Muon Monte Carlo: a new high-precision tool for muonpropagation through matter

Dmitry Chirkin��� �

, Wolfgang Rhode�

chirkin@physics.berkeley.edurhode@wpos7.physik.uni-wuppertal.de

�University of California at Berkeley, USA�

Bergische Universitat Gesamthochschule Wuppertal, Germany

Abstract

Propagation of muons through large amounts of matter is a crucial necessity for analysis ofdata produced by muon/neutrino underground experiments. A muon may sustain hundreds ofinteractions before it is seen by the experiment. Since a small uncertainty, introduced hundredsof times may lead to sizable errors, requirements on the precision of the muon propagation codeare very stringent. A new tool for propagating muon and tau charged leptons through matter thatis believed to meet these requirements is presented here. The latest formulae available for thecross sections were used and the reduction of the calculational errors to a minimum was our toppriority. The tool is a very versatile program written in an object-oriented language environment(Java). It supports many different optimization (parametrization) levels. The fully parametrizedversion is as fast or even faster than the competition. On the other hand, the slowest version of theprogram that does not make use of parametrizations, is fast enough for many tasks if queuing orSYMPHONY environments with large number of connected computers are used. An overviewof the program is given and some results of its application are discussed.

mmc code homepage ishttp://area51.berkeley.edu/˜dima/work/MUONPR/

mmc code available athttp://area51.berkeley.edu/˜dima/work/MUONPR/BKP/mmc.tgz

1 IntroductionIn order to observe atmospheric and cosmic neutrinos with a large underground detector (e.g.

AMANDA [1]), one needs to isolate their signal from the 3-5 orders of magnitude larger signal fromthe background of atmospheric muons. Methods that do this have been designed and proven viable[2]. In order to prove that these methods work and to derive indirect results such as the spectral indexof atmospheric muons, one needs to compare data to the results of the computer simulation. Such asimulation normally contains three parts: propagation of the measured flux of the cosmic particlesfrom the top of the atmosphere down to the surface of the ground (ice, water); propagation of theatmospheric muons from the surface down to and through the detector; generation of the Cerenkovphotons emanating from the muon tracks in the vicinity of the detector and their interaction withthe detector components. The first part is normally called generator, since it generates muon flux atthe ground surface; the second is propagator; and the third simulates the detector interaction withthe passing muons. Mainly two generators were used so far (by AMANDA): basiev and CORSIKA[3]. Results and method of using CORSIKA as a generator in a neutrino detector (AMANDA)were discussed in our previous contribution [4]. Several muon propagation Monte Carlo programswere used with different degrees of success as propagators. Some are not suited for applicationswhich require the code to propagate muons in a large energy range (e.g. mudedx), the others seemto work in only some of the interesting energy range ( � � � TeV, propmu) [5]. Most of theprograms use cross section formulae, whose precision has been improved since their writing. Forsome applications, one would also like to use the code for the propagation of muons that contain��������������� interactions along their track, so the precision of each step should be sufficiently high andthe computational errors should accumulate as slowly as possible. Significant discrepancies betweenthe muon propagation codes we tested were observed, believed to be mostly due to algorithm errors.This motivated writing of a new code that would reduce calculational errors to minimum, leavingonly those uncertainties that come from the imperfect knowledge of the cross sections. Here wepresent a new tool (Muon Monte Carlo: MMC), designed to meet this goal.

2 Description of the codeThe primary design goals of MMC were uncompromising computational precision and code clar-

ity. It was decided that the program should be written in JAVA, since JAVA is an object-oriented pro-gramming language (for best code readability) and has consistent behavior across many platforms.MMC consists of pieces of code (classes), each contained in a separate file. These pieces fulfill theirseparate tasks and are combined in a structured way (Fig. 1). This simplifies code maintenance andintroduction of changes/corrections to the cross section formulae. It is also very straightforward toeven “plug in“ new cross sections, if necessary. Writing in an object-oriented language allows sev-eral instances of the program to be created and accessed simultaneously. This is useful for simulatingthe behavior of the e.g. neutrino detectors, where different conditions apply above, inside and belowthe detector.

The code evaluates many cross-section integrals, as well as two tracking integrals. All integralevaluations are done by the Romberg method of the 5th order (by default) [6] with a variable sub-stitution (mostly log-exp). If an upper limit of an integral is an unknown (that depends on a randomnumber), an approximation to that limit is found during normalization integral evaluation, and thenrefined by Newton-Raphson method combined with bisection [6].

Propagate

Particle

Medium

Cross Sections

Decay

Ionization

Photonuclear

Epair production

i, b, p or e

continuous

stochastic

Bremsstrahlung

Output

Scattering Energy2LossE

X

Applications

Amanda Frejus

Applets MMC

Integral

Interpolate

FindRoot

StandardNormal

mmc

Physics model Math. model

Fig. 1: MMC structure

Originally, the program was designed to be used in the Massively Parallel Network Computing(SYMPHONY) [10] framework, therefore computational speed was considered only a secondaryissue. However, parametrization and interpolation routines were implemented for all integrals. Theseare both polynomial and rational function interpolation routines spanned over varying number ofpoints (5 by default) [6]. Inverse interpolation is implemented for root finding (i.e. when � �����

isinterpolated to solve

��� � ���). Two dimensional interpolations are implemented as two consecutive

one-dimensional ones. It is possible to turn parametrizations on or off for each integral separatelyat program initialization. The default energy range in which parametrized formulae will work waschosen to be from 105.7 MeV (the muon rest mass) to ������ MeV and the program was tested to workwith much higher setting for the higher energy cutoff. With full optimization (parametrizations) thiscode is at least as fast or even faster than the competition.

Generally, as a muon travels through matter, it loses energy due to ionization losses, bremsstrah-lung, photo-nuclear interaction and pair production. Formulae for the cross sections were taken fromthe recent contribution [7]. These formulae are claimed to be valid to within about 1%. All of theenergy losses have continuous and stochastic components, the division between which is superficialand is chosen in the program by selecting an energy cut ( ������� ) or a relative energy loss cut ( ������� ).In the following text ������� and ������� are considered to be interchangable and related by ������� � ������� �(even though only one of them is a constant). Ideally, all losses should be treated stochastically.However, that would bring the number of separate energy loss events to a very large value, sincethe probability of such events to occur diverges as ��� ��� �"!#� for the bremsstrahlung losses, as the lostenergy approaches zero, and even faster than that for the other losses. In fact, the only reason thislarge number is not infinity is existence of kinematic cutoffs (larger than some ��$ ) for all divergingcross sections. A good choice of �%����� for the propagation of atmospheric muons should lie in the

range ( ��� � � - ��� � ) [8]. For monoenergetic beams of muons, � ����� may have to be chosen to be high as������� � ����� .

