Transcript Document 7405929

Open Questions in
Jet Quenching Theory
Ivan Vitev
QCD Workshop, Brookhaven National Laboratory
July 17-21, 2006 , Upton, NY
1
Outline of the Talk
Based on:
I.V., Phys.Lett.B 639 (2006), I.V. in preparation
I.V., T.Goldman, M.B.Johnson, J.W.Qiu, hep-ph/0605200
 Final state interactions in the QGP:
 Radiative energy loss
 Recursive solutions for multiple parton scattering
 Energy, system size dependence and QGP properties
 Initial state energy loss:
 Evidence for energy loss in cold nuclei in p+A
 Differential distributions for medium-induced initial state
gluon bremsstrahlung
 Phenomenological implications
 Heavy versus light quarks in p+p and p+A:
 Heavy quark correlations
 Cold nuclear matter effects for heavy versus light quarks
 Conclusions:
2
In-Medium Modification
of the PQCD Cross Sections
• The way to understand medium effects on hadron cross sections in the framework of PQCD
is to follow the history of a parton from the IS nucleon wave function (PDF) to the FS hadron
wave function (FF)
1
  el 2 (q )
2

d  el
d 2 q

1
  el 2 (q )
2

Scattering in the medium
2 / 9 
4 s2


 1/ 2 
2
2 2
9 / 8   q   


Range of the interaction in matter

n 
f
2
2 2
QGP: D  g T 1  6 


D ~
1
D
Cold nuclear matter: r0 ~ 1.2 fm
Calculated in the Born approximation
3
Understanding the LPM Effect
• Bremsstrahlung is the most efficient way to lose energy since it carries a fraction of the energy
f
k   xp 
LPM = LandauPomeranchuk-Migdal
k
p
1 k  1 ( xp  ) 2
y  ln   ln
2 k
2
k 2
g
q1 q1
q2
q3
q4  q4 q  q
5
5
q6  q6
q7
• Acceleration: radiation
H
• Formation time: coherence effects
k  qi1  ...  qin
k
,
C

i1 ...im
k 2
(k  qi1  ...  qim ) 2
B(i1 ...im )( j1 ... jn ) 
k  qi1  ...  qim
(k  qi1  ...  qim ) 2

k  q j1  ...  q jn
(k  q j1  ...  q jn ) 2
• Onset of coherence
 f  g 
k 2
(k  q ) 2
1
0     , i   f 
k
k
2
(
k

q

...

q
)

i
i
1
m
i1 ...im   f 1 

k
1
f
• Full coherence
1
 f  L  g 
D
4
1
D
Building up Multiple Scattering
Approximations that allow to treat
many scatterings:
p

k

(k  qi1  ...  qin )
(k  qi1  ...  qin )
k
p
Double Born scattering
Single Born scattering
C  CA
Vˆn Ai1 ...in ( x, k ; c)   R
Ai1 ...in1 ( x, k ; c)
2
-ei (0 n ) zn an Ai1 ...in1 ( x, k  qn ; [c, an ])
Dˆ n Ai1 ...in ( x, k ; c)  an Ai1 ...in1 ( x, k ; c)
 ei (0 n ) zn Ai1 ...in1 ( x, k  qn ; [c, an ])
N
v
 1  i0 zn
    e Bn [c, an ]Tel ( Ai1 ...in1 )
 2
 1
-  
 2
5
Nv
C A i0 zn
e Bn cTel ( Ai1 ...in1 )
2
Medium-Induced Radiation in the Final State
• Includes interference with the radiation from hard scattering
k
Rˆ =
zn

dN gn

2
dk d k
i1 ... in
 Ai1 ...in1  D† D  V †  V  Ai1 ...in1  Ai1 ...in1 Rˆ Ai1 ...in1
zn
zn
zn
+ q , a
qn , an
 Tr  Ai1 ...in Ai1 ...in
qn , an
n
n
qn , an
+
qn , an
qn , an
Momentum transfers
Number of scatterings
M.Gyulassy,P.Levai,I.V., Nucl.Phys.B 594 (2001)
n
 1 d el

CR s  n L  ji1 z j d zi
2
2
k
 k

d qi 
  (qi )  

