Osaka,2008QCD and Heavy-ion Collisions

• Introduction:– QCD and high-energy heavy-ion collisions

• Fragmentation, parton distribution functions and factorization.– A primer on pQCD

• Initial Conditions in Heavy-ion Collisions– Mini-jet production, multiple scattering in nuclei,

parton distribution in nuclei-parton shadowing and saturation.

• Hard Probes of Dense Matter– Final state scattering (in QGP), parton energy loss, jet

quenching as probes of dense matter in heavy-ion collisions.

Osaka,2008Introduction

QCD and Heavy-ion Collisions

Osaka,2008QCD Theory

• For strong interaction

– SU(3) gauge symmetry (non-Abelian)• Confinement at long distance

• Asymptotic freedom at short distance

– (approx.)Chiral symmetry and its spontaneous breaking• Goldstone boson and chiral condensate

– Scale and UA(1) anomaly

– …

,1

1( )

2 4

fna

QCD a a af a

L i gA m F F

Osaka,2008Lattice QCD results

MeVTc 8170 3/ 3.06.0 fmGeVc

F. Karsch ‘2001

Osaka,2008Confinement-deconfinement

Karsch, Laermann and Peikert 2001

( ) + V r rr

SU(3) non-Abelian gauge interaction confinement

Heavy quark potential:

Osaka,2008Heavy-ion Collisions

RHIC BNL

Au+Au up to 200 GeV/n

Osaka,2008What is “QGP” in AA Collision?

• Criteria:– High density: >>c

– Large volume: V>> (mean-free-path)

– Long life-time: t>>– Local thermal equilibration (interaction)

• Deconfinement – Debye screening of strong interaction: J/ suppression

• High-density:

– Initial condition

– Probes of dense matter: propagation of high-energy partons in medium

Expanding, short-lived, small volume QGP?

Osaka,2008Asymptotic freedom

S Bethke J.Phys. G26 (2000) R27

22 2 23

4( )

(11 ) ln( / )sf QCD

Qn Q

Gross,Wilczek;Politzer (73)

Non-abelian interaction

Anti-screening of color

Asymptotic freedom

3

2

2(11 )

3 16f

g gQ n

Q

Osaka,2008

Jet production in high-energy collisions

2p p jets X e+e- 3 jets

Osaka,2008pQCD Works!

p-p PRL 91 (2003) 241803

Good agreementwith pQCD

Osaka,2008Hard Probes

• Hard processes: occur early in the stage t~1/Q

• Production rate: Calculable in pQCD

– Heavy quark

– Charmonium: breakup deconfinement

– Thermal photon and dilepton

– Large pT jets

• Jet Tomography of dense matter:

jet

density

geometry

Osaka,2008

Fragmentation, parton distribution and factorization

-A primer on perturbative QCD

• Hard Processes in QCD: Factorization– e+e- hadrons: fragmentation function– DIS: parton distribution function

1 2[ ] ( ) ( ) ( )a b ab cd c h hd d x f x f x d D z

Osaka,2008Feynman rules

Quark propagator

Gluon propagator

ij

i

k i

2

( )ab

id k

k i

Quark-gluon vertexa

iji T

ki j

3-luon vertex

2 3

3 1

1 2

( )

( )

( )

abc

bca

cab

f k k g

f k k g

f k k g

i

aj

a

c

b

k1k2

k3

a bk

4-gluon vertex

Osaka,2008Optical Theorem and cuts

ki

2

3

30

1([ ]) *([ ])

2 (2 ) 2

1 Im ( , 0)

2

itot i i

i i

d kT k T k

s k

A s ts

A(s,t) – forward scattering amplitude

Cut quark line

Cut gluon line

k 22 ( )k k

2( )2 ( )d k k

imk

kd

mkkd

k

kd

ii

i

iii

i

i

224

4

224

4

03

3

1Im

)2(

)(2)2(2)2(

=T T*

Osaka,2008 e+e- hadrons

4

4

1( , ) ( )

2e e htot

eL p q W q

s q

1 2 1 2 2 1 1 2

leptonic tensor(LO in )

1 ( , ) = Tr = +

4

em

L p q p p p p p p g p p

hadronic tensor

0)0(),(0Im

)()2(0)0()0(0)(

4

44

JyJTyedi

pqJXXJqW

yiq

XX

)(qW

q q

( ) ( ) ( ) Hadronic EM currentq q qq

J y e y y

Osaka,2008e+e- hadrons: Total cross section

e e hadrons e e partonstot tot

)(qW

q q

Partons and hadrons each constituent a complete set of states

4 4

4

( ) 0 (0) (0) 0 (2 ) ( )

Im 0 ( ), (0) 0

XX

iq y

W q J X X J q p

i d ye T J y J

Osaka,2008LO and NLO total cross section

q q

( )

2

2

42 2 2

4

4

4

2

4

4

1( , )

2

1( , )

2

( )

2 ( ) 2 Tr ( ) (2 )

4

3

e e LOtot

e

c qq

c qq

m

W q

d kN e q k

eL p q

s q

eL p q

s qk

q

k q k

N e

0

2( ) 2 2

2

4 11 ( )

3e e NLO emtot c q s

q

N e qq

IR and collinear safeFields: Application of Perturbative QCD

q q

k

q-k

Osaka,2008Field Theory Basics

2( ) † ( )

3( ) ( ) ( ) ( ) ( )

(2 ) 2ik y ik y

q

dk d ky b k u k e d k v k e

k

2† ( ) ( )

3( ) ( ) ( ) ( ) ( )

(2 ) 2ik y ik y

q

dk d ky b k u k e d k v k e

k

( ) ( ) ( ) ( )( ) ( ) ( ) ( )u k u k v k v k k

2†

3( ) ( ) ( ) ( )

(2 ) 2a a ik y ik ydk d k

A y t k a k e a k ek

†( ) ( ) ( ) polarization tensork k d k

( )n k n k

d k gn k

+in axial gauge: 0, n=[0 ,1,0 ]n A

Osaka,2008Single inclusive hadron distribution

4( ) 0 ( ) (0) 0iq y

X

W q d ye J y X JX

3 4

2,3 4

2ˆ( ) ( , )(2 )

( ) 2 (2 (2 )

)hc q q q h

q h

d p d kW q N e T q k qp kr T k

E

(LO, in pQ D, C )hX q k p S q

ph

Sk

q-k

( ) ( ) ( ) Hadronic EM currentq q qq

J y e y y

, ,hS p q k( ) ( )q qy y (0) (0)q q , , hq k p S

ˆ ( , )H k q

Osaka,2008Collinear Approximation

0)(,,)0(0),(ˆ 4 ySpSpyedpkT qhhS

qyik

hq

0)(,,)0(0),(ˆ 4 ySpSpyedpkT qhhS

qyik

hq

/ˆ ˆ ˆ( , ) ( / , ) ( / ) ( , ) |

h hh h h h k k p zH k q H p z q k p z H k q

Osaka,2008Collinear Approximation

0)(,,)0(0),(ˆ 4 ySpSpyedpkT qhhS

qyik

hq

4

( ) ( )4ˆ ( , ) ( ) ( ) (higher twist)

(2 ) 2hh

h q q h h h q q h

ppd kdz T k p z dz T z

k

)(insert

k

pzdz h

hh

4

( ) ( )4

1 ˆ( ) ( ) ( , )(2 ) 2

hq q h h q q h

h

pd kT z z Tr T k p

k p

/2

10 (0) , , ( ) 0

2 2h hip y z

q h h qSh

dye Tr p S p S y

z

Osaka,2008

3

2 2( ) 3

1( ) ( ) 2 ( ) ( )

(2 ) 2 2h

c q h q q h hq h

d pW q N e dz T z q k Tr p q k

E

3 4

2 2,3 4

ˆ( ) 2 ( ) ( , ) ( ) (2 ) 2 (2 )

hc q q q h

q h

d p d kW q N e q k Tr T k p q k

E

4

( ) ( )4

/

ˆ ( , ) ( ) ( ) (higher

ˆ ˆ ˆ( , ) ( / , )

twist)(2 ) 2

( / ) ( , ) |h hh

hhh q q

h h h k

h h q

p

h

k z

h q

H k q H p z q k p z H

ppd kdz T k p z dz z

k

k

T

q

3 42 2 2

, 4( ) ( ) 2 ( )2 ( ) ( )

2 (2 )h

c q h q q hq

z d kW q N e dz T z k q k Tr k q k

4 42 2 2

4 4( ) ( )

(2 ) (2 )h

h h

d p d kp z k

( )h hp z k

)2

()(22

2

q

pqz

q

zkq h

hh

22 h

hq pz

q

Collinear Approximation

Osaka,2008Fragmentation function

4

4

34 42 2 2

,4 4

1( , ) ( )

