modified fragmentation function in strong interaction matter
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Enke Wang (Institute of Particle Physics, Huazhong Normal University)
I. Jet Quenching in QCD-based Model
II. Jet Quenching in High-Twist pQCD
III. Jet Tomography of Hot and Cold Strong Interaction Matter
IV. Modification of Dihadron Frag. Function
Modified Fragmentation Function in Strong Interaction Matter
hadrons
q
q
hadrons
leadingparticle
leading particle
p-p collision
hadrons
q
q
hadrons
Leading particle suppressed
leading particle suppressedA-A collision
Jet Quenching:
EEE '
E
Fragmentation Function:
p
hp
S~
q
),( 2QzD hhq DGLAP Equation
p
pz h
h
hhh zzz
),(),(),(~ 222 QzDQzDQzD hhhqhhq
I. Jet Quenching in QCD-based Model
G-W (M. Gyulassy, X. –N. Wang) Model:
Static Color-Screened Yukawa Potential
Feynman Rule:
q
p
p-q )2()2( 00 qpiqpiD
ip
ipi
2)(
p
pp+k
,)2( ckpigs
cTc
k
ik
gikig
2)(
0
k-q,a k,c
q,b })]()[(
)()]([{0
00
kqkg
qkgqkqgf abc
Opacity Expansion Formulism (GLV)
Double Born Scattering
GLV, Phys. Rev. Lett. 85 (2000) 5535; Nucl. Phys. B594 (2001) 371
Elastic Scattering
Assumption
• The distance between the source and the scattering center are large compaired to the interaction range:
• The packet j(p) varies slowly over the range of the momentum transfer supplied by the potential:
•The targets are distributed with the density:
Opacity: Mean number of the collision in the medium
1
0 zzi
)()( qpjpj
),,,(),,,( 2121 NN zzzA
Nxxx
)(/
121 )(
)(),,,( NLz
e
jN
jN
ejeNL
zzzz
jjj zzz 1
1)(
N
LNLe
el 1
A
NLn el
First Order in opacity Correction
First Order in opacity Correction
Medium-induced radiation intensity distribution:
Induced radiative energy loss:
Induced gluon number distribution:
)cos(1)2)(( 111122
22
)1(
zBCqvqdLC
kdxd
dNx
g
sR
Non-Abelian LPM Effect
2)1( LE LE )1(
QCD:
QED:
Higher order in Opacity
Reaction Operator Approach: (GLV)
Induced gluon number distribution:Non-Abelian LPM Effect
Radiated Energy Loss vs. Opacity
First order in opacity correction is dominant!
Detailed Balance Formulism (WW)
E. Wang & X.-N. Wang, Phys. Rev. Lett.87 (2001) 142301
k
x0 p
k
x0 p
Stimulated Emission Thermal Absorption
B-E Enhancement Factor
1+N(k)
Thermal Distribution Func.
N(k)
Final-state Radiation
k
x0 p
k
x0 p
Energy loss induced by thermal medium:
0
)0()0(
)0(
T
abs d
dp
d
dpdE
22
2 )2('62
4ln
3
E
FsET
E
TC=
Net contribution: Energy gain
Stimulated emission increase E loss Thermal absorption decrease E loss
First Order in Opacity Correction
Single direct rescattering:
k
y0 y1 p
k
y0 y1 p
y0 y1 p
k
Double Born virtual interaction:
k
y0 y1 y1 p
y0 y1 y1 p
k
k
y0 y1 y1 p
y0 y1 y1 p
k
Key Point: Non-Abelian LPM Effect—destructive Interference!
