Small-x physics

2- The saturation regime of QCD and the Color Glass Condensate

Cyrille Marquet

Columbia University

The saturation regime of QCD

gluon kinematics

recombination cross-section

gluon density per unit area

• gluon recombination in the hadronic wave function

recombinations important when

• the BFKL approximationsumming terms leads to a growth of the gluon density

in the hadronic wave function but high-density effects are missing

when the gluon density is large enough, gluon recombination is important

the saturation regime: for with

magnitude of Qs x dependencethis regime is non-linear,yet weakly coupled

the idea of the CGC is to take into account this effect via strong classical fields

the CGC is moving in the direction and the gauge isthe current is

solving Yang-Mills equations gives

The Color Glass Condensate

gggggqqqqqqgqqq .........hadron

separation between the long-lived high-x

partons and the short-lived low-x gluons

CGC wave function

CGC][hadron xD

the BFKL equation is recovered in the low-density regime

short-lived fluctuations

• effective description of a high-energy hadron/nucleus

][][2

fDf xx in practice we deal with CGC averages such as

we are interested in the evolution of with x, which sums both

2][x

Outline of the second lecture

• The evolution of the CGC wave functionthe JIMWLK equation and the Balitsky hierarchy

• A mean-field approximation: the BK equationsolutions: QCD traveling wavesthe saturation scale and geometric scaling

• Beyond the mean field approximationstochastic evolution and diffusive scaling

• Computing observables in the CGC frameworksolving evolution equation vs using dipole models

The evolution of theCGC wave function:

the B-JIMWLK equations

The JIMWLK equationJalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner

22][][

)/1ln( x

JIMWLKx H

xdd

2][x• a functional equation for the x evolution of :

with

the JIMWLK equation gives the evolution of the wave function for small enough x

the equivalent Balitsky equations are obtained by consideringthe scattering of simple test projectiles (dipoles) off the CGC

the Wilson lines sum powers of

adjoint representation

Scattering off the CGCtheir interaction should conserve their spin, polarisation, color, momentum …

in the high-energy limit, the eigenstates are simple when the partonstransverse momenta are Fourier transformed to transverse coordinates

x, y : transverse coordinates

in the large-Nc limit, further simplification: the eigenstates are only made of dipoles

),( yxother quantum numbers aren’t explicitely written, the

eigenvalues depend only on the transverse coordinates),();...;,( 11 NN yxyx

• eigenstates ?

xyxy ST 1][][

2 SDS xCGC

qq

),();...;,(...),();...;,(ˆ1111 11 NNCGCNN nn

SSS yxyxyxyx yxyx

• eigenvalues

scattering amplitude off the (dense) target CGC

the interaction with the CGC conserves transverse position

Dipoles as test projectiles

))()((11][ uvuv FFc

WWTrN

T

][][][nn11nn11

2 vuvuvuvu TTDTT xx

u : quark space transverse coordinatev : antiquark space transverse coordinate

the dipole:qq

JIMWLK equation → evolution equation for the dipole correlators

scattering of the quark:

• dipoles are ideal projectiles to probe small distances

dependence kept implicit in the following

The Balitsky hierarchy

an hierarchy of equation involving correlators with more and more dipoles

Balitsky (1996)

YYYYYYTfTTTTT

zdT

dYd )(

)()()(

2 22

22

uvzvuzuvzvuzuv vzzuvu

2

1)()(c

YYY NOTfTTTfTT

dYd

zvuzzvuzzvuz

• equations for dipoles scattering of the CGC

the BFKL kernel

BFKL saturation

general structure:

we will now derive the first equation of the hierarchy

solving the B-JIMWLK equations gives , …which can then be used to compute observables

YTT zyxz

YTxy

• in the large Nc limit, the hierarchy is restricted to dipoles

• the valence component (using the mixed space):

Recall the dipole wave function

)();()(~22 yxyx qqyxddd

x : quark space transverse coordinatey : antiquark space transverse coordinate

other degrees of freedom are not explicitely written

qqd

such that only perturbative sizes r << 1/ QCD are included

)(~ r

22 )()( zyzy

zxzx

)();(),,(1)(~ 2/1

22222 yxzyxyx qqzdNCgyxddd cFS

),();();(),,(2 agqqTzdg a

S zyxzyx

from QCD at order gS

colorindex

gqqqqd

the dipole rapidity:

