Transcript Rapidity Asymmetry in d+A Collisions

Modified Fragmentation Function
in Strong Interaction Matter
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
Jet Quenching:
hadrons
hadrons
q
E
q
q
hadrons
leading
particle
E '  E  E
Leading
particle
suppressed
leading
particle
q
hadrons
leading
particle
suppressed
A-A collision
p-p collision
Fragmentation Function:
2
Dq h ( zh , Q )
ph
DGLAP Equation

zh 
ph
p

z h  z h   z h
~
2
2
2
D q  h ( z h , Q )  D q  h ( z h , Q )   D ( z h , Q )
q
p
~
S
I. Jet Quenching in QCD-based Model
G-W (M. Gyulassy, X. –N. Wang) Model:
Static Color-Screened Yukawa Potential
Feynman Rule:
  iD ( 2 p  q )   i ( 2 p  q )
0
p-q
p
q
 i ( p ) 
p
i
p  i
2

  ig s ( 2 p  k ) c ,
p+k



k,c 
k-q,a
 0
c  Tc
p
k

0
q,b
  ig

 f
{g
abc
 (k )  i
0
g

k  i
2

[ q  ( k  q )]  g
 g

0
(k  q)

[( k  q )  (  k )] }
0
Opacity Expansion Formulism (GLV)
GLV, Phys. Rev. Lett. 85 (2000) 5535; Nucl. Phys. B594 (2001) 371
Elastic Scattering
Double Born
Scattering
Assumption
• The distance between the source and the scattering center are large
compaired to the interaction range:
z i  z 0 
1

• The packet j(p) varies slowly over the range of the momentum
transfer supplied by the potential:
j( p )  j( p  q)
•The targets are distributed with the density:
 

N
 ( x1 , x 2 ,  , x N ) 
 ( z1 , z 2 ,  , z N )
A
N
 (  z j )   z j / Le ( N )
 ( z1 , z 2 ,  , z N )  
e
j 1 L e ( N )
 z j  z j 1  z j
L
Le ( N ) 
N 1
Opacity: Mean number of the collision in the medium
n 
L


N  el
A
 
1

el
First Order in opacity Correction
First Order in opacity Correction
Induced gluon number distribution:
C R s L

x

2
2

g
dxd k 
dN
(1 )
Non-Abelian
LPM Effect
 2 
 d q  v ( q  )(  2 C 1  B1 ) 1  cos(  1  z1 )
2
Medium-induced radiation intensity distribution:
Induced radiative energy loss:
QCD:  E (1 )  L2
QED:  E
(1 )
L
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
B-E Enhancement Factor
Thermal Distribution Func.
1+N(k)
N(k)
k
k
x0
p
x0
Stimulated Emission
p
Thermal Absorption
Final-state Radiation
k
x0
k
x0
p
p
Energy loss induced by thermal medium:
 dp
dp
  d  

d
 d
(0)
 E abs
(0)
(0)


T 0 
 s C F T  4 ET
6 ' ( 2 ) 
 2E 
2
2
 ln

3
E 



2
=
Net contribution:
Energy gain
Stimulated emission increase E loss
Thermal absorption decrease E loss
First Order in Opacity Correction
Single direct rescattering:
k
y0
k
y1
p
y0
y1
p
y0
y1
p
k
Double Born virtual interaction:
k
k
y0
y0
y0
y1
y1
y1
y1
y1
y1
p
p
p
k
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:
E
  E rad   E abs
Take limit:
E  
EL  1
(1 )
(1 )
(1 )
Zero Temperature Part:
(0)
dp
(1 )
 E rad   d 
d

2
L
T 0
T
2
 sC F  
 dp
dp
  d  

d
 d
2
2
 s C F LT   L

Energy gain
3
 g E 
2E
GLV Result
(1 )
 E abs
 L

 ln 2  0 . 048 
4 g   L

       
Temperature-dependent Part:
(1 )

