Transcript Jet transport and gluon saturation in medium Xin-Nian Wang

Workshop on Structure of hadrons and nuclei at an
Electron Ion Collider, Trento, July 13-18, 2008
Jet transport and gluon saturation in medium
Xin-Nian Wang
Lawrence Berkeley National Laboratory
Hard Probes & Structure of Dense Matter
eJet quenching
e-
D( z, k )
dE
kT broadneing dx
q̂
W (q) 
1
4
iq x
em
em
d
xe
A
j
(0)
j

 ( x) A

4
 eT F1 ( xB )  eL F2 ( xB )
Q2
xB 
2pq
Quark Propagation: Jet Quenching &
Broadening
parton
E
Dh/a(z)=dN/dz (z=ph/E)
hadrons
ph
Fragmentation Function
dN/d2kT
Angular distribution
<k2T>  jet broadening
dE/dx  modified frag. functions
Dh0 a ( z)  Dh a ( z, E ),
Suppression of leading
particles
Jet Quenching phenomena at RHIC
Pedestal&flow
subtracted
STAR Preliminary
DIS off a large nucleus
e-

p  [ p , 0, 0 ] momentum per nucleon


q  [ xB p , q , 0 ], xB  q / 2q  p
2
Loosely bound nucleus (p+, q- >> binding energy)
A
1

 A ( N ) N ( p)

2p
N ( p)
DGLAP Evolution
q
q
k1
k2
p
S
Dqh ( zh ) 
2
2
p
d 2 dz 
 zh 
 z h 
 2 z z Pqqg ( z) Dqh  z   Pqqg (1  z) Dqh  z 
h
Splitting function
1
 1 z2

3
Pqqg ( z )  CF 
  (1  z )
 (1  z )  2

Induced gluon emission in twist expansion
q
q
xp
Ap
x1p+kT
xp
Ap
WD   d 2 kT eik ( y1  y2 ) H D ( p, q, kT ) A   A ( y1 ) A ( y2 ) A
Collinear expansion:
D
D
D
D
H 
( p, q, kT )  H 
( p, q,0)   kT H 
( p, q,0)kT   2kT H 
( p, q,0)kT2 
D
H 
( p, q,0) 
Eikonal contribution to vacuum brems.
Double scattering
D
D
W
  2kT H 
( p, q, kT  0) A    F  F  A
Different cut-diagrams
+ …..
Eikonal contribution
central-cut = right-cut = left-cut in the collinear limit
 dx1dx2 dxe
  dxeixp
 
y
ix1 p  y2  ixp  ( y1  y2 )  ix2 p  ( y   y1 )
s
(0)
H  ( xp, q)
2
2

0
d
2
T
2
T
(2) R ,C , L
H 
( x1 p, x2 p, xp, q, z )
Pq qg ( z )
  ( y   y1 ) ( y1  y2 )   ( y   y1 ) ( y2 )   ( y2 ) ( y2  y1 ) 
(2)
dW
dzh

 e
2
q
q

dy ixp  y 
e
2
 dxH 
(0 )
zh  s
dz
( xp, q, z )  Dq h ( )
z
z 2
2

0
d
2
T
2
T
Pq qg ( z )
y1
y

 
2

 



A  (0)
 (ig )  dy1  dy2 A ( y2 ) A ( y1 )  ( y  ) A
2 

0
0




LPM Interference
 [  , zq  ,
xB  xL
xL 
x2  0
D
(0 )
 2kT H 
( p, q, k ) |k 0  H 
Formation time
d
N
qg
xB
x2  xL


T
]
2
T
2 p  q  z (1  z )
1  z 2  s2
 ixL p  y2
ixL p  ( y1  y _ )
1 e
1 e
4
(1  z ) T
1
f 
xL p 
1  z 2  s2
xLGN ( xL )dzd
4
(1  z )  T
2

xL p  
2
T
2q  z (1  z )
Quark-gluon Compton scattering

Modified Fragmentation
Q2
S d
Dqh ( zh , Q ) 
2 0
Modified splitting functions
2 1

4
 zh
2

dz
z
Guo & XNW’00

 zh 
  ( z, xL ) Dq h  z  
 




A
1  z 2 Tqg ( x, xL ) CA 2 S
 ( z, xL ) 
 (virtual)
A
(1  z ) f q ( x)
Nc
Two-parton correlation:


dy

TqgA ( x, xL )  
dy1 dy2 eix p y A  (0) F ( y1 ) F  ( y2 ) ( y  ) A
2
2
 
B

 1 e
 ixL p  y2
1  e
ixL p  ( y1  y _ )

A
2 s Tqg ( x, xL )

 
ˆ
ˆ

d

q
(

,
0)

q
(

,
x
)
1

cos(
x
p
N )


N
N
N
L
L
A

Nc
f q ( x)

 RA  A1/ 3

Quadratic Nuclear Size Dependence
d D  s2 dy  dy1 dy2 ixB p y ixT p ( y1  y2 )
     