2

( ) 2 ( )2 ( ) ( ) ( , )2 2 (2 )

e e htot

c hq h q q h

q

eL p q W q

s q

N ze d ke dz T z k q k Tr k q k L p q

s q

( / )h hk p z

0 ( ) ( )e e

qq h h q h h

qh

dD z D z

dz

2

20 2

4

3q em

q ce Nq

)(2

)( )(

3

)( hqqh

hhqq zTz

zD 22 h

hq pz

q

4 4

2 2 20 4 4

2 ( ) 2 Tr ( ) ( , )2 (2 )

cq

q

N e d ke q k k k q k L p q

s q

Osaka,2008Factorization

q S

diagrammatically

Time-like cut-vertex

=

q-ph/zh

q1

2 hp

2 2( ) 2 [( ) ]2

hq

q

pH e Tr q k q k

k

pz

ph

hh

2

ph phS

kk

)(2 )(

3

hqqh zT

z

Hard subprocess

( , )hH q p

Osaka,2008Cut-vertex for quark frag. function

3 44

4

/

( ) ( ) 0 (0) , , ( ) 02 (2 ) 2

0 (0) , , ( ) 02 2 2

h h

ik yh hq h h h q h h q

S h

ip y zhq h h q

S

z pd yD z d ke z Tr p S p S y

k p

z dye Tr p S p S y

Quark fragmentation function

Osaka,2008Gauge Invariance & Higher Twist

0

34

)

4

(4

( ) ( ) 0 (0) , , ( ) 02 (2 ) 2

y

ik yh hq h h h

ig dz n

q h

A

qS

z

hh

z pd yD z d ke z Tr p S p

pe S y

k

q

ph

Sk

q-k

Soft (collinear) gluons

Physical (transverse) gluons(+higher twist in the colllinear expansion)

2

10 (0) , , ( ) 0q h h q

S

Tr D p S p S D yQ

Osaka,2008Radiative Corrections

q-ph/zh

q

1

2 hp

k

pz

ph

hh

2

ph phS

kk

)(2 )(

3

hqqh zT

z

=

k

pz

ph

hh

2

ph phS

kk

)(2 )(

3

hqqh zT

z

q-ph/zh

q

1

2 hp

+

phq S q

Osaka,2008Radiative Corrections (ii)

32 2

( )

1 1ˆ ( , ) ( )2 (( ) ) ( )2 2

h hR F h

h

z pH l k C g d k l k l z

k p k k

1

2 hp

1

4( )( ) ( ) ( ) 4

/

ˆ( ) ( )Tr ( , )2 (2 )

h h

hrq q h h h q q h R

l p z

p d kdD z dz T z H l k

=

2h

hh

pz

p l

ph phS

ll

ph ph

S

kk

l k-l

k

l k-l

k

pz

ph

hh

2

3

2hz

1

12 2( )( ) ( )2

1( ) ( / )

2 1h

R sq q h h q q h h F

z

dk dz zdD z D z z C

k z z

/ /h hz z z l k splitting function ( )q qgP z k l

k-l

Osaka,2008Gluon fragmentation function

2 44

4

2/

( ) ( ) ( ) 0 (0) , , ( ) 02 (2 )

0 (0) , , ( ) 02 2

h h

ik yh hg h h h h h

S

ip y zhh h

Sh

z pd yD z d k e z d k A p S p S A y

k

z dye F p S p S F y

p

44

4( ) ( ) 0 (0) , , ( ) 0

(2 )ik y h

h h h hS

pd kT z d y e z A p S p S A y

k

( ) ( ) (higher twist)h h g h hT z z d T z

2

( ) ( ) ( )2h

g h h h

zD z d k T z

Osaka,2008Radiative correction (iii)

2

12 2( )( ) 2

1 (1 )( ) ( / )

2h

R sq q h h g h h F

z

dk dz zdD z D z z C

k z z

( )q gqP z

=

2

( )2h h

h

z pd l z

l

phph

S

ll

ph ph

S

kk

l k-l k

l k-l

k

pz

ph

hh

2

3

2hz

( )d l

Osaka,2008Virtual correction

(v) 2ˆ ˆ( ) ( )

h

iH l l

lp

(v)( ) ( ) (v)

/

31 ˆ( ) ( ) Tr ( )2

( )2h h

q q h h h q q h hh h

hl p z

z pzdD z dz H

lT z p l

ˆ ˆ( ) Im ( )i

l ll l

k

l l-k

1

2 hp

l=

2h

hh

pz

p l

ph phS

ll

ph phS

kl

2h

hh

pz

p l

3

2hz

ll

1ˆ ( ) ( ) (leading log approximation)l il l

Osaka,2008Virtual correction (ii)

kl

l-kl

42

4 2

1 ( )ˆ ( ) (2 )F

d k d kl C g

l k k

42 2

4

1 1ˆIm ( ) ( )2 ( )(2 )F

d kl C g d k k

l l k l k

k l k 4

(v) 2( 4

2) ( ) ( ) Tr

(2 ) 2

1 1( )2 (( ) )

2h

q q h h hh

F d k l k lk p k

pd kdD z D z C g

12 2

( ) 20

1( )

2 1s

q q h h F

dk zD z C dz

k z

Similar to real correction

32

(2

)

1 1( )2 (( )ˆ ( , ) ( )

2)

2h h

R F hh

z pH l k d k l k l

k p kC g z

k

(v)( ) ( )

1 1 ˆ( ) ( ) Tr Im ( )2 2q q h h q q h h h

h

dD z D z p lp l

Osaka,2008Real + virtual correction

1 12 2 2

( ) ( )20

1 1( / ) ( )

2 1 1h

sF q q h h q q h h

z

dk dz z zC D z z D z dz

k z z z

1( ) (v)( ) ( ) ( )( ) ( ) ( )r

q q h h q q h h q q h hdD z dD z dD z

)1(2

3

)1(

1)(

2

zz

zCzP Fqgq

12

( )2( / ) ( )

2h

sq q h h q qg

z

dk dzD z z P z

k z

1 1

0 0

( ) ( ) (1)

(1 ) 1

F z F z Fdz dz

z z

+ function

Infrared finite

collinear divergent

Osaka,2008DGLAP Evolution Equation

2

20

12

2( ) ( ) ( )

2h

S h hq h h q qg q h q gq g h

z

z zdk dzD z P z D P z D

k z z z

2

20

2 2

2 20

ln2 2

S Sdk

k

Resummation and DGLAP Evolution Eq.

2 12 2

2

( , )( ) , ( ) ,

ln 2h

q h S h hq qg q h q gq g h

z

D z z zdzP z D P z D

z z z

LLA

2 2 20( , ) ( , ) ( , )q h q h q h hD z D z D z

Osaka,2008Coupled DGLAP evolution equations

2 12 2

2,

( , )( ) , ( ) ,

ln 2h

g h S h hg gg g h g qq q h

q qz

D z z zdzP z D P z D

z z z

phph

S

ll

phph

S

ll

ph ph

S

kk

lk-l k-l

ph ph

S

kk

l

2 12 2

2

( , )( ) , ( ) ,

ln 2h

q h S h hq qg q h q gq g h

z

D z z zdzP z D P z D

z z z

1 1 2( ) 6 (1 ) 11 (1 )

(1 ) 12 3g gg f

z zP z z z n z

z z

2 21( ) (1 )

2g qqP z z z

Osaka,2008Exercise : I

1),( 21

0

QzDzdz hah

12

0

( , ) 0a hh

dz zdD z Q

General proof:

Perturabtive:

From operator definition: ( ) ( 1)a aD z z

2

3,

ˆ , ( ) , ( )(2 ) 2X h

dk d kk X h k X h k

k

Hints:

Osaka,2008Deep inelastic scattering (DIS)

1 22 2 2

14

DISq q p q p qW g F p q p q F

q p q q q

34

4 3

1( , ) ( , )

2 (2 ) 2e

DIS DISe

d Led L p q W q p

s q E

pyJJpyedpqW yiqDIS )(),0( ),( 4

qp

L’e

1 2, nucleon structure functionsF F

EM current conservation: ( , ) 0DISq W q p

222 2 2 2

2 12 4( , ) cos 2 ( , )sin

4 sin ( / 2) 2 2N

N

mdF x Q F x Q

d dE m E p q

Osaka,2008Collinear approximation (LO)

pyJJpyedpqW yiqDIS )(),0( ),( 4

q

pk

q+k

)(insert

p

kxdx

4

4

1ˆ ( , ) ( ) ( ) (higher twist)(2 ) 2

d k kk p x q x p

p

4

2 24

ˆ2 ( ) ( , ) ( ) (2 )q

q

d ke q k Tr k p q k

4ˆ ( , ) (0) ( )ik yk p d ye p y p

2 2 1( , ) ( )2 ( ) ( )