2
dk  d 2 k n 1 dk  d 2 k n 1  2  i 1 0
g ( zi ) 

d
q
i
 el
 
n
m
m


 -2C1...n    B m 1...n  m...n  cos  k  2  k ...n  zk  cos  k 1  k ...n  zk 
m 1



dN g


dN n g

 
Color current propagators
6


Coherence phases
(LPM effect)

Analytic Approximations and Numerical Results
0-10%, 20-30% and 60-80% Au+Au,
Cu+Cu and central Pb+Pb
L L
 2 L2
E  

 lf

mean number Landau-Pomeranchuk
of scatterings -Migdal (LPM) effect
CRs 2L2
2E
E 
Log 2
 ... ,
4 g
 (L)L
(1)

Static medium
3
9CRs
1 dNg
2E
E 
L
Log 2
 ... ,
4
A  dy
 (L)L
(1)

1+1D Bjorken
 ( )   ( )
0

J.D.Bjorken, Phys.Rev.D 27 (1983)
7
Jets and Hadrons from PQCD
Can also incorporate Cronin effect:
d k
2
T
f med (kT )
d
P
P’
pd
zd
{
xaP
Pd / zd
xbP’
Pc / zc
Pc
dy 1d 2 pT 1
=
å
abcd
d s Nh1Nh2
2
2
dy 1dy 2d pT 1d pT 2
1
1
a s2
ò dxa ò dxbf (xa )f (xb ) (xa x S )2 M
b
x min
x min
a
b
d(D j - p )
=
å
pT 1pT 2 abcd
1
ò
z1 min
dz 1
Dh
1
/c
(z 1 )
Dh
2
ab® cd
Dh
1
/c
8
0
Nuclear
medium
(z 1 )
z1
f (x a )f (x b ) a s2
(z 2 )
M
/d
2
xa xb
S2
z1
Pd
X
X
h1
d s NN
0
2
ab® cd
Kinematic
modifications
System Size Dependence of Jet Quenching
A A 0  X
Reduction of the hard scattering cross section
E  i


E
i 1 E
Probability density -
pc  pc (1   ), zc 
zc
(1   )
1
D h / q ( z , Q )   d  P( )
'
2
0
P ( )
1
 z

D vac h / q 
, Q2 
1 
 1 

dN g
1
z

+  d
( ) D vac h / g  , Q 2 
d



0
1
• Absolute scale comparisons can and
should be done at large pT
• Similar pT dependence (flat) in Au+Au
and Cu+Cu
• In classes with the same N part we find
numerically the same suppression RAA
(For example central Cu+Cu and mid central Au+Au)
I.V., Phys.Lett.B 639 (2006)
9
Tomographic Summary
dE
dz
SPS
RHIC
LHC
*
 GeV 
 fm 


 0 [ fm]
T [ MeV ]
 [GeV / fm3 ]
 tot [ fm]
dN g / dy
2-3.5
0.8
210-240
1.5-2.5
1.4-2
200-350
7-10
0.6
380-400
14-20
6-7
800-1200
17-28
0.2
710-850
190-400
18-23
2000-3500
SPS RHIC
LHC
F.Karsch, Nucl.Phys.A698 (2002)
RAu  6.8 fm, Tc  175 MeV ,  c  1 GeV / fm3
10
D. d’Enterria, Eur.Phys.J C (2005)
Transport Coefficients in Thermalized QGP
• Experimental: Bjorken expansion
exp ( ) 
g
1 dN
,
A dy
• Theoretical: Gluon dominated plasma
dN g
 1200
dy
A  120 fm2
 0  0.6 fm
theory (T )  # DoF 

0
1
e p /T
4 p 2 dp # DoF

 [3]  T 3
3
2
 1 (2 )

where # DoF  2( polarization)  8(color ),  [3]  1.2
 exp ( 0 )  17 fm3
T  400 MeV
• Energy density
4
 theory (T ) 
 theory (T )  T
30 [3]
 exp ( 0 )  18 GeV . fm3  100  0.14 GeV . fm3
• Transport coefficients (not a good measure for expanding medium)
g2
 D  gT , g  2  2.5 ( s 
 0.3  0.5)
4
 D2 9 s2
2
1
ˆ
ˆ
2
q

