Lecture III

5. The Balitsky-Kovchegov equation

Properties 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 Q 2 < Qs 4 /  2 - Phenomenological success  The CGC fit for F 2 at HERA

Saturation scale from linear regime Matching 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 diffusion BFKL equation has the infrared diffusion problem: even if one starts from the initial 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. Numerical solution to the BK equation shows the geometric scaling and its violation scaling variable ~ F.T. of N(x)

Geometric Scaling above Qs

“Phase diagram” as a summary Energy (low  high) Transverse resolution (low  high) BFKL Parton gas BFKL, BK DGLAP

Froissart 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 saturation

Coefficient in front of ln 2 s  ~ B ln 2 s -- Froissart Martin bound B =  /m  2 = 62 mb -- Experimental data (COMPETE) B = mb -- CGC + confinement initial condition LO BFKL B = 2.09 ~ 8.68 mb (  S =0.1 ~ 0.2) rNLO BFKL B = mb (  S =0.1 )

6. Recent progress in phenomenology 6. Recent progress in phenomenology HERA (Lecture III) RHIC AuAu (Lecture IV) RHIC dAu (Lecture IV)

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

Attempts with saturation (II) Improvements of the GBW model

Attempts with saturation (III) our approach

Geometric scaling and its violation Total  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) our approach

The 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) Vector meson production, F 2 Diff Forshaw, et al. PRD69(04) hep-ph/

Other observables (II) F L Goncalves and Machado, hep-ph/

Summary for lecture III The Balitsky-Kovchegov equation is the evolution equation for the change of 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 confinement. HERA data at small x is well described by the CGC fit.