� � � � � ��� � � �� � � � � � ��� � ������� � ��� � �� � � � � � ��� � � ����� � � ��� � � ����� �� � ������� "!#�$��%&�('*),+.-0/21 35476985:;'<8 = !<3&><?@-)BADCFEHGJILKNMOE$P(Q RTS UWVYXZE.S[R\K]C;E^V VJP*_`ILKOabIcXdILQfe�Ahg�i- _jILKlkJEHM[m]noI ADCFEpGYILKNMOE$PbqVJP`rsI]M[V\t Ppn�R�m]nuMvm]nw�XxI y w�z{z;IcKLP9VYX|e`rsILK]t VYC;Q

}�~[���Z�����,���Y� �\���T�������������d�������@����~Continuous part of the energy losses: �f����Z�B�5� �¢¡�£

x xdxi f

The stochastic part: ¤p¥ �¢¡��§¦^¨.£©£]�<ª �¢¡\�«¦(¨.£©£ ¤ ¦ . Probability to suffera catastrophic loss on ¤ ¦ at ¦9¬ is�©­ �®¤p¥ �¢¡\�«¦(¨¯£©£°£²±H³´³µ³d±H�©­ �®¤p¥ �¢¡\�«¦ ¬ £©£©£¶± ¤H¥ �¢¡\�«¦ ¬ £©£�¸·º¹H»l� �[¤p¥ �¢¡��§¦(¨¯£©£©£¶±H³´³´³¼±�·º¹H»l� �[¤p¥ �¢¡\�«¦ ¬ £°£©£²± ¤p¥ �¢¡\�«¦ ¬ £©£�®·º¹H»�½ ��¾ �¶¿�ÁÀ ¤p¥ �¢¡\�«¦Â£°£.Ãı ¤p¥ �¢¡\�«¦ ¬ £©£� ¤ ¬ ½ � ·º¹H»Â� � ¾ � ¿� À ªT�¢¡�£� � �¢¡�£ ± ¤ ¡�£ à � ¤ � �LÅ £ÇÆ ÅfÈ �«ÉHÊZ­ºËThe above equation can be solved for ¡[¬ :¾ � ¿� À ªT�¢¡�£� � �¢¡�£ ± ¤ ¡<� �@ÌÎÍ�Ï � Å £ (energy integral),

and then the corresponding displacement is found:

¦l¬f�¸¦(¨ � ¾ �²¿�9À ¤ ¡� �¢¡�£ (tracking integral).Ð �\�Y�,���{��\�@ÑÒ���Ó�@Ô��\����Õc���,�Ö�\�choose ¤ ¦ so small that ×ÂØ �ÚÙÜÛ°ÝÎÞ¯ßÛ°à × �§á�ʺ¡�£ ¤ á]± ¤ ¦bâ�­ã á�äc� ¾ Û ÝåÞ¯ßÛ à áL± × �§á¼Êº¡�£ ¤ áL± ¤ ¦ ã á�æLäc� ¾ Û ÝåÞ¯ßÛ à á�æ�± × �§á¼Êº¡�£ ¤ áN± ¤ ¦ã �§ç�áè£ æ äc� ã á æ ä � ã á�ä æFor small á�éëêèì , × �§á¼Êº¡�£Bí × �§á¼Êº¡O¨¯£fî distributions × �§á¼Êº¡�£ are thesame î central limit theorem can be appliedã �§ç\�§ç�¡�£©£ æ äc�Úï�ðòñ ã á æð ä � ã á ð ä æ|ó �ï�ðõôHö ¾ Û ÝåÞ¯ßÛ÷à á æð ± × �§á ð ʺ¡N¨¢£ ¤ á ðHø ¤ ¦ ð � ö ¾ Û ÝåÞ¯ßÛ÷à á ð ± × �§á ð ʺ¡N¨¯£ ¤ á ðHø æ ¤ ¦ æð¼ùí ¾ �x¿��À ¤ ¦²± ô7ö ¾ Û÷ÝåÞ¯ßÛ÷à á�æ�± × �§á¼Êº¡\�«¦Â£©£ ¤ á ø � ö ¾ Û÷ÝåÞ¯ßÛ÷à á]± × �§á¼Êº¡\�«¦Â£©£ ¤ á ø æ ¤ ¦ ùHere ¡N¨ was replaced with average expectation value of energy at¦ , ¡��«¦9£ . As ¤ ¦bî`É , the second term disappears. The lower limit ofthe integral over á can be replaced with zero, since all of the crosssections diverge slower than ­�údáèû . Then,ã �§ç\�§ç�¡�£°£.æLäcí ¾ � ¿� À ¤ ¡� �ü�¯¡�£ ± ö ¾ Û ÝåÞ¯ßØ á�æ�± × �§á¼Êº¡�£ ¤ á ø