1

2.5
GeV
.
fm
q



9 s
1
 D  0.8  1 GeV
g
2
 gg 
,


g
2
gg
2 D
 
g  0.75  0.42 fm

• Define the average for Bjorken
qˆ
2

( L  z0 ) 2
11

L
z0
qˆ ( z ) zdz
qˆ  0.35  0.85 GeV 2 . fm 1
New Directions for Energy Loss Calculations
• One possible discussion
Provide simulations including the geometry, combination of elastic and
radiative E-loss, jet topologies …
• A new direction: energy loss in cold nuclear matter, initial state
• It is a challenging theoretical
problem that has not been solved
(you will see the solution)
d  Au    X
0
• Of immediate relevance to pQCD
effects in cold nuclear matter p+A
• High twist shadowing only
• Initial state energy loss + HTS
Implementation of initial state E-loss
 x

, Q2 
 1 

 ( x, Q 2 )   
  E / E  kA1/ 3 , kmin
S.S.Adler et al., nucl-ex/0603017
12
bias
 0.0175
High Twist Shadowing in DIS
Final state coherent scattering
2
2 1/3
• Dynamical parton mass (QED analogy): mdyn   A
x = energy = mass
2
  mdyn
 2

x 2 ( A1/ 3  1) 2 
( LT )
FTA ( x, Q 2 )  A FT( LT )  x 
,
Q
=
A
F
x
1

 
 , Q 

T
2
2

Q
Q



 

13
J.W.Qiu, I.V., Phys.Rev.Lett. 93 (2004)
Cold Nuclear Matter Energy Loss
• Shadowing parameterizations:
(not)
S LT  S LT ( x, Q 2 )
• Dynamical calculations of high
twist shadowing: (not)
S HT  S HT (q ( g ); tˆ( z1 , ( z2 )))
• Energy loss: in combination
with HTS (yes)
T.Alber et al., E.Phys.J.C 2 (1998)
• Circular arguments should be avoided
• Nigh statistics 200 GeV p+A measurements will certainly reduce error
bars, however …
• Most useful measurement – low energy p+A run (only at RHIC II)
14
Medium-Induced Radiation in the Initial State
Asymptotic
Asymptotic
t  
t  , t  
Large Q2
t  zL  L
• Bertsch-Gunion case with interference
n
 1 d el

CR s  n L  ji1 z j d zi
2
2
k
 k

d qi 
  (qi )  

2
dk  d 2 k n 1 dk  d 2 k n 1  2  i 1 0
g ( zi ) 

d
q
i
 el
 
n
m


  B 2...n 1...n   B 2...n 1...n   2 B 2...n 1...n    B m 1...n  m...n  cos  k  2  k ...n  zk 
m 2



dN g


dN n g


 
• Realistic initial state medium induced radiation
n
CR s  n L  ji1z j d zi
k
 k


dk  d 2 k n 1 dk  d 2 k n 1  2  i 1 0
g ( zi )

dN g


dN n g

 1 d el

2
 d qi   el d 2 qi   (qi ) 

2
 
n

  B 2...n 1...n   B 2...n 1...n   2 B 2...n 1...n    B m 1...n  m...n  cos
m2

 
 2 H  B 2...n 1...n  cos
I.V. in preparation
n 1
k 2

 k ...n  zk 
15
m
k 2
 k ...n  zk

Energy Loss to First Order in Opacity
• Bertsch-Gunion Energy Loss
CR s L
 dN g

d d 2 k
 2 g

s/4
0
eff2
q2
d q 2
(q   2 ) 2 k2 (k  q ) 2
• Initial-State Energy Loss
CR s
 dN g

d  d 2 k
2

s/4
0
eff2
d q 2
(q   2 ) 2
2
L
q2

2
2
 g k (k  q )
q2  2k  q k 
k2 L 
2 2
sin  
k (k  q ) 2 k2 g
k 
New
• Final-State Energy Loss
CR s
 dN g

d d 2 k
2
Old

s/4
0
 2k  q
 L


 2
2 
k
(
k

q
)
     g
2
k  -q   L  

k


sin
2


k
 k  -q   g

eff2
d q 2
(q   2 ) 2
2
Qualitatively
2
16
E
L Q
 ln 0 const (1)
E
g 
E
L Q
 ln 0 const (2)
E
g 
const (2)
const (1)
E  2 L2 ln E / E0

