Transcript Saturation Physics at Forward Rapidity at RHIC

Small-x Theory Overview
Yuri Kovchegov
The Ohio State University
Columbus, OH
Outline



A brief review of Saturation/Color Glass Physics
Hadron production in p(d)A collisions: going from
mid- to forward rapidity at RHIC, transition from
Cronin enhancement to suppression.
Open questions: what needs to be measured?
 Di-leptons in p(d)A.
 Back-to-back jet correlations.
 Is suppression tied to fragmentation region?
 Open charm production in p(d)A.
Nuclear/Hadronic Wave Function
Imagine an UR nucleus
or hadron with valence
quarks and sea gluons
and quarks. Define
Bjorken (Feynman) x as
p
xBj
k

p
k
Rest frame of the hadron/nucleus
Longitudinal coherence length of a gluon is
1
1
1
lcoh ~
~
~
k
xBj p
xBj mN
such that for small enough xBj we get
with R the nuclear radius.
(e.g. for x=10-3 get lcoh=100 fm)
lcoh
1

 R
2 mN xBj
Color Charge Density
Small-x gluon “sees” the whole nucleus coherently
in the longitudinal direction! It “sees” many color charges which
form a net effective color charge Q = g (# charges)1/2, such
that Q2 = g2 #charges (random walk). Define color charge
McLerran
density 2 Q 2
g 2 # charges
2 A
1/ 3
 
S

S
~g
S
~A
Venugopalan
’93-’94
such that for a large nucleus (A>>1)
 ~
2
2
QCD
A  
1/ 3
2
QCD
 S ( )  1
2
Nuclear small-x wave function is perturbative!!!
McLerran-Venugopalan Model
 The density of partons in the nucleus (number of partons per
unit transverse area) is given by the scale 2 ~ A/pR2.
 This scale is large,  >> ΛQCD , so that the strong coupling
constant is small, αS () << 1.
 Leading gluon field is classical! To find the classical gluon field
Aμ of the nucleus one has to solve the Yang-Mills equations,
with the nucleus as a source
of color charge:
D F

J

Yu. K. ’96
J. Jalilian-Marian et al, ‘96
Shadowing Ratio
Defining the shadowing ratio
for the unintegrated gluon
distributions
we plot it for the
distribution found
2

(
x
,
k
)
2
A
RA ( x, k ) 
Ap ( x, k 2 )
Enhancement
(anti-shadowing),
most glue are here
Shadowing!
(small k  small x)
 Gluons are redistributed from low to high pT.
DIS in the Classical Approximation
The DIS process in the rest frame of the target is shown below.
It factorizes into
 *A
 tot
( xBj , Q 2 )   *q q  N ( x , Y  ln(1 / xBj ))
with rapidity Y=ln(1/x)
DIS in the Classical Approximation
The dipole-nucleus amplitude in
the classical approximation is
 x2 QS2
1 
N ( x , Y )  1  exp
ln

4
x




A.H. Mueller, ‘90
Black disk
limit,
 tot  2p R2
Color
transparency
1/QS
Quantum Evolution
As energy increases
the higher Fock states
including gluons on top
of the quark-antiquark
pair become important.
They generate a
cascade of gluons.
These extra gluons bring in powers of S ln s, such that
when S << 1 and ln s >>1 this parameter is S ln s ~ 1.
BFKL Equation
Balitsky, Fadin, Kuraev, Lipatov ‘78
Start with N particles in the proton’s wave function. As we increase
the energy a new particle can be emitted by either one of the N
particles. The number of newly emitted particles is proportional to N.
The BFKL equation for the number of partons N reads:

N ( x, Q 2 )   S K BFKL  N ( x, Q 2 )
 ln(1 / x )
Nonlinear Equation
At very high energy parton recombination becomes important. Partons not
only split into more partons, but also recombine. Recombination reduces
the number of partons in the wave function.
 N ( x, k 2 )
  s K BFKL  N ( x, k 2 )   s [ N ( x, k 2 )]2
 ln(1 / x)
Number of parton pairs ~ N 2
Yu. K. ’99 (large NC QCD)
I. Balitsky ’96 (effective lagrangian)
Nonlinear Equation: Saturation
Gluon recombination tries to reduce the number of gluons in the wave
function. At very high energy recombination begins to compensate gluon
splitting. Gluon density reaches a limit and does not grow anymore. So do
total DIS cross sections. Unitarity is restored!
Nonlinear Evolution at Work
 First partons are produced
overlapping each other, all of them
about the same size.
 When some critical density is
reached no more partons of given
size can fit in the wave function.
The proton starts producing smaller
partons to fit them in.
 The story repeats itself for smaller
partons. The picture is similar to
Fermi statistics.This way some Color
critical density is never exceeded.
Proton
Glass Condensate
Phase Diagram of High Energy QCD
Saturation physics allows us
to study regions of high
parton density in the small
coupling regime, where
calculations are still
under control!
Transition to saturation region is
characterized by the saturation scale
QS
(or pT2)
We are making real progress in understanding proton/nuclear structure!
What Happens to Gluon
Distributions?
With the onset of evolution, as energy/rapidity
increases, the shadowing ratio starts to decrease:
Anti-shadowing
Suppression!
A new
phenomenon.
Shadowing
(Toy model picture)
What Happens to Gluon
Distributions?
At very high energy “anti-shadowing” disappears:
Suppression
everywhere!
(Toy model picture)
Why Suppression?
In general one can write:
Q
 ( k , y ) ~ 
 k
2
S
2





Without quantum evolution =1 and there is no suppression:
A
R ~

1/ 3
A
QS2
~ 1/ 3 ~ 1
A
Quantum evolution leads to =1/2, such that
leading to
QS
 (k , y ) ~
k
QS
R ~ 1/ 3 ~ A1/ 6  1
A
A
Kharzeev, Levin, McLerran,
hep-ph/0210332
Hadron Production
Let’s start with gluon production, it will have all the essential
features, and quark production will become clear after that.
How to Calculate Observables

Start by finding the classical field of the
McLerran-Venugopalan model.

