tau neutrino propagation and tau energy loss

PHYSICAL REVIEW D 72, 013005 (2005)

Tau neutrino propagation and tau energy loss

Sharada Iyer DuttaDepartment of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, New York 11794 U.S.A.

Yiwen Huang and Mary Hall RenoDepartment of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242 U.S.A.

(Received 25 April 2005; published 14 July 2005)

1550-7998=20

Electromagnetic energy loss of tau leptons is an important ingredient for eventual tau neutrino detectionfrom high energy astrophysical sources. Proposals have been made to use mountains as neutrinoconverters, in which the emerging tau decays in an air shower. We use a stochastic evaluation of bothtau neutrino conversion to taus and of tau electromagnetic energy loss. We examine the effects of thepropagation for monoenergetic incident tau neutrinos as well as for several neutrino power-law spectra.Our main result is a parameterization of the tau electromagnetic energy loss parameter �. We compare theresults from the analytic expression for the tau flux using this � with other parameterizations of �.

DOI: 10.1103/PhysRevD.72.013005 PACS numbers: 13.15.+g, 96.40.Tv

I. INTRODUCTION

The observed neutrino fluxes in solar neutrino experi-ments [1] and in atmospheric neutrino experiments [2] leadto a picture of neutrino oscillations with substantial �� !

�� mixing [3]. For high energies over astronomical dis-tances, astrophysical sources of �e and �� can becomesources of �� as well. One method of detecting �� fromastrophysical sources is to look for the charged particlesthey produced in or near neutrino telescopes.

Because of the short lifetime of the tau, �� ! � ! ��regeneration can be an important effect as the �� passesthrough a significant column depth through the Earth [4–8]. This is typically relevant for neutrino energies belowE� � 108 GeV. At higher energies, one can imagine usingthe Earth as a tau neutrino converter [8–19]. Because of therelatively short interaction length of the neutrino at highenergies, column depths on the order of the distancethrough a mountain or from small skimming angles relativeto the horizon are most important. Following the tau neu-trino conversion, a tau is detected by its decay e.g., in icewith IceCube, RICE, or the proposed ANITA detectors, orafter it emerges into the air. The possibility of detectablesignals depends on the incident fluxes, of which there are avariety of models [20], and on the detectors and theirsensitivity to various energy regimes.

Theoretical calculations often rely on approximating theneutrino interaction and tau energy loss in terms of analyticfunctions of energy and column depth [11,12]. Theseanalytic tools are important in exploring the possibilityfor tau neutrino signals in a variety of detectors, eventhough ultimately a full simulation of tau production andenergy loss is required. In the energy regime where E� �108 GeV Huang, Tseng, and Lin have made an evaluationof the smearing of energy due to tau propagation by doingan approximate stochastic evaluation of electromagneticenergy loss of the tau [19]. Looking at fixed incidentneutrino energies, they find that the relative energy fluc-

05=72(1)=013005(10)$23.00 013005

tuation of the tau can become large at high energies,especially for larger column depths. This has the potentialto make interpretation of fluxes difficult at high energies.

In this paper, we evaluate tau neutrino and tau propaga-tion using a stochastic evaluation of tau electromagneticenergy loss based on the work of Ref. [21]. We find theapproximate evaluation of Ref. [19] reliable. In addition toconsidering monoenergetic incident neutrinos, we alsoconsider incident neutrino fluxes with a power-law behav-ior, E��

� . We compare the flux of taus following propaga-tion through 10 km rock and 100 km rock using thestochastic evaluation with estimates of the emerging fluxof taus using analytic approximations. We refine somerecent parameterizations of the tau energy loss as a func-tion of tau energy [11,12] to better approximate the sto-chastic result for the flux of emerging taus using analyticapproximations. The best parameterization of the electro-magnetic energy loss parameter � is with a logarithmicdependence on tau energy.

In the next section, we describe the models for the highenergy neutrino-nucleon cross section and the tau energyloss used in our Monte Carlo simulation. Results of thesimulation and analytic approximations for the tau energyas a function of distance traveled, for monoenergetic neu-trinos, appear in Sec. III. Section IV includes results andanalytic approximations for power-law incident neutrinospectra. A summary of our results and a comparison with asimplified analytic approach are in the final section.

