high energy leptons from muons in transit

High energy leptons from muons in transit

Alexander Bulmahn and Mary Hall Reno

Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA(Received 11 December 2009; published 4 March 2010)

The differential energy distribution for electrons and taus produced from lepton pair production from

muons in transit through materials is numerically evaluated. We use the differential cross section to

calculate underground lepton fluxes from an incident atmospheric muon flux, considering contributions

from both conventional and prompt fluxes. An approximate form for the charged current differential

neutrino cross section is provided and used to calculate single lepton production from atmospheric

neutrinos. We compare the fluxes of underground leptons produced from incident muons with those

produced from incident neutrinos and photons from muon bremsstrahlung. We discuss their relevance for

underground detectors.

DOI: 10.1103/PhysRevD.81.053003 PACS numbers: 12.20.�m, 13.15.+g, 13.60.�r, 13.85.Tp

I. INTRODUCTION

Atmospheric muon fluxes at sea level have pushed neu-trino experiments underground or under ice, however, evenat a depth of more than 1.5 km under ice, the muon to muonneutrino induced muon ratio is on the order of 106. Muchof the work in detectors like AMANDA [1] and IceCube[2] has concentrated on upward �� ! � conversions to

avoid the downward muon background. The newDeepCore module [3], with a threshold of �10 GeV andsituated within IceCube, is designed to allow some of thedownward muon neutrino flux to be studied where theIceCube detector acts as a muon veto for the backgroundto muons produced in the DeepCore detector. TheDeepCore module also offers the potential to measure�� ! � and �e ! e conversions [3].

The downward atmospheric muon flux itself [4–7]presents an opportunity to explore the predictions of quan-tum electrodynamics. A well-known process is chargedlepton pair production [8–13], e.g.,

�A ! �eþe�X:

This process is the primary contributor to the muon energyloss, along with bremsstrahlung, photonuclear, and ioniza-tion interactions in transit through materials [14]. Most ofthe energy loss in electron-positron pair production occursthrough small changes in muon energy. Nevertheless, occa-sionally the charged lepton produced carries a large energy.Given that there are so many downward-going muons, onemay have the opportunity through rare high energy leptonproduction to explore physics at high energy scales.

Through charged lepton pair production, one has thepotential to see evidence for the onset of charm particleproduction of muons in the atmosphere [15–17]. At highenergies, the long decay length of light mesons does notallow them to decay before the surface of the Earth.Studying charged lepton signals underground presents an-other opportunity for determining the contributions fromcharm production to the atmospheric muon flux.

Charged tau pair production is also of interest.Atmospheric production of tau neutrinos is quite low, sincetau production comes from Ds and b meson production incosmic ray interactions with the air nuclei [18,19]. At highenergies, neutrino oscillations for �� ! �� are also sup-

pressed. Atmospheric muon production of tau pairs mayproduce a tau signal (accompanied by a muon) or ulti-mately a tau neutrino flux in an energy regime where feware expected.In Sec. II, we describe how our prior work for calculat-

ing the cross section for lepton pair production frommuonsscattering with atomic targets can be applied to numericalcalculations of the energy distribution of the leptons pro-duced for fixed muon energy [8]. The approximate analyticform of Tannenbaum [20] can be translated to an electronenergy distribution which agrees well with our numericalresults. The tau energy distribution must be evaluatednumerically. To compare the fluxes of underground leptonsproduced from incident atmospheric muons with thoseproduced from incident neutrinos, we present an approxi-mate form of the neutrino-nucleon differential cross sec-tion that agrees well with numerical evaluations for a largerange of incident neutrino energies and energy transfers tothe produced charged lepton. We review in Sec. III thesteps to go from a muon or neutrino flux to an electron ortau flux, and we describe the parametrizations for atmos-pheric muon and neutrino fluxes used here.Our calculation can be applied to many detector geome-

tries. In Sec. IV, we have focused our calculation of under-ground electron production for the large undergroundCherenkov detector IceCube. Because electromagneticshowers produced by electrons are difficult to distinguishfrom those produced by photons, we compare the flux ofhigh energy electrons produced via pair production withthat of high energy photons produced by muon bremsstrah-lung. In addition to electron signals, we also calculate thefluxes of underground taus produced by incident muonsand tau neutrinos, and we explore the possibility of � pairproduction signals at IceCube. In addition to IceCube, we

PHYSICAL REVIEW D 81, 053003 (2010)

1550-7998=2010=81(5)=053003(11) 053003-1 � 2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.81.053003

also explore the possibility of taus produced from incidentmuons and tau neutrinos in the mountains surrounding theHigh Altitude Water Cherenkov (HAWC) detector [21].HAWC has the potential to see the subsequent tau decayshowers. In the final section, we discuss our results.

II. PRODUCTION OF CHARGED LEPTONS

With our focus on lepton pair production by muons intransit, we begin our discussion with our results for theelectron or tau energy distribution for a fixed incidentmuon energy. Another source of electrons and taus, inthis case single leptons, is from neutrino charged currentinteractions. We discuss this below.

A. Lepton pair production

We have presented in Ref. [8] the formulas to evaluatethe differential cross section for a charged muon to scatterwith a nucleus of charge Z and atomic number A. Theseformulas extend the work of Kel’ner [9] and Akhundovet al. [12] and others [13] so that they are applicable to bothelectron-positron pair production and tau pair production.We have not done the full calculation required for �þ��production because of the complication of identical parti-cles in the final state.

As we showed in Ref. [8], for electron-positron pairproduction, the differential cross section is dominated byvery low momentum transfers to the target. Consequently,coherent scattering with the nucleus dominates the crosssection, and the static nucleus approximation of Kel’ner [9]is quite good. In the numerical work below, we use thedifferential cross section as a function of Ee, the energy ofthe positron or electron. Our numerical results for theelectron energy distribution for incident muon energiesof 103 GeV, 106 GeV, and 109 GeV are shown by thesolid lines in Fig. 1.

