dn ␶ →e
共 E ␶ ,E e 兲 .
dE e

共8兲

The other contribution comes from the taus that are produced
and decay electromagnetically inside the detector. In
IceCube these correspond to ‘‘double bang’’ events. The distribution is obtained from
em,2
F sh,
␶ 共 Ee兲⫽

冕 冕 冕
d␳

d␳

dx

dy

0

x

⫻e ⫺xN A ␴

IV. SHOWERS

The above distributions illustrate the effects of the propagation through matter on the initial neutrino fluxes. We now
turn to the experimentally relevant observables, which are
the showers.
We consider hadronic showers produced in ␯ ␶ interactions, as well as electromagnetic and hadronic showers from␶ decays in ice. All other flavors of neutrinos will contributeto the hadronic showers, while ␯ e will also contribute toelectromagnetic showers. The shower distributions will bedifferent for each type of experiment, depending on the detector setup.
We start by considering contributions to electromagnetic
and hadronic showers in a depth of ⬃1 km of ice, relevant
for experiments like IceCube and RICE.
We use Eqs. 共1兲–共3兲 and the corresponding ones for e and␮ flavors to obtain the tau and neutrino distributions 关 F ␶ (E ␶ )and F ␯ (E ␯ )] after propagation up to the last few kilometersof ice. We then estimate the shower distributions that are
produced over the distance d⬃ few km. The contributions ofneutral currents and regeneration over a few kilometers issmall and here we neglect them for the last part of the trajectory, which is where the showering takes place. This canno longer be done when showers are detected over long dis-冕

Ee

CC (E )
␯dec⫻e ⫺y/␭ ␶
⫻

E max

NA

(E ␶ e ⫺ ␤ y )dE ␯ ␶

冕E␯

EedE ␶ F ␯ ␶ 共 E ␯ ␶ 兲d ␴ CC共 E ␯ ␶ ,E ␶ 兲dE ␶1␭ ␶dec 共 E ␶ e ⫺ ␤ y 兲

dn ␶ →e
共 E e ,E ␶ e ⫺ ␤ y 兲 .
dE e

共9兲

When energy loss is negligible, this becomes
em,2
F sh,
␶ 共Ee兲

033004-5

⫽

冕

E max

Ee

⫻

dE ␯ ␶

冕E␯

EedE ␶ F ␯ ␶ 共 E ␯ ␶ 兲 N A

冉1
dn ␶ →e
共 E e ,E ␶ 兲
CCdE e
N A ␴ 共 E ␯ ␶ 兲 ⫹1/␭ ␶dec 共 E ␶ 兲⫻ 共 1⫺e ⫺d ␳ [N A ␴⫺d ␴ CC共 E ␯ ␶ ,E ␶ 兲
dE ␶

1
N A␴

CC

CC (E

dec
␯ ␶ )⫹1/␭ ␶ (E ␶ )]dec

共 E ␯␶兲e ⫺d ␳ /␭ ␶

(E ␶ )

兲共 1⫺e ⫺d ␳ N A ␴

CC (E

␯␶)

冊

兲 .

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The main source of electromagnetic showers comes fromthe ␯ e charged current interactions given by
em
F sh,
␯⫽

冕

E max

Ee

⫻

dE ␯ F ␯ 共 E ␯ 兲共 1⫺e ⫺d ␳ N A ␴

CC (E )
␯d ␴ CC共 E ␯ ,E e 兲 .
␴ CC 共 E ␯ 兲 dE e
1

兲

共11兲

Hadronic showers from tau decays are given by Eq. 共8兲 withthe electrons replaced by hadrons.
The hadronic showers from ␯ ␶ interactions are given by
h
F sh,
␯⫽

冕

E max

Eh

⫻

dE ␯ F ␯ 共 E ␯ 兲共 1⫺e ⫺d ␳ N A ␴ (E ␯ ) 兲d␴
1
共 E ,E 兲 .
␴ 共 E ␯ 兲 dE h ␯ h

共12兲

In Eq. 共12兲, ␴ is the total cross section and the hadronicenergy comes from the energy transfer to the target nucleus.
Hadronic showers are also produced in ␯ ␮ and ␯ e neutral
current and charged current interactions. These are also obtained from Eq. 共12兲, using the corresponding fluxes.
We briefly describe below all types of interactions andtheir signatures in the various experiments. We then show
shower distributions relevant in each case.
A. Showers in kilometer size detectors: IceCube, RICE