II. NEUTRINO INTERACTIONS ANDTAU ENERGY LOSS

Our evaluation of neutrino interactions and tau energyloss is performed using a one-dimensional Monte Carlocomputer simulation. The neutrino charged current andneutral current interactions are calculated based onneutrino-isoscalar nucleon cross sections [22,23] usingthe CTEQ6 parton distribution functions (PDFs)[24].

-1 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.013005

SHARADA IYER DUTTA, YIWEN HUANG, AND MARY HALL RENO PHYSICAL REVIEW D 72, 013005 (2005)

At ultrahigh energies, there is a competing effect in theneutrino-nucleon cross section as a function of Q2 ��q2 > 0, the quantity describing the four-momentumsquared of the vector boson. PDFs increase as a functionof lnQ2 [25], however, the vector boson propagator effec-tively cuts off the growth in Q2 at the vector boson mass.For charged-current scattering, this means that the value ofparton x � Q2=2p � q for nucleon four-momentum p isapproximately

x�M2

W

ME�(1)

in terms of the incident neutrino energy E�, nucleon massM and W boson mass MW . For E� � 1012 GeV, one findsthat x� 10�8.

The CTEQ6 PDFs are parameterized for 10�6 < x< 1,so one must extrapolate for smaller values of x. The PDFsat small values of parton x (x < 10�6) are extrapolated hereusing a power law xq�x;Q2 � x�� matched to DGLAPevolved PDFs at xmin � 10�6:

xq�xmin;M2W � Ax��

min: (2)

This approach is based on the result [26] that for �� 0:3 �0:4, the gluon PDF has the approximate form of xg�x;Q �x��, and g ! q �q splitting is responsible for the sea quarkdistributions that dominate the cross section at ultrahighenergies. We show in Fig. 1 the neutrino interaction lengthin rock. Other approaches to the small x extrapolations forultrahigh energy neutrino cross sections yield similar in-teraction lengths [27], to within a factor of approximately21 at 1012 GeV.

FIG. 1. Neutrino interaction length Lint in rock as a function ofneutrino energy.

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The differential distribution for neutrino-nucleon scat-tering in terms of the lepton inelasticity y,

y �E� � Ei

�

E�; (3)

is evaluated similarly. We indicate the energy of the tau atthe point of its production by Ei

�. The neutrino interactionprobability and resulting tau energy is evaluated using aone-dimensional Monte Carlo program. In our analyticapproximations described below, we use hyi ’ 0:2, whichis approximately correct at ultrahigh energies [22]. Anindication of this is shown by the solid line in Fig. 2, whichshows the ratio of the average tau energy to the initialneutrino energy, as a function of neutrino energy, in theabsence of electromagnetic energy loss corrections.

Our evaluation of the tau electromagnetic energy lossfollows the procedure described in Ref. [21]. Based on thescheme outlined by Lipari and Stanev [28] and others [29],we have incorporated the electromagnetic mechanisms ofionization, bremsstrahlung (brem), e�e� pair production(pair) and photonuclear (nuc) scattering. The tau lifetime istaken into account.

The average energy lost per unit distance is commonlydescribed by the formula�

dE�

dz

�� E��: (4)

In this expression, z is the distance the tau travels. Thequantity � ’ 2 � 10�3 GeV cm2=g accounts for ioniza-

FIG. 2. Average tau energy scaled by the energy of incidentneutrinos that produced the taus, without energy loss (solid), andwith energy loss for taus produced in 10 km and 100 km rock(dashed).

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FIG. 3. Average distance the tau travels after production whichsurvives to emerge from 10 km rock (solid), 50 km rock (dashed)and 100 km rock (dashed-dotted), as a function of incidentneutrino energy.

TAU NEUTRINO PROPAGATION AND TAU ENERGY LOSS PHYSICAL REVIEW D 72, 013005 (2005)

tion energy loss [30]. Here, � is

� � �brem � �pair � �nuc �Xi

NA

A

Zdyy

d�i

dy; (5)

for Avogadro’s number NA, atomic mass number A, tauinelasticity y and i � brem; pair, and nuc.