2
(1 )
ln
T
T 0



1  E 
6 ' ( 2 ) 
2



Numerical Result for Energy Loss
• Intemediate large E,
absorption is important
•Energy dependence
becomes strong
•Very high energy E,
net energy gain can be
neglected
 S  0 .3
 E   E abs   E abs   E rad
(0)
(1 )
(1 )
Parameterization of Jet Quenching
with Detailed Balance Effect
Average parton energy loss in medium at formation time:
dE
dL
  0 ( E /  0  1 .6 )
1 .2
/( 7 . 5  E /  0 )
1d
Energy loss parameter
proportional to the initial gluon density
0 
dN
1
d   0 R A
2
Modified Fragmentation Function (FF)
D h / c ( z c ,  ,  E c )  (1  e
2
e
 L/
Dh /c ( zc ,  )
0
'
2
 L/
'
'
)[
zc
zc
Dh /c ( zc ,  )  L / 
0
'
2
zg
zc
D h / g ( z g ,  )]
0
2
(X. -N. Wang , PRC70(2004)031901)
z c  p T /( p Tc   E c ), z g  L /  p T /  E c ,
'
'
'
Comparison with PHENIX Data
PHENIX,
Nucl. Phys. A757
(2005) 184
DGLAP Equation at Finite Temperature
J. 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-
dW  

dzh
fq ( xB ) 
dy
 2

e
ixp

y
  d x f ( x ) H  ( x , p , q ) D
( zh )
q

1


p  ( 0 )  ( y ) p
2
H  ( x , p , q )  e q
2
Frag. Func.
q h
1
2
Dq h ( zh ) 

Tr   p     ( q  xp )   2 ( q  xp )
zh
2

dy

2

e

 ip h y / z h
2

 
 T r  2 0  q (0) p h , S
S



ph , S  q ( y ) 0 

Modified Fragmentation Function
D ( zh , Q )  D ( zh , Q )  D ( zh , Q )
2
2
2
Cold nuclear matter or hot QGP medium lead to
the modification of fragmentation function
Jet Quenching in e-A DIS
X.-N. Wang, X. Guo, NPA696 (2001); PRL85 (2000) 3591
e-
Modified Frag. Function in Cold Nuclear Matter
D ( zh , Q )  D ( zh , Q )  D ( zh , Q )
2
2
Dq h ( zh , Q ) 
2
S
2
Q
2

2
2 1

d
4

0

zh
dz 
 zh 


(
z
,
x
)
D
L
q h 


z 
 z 



Modified splitting functions
 ( z, xL ) 
1 z
A
T qg ( x , x L ) C A 2  S
2
(1  z ) 
A
fq ( x)

( virtual)
Nc
Two-parton correlation:
Tqg ( x , x L ) 
A

dy

2


dy1 dy 2 e

 1 e
LPM

 ix B p y


A  (0)


2


 ix L p y 2

1 e


F ( y1 ) F

_
ix L p ( y1  y )





( y 2 ) ( y ) A


 (  y 2 ) ( y  y1 )
Modified Frag. Function in Cold Nuclear Matter
Fragmentation function without medium effect:
0
Dh a (z)
are measured, and its QCD evolution
tested in e+e-, ep and pp collisions
Fragmentation function with medium effect:
parton
E
hadrons
ph
D ( z )  D h a ( z ,  E ),
0
h a
Suppression of leading
particles
Dh a ( z, E ) 
1
1  z
D
0
h a
(
z
1  z
),
Heavy Quark Energy Loss in Nuclear Medium
B. Zhang, E. Wang, X.-N. Wang, PRL93 (2004) 072301; NPA757 (2005) 493
Mass effects:
1) Formation time of gluon radiation time become shorter
f 
2 z (1  z ) q

l T  (1  z ) M
2
2
2
LPM effect is significantly reduced for heavy quark
2) Induced gluon spectra from heavy quark is suppressed by
“dead cone” effect
2
2
lT