While we use our numerical results for the differentialelectron energy distribution, we note that Tannenbaum inRef. [20] has an approximate expression for the differentialcross section as a function of v ¼ ðE� � E0

�Þ=E� for an

initial muon energy E� and final muon energy E0�. In the

limit of low momentum transfer to the nucleus, E�v ’Ee þ E �e, the sum of energies of the charged lepton pair. Tofirst approximation, Ee ’ vE�=2. In terms of v,

Tannenbaum has [20]

vd�

dv¼ 28

9�ZðZþ 1Þð�rlÞ2

��ð1þ z2Þ ln

�1þ 1

z2

�� 1

�fðe; vÞ: (1)

Here rl is the classical radius of the lepton produced andz ¼ vm�=4ml. The function fðe; vÞ can be expressed in

two limiting regions as

fðe; vÞ ¼�lnðvE�=6:67mlÞ unscreened

lnð184:15ml=meZÞ fully screened:(2)

In comparing our numerical result to those found usingEq. (1), we have taken the lower of the two values offðe; vÞ. We have modified v to access the high energytail of the distribution, and with v ’ 2Ee=ðE� � 1

2EeÞ,we can reproduce our numerical results as long as v < 4.This gives

Ee ¼ 2v

4þ vE�; v < 4

ð1�m�=E�Þð1þm�=E�Þ ; (3)

neglecting me compared to m�. The dashed lines in Fig. 1

show the differential cross section for muon production ofan electron of energy Ee for E� ¼ 103 GeV, 106 GeV, and

109 GeV with the approximation of Tannenbaum usingEq. (3).The approximate form for the differential energy distri-

bution of the produced electron does not have an easycorrespondence to tau pair production [8]. For tau pairs,there is a significant contribution from inelastic scatteringof the muon with the target and higher momentum transfersneed to be taken into account. For tau pairs, we can onlyuse our numerical differential cross section. Figure 2 showsthe scattering contributions to the differential distributionof the � energy for �þ�� production by muons in transitthrough rock with an initial muon energy E� ¼ 106 GeV.

The contributions in Fig. 2 are the coherent scatteringwith the nucleus (coh), deep inelastic scattering (DIS), andscattering with individual nucleons (incoh-N) and elec-trons (incoh-e). The curve showing incoherent scatteringwith atomic electrons demonstrates the threshold behaviorfor �þ�� production.

FIG. 1. Differential cross section as a function of electronenergy for �A ! �eþe�X for fixed muon energy. Here A ¼14:3 and Z ¼ 7:23 for water. The solid line shows our numericalresult from Ref. [8] and the dashed lines show the approximationof Tannenbaum [20] using v ¼ ðE� � E0

�Þ=E� ’ 2Ee=ðE� �Ee=2Þ.

ALEXANDER BULMAHN AND MARY HALL RENO PHYSICAL REVIEW D 81, 053003 (2010)

053003-2

In what follows, we use the numerical differential crosssections for ‘ ¼ e or ‘ ¼ � production by � ! �‘þ‘�X,where the incident muons are produced by cosmic rayinteractions in the atmosphere.

B. Neutrino charged current interactions

Neutrino production of charged leptons is the primaryfocus of IceCube and other underground experiments. Tocompare charged lepton pair production by atmosphericmuons to neutrino production of a single charged lepton ininteractions with nucleons, it is helpful to have an approxi-mate differential cross section for charged current neutrinointeractions.

In principle, the neutrino charged current cross sectiondepends on the mass of the charged lepton produced. AtE� ’ 100 GeV, the ��N charged current cross section, forisoscalar nucleonN, is about 80% that for ��N [22,23]. By

E� ¼ 1 TeV, the ratio of the ��N to ��N charged current

cross sections is about 0.95. We neglect the lepton masscorrections in what follows.

An approximate form for the charged current differentialneutrino cross section at low energies can be found, e.g., inRef. [24],

d��

dy¼ 2mpG

2fE�

�ð0:2þ 0:05ð1� yÞ2Þ (4)

d� ��

dy¼ 2mpG

2fE�

�ð0:05þ 0:2ð1� yÞ2Þ (5)

with inelasticity y ¼ 1� E‘=E�. While reasonable ap-proximations to the neutrino-nucleon differential crosssection at E� � 1 TeV, Eqs. (4) and (5) do not account

for the y dependence and energy dependence that comefrom increased contributions from sea quarks as neutrinoenergies increase. They do not reflect the violation ofBjorken scaling as the momentum transfer increases.Given our interest in high energies, we calculated the

neutrino isoscalar nucleon cross section using the CTEQ6parton distribution functions [25] for E� ¼ 50–1012 GeV[26,27]. The numerical results use a small Bjorken xextrapolation at very small x according to a power lawxqðx;Q2Þ � x�� [26,27]. We use the following parametri-zation for the differential cross section,

d�cc

dy¼ 2mpG

2fE

�ðaðEÞ þ bðEÞð1� yÞ2Þ 1

ycðEÞ: (6)

For neutrino scattering, we have two energy regimes splitFIG. 2. Contributions to the differential cross section as afunction of tau energy for �A ! ��þ��X for fixed muonenergy, E� ¼ 106 GeV. Here A ¼ 22 and Z ¼ 11 is used for

standard rock. Indicated are the coherent scattering contributions(coh), deep inelastic scattering (DIS) and scattering with indi-vidual nucleons (incoh-N) and electrons (incoh-e).

