Workshop on Structure of hadrons and nuclei at an Electron Ion Collider, Trento, July 13-18, 2008
Xin-Nian WangLawrence Berkeley National Laboratory
Jet transport and gluon saturation in medium
Hard Probes & Structure of Dense Matter
41( ) (0) ( )
4iq x em emW q d xe A j j x A 1 2( ) ( )T B L Be F x e F x
2
2B
Qx
p q
e-
e-
dE
dx
q̂
( , )D z k
Jet quenching
kT broadneing
Quark Propagation: Jet Quenching & Broadening
dE/dx modified frag. functions
hadrons
ph
parton
E
Dh/a(z)=dN/dz (z=ph/E)
),,()(0 EzDzD ahah
Suppression of leading particles
Fragmentation Function
Angular distribution
dN/d2kT
<k2T> jet broadening
Jet Quenching phenomena at RHIC
Pedestal&flow subtracted
STAR Preliminary
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 NAN p N p
p
Loosely bound nucleus (p+, q- >> binding energy)
e-
DGLAP Evolution
z
zDzP
z
zDzP
z
dzdzD h
hqqgqh
hqqgq
z
Shhq
h
)1()(2
)(1
2
22
)1(2
3
)1(
1)(
2
zz
zCzP Fqgq Splitting function
q
p
k1 k2
p
q
Induced gluon emission in twist expansion
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
q
Apxp
xp
Ap
q
x1p+kT
( , ,0)DH p q Eikonal contribution to vacuum brems.
Different cut-diagrams
+ …..
Eikonal contribution
1 2 1 2 2 1
2
(2) , ,1 2
2(0)
( )
2
( )1 2
1 1 2 1 2 2 2 1
0
( , , , , )
( , ) ( )2
( ) ( ) ( ) ( ) ( ) ( )
R C L
s Tq
ix p y ixp y y ix p y y
iqg
T
xp y
dx dx dxe
dxe
y
H x p x p xp
y
q
y y y y y
z
dH xp q P z
y y y
2
1
(2) 22 (0
21 2 2 1
2
0
0
0
) ( ,
( )
, ) ( ) ( )2
(0) ( )2 2
( ) ( )
h s Tq q h q qg
qh T
ixp yyy
dW zdz de dxH xp q z D P z
dz z z
dig
ye A dy dy A y y yA A
central-cut = right-cut = left-cut in the collinear limit
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 e
z
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 dzd
z
Quark-gluon Compton scattering
Modified Fragmentation
2 122
40
( , ) ( , )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
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 ydy
T x x dy dy A A
e
e
e
y
Two-parton correlation:
ˆ ˆ( ,0) (( , )
1 cos(2
( ), ))
Aqg Ls
N Nc q
L NN LA
T x xd
N fq
xx pq x
1/3AR A
Quadratic Nuclear Size Dependence
02 2
~ ( )qsN B
dAf x
d
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 yd
A
2
04/3
4~ [ ( )]( )
T
qN B
sT N T xx G xA f x
1/32 2
02 2
~ ( )[1 ( ) ) ](SDT T
sq B
dc A x G x
df
dA x
d
3/1
22 LPM
A
Q 2
3/2
1Q
Ac
Validity of collinear expansion
2
2 (1 )LPM limits A
z z EL
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:
2 2Results good for k
2 2For k One has to re-sum higher-twist terms Or model the behavior of small lT behavior
22
2
( , ) ( , )( , ), ,
A Aqg B L qg B LA
qg B L L LL L
T x x T x xT x x x x
x x
Need to include all:
Gauge Invariance
2
1 2 1 2( ) ( )
22 12
2( )
2 2
kik y y i y ype
dk i y y ek p k i p
Expansion in kT k i
i gA iD
One should also consider A
1 1 2 2 2 3( ) ( , ) ( ) ( , )D y y y D y y y L LFinal matrix elements should contain:
k
p
TMD factorization
Collinear Expansion
( ) 4 ˆTr ( )ni
i
W d k H k A A A A
pAA
pA
ˆ (0)ˆ ( ) ˆ( ) ( ) |k k xpk xpH k H kH
Collinear
expansion:
( )k k xp
Collinear Expansion
pAA
pA
ˆ (0)ˆ ( ) ˆ( ) ( ) |k k xpk xpH k H kH
Collinear
expansion:
( )k k xp
(1) (0)ˆ ˆ( ) ( )p H x H x Ward identities
(0) (1) (0)ˆ ˆ ˆ( ) ( ) ( )(1 )H x igA p H x H x igA
(0) (1)ˆ ˆ( ) ( , )k H x H x x
(0) (1) (1)ˆ ˆ ˆ( ) ( ) ( ) ( , ) ( )kk xp H k A H k H x x igA
D
Collinear Expansion (cont’d)
(0) 4ˆ ( ) (0) (0, ) ( )ik yk d ye A y y A L
(0) 4(2)
2(0) (
40)1
( )Tr[ ]2 (
ˆ ( ) ˆ (2
))
dW d kH x kk