const
E
g
E
Numerical Results For Quark Energy Loss
dN g (i )
0

2
i 1 dyd k 
n
At any order in opacity we require
• Energetic quark jets can easily lose 20-30%
of their energy, gluon jets x
• Initial state E-loss is much smaller than
the incoherent Bertsch-Gunion limit
• Initial state E-loss is much larger than
C A / CF  9 / 4
final state energy loss in cold nuclei
• Coherence effects lead to cancellation of
the medium-induced radiation
k2  k2  Q02  x2 M q2
M.Djordgevic, M.Gyulassy, Nucl.Phys.A (2004)
Fractional energy loss
dN g
x  1 contribution to x
dx
Radiation intensity
17
Path Length Dependence of E-Loss
• Bertsch-Gunion – linear dependence on L by definition
• Final state E-loss – approaches quadratic dependence on L, important for
the centrality dependence and elliptic flow
• Initial state E-loss – approaches linear dependence on L, important for the
centrality dependence in p+A reactions
18
pQCD Calculations of Heavy Quarks
Schematic NLO and NNL
LO, NLO, NNLO expansion
A(m), B(m)  coefficient functions
LL, NLL, NNLL expansion
 1,
m
0
pT
m/pT, (m/pT)2 power corrections
Will return to power corrections
• The new scale, mass, implies large
logarithms, but …
• The contribution of logarithms is small in
measurable pT ranges
• The quarks are treated as “heavy” – in the fixed
order calculation. Implies that NLO generates
the PDF for charm and bottom (mostly)
M.Cacciari, P.Nason, JHEP 9805 (1998)
19
Phenomenological Results
Comparison to the Tevatron data
Comparison to the RHIC data
Scales:
mT / 2, mT , 2mT
R.Vogt et al., Phys.Rev.Lett.95 (2005)
M.Cacciari, P.Nason, JHEP 0309 (2003)



  H ( s   ) (  )   0  ln H ( s   )  ln  (  )




• Description of open charm at the Tevatron is within uncertainties but not perfect
• Residual large scale uncertainties – should be careful with consistent choices
• At RHIC perturbative calculations under predict the data by factor of 2 – 4. Whether
it is experimental systematic, incomplete theory or both – open question
20
Numerical Results and Partonic Sub-Processes
Partonic sub-processes
FFs: Braaten et al., Phys.Rev.D51 (1995)
PDFs: CTEQ 6.1 LO, J.Pumplin et al., JHEP 207 (2002)
• Meaningful K-factors (otherwise K>4)
• Anti-correlation between K and the
•
21
hardness of fragmentation r
If (LO,c-PDF) ~(NLO,no c-PDF) what are
the corrections from (NLO,c-PDF)?
Hadron Composition of C (B) Triggered Jets
Possibility for new measurements of heavy flavor production at RHIC
~ DD / c ( z, Q2 ) / Dh / q, g ( z, Q2 )
D (B) meson
D (B) meson
“Few” hadrons
• Can clarify the underlying hard scattering
processes and open charm production
mechanisms
“Many” soft hadrons
• Can constrain the hardness of D and B
meson fragmentation
Robust
22
HTS for Light Hadrons and Open Charm
Single inclusive particles
Away-side correlations
• Very similar dynamical shadowing
f (x b )
F (x b ) =
M ab2® cd
xb


