# Presentation: 370009

## Lecture II

## 3. Growth of the gluon distribution and unitarity violation

## Solution to the BFKL equation

*Solution to the BFKL equation*

- Coordinate space representation *** = ln 1/x*
- Mellin transform + saddle point approximation
- Asymptotic solution at high energy
-                                                                       dominant energy dep.
-                                                                                              is given by exp{ ***S*** }
-                                         ** = 4 ln 2 =2.8

- 

## Can BFKL explain the rise of F2 ?

## Can BFKL explain the rise of F2 ?

- Actually, exponent is too large  ** = 4ln2 = 2.8
- NLO analysis necessary!
- But! The NLO correction is too large and the exponent becomes **NEGATIVE**!
- Resummation tried
- Marginally consistent
-     with the data
-    (but power behavior always has
-      a problem cf: soft Pomeron)

- exponent

## High energy behavior of the hadronic cross sections – Froissart bound

- Intuitive derivation of the Froissart bound  ( by Heisenberg)
- BFKL solution violates the unitarity bound.

- Total energy

- Saturation is implicit

## 4. Color Glass Condensate

## Saturation & Quantum Evolution - overview

## Population growth

*Population growth*

- 1. Exp-growth is tamed by **nonlinear** term   **saturation** **!!** (balanced)
- 2. Initial condition dependence disappears
-     at late time *dN/dt *=0  **universal !**
- 3. In QCD, *N**2* is from the gluon recombination *gg**g*.

- linear regime         non-linear
- exp   growth          saturation
-                               universal

-  Time (energy)

- -- *ignoring transverse dynamics* --

## McLerran-Venugopalan model

*McLerran-Venugopalan model*

- (Primitive) Effective theory of saturated gluons with high occupation number (sometimes called classical saturation model)
-  _**Separation of degrees of freedom**_ in a fast moving hadron
-  Large x partons    slowly moving in transverse plane  random source,  Gaussian weight function
-  Small x partons    classical gluon field induced by the source
-    LC gauge
-    (*A*+=0)
-  Effective at fixed x,
- no energy dependence in 
- Result is the same as independent
- multiple interactions (Glauber).

## Color Glass Condensate

**Color**  : gluons have “color” in QCD.
**Glass**** ** : the small *x* gluons are created by slowly moving valence-like partons (with large *x*) which are distributed randomly over the transverse plane
-      almost frozen over the natural time scale of scattering
-     This is very similar to the spin glass, where the spins are distributed randomly, and moves very slowly.
**Condensate**: It’s a dense matter of gluons. _Coherent state_ with high occupancy (~1/s at saturation). Can be better described as a _field_ rather than as a point particle.

## CGC as quantum evolution of MV

*CGC as quantum evolution of MV*

- Include quantum evolution wrt   = ln 1/x into MV model
- - Higher energy  new distribution *W*[]
- - Renormalization group equation is a linear functional differential equation for *W*[], but nonlinear wrt .
- - Reproduces the Balitsky equation
- - Can be formulated for (x) (gauge field)
-  through the Yang-Mills eq.
-       [*D* , *F*] = ***x*T
- *****x*T***x*T
- is a covariant gauge source
-      JIMWLK equation

- 

## JIMWLK equation

- Evolution equation for *W****[****]**,  *wrt*  *rapidity ***= ln 1/x*
-                   Wilson line in the adjoint representation gluon propagator
- Evolution equation for an operator *O*

## JIMWLK eq. as Fokker-Planck eq.

- The probability density *P(x,t)* to find a stochastic particle at point x at time t obeys the _Fokker-Planck equation_
-   *D* is the _diffusion coefficient_, and *F**i**(x)* is the _external force_.
- When *F**i**(x)* =0, the equation is just a diffusion equation and its solution is given by the Gaussian:
- JIMWLK eq. has a similar expression, but in a functional form
- Gaussian (MV model) is a solution when the second term is absent.