FIG. 3. Differential neutrino-nucleon cross section defined byEq. (6) with parameters from Eqs. (7) and (8). The solid linesrepresent numerical results using the CTEQ6 parton distributionfunctions [25] and the dashed lines are our approximate analyticformula. Figure (a) is the differential cross section for E� �3:5� 104 with the fit parameters defined in Eq. (7). The curvesrepresent E� ¼ 50, 100, 500, 1000, 5000, 104 GeV from bottomto top. Figure (b) is the differential cross section with parametersdefined in Eq. (8). The curves represent incident neutrino en-ergies E� ¼ 105, 106, 108, 1010, 1012 GeV from bottom to top.

HIGH ENERGY LEPTONS FROM MUONS IN TRANSIT PHYSICAL REVIEW D 81, 053003 (2010)

053003-3

by neutrino energy E�c ¼ 3:5� 104 GeV, with

a� ¼ 0:19� 0:0265ð2:214� logðEc=EÞÞ2b� ¼ 0:036� 0:0344ð1:994� logðEc=EÞÞ2c� ¼ 2:3� 10�2 E< E�

c ;

(7)

and for higher energies

a� ¼ 0:060ðEc=EÞ0:675 b� ¼ 0:169ðEc=EÞ0:73c� ¼ 0:66� 10p p ¼ 1:453ðlogðEcÞ= logðEÞÞ6:24E> E�

c : (8)

Figure 3 compares the approximate form of the neutrino-nucleon differential cross section with Eq. (6) and parame-ters from Eqs. (7) and (8) with numerical results.

Because the differential cross section is dominated bythe contribution from sea quarks at high neutrino energy,we use the same high energy fit for antineutrino-nucleonscattering above E ��

c ¼ 106 GeV. The corresponding pa-rameters for lower energy antineutrino-nucleon scatteringare

a �� ¼ 4:89� 10�2� 10pa pa ¼�6:31� 10�4 logðEÞ4:05b �� ¼ 0:177� 10pb pb ¼�2:78� 10�5 logðEÞ5:9c �� ¼ 4:4� 10�3E0:32 E<E ��

c : (9)

Figure 4 shows the approximate form for antineutrinos forE �� 105 GeV with the parameters from Eq. (9).

The parametrizations for the y distribution in neutrinoand antineutrino scattering with isoscalar nucleons arewithin about 15% of the result using the CTEQ6 partondistribution functions [25].

III. FORMALISM FOR UNDERGROUNDPRODUCTION OF LEPTONS

A. Electron production

The probability for a muon to produce an electron viapair production as a function of muon energy (E�) and

emerging electron energy (Eie), in a depth interval d‘, can

be written as [28,29]

Pprodð� ! e; E�; EieÞ ¼ d‘dEi

e

NA

A�d�pairðE�; E

ieÞ

dEie

(10)

where NA ¼ 6:022� 1023 is Avogadro’s number, A is theatomic mass, and � is the density of the material the muonis traversing. For electrons produced in a detector (con-tained) which begins at depth D and extends to depth DþLmax, the event rate is given by

dN

dEfe

¼Z Lmax

0d‘

Z 1

Eie

dE�

dPprod

d‘�þ ��ðE�;Dþ ‘; Þ

�Z

dEie�ðEf

e � EieÞ

¼Z Lmax

0d‘

Z 1

Efe

dE�

NA


feÞ

dEfe

��þ ��ðE�;Dþ ‘; Þ; (11)

where we identify the initial electron energy Eie with the

electron energy detected Efe . (This correspondence cannot

be made for high energy taus.) In the above equation,�ðE�;Dþ ‘; cosÞ is the differential muon flux at depth

Dþ ‘ as a function of energy and zenith angle .The same formalism can be applied to production of

electrons via charged current electron neutrino interac-tions. For production of single electrons or positronsfrom an incident neutrino or antineutrino, the productionprobability has the same form with the replacementd�pairðE�; E

ieÞ=dEi

e ! d�ccðE�; EieÞ=dEi

e. The incident

differential muon flux at depth also needs to be replacedwith an incident differential neutrino flux.

B. Tau production

Electrons and positrons must be produced in the under-ground detector to be observed, however, very high energytaus can persist over long distances, although the tau losesenergy in transit. A tau with E� ¼ 1 PeV has a decaylength of �c�� 50 m. The tau’s electromagnetic energyloss over that distance is governed by � which is a factorof m�=m� smaller than the corresponding energy loss

parameter for muons. In the high energy limit (Ei� >

1 TeV) with continuous energy loss,�dE

dz

�’ dE

dz’ � �E: (12)

The quantity z is the column depth. In the constant �

limit, the relation between tau initial energy Ei� and its

FIG. 4. Differential antineutrino-nucleon cross section definedby Eq. (6) with parameters from Eq. (9). The solid lines representnumerical results using the CTEQ6 parton distribution functions[25] and the dashed lines are the results of our approximateanalytic formula. The curves represent incident antineutrinoenergies of E �� ¼ 100, 103, 104, 105 GeV from bottom to top.


053003-4

energy Ef� after traveling a distance ‘ in a material with a

constant density � is

Ef� ¼ Ei

� expð� ��‘Þ; (13)

since z ¼ �‘. We approximate � ¼ 8:5� 10�7 cm2=gfor standard rock [28,30].

The tau survival probability, accounting for tau energyloss and its finite lifetime, is [28–30]

PsurvðEf�; Ei

�Þ ¼ exp

�m�

c��

�1

Ei�

� 1

Ef�

��: (14)

The short lifetime of the tau means that the survival proba-bility goes to zero for low energy taus traversing anyconsiderable distance. Because of this, we will focus ourcalculation in the high energy limit.

