Color Glass Condensate

HIM MEETING( 광주 )Dec. 4, 2004

0. Introduction

1) What is CGC?

2) Physics related to the CGC

3) Classical Theory(MV)

4) Quantum Theory

5) RGE

<<< HEAVYLY BORROWED >>>From IANCU, LEONIDOV & McLERRAN

(hep-ph/0202270)Try McLerran or A. Mueller for general references

Notation:

Light Cone(LC) coordinate:

Light Cone gauge:

Rapidity:

Pseudorapidity:

Feynman, Bjorken x:

I. What is CGC ?

- Color: colored particles

- Glass: a) disordered

b) Solid in short time scale but

Liquid in long time scale.

- Condensate: highly dense, ~ 1/alpha_s

II. Physics Related to the CGC

A. Total Cross Section of Hadron-Hadron:

1) Intrinsically non-perturbative even though E increase

2) Regge theory in 70s but not so satisfactory

3) Froissart bound:

Unitarity of scattering matrix

4) Why the unitarity saturated.

5) Is this understandable from QCD?

B. Particle Production

1) Can we compute dN/dy even at y=0?

2) Feynman scaling:

Cut the green in half (low energy) and move to red

(high energy) distribution. They match!!! i.e.

independent of energy

As E goes up, the small x degree of freedom fills

in while the large x held fixed

Some kind of RG property shows up.

C. DIS(Electron-Hadron)

1) Electron emits virtual photon

fluctuate into quark-antiquark

interact with partons in hadron

2) The rapidity distribution for gluons of hadron is almost

same as the distribution for produced hadrons after

hadron-hadron collision(or heavy ion collision)

3) However, ZEUS data show sharp increase in rapidity

distribution SMALL-x problem.

4) The saturation must be shown up!!!

the saturation momentum:

5) The saturation momentum increase as E increases

Once

Weak coupling:

Hope gluon, quark distribution can be calculable.

However, it is highly dense so that perturbation

cannot be applicable

D. Geometrical ScalingStructure functions from DIS depend on (in general) (x, Q^2/Lambda^2), but the function depend only on

at x < 0.01 (HERA), i.e. independent of x.

Some kind of scaling at small x.

E. Universality

1) Weak universality:

At sufficient high energy, physics must depend on the specific properties of hadron(nuclei) via saturation scale Q_s(x,A), i.e. With same Q_s, they have to have same properties.

2) Strong universality:

Statistical sense. The behavior of correlation function

is given by universal critical exponents.

III. Classical Theory (MV)

A. Basics

WHAT WE WILL SOLVE:

: Classical soft gluon field

: Color current of fast partons in x^+

direction sitting at x^- = 0

- Softer partons have larger energies(note that )

thus shorter life-times

time-independent color source in view of soft field:

treat the density as a classical random variable

with some probability, weight function, as functional:

Using ,

- The Gluon correlation functions at scale

for example,

- Gluon distributions:

: # of gluons having longitudinal momentum between and and transverse size

The gluon distribution with transverse momenta

per unit rapidity:

To get the readable form:

Consider the equation of motion in light cone gauge without source:

for + component,

therefore,

The quantization:

with the commutation relation,

The gluon density with Fock space operators,

Using that

In coordinate space,

This form is gauge dependent!!!!!

Now inserting WILSON LINES:

where the Wilson line is

We choose the path such that

along the horizontal line,

along the vertical line, (retarded condition)

IN RESULT,

Another interesting quantity: number of gluons per unit rapidity and transverse phase space:

And for homogeneous transverse space,

B. Formal Solution of

가 . Anticipating a solution,

so A^i are pure gauge,

나 . Now choose Covariant gauge:

( 가 ) 항을 이용 ,

다 . Formal Solution

Thus,

라 . Gauge transforming to LC-gauge

Since , we get the equation with solution,

마 . Classical source is dummy and ‘measure’ and ‘weight function’ are gauge invariant so that

C. MV approximation

Independent color sources: A*N_c valence quarks

Consider a tube of transverse area:

The number of valence quarks in the tube:

따라서 ,

한편 , 위의 correlator 들은 ( 마 ) 항을 이용할 때 다음의 weight function 을 사용하면 유도될 수 있다 ,

Linearized Solution:

gluon field 에 대한 운동방정식에서 선형항만 사용하여 풀면 ,

Exact and non-linear solution:이 경우 다음의 correlator 계산이 필수 ,

여기서 기본 propagator 는

위의 correlator 를 전개한 후 위의 propagator 을 이용하여 contraction 을 계산하면 ,

여기서 ,

using

Final Solution:

Quantum Theory and RGE

can be presented by someone

who is an expert on the subject !!

THANKS