Continue by including the quantum corrections
of the nonlinear evolution equation.

Works for structure functions of hadrons and
nuclei, as well as for gluon production in
various hadronic collisions. Let us consider pA
collisions first.
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
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!
Understanding Cronin Effect
It is easier to understand this enhancement if we note that,
amazingly, one can rewrite gluon production cross section in a
kT –factorized form in terms of gluon distributions:
d  pA 2  S 1
2

d
q  p ( q, Y  y ) A (k  q, y )
2
2 
d k dy CF k
Since for nuclear unintegrated
gluon distribution we had
enhancement at high pT, it
leads to enhancement in RpA .
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. It resums powers of  ln 1/x =  Y.
(Yu. K., K. Tuchin, ’01)
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
q  p ( q, Y  y ) A (k  q, y )
2
2 
d k dy CF k
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.
Understanding Suppression
Again, using the kT-factorization formula
d  pA 2  S 1
2

d
q  p ( q, Y  y ) A (k  q, y )
2
2 
d k dy CF k
And remembering that the
unintegrated nuclear gluon
distribution at high (forward)
rapidity y is suppressed at all
pT, we conclude that RpA is
also suppressed.
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
Future Experimental Tests
Di-lepton Production
 To calculate hadron production one always needs to
convolute quark and gluon production cross sections with the
fragmentation functions. Since fragmentation functions are
impossible to calculate and are poorly known in general, they
introduce a big theoretical uncertainty.
 Di-lepton production involves no fragmentation functions.
It is, therefore, a much cleaner probe of the collision
dynamics.
 Theoretical calculation for di-lepton production in dAu is
pretty straightforward.
Di-lepton Production
from J. Jalilian-Marian, hep-ph/0402014
M2 is the photon’s invariant mass, kT and qT are total and
relative transverse momenta of the lepton pair.
 The photon does not interact (while everything else is just
like for gluons): theoretical calculations are simpler! They were
first performed by Kopeliovich, Schafer, and Tarasov in ’98.
Di-lepton Production
Again one should get suppression at forward rapidities at RHIC.
Here we plot Rp(d)A
integrated over kT and
qT , for both pA and dA,
for y=2.2 as a function
of M.
from J. Jalilian-Marian, hep-ph/0402014
Di-lepton Production
The suppression at forward rapidities at RHIC can also be seen
as a function of kT:
RpA as a function of kT
for M=2GeV for
y=1.5 (short-dashed)
and y=3 (dashed),
as well as for M=4GeV
and y=3 (lower solid
line).
from Baier, Mueller, Schiff, hep-ph/0403201
Di-lepton Production
and as a function of centrality:
RpA as a function of A
for kT=5 GeV, M=2GeV
and y=3 for the saturation
model (solid curve) and for
analytical estimate
RpA ~ A-0.124.
from Baier, Mueller, Schiff, hep-ph/0403201
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:
Is Suppression due to Proximity to
Fragmentation Region?
Kopeliovich et al suggest that observed suppression may
not be due to small-x evolution in the nuclear wave function,
but is due to large-x effects in the deuteron.
To resolve this issue one would ideally want to have d+Au
collisions at higher energy, where we could test the same
small-x region of the nuclear wave function for which x of the
deuteron is not that large anymore – CGC would expect
suppression, Kopeliovich no suppression.
Alternatively one can run dAu at lower energy: large-x effects
would stay the same, nuclear x would increase: CGC would
expect less suppression, Kopeliovich would expect the same
amount of suppression.
Open Charm Production
Kharzeev and Tuchin model open charm production in the
saturation/Color Glass formalism by the following model:
It results in the spectra which fall
off slower than PYTHIA:
Caution: only a model, for exact calculation see Tuchin ’04 and Blaizot et al, ’04.
Open Charm Production
Similar suppression applies to heavy flavors, in particular to
open charm. The figure below demonstrates that in going from
mid-rapidity to forward rapidity open charm production should
start scaling slower than Ncoll , which indicates suppression.
Charmed
Meson
Yield
from D. Kharzeev and
K. Tuchin,
hep-ph\0310358
Ncoll
Summary
A lot of interesting new information about
saturation/Color Glass dynamics can be found by
studying d+Au collisions at RHIC II.
Things to look at are:
 RpA of hadrons in forward direction (suppression?).
 RpA of open charm and J/y (forward suppression?).
 Di-leptons: cleaner signal, expect suppression in
forward direction as well.
 Various back-to-back correlations: broadening,
disappearance...

Summary

Understanding nuclear and hadronic wave
functions reveals interesting physics of
strong gluonic fields and complicated
interactions.