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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
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 ( , , ))
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
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
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
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 ?