~
e
A

(0)
F
(
y
)
F
(
y
)

(
y
)A

1
2
2
4 
d 
2 2 2
2

~ A4 / 3 f Nq ( xB )
 s2
4

[ xT GN ( xT )]xT 0
d 0  s
q
~
Af
N (x B )
2
2
d 

d 0 d D  s

~ 2 Af q ( x B )[1  c(
2
2
d  d 

2
Q
LPM    1/ 3
A
2

)
S
2

A1/ 3 xT G ( xT ) 
A2 / 3
1 c 2
Q
]
Validity of collinear expansion
Collinear expansion:
D
D
D
D
H 
( p, q, kT )  H 
( p, q,0)   kT H 
( p, q,0)kT   2kT H 
( p, q,0)kT2 
Results good for
For
2

k2
2

k2
One has to re-sum higher-twist terms
Or model the behavior of small lT behavior
A
Need to include all: Tqg ( xB , xL ), xL
LPM limits
LA 
2 z (1  z ) E
2

TqgA ( xB , xL )
xL
, xL2
 2TqgA ( xB , xL )
 2 xL
Gauge Invariance
k2
ik  ( y1  y2 )
i  ( y1  y2 )
e
2



2p
dk


i

(
y

y
)
e
2
1
 2k  p   k2  i
2 p
Expansion in kT
TMD factorization

p
k
k   i 
One should also consider A
Final matrix elements
should contain:

i   gA  iD
D( y1 )L ( y1 , y2 ) D( y2 )L ( y2 , y3 )
Collinear Expansion

W( n)    d 4 ki Tr  Hˆ 
(k ) A  A A
i
Collinear
expansion:
 A 
Hˆ  (k )  Hˆ  (0)  (k  xp)   k Hˆ  (k ) |k  xp 
p 
A   A   A
p
 k  (k  xp)
Collinear Expansion
Collinear
expansion:
Ward identities
Hˆ  (k )  Hˆ  (0)  (k  xp)   k Hˆ  (k ) |k  xp 
p 
A   A   A
p
 k  (k  xp)
(1)
(0)
p Hˆ 
( x)  Hˆ 
( x)
(0)
(1)
Hˆ 
( x)  igA p Hˆ 
( x) 
(0)
 Hˆ 
( x)(1  igA  )
(0)
(1)
k Hˆ 
( x)  Hˆ 
( x, x)
(0)
(1)
(1)
(k  xp)   k Hˆ 
(k )   A Hˆ 
(k )  Hˆ 
( x, x) (   igA )
D
Collinear Expansion (cont’d)
dW(0)
d
2

1

2
d 4 k (2)
 (2 )4  (

q
ˆ (k )]
 k )Tr[ Hˆ  ( x)
(0)
( 0)
xp
‘Twist-2’ unintegrated quark distribution
ˆ (0) (k )  d 4 yeik  y A  (0)L (0, y ) ( y ) A


dW(1)
d2

1

2
d 4 k d 4 k1
 (2 )4 (2 )4

c L,R
 (2) (
ˆ (1,c )  ( x, x )  '
ˆ (1) (k , k )]

k
)Tr[
H

c

1

'
1
q
‘Twist-3’ unintegrated quark distribution
xp x1
p
ik  y  ik1  y1
4
4
ˆ (1)

(
k
,
k
)

d
yd
y
e
A  (0)L (0, y1 ) D ( y1 ) L ( y1 , y ) ( y ) A

1
1

Liang & XNW’06
TMD (unintegrated) quark distribution
ˆ (0) (k )  d 4 yeik  y A  (0)L (0, y ) ( y ) A


1
ˆ (0) (k , s)   p f Aq (k ) (k  xp) f (k )    p k s f 
Tr   

A
   1T
2
4
ik  y
ˆ (1)

A  (0)L (0, y ) D ( y ) ( y ) A
 ( k )   d ye
Contribute to azimuthal and single spin asymmetry
f Aq ( x)   dk  d 2 k f Aq (k )
Twist-two integrated quark distribution
TMD (unintegrated) quark distribution
1
ˆ (0) (k )  ]
f ( x, k )   dk Tr[
2
dy  d 2 y ixp y  ik  y


e
A

(0)

L (0; y ) ( y) A
2
4 (2 )
q
A
L (0; y)  L † (, 0; 0 )L† (; y , 0 )L (, y  ; y )





L (, y ; y )  P exp  ig  d A ( , y ) 


y


y


L (; y , 0 )  P exp  ig  d  A (,  ) 
 0

y
0
Longitudinal gauge link
Transverse gauge link
Belitsky, Ji & Yuan’97
Transport Operator
Taylor expansion
d 2 y ik  y
(2)


e
F
(
y
)

exp
i


F
(0)