A km3 Cherenkov detector in ice, IceCube 关7兴 has sensitivity to high energy neutrinos up to ⬃108 –109 GeV, theenergy range where distinguishing the effects of tau neutrinos from muon and electron neutrinos could be possible.
Neutral current interactions of all types of neutrinos produce hadronic showers. Charged current interactions of ␯ econtribute to both hadronic and electromagnetic showers. It
is likely that IceCube would observe a big shower with thetotal energy of the incoming neutrino rather than being able
to separate the two components 共hadronic and electromagnetic兲. Charged current interactions of ␯ ␮ give a hadronic
shower and a muon. The muon track is the main signal in
IceCube, so these events are easily identifiable, containing ashower and a long track emerging from it. At high energies,
however, the same signal could be produced by a ␯ ␶ interaction, since the decay length of the tau becomes longer than
the size of the detector for energies above a few times
107 GeV.
Charged current ␯ ␶ interactions can have very different
signatures depending on energy. At 106 GeV, the tau decay
length is ⬃50 m, and the shower 共hadronic or electromagnetic兲 from the tau decay cannot be separated from the hadronic shower from the initial ␯ ␶ interaction. At a few times
106 GeV the range of the tau becomes a few hundred metersand can give the characteristic signal of the ‘‘double bang’’
events 关25兴. These are events where the shower from theneutrino interaction and the shower from the tau decay canbe separated and are both observed in the detector, together
with the tau track. At a few times 107 GeV the tau decay

FIG. 8. 共Color online兲 Electromagnetic shower distributions for
GZK neutrinos, at a nadir angle of 85° for a kilometer size detector.
The ␶ decay curve is the sum of the lollipop and double bang
curves.

length is already longer than 1 km, and the taus look likemuons. What is seen is the shower from the neutrino interaction and then a track.
Taus produced in neutrino interactions outside the detector can generate ‘‘lollipop’’ events. The initial shower fromthe neutrino interaction that produces the tau is missed, andwhat is seen is the track of the tau and the shower from its
decay, which IceCube can identify as electromagnetic or
hadronic from the existence of muon tracks in the shower.
RICE 关9兴 uses dipole antennas in the Antarctic ice to measure radio frequency Cherenkov radiation from high energy
showers. For energies around 109 GeV the effective volumeof the detector is ⬃15 km3 sr. At present, RICE limits thefluxes of downward ␯ e with energies between a few times
107 and 1012 GeV 关9兴. Interactions modeled by a Monte
Carlo simulation at an energy of 109 GeV are detected fromwithin a depth of 1 km from the surface and out to a radial
distance of about 4 km. The effective volume of the detector
has a strong dependence on energy below 109 GeV. Hadronic and electromagnetic showers can be separately identified, with somewhat different effective volumes, depending
on energy. The RICE experiment can therefore measure electromagnetic showers from ␯ e interactions, hadronic showers
from the charged and neutral current interactions of all flavors of neutrinos, and electromagnetic and hadronic showers
from tau decays.
Figure 8 shows the electromagnetic shower distributions
at a nadir angle of 85°. In the absence of tau neutrinos, only
␯ e interactions lead to electromagnetic showers. These ␯ e→e CC conversions still dominate at high energies, evenwhen tau neutrinos produce tau leptons that decay electromagnetically. At lower energies, however, the taus add a significant contribution to the electromagnetic shower signal.
Tau decays give their most important relative contribution toelectromagnetic showers at electron energies of a few times
107 GeV. These decays are separated into ‘‘double bang’’and ‘‘lollipop’’ events on the same plot. It can be seen thatthe contribution of the ‘‘double bang’’ is relatively small.

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FIG. 9. 共Color online兲 Hadronic shower distributions for GZK
neutrinos, at a nadir angle of 85° for a kilometer size detector.