2.1 Tracking formulae Let the continuous part of the energy losses (a sum of all energylosses, integrated from zero to � ����� ) be described by a function f(E):

���� �

� ����

� �x xdxi f

Fig. 2: derivation of tracking formulae

The stochastic part of the losses is described by the function � ��

�, which is a probability for

any energy loss event (with lost energy �������� ) to occur along a path of 1 cm. Consider the particlepath from one interaction to the next consisting of small intervals (Fig. 2). On each of these smallintervals probability of interaction is

�� ��

� �� � � � � ��

� �� � � � � . It is easy to derive an expressionfor the final energy on this step as a function of the random number . Probability to completelyavoid stochastic processes on an interval ( �� ; ��� ) and then suffer a catastrophic loss on

� � at ��� is

�� �

�� ��

� �� � � ��� ����� � �� �

�� ��

� ��� � � ��� �� ��

� ��� � �������� �

��� �

�� �� � � ��� ����� ������� �

��� �

�� ��� � � ��� �� �

�� ��� � �

������������! �" �#

�� ��

� � � �%$&� �� ��

� ��� � �

� � � � � �'��� ���� " �#

� ��

��

����

� � � � �%$ � � ��( �') +* �

��, �.-To find the final energy on each step the above equation is solved for �/� :

�& " �#� ��

��

����

� � � � ��103254 � �

(energy integral).

This equation has a solution if

�! �$ ���'��������& �#687 9;: � �

��

� ��

� � � � $ �Here ��� �8< is a low energy cutoff, below which the muon is considered to be lost. Please note, that� ��

�is always positive due to ionization losses (unless �������>=@? �BA��

). � ��

�is also always positive

because it includes the positive decay probability. If DCE �$ , the particle is stopped and its energy isset to � � �8< . The corresponding displacement for all can be found from

��� � �� �F�& �" �#��� ��

� (tracking integral).

2.2 Continuous randomization It was found that for higher �%����� muon spectra do notlook compact (Fig. 3). In fact, there is a large peak (at ��� 6���� ) that collects all particles that did notsuffer stochastic losses followed by the main spectrum distribution separated from the peak by at leastthe value of ������� ��� 6���� (the smallest stochastic loss). The appearance of the peak and its prominenceare governed by ������� , initial energy � propagation distance ratio and the binning of the final energyspectrum histogram. In order to be able to approximate the real spectra with even large � ����� and tostudy the systematic effect at a large ������� , a “continuous randomization” feature was introduced.

For a fixed ������� or ������� a particle is propagated until the algorithm discussed above finds an in-teraction point, i.e. a point where the particle loses more than the cutoff energy. The average valueof the energy decrease due to continuous energy losses is evaluated according to the energy integralformula above. There will be some fluctuations in this energy loss, which are discarded by the for-mula. Let’s assume there is a cutoff for all processes at some small ��$�� ������� . Then the probability � �5, � �

for a process with � $ C ��� �"!�� C ������� on the distance� � is normalizable to 1. Let’s choose

� �so small that $ � � 6� ���6�� � �5, � � � � � � ��� �

Then the probability to not have any losses is � � $ , and the probability to have two or more separatelosses is negligible. The standard deviation of the energy loss on

� � from the average value

C � � � � 6 ����6 � � � � �5, � � � � � � �is then C ��� � ���

�� C � � � � C � � � , where

C � � � � � 6�����6�� � � � � �5, � � � � � � �If ������� or ������� used for the calculation is sufficiently small, the distance ��� � �� determined by

the energy and tracking integrals is so small that the average energy loss � � � � is also small (ascompared to the initial energy � ). We therefore may assume � �5, � ��� � �5, � � , i.e. the energy lossdistributions on the small intervals

� ��� that sum up to the � � � �� , is the same for all intervals. Sincethe total energy loss � � � � ��� ��� , the central limit theorem can be applied, and the final energyloss distribution will be Gaussian with the average

��

�� � � � and widthC ��� ���

�� � �

���� � � C � �� � � C ��� � �!

� � �" � � 6� ���6 � � �� � � ����, �( � � ��� $ � �#� � � � 6�����6 � ��� � � ����, � � � ��� $ � � � ��%$� �'&�"&'#

� � ��� � 6 ����6 � � � � � �5, � � � � � � � $ � �(&�"&'#� � ��� � 6 ���6 � � � � �5, � � � � � � � $ � � �

Here �( was replaced with average expectation value of energy at � , �� � �

. As� �*) � , the second

term disappears. The lower limit of the integral over � can be replaced with zero, since all of thecross sections diverge slower than � ��� � . Then,

C ��� ����

� � ��� � &�"&'#

��

�� ��

� ��� � 6 ���$ � � � � �5, � � � � $

This formula is applicable for small ������� , as seen from the derivation. Energy spectra calculated with“continuous randomization” converge faster than those without (see Fig. 4-5).

energy [TeV]

muo

ns w

ith th

at e

nerg

y

10 2

10 3

10 4

10 5

10 6

0 10 20 30 40 50 60 70 80 90 100energy [TeV]

muo

ns w

ith th

at e

nerg

y

10 2

10 3

10 4

10 5

10 6

70 75 80 85 90 95 100

Fig. 3: Distribution of the final energy of themuons that crossed 300 m of Frejus Rock withinitial energy 100 TeV: � ������� �������

(solid),� ������� � � (dashed-dotted), � ������ �������

and“cont” option (dotted)

Fig. 4: A close-up on the Fig. 3: � ������� �������(dashed), � ����� � ����� (solid), � ������� � ���(dotted), � ������� � � (dashed-dotted)

energy [TeV]

muo

ns w

ith th

at e

nerg

y

10 2

10 3

10 4

10 5

10 6

70 75 80 85 90 95 100

energy [TeV]

muo

ns w

ith th

at e

nerg

y

1

10

10 2

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10 4

10 5

0 10 20 30 40 50 60 70 80 90 100

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60 70 80 90 100

Fig. 5: Same as in Fig. 4, but with “cont” optionenabled

Fig. 6: Comparison of paramerized (dashed-dotted) with exact (non-parametrized, dotted)versions for � ������� ����� . Also shown is the rela-tive difference of the curves.