2DIS qq

W q p e dx q x q k Tr p q k

k xp

Osaka,2008Collinear factorization

2 2 12 ( ) ( )

2( , ) ( )DIS q q

q

W q p e dx f q k kx Tr p q

k xp

44

4( ) ( ) (0) ( )

(2 ) 2ik yd k k

q x d ye x p y pp p

q

pk

q+k

p

k

p

kx

p

2q

k

q+k

1

2p

=

Hard part: H(q,k) Space-time cut-vertex

2 2( , ) ( ) 2 ( )2q

q

pH q p e Tr q xp q xp

Osaka,2008

1( ) (0) ( )

2 2ixp y

B

dyq x e p y p

quark distr.

function

Quark Distributions

22212

14 Fq

q

qppq

q

qpp

qpF

q

qqgWDIS

2( ) 2 ( )Bq xp p q x x qp

qxB

2

2

2 2 1( , ) ( )2 ( ) ( )

2DIS q qq

W q p e dx f x q xp Tr p q xp

21

1 ( )

2 q Bq

F e q x 22 ( )B q B

q

F x e q x

Osaka,2008Exercise: II

44

4( ) ( ) (0) ( )

(2 ) 2ik yd k k

q x d ye x p y pp p

single quark statep ( ) ( 1)q x x

2 2 2

( 1) 1( 1)

2B

B B

q q x p q p qg p q p q x x

q p q q q

1)

q

p

L

e

L’

e

p’Electron-quark scattering

34

4 3

1( , ) ( , )

2 (2 ) 2e

DIS qe

d Led L p q W q p

s q E

2( , ) 1 1Tr ( ) 2 ( )

4 4 2qW q p

p q p p

2)

qp

qxB

2

2

Osaka,2008Radiative Correction

2

kx

p p

2 2( )

1 1ˆ ( , ) ( )2 (( ) ) ( )2R FH l k C g d k l k l

kx

k kp p

2

ly

p p

4

( ) 4ˆ( ) ( )Tr ( , )

2 (2 )R

l yp

p d kdq x dy q y H l k

=p p

l

S

l

2 h

kx

p p

1

2p

k

l

k-l

p p

kk

l k-lS

12( 1)

2( ) ( ) ( )

2R s

q qg

x

dk dy xdq x q y P

k y y

21( )

1q qg F

zP z C

z

Osaka,2008Gluon distribution

44

4( ) ( ) ( ) (0) ( )

(2 )ik y a

a

d k k kG x d ye d k x p A A y p

p p

4

4( , ) ( ) ( ) ( ) (higher twist)

(2 ) 2

d k k pT k p x G x d k

p k

4( , ) (0) ( )ik y aaT k p d ye p A A y p

1( ) (0) ( )

2ixp y a

a

dyG x e p F F y p

xp

(partial integral)

p

k ( )k

xd k xp

Osaka,2008Radiative correction (ii)

12( 2)

2( ) ( ) ( )

2R s

g qq

x

dk dy xdq x G y P

k y y

2 21

( ) (1 )2g qqP z z z

=

p p

k

lk-l

S

( )l

yd l yp

p pS

l lkk

lk-l

l

( )2 2h

k x kx

p p l y l

( )

2

d l

2 2( )

1 1 1( , ) Tr ( ) 2 (( ) )

2( )

22R

dH l k g l

x k

l y lk k l

k k

Osaka,2008Virtual correction

(v)ˆ ˆ( ) ( )

2H l

p

il

l

(v)(v)

ˆ( ) ( )Tr ( ) ( )2

l xp

p ldq x dy q y H l x

p

ˆ ˆ( ) Im ( )i

l ll l

2

ly

p p

2

lx

p p

k

l

l-k

1

2p

l=

pp

kl

S

l

l-kp p

l

S

l

12(v)

20

( ) ( ) ( )2

sq qg

dkdq x q x dzP z

k

Osaka,2008Real + virtual correction

1 1 12 2 2 2

20

1 ( / ) 1 1 (1 / )( ) ( ) ( )

2 1 ( / ) 1 ( / )s

F

x x

dk dy x y z dy x yC q y q x dz G y

k y x y z y x y

( 1) ( 2) ( )( ) ( ) ( ) ( )R R Vdq x dq x dq x dq x

)1(2

3

)1(

1)(

2

zz

zCzP Fqgq

1 12

2( ) ( ) ( ) ( )

2s

q qg g qq

x x

dk dy x dy xq y P G y P

k y y y y

2 21( ) (1 )

2g qqP z z z

Osaka,2008

DGLAP evolution for parton distributions

k-l

pp

k

l S

p p

k

l Sk-l

p p

ll

S

p p

ll

S

12

2 22

,

( , )( ) , ( ) ,

ln 2S

g gg q gqq qx

G x dy x xP G y P q y

y y y

12

2 22

( , )( ) , ( ) ,

ln 2S

q qg g qq

x

q x dy x xP q y P G y

y y y

1 1 2( ) 6 (1 ) 11 (1 )

(1 ) 12 3g gg f

z zP z z z n z

z z

21 (1 )

( )q gq F

zP z C

z

Osaka,2008Exercise : III

12 2 2

0

( , ) ( , ) ( , ) 0q

dx dq x Q dq x Q dG x Q x

General proof:

Perturabtive:

12 2 2

0

( , ) ( , ) ( , ) 1q

dx q x Q q x Q G x Q x

Osaka,2008Factorization in semi-inclusive DIS

Semi-inclusive DIS cross section

21( ,

2)

1) (

)(B h

q B q h hqh

q xdF x z

ed

D zz

q+k

ph

q

k

q+k

Hard process

Fragmentation func.

p

k

Parton distribution

q+k

q

pk

ph

)2 ( 21( ) 2 (( ) )(

2) )(

h h

q hDI

h

p z q k

k x

S

pq hq

W e dxq Tr p qx k xz pdz D q

Osaka,2008Initial Conditions in AA

• Semi-hard processes in pp collisions:

– Perturbative vs non-perturbative

– Eikonal formalism: unitarization

• Hard scattering in p+A and AA collisions:

– Multiple scattering

– Nuclear shadowing, parton saturation

-- Initial (semi-hard) particle production

Osaka,2008Jet production in pp Collisions

p+pjet +X

xa

xb

Defining a jet222 )()( Ryy cc

)ˆˆˆ(ˆˆ

ˆˆˆ

),(),( 2/

2/3

3

utstd

d

td

dsxfxfdxdx

pd

dE

cdeabcdab

bpbapaabcde

ba

2S 3

S

)ˆˆˆ(),(

ˆˆˆ

),(),( 22

2/

2/3

3

utszDz

s

td

dxfxfdxdx

pd

dE hhc

h

cdab

bpbapaabcd

bah

h

Inclusive hadron cross section

Osaka,2008Jet and high pT hadron production

Soper et al

Zhang, Owen, Wang & XNW’07

Osaka,2008Mini-jet production

Minijet? ET~ few GeV

Osaka,2008Hard versus soft

• pQCD fails at small momentum scale– s large, factorization breaks down, higher-twists important …

• How to incorporate mini-jets into pp total cross section?

T

s

p

Tjet pd

dpdp

2

2/2

0

0

)( 0for diverges

)(

),(0

0 ps

spN

ppin

jetjet

factorize

“Pomeron”

Osaka,2008Eikonal Approximaton

),(1

2),(1

qsA

si

itsf

Elastic amplitude q 2 qt

)()2)(,(1

),(1

)2()2(2!2

1),( 21

)2(2212

22

21

2

2

qqqqsA

siqsA

si

qdqditsf

biqebsbdi

),(

!2

1

222

),( )2(

),(2

2

qsAe

qd

s

ibs biq

1q 1qq

1/2 sq Neglect phase shift coherent

Eikonal function

Osaka,2008Unitarization

biqbs

nn eebd

itsftsf 1

2),(),( ),(2

biqnn ebsbd

n

itsf

),(

!

1

2),( 2

1q …

12)0,(Im4 ),(2 bstot ebdtsf

),( tsfdt

d el

22 ( , )1 s bel d b e

tbJdbbed biq 0

22

)(2

00 bbb

tbJtbJdt 1 ),(22 bseltotin ebd

Osaka,2008Two-component model

Additive quark model2

),()0,(),( sqsaqsA

2),(

4

1 qsa

sdt

d q

)0,(Im2

sas

q

),(2

1),(

)2(2

1),(

2

2

2

bsTsqeqd

bs Nqbiq

q

1),( 2 bsTbd N

nucleon overlap function

0)0,(Re assume sa

q-wave function

),()(2

1),( 0 bsTpbs Nsoftjet softjetq p )( 0

Fixing p0 by expt data )( ),(),(

0 sb

bbs

175.0tot

el

Geometrical scalingat low energies

Osaka,2008Onset of mini-jet production

(XNW&Gyulassy’91)jethardsoft

nnndy

dN

in

jet

jet

n

Osaka,2008Multiple Mini-jet production

)(2

!