These events do not contribute significantly to total event
rates, but even in very small numbers they are important as acharacteristic signature for taus. Secondary ␯ e interactions
also produce electromagnetic showers, but it can be seen that
their contribution is very small.
Figure 9 shows the hadronic shower distributions. The
features seen in the neutrino fluxes are recovered in the hadronic showers as well. The shower rates from ␯ ␶ , ␯ ␮ , and ␯ e
are nearly identical at shower energies above 108 GeV. The
tau neutrino pileup is important below 108 GeV. Between
107 and 108 GeV, tau decays give the main contribution to
the hadronic shower rate. Showers from ␯ e and ␯ ␮ secondary
neutrinos are also shown and their contribution is negligible.
Figure 10 shows the ratio of the electromagnetic shower
rates at nadir angle 85° in the presence and absence of oscillations for the GZK and Z burst neutrino fluxes. This ratio
illustrates the effect of the oscillations on the signal and inparticular the possible enhancements due to tau pileup. In theabsence of oscillations, the only contribution to electromagnetic showers comes from ␯ e interactions. In the presence of␯ ␮ → ␯ ␶ oscillations, electromagnetic decays of taus from tau

FIG. 10. 共Color online兲 Ratio of electromagnetic shower rates in
the presence and absence of ␯ ␮ → ␯ ␶ oscillations for GZK and 1/E
neutrino spectra for a nadir angle 85° for a kilometer size detector.

FIG. 11. 共Color online兲 Ratio of hadronic shower rates in thepresence and absence of ␯ ␮ → ␯ ␶ oscillations for GZK and 1/E
fluxes for nadir angle 85° for a kilometer size detector.

neutrinos add significant contributions to these rates at energies below 108 GeV. At the same time, for the GZK flux,␯ e → ␯ ␮ , ␶ oscillations reduce the number of ␯ e ’s at low energy, such that below a few times 106 GeV there are fewer
electromagnetic showers than in the absence of oscillations.
Figure 11 shows the ratio of the hadronic shower distributions in the presence or absence of oscillations for thesame fluxes and nadir angle. Without oscillations, the hadronic showers come from ␯ e and ␯ ␮ interactions. In the presence of oscillations ␯ ␶ and the ␯ e and ␯ ␮ secondaries fromtau decay contribute as described above. For hadronic showers the enhancement due to the presence of ␯ ␶ is ⬃50% atenergies of a few times 107 GeV. For hadrons there is also
some small enhancement for shower energies between 108
and 109 GeV, which is not present for the electromagnetic
showers.
The tau contribution can be even more clearly seen if
some of the other flavors can be separately identified. As
previously discussed, IceCube can identify showers from ␯ ␮and ␯ ␶ charged current interactions by the ␮ or ␶ tracks thatemerge and exit the detector. Once these can be removed,
what remains are hadronic showers from neutral current interactions of all flavors of neutrinos, ␯ e CC interactions in
the detector, as well as the hadronic showers from ␶ decays
in the detector 共both from taus produced within the detector
and from taus produced outside the detector that propagate inand then decay兲. The enhancement in the ratio is even higher
in this case, about 80%.
Figure 12 also shows the hadronic shower distributions
for GZK neutrinos, but at a nadir angle of 89°. As expected,
there are more showers for this distance than for the longer
one, but the relative contribution of the tau is smaller in thiscase. This is because both the attenuation and the regeneration effects are much smaller for smaller column densities.
The experimental angular resolution for high energy
showers is about 10° to 25°. Averaging over such angles, theeffect of the pileup is somewhat reduced 共to ⬃10% of theevent rate兲, since the total rates are dominated by the trajectories that go through a small amount of matter. However,rates can get significant enhancements at low energies where

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FIG. 12. 共Color online兲 Hadronic shower distributions for GZK
neutrinos, at a nadir angle of 89° for a kilometer size detector.