• the dipole dressed with one gluon

Elastic scattering of the dipole

)();())()(()();(ˆ yxxyyx qqWWTrqqS FF

scattering of the quarkscattering of the antiquark

for the qqg component:

let’s compute the elastic scattering amplitude Ael(Y ) in the

dipole-CGC collision where is the total rapidityY = YCGC + Yd

• the dipole scattering is described by Wilson lines

)(xFW )(yFW )(zAW

][][)(~)(2222 xyyx TDyxddYiA Yel

• in the frame in which and

))()((11][ xyxy FFc

WWTrN

T Y

Tyxdd xyyx 222 )(~

Frame independence of Ael(Y )

CGCYel TyxddYiA xyyx222 )(~)(

using the following identity to get rid of the adjoint Wilson line

))()((2

1))()(())()((21)())()(( xyxzzyzxy FF

cFFFF

abA

bF

aF WWTr

NWWTrWWTrWTWTWTr

= 1/2 -1/(2Nc)gqq qqqq qq

• in the frame in which andgqqqqd

one obtains

our two expressions for Ael(Y )should be identical, thisrequirement in the limitY - YCGC dY 0 gives

the first Balitsky equation

• frame independence:

CGCCGCCGCCGC YYYYCGC TTTTT

zdYY zyxzxyzyxzyzzx

yx22

22

)()()(

2)(

A mean field approximation: the Balitsky-Kovchegov equation

The BK equation

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

Txy

YYYYYY

TTTTTzd

TdYd zyxzxyzyxzxy yzzx

yx

22

22

)()()(

2

let’s consider impact-parameter independent solutions

as gets close to 1 (the stable fixed point of the equation), the non-linear term becomes

important, and , saturates at 1

with increasing Y, the unitarization scale get bigger

• solutions: qualitative behavior

• obtain by neglecting correlations

at small Y, is small, and the quadratic term can be neglected, the equationreduces then to the linear BFKL equation and rises exponentially with Y

r = dipole size

Coordinate vs momentum space

both equation are in the same universality class as the F-KPP equation

unintegrated gluon distributiondipole scattering amplitude

• let’s go to momentum space

coordinate space momentum space

due to conformal invariance, the linear part of the equation is the same for and

genuine saturation: no real saturation:

linear BFKL equation linear BFKL equation

The F-KPP equation• same features as the BK equation

the precise forms of the space derivatives and of the non-linear term doesn’t matter

Fisher, Kolmogorov, Petrovsky, Piscounov

when expanding to second order,the equations are the same (in momentum space)in spite of small difference, same universality class

• dictionary F-KPP → BK

time derivative space derivative linear term meaning exponential growth

non linear term meaning saturation

F-KPP and BK asympotic (in t or Y) solutions are the same: traveling waves

these equations belong to the same universality class

position:

Traveling wave solutions

the speed of the wave is determinedonly by the linear term of the equation

• what is a traveling wave

saturationregion

wave traveling at speed v,independently of the initial condition

provided it is steep enough

there the initial conditionhas not been erased yet

BK solutions: same rapidity evolutionquantitative features can be derived

• the formation of the traveling wave

Recall the BFKL solutions

initial condition in Mellin space• the minimal speed

• a superposition of waves with speeds

is obtained for and its value is

this will be the speed of theasymptotic traveling wave

• the saturation exponent

for the traveling wave to form, the initial condition must feature

Y0

Y >Y0

QCD traveling waves• the initial condition is steep enough Munier and Peschanski (2004)

• asymptotic solutions of BK

in QCD

same features for exceptin the saturation region

• the saturation scale

with then

called geometric scaling

Numerical solutions

sub-asymptotic corrections (≡ geometric scaling violations) are also known:

• numerical simulations confirm the results

for the scattering amplitude for the saturation scale

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

Beyond the mean fieldapproximation

Recall the Balitsky hierarchy• equations for dipole correlators ][][][

nn11nn11

2 vuvuvuvu TTDTT xx

it describes the so-called dense regime where

YYYYYYTfTTTTT

zdT

dYd )(

)()()(

2 22

22

uvzvuzuvzvuzuv vzzuvu

21)()( cYYYNOTfTTTfTT

dYd

zvuzzvuzzvuz

general structureat large Nc

the BK equation is agood approximation

the equation for is fine, its dilute limit is the BFKL equation, however the other equationsare incomplete, they do not agree with Mueller’s dipole model in the limit