We note that at high energies where Psurv ’ 1, a tau trackwithout a tau decay will mimic that of a muon with lowerenergy. The electromagnetic energy loss of the tau scaleswith energy according to Eq. (12), so �E=�z ’ � �E�.Since E� is not know a priori, a muon with E� ’ �E�= �

will show the same �E=�z. Taus that do not decay in adetector will be difficult to distinguish from lower energymuons.

The flux of taus produced from the atmospheric muonflux entering rock at sea level and which emerge from rockof a total thickness D is, including the survival probabilityof the tau,

dN

dEf�

¼Z D

0d‘

Z Ei� expð ��ðD�‘ÞÞ

Ei�

dEi0�

Z 1

Ei0�

dE�

��þ ��ðE�; ‘; cosÞdPprod

d‘PsurvðEf

�; Ei0� Þ

� �ðEf� � Ei0

� expð� �ðD� ‘ÞÞÞ

¼Z D

0d‘

Z 1

Ei�

dE�

NA


i�Þ

dEi�

� exp

�m�

c� ��

�expð� ��ðD� ‘ÞÞ

Ef�

� 1

Ef�

��

��þ ��ðE�; ‘; cosÞ: (15)

The delta function in the above equation explicitly enfor-ces the energy loss relation found in Eq. (12).

Again, the same formalism for the production of taupairs from an incident muon flux can be applied to theproduction of a single tau (antitau) particle from an inci-dent tau neutrino (antineutrino) by replacing the incidentmuon flux with an incident tau neutrino flux. One alsoneeds to replace the differential pair production crosssection d�pair=dE� with the differential charged current

neutrino cross section d�cc=dE�, namely, Eq. (6).

C. Atmospheric lepton fluxes

Atmospheric lepton fluxes come from cosmic ray inter-actions in the atmosphere, producing mesons which decay

to leptons [4–7,15–18]. For the differential muon flux frompion and kaon decay (the ‘‘conventional flux’’) at sea level,we use the analytical form from Ref. [31] which can bewritten as a function of energy and zenith angle,�þ ��ðE�; d; Þ, for depth d ¼ 0 at sea level:

�þ ��ðE�; 0; Þ ¼ 0:175 ðGeV cm2 sr sÞ�1

ðE�=GeVÞ2:72

��

1

1þ E� cos��=103 GeV

þ 0:037


�: (16)

Here 103 and 810 GeV are the pion and kaon criticalenergies, respectively, which separate the high and lowenergy contributions to the atmospheric flux. The effectivecosine, cos��, takes into account the spherical geometryof the atmosphere and is given by [31]

cos�� ¼ SðÞ cos� (17)

SðÞ ¼ 0:986þ 0:014 sec: (18)

The parametrization for cos� can be found in Appendix Aof Ref. [31].An additional contribution to the atmospheric flux

comes from heavy flavor particle production and decayin the atmosphere, the so-called prompt flux. For theprompt muon flux, charmed meson production dominates.There are a number of predictions for the prompt muonflux [15–19]. The vertical prompt flux can be predictedfrom a perturbative QCD calculation which can be parame-trized by [15]

logðE3��þ ��ðE; 0; ÞÞ ’ �5:37þ 0:0191xþ 0:156x2

� 0:0153x3 (19)

for x ¼ logðE�=GeVÞ. The results for atmospheric charm

using a dipole model evaluation of the c �c cross sectionfrom Ref. [17] gives a lower prompt flux prediction. Theapproximate form for the sum of �þ �� at sea level fromRef. [17] is

�þ ��ðE�; 0; Þ ’ 2:33� 10�6 ðGeV cm2 sr sÞ�1

ðE�=GeVÞ2:53ð1þ E� cos��=E0Þ(20)

where E0 ¼ 3:08� 106 GeV. Both of the prompt fluxformulas are the same for the �þ ��, �� þ ��, and �e þ��e fluxes, since to first approximation, the charmed mesonsdecay to electronic and muonic channels with equalbranching fractions and the energy distribution of themuon and the muon neutrino is about the same.The atmospheric muon flux at depth in rock or ice

depends on electromagnetic energy losses of the muon asit passes through the material [14,32]. Using the energy


053003-5

loss formula

��dE

dz

�¼ �þ E; (21)

the muon flux at depth d in a material of constant density �can be written, to first approximation assuming continuousenergy losses (cl), by

cl�þ ��ðE�; d; Þ ’ �þ ��ðEs

�; 0; Þ expð �dÞ: (22)

Here, the muon energy E� at depth d is related to the

surface muon energy by

Es� ¼ expð �dÞE� þ ðexpð �dÞ � 1Þ�= : (23)

The exponential factor in Eq. (22) comes from

dEs�

dE�¼ expð �dÞ: (24)

The above formulas are valid for continuous energy losses.Fluctuations in energy loss have the effect of increasing thedown-going muon flux underground [32], however, thesecorrections amount to 5%–10% enhancements for a depthof 1 km water equivalent distance for muon energiesbetween 100 GeV–1 TeV [31]. For our evaluations here,we neglect the corrections to the underground muon fluxdue to fluctuations in energy loss.