4
4
0
fQ /q  [ 2
]  [1  2 ]
2
2
lT  z M

0 
M
 
lT
q


q z
Dead cone Suppresses gluon radiation amplitude at   
0
Heavy Quark Energy Loss in Nuclear Medium
zg ( xB , Q ) 
Q
2
~
2
C C A s x B
2
N cQ x A
1
(~
xL  ~
xM )
~
dz
d
x
0
 L
4
~
z (1  z ) ~x
xL
1 z
1
2
~
xL
~
xB M
xA
x AQ
2
2
1
xA 
,
mN RA
1) Larg x or small
Q
B
zg
Q
~
Nc
2) Larg
zg
Q
~ 2
C AC  S
~
Q
2
xB
x AQ
2
2
:
 RA
or small x :
B
~ 2
C AC  S
xB
Nc
x AQ
2
 RA
2
2
2
LPM Effect
M
{ c 3 ( z , l T , M )  (1  e
2
2
~
x
2
~
xL / x A
2
2
)[ c 1 ( z , l T , M )  c 2 ( z , l T , M )]}
2
2
2
2
Heavy Quark Energy Loss in Nuclear Medium
The Q dependence of
the ratio between charm
quark and light quark
energy loss in a large
nucleus
2
The x dependence of
the ratio between charm
quark and light quark
energy loss in a large
nucleus
B
III. Jet Tomography of Hot and Cold Strong
Interaction Matter
E. Wang, X.-N. Wang, Phys. Rev. Lett. 89 (2002) 162301
Cold Nuclear Matter:
Quark energy loss = energy carried by radiated gluon
Q
zg 
2

0
d
2 1
T
s
0
2
2
T
 dz
Q
z ( z, xL ) 
2

0
1
d
2
T
 dz
0
1  (1  z )
2
T

2
T
A
2
 kT
2

2
C A s Tqg ( x , x L )
Nc
A
fq ( x)
Energy loss
E  C
2
s
CA
Nc
2
A
m N R 3 ln
1
2 xB
E  A
2/3
Comparison with HERMES Data
~ 2
2
C ( Q )  0 . 0060 GeV ,
 s ( Q )  0 . 33, Q  3GeV
2
2
HERMES Data: Eur. Phys. J. C20 (2001) 479
2
Expanding Hot Quark Gluon Medium
Tqg ( x , x L ) 
A

dy

2


dy1 dy 2 e

 1 e
A
T qg ( x , x L )
A
fq ( x)

 ix B p y

A  (0)




 ix L p y 2

1 e


_
ix L p ( y1  y )

y
~  dy   g  ( y ) 1  cos
f

 2E 
 E    d   ( ) ln  2 
  
0

F ( y1 ) F
2
2
R
3
s









( y 2 ) ( y ) A


 (  y 2 ) ( y  y1 )
R. Baier et al
Initial Parton Density and Energy Loss
 2E 
 E    d   ( ) ln  2 
  
0
R
3
s
 E1 d   E 0
2 0
RA
jet1
 E 0 : Initial energy loss in
jet2
 ( )   0
0

 (R  r)
 0  0.1 fm
 dE 

  0 .5 G eV /fm
 d x 1 d
a static medium with
density  0
RA 
15
2 0
 dE 

  14 . 6 GeV/fm
 dx  0
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
A. Majumder, Enke Wang, X. –N. Wang, Phys. Rev. Lett. 99 (2007) 152301
Dihadron fragmentation:
h1
h1
h2
jet
h2
DGLAP for Dihadron Fragmentation
h1
h1
h2
h2
h1
h2
 D h1 h2 ( z1 , z 2 , Q )
q
2
 ln Q
2


z1  z 2
1 z 2

1

z1
dy
dy
y
2
Pq  q g ( y ) D
q
h1 h2
z1 z 2
2
( , , Q )  ( g  h1 h2 )
y y
z
q z
2
g
2
Pˆq  q g ( y ) D h1 ( 1 , Q ) D h2 ( 2 , Q )  ( q  g )
y (1  y )
y
1 y
Evolution of Dihadron Frag. Function
Evolution of Dihadron Frag. Function
D
h1 h 2
q
( z1 , z 2 )  D ( z1 ) D
h1
q
h2
q
( z2 )
Medium Modi. of Dihadron Frag. Function
Nuclear Modification of Dihadron Frag. Func.
R2h ( z2 ) 
N
N
A
2h
1
2h
( z2 )
( z2 )
e-A DIS
Hot Medium Modification
Thank You