Lecture IILecture II

3. Growth of the gluon distribution3. Growth of the gluon distribution and unitarity violation and unitarity violation

Solution to the BFKL equationSolution to the BFKL equation• Coordinate space representation = ln 1/x

• Mellin transform + saddle point approximation

• Asymptotic solution at high energy

dominant energy dep.

is given by exp{ S }

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 = 4 ln 2 =2.8

Can BFKL explain the rise of FCan BFKL explain the rise of F22 ? ?

Can BFKL explain the rise of FCan BFKL explain the rise of F22 ? ?

Actually, exponent is too large = 4ln2 = 2.8

NLO analysis necessary! But! The NLO correction is too large and the exponent becomes

NEGATIVE!

Resummation tried Marginally consistent

with the data

(but power behavior always has

a problem cf: soft Pomeron)

exp

one

nt

High energy behavior of the hadronic High energy behavior of the hadronic cross sections – Froissart boundcross sections – Froissart bound

Intuitive derivation of the Froissart bound ( by Heisenberg)

BFKL solution violates the unitarity bound.

Total energy

Saturation is implicit

4. Color Glass Condensate4. Color Glass Condensate

Saturation & Quantum Evolution - overviewSaturation & Quantum Evolution - overview

dilute

Low energyBFKL eq. [Balitsky, Fadin,Kraev,Lipatov ‘78]

N : scattering amp. ~ gluon number: rapidity = ln 1/x ~ ln s

exponential growth of gluon number violation of unitarity

High energy

dense,saturated, random

Balitsky-Kovchegov eq.

Gluon recombination nonlinearity saturation, unitarization, universality

[Balitsky ‘96, Kovchegov ’99]

Population growthPopulation growth

Solution population explosion

N : polulation densityT.R.Malthus (1798)Growth rate is proportional to the population at that time.

P.F.Verhulst (1838) Growth constant decreases as N increases.(due to lack of food, limit of area, etc)

1. Exp-growth is tamed by nonlinear term saturation !! (balanced)

2. Initial condition dependence disappears at late time dN/dt =0 universal !3. In QCD, N2 is from the gluon recombinat

ion ggg.

Logistic equationlinear regime non-linear exp growth saturation

universal

Time (energy)

-- ignoring transverse dynamics --

McLerran-Venugopalan modelMcLerran-Venugopalan model(Primitive) Effective theory of saturated gluons with high occu

pation number (sometimes called classical saturation model)

Separation of degrees of freedom in a fast moving hadron

Large x partons slowly moving in transverse plane random source, Gaussian weight function

Small x partons classical gluon field induced by the source

LC gauge

(A+=0)

Effective at fixed x,

no energy dependence in

Result is the same as independent

multiple interactions (Glauber).

Color Glass CondensateColor Glass CondensateColor : gluons have “color” in QCD.

Glass : the small x gluons are created by slowly moving valence-like partons (with large x) which are distributed randomly over the transverse plane

almost frozen over the natural time scale of scattering

This is very similar to the spin glass, where the spins are distributed randomly, and moves very slowly.

Condensate: It’s a dense matter of gluons. Coherent state with high occupancy (~1/s at saturation). Can be better described as a field rather than as a point particle.

CGC as quantum evolution of MVCGC as quantum evolution of MVInclude quantum evolution wrt = ln 1/x into MV model

- Higher energy new distribution W[]

- Renormalization group equation is a linear functional differential equation for W[], but nonlinear wrt .

- Reproduces the Balitsky equation

- Can be formulated for (x) (gauge field)

through the Yang-Mills eq.

[D , F] = xT

xTxT

is a covariant gauge source

JIMWLK equation

JIMWLK equationJIMWLK equationEvolution equation for W[], wrt rapidity 　 = ln 1/x

Wilson line in the adjoint representation gluon propagator

Evolution equation for an operator O

JIMWLK eq. as Fokker-Planck eq.JIMWLK eq. as Fokker-Planck eq.The probability density P(x,t) to find a stochastic particle at point x

at time t obeys the Fokker-Planck equation

D is the diffusion coefficient, and Fi(x) is the external force.

