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3. Growth of the gluon distribution and unitarity violation

Solution to the BFKL equation

- Coordinate space representation t = ln 1/x
- Mellin transform + saddle point approximation
- Asymptotic solution at high energy
dominant energy dep.

is given by exp{ w aS t }

w = 4 ln 2 =2.8

Can BFKL explain the rise of F2 ?

Actually, exponent is too largew = 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)

exponent

High energy behavior of the hadronic cross 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 Condensate Froissart bound

Low energy Froissart bound

BFKL eq.[Balitsky, Fadin,Kraev,Lipatov ‘78]

dilute

N :scattering amp. ~ gluon number

t : rapidity t = ln 1/x ~ ln s

exponential growth of gluon number

violation of unitarity

[Balitsky ‘96,

Kovchegov ’99]

Balitsky-Kovchegov eq.

dense,

saturated,

random

Gluon recombination

nonlinearity saturation,

unitarization,

universality

High energy

Saturation & Quantum Evolution - overviewT.R.Malthus Froissart bound (1798)

N:polulation density

Growth rate is proportional to the population at that time.

Solutionpopulation explosion

P.F.Verhulst (1838)

Growth constant k decreases as N increases.

(due to lack of food, limit of area, etc)

Logistic equation

-- ignoring transverse dynamics --

Population growthlinear regime non-linear

exp growth saturation

universal

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 recombination ggg.

Time (energy)

McLerran-Venugopalan model Froissart bound

(Primitive) Effective theory of saturated gluons with high occupation 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 m

Result is the same as independent

multiple interactions (Glauber).

Color Glass Condensate Froissart bound

Color : 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/as at saturation). Can be better described as a field rather than as a point particle.

CGC as quantum evolution of MV Froissart bound

Include quantum evolution wrt t = ln 1/x into MV model

- Higher energy new distributionWt[r]

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

- Reproduces the Balitsky equation

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

through the Yang-Mills eq.

[Dn , Fnm] = dm+r (xT)

- T2 a(xT) = r(xT)

(r is a covariant gauge source)

JIMWLK equation

D

JIMWLK equation Froissart bound

Evolution equation for Wt [a], wrtrapidityt = ln 1/x

Wilson line in the adjoint representation gluon propagator

Evolution equation for an operator O

JIMWLK eq. as Fokker-Planck eq. Froissart bound

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 formalism Froissart bound

_

Life time of qq fluctuation is very long >> proton size

This is a bare dipole (onium).

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

Dipole factorization

DIS at small x : dipole formalism Froissart bound

N: Scattering amplitude

S-matrix in DIS at small x Froissart bound

Dipole-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

~ (a(x)-a(y))2

stay at the same transverse positions

The Balitsky equation Froissart bound

Take 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) Froissart bound

The Balitsky equation

The Balitsky-Kovchegov equation

A closed equationfor<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)

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

The Balitsky-Kovchegov equation (II) Froissart bound

Evolution eq. for the onium (color dipole) scattering amplitude

- evolution under the change of scattering energys (not Q2)

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

-nonlineardifferential 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

t = ln 1/x ~ ln s is the rapidity

- Energy and nuclear Froissart boundA dependences

LO BFKL

NLO BFKL

R

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

[Triantafyllopoulos, ’03]

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

Saturation scale1/QS(x) : transverse size of gluons when the transverse

plane of a hadron/nucleus is filled by gluons

- 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)

= Froissart bound

Qs(x)/Q=1

Saturation scale from the data

consistent with theoretical results

Geometric scalingDIS cross section s(x,Q) depends only on Qs(x)/Q at small x

[Stasto,Golec-Biernat,Kwiecinski,’01]

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

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

Geometric scaling!!

Geometric scaling persists even outside of CGC!!

“Scaling window”[Iancu,Itakura,McLerran,’02]

Scaling window = BFKL window

Summary for lecture II Froissart bound

- BFKL gives increasing gluon density at high energy, which however contradicts 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.

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