incorrect results for have consequences for

YTxy

1Y

TT zyxz

• still there is something missing

describing properly the dilute regime is also important

the B-JIMWLK equations do not describe it properly

YTxy

A new hierarchy of equations• the hierarchy must be completed

BFKL fluctuations (included to recover Mueller’sdipole model in the dilute limit) important when

saturationimportant when

Iancu and Triantafyllopoulos (2005)

this QCD evolution is equivalent to a stochastic process

• reformulation into a Langevin problem

average over therealisations of the stochastic process

the correlators can all be obtained from a singlestochastic quantity which obeys a Langevin equation

average over theCGC wave function

The stochastic F-KPP equation

high-energy QCD evolution = stochastic process in the universalityclass of reaction-diffusion processes, of the sF-KPP equation

Iancu, Mueller and Munier (2005)

• the QCD Langevin equation

),,,(),()()()()()(

ln21622

222

222222 YTvudzdd Y

S zvuvuzuvu

zyuxuyzx

vu

deterministic part of the equation: BK !

),(),(),(),(),()()()(

2),( 22

22yzzxyxyzzxyzzx

yxyx YYYYYY TTTTTzd

TdYd

noise

the reduction to one dimension introduces the noise strength parameter

noise

r = dipole size

• the stochastic F-KPP equation

A stochastic saturation scale

equivalent to a stochastic saturation scale Qs

the dispersion coefficient: D

• the stochastic evolutionthe Langevin evolution generates, at each rapidity, a stochastic ensemble of solutions for

the average speed of the waves:

(for ) DYSS 22 Q/Qln

DYDYP S

)Q/Q²(lnexp1)Q(ln

22S2

SQs is a stochastic variable distributedaccording to a Gaussian probability law:

corrections to the Gaussian law for improbable fluctuationsalso known, picture confirmed by numerical simulations C.M., Soyez and Xiao

average saturation scale:

• the probability distribution of Qs

analytic expressions for v and Dare known only for very small

A new scaling law

: the diffusion is negligible and )(Q)( YrTrT SY

one recovers geometric scaling

DYYrTrT SY)(Qln)( 22: the diffusion is important and

new regime: diffusive scaling

• the average dipole scattering amplitude

very high energies

The diffusive scaling regime

in the diffusive scaling regime:

the amplitudes are dominated by events thatfeature the hardest fluctuation of

in average the scattering is weak, yet saturation is the relevant physics

• important properties

• running coupling corrections

• conclusion at the moment

push the diffusive scaling regime up to asymptotically high energies

Dumitru et al.

using the mean field approximation is good enough for phenomenology

Computing observablesin the CGC framework

The CGC averages][][

2 SDS x

CGC

x

the energy evolution of cross-sections is encoded in the evolution of 2][x

in the CGC framework, any cross-section is determined by colorlesscombinations of Wilson lines, averaged over the CGC wave function

22][][

)/1ln( x

JIMWLKx H

xdd

• the Wilson line correlators

Balitsky/BK equation for the correlators

this wave function is mainly non-perturbative, but its evolution is known

xFF WWTr ))()(( xy

this is the most common average, (for instance it determines deep inelastic scattering)its Fourier transform is the unintegrated gluon distribution, it obeys the BK equation

• the 2-point function or dipole amplitude

for 2-particle correlations

• more complicated correlators

for instance

The strategy

the same correlators enter in the formulation of very different processesthe more exclusive the final state is, the more complicated the correlators are

• what should be done in principle

- express the cross-sections in terms of correlators of Wilson lines

use appropriate initial conditionsinclude non-perturbative effects such as the impact parameter dependencerunning coupling effects should also be included

- use B-JIMWLK or BK equation to compute the correlators

many processes have been studied in the CGC framework using dipole models

• what is done in practicesolving evolution equations with running coupling is not the favored approach

instead one uses parametrizations with features inspired from theoretical results

it makes it easier to deal with impact parameter dependence of the dipole amplitude

famous dipole models: GBW, IIM, IPsat

Outline of the third lecture• The hadron wave function

summary of what we have learned

• The saturation modelsfrom GBW to the latest ones

• Deep inelastic scattering (DIS)the cleanest way to probe the CGC/saturationallows to fix the model parameters

• Diffractive DIS and other DIS processesthese observables are predicted

• Forward particle production in pA collisionsand the success of the CGC picture at RHIC