Transcript Saturation Physics at Forward Rapidity at RHIC

QCD Phenomenology and
Heavy Ion Physics
Yuri Kovchegov
The Ohio State University
Outline
We’ll describe application of Saturation/Color Glass
Condensate physics to Heavy Ion Collisions,
concentrating on:
 Multiplicity vs. Centrality and vs. Energy,
dN/dη vs. rapidity η
 Hadron production in p(d)A collisions: going from
mid- to forward rapidity at RHIC, transition from
Cronin enhancement to suppression.
 Two-particle correlations, back-to-back jets.
Multiplicity
Particle Multiplicity
In Saturation/Color Glass Physics one has
 only one scale in the problem – the saturation
scale QS .
 the leading fields are classical: A ~ 1 / g
The resulting gluon multiplicity is given by
dN
1  kT 

~ A A ~
f 
2
2
d k d b dy
 S  QS 
dN
1
such that
~
QS2 R 2
dy
S
since d2b ~ S ~ p R2 , with R the nuclear radius.
Particle Multiplicity vs. Centrality
dN
1
~
QS2 R 2
dy
S
Since
2
2/3
R ~A
2
S
1/ 3
Q ~A
we get :
~N
~N
2/3
part
1/ 3
part
and
2
S
2
Q R ~ N part
1 dN
1
~
N part dy
 S (QS )
which is not a constant due to running of the coupling:
1
1
Thus
 S (QS ) ~
~
ln QS ln N part
1 dN
~ ln N part
N part dy
Particle Multiplicity vs. Centrality
This simple reasoning
allowed D. Kharzeev
and E. Levin to fit
multiplicity as a function
of centrality.
(from nucl-th/0108006)
Particle Multiplicity vs. Energy
Let’s try to use the same simple
formula to check the
energy dependence of multiplicity.
Start with
dN
1
~
QS2 R 2
dy
S
From saturation models of HERA DIS data we know that
 s 
Q ( s )  Q ( s0 )
s 

 0
2
S
2
S
 /2
with
  0.25  0.3
Therefore we write
dN
dN
 200 
( s  200GeV ) 
( s  130GeV )

dy
dy
 130 
obtaining dN
dy
( s  200GeV )  616  634

Kharzeev,
Levin ‘01
Particle Multiplicity vs. Energy
Using the known multiplicity at 130 GeV Kharzeev and Levin
predicted multiplicity at 200 GeV using the above model:
dN
( s  200GeV )  616  634
dy
The result agreed nicely with the data:
dN
( s  200GeV )  650  35( syst )
dy
 Energy dependence works too!
(PHOBOS)
dN/dη
To understand the rapidity dependence one has to make
a few more steps. Starting with factorization assumption
dN
S
~ 2
2
d k dy k
2
d
 q  p ( q, Y  y )  A ( k  q, y )
inspired by the production diagram,
and assuming a saturation/CGC form of
the unintegrated gluon distribution :
 S
~ 2 , k T  QS

 k
2
A ( x, k )   S
 ~  , kT  QS

 S
dN/dη
Kharzeev and Levin
obtained a successfull fit of
the pseudo-rapidity
distribution of charged
particles in AA:
The value of the saturation
scale turned out to be
Q  1  2 GeV
2
S
2
(see also Kharzeev & Nardi ’00, Kharzeev, Levin, Nardi ’01)
dN/dη in dAu
The same approach
works for pseudo-rapidity
distribution of total charged
multiplicity in dAu
collisions:
(from Kharzeev, Levin, Nardi, hep-ph/0212316)
Thermalization: Bottom-Up Scenario
Baier, Mueller, Schiff, Son ‘00
Includes 2 → 3 and 3 → 2 rescattering processes with the
LPM effect due to interactions with CGC medium.
Does not introduce any new scale, one still has QS only, with
dN
1
~ 7 / 5 QS2 R 2
dy
S
Can fit the multiplicity data assuming that less particles were
produced initially (smaller QS) but their numbers increased
during thermalization. 2 dN
N part d
Baier, Mueller,
Schiff, Son ‘02
N part
Bottom-Up Scenario: Questions
 Instabilities!!! Evolution of the system may develop
instabilities. (Mrowczynski, Arnold, Lenaghan, Moore,
Romatschke, Strickland, Yaffe) However, it is not clear whether
instabilities would speed up the thermalization process and how
to interpret them diagrammatically .
Another problem is that since
dN
1
~ 7 / 5 QS2 R 2
dy
S
It appears that
and
1
1
A ~ 7 / 5 
g
g
dN
~ A A
dy
Stronger than classical
field? Stronger than any
QCD gluon field?
Hadron Spectra
Let’s consider gluon production, it will have all the essential
features, and quark production could be done by analogy.
Gluon Production in Proton-Nucleus
Collisions (pA): Classical Field
To find the gluon production
cross section in pA one
has to solve the same
classical Yang-Mills
equations