3 ErrorsAll cross-section integrals are evaluated to the relative precision of ����� � , the tracking integrals

are functions of these, so their precision was set to a higher value of � ��� � . To check the precision

of interpolation routines, results of running with parametrizations enabled were compared to thosewith parametrizations disabled. Fig. 7 shows relative energy losses for ice due to different mech-anisms. Decay energy loss is shown here only for comparison and is evaluated by multiplying theprobability of decay by the energy of the particle. In the region below 1 GeV bremsstrahlung energyloss has a double cutoff structure. This is due to a difference in the kinematic restrictions for muoninteraction with oxygen and hydrogen atoms. A cutoff (for any process) is a complicated structureto parametrize and with only a few parametrization grid points in the cutoff region, interpolationerrors

� � � � � ��� � � ��� � � may become quite high, reaching 100% right below the cutoff, where theinterpolation routines give non-zero values, whereas the exact values are zero. But since the energylosses due to either bremsstrahlung, photonuclear process or pair production are very small nearthe cutoff in comparison to the sum of all losses (mostly ionization energy loss), this big relativeerror results in a much smaller increase of the relative error of the total energy losses (Fig. 8). Be-cause of that, parametrization errors never exceed ��� � - ��� ��� , for the most part being even muchsmaller ( ����� � - ����� � ), as one can estimate from the plot. These errors are much smaller than theuncertainties in the formulae for the cross sections. Now the question arises whether this precisionis sufficient to propagate muons with hundreds of interactions along their way. Fig. 6 is one ofthe examples that demonstrate that it is sufficient: the final energy distribution did not change afterenabling parametrizations.

ionizbremsphotoepairdecay

energy [GeV]

ener

gy lo

sses

[GeV

g-1

cm

2]

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

11010 210 310 410 510 6

10-1

1 10 10210

310

410

510

610

710

810

910

1010

11

energy [GeV]

prec

isio

n of

par

amet

rizat

ion

(tot

al)

10-10

10-9

10-8

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10-3

10-2

10-1

1

10-1

1 10 10210

310

410

510

610

710

810

910

1010

11

Fig. 7: ioniz (upper solid curve), brems (dashed),photo (dotted), epair (dashed-dotted) and decay(lower solid curve) losses in ice

Fig. 8: Interpolation precision ��� � ��� � � ����� � � �MMC has a low energy cutoff � � �8< below which the muon is considered to be lost. By default

it is equal to the mass of the muon, but can be changed to any higher value. This cutoff entersthe calculation in several places, most notably in the initial evaluation of the energy integral. Todetermine the random number �$ below which the particle is considered stopped, the energy integralis first evaluated from �( to ��� �8< . It is also used in the parametrization of the energy and trackingintegrals, since they are evaluated from this value to � and � � , and then the interpolated value for� � is subtracted from that for � . Fig. 9 demonstrates the independence of MMC from the value of

��� �8< . For the curve with � � �8< ��� � integrals are evaluated in the range 105.7 MeV � 100 TeV, i.e.over six orders of magnitude, and they are as precise as those calculated for the curve with ��� �8< =10TeV, with integrals being evaluated over only one order of magnitude.

energy [TeV]

muo

ns w

ith th

at e

nerg

y

1

10

10 2

10 3

10 4

10 5

0 10 20 30 40 50 60 70 80 90 100

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60 70 80 90 100 ionizbremsphotoepair

energy [GeV]

seco

ndar

ies

with

that

ene

rgy

1

10

10 2

10 3

10 4

10 5

1 10 102

103

104

Fig. 9: Comparison of � � �8< � � � (dotted-dashed) with � � �8< =10 TeV (dotted). Also shownis the relative difference of the curves.

Fig. 10: ioniz (upper solid curve), brems(dashed), photo (dotted), epair (dashed-dotted)spectra for

� � =10 TeV in the Frejus rock

Fig. 10 demonstrates the spectra of secondaries (delta electrons, bremsstrahlung photons, excitednuclei and electron pairs) produced by the muon, which energy is kept constant at 10 TeV. Thethin lines behind the histograms are the probability functions (roughly cross sections) used in thecalculation. They have been corrected to fit the logarithmically binned histograms (multiplied by thesize of the bin which is proportional to abscissa, i.e. energy). While the agreement is trivial from theMonte Carlo point of view, it demonstrates that the computational algorithm is correct.

1 TeV

vcut

-0.8-0.6-0.4-0.2

00.20.40.60.8

x 10-3

10-3

10-2

10-1

100 TeV

vcut

-0.8-0.6-0.4-0.2

00.20.40.60.8

x 10-3

10-3

10-2

10-1

Fig. 11

Fig. 11 shows the relative deviation of the average final energy of the � � ��� � 1 TeV and 100TeV muons propagated through 100 m of Fr ejus rock with the abscissa setting for � ����� , from thefinal energy obtained with ������� �

� . Just like in [8] the distance was chosen small enough so thatonly a negligible number of muons stop, while big enough so that the muon suffers a big numberof stochastic losses ( � ��� for ��������� � ����� ). All points should agree with the result for �%����� �

� ,

since it should be equal to the integral of all energy losses, and averaging over the energy losses for������� C � is evaluating such an integral with the Monte Carlo method. There is a visible systematicshift � � ��� � ����� (similar for other muon energies), which can be considered as another measureof the algorithm accuracy [8].

energy [GeV]

muo

ns s

urvi

ved

1

10

10 2

10 3

10-1

1 10 102

103

distance traveled [km]

muo

ns [x

103 ]

2468

101214

1 2 3 4 5 6 7 8 9 10

Fig. 12: � � muons with energy 9 TeV propagated through 10 km of water: regular (dashed) vs.“cont” (dotted)

In case when almost all muons stop before passing the requested distance (see Fig. 12), evensmall algorithm errors may affect survival probabilities by a lot. The following table summarizes thesurvival probabilities of monochromatic muon beam of ���

�muons with three initial energies (1 TeV,

9 TeV and ����

TeV) going through three distances (3 km, 10 km and 40 km) in water. One shouldnote that these numbers are very sensitive to the formulae of cross sections used in the calculation;e.g. for the muons with energy ����� GeV propagated through 40 km the rates decrease 30 % whenthe default photonuclear cross section is replaced with the ZEUS parametrization (case number fourfrom Sec. 6.3). However, the same set of formulae was used throughout the calculation. The errorsof the values in the table are statistical and are ��� ��� � � � .

