To Those Who Perished in the 5.12 Earth Quake in China
Xin-Nian WangLBNL
4Th Electron-Ion Collider WorkshopHampton, May 19-23, 2008
Quark Propagation in a Nuclear Medium
Medium Response Function
41( ) (0) ( )4
iq x em emW q d xe A j j x A
1 22 2 2
1( ) ( )DIS B Bq q p q p qW g F x p q p q F x
q p q q q
qpqxB
2
2
Dynamic System: Photon or dilepton emission
J/ suppression
( ) ( )emj x j x QCD Response: Quark-medium scattering
Quark Propagation: Jet Quenching & Broadening
dE/dx modified frag. functions
hadronsph
parton
E
Dh/a(z)=dN/dz (z=ph/E)
),,()(0 EzDzD ahah
Suppression of leading particles
Fragmentation Function
Angular distributiondN/d2kT
<k2T> jet broadening
DIS off a large nucleus
[ ,0,0 ] momentum per nucleonp p
2[ , ,0 ], / 2B Bq x p q x q q p
1 ( ) ( ) ( )2 A NA
N p N pp
Loosely bound nucleus (p+, q- >> binding energy)
e-
Collinear Expansion
( ) 4 ˆTr ( )ni
i
W d k H k A A A A
p AAp
A
ˆ (0)ˆ ( ) ˆ( ) ( ) |k k xpk xpH k H kH
Collinear expansion:
Ward identities(1) (0)ˆ ˆ( ) ( )p H x H x
(0)
(1)ˆ ( ) ˆ ( , )
H xH x x
k
( )igA D
Collinear Expansion (cont’d)
(0) 4ˆ ( ) (0) (0, ) ( )ik yk d ye A L y y A
(0) 4(2)
2(0) (
40)1 ( )Tr[ ]
2 (ˆ ( ) ˆ (
2)
)dW d k H x kkd
‘Twist-2’ unintegrated quark distribution
q
xp
=
(1) 4 4(1, )(2)1
2 4 4 1,
' (1)' 1
1 ( )Tr[ ]2 (2 ) (
ˆ ( )2
ˆ ())
,,cc
c
L R
kdW d H x xk d k kd
k
1 1(1) 4 41 1 1 1 1
ˆ ( , ) (0) (0, ) ( ) ( , ) ( )ik y ik yk k d yd y e A L y D y L y y y A
‘Twist-3’ unintegrated quark distribution x1p
q
xp
TMD (unintegrated) quark distribution
(0)
2
2
1 ˆ( , ) [ ( ) ]2
( (0) ( )4
;(2 )
0 )
qA
ixp y ik y
f x k dk Tr k
d L yy d y e A y A
† †( ,0;0 ) (( ; ,0 )0; ,) ; )( L L yL L yy y
Longitudinal gauge link( , ; ) exp ( , )y
L y y P ig d A y
Belitsky, Ji & Yuan’970
( ; ,0 ) exp ( , )y
L y P ig d A
Transverse gauge link
Transport Operator
†(0, ) (0, ( ) ( ; ) ( ) ( ; )) |y
yy iD y g d L yi L y L F L yy
(2)( , ) (0) (0; , ( ,0 ) exp[ ] ( ,0 ) ( )4
0 )qA k
ixp y Wdyf x k e A L y y A ky
All in terms of collinear quark-gluon matrix elements
Liang, XNW & Zhou’08
2(2)
2 ( ) exp (0) ( )(2 )
ik yk y
d y e F y i F k
Taylor expansion
( , )W y y
Transport operator
Maximal Two-gluon Correlation
†0( ,0 ) ( ; ) ( ) ( ;( ) ) |
y
yW y g d L yi Fy LD y
22 (0) (0; ) ( )4
( )nn ixp y
kdyM e A L y W y y A
2
2( )
nn
A Nn A
AF N F xG xF N
p
1 2
4/31
2 1
2 2
(0) ( )
(
( ) ( )
() ) ( )qN N A N N
dy dy d F y F y
d x G
y A y A
Af xx A
1 2 12 (( ) ( ()0) )Ddy dy dy A y Ay D y A
Nuclear Broadening & Gluon Saturation
2(2)
2
ˆ( , ) ( ) exp ( ) ( )4
1 ( ) exp
kq qA N N N
qN
f x k Af x d q k
kAf x
2 ˆ( )N Nk d q
2
02
4ˆ( ) ) ) |1
( (A N N xs F
Nc
CN
xq xG
Quark jet transport parameter
22 2
02
4ˆ ( ) ( ) ( , ) |1
s Asat A N A N N sat x
c
CQ q xG x QN
Liang, XNW & Zhou’08
Coherence effect correlation between different nucleons
Gluon saturationKochegov & Mueller’98
McLerran & Venugapolan’95
SIDIS off Nuclei
e-
(0) ( ),) (( , ) q h hqh
f x H xdW
dxdz
p Dq z
pypedyxf yixpBq )()0(
21
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.
