Workshop on Structure of hadrons and nuclei at an Electron Ion Collider, Trento, July 13-18, 2008 Xin-Nian Wang Lawrence Berkeley National Laboratory Jet transport and gluon saturation in medium
Feb 02, 2016
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