������� “cont” 1 TeV 3 km 9 TeV 10 km ����

TeV 40 km0.2 no 0 0 0.1530.2 yes 0.010 0.057 0.1770.05 no 0 0.035 0.1430.05 yes 0.045 0.039 0.1390.01 no 0.030 0.037 0.1420.01 yes 0.034 0.037 0.139� ����� no 0.034 0.037 0.140� ����� yes 0.034 0.037 0.135

The survival probabilities converge on the final value for �%������� ��� � � in the first two columns. Usingthe “cont” version helped the convergence in the first column. However, the “cont” values departedfrom regular values more in the third column. The relative deviation (3.5%) can be used as anestimate of the continuous randomization algorithm precision (not calculational errors) in this case.One should note, however, that with the number of interactions � ����� the continuous randomizationapproximation formula was applied � ��� � times. It explains why the value of “cont” version for������� �

��� � � is closer to the converged value of the regular version than for � ����� �������� .

4 ResultsThe code was incorporated into the Monte Carlo chains of two detectors: Fr ejus [9] and AMANDA

[5]. In this section some general results are presented.

The energy losses plot was fitted to the function�� � � � � �����

� (Fig. 13). The first twoformulae for the photonuclear cross section (Sec. 6.3) can be fitted the best, all others lead to energylosses deviating more at higher energies from this simple linear formula; therefore the numbers givenwere evaluated using the first photonuclear cross section formula. In order to choose low and highenergy limits correctly (to cover the maximum possible range of energies that could be comfortablyfitted with a line), a � � plot was generated and analysed (Fig. 14). It can be seen that � � plot atthe low energies goes down sharply, then levels out. This corresponds to the point where linearapproximation starts to work. At high energies � � rises monotonically. This means that a linearapproximation, though valid, has to describe a growing energy range. An interval of energies from20 GeV to ��� �"� GeV is chosen for the fit. The following table summarizes the found fits to a and b:

medium a, � 6�� < 6 b, � $� � < 6 av. dev. max. dev.ice 0.259 0.357 3.7% 6.6%fr. rock 0.231 0.429 3.0% 5.1%

The errors in the evaluation of a and b are in the last digit of the given number. However, if the lowerenergy boundary of the fitted region is raised and/or the upper energy boundary is lowered, each byan order of magnitude, a and b change by about 1%.

dE/dx=a+bEa=0.259425 [ GeV/mwe ],b=0.000357204 [ 1/mwe ]

energy [ GeV ]

tota

l ene

rgy

loss

es [G

eV g

-1 c

m2]

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Fig. 13: Fit to the energy losses in ice Fig. 14: � � plot for energy losses in ice

To investigate the effect of stochastic processes, muons with energies 105.7 MeV - ��� �"� GeV werepropagated to the point of their disappearance. �%����� � � � ��� ��� was used in this calculation; using theversion with the continuous randomization did not change the final numbers. Average final distance(range) for each energy was fitted to the solution of the energy loss equation

�� � � � �������

� :

��� � 0�254 ���� ��� � � � � �

(Fig. 15). The same analysis of the � � plot as above was done in this case (Fig. 16). A region ofinitial energies from 20 GeV to � � �"� GeV was chosen for the fit. The following table summarizesthe results of these fits:

medium a, � 6�� < 6 b, � $ � < 6 av. dev.ice 0.268 0.470 3.0%fr ejus rock 0.218 0.520 2.8%

dE/dx=a+bEa=0.217798 [ GeV/mwe ],b=0.000520421 [ 1/mwe ]

energy [ GeV ]

aver

age

disp

lace

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t [ m

we

]

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Fig. 15: Fit to the average range in Frejus rock Fig. 16: � � plot for average range in Frejus rock

As the energy of the muon increases, it suffers more interactions before it is lost and the rangedistribution becomes more Gaussian-like (Fig. 17). It is obvious that the inclusion of stochasticprocesses into consideration leads in general to larger energy losses at higher energies than with onlycontinuous processes and the center of gravity of the muon beam travels to a smaller distance.

5 ConclusionsA very versatile, clear-coded and easy-to-use Muon propagation Monte Carlo program (MMC) is

presented. It is capable of propagating muon and tau leptons of energies from 105.7 MeV (muon restmass, higher for tau) to ��� �"� GeV (or higher), which should be sufficient for the use as propagatorin the simulations of the modern neutrino detectors. A very straightforward error control model isimplemented, which results in computational errors being much smaller than uncertainties in the for-mulae used for evaluation of cross sections. It is very easy to “plug in” cross sections, modify them,or test their performance. The program was extended on many occasions to include new formulae oreffects. MMC does all calculations and checks in three dimensions and takes into account Molierescattering on the atomic centers, which could be considered as the zeroth order approximation totrue muon scattering since bremsstrahlung and pair production are effects that appear on top of suchscattering. A more advanced angular dependence of the cross sections can be inserted at a later date,if necessary.

The MMC program was successfully incorporated into and used in the Monte Carlo chains ofAMANDA and Fr ejus experiments. We hope that the combination of precision, code clarity, speedand stability will make this program a useful tool in the research connected with high energy particlespropagating through matter.

Also, a calculation of coefficients in the energy loss formula�� � � � � � � �

� is presentedfor both continuous and full (continuous and stochastic) energy loss treatments. The calculatedcoefficients apply in the energy range from 20 GeV to �����"� GeV with an average deviation from thelinear formula of 3.7% and maximum of 6.6%.

30 GeV

entr

ies 7 ⋅ 103 GeV

1.7 ⋅ 106 GeV 4 ⋅ 108 GeV

distance [ m ]

20 30 40 50 60 0 2000 4000

0 5000 10000 0 10000 20000

Fig. 17: Range distributions in Frejus rock: solid line designates the value of the range evalu-ated with the second table (continuous and stochastic losses) and the broken line shows the rangeevaluated with the first table (continuous losses only).

6 FormulaeThis section summarizes cross sections formulae used in MMC. In the formulae below � is the

energy of the incident muon, while � � � � is the energy of the secondary particle: knock on electronfor ionization, photon for bremsstrahlung, virtual photon for photonuclear process and electron pairfor the pair production.

