# Presentation: 122339

## Kazunori Itakura 
IPNS, KEK

At the RBRC Alumni workshop
19th June, 2006

**Kazunori Itakura **
**IPNS, KEK**
**At the RBRC Alumni workshop**
**19****th**** June, 2006**

**Some topics about ****Color Glass Condensate**

## Plan

**Plan**

**Introduction**
**Theoretical framework for the CGC**
**Mean-field picture**
-       **-- The Balitsky-Kovchegov equation **
**       ****-- Reaction-diffusion dynamics (F-KPP equation)**
**Beyond mean-field picture**
-       **-- Reaction-diffusion dynamics: exact implementation**
**       ****-- Fluctuation & Stochastic F-KPP equation**
**       ****-- Back to the saturation physics**
**Summary**

## Introduction

**Introduction**

_**High-energy limit of QCD is the **__**Color**__** **__**Glass**__** **__**Condensate**__** **__(__**C**__**G**__**C**__)__**!!**_

**                                                                                                 ****Named by Iancu, Leonidov, & McLerran (2000)**
**Color :  ****A matter made of gluons with colors.**
**Glass:  **** Almost  “frozen” random color source creates gluon fields**
**Condensate:  ****High density. Occupation number** ~ ***O***( **1***/****s *)

_**Experimental supports:**_
**    ****Enhancement of gluon distribution in a proton at small *****x***
**    ****Geometric scaling at DIS and eA**
**    ****Suppression of *****R*****dAu**** at forward rapidity in deuteron-Au collisions, etc,...**

## Theoretical framework for the CGC

**Theoretical framework for the CGC**

## Theoretical framework for the CGC (1/2) 

** ****Large *****x***** partons**
**  ****(mostly quarks)**
- Large longitudinal mom. *p***+**
-   small LC energy *E*LC
-   long life time ~ 1/ *E*LC

**Small *****x***** partons**
** ****(mostly gluons)**
- Small longitudinal mom. *p***+**
-   large LC energy *E*LC
-   short life time ~ 1/ *E*LC

**Theoretical framework for the CGC (1/2)**** **

**Effective theory of a fast moving hadron**
**                                                                                                                                                                     **McLerran-Venugopalan picture

## Theoretical framework for the CGC (2/2)

**Theoretical framework for the CGC (2/2)**

**The JIMWLK equation**  (Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner)

**Renormalization group equation**

- 0

- 1

*x*0

**

*A*

*W**x*0[**]

*W**x*1[**]

-     **weight function for random source**

** ****Only weight func.  **
**  ****gets modified at **
**  ****ln (1/x) accuracy**
**         ******** ****RG for *****W***

*x*0 > *x*1

*x*

## Mean-field picture

**Mean-field picture**

## Mean-field picture (1/4) 

**Mean-field picture (1/4)**** **

_**Gluon number**_** ~ 2pt function of Wilson lines**
**Evolution equation for 2pt operator **
**     ****contains 4pt function **
**    ******** ****Mean-field approx.**** : necessary to obtain a closed equation**
**The Balitsky-Kovchegov equation**

-  **random gauge field**

## Mean-field picture (2/4)

**Mean-field picture (2/4)**

**BFKL + non-linear term **
**      ********            ****saturates (unitarizes) at fixed *****b *****= (*****x*****+*****y*****)/**2** :**
**Saturation scale *****Q******s*****(*****Y *****) **
**      ******** ****typical     size of gluons when     area is occupied: 1/*****Q******s*****(*****Y *****) **
**      ******** ****increases with rapidity *****Y*****  :   *****Q******s*****2****(*****Y *****) ~ **e*Y*
**Geometric Scaling**
**      ******** ****amplitude             is a function of ****(*****x***-***y*****)*****Q******s*****(*****Y *****)**
**Approximate scaling persists even outside of the CGC regime**

**The Balitsky-Kovchegov equation**

_**Consequences**_

## Mean-field picture (3/4)

**Mean-field picture (****3****/4)**

**Emerging picture**

- Non-perturbative (Regge)