In our treatment here, as in Ref. [21] we separate � intocontinuous and stochastic terms:

� � �cont � �stoc

�NA

A

Z 10�3

0dyy

d�i

dy�NA

A

Z 1

10�3dyy

d�i

dy: (6)

We treat the tau propagation stochastically, with a series ofelectromagnetic interactions occurring according to prob-abilities based on d�i=dy for y > 10�3 and the decayprobability. For each distance between interactions in theMonte Carlo program, the tau energy loss calculated fromEq. (3) is supplemented by Eq. (4) with the substitution of� ! �cont. By doing it this way, we account for the fullelectromagnetic energy loss, but we limit computer timespent where the cross section is large, but the energy loss issmall (y < 10�3). The detailed formulas for the brems-strahlung [31], pair production [32] and photonuclear[33] processes are discussed and collected in Ref. [21].See also Ref. [34] for the muon case.

For the muon, � is dominated by pair production andbremsstrahlung for E� � 103 � 107 GeV. At higher ener-gies, � grows slowly with energy due to the photonuclearprocess which contributes a larger fraction for E� >107 GeV. For the tau, the photonuclear process is compa-rable to pair production, while the bremsstrahlung contri-bution is suppressed, leading to an increase in � withenergy for E� > 103 GeV.

In the analytic approximations discussed below, we willconsider several forms of � as a function of E�, and we willtreat

�dE�

dz

�’dE�

dz: (7)

To guide our discussion, it is useful to compare the relevantconstants and distance scales. First, at 108 GeV,

�E� ’ 60cm2 GeV

g

�E�

108 GeV

�(8)

is much larger than �, so we can essentially ignore �. Thetau range is then characterized roughly by

1

�� 6 km (9)

in standard rock (� � 2:65 g=cm3). In fact, the range issomewhat larger than this estimate. The decay length of thetau is

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�c� ’ 5 km�

E�

108 GeV

�: (10)

It is in the energy regime near 108 GeV that the transitionfrom the lifetime dominated to the energy loss dominatedrange for the tau occurs.

III. MONO-ENERGETIC NEUTRINOS

A. Results from Monte Carlo simulation

We begin our discussion by considering single incidentneutrino energies to see the effect of energy loss combinedwith decay probabilities for two different trajectories:through 10 km rock (26.5 km.w.e.) and through 100 kmrock. Figure 2 shows the average tau energy divided by theincident neutrino energy as a function of incident neutrinoenergy. The average distance that the tau travels afterproduction which survives to emerge from the rock (10,50, and 100 km) is shown in Fig. 3.

For 10 km of rock, above 109 GeV, the average distancethe tau travels is 5 km. The neutrino interaction probabilityis equal anywhere in the 10 km path, and the energies arelarge enough that the decay length is long compared to thepath. For 50 km and 100 km, Fig. 3 shows that the last�15 km are the most important for the emerging taus.

The standard deviation, normalized by the average tauenergy,

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FIG. 4. Standard deviation � scaled by the average tau energyas a function of the incident neutrino energy without energy loss(solid) and for 10 km and 100 km of rock (dashed).


�

Eavg�

�

��hE2

�i � hE�i2

phE�i

(11)

FIG. 5. Average energy of emerging tau as a function of itsproduction point for a total distance of 100 km of rock, for fixedinitial neutrino energies between 108–1012 GeV.

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is shown is Fig. 4. Higher energies and larger distances leadto larger fluctuations. Huang et al.’s results in Ref. [19]correspond to our Figs. 2 and 4 and are in good agreement.We comment that by normalizing the standard deviation tothe incident neutrino energy rather than average energy ofthe emerging tau, one finds a more stable result for theratio. The standard deviation scales approximately as�0:20 � 0:25 � E� for 10 km rock over the same energyrange as the figure, and with a factor of �0:13 � 0:25 � E�for 100 km rock.

A final result for monoenergetic neutrinos is shown inFig. 5. Each histogram shows the average energy of theemerging tau as a function of the production point z for atotal depth of 100 km and at fixed initial neutrino energy.The energies of the incident neutrinos are E� � 10n GeVfor n � 8; 9; . . . ; 12 (lower to upper curves).