 2 1/ 3
F ( xb )  F  xb  xbCd
( A  1) 
2
t  md


I.V., T.Goldman, M.B.Johnson, J.W.Qiu, hep-ph/0605200
J.W.Qiu, I.V., Phys.Lett.B632 (2006)
23
for light hadrons and heavy quarks
S HT  S HT (q ( g ); tˆ( z1 , ( z2 )))
• Insufficient to explain the forward
rapidity data
• Single and double inclusive cross
sections are similarly suppressed
Energy Loss and High Twist Shadowing
Double inclusive yields (away-side)
Single inclusive particles
I.V., T.Goldman, M.B.Johnson, J.W.Qiu, hep-ph/0605200
• Main difference is much more pT independent suppression as compared to high twist
shadowing
Same
• Very similar e-loss effects for light hadron and heavy quark spectra
• Single and double inclusive cross sections are similarly suppressed
24
Effects of Medium-Induced Radiation
I.V., Phys.Lett.B630 (2005)
• Cancellation of collinear radiation – large angle soft gluons and
correspondingly soft hadrons
• Beyond the cancellation region - well defined power
dependence
• The importance – hard scattering has the
same power dependence
25
d
1
~ 4
dt
p
dN g
1
~
dyd 2 k k4
Phenomenological Implications
Correlated!
PHENIX Collaboration, Phys.Rev.Lett. (2005)
• Suppression at forward rapidity – from energy loss of the incoming partons
• Enhancement at backward rapidity – comes from the redistribution of the lost
energy
• Consistent pQCD code is still to be developed
26
Conclusions
 In-medim interactions can be understood
following the history of a jet in a hard scatter:
 Radiative energy loss in the QGP:
 Predicted supession for Cu+Cu versus centrality and pT
 QGP suppression is consistent with perturbative interaction of jets
in the medium
 Coherent final state interactions:
 Shadowing is dynamically generated and arises from the final state
 Shadowing for D mesons and light pions is similar
 Initial state interactions:
 Transverse momentum diffusion and Cronin effect
 Energy loss and rapidity asymmetry in p+A – new theoretical results
 Modification of di-jets:
 Gluon feedback is important for di-hadrons at large angle
 Flow leads to deflection of the jet+gluons, so be exp. determined
27
Initial State Elastic Scatterings
a) Initial state elastic scattering
Unitarization of multiple scattering
zn zn
ˆ n  Vˆ n † =
ˆ n † Dˆ n  V
R̂ n  D
qn , an
zn
+
qn , an qn , an
Reaction Operator = all possible on-shell
interaction with the propagating system
¥
f
dN ( p) =
å
e
-c
n= 0
i
zn
+
qn , an
t=¥
n
cn
1 d s el
2
i
d
q
d
N
( p - q1 - .. . - qn )
Õ
i
2
ò
n ! i= 1
s el d qi k
dN (k ^ ) = d (k ^ )
^
2
2
1 e cm x
dN (k ^ ) =
,
2p c m2x
f
Initial condition
=
- D kP = 2c m2 x
(Neglect
2 / 9 
4 s2


1/
2


2
2 2
 9 / 8   q   


1
2kP
c = L/ l
Mean number of scatterings
Solution
The approximate solution is that of a 2D diffusion
For D k ^ 2 = 2c m2 x,
qn , an
cuts through a new Double Born
-
2
qn , an
p
and O ( ( k  k )3 ) )
Elastic scattering cross section
Implemented in the PQCD approach as
k broadening of the initial state partons
28
Cronin Effect
p W     X
p  Be     X
Good description at mid rapidity
Default
d  Au   0  X
Wrong sign at forward rapidity
Cronin effect: enhancement of cross sections
at intermediate transverse momenta relative
to the binary scaled p+p
I.V., Phys.Lett.B526 (2003)
A.Accardi, CERN yellow report, references therein
29
Data
Heavy Quarks in p+p and p+A
New possibility: hadron composition
of heavy quark triggered jets
Robust
FFs: Braaten et al., Phys.Rev.D51 (1995)
PDFs: CTEQ 6.1 LO, J.Pumplin et al., JHEP 207 (2002)
• Anti-correlation between K and the
hardness of fragmentation r
• Non-trivial hadron composition of c and b
triggered jets
30
In-Medium Modification of the PQCD Cross Sections
• The way to understand medium effects on hadron cross sections in the framework of PQCD
is to follow the history of a parton from the IS nucleon wave function (PDF) to the FS hadron
wave function (FF)
a) Initial state interactions
Elastic scattering and Cronin effect
d) Final state interactions in the QGP
Jet quenching
b) Initial state interactions
Energy loss and forward Y suppression
c) Final state interactions
Dynamical shadowing, Generalization
to heavy quarks
e) Final state interactions in the QGP
Large angle correlations, Di-Jet
suppression, Deflection of jets by flow
Cold nuclear matter effects are present at times
• Jet interactions in the medium result in kinematic modifications to the hard scattering cross
section that are process dependent
31
Process Dependence of Power Corrections
Suppression (
tˆ  0
(For example forward rapidity)
)
• The function F(xb) contains the small xb dependence
Enhancement (
sˆ  0
)
(For example DY)
• Power corrections are process dependent and not separable in PDFs and FFs
S.Brodsky et al, Phys.Rev.D65 (2002)
• Similar process dependence in single spin asymmetries
S.Brodsky et al, Phys.Lett.B530 (2002)
• Shadowing is dynamically generated in the hadronic collision
32
Universal Features of Jet Quenching
Approximately universal behavior
d
a
a