the regeneration from tau decays adds an important contribution even for longer trajectories. Consequently, an experiment with a low energy threshold has a better chance ofdetecting the effect. IceCube has a low energy threshold andan energy resolution of ⬃10%, so it is in a good position tolook for pileup effects, given a high enough neutrino flux.
The same general features remain true for different initial
fluxes. However, the energy distributions are different for
each, and the enhancement due to regeneration appears at
somewhat different energies. The regeneration effects arealso smaller for fluxes steeper at high energy. For example,
from Figs. 10 and 11 it can be seen that for the 1/E flux,
predicted, for example, for Z burst 共ZB兲 models 关14兴, thepileup is slightly bigger, but at a lower energy than for the
GZK neutrino flux. This is just as expected, because the
GZK flux is steeper at higher energy, but does not fall as
rapidly at intermediate energies.
Figure 13 shows the electromagnetic shower distributions
for a 1/E flux, for example, predicted for Z burst, and for a
1/E 2 generic flux for a nadir angle of 85°. The 1/E 2 distri-

FIG. 13. 共Color online兲 Electromagnetic shower distributions
for a nadir angle of 85° for 1/E and 1/E 2 characteristic fluxes for akilometer size detector from ␯ e interactions and from ␶ decays.

FIG. 14. 共Color online兲 Electromagnetic shower distributions
for a nadir angle of 89° for 1/E and 1/E 2 characteristic fluxes for akilometer size detector.

bution has a normalization of 1010 GeV⫺1 km⫺2 yr⫺1 sr⫺1 ,about an order of magnitude below the present AMANDA
limit 关26兴. For the ZB flux the normalization is
1 GeV⫺1 km⫺2 yr⫺1 sr⫺1 up to 2.5⫻1012 GeV. Above thisenergy the flux is cut off and drops as 1/E 3 . The steepness ofthe 1/E 2 flux results in a small pileup and thus the relative
contribution of taus is much smaller in this case. Figure 14
shows the same shower distributions for a nadir angle of89°. As for the GZK flux, the attenuation in this case issmaller than for 85°, particularly at high energy. The effectof attenuation at high energy is striking, especially for the Z
burst flux. At 85° there is almost no flux left at energies
above 1010 GeV, while at 89° the neutrino flux is almost
unattenuated and still has the 1/E shape, being orders ofmagnitude higher than for the longer path length. However,
the effects of regeneration become smaller for 89° even forthe less steep Z burst flux. It can be clearly seen that, depending on the energy threshold of the detectors, the contribution
to event rates comes from different trajectories. At high energies the paths that go through the material with small column density will dominate event rates. However, at energies
below 108 GeV trajectories that go through 10–15 times
more material can contribute equally due to the pileup andadditional tau decays.
Figure 15 shows the hadronic shower distributions for thefluxes used in the previous figures, for a nadir angle of 85°.
The energy distribution of the hadronic showers is different
from that of electromagnetic ones due to the different decay
distributions. Taus decay mostly to hadrons and these hadrons carry most of the energy of the taus. Consequently, thetau contribution to hadronic showers is much higher, bigger
than the ␯ ␶ contribution at energies around a few times
107 GeV and extends to higher energies.
Figure 16 shows the electromagnetic distributions for the
AGN flux in 关16兴 for nadir angles of 85° and 89°. This fluxis also steep at high energy and consequently the regeneration effects are small. The relative contribution of the taus isbigger for 85°, but the overall rates are higher for 89°.

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FIG. 15. 共Color online兲 Hadronic shower distributions for a nadir angle of 85° for 1/E and 1/E 2 characteristic fluxes for a kilometer size detector from neutrino interactions and tau decays. The␯ e distributions are identical to the ␯ ␮ distributions. For the 1/E 2flux all neutrinos distributions are almost the same.

The energy threshold of RICE is high and for the energies
where the experiment has good sensitivity the ␯ ␶ enhancement is limited by the long lifetime of the tau lepton and by
its energy loss, such that it cannot be observed. We want toinvestigate if this remains true in the case of ANITA, which
has a much larger effective volume and consequently has thepotential to detect many more taus decaying over long trajectories.
B. Showers in ANITA

The ANITA experiment also uses the ice as a neutrino
converter 关10兴. The long duration balloon missions will
monitor the ice sheet from 40 km in altitude to a horizon
approaching 700 km for refracted radio frequency signals
with an effective telescope area of 1⫻106 km2 . The geometry of the experiment is rather complicated, as it has to take
into account the Cherenkov angle of the radio emission with

FIG. 16. 共Color online兲 Electromagnetic shower distributions
for the Mannheim-Protheroe-Rachen AGN model 关16兴 for a kilometer size detector.

