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Particle Production and
Correlations in p(d)A
Collisions
Yuri Kovchegov
The Ohio State University
Outline
We’ll talk about single-particle production and twoparticle correlations in p(d)A collisions:

Hadron production in p(d)A collisions: going from
mid- to forward rapidity at RHIC, transition from
Cronin enhancement to suppression.

Two-particle correlations (first-ever exact
calculation!), back-to-back jets.
Hadron Spectra in p(d)A
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 )  CF 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 (2 )
 x y


proton' s
wave function
With the gluon-gluon dipole-nucleus
forward scattering amplitude
NG ( 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 ( 2 )
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 h3.2
dAu at RHIC.
Rd(p)Au
pT
from D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0405045
Two-Particle Correlations
The Problem
We want to calculate 2-gluon production cross section in pA
scattering or DIS including all multiple rescatterings and
small-x evolution.
For simplicity we assume that the two gluons are ordered in
rapidity.
y2  y1
k2 ~ k1
Classical Approximation
We begin by resumming all multiple rescatterings of the system
on the nucleons in the nucleus.
t=-∞
t=+∞
t=0
t=-∞
double lines
= gluons
t=0
t=+∞
2
1
Classical Approximation
Emission of the gluon #2 would then look like:
yielding
d
S

2
2
d k1dy1d k2dy2 (2 )3
2
i k 2  x 22'
2
d
B
d
x
d
x
e
2
2'

2
 x 20 x 2~0   x 2'0 x 2 ' ~0 

  2  2    2  2  M ( x 2 , x 2' , x ~0 ; k 1 ) S ( x 2 , x 2' )  other graphs 
 x20 x2~0   x2'0 x2' ~0 

with M being the probability to emit gluon #1 in “dipole” 22’õ,
and S the S-matrix of the dipole 22’ scattering on the target.
Classical Approximation
Emission of the gluon #1 in a “dipole” 22’õ is given by the
following graphs:
leading to the cross section M expressed in terms of
color quadrupoles , along with dipoles, interacting with the
target nucleus. (In diagram A one has a quadrupole 22’11’ and
a dipole 11’ interacting with the target.)
In the end one gets a messy, but finite expression
for production cross section. J. Jalilian-Marian and Yu.K., ’04
Quantum Evolution
To include quantum evolution one has to write down an
evolution equation for M:
For instance, in diagram A gluon #1 can be emitted in the dipole
4õ or in “dipole” 22’4, etc.
Quantum Evolution
 In the end one gets an expression for the two-gluon production
cross section which can be represented diagrammatically by
(J. Jalilian-Marian and Yu.K., ’04)
One has to solve 6 integral equations to get the answer
(though only one of them is nonlinear).
Diagram C violates AGK cutting rules. This is the first example
of AGK violation in QCD that I’m aware of!
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
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:
Conclusions
• 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 

1
1
N ( z, y )  
C ( z QS 0 ) e

(

)


(1)


(

)

 (1   )
where
2 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

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  kgeom
(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
S0 

with
 NC


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.)