)]([ bNNTjetn

NNjetn e

n

bTbd

njetTnTnT fpfpp )1(

0

fn fraction of n from minijets

Osaka,2008Inclusive spectra

XNW&Gyulassy’91

Osaka,2008pA and AA Collisions

• Multiple parton scattering: coherence

• Nuclear modification of parton distribution function

• Initial parton production in AA

q

Glauber-Gribov Multiple Scattering

clE

qE

1

2

2

3/10length coherence ArRl Ac

Multiple scattering coherent different A-dependence

Osaka,2008Nuclear wave function

Ai

iAAA

A

i

i kkkPkkkd

A 1)3(3

11

3

3

)2)(()2(

i

ii xki

AA

A

i

iAA ekk

kdxx )(

)2()( 1

1

13

3

11

2

1111 )()( AAA xxxx 1)( 11

1

1

3

A

A

ii xxxd

)()()()( 111

1

1

31 nA

A

niinn xxxxxdxx

( ) ( , )t b dz z b Nuclear thickness function

Osaka,2008Multiple Scattering

qPPMn

AAhhA AAn

A

n

)3(3

1

)2(

iin

hnnn

xqi

i

n

i

in qqqqfxxexd

qdM

n

iii

)()2)(()( )2(

)3(311

3

13

31

1q nq

1

12111 )()()()()(),(

n

inin

hn qfqpDqfqpDqfqqf

1q

p p+q1

131 1 1( ) ( )iqxM d x e x f q

24 (0)( )

4iq bE f

d be t bi iE

( ) : peaked at 0, 0zf q q q

Osaka,2008Example: n=2

1q 2q

p+q1p p+q

ii

hxqi

qqqqfxxexdxdqdqd

M iii

)()2)(,(),( )2()2(

)3(32122122

31

33

23

31

3

2

2

1

),()()( )2( 21221

)(2

31

33

13

2112 qqfxxexdxdqd hxxiqiqx

)()(

1)(),( 222

11212 qf

imqpqfqqf h

zqizziq

zz

z

eE

i

imqp

edq

4)( 22

1

)( 12

E

qEqEqz 2

212

12

22

/ 2( )22

( , ) ( ) ( ) (0) (2 )

z Aq Ri qz iq x bA

d qd xdz z x f q e t b f e

assume: ( ) Gaussian and ( ) : peaked at 0x f q q

Osaka,2008Incoherent scattering

2

2/22

2

)(4

)0(

4

Az Rqbiq ebt

iE

febd

i

EM

n

i

Rqbiqn

AziebtiE

febd

i

EM

1

2/2 2

)(4

)0(

4 E

qqq i

iz 2

211

For large angle (hard) scattering 12

2

AAiz R

E

qRq

2 ( )NA N NA d b t b A

Incoherent scattering

NNfE

)0(Im4

1

11

A

nn

AM M AM

n

21

2z

qq

E

Osaka,2008Coherent scattering

1 Aiz Rq Coherent scattering

NNfE

)0(Im4

1

)()( bAtbTA

AN

biqn

A

n

btebdiEMn

AM )(11 4 2

1

)(2 1 4 bTbiq ANeebdiE

)(2 1 bTNA

ANeebd Glauber-Gribov

n

i

Rqbiqn

AziebtiE

febd

i

EM

1

2/2 2

)(4

)0(

4 E

qqq i

iz 2

211

Re (0) 0f

Osaka,2008Nuclear shadowing

*2 2

2 ( , ) ( , )AB qqA BF x Q x Q

Coherent length:21 1

2 2 2 2Nz

Bc

mq Qx

l

Coherent scattering Ac Rl AN

B Rmx

2

)()1()1( 222)/( 2

btiAAbdeA NqqlR

NqqAqqcA

Nqq

Nqq

f

f

Im

Re

Anti-shadowingAN

c Rmx

2

)(

)(

2

2

xAF

xFN

A

1

x

shadowing

anti-shadowing

Osaka,2008Parameterization of Shadowing

Gluon shadowing

Osaka,2008Particle Production in AA

• Soft processes: coherent interaction

– Cross section ~ surface of nuclei

– Particle production ~ number of participants

• Hard processes: Incoherent interaction

– Cross section ~ volume of nuclei

– Particle production ~ number of binary collisions

– Parton distribution nuclear shadowing

)()( bTbN ABNNbinary

)()(2 1)(1)()( sTB

bsTApart

BNNBNN ebsTesTsdbN

b

Osaka,2008Energy dependence

in

jet

hardsoft

nndy

dN

ppin

ppjet

hardBinarysoftpart

AAch nNnN

d

dN

5.0

AA collisions

4.7 (b=0)0.5

Binary

part

N

N

pp collisions

ppin

ppjet

hardpart

Binary

soft

AAch

part

nN

Nn

d

dN

N

22

shadowing

Li & XNW’01

nsoft

Osaka,2008Centrality dependence

x

z

y

EZDC

ET

CentralityNpart Li & XNW’01

Osaka,2008Jet tomography of dense matter

• Studying structure of matter:– Electromagnetic (Weak) Probe:

• Quark distribution, Spin, CP violation etc

– Hadronic (Strong) Probe:• Density, Temperature, Confinement

• Interaction of a fast quark with matter:– Difference between cold & hot, confined

& deconfined matter

– How strong is the interaction?

– Thermalization of the dense matter Strong interactionleads to parton energy loss

Leading hadrons

e-

Nucleus

QGP

Osaka,2008Calibration of parton beams

• Luminosity and initial energy

– Calibration by experiments:• DIS: quark initial energy=energy transfer from lepton

• +jet: Ejet=Emomentum conservation

– Calibration by pQCD• Initial jet production rates by pQCD

• Calibrated by pp collisions

• Need to understand nuclear effect on initial production

• Parton energy loss

– Depends on density of the medium

– Size and geometry of system

– Dynamic evolution: expansion

Osaka,2008A Brief Overview of parton energy loss

• Elastic energy loss: – Bjorken ( 82), Thoma and Gyulassy (91), Mrowczynski (91), Koike and Matsui (92),…

el22el E

dq

ddq

dx

dE

k

qE

2

2

el

E E’

kq qkE ,

22

2s

2 2

3ln

2

3

ET

TC

dEel/dx ~ 0.1 - 0.2 GeV/fm T=300 MeV, s=0.3

2

21

dq

ddq

1)/exp(

][

)2()(

03

3

Tk

kdk

Osaka,2008

• Radiative energy loss:– Brodsky and Hoyer (93):

– Gyulassy and XNW (94,95)

– Baier, Dokshitzer, Mueller, Peigne Schiff (BDMPS)(95,97)

– Zakharov(97), Levin (96),

– Thin plasma: Gyulassy, Vitev, Wiedmann’2000

– Modified fragmentation: Huang, Guo, XNW, 96, 2000

– The saga continues …

LpC

dx

dET

sA 2

8

)2

ln(222

Er

qC

dx

dET

sA

Overview: cont’d

2g/k2T< dE/dx < 1/2< k2

T>

Osaka,2008

QED: A Classical Case

• Classical radiation of a point charge • (Jackson, p671)

• Bethe-Heitler (1934) radiation spectrum (for a relativistic particle)

• Lorentz Invariant form:

2)(

2

222

)(4

rktiedrnn

e

dd

Id

0 t

0 t)(

tE

ptv

tEp

tvtr

ff

ii

2

2

22

4

f

f

i

i

vk

vk

vk

vke

dd

Id

2

3

2

3

3

)()()2(2

kJke

kd

Id

f

f

i

i

pk

p

pk

pkJ

)(

1

)1(2ln2

2

2

m

vvE

d

dI fi

EM current of charged particle through a sudden acceleration

Osaka,2008

LPM Effect: A New Twist

• Induced Radiation via multiple scattering

• Factorization Limit:

• Bethe Heitler Limit(large relative phases)

i

j

ixik

i i

i

i

i epk

p

pk

pkJ

)(1

1

)(kJ i

,i ij',

)('

2

,3

3

))((Re2

ji xxikji

ii eJJJ

kd

Id

BH

2 1

d

dI

dzd

Id

1~)( ji xxike N

N

pk

p

pk

pkJ

1

1)( Like a single scattering

Osaka,2008

,i ij',

)('