Figure 5 compares the contributions to the undergroundmuon flux at a depth of d ¼ 1:5 km in ice (A ¼ 14:3). Weuse the two-slope parametrization of Ref. [31] for theenergy loss, with

� ¼ 2:67� 10�3 GeV cm2=g

¼ 2:4� 10�6 cm2=g(25)

for E� � 3:53� 104 GeV, and

� ¼ �6:5� 10�3 GeV cm2=g

¼ 3:66� 10�6 cm2=g(26)

at higher energies. We show the contributions from thevertical conventional flux given in Eq. (16) as well as bothof the prompt parametrizations given in Eqs. (19) (upper)and (20) (lower prompt curve). The prompt flux becomesthe dominant contribution for muon energies E� �106 GeV. This is due to the fact that charmed mesonsdecay more quickly than pions and kaons. The probabilityfor pion and kaon decays introduces a factor of 1=Erelative to the probability for charm decays in the atmo-sphere for E & 108 GeV.For the muon produced tau lepton pairs for a detector

array like HAWC, we need the atmospheric muon flux atan altitude of 4.1 km. The high energy atmospheric muonflux at this altitude is approximately the same muon flux asat sea level. This is because the majority of the muons areproduced at an altitude of about 15 km [33]. For theenergies considered here, at altitudes between 15 and4 km, pion and kaon energy loss through interactionswith air nuclei are favored over meson decays.Finally, to compare the electron and tau pair production

rates from muons in transit to the rate for single electronand single tau production by electron neutrinos and tauneutrinos, respectively, we need the atmospheric electronand tau neutrino fluxes. At the energies considered here,E> 100 GeV, the conventional electron neutrino flux isapproximately a factor of 135 smaller than the conven-tional muon flux [5]. For our calculations here, we use

�eþ ��eðE�; 0; Þ ¼ 1:30� 10�3 ðGeV cm2 sr sÞ�1

ðE�=GeVÞ2:72

��

1


þ 0:037


�: (27)

The conventional flux of electron neutrinos has an approxi-mate 60:40 ratio of �e: ��e at E� ¼ 1 TeV [5], a ratio we usehere for the full energy range. For the prompt atmospheric�e þ ��e flux, we use Eqs. (19) and (20) as two representa-tive fluxes. The prompt neutrino to antineutrino ratio is50:50.The tau neutrino flux comes from two sources, oscilla-

tions from conventional neutrinos, primarily ��, and from

prompt decays of the Ds and b mesons and subsequent taudecays [18,19]. The prompt tau neutrino flux fromRef. [18] can be written approximately as

FIG. 5. Contributions to the underground muon flux fromatmospheric conventional and prompt fluxes. The solid linerepresents the contribution from the conventional atmosphericflux given in Eq. (16). The dashed curve represents the contri-bution from the atmospheric prompt flux given by Eq. (19) whilethe dot-dashed curve is for Eq. (20). These contributions are for adepth of d ¼ 1:5 km in ice in the vertical direction.


053003-6

��þ ��ðE;0;Þ ¼ 1� 10�7E0:5 ðGeVcm2 sr sÞ�1

ðE=GeVÞ3

��

1

1þðE=1� 106Þ0:7þðE=4� 106Þ1:5�:

(28)

The average height of production of atmospheric leptons isat an altitude of �15 km [33]. We are considering the tauneutrino downward flux or flux at 45� zenith angle atenergies above �50 GeV. The oscillation of �� ! ��

does not contribute significantly to the tau neutrino fluxover these distances at the energies of interest, so we do notinclude it in our calculation.

IV. RESULTS

A. Underground electrons in IceCube

We begin with our results for the rate of electron pro-duction in large underground detectors. For undergroundelectrons, we have focused our calculation on large under-ground Cherenkov detectors such as IceCube. We comparethe flux of electrons produced via pair production bymuons in transit through the detector with the flux pro-duced by electron neutrinos or antineutrinos interacting viacharged current interactions. We then show the contribu-tions from muon bremsstrahlung which also produce anelectromagnetic shower.

Figure 6 shows the differential flux of electrons andpositrons as a function of electron energy produced in ice

between the vertical depths of 1:5 km � d � 2:5 km. Formuon induced events, we compare the contributions fromincident conventional and prompt atmospheric muonfluxes, labeled � ! e in the plot. It is important to notethat because the production mechanism is �A !�eþe�X, the total number of high energy events comesfrom the sum of the �þ �� atmospheric flux.Accompanying the electron is a positron, a muon, andpossibly evidence of the interaction with the target. Forcomparison, we show the vertical conventional � ! e rateusing the Tannenbaum approximation to d�pair=dEe with

the dashed line.We also show the contributions to the underground

differential electron flux from an incident flux of electronneutrinos and antineutrinos, again in the vertical direction.For neutrino induced events, there is only one chargedlepton produced, so the electron flux comes purely fromthe incident neutrino flux while the positron flux comesfrom the incident antineutrino flux. Because detectors likeIceCube have no way of measuring charge, we havesummed the rates electron and positron events together toshow the total number of high energy events producedfrom incident neutrinos and antineutrinos to better com-pare with the number of events expected from atmosphericmuon production.Roughly speaking, the conventional atmospheric muon

flux falls approximately as �E�4� at high energies and the

prompt flux falls as �E�3� . With the steeply falling muon

flux, the production rate of electrons is dominated by thehigh energy tail of d�pair=dEe. As can be seen in Fig. 6, thevertical underground electron flux is dominated by theconventional atmospheric muon flux for electron energiesEe < 104 GeV. Around Ee ’ 105 GeV the contributionsfrom the prompt flux start to take over. This is in contrastto the crossover point for the muon flux, which occurs at anenergy an order of magnitude higher, at �106 GeV. Theelectron energy distribution, with the electron accompa-nied by a muon, may augment efforts to measure the onsetof the prompt muon flux.For the downward electron neutrino and antineutrino

fluxes, it is not until Ee ’ 107 GeV that the contributionsfrom incident neutrinos start to be comparable to thosefrom incident muons. The crossover between the verticalconventional and prompt electron fluxes occurs at Ee �104 GeV. The electron production rate from �e ! e isseveral orders of magnitude lower than from � ! e, ex-cept at the highest energies considered here.Although our present analysis only considers atmos-

pheric muon events in the vertical direction, it can begeneralized to other zenith angles. There are several fea-tures to consider with increased zenith angle. First, thesurface conventional atmospheric muon flux increaseswith angle, for example, by a factor �2 for E� ¼ 106

when the zenith angle increases from 0� ! 60�. The sur-face conventional atmospheric electron neutrino flux alsoincreases with zenith angle.

