Forward particle productionin d+Au collisions

in the CGC framework

Cyrille Marquet

Institut de Physique Théorique, CEA/Saclay

the spectrum and

Motivation- after the first d+Au run at RHIC, there was a lot of new results on

single inclusive particle production at forward rapidities

kdyddN

kdyddN

NR

hXpphXdA

colldA 22

1

the suppressed production (RdA < 1) was predicted in the Color Glass Condensate picture of the high-energy nucleus

d Au → h X

y increases

the modification factor were studied

- but single particle production probes limited information about the CGC(only the 2-point function)to strengthen the evidence, we need to study

more complex observables to be measured with the new d+Au run

- the experimental focus has been on IdA

a correlation measurement sensitive to possible modificationsof the back-to-back emission pattern in a hard process d Au → h1 h2 X

Outline

• Saturation and the Color Glass Condensatethe unintegrated gluon distribution and the BK equationmulti-parton distributions in the nuclear wave function

• Single particle production at forward rapiditiesdifferent parametrizations of the unintegrated gluon distributionRdA and the success of the CGCrunning coupling corrections to the BK equation

• Probing small x with two-particle correlationsthe ideal final-state kinematicscorrelations in azimuthal angle and IdA

some results of CGC calculations

Saturation and theColor Glass Condensate

Gluon saturationx : parton longitudinal momentum fraction

kT : parton transverse momentum

the distribution of partons

as a function of x and kT :

dilute/dense separation characterized by the saturation scale Qs(x)

QCD linear evolutions:

DGLAP evolution to larger kT (and a more dilute hadron)BFKL evolution to smaller x (and denser hadron)

QCD non-linear evolution: meaning

recombination cross-section

gluon density per unit areait grows with decreasing x

recombinations important when

the saturation regime: for with

this regime is non-linearyet weakly coupled

The Color Glass Condensatethe idea of the CGC is to describe the saturation regime with strong classical fields

McLerran and Venugopalan (1994)

lifetime of the fluctuations

in the wave function ~

high-x partons ≡ static sources

low-x partons ≡ dynamical fields

small x gluons as radiation field

),(,

z zFD cc

valence partonsas static random

color source separation between

the long-lived high-x partons

and the short-lived low-x gluons

CGC wave function

classical Yang-Mills equations

• an effective theory to describe the saturation regime

gggggqqqqqqgqqq .........hadron CGC][hadron xD

from , one can obtainthe unintegrated gluon distribution,

as well as any n-parton distributions

2][x

in the A+=0 gauge

The small-x evolution

the solution gives 3.03/12 ~),(Q xAAxs

the evolution of with x is a renormalization-group equation2

][x

for a given value of k², the saturation regime in a nuclear wave functionextends to a higher value of x compared to a hadronic wave function

22][][

)/1ln( x

JIMWLKx H

xd

d

• the JIMWLK equation

is mainly non-perturbative, but its evolution is known

Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner

2][x

the energy evolution of cross-sections is encoded in the evolution of2

][x

in the CGC framework, any cross-section is determined by colorless combinations ofWilson lines , averaged over the CGC wave function

][][2

SDS xx ][S

• Observables

Scattering off the CGC

scattering of a quark:

• this is described by Wilson lines

dependence kept implicit in the following

))()((1

1][ xyxy FFc

WWTrN

T

x : quark space transverse coordinatey : antiquark space transverse coordinate

the dipole scattering amplitude:qq

this is the most common averagefor instance it determines deep inelastic scattering

• the 2-point function or dipole amplitude

xTxy

it is used in many CGC calculations without precaution

when only the two-point function enters in the formulation of

a cross-section, the so-called kT-factorization is applicable

• more complicated correlators for less inclusive observables

The Balitsky-Kovchegov equation

YYYTTTT zyxzzyxz the BK equation is a closed equation for obtained by assuming

YTxy

YYYYYY

TTTTTzd

TdYd

zyxzxyzyxzxy yzzxyx

22

22

)()()(

2

robust only for impact-parameter independent solutions

• the BK equation

r = dipole size• the unintegrated gluon distribution

• modeling the unintegrated gluon distribution

the numerical solution of the BK equation is not useful for

phenomenology, because this is a leading-order calculation

instead, CGC-inspired parameterizations are used for ,

with a few parameters adjusted to reproduce the data

Single particle productionat forward rapidities

Forward particle production

),(),( 22

212

2TT

TT kxfkxg

dykd

dk

kT , y

yT eksx 1

transverse momentum kT, rapidity y > 0

yT eksx 2

• forward rapidities probe small values of x

the large-x hadron should be described by

standard leading-twist parton distributions

the small-x hadron/nucleus should be

described by CGC-averaged correlators

values of x probed in the process:

the cross-section:single gluon production

probes only the unintegrated

gluon distribution (2-point function)