(k )

k
y
 (2 )2
 
 

dy
q
ixp  y 
f A ( x, k  )  
e
A  (0)  L (0; y  ,0 ) exp[W ( y  ,0 ) k ] ( y  ,0 ) A  (2) (k )
4
y


 †
i y L (0, y)  L (0, y) iD ( y)  g  d L ( ; y) F ( )L ( ; y) |  y 




W ( y  , y )
Color Lorentz force:
Transport operator
dp
 gF  v 
d
All info in terms of collinear quark-gluon matrix elements
Liang, XNW & Zhou’08
Maximal Two-gluon Correlation
y
W ( y  , 0 )  iD ( y )  g  d  L † ( ; y ) F ( ) L ( ; y) |  y 0

M
2n
dy ixp

e
4

 
y
2n
A  (0) L (0; y ) W ( y  ) k   ( y  ) A



d
y
 A  (0)D ( y) D ( y) ( y) A  A







dy
d

d

A

(0)
F
(

)
F
(

1
2
2

1 ) ( y ) A

 Af Nq ( x1 )  d N  A ( N ) x2GN ( x2 )  A4 / 3
W
2n
A


 A ( y)
dy
N
F
F
N
 



2
p


n
 dy A ( y ) xGN ( x) 


n
Nuclear Broadening
2

 q

k
q

f A ( x, k )  A exp   d N qˆ ( N )
 f N (k )
4 

2

 q
A
(
k

q
)
2



d q exp  
 f N (q )





4 2 s CF
qˆ ( N ) 
 A ( N ) xGN ( x) |x 0
2
Nc  1
Liang, XNW & Zhou’08
Majumder & Muller’07
Kovner & Wiedemann’01
  k2   d  N qˆ ( N )
Jet transport parameter
d  ixp 
A



xGN ( x)  
e
N
F
(0)
L
(0,

)
F
(

) N



2 p
Solution of diffusion eq.
dp
 gF  v 
d
Extended maximal two-gluon correlation
y
W ( y  , y )  iD ( y )  g  d   L † ( ; y ) F ( ) L ( ; y ) |  y

W2 n ( y  )
C
n
g 2  d1d2 F (1 )W2( n1) ( y) F (2 )
C
d  ixp 
A

A





xGN ( x, y )  
e
N
F
(
0
)
L
(0,

)
exp

iy

W
(

)
F
(

) N







2 p
2

2
4

 s CF
2
qˆ ( N , y2 ) 

(

)
xG
(
x
,
y
A
N
N
 ) | x  0 Scale dependent qhat
2
Nc  1
Non-Gaussian distr. contains information about multi-gluon correlation in N
2

y
ik  y q
q
2
q


2 

f A ( x, y )   d ke
f A ( x, k )  Af N ( x, y ) exp   d N qˆ ( N , y ) 
 4

Jet transport parameter & Saturation
4 2 s CF
qˆ F ( N , x) 
 A ( N ) xGN ( x)
2
Nc  1
Multi-gluon correlation:
qˆ (Q 2 )  xGN ( x, Q 2 )
Gluon saturation
2
qˆ A LA  Qsat
Kochegov & Mueller’98
McLerran & Venugapolan’95
4 2 s C A
2

L

(

)
xG
(
x
,
Q
A A
N
N
sat ) | x  0
2
Nc  1
Casalderrey-Salana, &
XNW’07
Conformal or not
Casalderrey-Salana, XNW’07
Gluon distr. from HTL at finite-T (gluon gas)
q̂
 s  2  3  2 1 D 
xGN ( x,  )  C A
 ln 2  ln
 12 (3)  2 D 3 xT 
2
DGLAP
DGLAP evolution in linearized regime
Qs2 ( x) 
4  s C A
 A ( N ) xGN ( x, Qs2 ) min( L, Lc )
2
Nc  1
2
DGLAP with
fixed s:
CR 3T 2 E
qˆ 
,
Nc 2
Strong coupling SYM:
( Lc  1/ xT )
2N c s

xGN ( x, Q 2 )  0, x  xs
qˆ
T 3 E / M
Qs2
1
T /Q
Hatta, Iancu & Mueller’08
Gubser 07,
Casaderrey-Salana & Teaney’07
Measuring qhat
Direct measurement:
Measuring parton energy loss
or modified fragmentation function
q
q
xp
Ap
x1p+kT
xp
Ap
GW:Gyulassy & XNW’04
BDMPS’96
LCPI:Zakharov’96
GLV: Gyulassy, Levai & Vitev’01
ASW: Wiedemann’00
HT: Guo & XNW’00
AMY: Arnold, Moore & Yaffe’03
Summary
• Jet transverse momentum broadening provides a
lot of information about the medium: gluon
density, gluon correlations, etc, all characterized
by jet transport parameter qhat
• Jet quenching provided an indirect measurement
of qhat
• Jet quenching phenomenology has advanced to
more quantitative analysis
• More exclusive studies such as gamma-jet and
medium excitation are necessary