� � � ��� and � � �� �

� � � � ��� � . � is muon mass,� � � 6 is electron mass

and � is proton mass. Please refer to the next section for values of any constants appearing below.Most of the formulae in Sec. 6.1- 6.4 are taken directly from [7]

6.1 Ionization A standard Bethe-Bloch equation given in [11] was modified for muon andtau charged leptons (massive particles with spin 1/2 different from electron) following the procedureoutlined in [12]. The result is given below:

���� �

����� � A� � � " �� ��� � � � 6 � � � � � upper? �8A � � $�

� ����� �

upper�max

$�� ��� �

upper� � �

����� � � $ � �

� $where �

max� � � 6 � � � � �

��� � � �� � � ���� � and �

upper� ��� � � �

cut) �

max�

The density correction

is computed as for nonconductors: ��)

if � C�� $ � � � 0�� ��� � � ��� ��� � � � ��� � )if � $ ��� C�� � � � � 0�� ��� � � ��� )

if ����� � where � � ������ $

� � � �� � �� � � �� ����� � A� �� � �� � " � �

� � ��max

� ��� ��

������ � � $ � $

This formula, integrated from � � � �� � �"!$#&%(')+* � � to � ��� � 6-, , gives the expression for energy loss

above, less the density correction and� �

terms (plus two more terms which vanish if � � � � ��� � 6-, ).6.2 Bremsstrahlung According to [13], bremsstrahlung cross section may be representedby the sum of elastic component ( � 6 � , discussed in [14, 15]) and two inelastic components (

� � ��/. � )� � � 6 � �(� � �� �'� � ��6.2.1 Elastic Bremsstrahlung

� 6 � � � ) � � �10�� � A � �32 6 $ � � � 4 � � 4 � � � � $ 5 0��76 � 98 � �

� �� 6 �� � � 6 ��;:

where=< � �?>

� � �� �

> �is the minimum momentum transfer. The formfactors (atomic

� 6 �� and nuclear� 6 �� ) are� 6 �� � � � 0�� 5

�� �A@ �CB A � ��� � � � :� 6 �� � � � 0�� 5 D ��� � D � @ � � � � ��� : , D � �

�5� � � � $FE �HG

6.2.2 Inelastic Bremsstrahlung The effect of nucleus excitation can be evaluated as� �� � �A � 6 �� , �8A����

�

Bremsstrahlung on the atomic electrons can be described by the following diagrams:

e-diagram is included with ionization losses (because of its sharp ����� � energy loss spectrum), asdescribed in [16] � � �$�� � � �

� � � � �� � � �$ ! � � 0

���� � � � � � � �

�� � �

� � 03254 ��� � � � � 6 �') � � 03254 � �

� � � � � � & � � �� � � � � � �')

� � 032 4 � � � � �� � � � � � � 6 � � � � � � � � � �

Maximum energy lost by a muon is the same as in the pure ionization (knock-on) energy losses.Minimum energy is taken as � � � ? �8A��

. In the above formula � is the energy lost by the muon,i.e. the sum of energies transferred to both electron and photon. On the output all of this energy isassigned to the electron.

The contribution of the � -diagram (included with bremsstrahlung) is discussed in [13]:� � �� ��) � � < 0

�� � A � � 2 6 $ � � � 4 � � 4 � � � � $ � ��� �� < �A��� �� � �

with �� �� � � � 0�� 5 � � � � � � � @ � : � 0 � 5

�� � @ ��B�� A � � � � :B � =1429 for

A � � and B � =446 for Z=1.

The maximum energy, transferred to the photon is

� � & � � �� � � �

��� � � � �

On the output all of the energy lost by a muon is assigned to the bremsstrahlung photon.

6.3 Photonuclear Interactions

Photonuclear cross section is used as parametrized in [17]:� �� �� 0

���� � *�� �

������ ��� � � � 5�� 0�� � � � � � � $

�� � �

�� ��� � ��� � : �

���� � � 5 � 0 � � � � � �� $

� � �� : � � �

� 5���� ��� � � � � �

�� ��� � ��� � � � �� 0�� � � �

� �� $ : �where

��� � � & � � � � �� � �

) � �� � ��

� �� � ) � �

����� � � GeV

� )and

� �� ��5��� GeV

�Nucleon shadowing is taken care of by� *�� � � � � � � *�� � � ���

����� ��� � � � ���� � ���

with� � � � �

4� � � � �

� � �� � � & �

�� � � $ )

forA���

� , and� � � ���

� for Z=1

� � � � ��� � ���� ��������� ���� � *�� � � �

Several parametrization schemes for the photon-nucleon cross section are implemented. The first is� *�� � � � � ��� � � � ���@ �)

for � � � � GeV� *�� � � � ���� � � 4 � �5� � �!� 0�� � " ��� � � � 4 � - � b

)for � * "

� � ) � ��� GeV - [17]� *�� � � � � � � � � � ���5� � 0 � " � - � � � �5��� � @ � � b)

above 200 GeV [18]

The second is based on the table parametrization of [19] below 17 GeV. Since the second formulafrom above is valid for energies up to � �

�GeV, it is taken to describe the whole energy range alone

as the third case. Formula [20]� *�� � � ���#� 4 � ��$ $FE $ � G � � � ��$ � $FE � � b with$ � �A� �

can also be used in the whole energy range, representing the forth case (see Fig. 18). Finally, theALLM parametrization (discussed in Sec. 6.8) can be enabled. It does not rely on the assumption thatthe virtual photon can be considered as real and involves integration over the square of the photon4-momentum (

� �).

Integration limits used for the photonuclear cross section are (kinematic limits for� �

are usedfor the ALLM cross section)

�&% � � �%�A� C � C � � �

�� �

�� � ��� � $

� �� � �� � � �

� �� � � � C � � C �A� � � � �&% �

�� �% )

� ��� � �

6.4 Electron Pair Production Two out of four diagrams describing pair production areshown below. These describe the dominant “electron” term. The other two have muon interactingwith the atom and represent the “muon” term. The cross section formulae used here were first derivedin [21, 22, 23].