FIG. 6. The differential underground electron flux scaled bythe square of the electron energy for electrons produced in icebetween the vertical depths 1:5 � d � 2:5 km. The solid curvesrepresent the electron flux produced by incident vertical con-ventional fluxes of muons and neutrinos given by Eqs. (16) and(27). The dashed curves labeled prompt represent the contribu-tion from an initial prompt flux given by Eq. (19) while the dot-dashed curve is from the prompt flux given in Eq. (20). Thedashed curve following the conventional � ! e curve wascalculated using the Tannenbaum approximation to d�pair=dEe.


053003-7

At depth, muon and electron neutrino fluxes are affecteddifferently. Downward neutrinos experience little attenu-ation. Even at the highest energies, the neutrino interactionlength from interactions with nucleons is larger than105 km of ice [26]. Muon energy loss from the surface tothe detector is an important feature. The muon flux relativeto the surface value reduces according to Eq. (22) in thelimit of continuous energy loss. The net effect is that eventhough the surface muon flux at nonzero zenith angles islarger than the vertical muon flux, the flux for a detector atdepth D (and slant depth d�D= cos) is decreased rela-tive to the vertical muon flux at depth D. The undergroundflux of electrons produced in IceCube from an incidentconventional atmospheric muon flux at a zenith angle of30� is reduced to �80% of the flux produced from muonsin the vertical direction for electron energy Ee ¼ 10 GeV.For electron energy Ee ¼ 109 GeV, the electron flux pro-duced by atmospheric muons with a zenith angle of 30� is�90% relative to the flux produced by vertical muons.

At IceCube, the electromagnetic showers produced byphotons look the same as the electromagnetic showersproduced by electrons. Muon bremsstrahlung in the detec-tor is therefore another source of electromagnetic showers.In Fig. 7(a), we show with dashed lines the flux of photonsaccompanied by a muon from�A ! ��X as a function ofphoton energy. This flux is evaluated using the analyticformula of Ref. [34] for the muon bremsstrahlung differ-ential cross section. On the scale of the figure, the resultsare not much changed by including the more precise scal-ing of Ref. [35].

Above E ’ 20 GeV, the bremsstrahlung contributiondominates the electromagnetic signal. This can be under-stood by the characteristic behavior of the energy distribu-tion of photons versus electrons. The electron energydistribution from muon pair production falls more rapidlywith Ee than d�brem=dE� � E�1

� , so when convoluted with

the steeply falling atmospheric flux, the bremsstrahlungcontribution dominates. The crossover between promptand conventional photon signals moves to higher energiesthan the electron signals, but it is still less than the energyof the crossover of the muon flux itself.

Finally, we remark that the muon produced photon fluxhas a different relation between initial muon energy andphoton energy than the corresponding muon and electronenergies. This comes from the E� scaling behavior which

favors v� ¼ E�=E� > ve ¼ Ee=E� for fixed E�. If one

could correlate the incident E� to the outgoing E� or Ee

and consider veð�Þ < vmaxeð�Þ ¼ 0:01, the electron flux from

conventional muons would dominate the photon flux fromconventional muons. The differential flux of photons andelectrons with this scaling restriction is shown in Fig. 7(b).For such a restriction in veð�Þ, the electron flux from

conventional muons drops by a factor of �102, however,the photon flux from conventional muons would be re-duced by a factor of �105. The electron signal dominates

the bremsstrahlung signal by about a factor of 10 with thisrestriction. Electromagnetic showers from muons stilldominate the electron neutrino induced electromagneticshowers.

B. PeV taus at IceCube

Muon production of high energy electrons and positronsmay be a background to searches for taus via ‘‘lollipop’’events [36]. At these energies, muons do not decay, but tausdo, yielding an event with a ‘‘muonlike’’ track which has asplash of energy from the tau decay. The tau track, beforethe decay, appears as a muonlike track. The high energyelectron from �A ! �eþe�X could also leave an energysplash, with a continuing muon as part of the event con-

FIG. 7. The differential underground electron and photonfluxes scaled by the square of the electron or photon energyfor particles produced in ice (A ¼ 14:3) between the verticaldepths 1:5 � d � 2:5 km. Figure (a) shows the total differentialflux calculated using Eq. (11). The solid curves represent theelectron flux produced by incident vertical conventional andprompt (Enberg et al.) fluxes of muons. The dashed curvesshow the conventional and prompt � ! � contribution.Figure (b) represents the differential flux calculating by settingthe upper bound on ve;� to vmax

e;� ¼ 0:01 (Emin� � 100Ee;�). The

curves are the same as in Fig. (a).


053003-8

figuration. Furthermore, there is the potential to producetau pairs by �A ! ��þ��X. Some of these taus willappear at the edge of the detector, and some of them willdecay in the detector.

To explore tau production by atmospheric muons andatmospheric tau neutrinos and antineutrinos, we have cal-culated the vertical tau flux entering the IceCube detectorat a depth of 1.5 km in ice for tau energies between 1–1000 PeV. Figure 8 shows the different contributions to thetau flux from incident atmospheric lepton fluxes. At a depthof 1.5 km, the tau events are dominated by the chargedcurrent production from the incident prompt tau neutrinoflux given in Eq. (28). In this energy range, the contributionto the underground tau flux from an incident atmosphericmuon comes mainly from the prompt flux.