FIG. 17. 共Color online兲 Electromagnetic shower distributions
for detection over 222 km of ice.

respect to the particle trajectory, its refraction at the ice surface, and the position of the balloon. The huge volume covered gives ANITA remarkable sensitivity for detecting very
high energy neutrinos. While IceCube and RICE can only
detect showers produced in a km3 volume, ANITA can detect
all the showers produced over long distances in ice.
For example, for a nadir angle of 89°, the entire trajectory(⬃222 km) of the neutrino is in ice, at less than 1 km depth,
and all showers produced over this distance can be observed.
For a nadir angle of 85°, observable showers 共at less than 1
km depth兲 could be produced over a distance of ⬃12 km.
Given the very large detector area, a trajectory is not fully
defined by the nadir angle of the incident neutrino at theentrance point, and one also needs the position of theentrance/exit point with respect to the icecap and the balloon.
The trajectories that maximize the path through ice, rather
than the ones that combine ice and rock, are the ones likely
to dominate event rates because of the much smaller attenuation. Those going through more rock give larger relative
enhancements due to regeneration. They could in principle
contribute as much as the others in the region where theregeneration is effective.
As previously discussed, due to the showering over entire
trajectories, we can no longer use Eqs. 共8兲–共12兲, but rather
we have to combine the propagation and showering from the
beginning in order to correctly take into account neutral current interactions, energy loss of tau leptons, and neutrino
regeneration from tau decay.
Figure 17 shows the electromagnetic shower distributions
for the GZK and ZB initial fluxes over a trajectory of 222 km
in ice. Qualitatively, these are similar to the showers obtained in kilometer size detectors for the same trajectory.
However, in this case the rates are larger by up to threeorders of magnitude, depending on energy, due to the muchlarger surface of the detector. The energy distribution issomewhat different, being more spread out toward higher
energies. Due to the longer detectable path length, higher
energy taus, with decay length much longer than 1 km, cannow decay producing observable showers.

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V. CONCLUSIONS

FIG. 18. 共Color online兲 Hadronic shower distributions for detection over 222 km of ice. The ␯ ␮ distribution for GZK neutrinos is
the same as that of ␯ ␶ . For the 1/E flux all neutrino distributions arealmost the same.

Figure 18 shows the hadronic shower distributions for thesame initial fluxes and trajectory. The regeneration of the ␯ ␶neutrinos in this case is very small and ␯ ␮ distributions arealmost the same as the ␯ ␶ ones. For the ZB flux, the ␯ edistribution is the same as for ␯ ␮ , while for the GZK neutrinos it is different due to the difference in the initial fluxes.
Figure 19 shows the ratio of electromagnetic and hadronic
shower rates in the presence and absence of ␯ ␮ → ␯ ␶ oscillations corresponding to the distributions in the previous figures. The maximum enhancement due to the presence of ␯ ␶
is about 40% at this angle, as expected since this trajectoryhas low column density. However, as previously discussed,
the enhancement occurs in a larger energy range and it peaks
at higher energy than in the case of small size detectors. In 1
km only taus with energies below a few times 107 GeV havea significant probability of decay, while much higher energytaus can decay over the total distance of more than 200 km.

FIG. 19. 共Color online兲 Ratio of electromagnetic and hadronic
shower rates in the presence and absence of ␯ ␮ → ␯ ␶ oscillations for
GZK and 1/E fluxes of neutrinos for detection over 222 km of ice.