2

,3

3

))((Re2

ji xxikji

ii eJJJ

kd

Id

)(/)( kzxxk m

j

immji

22

22)(

mmm uk

k

Formation time

E

quu

ku m

mmm

m

1 ,

Random walk ijNE

qNxxk ji

;4

)(2

2

2

2

2

coh 2

q

ENN

2

2

cohBH

2

)(

11

E

q

Nd

dI

dzd

Id

Nco

h

Nco

h

Nco

h

Landau-Pomeranchuk-Midgal Spectrum (1953)

Osaka,2008SLAC Expt:Confirmation

Anthony, et al.Phys. Rev. D56(97)1373

2

2

coh 2

q

EN

2EBH fact BH

• SLAC E146 Experiment: 40+ years after LPM

Osaka,2008Coherence: Feynman’s Interpretation

pi pf

k

x1 x2

pi pf

k

21)1( xik

f

fxik

i

iD e

pk

pe

pk

pR

12)2( xikxikD ee

pk

pR

pi pf

kp

kpi pf

p

21 xik

f

fxik

i

iD e

pk

p

pk

pe

pk

p

pk

pR

1)()( 12 kxxxk

22

sD RR

pi pf

k

pi pf

k

pi pf

k

x1 x2

Factorization limit

Osaka,2008

Radiation in QCD: Colors Make the Difference

QED: )1(0s

Opk

p

pk

pR

f

f

i

iS

QCD: gluons carry color: interference incomplete

dy

dN

y0

accaS TTTTk

kR

2

)1( 2

caS TTkq

kqR ,

)(

)(22

)2(

pi pf

ka

c

22

2

22 )(

kqk

qN

kdyd

dnscg

pf

pi

dy

dN

y0pfpi

pi pf

k

k

Osaka,2008LPM in QCD:Remembering History

Energy loss/scattering (Gyulassy and XNW’94)

))(( QCD22 k

dykd

dndykdE

2

qNE cs

More Complete QCD Analysis (Baier,Dokshitzer,Mueller,Peigne,Schiff ‘96)

2q LQ 22

Accumulated qT transfer

LN

dx

dE cs

2

8

QCD multiple scattering

For a fast quark

2

ˆ jet transport coefficientq

Osaka,2008

Phenomenology:How to measure dE/dx in QCD?

• How to measure dE/dx ?– Modification of fragmentation function

hadrons

ph

parton

E

),,()(0 EzDzD ahah

)(0 zD ahare measured, and its QCD evolution

tested in e+e-, ep and pp collisions

Dh/a(z)=dN/dz (z=ph/E)

Suppression of leading particles (Huang,

XNW’96)

Fragmentation Function

Osaka,2008Modified Jet Fragmentation

• Generalized factorization at higher twist

• Multiple parton scattering as higher twist contribution

• Multiple parton scattering and transverse momentum broadening

• Gluon bremsstrahlung induced by multiple parton scattering

• Modified fragmentation functions

Osaka,2008

Osaka,2008Kinematics in DIS

34

4 3

1( , ) ( , )

2 (2 ) 2e

DIS DISe

d Led L p q W q p

s q E

q

p

L’e

2

, ,0 , ,02 T B T

Qq q x p q

q

1,0,0]n ,0 ,0Tp p 0,1,0n n A A

is a nuclear state with each nucleon carrying momentum A p

q

xp

q+xpq

xp[0, ,0]q xp q

Osaka,2008

Multiple scattering and twist expansion

Generalized factorization:Luo,Qiu & Sterman (94)

AAAAkqpHkdW TD

TD

),,(2

22 )0,,(

)0,,()0,,(),,(

TTD

k

TTD

kTD

TD

kkqpH

kkqpHkqpHkqpH

T

T

Collinear expansion:

First term Eikonal

(0) exp ( )A ig dzA z A

AFFAkqpHW TD

kD

T

)0,,(2

Double scattering

Osaka,2008Leading order

4(0) 2 (0)

4(0) ˆTr ( , )

(2, )

)ˆ (q

q

Hd k

W e k q k p

q

p

k

q+k

(0) 4ˆ ( , ) (0) ( )iy kk p d ye A y A (0) 2ˆ ( , ) ( ) 2 ( )H k q q k q k

(0) (0) (0)ˆ ˆ ˆ( , ) ( , ) ( , )k xp

H k q H xp q k H k qk

collinear expansion

( )k k xp g n n

Osaka,2008

(0 0(0) )) (2 ˆ T ˆ (( )r ),qq

W e xx H xp qd (0)2 (0)ˆ ( , )ˆ (r )T

k xq

q p

H k qk

xe dx

(0)ˆ ( ) (0) ( )2

ixp yp dyx e A y A

(0)ˆ ( ) (2

) )((0)ixy

p yp dyx e iA y A

Osaka,2008NLO: Soft gluon interaction

1 1 2 1( )(1) 4 41 2 1 1

ˆ ( , , ) (0) ( ) ( )ik y ik y yk k p d yd y e A gA y y A

(14

(1)1 2

)1 2

4(1) 2 1 2

4 4ˆ (Tr

(2 ), , ) ˆ ( , , )

(2 )qq

Hd k d

k k pW qk

e k k

2(1) 21 2 1 12

2

1 22 22

1

( )ˆ ( , , ) ( ) 2 ( )( )

( ) ( ) 2 ( )

( )

q kH k k q q k q k

q k i

q kq k q k

q k i

q

p

k1

q+k1

k2

q+k2

p

q

Osaka,2008

Collinear expansion: (1) (1)1 2 1 2

ˆ ˆ( , , ) ( , , )H k k q H x p x p q

pA A A

p

(1)1 2

(

(1) (1)1 2

(

21

1)1

2

1)1 22

Tr

Tr

ˆ ( , , )

ˆ

ˆ

( , ,

( , , )

ˆ ( , , ))

qq

x x p

x x

W e dx d

p

p H x p x p q

H x p x p q

x

1 1 2 1( )(1) 11 2 1

1ˆ ( , , ) (0) ( ) ( )2 2

ix p y ix p y yp dy p dyx x p e A gA y y A

p

1 1 2 1( )(1) 11 12 (ˆ ( , , ) (0) ( ))

2 2ix p y ix p y yp dy p dy

x x p e gAA y Ay

Osaka,2008Exercise IV

2(1) 21 2 1 12

2

1 22 22

1

( )ˆ ( , , ) ( ) 2 ( )( )

( ) ( ) 2 ( )

( )

q kH k k q q k q k

q k i

q kq k q k

q k i

(0) 2ˆ ( , ) ( ) 2 ( )H k q q k q k q

k

q+kq

k

k1

q+k1

k2

q+k2q q

(0) (0)1 2(1)

1 22 1

ˆ ˆ( , ) ( , )ˆ ( , , )B B

H x p q H x p qp H x p x p q

x x i x x i

(0) (1)ˆ ( , ) ( , , )k xp

H k q H xp xp qk

(2) Generalized Ward Identity:

(1) Order reduction:2

2B

qx

p q

22

1Hint : 2 ( ) Im

( )q k

q k i

Osaka,2008Exercise IV

2 1

1

2 1

1

( )( )

2 12

1 11

2 ( );

2 ( )

B

B

ix p y yix p y y

B

ix p yix p y

B

edx ie y y

x x i

edx ie y

x x i

2(2) 21 2 1 12 2

2

( )ˆ ( , , , ) ( ) 2 ( )( ) ( )

q k q kH k k k q q k q k

q k i q k i

q

p

k1

q+k1

k2

q+k2

p

q

k3 k4

q+k

Osaka,2008

(2)1 2

(0) (0) (0)1 2

2 1 2 1

( , , , )

( , ) ( , ) ( , )

( )( ) ( )( ) ( )( )B B B B B B

p p H x p x p xp q

H x p q H xp q H x p q

x x i x x i x x i x x i x x i x x i

1

1 2 1 1 2 1 2 2 2 1

1 2

0 0

( ) ( ) ( ) ( ) ( ) ( )

yy

dy dy y y y y y y y y y y

dy dy

1 2 1 2 2 1( ) ( )21

1 1 22

( ) ( ) ( ); 2 2 ( )( )

B

ix p y ixp y y ix p y yix p y

B B

dx dx ei e y y y y

x x i x x i

Osaka,2008

(0) (0)1 2(1)

2 1

( , ) ( , )

B B

H x p q H x p qp H

x x i x x i

2 1 2 1

1 1

( )( )