d
‘Twist-2’ unintegrated quark distribution
q
xp
(0) 4ˆ ( ) (0) (0, ) ( )ik yk d ye A y y A L(0) 4ˆ ( ) (0) (0, ) ( )ik yk d ye A y y A L
Liang & XNW’06
(1) 4 4(1, )(2)1
2 4 4 1,
' (1)' 1
1( )Tr[ ]
2 (2 ) (ˆ ( )
2ˆ ()
),,c
c
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 y D y y y y A L L
‘Twist-3’ unintegrated quark distribution x1p
q
xp
TMD (unintegrated) quark distribution
(0)1 ˆTr ( , )2
( )qAk s p f k
(0) 4ˆ ( ) (0) (0, ) ( )ik yk d ye A y y A L
1( ) ( )A Tk xp f k p k s f
(1) 4ˆ ( ) (0) (0, ) ( ) ( )ik yk d ye A y D y y A L
Contribute to azimuthal and single spin asymmetry
2( ) ( )q qA Af x dk d k f k
Twist-two integrated quark distribution
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
dy d yy Aye A
L
Longitudinal gauge link( , ; ) exp ( , )y
y y P ig d A y
L
Belitsky, Ji & Yuan’970
( ; ,0 ) exp ( , )y
y P ig d A
L Transverse gauge link
† †( ,0;(0; ( ; ,0 ) ;) 0 ) ( , )yy yy
LL L L
0y
Transport Operator
†(0, ) (0, ) ( ) ( ; ) ( ) ( ; ) |y
yy iD y g d yi y F yy
L LL L
(2)( , ) (0) (0; ,0 )exp[ ] ( ,0 ) ( )(4
,0 ) kq ixp y
A
dyf x k e A y y kW Ay
L
All info in terms of collinear quark-gluon matrix elements
Liang, XNW & Zhou’08
2(2)
2( ) exp (0) ( )
(2 )ik y
k y
d ye F y i F k
Taylor expansion
( , )W y y
Transport operator
dpgF v
d
Color Lorentz force:
Maximal Two-gluon Correlation
†0
( ,0 ) ( ; ) ( ) ( ; )( ) |y
yW y g d yi y yD F
L L
22 (0) (0; ( )) ( )4
ni yk
n xpdyM e A y y AW y
L
1 2
4/31
2 1
2 2
(0) ( )
(
( ) ( )
() ) ( )qN N A N N
dy d d F F
d x G x
A y A
Af x A
( )( ( )( )0)D y Dd A y A Ayy
2 ( ) ( )( )
2
nnn A
AA N
yW dy N F dy y xF N G x
p
Nuclear Broadening
2
22
ˆ( , ) exp ( ) ( )4
( ) exp ( )
kq qA N N N
qN
f x k A d q f k
A k qd q f q
2 ˆ( )N Nk d q
Liang, XNW & Zhou’08Majumder & Muller’07Kovner & Wiedemann’01
2
02
4ˆ( ) ) ) |
1( (A N N x
s FN
c
C
Nxq xG
Jet transport parameter
( ) (0) (0, ) ( )2
ixp AN
dxG x e N F F N
p
L
dpgF v
d
Solution of diffusion eq.
Extended maximal two-gluon correlation
†( , ) ( ) ( ; ) ( ) ( ; ) |y
yW y y iD y g d y F y
L L
2 2 2( 1)1 21 2 (( ) ) ( )( )n n
C CW y n g d F Fd W y
2( , ) (0, ) exp (( ) )2
0 ) (ixp AN
AdxG x y e N NF Fiy W
p
L
2 20
2
2ˆ( , ) ( ) ( ,
4
1) |N A N N
s F
cxq y xG x y
C
N
Scale dependent qhat
22 2( , ) ( , ) ( , ) exp ˆ( , )
4ik yq q q
A A N NN
yf x y d k e f x k Af q yx y d
Non-Gaussian distr. contains information about multi-gluon correlation in N
Jet transport parameter & Saturation
Gluon saturation2
2 202
4ˆ ( ) ( , ) |
1s A
A A sat A A N N sat xc
Cq L Q L xG x Q
N
Kochegov & Mueller’98McLerran & Venugapolan’95
2
2
4ˆ ( , ) )
1( ( )A N N
s FF N
c
Cq x
NxG x
Multi-gluon correlation:2 2ˆ( ) ( , )Nq Q xG x Q
Casalderrey-Salana, &XNW’07
Conformal or not
2 22
2
3 1( , ) ln ln
12 (3) 2 3s D
N AD
xG x CxT
Gluon distr. from HTL at finite-T (gluon gas)
DGLAP evolution in linearized regime2
222
( ) (4
( )1
, ) min( , ) ( 1/ )A N N ss A
sc
c c
CQ xG x Q L L L xTx
N
Casalderrey-Salana, XNW’07
Strong coupling SYM: 2( , ) 0, /N sxG x Q x x T Q Hatta, Iancu & Mueller’08
3ˆ /q T E MGubser 07, Casaderrey-Salana & Teaney’07
q̂̂q
2sQ
DGLAP
DGLAP withfixed s:
2c2N3
ˆ , 12
sR
c
C T Eq
N
Measuring qhat
Direct measurement:
or modified fragmentation function
Measuring parton energy loss
GW:Gyulassy & XNW’04BDMPS’96LCPI:Zakharov’96GLV: Gyulassy, Levai & Vitev’01ASW: Wiedemann’00HT: Guo & XNW’00AMY: Arnold, Moore & Yaffe’03
q
Apxp
xp
Ap
q
x1p+kT
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