Lecture IIILecture III

5. The Balitsky-Kovchegov 5. The Balitsky-Kovchegov equationequation

Properties of the BK equationProperties of the BK equation

The basic equation of the Color Glass Condensate

- Rapid growth of the gluon density when the field is weak, and saturation of the scattering amplitude N(x,y) 1 as s infinity also unitarity is restored

- Energy dependence of saturation scale computable from linear + saturation.

Known up to NLO BFKL

Qs large as s large

- Absence of infrared diffusion problem (cf BFKL)

- Geometric scaling exists in a wide region Q2 < Qs4/2

- Phenomenological success The CGC fit for F2 at HERA

Saturation scale from linear regimeSaturation scale from linear regimeMatching the linear solution to saturated regime

The BK equation is known only at the LO level, but one can compute Qs(x) up to (resummed) NLO level by using this technique.

Absence of Infrared diffusionAbsence of Infrared diffusion• BFKL equation has the infrared diffusion problem: even if one starts from the init

ial condition well localized around hard scale, eventually after the evolution, the solution enters the nonperturbative regime. Thus, BFKL evolution is not consistent with the perturbative treatment.

• However, there is no infrared diffusion

problem in the BK eq.

• Most of the gluons are around Qs(x).

Justifies perturbative treatment

S(Q=Qs(x))

Geometric Scaling from the BK eq.Geometric Scaling from the BK eq.

• Numerical solution to the BK equation shows the geometric scaling and its violation

scaling variable

~ F

.T. o

f N(x

)

Geometric Scaling above QsGeometric Scaling above Qs

““Phase diagram” as a summaryPhase diagram” as a summary

En

ergy

(lo

w

hig

h)

Transverse resolution (low high)

BFKL

Parton gas

BFKL, BK

DGLAP

Froissart bound from gluon saturationFroissart bound from gluon saturation

BK equation gives unitarization of the scattering amplitude at fixed impact parameter b.

However, the physical cross section is obtained after the integration over the impact parameter b.

The Froissart bound is a limitation for the physical cross section, and it is highly nontrivial if this is indeed satisfied or not.

Froissart bound from gluon saturationFroissart bound from gluon saturation

Froissart bound from gluon saturationFroissart bound from gluon saturation

Froissart bound from gluon saturationFroissart bound from gluon saturation

Coefficient in front of lnCoefficient in front of ln22 s s

~ B ln2 s

-- Froissart Martin bound

B = /m2 = 62 mb

-- Experimental data (COMPETE) B = 0.3152 mb

-- CGC + confinement initial condition

LO BFKL B = 2.09 ~ 8.68 mb (S=0.1 ~ 0.2)

rNLO BFKL B = 0.446 mb (S=0.1 )

6. Recent progress in ph6. Recent progress in phenomenologyenomenology

HERA (Lecture III)RHIC AuAu (Lecture IV) RH

IC dAu (Lecture IV)

Attempts with saturation (I)Attempts with saturation (I)Golec-Biernat, Wusthoff modelGolec-Biernat, Wusthoff model

Attempts with saturation (II)Attempts with saturation (II)Improvements of the GBW modelImprovements of the GBW model

Attempts with saturation (III)Attempts with saturation (III)our approachour approach

Geometric scaling and its Geometric scaling and its violationviolationTotal p cross section (Stasto,Kwiecinski,Golec-Biernat)

in log-log scale deviation from the pure scaling in linear scale

Figure by S.Munier

Attempts with saturation (III)Attempts with saturation (III)our approachour approach

The CGC fitThe CGC fit

The CGC fitThe CGC fit

DGLAP regime

Effects of charm (not shown in the paper)

• Performed the fit with charm included (for example ) • Still have a good fit (=0.78), but saturation scale becomes smaller

Other observables (I)Other observables (I)Vector meson production, F2

Diff

Forshaw, et al. PRD69(04)094013 hep-ph/0404192

Other observables (II)Other observables (II)• FL Goncalves and Machado, hep-ph/0406230

Summary for lecture III Summary for lecture III The Balitsky-Kovchegov equation is the evolution equation for the change o

f scattering energy when it is large enough. It is a nonlinear equation, and leads to

-- saturation (unitarization) of the scattering amplitude -- geometric scaling and its violation also free from the infrared diffusion problem.

One can compute the cross section and its increase as increasing energy. Froissart bound is satisfied if one adds the information of the confinemen

t.

HERA data at small x is well described by the CGC fit.