� � ��) � )�� �� � � � � �4

�

A �8A ��� � � 0 2 6 � � � � ��

� � 6 � � �� � � � $

� � ����� ��� � � � � ) � � ������

� � � � �� 6 � � " � � ��� � � �

�� � � � � 4 ��� � � - 0�� � � � � $ � � �

� ��

�

�� �

� 4 ��� � � � 6� � � � 5 �

���� � � �

��

4�� $

� � ��� � � � �

� �� � � : 0 � �

�� � �

� �� �

� ��

� ��� � �

�� � � � �

� �� � � � �

6 � 0 � �� B A � ��� �� ��� � �

���� 6 �

�� � �� 6�� % ��� � # � ��� ' # � ��� '

�� # � � ��� '��� �� 0��

"�� � 4 �

� �A ��� � $ � �

�� � �

���� 6 � $

� � 0 � �� �� � B A � � � ��� � � 6 � % ��� � # � ��� ' # � ���"! '

�� # � � � � '��

� 6 � � � � � � � � ����� � �

� ��� 4 � � 0�� � 4 �

��� ��

� � � � � � � � � � �� � � � ��� � � 4 � �

���� ���

����� � � � 4 � � � � � � 0 � � 4 � � �

� � �� � �� � � �

� �� � � � ) � � ���

� � � � � � � �� � �

� � � ,� �"!�! ��) A�� = ��� ��� 4 0�� � � ��

� * � % � � � � � � � ��� � ���� � � �>0�� � � ��

� * � % ��� � � � � � ��� � �� �

��5� � � ��� � � and � � � � � 4 ��� � � for

A����

� �� ��� � ��� � � and � � � � ��� ��� � � for

A ��

Integration limits for this cross section are

� ��

� � � � � � � � & �� �

4 @ ��

��A ��� �

� � � � � � � � & ���� � �

�� �

5� �

� � �� � �

� � � � :Muon pair production is discussed in detail in [24] and is not considered by MMC. Its cross sectionis estimated to be = � � ���� times smaller than the direct electron pair production cross sectiondiscussed above.

6.5 Muon decay Muon decay probability is calculated according to� �� �� �

� � ���The energy of the outgoing electron is evaluated as

� 6 � � � � , 6 !#� � ��� � �, 6 !#� � � �6�� 2 ����� $� 2 �����

is distributed uniformly on�� �)�

�and � , 6 !#� is determined at random from the distribution� �� �

� � � � ��� � � � � 4

� ��� � � � , � � �� � & with � � � � 6 and � � & � � � � � �6

���6.6 Moliere scattering After passing through distance x angle distribution is assumed Gaus-sian with a width

@ � � $ [11]

� $ � �4 � � ��� � � � � � � $ " � � ��� � 4 �>0�� � � � � $ � -

� $ is evaluated as � $ � 5 ��� , 6 ! ���� �� �

��� �� : � �for ��� �� = ���

� $ �� x

y

θDeviations in two directions, perpendicular to the muon track are independent, but for each directionexit angle and actual deviation are correlated:

��� � � 6 � �� � � $�� @ � � ��� � � � $ � � and

� ��� � � 6 � � � � $for independent standard Gaussian random variables (

�� ,

� � ). A more precise treatment should takethe finite size of nuclei into account as described in [25]. See Fig. 22 for an example of Molierescattering of a high energy muon.

6.7 Landau-Pomeranchuk-Migdal and Ter-Mikaelian effects These affect brems-strahlung and pair production. See Fig. 21 for the combined effect in ice and Fr ejus rock.6.7.1 LPM suppression of the bremsstrahlung cross section Bremsstrahlung cross section ismodified as follows [26, 27, 28]:

� 4 �� � � � � � � ) � $ �4 � � � � � $ � � � " � � �

� � � � � -�� � $ � The regions of the following expressions for � � $ �

and� � $ �

were chosen to represent the best con-tinuous approximation to the actual functions.

� � $ � �� �

�'��� ��� $ " � �� 4

� �� $ - � $ �

��� � � 4 � ����� ��� $ � ��� � � � $ � $ for$ C �5� � � � � �

� � $ � �� � ��� � � �%� $ for

$ � � � � � � � �� � $ � �

� ��'��� �

� � $ � � $ ��� 4 � � 4 � $ � ��� � � $ � � ��� � ��$ � � ��� � � $

$� � $ � � 4 � � $ �� ��� � $ �

for$ C ����� ��� 4 � �� � $ � � 4 � $ � � � 4 � $ � � �

�for ����� ��� 4 � � � $ C ��� � � � � � �� � $ � �

� � ��� ��� �%� $ for$ � ��� � � � � � �

Here the SEB scheme [29] is employed for evaluation of � � $ �and

� � $ �and � $ �

below: � $�� � � for

$� C $

� � $�� �

����

� ��� ��� �� �

� � "� �

�� �

� � � -0�� $�

for$

� �$� C � � $

�� �

� for$� � �

����� � 0 � � � ��� � � $� � � �� $ is the same as in Sec. 6.6. Here are the rest of the definitions:$ � $

�@ $�

� @�A ��� � D �B � 6

�$�

� ����� ���� �� � � � � � 0�� $

�0�� $�

6.7.2 Dialectric (Longitudinal) suppression effect In addition to the above change of the brems-strahlung cross section, s is replaced by � � $ and functions � $ �

, � � $ �and

� � $ �are scaled as [28] � $ � ) � � $ � � � $ � ) � � � $ � �� � � $ � ) � � � $ � ��� �

Therefore the first formula in the previous section is modified as

� 4 �� � � � � � � ) � � $ �4 � � � � � � $ �

� � � � " � � �� � � � � - � � � $ �

�$

� is defined as � ��� �

� � � > �� �

$ �where

> � � � � ��A � � � � is plasma frequency of the medium and � � is the photon energy. Thedialectric suppression affects only processes with small photon transfer energy, therefore it is notdirectly applicable to the direct pair production suppression.

6.7.3 LPM suppression of the direct pair production cross section� 6 from the pair produc-

tion cross section is modified as follows [28, 30]:

� 6 ) � ��� � � � � � "

���� � -�B � � � �-� � "

���� � - D � � �

� �� � �

� � 6

$ � ��

������ �

�� �� �

� � �� ���� energy definition is different than in the bremsstrahlung case:

� ���� � � � � � 0 � � A � where � 0 � � 4 � � � B A � ��� � �

Functions� � $ ) �

, B � $ ) �,� � $ ) �

and D � $ ) �are based on the approximation formulae

� � $ ��� � $� $ � �and

� � $ � � � � $ ������ $ � � � �

and are given below� � $ ) � ��� ��

��� � � � � 0�� 4 � $ ���

�� � � � �

�4 � $ � � � � � � � � $ � � � 4 � $ � � �4 � $ � � � � $ � ��� ��� � � � � $ " � � �.- � � � � �B � $ ) � � � � �

�� � � � 0�� � $ �

�� � � �

�� $ � � �� � $ ) � � �� � � � 0 � 4 � $ � �

�� � ��� �

�4 � $ � � � � � �� ��� 4 � $ � � �

�� $ � � ��� ��� � � ��� $ " � � �.- � � � � �D � $ ) � � � � � � � � 0 � � $ ��� � � �

�� $ ��

� $ ) � � ��� $ � ��� ��� � � � � $ " � � �.- � � � � �

6.8 The Abramowicz Levin Levy Maor (ALLM) parametrization of the pho-tonuclear cross section The ALLM formula is based on the parametrization [31, 32, 33]� � � � )�� � �� � � � � � � � 0 ��

� ��

5� � � � � � �

� �� �

� � ����� � $ � ���

�� � � � � � � � ���� ��� � � � :

� � � ��A� � �

The limits of integration over� �

are given in the section for photonuclear cross section.