B. Analytic approximations

Standard approaches to parameterizing the final energyin terms of the initial energy begin with

dE�

dz’ ��E�: (12)

The solution depends on the energy dependence of �.Several parameterizations have been used, including alinear dependence on energy [12] and a logarithmic energydependence [8]. We consider three cases:

�I E� � Ei�e

��z0 ; � � const.;

�II E� �Ei��e��z0

�� Ei��1 � e��z

0; � � �� E�;

�II E� � exp��0

�1�1 � e��1�z0 � ln�Ei

�=E0e��1�z0

E0;

� � �0 � �1 ln�E=E0: (13)

The quantity z0 is the distance traveled by the tau after itsproduction. None of the choices for � is entirely satisfac-tory. Case (III) is the best approximation to the energycurves of Fig. 6. For case (I),

� � 0:85 � 10�6 cm2=g (14)

works moderately well for E� � 1010 GeV, but poorly forhigher and lower energies. We reproduce the histograms ofthe stochastic results from Fig. 5 showing the last 20 km inFig. 6 together with parameterizations of �. In all cases ofthe parameterized energy, we take Ei

� � 0:8E�. Thedashed-dotted lines show the parameterization of case (I).

Case (II) is used by Aramo et al. in Ref. [12] with

�� 0:71 � 10�6 cm2=g;

�� 0:35 � 10�18 cm2=gGeV:(15)

The energy parameterization in this case is shown with thedashed lines in Fig. 6.

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FIG. 7. Flux ratio of tau flux to incident neutrino flux as afunction of energy for 10 km and 100 km of rock, for an incidentneutrino energy scaling like E�1

� . The dashed lines do not haveenergy loss or decay included. The histograms include thestochastic treatment of the neutrino interaction and tau energyloss and decay.

FIG. 6. Average energy of emerging tau as a function of itsproduction point for a total distance of 100 km of rock, for fixedinitial neutrino energies between 108–1012 GeV from MonteCarlo simulation (histograms), and parameterization of � as incase (I) (dashed-dotted), case (II) (dashed), and case (III) (solid).


The parameterization using a logarithmic dependenceon energy for � does the best at reproducing the energybehavior of the emerging taus. This is shown with solidlines in Fig. 6, with

�0 � 1:2 � 10�6 cm2=g; �1 � 0:16 � 10�6 cm2=g;

E0 � 1010 GeV: (16)

Not surprisingly, this energy dependent � will give the bestrepresentation of the Monte Carlo simulation results for theemerging tau fluxes from power-law incident neutrinofluxes. We note that these values of � are larger thanwhat one would extract from the direct calculation of �[21] as in Eq. (6). This is because Eq. (7) is only anapproximate equality.

IV. POWER-LAW NEUTRINO SPECTRA

A. Results from Monte Carlo simulation

In this section, we consider cases of incident neutrinofluxes that depend on energy. Rather than use specific fluxmodels, we consider power-law fluxes F� � E�n

� , for n �1; 2; 3, to see qualitatively and quantitatively how fluxpropagation is affected by tau energy loss.

In Fig. 7, we show the results of the stochastic propaga-tion of the incident tau neutrino, which converts to a tauand continues through the depth D of rock, for D � 10 kmand 100 km and an incident neutrino flux scaling like E�1

� .All of the fluxes shown in this section have a sharp cutoff at

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E� � 1012 GeV. The two upper dashed curves show theoutgoing tau flux to the incoming neutrino flux neglectingtau energy loss and decay. Because of the larger columndepth for 100 km of rock, the ratio of the tau to neutrinoflux is 10 times larger than for 10 km. The flux ratioincreases with energy to scale with the neutrino crosssection.

Our stochastic propagation including energy loss anddecay is shown by the histograms in Fig. 7. Energy loss ismuch stronger for the 100 km depth, so the ratio of out-going tau flux to incident neutrino flux is smaller for100 km than 10 km for E> 109 GeV, despite the increasednumber of targets for the neutrinos. A qualitative discus-sion of the results for Fig. 7 is in Sec. IV B.

