Forward particle production in proton-nucleus collisions

Cyrille MarquetInstitut de Physique Théorique – CEA/Saclay

C. Marquet, Nucl. Phys. B705 (2005) 319C. Marquet, Nucl. Phys. A796 (2007) 41

C. Marquet and J. Albacete, in preparation+ work in progress

The hadron wavefunction in QCDgggggqqqqqqgqqq .........hadron

non-perturbative

regime: soft QCD

1, 1, ~hadron xkxkk QCDTQCDTQCDT

relevant for instance for

the total cross-section in

hadron-hadron collisions

perturbative regime,

dilute system of partons:

hard QCD (leading-twist

approximation)

relevant for instance for

top quark production

Three types of states:

S (kT ) << 1

weakly-coupled regime,

effective coupling constant:

dense system of partons

mainly gluons (small-x gluons):

the saturation regime of QCD

not relevant for experiments

until the mid 90’s

with HERA and RHIC: recent gain of interest for saturation physics

)/1ln( xS

The dilute regime1, 1, ~hadron xkxkk QCDTQCDTQCDT

1T

QCD

kThe dilute (leading-twist) regime:

hadron = a dilute system of partons which interact incoherently

)Q,/(ˆ)Q,()Q,( 22/

12 xxxdxx Bjapa

apartons x

BjDIS

Bj

for instance, the total cross-section in DIS

partonic cross-sectionparton density

leading-twistregime

1/kT ~ parton transverse size

as kT increases, the hadron gets more dilute

Dokshitzer GribovLipatov Altarelli Parisi

transverse view of the hadron

The saturation regime1, 1, ~hadron xkxkk QCDTQCDTQCDT

The saturation regime of QCD:the weakly-coupled regime that describes the collective behavior of quarks and gluons inside a high-energy hadron

1~)(Q

, 1T

s

T

QCD

kx

kThe saturation regime:

hadron = a dense system of partons,responsible for collective phenomena

the separation between the dilute and dense

regimes is caracterized by a momentum scale:

the saturation scale Qs(x)

Balitsky Fadin Kuraev Lipatov

as x decreases, the hadron gets more dense

• deep inelastic scattering at small xBj :

• particle production at forward rapidities y :

When is saturation relevant ?In processes that are sensitive to the small-x part of the hadron wavefunction

22

2

Q

Q

WxBj

in DIS small x corresponds to high energy

saturation relevant for inclusive,diffractive, exclusive events

pT , y

yT epsx 2

yT epsx 1 in particle production, small x corresponds

to high energy and forward rapidities

saturation relevant for the production ofjets, pions, heavy flavours, dileptons

at HERA, xBj ~10-4 for Q² = 10 GeV²

at RHIC, x2 ~10-4 for pT ² = 10 GeV²

Geometric scaling in DISgeometric scaling can be easily understood as a consequence of large parton densities

the hadron in the (Q², x) plane:

0.3

Stasto, Golec-Biernat and Kwiecinski (2001)

x < 10-2

lines parallel to the saturation line are lines ofconstant densities along which scattering is constant

Contents

• The Color Glass Condensate formalism- effective description of the small-x gluons- the JIMWLK evolution equation- scattering off the CGC and n-point functions

• Single particle production at forward rapidities- probes the two-point functions- inclusive spectra and modification factors at RHIC- from qualitative to quantitative CGC description

• Two-particle production at forward rapidities- probes more information about the CGC- comparisons with recent RHIC data

The CGC formalism

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 coordinate

y : 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

Balitsky (1996), Kovchegov (1999)

BK evolution at NLO• running coupling (RC) corrections to the BK equation

taken into account by the substitution

Kovchegov

Weigert

Balitsky

RC corrections represent most of the NLO contribution

(2007)

• the begining of the NLO-CGC era

first numerical solution

first phenomenological implementation

Albacete and Kovchegov (2007)

to successfully describe the proton structure function F2 at small x

Albacete, Armesto, Milhano and Salgado (2009)

Single particle production

Forward particle production

),(),( 22

212

2TT

TT kxfkxg

dykd

dk

kT , y

yT eksx 1

transverse momentum kT, rapidity y > 0

yT eksx 2

the large-x hadron should be described by

standard leading-twist parton distributions

the small-x hadron/nucleus should be

described by all-twist parton distributions

values of x probed in the process:

the cross-section:single gluon production

probes only the unintegrated

gluon distribution (2-point function)

Kovner and Wiedemann (2001), Kovchegov and Tuchin (2002), Dumitru and McLerran (2002)Blaizot, Gélis and Venugopalan (2004), Marquet (2005), Gélis and Mehtar-Tani (2006)

if the emitted particle is a (valence) quark, involves

if the emitted particle is a gluon, involves

The suppression of RdA

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 we learned

if forward rapidity data are included in npdfs fit, the resulting gluon distribution is over suppressed

forward rapidities are needed to see the suppression 22 GeV 2~),01.0(Q Aus

RdA and forward pion spectrum

first comparisons to data:

Kharzeev, Kovchegov and Tuchin (2004)Kharzeev, Levin and Nardi (2005)