D F
J
for two sources – proton and
nucleus.
This classical field has been found by
Yu. K., A.H. Mueller in ‘98
Gluon Production in pA:
McLerran-Venugopalan model
The diagrams one has to resum are shown here: they resum
powers of
 A
2
S
1/ 3
Yu. K., A.H. Mueller,
hep-ph/9802440
Gluon Production in pA:
McLerran-Venugopalan model
Classical gluon production: we
need to resum only the
multiple rescatterings of the
gluon on nucleons. Here’s one
of the graphs considered.
Yu. K., A.H. Mueller,
hep-ph/9802440
Resulting inclusive gluon production cross section is given by
d
1
i k ( x  y )  C F x  y
2
2
2

d bd x d y e
NG ( x)  NG ( y)  NG ( x  y)
2
2 
2
2 2
d k dy (2p )
p x y


proton' s
wave function
With the gluon-gluon dipole-nucleus
forward scattering amplitude
N G ( x, Y  0)  1  e
 x 2Qs2 / 4

McLerran-Venugopalan model: Cronin Effect
To understand how the gluon production
in pA is different from independent
superposition of A proton-proton (pp)
collisions one constructs the quantity
Enhancement
(Cronin Effect)
d  pA
2
d
k dy
R pA 
d  pp
A 2
d k dy
We can plot it for the quasi-classical
cross section calculated before (Y.K., A. M. ‘98):
k4
R (kT )  4
QS
pA

2 k 2 / QS2
1 k 2 / QS2
 1


e

e

2
2
2
k
QS

 k

 k 2  
QS4



ln

Ei

2 2
 Q 2  
4

k
 S  


Kharzeev
Yu. K.
Tuchin ‘03
(see also Kopeliovich et al, ’02; Baier et al, ’03; Accardi and Gyulassy, ‘03)
Classical gluon production leads to Cronin effect!
Nucleus pushes gluons to higher transverse momentum!
Proof of Cronin Effect
 Plotting a curve is not a proof of
Cronin effect: one has to trust the
plotting routine.
 To prove that Cronin effect actually
does take place one has to study the
behavior of RpA at large kT
(cf. Dumitru, Gelis, Jalilian-Marian,
quark production, ’02-’03):
Note the sign!
2
2
3
Q
k
S
R pA (kT )  1 
ln 2   ,
2
2k

kT  
RpA approaches 1 from above at high pT  there is an enhancement!
Cronin Effect
2
2
3
Q
k
S
R pA (kT )  1 
ln 2   ,
2
2k

kT  
The position of the Cronin
maximum is given by
kT ~ QS ~ A1/6
as QS2 ~ A1/3.
Using the formula above we see
that the height of the Cronin
peak is
RpA (kT=QS) ~ ln QS ~ ln A.
 The height and position of the Cronin maximum are
increasing functions of centrality (A)!
Including Quantum Evolution
To understand the energy
dependence of particle
production in pA one needs to
include quantum evolution
resumming graphs like this one.
This resums powers of
 ln 1/x =  Y.
This has been done in Yu. K.,
K. Tuchin, hep-ph/0111362.
The rules accomplishing the inclusion of quantum corrections are
Proton’s
 Proton’s BFKL and N ( x, Y  0)  N ( x, Y )
LO wave function
wave function
where the dipole-nucleus amplitude N is to be found from (Balitsky, Yu. K.)
 N (Y , k 2 )
  s K BFKL  N (Y , k 2 )   s [ N (Y , k 2 )]2
Y
Including Quantum Evolution
Amazingly enough, gluon production cross section
reduces to kT –factorization expression (Yu. K., Tuchin, ‘01):
d  pA 2  S 1