Figures 8 and 9 show the ratio of the outgoing tau flux tothe incident tau neutrino flux for incident neutrino fluxesscaling as E�2

� and E�3� , respectively. The flux ratios in-

cluding decay and energy loss at E � 108 GeV for thesesteeper fluxes can be understood based only on the lifetime,since energy loss is not overwhelming at that energy andthere is little pile-up of taus from neutrino interactions athigher energies, followed by tau energy loss. The dashedlines have a factor of D to account for the column depth ofnucleon targets in the neutrino interaction probability.When decays are included, the relevant distance at E �108 GeV is d � E�c�=m�, so the neutrino interactionprobability is proportional to d rather than D. Neutrino

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� . The dashed lines do not haveenergy loss or decay included. The histograms include thestochastic treatment of the neutrino interaction and tau energyloss and decay.


� . The dashed histograms do nothave energy loss or decay included. The histograms include thestochastic treatment of the neutrino interaction and tau energyloss and decay.


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attenuation in 10 km and 100 km of rock at E� � 108 GeVis small, so the flux ratios in Figs. 8 and 9 are the same forthe two depths. At higher energies, attenuation and energyloss play a bigger role.

Our aim here is to compare the stochastic propagation toanalytic approaches. In the next section we outline theanalytic evaluation of the emerging tau flux and makequantitative comparisons with the histograms in Fig. 7–9.

B. Analytic approximations

The analytic approximation to the emerging tau flux as afunction of tau energy depends on a number of compo-nents. We review the discussion of Ref. [11] here, ex-panded to include two energy dependences of �, a lineardependence [12] and a logarithmic one [8].

The incident flux F��E�; 0 is attenuated along its tra-jectory a distance z through the rock with an approximatefactor

F��E�; z ’ exp��z�tot�E��NA�F��E�; 0: (17)

In our evaluation here, we are assuming a constant density(standard rock). Neutrino regeneration[4–8]. from neutralcurrent processes or tau decay is not important over theseshort distances.

The probability for neutrino conversion to taus in thedistance interval [z; z� dz] with energy in the interval�Ei

�; Ei� � dEi

�� is

PCC�� ! �; E�; Ei� � dzdEi

�NA�d�CC�E�; E

i�

dEi�

: (18)

As discussed above, we take Ei� � 0:8E�:

d�CC

dEi�

’ �CC�E�%�Ei� � �1 � hyiE�; (19)

hyi ’ 0:2: (20)

Writing the differential cross section this way ensures thatthe integral over tau energy yields the charged-current(CC) cross section.

In the infinite lifetime, no energy loss (� ! 1, � ! 0)limit, the emerging tau flux given incident tau neutrinos is

F��Ei��!1;�!0 �

Z D

0dz

ZdE�NA�

d�CC�E�; Ei�

dEi�

� exp��z�tot�E��NA�F��E�; 0

�Z D

0dz

dF0��Ei

�; zdz

(21)

for rock depth D.In the case where the energy loss is treated analytically,

the lifetime and energy loss are accounted by a factorPsurv�E

i�; E�; D� z as the tau travels over the remaining

distance from production point z to emerge from the rockafter distance D with outgoing energy E�. In fact, Ei

�, E�,and D� z are not all independent, as shown in Eq. (13).

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The survival probability as a function of energy comesfrom the solution to the equation

dPsurv

dz� �

Psurv

c�E�=m�(22)

together with the approximate dE�=dz formula of Eq. (12),leading to

dPsurv

dE�’

m�

c��E2�Psurv: (23)

The solution depends on the energy dependence of �. Forconstant �, the solution is

Psurv�E�; Ei� � exp

�

m�

c��

�1

E��

1

Ei�

�case �I:

(24)

The emerging tau flux is

F��E� �Z D

0dz

dF0�Ei0� ; z

dzPsurv�E�; Ei0

�

� %�Ei0� � Ei

��E�;D� zdEi0

�

dE�dEi0

�

�Z D

0dz

dF0�Ei0� ; z

dzPsurv�E�; Ei0

�

� %�E� � E��Ei0� ; D� zdEi0

� : (25)

The delta function explicitly enforces Eq. (13) rewrittenwith Ei

� in terms of E� and D� z. A factor of dEi0�=dE�

accounts for the fact that F��E� represents the number of�’s per tau energy interval, say for #E� � E2 � E1.Electromagnetic energy loss means that one must samplea much larger interval of initial tau energies, #Ei

� �E2 exp��D � E1. Combining this factor with the deltafunction leads to the factor of %�E� � Ei0