� � ��� �8A �� ��A�� � � �� Here

� � � ) � )�� � ��� � � � ) � �

� � � ) � � � � � $FE � for ��C ��� � � � �� � � ) � � � � $FE $ � ����� � � & � $FE $ � G for ��� ��� � � � ��C ��� � �� � � ) � � �� for �7� ��� � �

� �� � � )�� � � � � �� � � � �$ � � �� � ���� �� � � � ) � � � � � � � # �

� � � � � # for� � )��

For� � � � ) � � ) � � ) � � � � � � �

�� � � � �

For� � � � ) � � � � � � �

�� � �

� �� � � 5 �

�� � � � � :

� 0�� 0�� � � � � ��� �0�� � ��� ��� � � � � � �� � � � � ��� � � � � for

� � )���

is the invariant mass of the nucleus plus virtual photon [34]:� � � � � � �A� � � � � �

. Fig. 19compares ALLM-parametrized cross section with formulae of Bezrukov and Bugaev from Sec. 6.3.� � � )�� ���

is not very well known, although it has been measured for high x ( � � ��� � ) [35]and modeled for small x ( ��� � G C ��C ��� � , ��� � � GeV

� C � � C � � GeV�) [36]. It is of the order= ��� � � ��� 4 and even smaller for small

� �(behaves as �

� � � �). In the Fig. 20 three photonuclear

energy loss curves for R=0, 0.3 and 0.5 are shown. The difference between the curves never exceeds7%. In the absence of a convenient parametrization for R at the moment, it is set to zero in MMC.

1234

energy [GeV]

phot

on-n

ucle

on c

ross

sec

tion

[µb]

0

200

400

600

800

1000

10-210

-11 10 10

210

310

410

510

610

710

810

910

1010

11

1 BB2 BB3 BB4 BBALLM

energy [GeV]

phot

onuc

lear

ene

rgy

loss

es p

er e

nerg

y [g

-1 c

m2]

10-7

10-6

10-1

1 10 10210

310

410

510

610

710

810

910

10

Fig. 18: photon-nucleon cross sections, asdescribed in the text (Bezrukov Bugaevparametrization): 1 (solid), 2 (dashed), 3(dotted), 4(dashed-dotted)

Fig. 19: photonuclear energy losses (divided byenergy), according to different formulae. Desig-nations are the same as in Fig. 18, higher solidline is for ALLM parametrization

1 ALLM R=02 ALLM R=0.33 ALLM R=0.5

energy [GeV]

phot

onuc

lear

ene

rgy

loss

es p

er e

nerg

y [g

-1 c

m2]

10-7

10-6

10-1

1 10 10210

310

410

510

610

710

810

910

10

epair

brems

epair

brems

energy [GeV]en

ergy

loss

es p

er e

nerg

y [g

-1 c

m2]

10-10

10-9

10-8

10-7

10-6

10-5

10-1

10 103

105

107

10910

1110

1310

1510

1710

1910

21

Fig. 20: comparison of ALLM energy loss (di-vided by energy) for R=0 (dashed-dotted), R=0.3(dotted), R=0.5 (dashed)

Fig. 21: LPM effect in ice (higher plots) andFrejus rock (lower plots, multiplied by � ��� )

x deviation [cm]

y de

viat

ion

[cm

]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1distance traveled [km] of ice

devi

atio

n fr

om s

how

er a

xis

[cm

]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Fig. 22: Moli ere scattering of 100 10 TeV muons going straight down through ice

7 TablesAll cross sections were translated to units

"��� � � - via multiplying by number of molecules per vol-

ume. Unit conversions (like eV ) J) were achieved using values of 0 � � � � � � and 2 6 � � � � � 6 � � .7.1 Summary of physical constants employed by MMC0 1/137.03599976 2 6 ����� � � � � ����� � � ����� �;� cm� � � � ��� � � � � ��� � ��� � � 1/mol

�0.307075 ��� � � ��� � �

� � ��� � � � � � � � ��� � $ cm/s���

13.60569172 eV� 6 0.510998902 MeV� %

139.57018 Mev� � 938.271998 MeV� � 939.56533 MeV� � 105.658389 MeV � � ��� � � � � 4 � ��� � � s� � 1777.03 MeV � � � � ��� �/� ����� � � s

7.2 Media constants

MaterialA � ? , eV

� � � � $ � ��,� ��� � �

Water 1 + 1.00794 75.0 -3.5017 0.09116 3.477 0.240 2.8004 1.000Ice + 8 15.9994 75.0 -3.5017 0.09116 3.477 0.240 2.8004 0.917

Stand. Rock 11 22 136.4 -3.774 0.083 3.412 0.049 3.055 2.650Fr ejus Rock 10.12 20.34 149.0 -5.053 0.078 3.645 0.288 3.196 2.740

Iron 26 55.845 286.0 -4.291 0.147 2.963 -0.001 3.153 7.874Hydrogen 1 1.00794 21.8 -3.263 0.135 5.625 0.476 1.922 0.063

Lead 82 207.200 823.0 -6.202 0.094 3.161 0.378 3.807 11.350Uranium 92 238.0289 890.0 -5.869 0.197 2.817 0.226 3.372 18.950

7.3 Radiation logarithm constant�

(taken from [37])A B1 202.42 151.93 159.94 172.35 177.96 178.37 176.6

A B8 173.49 170.0

10 165.811 165.812 167.113 169.114 170.8

A B15 172.216 173.417 174.318 174.819 175.120 175.621 176.2

A B22 176.826 175.829 173.132 173.035 173.542 175.950 177.4

A B53 178.674 177.682 178.092 179.8

other 182.7

7.4 ALLM parameters (as in [32, 38])� � � -0.0808

� � � -0.44812� � � 1.1709� � � 0.58400

� � � 0.37888� � � 2.6063� � � ��� � � � � 4 � � � � �5� 4 � � � � � � � 1.8439� � � ��� ����� ��� � � � � �5� � 4 � � � � � � 0.49338