2
d k dy CF k 2
2
d
 q p (q,Y  y ) A (k  q, y )
with the proton and nucleus “unintegrated
distributions” defined by

p, A
CF
2
2
i k  x
2
p, A
(k , y ) 
d
b
d
x
e

N
x
G ( x , b, y )
3 
 S (2p )
with NGp,A the amplitude of a GG dipole on a p or A.
Our Prediction
Our analysis shows that as
energy/rapidity increases the
height of the Cronin peak
decreases. Cronin maximum
gets progressively lower and
eventually disappears.
• Corresponding RpA levels
off at roughly at
R
pA
RpA
Toy Model!
energy / rapidity
increases
1 / 6
~A
(Kharzeev, Levin, McLerran, ’02)
D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0307037; (see also numerical
simulations by Albacete, Armesto, Kovner, Salgado, Wiedemann,
hep-ph/0307179 and Baier, Kovner, Wiedemann hep-ph/0305265 v2.)
k / QS
At high energy / rapidity RpA at the Cronin peak becomes a decreasing
function of both energy and centrality.
Other Predictions
Color Glass Condensate /
Saturation physics predictions
are in sharp contrast with other
models.
The prediction presented here
uses a Glauber-like model for
dipole amplitude with energy
dependence in the exponent.
figure from I. Vitev, nucl-th/0302002,
see also a review by
M. Gyulassy, I. Vitev, X.-N. Wang,
B.-W. Zhang, nucl-th/0302077
RdAu at different rapidities
RdAu
RCP – central
to peripheral
ratio
Most recent data from BRAHMS Collaboration nucl-ex/0403005
Our prediction of suppression was confirmed!
Our Model
RdAu
pT
RCP
p
from D.T Kharzeev, Yu. K., K. Tuchin, hep-ph/0405045, where we construct a
model based on above physics + add valence quark contribution
Our Model
We can even make a prediction for LHC:
Dashed line is for mid-rapidity
pA run at LHC,
the solid line is for 3.2
dAu at RHIC.
Rd(p)Au
pT
from D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0405045
Two-Particle Correlations
Back-to-back Correlations
Saturation and small-x evolution effects may also deplete
back-to-back correlations of jets. Kharzeev, Levin and
McLerran came up with the model shown below (see also
Yu.K., Tuchin ’02) :
which leads to suppression of B2B
jets at mid-rapidity dAu (vs pp):
Back-to-back Correlations
and at forward rapidity:
from Kharzeev, Levin,
McLerran, hep-ph/0403271
Warning: only a model, for
exact analytical calculations
see J. Jalilian-Marian and
Yu.K., ’04.
Back-to-back Correlations
An interesting process to look at is when one jet is at forward
rapidity, while the other one is at mid-rapidity:
The evolution between the jets
makes the correlations disappear:
from Kharzeev, Levin, McLerran, hep-ph/0403271
Back-to-back Correlations
 Disappearance of back-to-back correlations in dAu collisions
predicted by KLM seems to be observed in preliminary STAR
data. (from the contribution of Ogawa to DIS2004 proceedings)
Back-to-back Correlations
 The observed data shows much less correlations for dAu than
predicted by models like HIJING:
Back-to-back Correlations
 However, KLM calculations are just a model. An exact
calculation of two-particle inclusive cross section in p(d)+A (or
DIS) has been performed in J. Jalilian-Marian and Yu.K., ’04.
 The resulting expression for the cross section is so horrible that
no sane person would show it in a talk. It won’t fit in the
PowerPoint format anyway.  Nevertheless it exists and can be
used to make numerical predictions, though after a lot of work.
(One has to solve 6 integral equations to get the answer.)
Conclusions
• Particle multiplicity in AuAu and dAu collisions
varies as a function of energy, centrality and rapidity
in apparent agreement with saturation/CGC
predictions.
• New RHIC dAu data at forward rapidity seem to
confirm expectations of Saturation / CGC physics: at
mid-rapidity we see Cronin enhancement, while at
forward rapidity we see suppression arising from the
small-x evolution.
• Back-to-back correlations seem to disappear in a
certain transverse momentum region in dAu, in
agreement with preliminary CGC expectations.
• Implications for AA collisions need to be understood.
Backup Slides
Extended Geometric Scaling
A general solution to BFKL equation can be written as
2 S NC
y  ( )
d
2 2 
p
1
1
N ( z, y )  
C ( z QS 0 ) e

