Transcript Lecture II

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 Q2 < Qs4/L2
- Phenomenological success The CGC fit for F2 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
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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
aS(Q=Qs(x))
Geometric Scaling from the BK eq.
~ F.T. of N(x)
• Numerical solution to the BK equation shows the
geometric scaling and its violation
scaling variable
Geometric Scaling above Qs
“Phase diagram” as a summary
Energy (low high)
BFKL
BFKL,
BK
Parton gas
Transverse resolution
(low high)
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
Froissart bound from gluon saturation
Froissart bound from gluon saturation
Coefficient in front of ln2 s
s ~ B ln2 s
-- Froissart Martin bound
B = p/mp2 = 62 mb
-- Experimental data (COMPETE)
B = 0.3152 mb
-- CGC + confinement initial condition
LO BFKL B = 2.09 ~ 8.68 mb (aS=0.1 ~ 0.2)
rNLO BFKL B = 0.446 mb (aS=0.1 )
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 g* 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
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 (c2=0.78), but saturation scale becomes smaller
Other observables (I)
Vector meson production,
Forshaw, et al. PRD69(04)094013
F2Diff
hep-ph/0404192
Other observables (II)
• FL Goncalves and Machado, hep-ph/0406230
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.