To convert to lollipop events, one needs to multiply thetau flux by the decay probability. Since the decay length at106 GeV is about 50 m, the decay probability over the 1 kmof the detector at that energy is Pdecay ¼ 1, and it decreases

to Pdecay ’ 0:2 at E� ¼ 108 GeV. The flux of decaying

taus, where the taus are produced by muons or tau neutri-nos, is quite small compared to the fluxes shown in Fig. 6.

While we have not done a full scale analysis of thenumber of events with decaying taus in the detectors, wecan use a characteristic

�þ �� ¼ 10�8 ðcm2 s sr GeVÞ�1ðGeV=EÞ2 (29)

isotropic neutrino flux and look at the relative normaliza-tions. At E� ¼ 105 GeV, Eq. (29) gives a flux that is afactor of about 30 larger than the prompt �� flux used inFig. 8. The downward tau flux produced near the edge of

the detector from this E�2 tau neutrino flux would remainat least a factor of 30 larger than the prompt �� ! �contribution shown. It is clear that the prompt tau neutrinoflux will be quite difficult to see in IceCube, as will be thehigh energy tau flux from muons in transit.

C. Tau production for HAWC

A different geometry for muon production of taus is touse a mountain as the muon conversion volume. For suffi-ciently high energies of the produced taus, the taus can exitthe mountainside and decay in the air. The proposedHAWC surface array has the potential to measure the taudecay air shower. This detector sits in a mountain saddle atan altitude of 4.1 km shielded by mountains on two sides.For our calculation, we have used a zenith angle of 45�

in the incident flux and 1 km water equivalent distance ofrock for the incident muons or tau neutrinos in transit.Figure 9 shows the energy distribution of taus emergingfrom the rock for both incident muon and prompt tauneutrino atmospheric fluxes. We also show the contribu-tions from an incident prompt muon atmospheric flux.As can be seen in the plot, at energies of E� < 105 GeV

the dominant contribution to the emerging tau flux comesfrom the conventional atmospheric muon flux. As notedabove, a tau energy of E� ¼ 105 GeV has a decay length of�5 m. Even at E� ¼ 107 GeV, the decay length of 500 mmay allow the tau decay to be measured by HAWC. Atenergies above E� ’ 106 GeV, the contribution of tausproduced by atmospheric muons provides about a 20%–30% contribution to the total ‘‘atmospheric’’ tau flux which

FIG. 8. Differential tau flux scaled by the square of the finaltau energy entering the detector at a depth of 1.5 km in ice. Thelower solid line corresponds to tau production from a verticalincident conventional muon flux given in Eq. (16). The dashedand dot-dashed curve represents the tau flux from a verticalincident prompt flux given in Eqs. (19) and (20), respectively.The top solid curve is for the tau flux produced with an incidentprompt tau neutrino flux given in Eq. (28).

FIG. 9. Differential tau flux scaled by the square of the finaltau energy emerging from 1 km water equivalent of rock. We usea zenith angle of 45� for our incident fluxes. The lower solid linecorresponds to tau production from an incident conventionalmuon flux given in Eq. (16). The dashed and dot-dashed curverepresents the tau flux from an incident prompt flux given inEqs. (19) and (20), respectively. The top solid curve is for the tauflux produced with an incident prompt tau neutrino flux given inEq. (28).


053003-9

is dominated by atmospheric tau neutrino conversions. Asis the case with taus in IceCube from atmospheric sources,the event rates in HAWC would be quite small.

V. DISCUSSION

Our evaluation of the pair production cross section inRef. [8] already showed that Tannenbaum’s parametriza-tion of the cross section [8] is a valuable shortcut. Here, wehave also shown that his differential distribution forelectron-positron pair production, with the identificationof v ¼ ðE� � E0

�Þ=E� ’ 2Ee=ðE� � Ee=2Þ, is a reason-

ably good representation of the numerical evaluation of theelectron energy distribution. Using the approximate formfrom Tannenbaum for the differential cross section yieldserrors that are � 30% in the calculation of undergroundelectron fluxes for electron energies 10 GeV � Ee �109 GeV.

Underground electron and photon production by atmos-pheric muons may aid in our understanding of the atmos-pheric lepton flux itself. The crossover point for promptversus conventional sources in terms of the electron energydistribution is at a lower energy than the crossover point formuons. Comparing Figs. 5 and 6 shows that the crossoverhappens 1 order of magnitude lower when looking at pairproduced electrons as opposed to muons for a verticaldepth of �1:5 km. Figure 7 shows that for the full brems-strahlung signal, the crossover point is intermediate be-tween electrons and the incident muons. The additionalinformation gained in studying underground electromag-netic signals may aid in the determination of the correctcharm production model.

The energy threshold for lollipop events from tau decaysin IceCube is �5 PeV. A 5 PeV tau will produce >200 mtrack length in the detector [36]. Although potentiallydifficult to see, the prompt �� ! � flux entering the detec-tor receives �20% contribution from high energy tausproduced by prompt muons in transit through ice. Theconventional � ! � flux is suppressed by about 2 ordersof magnitude in this energy regime relative to prompt �� !� flux.