dyd 2 pT ( p0  pT ) n pT n
Baseline:
2/ 3
Prediction: ln RAA   N part
Scalings:
L A
1/ 3
N
1/ 3
part
L dN g
2/3
,
 A2 / 3  N part
A dy
Natural variables
dN g
 A  N part
dy
Fractional energy loss:
E
L dN g
2/3


 A2 / 3  k ' N part
E
A dy
Suppression:
RAA
d ( pT (1   ) / dyd 2 pT

 (1   ) 2
2
d ( pT ) / dyd pT

1
1   ' N 
2/3
part
n2
I.V., Phys.Lett.B in press, hep-ph/0603010
33
Numerical Results for Jet E-Loss
P0 ( )  e
n
i
i 1
E
 
 Ng
…
M.Gyulassy, P.Levai, I.V., Phys.Lett.B (2002)
1
1
 d   P( ')  1,
 d    P( ') 
0
1
 d 
0
dN g
d
0-10%, 20-30% and 60-80% Au+Au,
Cu+Cu and central Pb+Pb
…
 ( )
0
( ')  N g ,
1
 d   
0
dN g
d
E
E
( ') 
E
E
• Small probability not to radiate
P0  e
 Ng
1
• Small fractional energy loss at large ET
dN g 3 d dN ch
• Scales in the QGP
dy
2 dy d
g
dN
Initial parameters
 1200
dy
 D  0.8  1 GeV
g  0.75  0.42 fm
qˆ  1  2.5 GeV . fm 1
34
System Size Dependence of Jet Quenching
• Absolute scale comparisons can and
should be done at large pT
• Similar pT dependence (flat) in Au+Au
and Cu+Cu
• In classes with the same N part we find
numerically the same suppression RAA
For example central Cu+Cu and mid
central Au+Au
A A 0  X
• Future tests of high energy nuclear
physics at the LHC
I.V., Phys.Lett.B in press, hep-ph/0603010
35
Energy Loss and Di-Jets
e) QGP effects on di-jet production
One way of incorporating energy loss:
• “Standard” quenching of leading
hadrons
• Redistribution of the lost energy
in “soft” hadrons
0
A+A
0
Satisfies the momentum sum rule
  E / E
Single inclusive particles
Away-side yields
RHIC
LHC
I.V., Phys.Lett.B630 (2005)
36
Radiation Distribution and Flow Effects
q0
Gluon number distribution without or with q0 = 1 GeV
• Mechanical analogy, Theoretical derivation
k  q0
k  q0
k  2q0
k
k
k  2q0
k  q0
k  q0
k  2q0
k
k
k  2q0
Problem
Solution: expand about k  q0
Show that O(q0 ) vanishes
• We cannot confirm the prescription
N.Armesto et al., Phys.Rev.C (2005)
1 d el
2

 el d 2 q  (  2  (q  q0 )2 )2
• Important for deflected jets, to be seen in
experiment
37
Result: same energy loss and
shifted reference frame

dN g med
1 d el 2  k  q0   q
2

d
q

d d 2 k 0
 el d 2 q ( k  q0   q ) 2

( k  q0   q ) 2 z 
 1  cos

2


38
Cancellation of collinear radiation
39
40
41
Energy Loss to First Order in Opacity
Qualitatively
• Bertsch-Gunion Energy Loss
C
 dN g
 R2 s
2
d d k


d z
L

g
0
C L
 dN g
 R2 s
2
d  d k
 g

s/4
0
d 2 q
eff2
2
B1
2
2 2
(q   )
2 L
E

const (1)
E
g
eff2
q2
d q 2
(q   2 )2 k2 (k  q ) 2
s/4
2
0
• Initial-State Energy Loss
C
 dN
 R2 s
2
d d k

g
C
 dN g
 R2 s
2
d d k

d z

L

s/4
g
0
0

s/4
0
E

E
eff2

k2 z 
2
2
d q 2
| B1 | 2 H  B1cos  
(q   2 )2 
k 
eff2
d q 2
(q   2 )2
2
L
q2
q2  2k  q k 
k2 L 
2 2
sin  