� exp��D�z that appears in the literature[11] for tau neutrinoproduction of taus for a constant beta:

F��E� �Z D

0dz

dF0�Ei0� ; z

dzPsurv�E�; Ei0

�

� %�E� � Ei0� exp��D� zdEi0

� : (26)

In the � ! 0 limit, the emerging tau flux is the expectedresult,

F��E� �Z D

0dz

dF0�Ei0� ; z

dzexp

��m��D� z

c�E�

�: (27)

Continuing with case (I) for nonzero �, explicit substi-tutions give

F��E��Z D

0dz

ZdE�exp��z�tot�E��NAF��E�;0

�NA��CC�E�%�E��0:8E�exp��D�z

�exp��m�

c��

�1

E��e��D�z

E�

�: (28)

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A useful limit to understand our results in Fig. 7 is thecase of constant � in the long lifetime limit. For simplicity,we set hyi � 0 so that Ei

� � E� [11]. If we also assumeD � Lint where Lint is the neutrino interaction length, wecan simplify Eq. (28) to

F��E� �Z D

0

dz

LCCint

ZdE�F��E�; 0%�E� � E�e��D�z

�1

E��LCCint

Z Emax

Emin

dE�F��E�; 0; (29)

for Lint approximately constant. The energy integral runsfrom Emin � E� to Emax � min�E� exp��D, 1012 GeV).At low energies and short distances, (e.g., E� � 108 GeVand D � 10 km), the maximum energy is E� exp��D.For F� � E�1

� , F� � �D=LCCint � E

�1� , while for F� � E�n

�

for n > 1, F� � 1=��n� 1��LCCint � E

�n� . This means that

even with strong energy loss effects, for an E�1� flux in the

appropriate energy regime, the total distance D determinesthe tau flux rather than the characteristic scale of the energyloss, 1=��. In the actual case of the �, the tau decay length�c� is comparable to the energy loss scale 1=�� at E �108 GeV. For the E�1

� flux, D is still the dominant scale,although the details including both energy loss and decayas well as energy dependent cross sections have an effect,as seen in Fig. 7. Figures 8 and 9 show that this is not thecase for E�2

� and E�3� .

Equation (25) is the master equation for all three casesdiscussed here for the three parameterizations of �. Theargument of the delta function comes from Eq. (13) for thespecific cases I–III. The survival probabilities come fromthe solution to Eq. (23) for the appropriate energy depen-dent �. This gives (see Ref. [12] for case (II))

Psurv�E�;Ei��

�Ei��E�

E��Ei�

�m��c��2

��

�exp�

m�

c��

�1

E��

1

Ei�

�case�II;

Psurv�E�;Ei�� exp

�m��1

c��20

1

E�

�1� ln

�E�

E0

�

�1

Ei�

�1� ln

Ei�

E0

��

exp�

m�

c��0�

�1

E��

1

Ei�

�case �III:

(30)

In case (III), an expansion in �1=�0 has been made inEq. (23). Numerically, we have kept terms through �3

1=�30,

however in Eq. (30), we show the result where only the firstterm has been kept.

None of the parameterizations can completely describethe Monte Carlo results. Based on Fig. 6, it is not surprisingthat the form of � which depends on logE�=E0 works thebest to describe the outgoing tau flux. In Figs. 10–12, weshow a comparison of the various parameterizations of theoutgoing tau flux, normalized to the incident neutrino flux.

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FIG. 11. Flux ratio of tau flux to incident neutrino flux as afunction of energy for 100 km of rock, for an incident neutrinoenergy scaling like E�2

� . The histograms include the stochastictreatment of the neutrino interaction and tau energy loss anddecay. The analytic approximations for cases (I, dashed-dotted),(II, dashed-dot-dot-dotted) and (III, dashed) are also shown.


Figure 10 shows the Monte Carlo results for 10 km ofrock and an incident flux with E�1

� energy behavior withthe histogram and the case (III) parameterization with thedashed line. The case (I) parameterization, with constant �appears as a dashed-dotted line, while the case (II) powerlaw is shown with a dashed-dot-dot-dotted line. One seesthe turnover in the flux ratio as the change is made fromEmax � E� exp��D to Emax � 1012 GeV for the E�1

�flux. The parameterization of � as a function oflog�E�=E0 [case (III)] gives results for the flux ratio towithin about 50% of the Monte Carlo result. The othertwo parameterizations do not do as well, underestimatingby a factor of �2 for E � 109 GeV and overestimating by�3 for E � 1011 GeV.