2 1 1 12 1

2 ( ); 2 ( )B B

ix p y y ix p yix p y y ix p y

B B

e edx ie y y dx ie y

x x i x x i

1 1 2 1(1) 11

(2 1

)1ˆ ( , , ) (0) ( ) ( )2 2

ix p y ix p y yp dy p dyx x p A ge A y y A

p

(1) (1)1 2

(1) (0)1 21 2

ˆ ˆ( , , ) ( , )Tr Tˆ ˆ( , , ) ( ,r )x x p x pp H x p x p q Hdx dx d x qx p

1 1(1)

0ˆ ( , ) (0 ( )) ( )

2i yxp yp dy

x p e ig dA y A y Ay

order reduction:

1 0

1 1 11 101( ) ( )y

ydy dy dy dyy y y

Osaka,2008

1 101ˆ ( , ) (0) ( )

2( )y yixp i

pg

dyx p e A y Ady A y

Twist-2 quark distribution

q

p

k

q+k

(0)(0 1) 2 ˆ ( , )ˆ ( , ) Trqq

W e qdx H xxp p

(0)ˆ ( ) (0) ( )2

ixp yp dyx e A y A

q

p

k1

q+k1

k2

q+k2

p

q

+

Osaka,2008Twist-3 parton distribution

(0) (1)ˆ ( , ) ( , , )k xp

H k q H xp xp qk

Using generalized Ward Identity:

1 2(0) (1)

1 2

(

(0) (1)1 2

(1)1 2

1)1 2

ˆ ˆ( ) ( , , )Tr Tr

ˆ ( , ,

ˆ ˆ( , ) ( , , )

ˆ ( , , ) r )T

k xp

H k q H x p x p qxdx dx dxk

H x p x pdx

x x p

x x pq

q

p

k

q+kq

p

k1

q+k1

k2

q+k2

p

q

(0 0) 0)) ( (ˆ (ˆ ˆ( , ) )) , ,(k xp

k H k qk

H k q H xp q

pA A

pA

Osaka,2008

Twist-3 quark distribution

1 1 2 1( )(1) 11 2 1

ˆ ( , , ) (0) (( ) )2 2

ix p y ix p y yp dy p dyx x e A Dp y Ay

1 1 2 1( )(1) 11 12 (ˆ ( , , ) (0) ( ))

2 2ix p y ix p y yp dy p dy

x x p e gAA y Ay

(0)ˆ ( ) (2

) )((0)ixy

p yp dyx e iA y A

(1)( 3) (1)11 2 21

22

ˆ Tr ˆ ( , , )( , , )twistq

q

H x p x p x x pqW e dx dx

Osaka,2008Double scattering

2(2) 21 2 1 12 2

2

( )ˆ ( , , , ) ( ) 2 ( )( ) ( )

q k q kH k k k q q k q k

q k i q k i

(2) (2)1 2 1 2

ˆ ˆ( , , , ) ( , , , )H k k k q H x p x p xp q

pA A

p

Collinear expansion:

1 2 1 2 2 1( ) ( )(2) 4 4 41 2 1 2

2 1

ˆ ( , , , )

(0) ( ) ( ) ( )

ik y ik y y ik y yk k k p d yd y d y e

A gA y gA y y A

4 4(2)

1

4(2) (

22 1 2

4 4 42)

1 2ˆ ( ,Tr

(2 ) (2 ) (2 )ˆ ( , , , ), , )q

q

H k kd k d k d k

W e k q k k k p

q

p

k1

q+k1

k2

q+k2

p

q

k3 k4

q+k

Osaka,2008

(2)1 2

(0) (0) (0)1 2

2 1 2 1

( , , , )

( , ) ( , ) ( , )

( )( ) ( )( ) ( )( )B B B B B B

p p H x p x p xp q

H x p q H xp q H x p q

x x i x x i x x i x x i x x i x x i

1 2 1 2 2 1( ) ( )(2) 2 11 2

2 1

ˆ ( , , , )2 2 2

(0) ( ) ( ) ( )

ix p y ixp y y ix p y yp dy dy dyx x x p e

A gA y gA y y A

(2)1

(2 (2) 22 21

)1 2

ˆ ( ˆ (, , ,T ,, ) , )rqq

W e dx dx p p H x p x p xp xq xd xx p

1

1 2 1 1 2 1 2 2 2 1

1 2

0 0

( ) ( ) ( ) ( ) ( ) ( )

yy

dy dy y y y y y y y y y y

dy dy

1 2 1 2 2 1( ) ( )21

1 1 22

( ) ( ) ( ); 2 2 ( )( )

B

ix p y ixp y y ix p y yix p y

B B

dx dx ei e y y y y

x x i x x i

Path-ordered integral

q

p p

q

Osaka,2008Gauge-link & gauge invariance

1

(2) 21 2 2 1

0 0

ˆ ( , ) (0) ( ) ( ) ( ) ( )2

yyixp yp dy

x p e A ig dy dy A y A y y A

(0) (2) 2( )2 ˆ ( , ) ,Tr ˆ ( )qq

H xp qe xW pdx

(0) (1 (0)) (2) 2 ˆˆ ( , )( , )Trqq

W W W W e d H xp x px q

1

0

21 1 1 2 1 20

0 0

( )

ˆ ( , ) (0) ( )2

(0) ( )2

1 ( ) ( ) ( ) ( )

y

ixp y

ixp y

yyy

ig dz A z

ig dy A y ig dy dy Ap dy

x p e A y A

p dye A A

y A y

e y

Gauge link1ˆ ( , ) ( )2 Ax p pq x

0( )

( ) (0) ( )2 2

y

ixp yA

ig dz A ze

dyq x e A y A

Leading twist:

Osaka,2008Twist-4 contributions

1 2 1 2 2 1( ) ( ) 4 /31 2 2 1 1 2 2(0) ( ) ( ) ( ) ( ) ( )

2ix p y ixp y y ix p y y

N N

p dydy dy e A F y F y y A A q x x G x

q

p

k1

q+k1

k2

q+k2

p

q

k3 k4

q

p

k1

q+k1

k2

q+k2

p

qq

p

k

q+k

1 2 1 2 2 1( ) ( )1 2 2 1(0) ( ) ( ) ( )

2ix p y ixp y y ix p y yp dy

dy dy e A D y D y y A A

Osaka,2008Twist-4 from double scattering

q

p

k1

q+k1

k2

q+k2

p

q

k3 k4

q+k

k=xp+kTpA A

p

1 2 1 2 2 1 1( ) ( )(2) 21 2 2 1 1

2

2 1

( , , , )

(0) ( ) ( ) ( )2

T Tix p y ixp y y ix p y y ik yT

c

x p x p k p dy dy dy d y e

gA A y A y y A

N

(2)1 2

2(2) 2

1(2)

12 22ˆ ( , ( , , ,T , , )r

(2 ) 2)T

qq

d kW e dx

pH x p x p k q p p x x k pdx dx

(2)

1 2ˆ ( , , , )H k k k q

(2) (2) (2)1 2 1 2 1 2

ˆ ˆ ˆ( , , , ) ( , , , ) ( ,

,

, )

i iT k

k xpH k k k q H x p x p xp q k H x p x p k q

2 1 2 1 2 1partial integral: ( ) ( ) ( ) ( ) ( ) ( )

2 2ij iji j i i

T Tk k A y A y A y A y F y F y

1

2i jT Tk k (2)

1 2ˆ ( , , , )i j

k kk xp

H x p x p k q

Osaka,2008(2)

1 2

(0) (0) (0)1 2

2 1 2 1

ˆ ( , , , )

ˆ ˆ ˆ( , ) (( ) , ) ( , )

( )( ) ( )( ) ( )( )T

B T B B B B B T

H x p x p k q

H x p q H x x p q H x p q

x x x i x x i x x i x x i x x i x x x i

q

p

k1

q+k1

k2

q+k2

p

q

k3 k4

q+k

left-cut central-cut right-cut

1 2 1 1 2 1 2 2 2 1( ) ( ) ( ) ( ) ( ) ( )dy dy y y y y y y y y y y 2 ( )Tq

2 ( )Tq 2 ( )T Tq k

2 2 2 41( ) ( ) ( ) ( )

2i j

T T T T T i j T Tq q k k k O k

contact-term

1

221 2 1 2

21 2

0 0

(( ) ) ( )) (( )yy

T T TTdy dy q dy dy y y yq k

1/3nulcear enhanced 1.12AR A

Osaka,2008Transverse momentum broadening

2 (2) 22 2 2

2(0) 1

Tr ( ( ,)ˆ ,( , )2 2

)4

Aqg Tq T

qT c

pH xp q

d W ge dx q T x

d q Nx p

2 (0)Tr ˆ ( , )2

( )qq

pHW e x q xd qxp

1 2( )1 2 2 1 1 2 ( , ) (0) ( ) ( ) ( ) ( ) ( )