ý ý Ð Ñü�T�����N�,��,~

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

þÿ~vÑÜ�@���TÑmedium a, � Û ���� Û b,

� Ø����� Û av. dev. max. dev.ice 0.259 0.357 3.7% 6.6%fr. rock 0.231 0.429 3.0% 5.1%

medium a, � Û ���� Û b,� Ø ������ Û av. dev.

ice 0.268 0.470 3.0%frejus rock 0.218 0.520 2.8%

� �Ö�f�������� @� ~[���,����Ñ1 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

Average final energy deviation from the purely “continuous” prediction

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

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

Survival probabilities� é©êèì “cont” 1 TeV 3 km 9 TeV 10 km ­|É � 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|É � û no 0.034 0.037 0.140­|É � û yes 0.034 0.037 0.135

� �Z�ÜÑÚ�T� �T�Ó�Ü�Ó�f~0��~[�,���`�Ö��ÑTÑ��Ó��Ô �L�\~[�����f~ ���f��f~dE/dx=a+bE

a=0.259425 [ GeV/mwe ],b=0.000357204 [ 1/mwe ]

energy [ GeV ]

tota

l ene

rgy

loss

es [G

eV g

-1 c

m2]

10-3

10-2

10-1

1

10

10 2

10 3

10 4

10 5

10 6

10-1

1 10 102

103

104

105

106

107

108

10910

10

energy [ GeV ]

χ2

10-2

10-1

1

10

1 10 102

103

104

105

106

107

108

10910

10

Fit to the energy losses in ice ��� plot for energy losses in ice

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

energy [ GeV ]

aver

age

disp

lace

men

t [ m

we

]

10-2

10-1

1

10

10 2

10 3

10 4

10 5

10-1

1 10 102

103

104

105

106

107

108

10910

10

energy [ GeV ]

χ2

10-2

10-1

1

10

1 10 102

103

104

105

106

107

108

10910

10

Fit to the average range in Frejusrock

� � plot for average range in Frejusrockþ �Ó��f~ Ô��ÖÑü�T���� �u�,���Y��Ñ

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

in Frejus rock: solid line designates the value of the range evaluated withthe second table (continuous and stochastic losses) and the broken lineshows the range evaluated with the first table (continuous losses only).

Parametrization errors:

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

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1

10-1

1 10 10210

310

410

510

610

710

810

910

1010

11

energy losses in ice Interpolation precision � Û ��� � Û � �"! # Û �$�Final energy distribution of the muons that crossed 300 m ofFrejus Rock with initial energy 100 TeV:

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

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

Comparison of Û % &('*) ��+ (dotted-dashed) with Û % &(' =10 TeV (dotted).Also shown is the relative differenceof the curves.

Comparison of paramerized(dashed-dotted) with exact (non-parametrized, dotted) versions for, ÝåÞ¯ß ) Ø�- Ø � .

Final energy distribution of the muons that crossed 300 m ofFrejus Rock with initial energy 100 TeV:

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

, ÝåÞ¯ß ) Ø�- Ø�. (solid), , ÝåÞ¯ß ) � Ø ��/(dashed-dotted), , ÝåÞ¯ß ) Ø�- Ø�. and“cont” option (dotted)

no “cont”: , ÝåÞ¯ß ) Ø�- Ø�. (dashed),, ÝÎÞ¯ß ) Ø- Ø � (solid), , ÝÎÞ¯ß ) � Ø �� (dot-ted), , ÝåÞ¯ß ) � Ø���/ (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

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

with “cont” option: , ÝåÞ¯ß ) Ø�- Ø�.(dashed), , ÝåÞ¯ß ) Ø�- Ø � (solid), , ÝåÞ¯ß )� Ø �� (dotted), , ÝåÞ¯ß ) � Ø ��/ (dashed-dotted)

ioniz (upper solid curve), brems(dashed), photo (dotted), epair(dashed-dotted) spectra for � + =10TeV in the Frejus rock

Photonuclear losses, LPM effect and Moliere scattering

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

different photon-nucleon crosssections implemented in MMC,Bezrukov Bugaev parametrization

photonuclear energy losses (di-vided by energy), according to dif-ferent formulae.

epair

brems

epair

brems

energy [GeV]

ener

gy lo

sses

per

ene

rgy

[g-1

cm

2]

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

distance 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

LPM effect in ice (higher plots) andFrejus rock (lower plots, multipliedby

� ��� ) Moliere scattering of 100 10 TeVmuons going straight down throughice