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Lecture II. 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.

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

Solution to the bfkl equation
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 f 21
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)


High energy behavior of the hadronic cross sections froissart bound
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
4. Color Glass Condensate Froissart bound

Saturation quantum evolution overview

Low energy Froissart bound

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


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.




Gluon recombination

 nonlinearity saturation,



High energy

Saturation & Quantum Evolution - overview

Population growth

T.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 growth

linear regime non-linear

exp growth saturation


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 ggg.

 Time (energy)

Mclerran venugopalan model
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


Effective at fixed x,

no energy dependence in m

Result is the same as independent

multiple interactions (Glauber).

Color glass condensate
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
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


Jimwlk equation
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
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
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 formalism1
DIS at small x : dipole formalism Froissart bound

N: Scattering amplitude

S matrix in dis at small x
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
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
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
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

Saturation scale

- Energy and nuclear Froissart boundA dependences




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

[Triantafyllopoulos, ’03]

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

Saturation scale

1/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)


Geometric scaling

= Froissart bound


Saturation scale from the data

consistent with theoretical results

Geometric scaling

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


  • 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
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.