Energy Loss in First Order of Opacity
Energy loss induced by rescattering in thermal medium: )1()1()1(
absradEEE
Take limit:
1EL E LT 2
Zero Temperature Part:
0
)0(
)1(
T
rad d
dpdE
048.0
2ln
4 2
2
L
EC
g
Fs
L2
GLV ResultTemperature-dependent Part:
0
)1()1(
)1(
T
abs d
dp
d
dpdE
2
22 )2('61ln
3
E
g
Fs
T
L
E
LTC
Energy gain
Numerical Result for Energy Loss
3.0S
)1()1()0(
radabsabsEEEE
• Intemediate large E, absorption is important
•Energy dependence becomes strong
•Very high energy E, net energy gain can be neglected
Parameterization of Jet Quenching with Detailed Balance Effect
)/5.7/()6.1/( 02.1
001
EEdL
dE
d
Average parton energy loss in medium at formation time:
Energy loss parameter proportional to the initial gluon density 2
00
1
ARd
dN
Modified Fragmentation Function (FF)
),(
)],(/),()[1(),,(
2'0/
/
2'0/
'2'0
/
'/2
/
cchL
gghc
gcch
c
cLccch
zDe
zDz
zLzD
z
zeEzD
(X. -N. Wang , PRC70(2004)031901)
,//),/( ''cTgcTcTc EpLzEppz
Comparison with PHENIX Data
PHENIX,
Nucl. Phys. A757 (2005) 184
DGLAP Equation at Finite TemperatureJ. A. Osborne, E. Wang, X.-N. Wang, Phys. Rev. D67 (2003) 094022
DGLAP Equation at Finite Temperature
Splitting function at finite temperature:
Quark Energy Loss from Splitting Function
The minus sign indicates that the absorptive processes
in the plasma overcome the emissive processes.
The net Contribution is energy loss!
II. Jet Quenching in High-Twist pQCD
e-
, )) (( ,( )qh
q h hHdW
d f x p q Dxd
zz
x
pypedy
xf yixpBq )()0(
2
1
2)(
/( ) 0 (0) , , ( ) 02 2 2
h hip y zhq h h q h h q
S
z dyD z e Tr p S p S y
Frag. Func.
22 )(2)(2
1),,( xpqxpqpTreqpxH q
Modified Fragmentation Function
2 2 2( , ) ( , ) ( , )h h hD z Q D z Q D z Q
Cold nuclear matter or hot QGP medium lead to the modification of fragmentation function
Jet Quenching in e-A DISX.-N. Wang, X. Guo, NPA696 (2001); PRL85 (2000) 3591
e-
Modified Frag. Function in Cold Nuclear Matter
2 2 2( , ) ( , ) ( , )h h hD z Q D z Q D z Q 2 12
24
0
( , ) ( , )2
h
Q
S hq h h L q h
z
zd dzD z Q z x D
z z
2 ( , ) 21( , ) (virtual)
(1 ) ( )
Aqg L A S
L Aq c
T x x Czz x
z f x N
Modified splitting functions
_2 1(
1 2 1 2
2)
1
( , ) (0) ( ) ( ) ( )2 2
( ) ( )1 1
B
L Lix p y ix
ix p yA
y
g
y
q L
pe
dyT x x dy dy e A F y F y y A
y y ye
Two-parton correlation:
LPM
Modified Frag. Function in Cold Nuclear Matter
hadrons
ph
parton
E
),,()(0 EzDzD ahah
)(0 zDah
are measured, and its QCD evolutiontested in e+e-, ep and pp collisions
Suppression of leading particles
Fragmentation function without medium effect:
Fragmentation function with medium effect:
),1
(1
1),( 0
z
zD
zEzD ahah
Heavy Quark Energy Loss in Nuclear MediumB. Zhang, E. Wang, X.-N. Wang, PRL93 (2004) 072301; NPA757 (2005) 493
Mass effects:
1) Formation time of gluon radiation time become shorter