We have studied in detail the propagation of all flavors ofneutrinos with very high energy (E⭓106 GeV) as they
traverse the Earth. Because of the high energies, we have
limited our consideration to nadir angles larger than 80°. We
are particularly interested in the contribution from tau neutrinos, produced in oscillations of extragalactic muon neutrinos as they travel large astrophysical distances. After propagation over very long distances, neutrino oscillations change
an initial 共source兲 flavor ratio of 1:2:0 to 1:1:1 because of
the maximal ␯ ␮ ↔ ␯ ␶ mixing. For GZK neutrinos, the flavor
ratio at the Earth deviates from 1:1:1 because the incident
fluxes are different. At lower energies and smaller nadir
angles, tau neutrino pileups from regeneration via ␯ ␶ → ␶→ ␯ ␶ 关3,4兴 enhanced electromagnetic and hadronic signals inkilometer sized detectors. Our aim here was to see if there
are similar effects at high energy.
In our propagation of neutrinos and charged leptons
through the Earth, we have focused on kilometer sized neutrino detectors, such as IceCube and RICE, and on a detector
with much larger effective area which uses Antarctic ice as aconverter, ANITA. Our study can easily be generalized toother experiments and to propagation in materials other thanice.
We have found that the ␯ ␶ flux above 108 GeV resembles
the ␯ ␮ flux. The lore that the Earth is transparent to tau
neutrinos is not applicable in the high energy regime. Tau
neutrino pileups at small angles with respect to the horizon
are significantly damped due to tau electromagnetic energy
loss above E ␶ ⬃108 GeV if the column depth is at least as
large as the neutrino interaction length.
At lower energies, E⭐108 GeV, regeneration of ␯ ␶ becomes important for trajectories where the other flavors ofneutrinos are strongly attenuated. The regeneration effect depends strongly on the shape of the initial flux and it is larger
for flatter fluxes. The enhancement due to regeneration also
depends on the amount of material traversed by neutrinos
and leptons, i.e., on nadir angle. For GZK neutrinos, we havefound that the enhancement peaks between 106 and a few
times 107 GeV depending on trajectory. For 85° the enhancement is about a factor of 2, while for 80° it is a factorof 3.
We have translated the neutrino fluxes and tau lepton
fluxes into rates for electromagnetic and hadronic showers atselected angles to see the effect of attenuation, regeneration,and the different energy dependences of the incident fluxes.
We have focused on comparing the ␯ ␶ contribution to the ␯ eand ␯ ␮ contributions to determine in what range, if any, ␯ ␶ ’senhance shower rates.
The ␯ ␶ flux enhancements depend on the shape of theinitial flux. The electromagnetic showers are more sensitive
to this shape than hadronic ones. The relative enhancement
in hadronic showers is also smaller than for the electromagnetic showers. This is because for the electromagnetic signal
the only contribution in the absence of taus is from electron
neutrinos, while for hadrons the tau contribution is compared
to a much larger signal, from the interactions of all flavors ofneutrinos. We have included contribution from secondary