When Fi(x) =0, the equation is just a diffusion equation and its solution is given by the Gaussian:

JIMWLK eq. has a similar expression, but in a functional form

Gaussian (MV model) is a solution when the second term is absent.

DIS at small x : dipole formalismDIS at small x : dipole formalism

Life time of qq fluctuation is very long >> proton sizeThis is a bare dipole (onium).

_

1/ Mp x 1/(Eqq-E*)

Dipole factorization

DIS at small x : dipole formalismDIS at small x : dipole formalism

N: Scattering amplitude

S-matrix in DIS at small xS-matrix in DIS at small xDipole-CGC scattering in eikonal approximation scattering of a dipole in one gauge configuration

Quark propagation in a background gauge field average over the random gauge field should be taken

in the weak field limit, this gives gluon distribution ~ ((x)-(y))2

stay at the same transverse positions

The Balitsky equationThe Balitsky equationTake O=tr(Vx

+Vy) as the operator

Vx+ is in the fundamental representation

The Balitsky equation

-- Originally derived by Balitsky (shock wave approximation in QCD) ’96

-- Two point function is coupled to 4 point function (product of 2pt fnc) Evolution of 4 pt fnc includes 6 pt fnc.

-- In general, CGC generates infinite series of evolution equations. The Balitsky equation is the first lowest equation of this hierarchy.

The Balitsky-Kovchegov equation (I)The Balitsky-Kovchegov equation (I)The Balitsky equation

The Balitsky-Kovchegov equation

A closed equation for <tr(V+V)> First derived by Kovchegov (99) by the independent multiple interaction Balitsky eq. Balitsky-Kovchegov eq. <tr(V+V) tr(V+V)> <tr(V+V)> <tr(V+V)> (large Nc? Large A)

N(x,y) = 1 - (Nc-1)<tr(Vx+Vy)>is the scattering amplitude

The Balitsky-Kovchegov equation (II)The Balitsky-Kovchegov equation (II)

Evolution eq. for the onium (color dipole) scattering amplitude - evolution under the change of scattering energy s (not Q2)

resummation of (s ln s)n necessary at high energy

- nonlinear differential equation

resummation of strong gluonic field of the target

- in the weak field limit

reproduces the BFKL equation (linear)

scattering amplitude becomes proportional to unintegrated gluon density of target

= ln 1/x ~ ln s is the rapidity

R

Saturation scaleSaturation scale

- Boundary between CGC and non-saturated regimes

- Similarity between HERA (x~10-4, A=1) and RHIC (x~10-2, A=200)

QS(HERA) ~ QS(RHIC)

- Energy and nuclear A dependences

LO BFKL

NLO BFKL

[Gribov,Levin,Ryskin 83, Mueller 99 ,Iancu,Itakura,McLerran’02]

[Triantafyllopoulos, ’03]

A dependence is modified in running coupling case [Al Mueller ’03]

1/QS(x) : transverse size of gluons when the transverse

plane of a hadron/nucleus is filled by gluons

Geometric scaling Geometric scaling

Geometric scaling persists even outside of CGC!! “Scaling window” [Iancu,Itakura,McLerran,’02]

DIS cross section x,Q) depends only on Qs(x)/Q at small x [Stasto,Golec-Biernat,Kwiecinski,’01]

Once transverse area is filled with gluons, the only relevant variable is “number of covering times”. Geometric scaling!!

=Qs(x)/Q=1

Natural interpretation in CGC Qs(x)/Q=(1/Q)/(1/Qs) : number of overlapping 1/Q: gluon size times

Scaling window = BFKL window

consistent with theoretical results

Saturation scale from the data

Summary for lecture II Summary for lecture II • BFKL gives increasing gluon density at high energy, which however con

tradicts with the unitarity bound.

• CGC is an effective theory of QCD at high energy

– describes evolution of the system under the change of energy

-- very nonlinear (due to )

-- derives a nonlinear evolution equation for 2 point function which corresponds to the unintegrated gluon distribution in the weak field limit

• Geometric scaling can be naturally understood within CGC framework.