When calculating the underground electron or positronfluxes from pair production, there should be rare events

where one of the leptons comes out with a significantfraction of the initial muon energy. In a detector likeIceCube, it may be difficult to identify the accompanyinglepton, as well as the muon after scattering. This type ofevent could fake a lollipop type signal when looking for tauneutrino induced tau events. A comparison of Figs. 6 and 7shows that the flux of electrons at an energy of a few PeV isabout a factor of �100 times the flux of atmospheric ��

induced taus. Muon bremsstrahlung contributions are evenlarger, potentially adding to a faked signal.While muon production of tau pairs in the PeV energy

range is less than the prompt atmospheric �� contribution,at lower energies there is the potential for downwardsecondary �� production from conventional muons via

�A ! ��þ��X � ! ��X; (30)

in, for example, the TeV to PeV energy range. In thisenergy range, the taus decay promptly.Our calculations have focused on large Cherenkov de-

tectors, but the same formalism could be applied for cal-culating underground lepton rates at the Indian NeutrinoObservatory (INO). Because of magnetization, INO willhave the capability of separating the electron (or ��) signalfrom that of the positron (or �þ). This will allow observa-tions at INO to determine the energy distribution of thecharged partner when looking at pair production events,something that could be predicted numerically with ourevaluation of the differential cross section.Electromagnetic interactions, in particular, lepton pair

production by muons in transit through materials, areinteresting in their own right, not just how they affect theenergy loss of muons. As neutrino telescopes and airshower detectors focus on neutrino induced signals,muon signals with high energy electrons or taus mayprovide interesting cross checks to neutrino signals.

ACKNOWLEDGMENTS

This research was supported by US Department ofEnergy Contract No. DE-FG02-91ER40664. We thankT. DeYoung for helpful conversations.

[1] A. Silvestri (AMANDA Collaboration), Int. J. Mod. Phys.A 20, 3096 (2005); T. DeYoung (IceCube Collaboration),J. Phys. Conf. Ser. 136, 022046 (2008), and referencestherein.

[2] P. Berghaus (IceCube Collaboration), Nucl. Phys. B, Proc.Suppl. 190, 127 (2009), and references therein.

[3] C. Wiebusch (IceCube Collaboration), arXiv:0907.2263.[4] See, e.g., P. Lipari, Astropart. Phys. 1, 195 (1993).

[5] V. Agrawal, T. K. Gaisser, P. Lipari, and T. Stanev, Phys.Rev. D 53, 1314 (1996).

[6] M. Honda, T. Kajita, K. Kasahara, and S. Midorikawa,Phys. Rev. D 70, 043008 (2004); M. Honda et al., Phys.Rev. D 75, 043006 (2007).

[7] E. V. Bugaev et al., Phys. Rev. D 58, 054001 (1998).[8] A. Bulmahn and M.H. Reno, Phys. Rev. D 79, 053008

(2009).


053003-10

[9] S. R. Kel’ner, Sov. J. Nucl. Phys. 5, 778 (1967).[10] S. R. Kel’ner, Yad. Fiz. 61, 511 (1998) [Phys. At. Nucl. 61,

448 (1998)].[11] S. R. Kel’ner and D.A. Timashkov, Yad. Fiz. 64, 1802

(2001) [Phys. At. Nucl. 64, 1722 (2001)].[12] A. A. Akhundov, D.Yu. Bardin, and N.M. Shumeiko, Sov.

J. Nucl. Phys. 32, 234 (1980).[13] R. P. Kokoulin and A.A. Petrukhin, in Proceedings of the

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

[14] D. E. Groom, N.V. Mokhov, and S. Striganov, Atom. DataNucl. Data Tables 78, 183 (2001).

[15] L. Pasquali, M. H. Reno, and I. Sarcevic, Phys. Rev. D 59,034020 (1999).

[16] P. Gondolo, G. Ingelman, and M. Thunman, Astropart.Phys. 5, 309 (1996).

[17] R. Enberg, M.H. Reno, and I. Sarcevic, Phys. Rev. D 78,043005 (2008).

[18] A. D. Martin, M.G. Ryskin, and A.M. Stasto, Acta Phys.Pol. B 34, 3273 (2003).

[19] L. Pasquali and M.H. Reno, Phys. Rev. D 59, 093003(1999).

[20] M.H. Tannenbaum, Nucl. Instrum. Methods Phys. Res.,Sect. A 300, 595 (1991).

[21] A. Sandoval, M.M. Gonzalez, and A. Carraminana(HAWC Collaboration), AIP Conf. Proc. 1085, 854

(2009).[22] S. Kretzer and M.H. Reno, Nucl. Phys. B, Proc. Suppl.

139, 134 (2005).[23] S. Kretzer and M.H. Reno, Phys. Rev. D 66, 113007

(2002).[24] A. Strumia and F. Vissani, arXiv:hep-ph/0606054.[25] J. Pumplin et al., J. High Energy Phys. 07 (2002) 012.[26] R. Gandhi, C. Quigg, M.H. Reno, and I. Sarcevic, Phys.

Rev. D 58, 093009 (1998); Astropart. Phys. 5, 81 (1996).[27] M.H. Reno, Nucl. Phys. B, Proc. Suppl. 143, 407 (2005).[28] S. I. Dutta, Y. Huang, and M.H. Reno, Phys. Rev. D 72,

013005 (2005).[29] J. L. Feng, P. Fisher, F. Wilczek, and T.M. Yu, Phys. Rev.

Lett. 88, 161102 (2002).[30] S. I. Dutta, M.H. Reno, I. Sarcevic, and D. Seckel, Phys.

Rev. D 63, 094020 (2001).[31] S. I. Klimushin, E. V. Bugaev, and I. A. Sokalski, Phys.

Rev. D 64, 014016 (2001).[32] P. Lipari and T. Stanev, Phys. Rev. D 44, 3543 (1991).[33] T.K. Gaisser and T. Stanev, Phys. Rev. D 57, 1977 (1998).[34] A. A. Petrukhin and V.V. Shestakov, Can. J. Phys. 46,

S377 (1968).[35] A. Van Ginneken, Nucl. Instrum. Methods Phys. Res.,

Sect. A 251, 21 (1986).[36] D. F. Cowen (IceCube Collaboration), J. Phys. Conf. Ser.

60, 227 (2007).


053003-11

high energy leptons from muons in transit

Documents