-    1*/x  in*
*log scale*

*Q*2  *in log scale*

**Parton gas**

**Extended **
**scaling **
**regime**

**CGC**

**Higher energies ******

**Fine transverse resolution ******

- BFKL
-    BK

- DGLAP

**QCD2

*Q**S*2(*x*) ~ 1/*x*: grows as *x * 0

*Q**S*4(*x*)/**QCD2

## Mean-field picture (4/4)

**Mean-field picture (****4****/4)**

**The Reaction-Diffusion dynamics**

- Munier & Peschanski (2003~)

**Within a reasonable approximation, the BK equation in momentum space is rewritten as **_**the F-KPP equation**_ (Fisher, Kolmogorov, Petrovsky, Piscounov)
-                                                   where   *t* ~ ***    x ~ *ln* k*2  and  *u*(*t, x*) ~ *N***(*k*).

## Beyond mean-field picture

**Beyond mean-field picture**

## Beyond mean-field picture (1/5)

**Beyond mean-field picture (1/5)**

**Lessons from the reaction-diffusion dynamics**

## Beyond mean-field picture (2/5)

**Beyond mean-field picture (2/5)**

**Reaction-diffusion dynamics: exact implementation**

_**Master equation**_

*    **n**i*(*t*)** = **number of particles at site* i*** ,  **{*n**i*}** = (***n**0*** , ***n**1***, ***n**2 *** .....)**** **configuration
*    **P*({*n**i*}; *t* ) : probability to have a particular configuration {*n**i*} at time* t*
- _Change of  __*P*__({__*n*__*i*__}; __*t*__ )__ _
-   1) **Diffusion** (hopping to the right *i*  *i*+1, and  to the left *i*  *i –*1* *) at rate ***D/h*****2**
-   2) **Splitting** (AAA) at each site *i* at rate ********s**
-   3) **Merging **(AAA) at each site *i* at rate ********m**

- loss

- gain

*h*

- gain

- loss

## Beyond mean-field picture (3/5)

**Beyond mean-field picture (3/5)**

**Field-theory representation**

## Beyond mean-field picture (4/5)

**Beyond mean-field picture (4/5)**

**Fluctuation & stochastic F-KPP equation**

- Systematic analysis of fluctuation possible  Front position as a collective coordinate for the F-KPP solution   (work in progress)

## Beyond mean-field picture (5/5)

**Beyond mean-field picture (5/5)**

**Back to saturation physics**

- Difference btw Balitsky and BK eqs. becomes significant
-       when gluon (dipole) number is small (high transverse momentum).
- Inclusion of full fluctuation replaces the F-KPP equation by
-       the **stochastic** F-KPP equation.
-     Even the Balitsky equation must be modified so that it contains
-         dipole splitting.
-     Pomeron loop
- 3.   Saturation scale becomes
-          **slowly increasing** due to _diffusion_ at the edge,
-          **stochastic variable** due to fluctuation term in sFKPP eq.

- By Mueller, Shoshi, Iancu, Munier, Tryantafyllopoulos, Soyez, .....

## Summary 

**Summary**** **

**Theoretical description of the Color Glass Condensate is improving.  Analogy with the reaction-diffusion dynamics is very useful. **
** **
**In particular, the effects of fluctuation beyond the mean-field BK picture have been recognized to be significant in dilute regime (at high transverse momentum) **
** ****Slowly-growing and stochastic saturation scale is obtained.**
**Deeper understanding of the reaction-diffusion system is necessary. The field-theory representation will be useful.**

## Thanks 

**Thanks**** **

**Stayed in BNL from June 2000 to May 2003. The first collision at RHIC was just after I started. Naturally (blindly?) entered CGC physics guided by Larry. Spent a great time with nice people and good physics. **
**Shared a great exciting moment with many experimental discoveries:  geometric scaling, high pt suppression of *****R*****AA ****and *****R*****dA ****, etc, etc...**
**BNL and RBRC are keeping the position as the center for the CGC physics in the world, still involving some younger Japanese people (Y. Hatta, K. Fukushima, and more?) **
**Many thanks to all the people in BNL during and after my stay. Always feel at home when I visit here. **
**Hope that the activity in RBRC continue in the future! **