2 2Tixp y ix p y yA

qg T

dyT x x dy dy e A y y A y y yF F y

2 (2) 22 2 2

2(0) ( , ,Tr

2ˆ ( , )

2)T T q

qT c

Aqg T

pH

d W gT x xd q q e dx

d q Np px q

2

22

( , )2

( )

Aq qg B T

qsT

c q A Bq

e T x x

qN e q x

2

2T

T

kx

q p

Osaka,2008Vacuum bremsstrahlung

2 12

2( ) ( ) ( )

2h

S h hq h h q qg q h q gq q h

z

z zd dzD z P z D P z D

z z z

Gluon radiation induced

by *q scattering

2 (0)( ) ( (, ) ),q q hq

e dxq x H x p q D z 2

2(1) (0)

20

( , , ) ( , ) ( )2

s Tq qg

T

dH xp q z H xp q P z

21( )

1q qg F

zP z C

z

(1)2 (1) (1)( ) ( ) ( , , , ) ( ) ( , , ,1 )h hq q h g h

qh

dW z zdze dxq x D H x p q z D H x p q z

dz z z z

Osaka,2008Induced gluon radiation

(2)1

22

2 2 2 21 1

2( , , ,( )1 ( ) 1

( ) ( ) ( ))

),

(T

q

q x p i q xH x p x p x

p i q xp k q z

p i q xp i

1 2 1 2 2 1

(2)(

2 (2)1

) ( )21 2 1 2

2 10

2

( )

( ) ( )

2

(0) (1

( , , , )2

,)4 T

T

ix p y ixp y y ix p y yhq q h

qh

k Tk

dW zdz dye D dx dx dx dy dy e

dz z z

H x p x p xp k q zA y y yF AF

2

1 2

1 1

1 ( ) ( ) 1L T B

B B L B B L

x x x x x

x x i x x x i x x i x x x i

2

2 (1 )T

Lxp qz z

2 2

2T T T

T

k kx

q pz

1zq

qp

qxB

2

2

central-cut case

Osaka,2008LPM interference

1 2 1 2 2 1( ) ( )

1 21 2

1 1 1

( )ix p y ixp y y ix p y yL T B

B B L B B L

e x x x x xdx dx dx

x x i x x x i x x i x x x i

2_

1 2 1( (12 2

) ( ) ) ( )1

1 ( )1B L T L Li x x p y ix p ixy y yp ix p

L

y ye e ex

y y y

[ , , ]Tzq

2 Tx xB Lx x 2 L Tx x x Bx

2

2 (1 )

TLx

p q z z

Formation time:2

f

1

2 (1 )T

Lx pq z z

Osaka,2008Different cut-diagrams

+ …..

Osaka,2008Eikonal 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

1

2(2) 22 (

21 2 2 1

0 0

0)2

0

( ) (

( ) (

) ( )

, , ) ( )2

(0) ( )2 2

h s Tq q h q qg

qh T

ixp yyy

dW zdz de D dxH xp q z P z

dz z z

dig

ye A dy dy A y y AA y

central-cut = right-cut = left-cut in the collinear limit

Osaka,2008Modified fragmentation

22 2

2 (0)4

0

21( , ) ( ) ( , )

2 1Ah s sT

q q h A qg Lq T c

zdz z de dxH xp q D C T x x

z z z N

1 2

_2 1

1 2 2 1( ) ( )

(1

)2

( , ) (0) ( ) ( ) ( )2 2

1 1 ( ) ( )

B L T

L L

i x x p y ix p y y

ix p y ix p

L

y y

Aqg

dyT x x dy dy A F y F y y Ae

e y y ye

22 2

40

( , )21( ) ( ) (virtual)

2 1 ( )

Aqg Lh s sT

q h q h AT c A

T x xzdz z dD z D C g h

z z z N q x

(2)2 (1) 2 (0)( ) ( ) ( , , , ) ( ) ( , , ) ( )hq A q h q A q h

q qh

dW zdze dxq x D H x p q z e dxq x H x p q D z

dz z z

dominant contribution

Osaka,2008Quark-quark scattering

Osaka,2008Quark-quark scattering

Osaka,2008

total 12 independent diagrams

Differ by color factor 4/9 and quark density from quark-gluonscattering

Osaka,2008Modified fragmentation in DIS

22 2

40

( , )21( ) ( ) (virtual)

2 1 ( )

Aqg Lh s sT

q h q h AT c A

T x xzdz z dD z D C g h

z z z N q x

( ) ( )

( )h h

h

D z D zR

D z

2 2/( , ) 11 ,

( )L A

Aqg L x x

N A AAq N A

T x xCm R e x

f x m R

Factorized approximation:

Osaka,2008Twist Expansion

)(~)(2

)0(2

~222 BqsypixsS xAfAyAe

dy

d

dB

AyyFyFAedydydy

d

d yypixypixsD TB )()()(2

)0(222

~ 21)(21

4221

)()(~ 3/44 TTsBqs xGxxfA

])(1)[(~ 3/12222

TTS

BqsDS xGxAcxAf

d

d

d

d

3/1

22 LPM

A

Q 2

3/2

1Q

Ac

Osaka,2008HERMES data

2 20.00065 GeVsC 0.5 GeV/fm

dE

dx

in Au nuclei

E. Wang & XNWPRL 2000

Osaka,2008Energy Dependence

Osaka,2008

Di-hadron fragmentation function

1 2 1 2 1 2 1 2( , ) 0 (0) , , ( ) 02q h h q h h h h q

S

D z z Tr p p S p p S y

h1 h2

jet

Majumder & XNW’04

Osaka,2008

DGLAP for Dihadron Fragmentation

2

1

1

1

2

2

2 11 2

1 222

21

2

( , , )( ) ( )

ln( , , )q

qh h

q q hg

z z

h

D z z Q dyP

z zD Q

y yy g h h

Q y

h1h2

h1h2

h1

h2

1

1 2

2

22

121ˆ ( ) (( , )

1)

(,

)( )

1q

z

q

z

hg hqgz

Dz

D Qy

dyP y q g

yQ

y y

Osaka,2008Medium Modified Dihadron

D(z1,z2)/D(z1)Triggering h1

Osaka,2008Single & Dihadron Frag. In AA

Dihadron frag. func Single hadron frag. func

Osaka,2008

Summary: Jet Quenching in Twist Expansion

2 1 22

40 0

1 (1 ) ( , )

( )

Q

s

Aqg L

s Aq

TT

E zd dz

E

T x x

f x

_2 1

( )1 2 1 2

( )2 1

( , ) (0) ( ) ( ) ( )2 2

1 ( ) ( )1

B L

L L

i x x p y

ix p y ix p y

Aqg L

y

dyT x x dy dy A y F y F y A

y

e

e e y y

2 122

40

( , ) ( , ) ( )2

H

Q

S hq h h L q h

z

zd dzD z Q z x D g h

z z

2 ( , ) 21( , ) (virtual)

(1 ) ( )

Aqg L A S

L Aq c

T x x Czz x

z f x N

2 ( , )2

( )

Aqg B Ts

Tc A B

T x xq

N q x

Osaka,2008Energy Loss in Twist Expansion

2 1 22

40 0

(1 (1 )(

,

()

)

)

Aqg

Q

q gL

sq s TT

Aq

E zdz P z d dz

E

T x x

f x

( ) ( ) ( ) ( ) 1( , )

( )Lix p y

N T N T L T N

Aqg L

TAq

Ldy y x G xT x x

x x ef

G xx

x

22 1 2( ) ( )(1 ) (1 )L LL LB Bi x x p y ix pix p ix p y yy ix py yee ee

1 2

_2 1

1 2 2 1( ) ( )

(1

)2

( , ) (0) ( ) ( ) ( )2 2

1 1 ( ) ( )

B L T

L L

i x x p y ix p y y

ix p y ix p

L

y y

Aqg

dyT x x dy dy A F y F y y Ae

e y y ye

1( ) (0) ( )

2ixp y a

N a

dyxG x e p F F y p

p

Osaka,2008Gluon distribution in hot medium

( , )

( )(1 ) [1 cos( )](0) ( )T Lix p i

Aqg L p

LAq

x F Fe e xT x x

dyx

pf

yd

3

3( )

(2 ) 2

d pO f p p O p

p

[ ( ) (( , )

( )[1 cos( )) ) ( )] ](

A

T T L T L Tqg L

Aq

L

T x xd x G x x py xy

fx x

xyx G

21( , ) (0) ( , )

2T Tixp i i

T T i T

dx q d e p F F p

p

q ξξ ξ

2

2( ) ( , )

(2 )T

T

d qxG x x q

2ˆ( , , ) ( ) ( ( ) )s

L T L T Lc

q E x y y x x G x xN

Generalized jettransport parameter

Osaka,2008Energy loss and pT broadening

1 2( )1 2 2 1 1 2 ( , ) (0) ( ) ( ) ( ) ( ) ( )

