6 mmc - icecube.wisc.edudima/work/BKP/DCS/PAPER_MMC/poster.pdf · ice 0.259 0.357 3.7% 6.6% fr....
Transcript of 6 mmc - icecube.wisc.edudima/work/BKP/DCS/PAPER_MMC/poster.pdf · ice 0.259 0.357 3.7% 6.6% fr....
� � � � � ��� � � �� � � � � � ��� � ������� � ��� � �� � � � � � ��� � � ����� � � ��� � � ����� �� � ������� "!#�$��%&�('*),+.-0/21 35476985:;'<8 = !<3&><?@-)BADCFEHGJILKNMOE$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