Dumitru, Hayashigaki andJalilian-Marian (2006)

more recent work:

from qualitative to quantitative agreement

shows the importance of both evolutions:

xA (CGC) and xd (DGLAP)

shows the dominance

of the valence quarks

RdA

pT - spectrum

New NLO-BK description

this fixes the two parameters of the theory:- the value of x at which one starts to trust (and therefore use) the CGC description- and the saturation scale at that value of x

in very forward particle production in p+p collisions at RHIC, (where NLO DGLAP fails) using the CGC to describe the (small-x) proton also works

Albacete and C.M, in preparation

Betemps, Goncalves, de Santana Amaral (2009)

the shapes and normalizations are wellreproduced, except the 0 normalization

the speed of the x evolution and of

the pT decrease are now predicted

Two particles at forward rapidities

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

the modification factor were studied

- my calculation: two-particle production at forward rapidities

- but single particle production probes limited information about the CGC(only the 2-point function)

to strengthen the evidence, we need to studymore complex observables to be measured with the next d-Au run

d Au → h1 h2 XI computed C. Marquet, NPA 796 (2007) 41

(probes up to a 6-point function)

Central/forward correlations• first measurements of azinuthal correlations

signal

STAR, PRL 97 (2006) 152302

PHENIX, PRL 96 (2006) 222301

coincidenceprobability

• difficult to make robust predictions

- the fragmentation of low energy particles is not well known(fragmentation functions are not constrained at low z)

- the values of xA are at the limit of the CGC applicability(trigger at central rapidity high x)

a measurement sensitive to possible modifications

of the back-to-back emission pattern in a hard process

moderate values of xd, typically 0.5

dominant partonic process :

Two particles at forward rapidities

feasible in d-Au collisions at RHIC

(or p-Pb at LHC, but then xp ~ 0.1,and or important)

|k1|, |k2| >> QCD collinearfactorization of the quark density

h+T h1+h2+X

y1 ~ y2 ~ 3 : both h1 and h2

in forward hemisphere

very low values of xA, typically < 10-4

need CGC resummation of large logarithms αS ln(1/xA) ~ 1 and large gS A ~ 1

the CGC cannot be describedby a single gluon distribution

The two-particle spectrum

collinear factorization of quark density in deuteron Fourier transform k┴ and q┴

into transverse coordinates

pQCD q → qg wavefunction

b: quark in the amplitudex: gluon in the amplitudeb’: quark in the comp. conj. amplitudex’: gluon in the comp. conj. amplitude

interaction with hadron 2 / CGCn-point functions that resums the powers of gS A and the powers of αS ln(1/xA)

Nikolaev, Schäfer, Zakharov and Zoller (2005)I obtain a formula similar to that of

2- 4- and 6-point functionsthe scattering off the CGC is expressed through the following correlators of Wilson lines:

if the gluon is emitted before the interaction, four partons scatter off the CGC

if the gluon is emitted after the interaction, only the quarks interact with the CGC

interference terms, the gluon interacts in the amplitude only (or c.c. amplitude only)

Blaizot, Gélis and Venugopalan (2004)

need more than the 2-point function: no kT factorization same conclusions in sea quark

production

and two-gluon productionusing Fierz identities that relate WA and WF, we recover the z → 0 (soft gluon) limit

Jalilian-Marian and Kovchegov (2004)

Baier, Kovner, Nardi and Wiedemann (2005)

we will now include the xA evolution

Performing the CGC average

characterizes the density of color charges along the projectile’s path

with this model for the CGC wavefunction squared, it is possible to compute n-point functions

• a Gaussian distribution of color sources

is the two-dimensional massless propagator

• applying Wick’s theorem

when expanding in powers of α and averaging,

all the field correlators can be expressed in terms of ),'(),( yx zz dc

the difficulty is to deal with the color structure

Fujii, Gelis and Venugopalan (2006)

MV model and BK evolution

in the large-Nc limitis related to in the following way

With this model for the CGC wavefunction squared, it is possible to compute then-point functions:

Blaizot, Gélis and Venugopalan (2004)

and obeys the BK equation:

we will use the MV initial condition: McLerran and Venugopalan (1994)

with the initial saturation scale

→

Final expression

quark density in dilute hadron

unintegrated gluon density of CGC(Fourier transform of 2-point function)

the final expression for the cross-section can be decomposed into three pieces:

modified q → qg vertexdue to multiple scattering

: pQCD q → qg wavefunction in momentum space

with zero quark masses, I reduces towith

goal: study the CGC evolution try to avoid the competition between the

xd (DGLAP) evolution of and the small xA evolution of and

Forward/forward correlations• the focus is on the away-side peak

where non-linearities have the biggest effect

• pT dependence

the away-side peak is restored at higher pT

typical coincidence probability

to calculate the near-side peak,one needs di-pion fragmentation functions

suppressed away-side peak

Centrality dependence• comparison with data for central collisions

there is a very good agreement with STAR data(an offset is needed to account for the background)

• the centrality dependence

this shows the qualitativebehavior of the correlation

for a given impact parameter,the initial saturation scale used is

Conclusions

• Forward particle production in d+Au collisions- the suppressed production at forward rapidities was predicted- there is a good agreement with CGC calculations- now that NLO-BK is known, one should stop using models

• Two-particle correlations at forward rapidities- probe the theory deeper than single particle measurements- forward/forward correlations probe x as small as in the RdA measurement- jet quenching seen in central d+Au collisions- first theory(CGC)/data comparison successful, more coming