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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/d η vs. rapidity η

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qcd phenomenology and heavy ion physics

QCD Phenomenology and Heavy Ion Physics

Yuri Kovchegov

The Ohio State University


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.
particle multiplicity
Particle Multiplicity

In Saturation/Color Glass Physics one has

  • only one scalein the problem – the saturation scale QS .
  • the leading fields are classical:

The resulting gluon multiplicity is given by

such that

since d2b ~ S ~ p R2 , with R the nuclear radius.

particle multiplicity vs centrality
Particle Multiplicity vs. Centrality

Since and

we get :

which is not a constant due to running of the coupling:


particle multiplicity vs centrality1
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
Particle Multiplicity vs. Energy

Let’s try to use the same simple formula to check the

energy dependence of multiplicity. Start with

From saturation models of HERA DIS data we know that


Therefore we write



Levin ‘01

particle multiplicity vs energy1
Particle Multiplicity vs. Energy

Using the known multiplicity at 130 GeV Kharzeev and Levin

predicted multiplicity at 200 GeV using the above model:

The result agreed nicely with the data:


  • Energy dependence works too!

To understand the rapidity dependence one has to make

a few more steps. Starting with factorization assumption

inspired by the production diagram,

and assuming a saturation/CGC form of

the unintegrated gluon distribution f:


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

(see also Kharzeev & Nardi ’00, Kharzeev, Levin, Nardi ’01)

dn d in dau
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
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
  • Can fit the multiplicity data assuming that less particles were produced initially (smaller QS) but their numbers increased during thermalization.

Baier, Mueller,

Schiff, Son ‘02

bottom up scenario questions
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


Stronger than classical

field? Stronger than any

QCD gluon field?

It appears that

hadron spectra
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
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


for two sources – proton and


This classical field has been found by

Yu. K., A.H. Mueller in ‘98

gluon production in pa mclerran venugopalan model
Gluon Production in pA: McLerran-Venugopalan model

The diagrams one has to resum are shown here: they resum

powers of

Yu. K., A.H. Mueller,


gluon production in pa mclerran venugopalan model1
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,


Resulting inclusive gluon production cross section is given by

With the gluon-gluon dipole-nucleus

forward scattering amplitude

mclerran venugopalan model cronin effect
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


(Cronin Effect)

We can plot it for the quasi-classical

cross section calculated before (Y.K., A. M. ‘98):


Yu. K.

Tuchin ‘03

Classical gluon production leads to Cronin effect!

Nucleus pushes gluons to higher transverse momentum!

(see also Kopeliovich et al, ’02; Baier et al, ’03; Accardi and Gyulassy, ‘03)

proof of cronin effect
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!

RpA approaches 1 from above at high pT  there is an enhancement!

cronin effect
Cronin Effect
  • The height and position of the Cronin maximum are increasing functions of centrality (A)!

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.

including quantum evolution
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

a ln 1/x = a Y.

This has been done in Yu. K.,

K. Tuchin, hep-ph/0111362.

The rules accomplishing the inclusion of quantum corrections are


LO wave function

Proton’s BFKL

wave function


where the dipole-nucleus amplitude N is to be found from (Balitsky, Yu. K.)

including quantum evolution1
Including Quantum Evolution

Amazingly enough, gluon production cross section

reduces to kT –factorization expression (Yu. K., Tuchin, ‘01):

with the proton and nucleus “unintegrated

distributions” defined by

with NGp,A the amplitude of a GG dipole on a p or A.

our prediction
Our Prediction

Toy Model!


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

energy / rapidity


(Kharzeev, Levin, McLerran, ’02)

k / QS

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.)

  • At high energy / rapidity RpAat the Cronin peak becomes a decreasing

function of both energy and centrality.

other predictions
Other Predictions

Color Glass Condensate /

Saturation physics predictions

are in sharp contrast with other


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

r dau at different rapidities
RdAu at different rapidities


RCP – central

to peripheral


Most recent data from BRAHMS Collaboration nucl-ex/0403005

Our prediction of suppression was confirmed!

our model
Our Model





from D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0405045, where we construct a

model based on above physics + add valence quark contribution

our model1
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.



from D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0405045

back to back 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 correlations1
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 correlations2
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 correlations3
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 correlations4
Back-to-back Correlations
  • The observed data shows much less correlations for dAu than predicted by models like HIJING:
back to back correlations5
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.)

  • 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.
extended geometric scaling
Extended Geometric Scaling

A general solution to BFKL equation can be written as


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

(geometric scaling):

  • Inside the saturation region, , where nonlinear
  • evolution dominates (Levin, Tuchin ‘99 )

(ii) In the extended geometric scaling region, where g≈1/2:

(Iancu, Itakura, McLerran ‘02)

geometric scaling in dis
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:

phase diagram of high energy qcd
“Phase Diagram” of High Energy QCD




High Energy or


kgeom = QS2 / QS0


Moderate Energy

or Rapidity


 pT2

Cronin effect and low-pT suppression

region i double logarithmic approximation
Region I: Double Logarithmic Approximation

At very high momenta, pT >> kgeom, the gluon production is given by the

double logarithmic approximation, resumming powers of

Resulting produced particle multiplicity scales as


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 L

leading to

as QS0 >> L.


Yu. K.

Tuchin ‘03

RpA < 1 in Region I  There is suppression in DLA region!

region ii anomalous dimension
Region II: Anomalous Dimension

At somewhat lower but still large momenta, QS < kT < kgeom, the BFKL

evolution introduces anomalous dimension for gluon distributions:

Kharzeev, Levin, McLerran,


with BFKL g=1/2 (DLA g=1)

The resulting gluon production cross section scales as (we loose one power of QS)

such that

For large enough nucleus RpA << 1 – high pT suppression!

How does energy dependence come into the game?

region ii anomalous dimension1
Region II: Anomalous Dimension

A more detailed analysis

gives the following ratio in

the extended geometric

scaling region – our region II:

RpA is also a decreasing function of energy,

leveling off to a constant RpA ~ A-1/6 at very high energy.

  • RpAis 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
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 –

fA (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

which leads to

Levin, Tuchin ’99

Iancu, Itakura, McLerran, ‘02

  • We do not need to know fA to determine how Cronin peak scales with
  • energy and centrality! (The constant carries no dynamical information.)