2
2
k (k  q ) 2 k2 g
k 
 g k (k  q )
2 L
const (2)
g
const (2)
const (1)
• Final-State Energy Loss
C
 dN g
 R2 s
2
d  d k

C
 dN g
 R2 s
2
d  d k


L
d z
0

s/4
0
g

s/4
0
2


eff2
k  -q   z  

 2C1  B1 1-cos
d q 2



(q   2 )2 
k


2


2
 2k  q
 L
k  -q   L  

k



 

d q 2
sin

(q   2 )2  k2 (k  q ) 2  g  k  -q  2 g
k



2

2
eff
42
E  2 L2 ln E / E0

const
E
g
E
Non-Perturbative Scales
• Chiral perturbation theory
Q 2 min ( pQCD)  Q 2 max (  PT )  4 f
Q 2 min ( pQCD) 1.15 GeV
f  92 MeV
• (Generalized) vector dominance model
Q 2 min ( pQCD)  max(mA2 , mV2 )
• Bertsch-Gunion
Q2min ( pQCD)  m2  0.6 GeV 2
• QCD evolution of FFs and PDFs
Q 2 min ( pQCD)  Q 2 0  0.4  2 GeV 2
• Coherent high twist shadowing
Q 2 min ( pQCD)  mN2
2
0.8 GeV 2
0.12 GeV 2
Implementation
k2  k2  Q02  x2 M q2
Q02  mN2  0.94 GeV 2  1/ 0.2 fm 
2
  0.35 GeV , g  1 fm
J.W.Qiu, I.V., Phys.Rev.Lett. 93 (2004)
43
Numerical Results For Quark Energy Loss
dN g (i )
0

2
i 1 dyd k 
n
At any order in opacity we require
• Energetic quark jets can easily lose 20-30%
the incoherent Bertsch-Gunion limit
• Initial state E-loss is much larger than
of their energy, gluon jets x C / C  9 / 4
A
F
final state energy loss in cold nuclei
• Coherence effects lead to cancellation of
dN g
x  1 contribution to x
dx
the medium-induced radiation
Fractional energy loss
• Initial state E-loss is much smaller than
44
Radiation intensity
II. Coherent Power Corrections
Deviation from A-scaling:
 A  A 
Shadowing
Longitudinal size:
If
1/ 2mN x
x  0.1 then z  r0
Transverse size:
1/ Q
If Q  mN then exceed
the parton size
What remains for theory:
power corrections in DIS - suppression
Data from: NMC
FSI are always present:
S.Brodsky et al.
Ivan Vitev, LANL
Medium-Induced Bremsstrahlung
• Calculating the multiple scatterings in the plasma
p
Calculate everything else
Vacuum DGLAP type
+
p
Example of hard
scattering
D ~
1
D

D  g 2T 2 1 

Potential
V~
nf 

6 
s
 (q0 )
q 2   D2
2
+
+
+
+
+
+
+
+
+
+ ...
Medium
M.Gyulassy, P.Levai, I.V., Nucl.Phys.B594 (2001)
Reaction operator: (Cross section level
S † S)
Advantage: applicable for elastic, inelastic and coherent
scattering
controlled approach to coherent radiation (LPM)
46
Comparison to Other Models
Strong coupling used as a parameter
• Find T = 370 MeV (OK)
S.Turbide et al., Phs.Rev.C. (2005)
2
• Find qˆ  14 GeV / fm (NOT OK)
G.Paic et al., Euro Phys.J C (2005)
K.Eskola et al., Phys.Rev.D (2005)
• Find dNg/dy = 1200 (OK)
dN g
dy
3 d dN ch
2 dy d
I.V., M.Gyulassy, Phys.Rev.Lett. (2002)
I.V. Phys.Lett.B in press
B.Cole, QM 2005 proceedings
• These are not equivalent descriptions – the medium properties differ by more than
an order of magnitude (sometimes close to two)
2
 14 GeV 2 / fm
• How do you build from T = 400 MeV qˆ 
g
2
 100 GeV 2 / fm
LHC: from T = 1 GeV qˆ 
g
47
The Source of the Problem
P0  e
 Ng
c ( L  5 fm)
R
qˆ200 GeV  14 GeV 2 /fm
350 GeV
~10000
qˆ5500 GeV  100 GeV 2 /fm
2650 GeV
~100000
qˆ200 GeV  0.4 GeV 2 /fm
11 GeV
~500
C.A.Salgado, U.Wiedeman, Phys.Rev.D (2003)
Problem
Realistic
GLV
A useful table
Typical gluon energy
c  qˆL2 / 2
R  c L
• Note that the region of PT at RHIC is
10-20 GeV and at the LHC 100-200 GeV
Negative gluon number and jet enhancement
from energy loss
Energy momentum
violation
Problem
Problem
Negative probability density
• Symptomatic of problems in the underlying model of energy loss
48
Analytic Limits of Delta E
• Controlled approach to coherence GLV
k q
• Average implementations in the large
number of scatterings limit
k, 
q
z z 
 ( z  z )(k  q ) 2 
cos  2 1   cos  2 1