(

)


(1)


(

)

 (1   )
where
2p i
2
2
It turns out that the full solution of nonlinear evolution equation
N(z,y) is a function of a single variable, N=N(z QS(y)), with
QS ( y )  QS 0 e
2 S NC
p
y
(geometric scaling):
(i) Inside the saturation region, k ~ 1/ z  QS ( y ) , where nonlinear
evolution dominates (Levin, Tuchin ‘99 )
(ii) In the extended geometric scaling region, where ≈1/2:
QS ( y )  k ~ 1/ z  QS2 ( y ) / QS 0  k geom
(Iancu, Itakura, McLerran ‘02)
Geometric Scaling in DIS
Geometric scaling has
been observed in DIS
data by
Stasto, Golec-Biernat,
Kwiecinski in ’00.
Here they plot the total
DIS cross section, which
is a function of 2 variables
- Q2 and x, as a function
of just one variable:
2
Q
 2
QS ( x )
“Phase Diagram” of High Energy QCD
III
II
QS
I
High Energy or
Rapidity
kgeom = QS2 / QS0
QS
Cronin effect and low-pT suppression
Moderate Energy
or Rapidity
 pT2
Region I: Double Logarithmic
Approximation
At very high momenta, pT >> kgeom , the gluon production is given by the
double logarithmic approximation, resumming powers of
1
pT2
 S ln ln 2
x

Resulting produced particle multiplicity scales as

d N pA QS20 2
k 


exp  2 2  y ln
2
4

d k dy
k
Q
S
0


with
 NC

p
where y=ln(1/x) is rapidity and QS0 ~ A1/6 is the saturation scale of
McLerran-Venugopalan model. For pp collisions QS0 is replaced by 
leading to
Kharzeev



k
k
pA
R  exp  2 2  y  ln
 ln    1 Yu. K.


Q

Tuchin ‘03
S
0



as QS0 >> .
RpA < 1 in Region I  There is suppression in DLA region!
Region II: Anomalous Dimension
At somewhat lower but still large momenta, QS < kT < kgeom , the BFKL
evolution introduces anomalous dimension for gluon distributions:
Q
 ( k , y ) ~ 
k
2
S
2





with BFKL =1/2 (DLA =1)
The resulting gluon production
cross section scales as (we loose
one power of QS)
such that
R
pA
QS
 (k , y ) ~
k
d N pA QS 0 2  P 1 y

e
2
3
d k dy
k
kT
1 / 6 kT
~
~A
QS 0

Kharzeev, Levin, McLerran,
hep-ph/0210332
For large enough nucleus RpA << 1 – high pT suppression!
 How does energy dependence come into the game?
Region II: Anomalous Dimension
A more detailed analysis
gives the following ratio in
the extended geometric
scaling region – our region II:
 2k
2 k 
ln

ln


QS 0 
pA
1 / 6
R ~ A exp 

 14  (3)  y 


RpA is also a decreasing function of energy,
leveling off to a constant RpA ~ A-1/6 at very high energy.
RpA is a decreasing function of both energy and centrality
at high energy / rapidity.
(D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0307037)
Region III: What Happens to Cronin Peak?
 The position of Cronin peak is given by saturation scale QS , such that the
height of the peak is given by RpA (kT = QS (y), y).
 It appears that to find out what happens to Cronin maximum we need to
know the gluon distribution function of the nucleus at the saturation scale –
A (kT = QS, y). For that we would have to solve nonlinear BK evolution
equation – a very difficult task.
 Instead we can use the scaling property of the solution of BK equation
 k 
 ,
 (k , y )   
 QS ( y ) 
A
A
which leads to
Levin, Tuchin ’99
Iancu, Itakura, McLerran, ‘02
k  k geom
 QS
 (k  QS , y )   
 QS
A
A

A



(1)  const


 We do not need to know A to determine how Cronin peak scales with
energy and centrality! (The constant carries no dynamical information.)