(0) 2 21( , , ) ( ) 2 ( )2qH x p q e Tr p q xp q xp
DGLAP Evolution
z
zDzPzzDzP
zdzdzD h
hqqgqh
hqqgqz
Shhq
h
)1()(2
)(1
2
22
)1(23
)1(1)(
2
zzzCzP Fqgq Splitting function
q
p
k1 k2
p
q
Induced gluon emission in twist expansionq
Apxp
xp
Ap
q
x1p+kT
1 2(1 2
)2 ( , , ) ( ) ( )ik y yDTT
DH p q kW d A y A yk e A A
2 2( , , ) ( , ,0)( , ,0) ( , ,0)TT
D D DT Tkk
DTH p q k H p q H p H p qq k k
Collinear expansion:
AFFAkqpHW TD
kD
T
)0,,(2
Double scattering
LPM Interference
[ , , ]Tzq
2 0x B Lx x2 Lx xBx
2
2 (1 )
TLx
p q z z
_2 1
2( )2 (0
0
2
4) 1
(1 )( , , ) | 1 1L L
T
ix p y ix p y yDk
s
TkH p q k H e ez
z
1f
Lx p Formation
time
2
2 (1 )T
Lx pq z z
222
4
1 ( )(1 )
N sqg L N L
T
zd x G x dzdz
Modified Fragmentation2 12
24
0
( , ) ( , )2
h
QS h
q h h L q hz
zd dzD z Q z x Dz z
2 ( , ) 21( , ) (virtual)(1 ) ( )
Aqg L A S
L Aq c
T x x Czz xz f x N
Modified splitting functions
Guo & XNW’00
_2 1(
1 22
)
1( , ) (0) ( )2 2
1
( ) ( )
1
B
L Lix p y ix p
ix p yAqg L
y y
F y F ydyT x x dy dy A A
e
e
e
y
Two-parton correlation:
ˆ ˆ( ,0) (( , )
1 cos(2( )
, ))A
qg LsN N
c qL NN LA
T x xd
N fq
xx pq x
1/3AR A
Quadratic Nuclear Size Dependence
02 2~ ( )qs
N Bd Af xd
1 2
2
1 22 4( )1 2~ (0) ( )
2 2( ) (
2)
2B Tix p y ix psD y ydyd Fd y F yy dy e A y
dA
2
04/3
4~ [ ( )]( )T
qN B
sT N T xx G xA f x
1/32 2
02 2~ ( )[1 ( ) ) ](SD
T Ts
q Bd c A x G xd fd
A xd
3/1
22 LPM
AQ
2
3/2
1QAc
HERMES data
2ˆ 0.01 GeV / fmq 0.5 GeV/fmdEdx
in Au nuclei
E. Wang & XNWPRL 2000
Energy Dependence
Jet quenching in heavy-ion collisionCentral Au+Au
single hadron
dihadron
2ˆ 1 2 GeV / fmq 2ˆ 0.01 GeV / fmq in cold nucleiin hot matter
Summary
• Quark transverse momentum distribution can be expressed in term collinear quark-gluon matrix elements
• quark propagation in nuclei leads to kT broadening gluon correlation function
• Jet transport parameter qhat saturation scale Qsat
2
• Modified fragmentation function qhat~0.01 GeV^2/fm in HERMES at HERA versus qhat~ 1-2 GeV^2/fm in QGP at RHIC
• qhat at small x and large Q^2 can reveal gluon dynamics (correlation, saturation) in nuclei