 l

2


 f 
k2 ~ n 2
Includes the fluctuations of the gluon
momentum and energy
• Calculate differential spectra in
k , 
E
(1)
CR s 2L2
2E

Log 2
 ... ,
4
g
 (L)L

• Calculate the energy loss
BDMPS, AMY
Static medium
3
E
(1)
Different dynamics REQUIRES
different solutions
9CR s
1 dNg
2E

L
Log 2
 ... ,
4
A  dy
 (L)L

1+1D Bjorken
M.Gyulassy, I.V., X.N.Wang , Phys.Rev.Lett.86 (2001)
Static:
qˆ   2 /  - transport coefficient
BJ expansion:
BJ+2D
g
dN / dy
- effective gluon
rapidity density
49
vc
 ( )   ( 0 )
0

Energy Loss and High Twist Shadowing
Double inclusive yields (away-side)
Single inclusive particles
I.V., T.Goldman, M.B.Johnson, J.W.Qiu, hep-ph/0605200
• Main difference is much more pT independent suppression as compared to high twist
shadowing
Same
• Very similar e-loss effects for light hadron and heavy quark spectra
• Single and double inclusive cross sections are similarly suppressed
50
Future Directions of Jet Interaction Studies
• Self
consistency of the description of interactions in cold nuclear matter
Cronin effect
Q2  2 2 L / g
Q2   2 A1/ 3
What is the energy loss for such momentum transfer from the medium?
• Regimes of
initial state energy loss
Is there a full Reaction Operator (GLV-like) expression via a formal
solution to recurrence relations?
I.V., in preparation
51
Energy Loss to First Order in Opacity
Meaning of the expansion in “n”
Qualitatively
• Bertsch-Gunion Energy Loss
C
 dN g
 R2 s
2
d d k


d z
L

g
0
C L
 dN g
 R2 s
2
d  d k
 g

s/4
0
d 2 q
eff2
2
B1
2
2 2
(q   )
New
eff2
q2
d q 2
(q   2 )2 k2 (k  q ) 2
s/4
2
0
• Initial-State Energy Loss
C
 dN
 R2 s
2
d d k

g
C
 dN g
 R2 s
2
d d k

d z

L

s/4
g
0
0
2 L
E

const (1)
E
g

s/4
0
E

E
eff2

k2 z 
2
2
d q 2
| B1 | 2 H  B1cos  
(q   2 )2 
k 
eff2
d q 2
(q   2 )2
2
L
q2
q2  2k  q k 
k2 L 
2 2
sin  

2
2
k (k  q ) 2 k2 g
k 
 g k (k  q )
2 L
const (2)
g
const (2)
const (1)
• Final-State Energy Loss
C
 dN g
 R2 s
2
d  d k

C
 dN g
 R2 s
2
d  d k


L
d z
0

s/4
0
g

s/4
0
2


eff2
k  -q   z  

 2C1  B1 1-cos
d q 2



(q   2 )2 
k


2


2
 2k  q
 L
k  -q   L  

k



 

d q 2
sin

(q   2 )2  k2 (k  q ) 2  g  k  -q  2 g
k



2

2
eff
52
E  2 L2 ln E / E0

const
E
g
E