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