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neutrinos, which we find to be relatively small for all fluxes.
For kilometer sized detectors, at, for example, a nadir
angle of 85°, the maximal enhancement due to ␯ ␶ contribution to electromagnetic shower rates for the GZK flux isabout 50% at 3⫻107 GeV, while for the 1/E flux, it is evenlarger, about 70%, at slightly lower energy. In the case of
hadronic showers for which the events identified by muontracks have been removed, the ␯ ␶ contribution peaks at about2⫻107 GeV and it gives an enhancement of about a factorof 1.8 for the GZK flux. These energy ranges are relevant for
IceCube, but not for RICE. For energies relevant to RICE,
tau neutrinos do not offer any appreciable gain in electromagnetic shower signals compared to ␯ e →e CC interactions,and they contribute at essentially the same level as ␯ ␮ tohadronic shower rates through neutral current 共NC兲 interactions.
One of the reasons that tau neutrinos do not contributelarge signals to kilometer-sized detectors at very high energies is that high energy tau decay lengths are very large, so
the probability of a tau decaying in the detector is low. For
detectors like ANITA which can sample long trajectories
through the ice one would expect a larger tau neutrino contribution to the signal from tau decay. Despite the long trajectory 共222 km with a maximum depth of 1 km for a neutrino incident at 89° nadir angle兲 the tau contributions to theelectromagnetic shower rate is quite small for fluxes expected to contribute in the ANITA signal. For hadronic showers, the suppression of ␶ decay to hadrons relative to ␯ e NC
interaction contributions is about the same as for electromagnetic showers compared to ␯ e →e. The ␯ ␶ contribution to thehadronic shower rate from interactions is the same as the ␯ econtribution. In summary, for ANITA, tau neutrinos do notgive any additional signal beyond what one would evaluate
based on no regeneration from ␯ ␶ → ␶ → ␯ ␶ due to tau electromagnetic energy loss at Eⲏ108 GeV.
Detection of GZK neutrinos would be an important confirmation of our current understanding of ultrahigh energy
cosmic rays. Because the high energy flux of neutrinos isrelatively small, detection of these neutrinos will be a greatchallenge. Our interest in this paper was to explore the possibility that the tau neutrino component, a result of neutrino
oscillations over astronomical distances, could enhance thenumber of observable GZK events. As we have shown in theprevious sections, the electromagnetic energy loss of highenergy tau leptons makes ␯ ␶ induced events at high energies
a small fraction of the total number of GZK neutrino events.
Computing accurate event rates for the ANITA detector isdifficult because of the details of the dependence of the acceptance on energy and angle. It is clear from Fig. 17, however, that at energies above E⬃108 GeV relevant for the
ANITA experiment, event rates are completely dominated bythe ␯ e flux. Estimates of the GZK ␯ e induced event rate areof the order of 1 event/10 days of flight for the standard
Engel et al. flux, enhanced by a factor of 4 for the strong
evolution model 关10兴. For RICE the present limit is about afactor of 50 above the GZK flux. In IceCube, because of thelower energy threshold, the relative ␯ ␶ contribution is potentially bigger. The total rates are, however, very small. For theflux in 关12兴 that we use, we obtain about
0.1 hadronic showers/km2 /yr. These are mostly from neutrinos coming from above, the contribution of those comingfrom below being smaller by about a factor of 50. These
values are consistent with those of Yoshida 关18兴. It is to be
noted, however, that the flux we use is the smallest one, andthe uncertainty in the normalization of the GZK flux is quite
large, such that all the above rates could be larger by factorsof 10 or even more. Detectors even larger than IceCube arealso envisioned, and the effects discussed here may become
observable in the future.
In addition to the experiments discussed here, there aremany studies concerning the possibility for detection of radio
Cherenkov emission from showers in materials other thanice. It has been noted 关27兴 that rock salt formations have
similar properties to the Antarctic ice and can therefore beused as large scale neutrino detectors. Salt has a higher density ( ␳ salt ⫽2.2 g/cm3 ) than ice ( ␳ ice ⫽0.9 g/cm3 ), so it ispossible to achieve an effective detection volume of several
hundred km3 water equivalent in salt. This is somewhat
larger than RICE, achieved with a much smaller actual detector size. The threshold for detecting the radio signal fromshowers in salt is of the order of ⬃107 GeV, similar to
RICE, but lower, such that detection of extra signals from ␯ ␶enhancements would be more promising.
Also proposed is LOFAR 关28兴, a digital telescope array
designed to detect radio Cherenkov emission in air showers.
LOFAR has sensitivity in an energy range of⬃105 –1011 GeV, so it can detect showers at much lower
energies than other radio Cherenkov experiments. LOFAR
will likely be configured to detect horizontal showers fromskimming neutrinos as well. With its low energy threshold,
LOFAR has an excellent opportunity to observe the shower
enhancement at lower energies due to ␯ ␶ regeneration andtau pileup, which is not easily accessible in ANITA.

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ACKNOWLEDGMENTS

This research was supported in part by the National
Science Foundation under Grant No. PHY99-07949 andunder DOE Contracts DE-FG02-91ER40664, DE-FG0295ER40906, and DE-FG02-93ER40792. I.M. and I.S. thank
KITP Santa Barbara for hospitality. M.H.R. and I.S. thankthe Aspen Center for Physics for hospitality while this work
was being completed.

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