## DIS at small x : dipole formalism

- Life time of qq fluctuation is very long >> proton size
- This is a bare dipole (onium).

- _

- 1/ *M*p *x**         * 1/(Eqq-E*)

-  Dipole factorization

## DIS at small x : dipole formalism

*N*: Scattering amplitude

##  S-matrix in DIS at small x

- Dipole-CGC scattering in _eikonal approximation_
-   scattering of a dipole in one gauge configuration
- Quark propagation in a background gauge field
-                                                             average over the random
-                                                                         gauge field should be taken
-                                                              in the weak field limit,
-                                                                           this gives gluon distribution
-                                                                           ~ ((x)-(y))2

- stay at the same transverse positions

## The Balitsky equation

- Take *O=*tr*(V**x**+**V**y**) *as the operator
*V**x**+ *is in the fundamental representation
_*The Balitsky equation*_
*-- *Originally derived by Balitsky (shock wave approximation in QCD)* ’96*
- -- Two point function is coupled to 4 point function (product of 2pt fnc)
-    Evolution of 4 pt fnc includes 6 pt fnc.
- -- In general, CGC generates infinite series of evolution equations.
-    The Balitsky equation is the first lowest equation of this hierarchy.

## The Balitsky-Kovchegov equation (I)

_*The Balitsky equation*_
_*The Balitsky-Kovchegov equation*_
-      A closed equation for* *<tr(V+V)>
-      First derived by Kovchegov (99) by the independent multiple interaction
-         Balitsky eq.                       Balitsky-Kovchegov eq.
-      <tr(V+V) tr(V+V)>                         <tr(V+V)> <tr(V+V)>   (large *Nc*? Large A)
*        **N****(x,y) = 1 - (Nc**-1**)<*tr*(V**x**+**V**y**)>***is the scattering amplitude

## The Balitsky-Kovchegov equation (II)

*The Balitsky-Kovchegov equation (II)*

- Evolution eq. for the _onium_ (color dipole) scattering amplitude
-  **- **evolution under the change of scattering energy *s *(not *Q*2)
-      resummation of (s ln *s*)n   necessary at high energy
-  **-** _**nonlinear**_** **differential equation
-      resummation of strong gluonic field of the target
-  **-** in the weak field limit
-      reproduces the **BFKL** equation (linear)
-      scattering amplitude becomes proportional to _unintegrated gluon density_ of target

-  = ln 1/x ~ ln s  is the rapidity

## Saturation scale

*Saturation scale*

- - Boundary between CGC and non-saturated regimes

- - Similarity between HERA (x~10-4, *A*=1) and RHIC (x~10-2, *A*=200)
-                               *Q**S*(HERA) ~ *Q**S*(RHIC)

- 1/*Q****S***(x) : transverse size of gluons when the transverse
-               plane of a hadron/nucleus is filled by gluons

## Geometric scaling 

**Geometric scaling persists even outside of CGC!!**
-   “Scaling window”  [Iancu,Itakura,McLerran,’02]

- DIS cross section *x,Q*) depends only on *Qs*(x)/*Q* at small *x*

-   [Stasto,Golec-Biernat,Kwiecinski,’01]

- Once transverse area is filled with gluons, the only relevant variable is “number of covering times”.
-  Geometric scaling!!

**Natural interpretation in CGC **
*Qs*(*x*)/*Q*=(1/*Q*)/(1/*Qs*) : number of overlapping
-  1/Q: gluon size                                                      times

- Scaling window = BFKL window

## Summary for lecture II 

- BFKL gives increasing gluon density at high energy, which however contradicts with the unitarity bound.
- CGC is an effective theory of QCD at high energy
-     – describes evolution of the system under the change of energy
-     -- very nonlinear (due to )
-     -- derives a nonlinear evolution equation for 2 point function which corresponds to the unintegrated gluon distribution in the weak field limit
- Geometric scaling can be naturally understood within CGC framework.