222 )1(
)1(2
Mzl
qzz
T
f
LPM effect is significantly reduced for heavy quark
2) Induced gluon spectra from heavy quark is suppressed by
“dead cone” effect
4
2
2
04
222
2
/]1[][
Mzl
lf
T
T
zq
l
q
M
T
0
Dead cone Suppresses gluon radiation amplitude at 0
Heavy Quark Energy Loss in Nuclear Medium
)]},,(),,()[1(),,(2
1{
~)~~(~
)1(
1~
),(
22
2
22
1
/~22
3
4
2~
~
1
0
2
2
2
2
22
MlzcMlzceMlzc
x
xxxd
zz
zdz
xQN
xCCQxz
TT
xx
T
L
ML
x
xL
Ac
BsA
B
Q
g
AL
M
LPM Effect
,~~
2
2
Qx
Mx
x
x
A
B
A
L
AN
A Rmx
1
1) Larg or small :
Bx
2Q
A
A
B
c
SAQ
gR
Qx
x
N
CCz
2
2~~
2) Larg or small :2Q
2
22
2~~
A
A
B
c
SAQ
gR
Qx
x
N
CCz
Bx
Heavy Quark Energy Loss in Nuclear Medium
The dependence of the ratio between charm quark and light quark energy loss in a large nucleus
2Q
The dependence of the ratio between charm quark and light quark energy loss in a large nucleus
Bx
III. Jet Tomography of Hot and Cold Strong Interaction Matter
E. Wang, X.-N. Wang, Phys. Rev. Lett. 89 (2002) 162301
2 21 1 22 2
22 2 2 2
0 0 0 0
1 (1 )( ,
()
, )
( )2
Q Qs A sT
g L T
Aqg
T cT T T
L
Aq
Cd zz dz z z x d dz
Nk
T x x
f x
Cold Nuclear Matter:Quark energy loss = energy carried by radiated gluon
2 2 13ln
2A
s N Ac B
CE C m R
N x
Energy loss
3/2AE
Comparison with HERMES Data
HERMES Data: Eur. Phys. J. C20 (2001) 479
22 0060.0)(~
GeVQC 33.0)( 2 Qs 22 3GeVQ , ,
Expanding Hot Quark Gluon Medium
_2 1(
1 2 1 2
2)
1
( , ) (0) ( ) ( ) ( )2 2
( ) ( )1 1
B
L Lix p y ix
ix p yA
y
g
y
q L
pe
dyT x x dy dy e A F y F y y A
y y ye
2( , )~ ( ) 1 cos
( )
Aqg L
gAq f
T x x ydy y
f x
0
32
( )2
lnR
s dE
E
R. Baier et al
Initial Parton Density and Energy Loss
jet1
jet2
0
32
2( ) ln
R
s
EE d
00( ) ( )R r
01 0
2d
A
E ER
Initial energy loss in a static medium with density 0
:0E
0 0.1 fm 015
2AR
1
0.5 GeV/fmd
dE
dx
6.140
dx
dEGeV/fm
Initial parton density (Energy loss ) is 15~30 times that in cold Au nuclei !
Comparison with STAR data
STAR, Phys. Rev. Lett. 91 (2003) 172302
d-Au Result
理论预言
实验结果
E. Wang, X.-N. Wang, Phys. Rev. Lett. 89 (2002) 162301
STAR, Phys. Rev. Lett. 91(2003) 072304
IV. Modification of Dihadron Frag. Function
h1 h2
jet
A. Majumder, Enke Wang, X. –N. Wang, Phys. Rev. Lett. 99 (2007) 152301
Dihadron fragmentation:
h1
h2
DGLAP for Dihadron Fragmentation
2
1
1
1
2
2
2 11 2
1 222
21
2
( , , )( ) ( )
ln( , , )q
qh h
q q hg
z z
h
D z z Q dyP
z zD Q
y yy g h h
Q y
h1h2
h1h2
h1
h2
1
1 2
2
22
121ˆ ( ) (( , )
1)
(,
)( )
1q
z
q
z
hg hqgz
Dz
D Qy
dyP y q g
yQ
y y
Evolution of Dihadron Frag. Function
Evolution of Dihadron Frag. Function
)()(),( 21212121 zDzDzzD h
qhq
hhq
Medium Modi. of Dihadron Frag. Function
Nuclear Modification of Dihadron Frag. Func.
)(
)()(
212
2222 zN
zNzR
h
Ah
h
e-A DIS
Hot Medium Modification
Thank YouThank You
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