The KKT parametrization• build to be used as an unintegrated gluon distribution

the idea is to play with the saturation exponentKovchegov, Kharzeev and Tuchin (2004)

• the DHJ version

• the BUW version

KKT modified to feature exact geometric scaling

Dumitru, Hayashigaki and Jalilian-Marian (2006)

Boer, Utermann and Wessels (2008)

in practice is always replaced by before the Fourier transformation

KKT modified to better account for geometric scaling violations

RdA and forward pion spectrum

Kharzeev, Kovchegov and Tuchin (2004)

RdA

• first comparison to data

qualitative agreement

with KKT parametrization

kdyddN

kdyddN

NR hXpp

hXdA

colldA

2

21

xA decreases(y increases)

• the suppression of RdA was predicted

in the absence of nucleareffects, meaning if the gluons in the

nucleus interact incoherently like in A protons

What about the large-x hadron?

Dumitru, Hayashigaki and Jalilian-Marian (2006)

shows the importance of both

evolutions: xA (CGC) and xd (DGLAP)

shows the dominance

of the valence quarks

for the pT – spectrum

with the DHJ model

• getting a quantitative agreement requires correct treatment

it has been proposed as an alternative explanation

pA collisions at the LHC would answer that

• suppression of RdA due to large-x effects?

both initial particles should not be described by a CGC, only the small-x hadron

Running coupling corrections• running coupling corrections to the BK equation

taken into account by the substitution

Kovchegov

Weigert

Balitsky

• consequences

similar to those first obtained by the simpler substitution

running coupling corrections slow down the increase of Qs with energy

also confirmed by numerical simulations, howeverthis asymptotic regime is reached for larger rapidities

Probing small x withtwo-particle correlations

Final-state kinematics

• the best situation

11 , yk 22 , yks

ekekx

yy

p

21 21

s

ekekx

yy

A

21 21

probes 2-, 4- and 6- point functions

final state :

one can test more information about the CGC compared to single particle production

at forward rapidities in order to probe small x

two hadrons close in rapidityboth in the same forward direction

xp ~ 1, xA << 1

xp ~ 1, xA ~ 1BFKL evolution ?

this increases xA a lot (~ a factor 20)probes initial condition, not evolution

• a large rapidity separation between the two particles ?

this does not probe the nuclear wavefunction at small-x

- doesn’t probe large parton densities- as much effect in pp as in d+Au- we know from Tevatron that for y < 5 there is no effect

C. Marquet, NPA 796 (2007) 41

RHIC d+Au measurements

PHENIX STAR

• central/forward correlation

trigger at central rapidity : high-x

correlation function coincidence probability

conditional yield

signal

STAR, PRL 97 (2006) 152302PHENIX, PRL 96 (2006) 222301

trigger at forward rapidity : low-x

transverse momentum range includes the region

• need to do forward/forward correlation

first measurments for x > 0.01problems to calculate the pp baseline

Status of CGC calculation• the d+Au part

results at parton level ready

but problems with pdf’s and fragmentation functions at low pT

(meaning pT < 1.5 GeV, which includes most experimental bins)

• the p+p part needed for IdA

to be computed in NLOQCD framework

potential problems for low pT bins

results at hadron level ready at high-pT

not yet ready to put numberson this plot, but hopefully soon

Conclusions• Forward particle production in d+Au collisions

- the suppressed production at forward rapidities was predicted- there is a good agreement with CGC calculations

• What we learned from single particle production- both d and Au should not be described by a CGC, the deuteron pdf is important- this only tests limited information about the CGC: 2-point function ~ gluon density- now that NLO-BK is known, one should stop using models for- if the suppression is due to (small)large-x effects, there will be (more)less suppression at the LHC

• Two-particle correlations- probe more than the 2-point function- no large rapidity interval is needed between the two particles, in fact this wouldn’t probe large parton densities- only forward/forward correlations will probe x as small as in the RdA measurement- CGC predictions almost ready