� � � 0.28067 � � � 0.22291 � � � 2.1979� � � 0.80107 � � � 0.97307 � � � 3.4942� �� � � � � � � � ��� � MeV

� ������ � � � ��� � ��� � MeV

� � �$ ��� 4 � � � � � � � � MeV�

� � � ��� � � � � � � ��� � MeV� � �$ � ��� ��� � � � � � � ��� � MeV

�

References[1] Andres, E. et al. (AMANDA Collaboration), Nature (2001) 441

[2] DeYoung, Tyce R., Dissertation, U. of Wisconsin-Madison, 2001

[3] Heck, D. et al., Report FKZA 6019, 1998

[4] Chirkin, D., Rhode, W., 26th ICRC, HE 3.1.07 Salt Lake City, 1999

[5] Desiati, P., Rhode, W., 27th ICRC, HE 205 Hamburg, 2001

[6] Numerical Recipes (W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling)

[7] Rhode, W., Carloganu, C., DESY-PROC-1999-01, 1999

[8] Bugaev, E., Sokalski, I., Klimushin, S., hep-ph/0010322, hep-ph/0010323, 2000

[9] Schroder, F., Rhode, W., Meyer, H., 27th ICRC, HE 2.2 Hamburg, 2001

[10] Winterer, V.-H., SYMPHONY (talk)

[11] The Europen Physics Journal C vol. 15, pp 163-173, 2000

[12] Rossi, B., High Energy Particles, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1952

[13] S.R. Kelner, R.P. Kokoulin, A.A. Petrukhin, About Cross Section for High Energy MuonBremsstrahlung, Preprint of Moscow Engineering Physics Inst., Moscow, 1995, no 024-95.

[14] Bethe, H., Heitler, W, Proc. Roy. Soc., A146, 83 (1934)

[15] Bethe, H., Proc. Cambr. Phil. Soc., 30, 524 (1934)

[16] S.R. Kelner, R.P. Kokoulin, A.A. Petrukhin, Bremsstrahlung from Muons Scattered by AtomicElectrons, Physics of Atomic Nuclei, Vol 60, No 4, 1997, 576-583.

[17] L.B. Bezrukov and E.V. Bugaev, Nucleon shadowing effects in photonuclear interactions, Sov.J. Nucl. Phys. 33(5), May 1981.

[18] R.P. Kokoulin, Nucl. Phys. B (Proc. Suppl.) 70 (1999) 475-479.

[19] W. Rhode, Diss. Univ. Wuppertal, WUB-DIS- 93-11 (1993); W. Rhode, Nucl. Phys. B. (Proc.Suppl.) 35 (1994), 250-253

[20] ZEUS Collaboration, Z. Phys. C, 63 (1994) 391.

[21] S.R. Kelner, Yu.D. Kotov, Muon energy loss to pair production, Soviet Journal of NuclearPhysics, Vol 7, No 2, 237, (1968).

[22] R.P. Kokoulin, A.A. Petrukhin, Analysis of the cross section of direct pair production by fastmuons, Proc. 11th Int. Conf. on Cosmic Ray, Budapest 1969.

[23] R.P. Kokoulin, A.A. Petrukhin, Influence of the nuclear form factor on the cross section ofelectron pair production by high energy muons, Proc. 12th Int. Conf. on Cosmic Rays, Hobart6 (1971), A 2436.

[24] Kelner, S., Kokoulin, R., Petrukhin, A., Direct Production of Muon Pairs by High-EnergyMuons, Phys. of Atomic Nuclei, Vol. 63, No 9 (2000) 1603

[25] Butkevitch, A., Kokoulin, R., Matushko, G., Mikheyev, S., Comments on multiple scatteringof high-energy muons in thick layers, hep-ph/0108016, 2001

[26] Klein, S., Suppression of bremsstrahlung and pair production due to environmental factors,Rev. Mod. Phys., Vol 71, No 5 (1999) 1501

[27] Migdal, A., Bremsstrahlung and Pair Production in Condensed Media at High Energies Phys.Rev., Vol. 103, No 6 (1956) 1856

[28] Polityko, S., Takahashi, N., Kato, M., Yamada, Y., Misaki, A., Muon’s Behaviors underBremsstrahlung with both the LPM effect and the Ter-Mikaelian effect and Direct Pair Pro-duction with the LPM effect, hep-ph/9911330, 1999

[29] Stanev, T., Vankov, Ch., Streitmatter, R., Ellsworth, R., Bowen, T., Development of ultrahigh-energy electromagnetic cascades in water and lead including the Landau-Pomeranchuk-Migdaleffect Phys. Rev. D 25 (1982) 1291

[30] Ternovskii, F., Effect of multiple scattering on pair production by high-energy particles in amedium, Sov. Phys. JETP, Vol 37(10), No 4 (1960) 718

[31] Abramowicz, H, Levin, E., Levy, A., Maor, U., A parametrization of ��� � � � �above the reso-

nance region for� � � � , Phys. Lett. B269 (1991) 465

[32] Abramowicz, H., Levy, A., The ALLM parametrization of � � ��� � � � �an update, hep-

ph/9712415, 1997

[33] Dutta, S., Reno, M., Sarcevic, I., Seckel, D., Propagation of Muons and Taus at High Energies,hep-ph/0012350, 2000

[34] Badelek, B., Kwiencinski, J., The low-� �

, low-x region in electroproduction, Rev. Mod. Phys.Vol 68 No 2 (1996) 445

[35] Whitlow, L., Rock, S., Bodek, A., Dasu, S., Riordan, E., A precise extraction of� � � � � ���

from a global analysis of the SLAC deep inelastic e-p and e-d scattering cross sections, Phys.Lett. B250 No1,2 (1990) 193

[36] Badelek, B., Kwiecinski, J., Stasto, A., A model for� � and

� � � � � � � at low x and low� �

,Z. Phys. C 74 (1997) 297

[37] Kelner, S., Kokoulin, R, Petrukhin, A., Radiation Logarithm in the Hartree-Fock Model, Phys.of Atomic Nuclei, Vol. 62, No. 11, (1999) 1894

[38] Abramowicz, H., private communication