2 2Tixp y ix p y yA

qg T

dyT x x dy dy e A y y A y y yF F y

2 ( , )ˆ( ,0,

( ))

2A

qg B TsT A

c q B

T xq

xq dy

N f xE y

( )( () )AqT Tx Gdy fy x x

2 2

24

ˆ ˆ( ,0,1 (1 )

[1 cos( )]) ( ,2 2 (1 )

, )c s TT

TLq E y q E x y

N z ydy d dz

Ez z

2 1 22

40 0

1 ( ( , )

( )

1 )( )

Q

Radq gq s

Aqg

T

Ls A

qT

E zdz P z d dz

T x x

E f x

Osaka,2008Jet transport parameter

2 2 2

2 2 ( ) ( , )

1 (2 )R T T

Tc

g C d q qdx x x q

N s

2 22

ˆ( ,0, ) ( ) qdq E y y dq q

dq

22

2

( )2

ˆ( ,0, ) 4 ( ) ( )1

s R

cTT T

Cq E y y

qx G x x

EN p

4 22 4

4 2

12 (( ) ) (0) ( )

2 (2 ) 1iqR

c

d q g Ck q d e p A A p

s N

k=E

pq

Osaka,2008Elastic versus radiative

[ , , ]Tzq

2 0Tx x B Lx x2 L Tx x x

Bx

2

2 (1 )

TLx

p q z z

( , )( )( ) ( ( )

( )1) L

Aqg L

s s N T N TAq

L T N L Tix p yx x G x x

T x xdy y x G x

fe

x

interference

1 22 2)( () (1 ) (1 )L LB LB L ix p y ix p yix p y ix p yi x yx p y ee ee

radiative elastic

Osaka,2008Elastic Energy Loss

2 12 22

40 0

1 (1 )( )( c )) 1 os(

Q

A sT L L

c Tg L

C zd dz x G xz x p yd y

Ny

0( ) ( ) 1

( , )( )

( )L

Aq ix p y

Ag

LA x

LL

q

dy y xGT

x ex x

x G xf x

22

2( )

11 cos( )

2Tg T L

Tdd

dz

zdy y dyx p y

E

2 22

4

1 (1 )2 ( )

2A

q a q g X s L a Lc

C d zdz x G x

N z

+….

Osaka,2008Elastic Energy Loss

2

2 /( ) ( ) ( 1)

2 1g L g LT

dy G x d x

e

3 11 cth( )ele

L

lETL

L TL

E

TLL

2

2 (1 )LxE z z

22

22

4 3 6lnˆ

12 ( ) 3 23el

Ls

Eq T

T

E

L

T

2

22

21 cos( )

1 cos( )12 (3)

( )2

ˆ( )

gel

Tg T

L

TLE x p ydy

dy

dy d

d

q y x p yT

21

2 12 (3)T

Osaka,2008

Interference effect in elastic energy loss

XNW

nucl-th/0604040

Osaka,2008

Elastic energy loss via uark-quark scattering

2 1 22 2

20 0

( )1 (1 )

1 cos(2

) ( )Q

Fs T Lq L

c T

C dz zdy yz d x p y

Nx

qf

z p

2

2 /( ) ( ) ( 1)

2 1q q L q LT

dy f x d x

e

2

2 (1 )LxE z z

2 22

6 11 -cos

3ln ec( )

6f

F sel TL

T

n TEC

LT

TL

E

L

2 2 2

2

2 1 (1 )( )

2 2sF

q a q q X q Lc

C d zdz f x

N p q z

Osaka,2008Radiative energy loss

2 22

4

1 (1 )[1 cos( )

)ˆ ) ]

2 (1(c s T

TT

NE z ydy d dz

E Ez zq y

( ) (( , )

( )(

) 1)

Lix p yA

Aqg L

T TA L Lq

T x xxdy G x

fy x e

xG x

3ln

8 1ˆ

1rad c sE N L

qE

LL

2

/ 22 2

2ˆ

Eg

g

dq dq q

dq

+….

Osaka,2008Radiative vs elastic energy loss

2 3 11 cth( )

12 (3)ˆelE

TLL TL

qT TL

3 ln

8ˆ

11 rad c sE N

qEL

LL

2

9 (3)ln 10

2 11rad c

sel

E N ELLT

E

For E=10 GeV, T=0.2 GeV, L=6 fm, s=0.3

Osaka,2008Evolution of qhat

2 2 22

2 2ˆ( , ) ( ) ( , )

1 (2 ) 2R T T

Tc

g C d q qq E dx x x q

N Ep

High energy jet small x

Large momentum transfer large scale

22

4 ( )1

s AT T

c

Cx G x

N

2 22

2

( , ) 1( , )

ln(1/ ) ln 2

xG xxG x

x

(DLA)

k=E

p

q

Osaka,2008Gluon Distribution from HTL

2 2 2

2 2ˆ ( ) ( , )

1 (2 ) 2R T T

Tc

g C d q qq dx x x q

N Ep

2222

( , )6 (3)

c sT T

Nx q q M

2

2 2 2 2 2

(1 )cos1

( ) (1 ) ( )q

D L q q D T q

xM

q x q x x

22 2

22

3 1( , ) ( , ) ln ln

4 12 (3) 2 3A sT D

TD

CdqxG x x q

xT

With HTL propagator:

k=E

pq

Osaka,2008Gluon Saturation in QGP

Evolution growth of gluon distribution at small x

Nonlinear effect (gluon fusion) willtame the growth of gluon distribution

Gluon Saturation2

2

2

22( , ) min

( ))

41

1( ,s s

ss

c

cc

QxG x Q L

N QL

N

QGP density is much larger than in nuclei

Saturation sets in at larger x and Qs2 is larger

E

p

q

Osaka,2008E-dependence of qhat

J. Casalderrey-Solana and XNW, arXiv:0705.1352 [hep-ph].

Nontrivial length dependence of qhat

Osaka,2008q-hat and Shear Viscosity

trC sT

2 22 2 2

ˆ1 4 2

9tr T Ttr cm T

d qdq q

E dq T

39ˆ2

TC

s q

Majumder, Muller and XNW (hep-ph/0703082)

Shear viscosity

1/ 3C

This relation is strongly violated for strongly coupled mediumWhere /s does reflect the transport of partons as quasi-particle

Jet quenching ˆ( ) q T ˆ( )q E

1

4s

Osaka,2008

Osaka,2008Alphabets of Jet Quenching

• Gyulassy-Levai-Vitev (GLV)– (Djordjevic, Wicks, Horowitz, …)

• Armesto-Salgado-Wiedeman (ASW)– (Dainess et al, Arleo, Renk, Eskola, …)

• Arnold-Moore-Yaffe (AMY)• High-Twist Expansion (HT)(Guo, E-K Wang, XNW)

– (Majumder, Osborne, Owens, B. Zhang, H. Zhang,…)

Comparative study of jet quenching schemes:A. Majumder QM06 (nucl-th/0702066 )

Osaka,2008NLO pQCD Calculation

Jet quenching in 2→3 processes

NLO (Next to Leading Order ):

Zhang, Owens, Enke Wang and XNW (nucl-th/0701045 )

Osaka,2008

Single and Dihadron hadron suppression at RHIC

00

zT=pTass/pT

trig

Zhang, Owens, E. Wang and XNW (nucl-th/0701045 )

Osaka,2008Sensitivity to initial density

2

20 0ˆ 1 2 GeVAq

2

/s = 0.1-0.2

Osaka,2008q-hat in a nucleus

2ˆ 0.01 GeV /Fq fm

e-

20ˆ 1.3 GeV / ( =1 fm)Fq fm

E. Wang & XNW PRL 89, 162301(2002)

Osaka,2008Surface emission?

Osaka,2008Surface vs. Volume

Osaka,2008A roadmap for future jet study

Re-constructed jets open up a whole new world for jet quenching study

Reconstructed jets: Single inclusive jets, di-jets or -jets

Osaka,2008(1) Direct measurement of qhat

2 ˆ( , )T q Edy yq

Osaka,2008(2) Measurements of jet shape

• RAA(ET) for jets:

1

Jet cone size R

RAA

For single hadrons

Osaka,2008

(3) Modified fragmentation functions

XNW, Huang and Sarcevic, Phys.Rev. Lett. 77, 231 (1996)

Osaka,2008A Mini-summary

• We have come a long way to this stage of study of jet quenching

• It reveals a dense and possible strongly coupling partonic matter

• Study of jet transport properties provides important information of the medium

• Reconstructed jet will usher in a new era

Osaka,2008Utopia?

polarization around jet ?