For 100 km of rock, with the same incident flux, the case(III) parameterization compares well (underestimates by�15%) the Monte Carlo result between 109 � 1011 GeV.By comparison, the other two parameterizations for100 km rock overestimate the tau flux by a factor of 1.3–1.6 and 1.8–2 at energies 1010 and 1011 GeV respectively.

For the E�2� incident neutrino flux, the logE parameteri-

zation of � does well for 100 km as seen in Fig. 11. Again,the dashed-dotted and dashed-dot-dot-dotted lines showcases (I) and (II), respectively. For incident neutrino fluxesscaling with E�3

� , the 10 km result is well represented bythe logE parameterization, as shown in Fig. 12. The over-estimate of the flux for cases I and II occurs because energy





013005-8


loss is underestimated at high energies with these param-eterizations of �.


� . The histogram includes the stochastictreatment of the neutrino interaction and tau energy loss anddecay. The analytic approximations using Eq. (28) with � ��0 � �1 ln�E�=E0 (dashed-dotted), using E��Ei

�;D� z forcase III but a simplified Psurv (triple dashed-dotted) and thefull case III result (dashed) also shown.

V. DISCUSSION

We have performed a one-dimensional Monte Carlosimulation of �� conversion to � and � propagation includ-ing electromagnetic energy loss over distances of 10–100 km in rock. Analytic approximations are useful toolsin exploring the possibility of detecting a variety of pre-dictions for incident neutrino fluxes [11]. Using monoen-ergetic sources of tau neutrinos, we modeled the tau energyloss with an energy loss parameter � depending onlog�E�=E0. In an analytic approximation for the outgoingtau flux, this model for energy loss does better at represent-ing the Monte Carlo results than two other choices for �: aconstant and a � increasing linearly with tau energy. Ourimproved parameterization of � will help refine theoreticalpredictions of tau neutrino induced events in neutrinotelescopes.

It is not difficult to numerically evaluate the integrals ofEq. (25) including the more complicated logarithmic de-pendence on energy of � [case (III)] for Psurv and E�,nevertheless, it is interesting to see the impact of themodifications from that energy dependence. Although for-mally inconsistent, one could simply use the flux calcula-tion for constant �, Eq. (26), and substitute� � �0 � �1 ln�E�=E0. This is shown in Fig. 13, for D �100 km and E�1

� , with the dashed-dotted line. As inFigs. 10–12, the dashed line comes from the full case III(logarithmic dependence) result, and the histogram repre-sents the Monte Carlo result. The dashed-dotted line isnearly identical to the complete case III evaluation. Thetriple dashed-dotted line uses the full expression for theenergy dependence on distance from Eq. (13), but it usesthe simplified form of the survival probability: Psurv �exp��m�=�c�� E

�1� � Ei�1

� �. This does as well inrepresenting the Monte Carlo result. Of the fluxes androck depths that we have discussed here, Fig. 13 is repre-sentative of the three curves incorporating to varying de-grees the logarithmic energy dependence of �.

In summary, our Monte Carlo results for tau neutrinointeraction and tau propagation through rock suggest thatthe tau electromagnetic energy loss parameter be parame-terized by � � �0 � �1 log�E=E0 (case III), where theparameters �0; �1 and E0 appear in Eq. (16). Using the

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analytic formula of Ref. [11], modified to account for thelogarithmic energy dependence, a reasonable agreementwith the Monte Carlo results is obtained. Using the for-mulae of Ref. [11] for constant �, with the small change oftaking hyi � 0:2 but substituting ��E�, also is in reason-able agreement with the Monte Carlo result.

ACKNOWLEDGMENTS

We thank I. Sarcevic and D. Seckel who were collabo-rators in our initial evaluation of tau electromagnetic en-ergy loss, and T. Gaisser, K. Gayley, Y. Meurice, andV. G. J. Rodgers. This work is funded in part by the U.S.Department of Energy Contract No. DE-FG02-91ER40664.

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tau neutrino propagation and tau energy loss

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