# Presentation: 505502

## A CLASSICAL ODDERON IN HIGH ENEGY QCD

- Raju  Venugopalan
- BNL

**RBRC Workshop, Sept. 27th-29th, 2005**

## Outline of talk:

** ****The ground state of a large nucleus at high energies**
** ****Random walks & Path Integrals for SU(N) quarks**
**  ****A “classical’’ Odderon **
**  ****Results **
** ****Summary and Outlook**

## Work in collaboration with Sangyong Jeon (McGill/RBRC)

** ****A CLASSICAL ODDERON IN QCD AT HIGH ENERGIES.**
-      _[Sangyong Jeon](http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+Jeon,+Sangyong)_,  Raju Venugopalan  **Phys.Rev.D71:125003,2005**

** ****RANDOM WALKS OF PARTONS IN SU(N(C)) AND CLASSICAL**
**     ****REPRESENTATIONS OF COLOR CHARGES IN QCD AT SMALL X.**
-      Sangyong Jeon,  Raju Venugopalan  **Phys.Rev.D70:105012,2004**

**Odderon paper inspired by**
** ****ODDERON IN THE COLOR GLASS CONDENSATE.**
-       Y. Hatta,  E. Iancu, K. Itakura,  L. McLerran
**      ****Nucl.Phys.A760:172-207,2005**

##  ground state of a nucleus at high energies 

**Consider large nucleus in the IMF frame:**

## Born-Oppenheimer: separation of large x and small x modes

## The effective action

- Generating functional:

**Focus in this talk on weight functional - for large nuclei**

## Coarse grained field theory: 

**Coarse grained field theory**:

**# of  random quarks in box of size**** **

## Well defined math problem:

- Given k non-interacting quarks belonging the the fundamental SU(N) representation:
- What is the distribution of degenerate irreducible representations?
- What is the most likely representation?
- Is it a classical representation ? N_c = Infinity is classical
-       even for k=1
**        **

**Random walk problem in space spanned**
**      ****by the n_c -1 Casimirs of SU(N)**

## QCD IN THE LARGE A ASYMPTOTICS:

**Evolution effects small =>**

**Kinematic region of applicability:**

**Kovchegov: Extend discussion to**** **

## An SU(2) Random walk

**Random walk of spin 1/2  partons:**

**In general,**

- =  **Multiplicity of representation s when k**
**    ****fundamental reps.  are multiplied**

## Binomial Coefficients satisfy:

**Multiplicity satisfies:**

**with**

**Using ****Stirling’s**** formula, k >> s >> 1:**

## Probability:

**degeneracy**
**of state**

**Mult. of**
**representation**

**# degrees**
**of freedom**

**Analogous to Maxwell-Boltzmann distribution**

**Casimir:**

**Average value:**

## PATH INTEGRAL:

**Coarse graining -> Box of size 1/p_t in transverse plane**

**Sum over spins in box:**

** **

**Classical color/spin density:**

**=>**

## Summing over all boxes  -> Classical path integral over 

**                                                ****SU(2) color charge density**

**Color charge squared per unit area- closely related **
**to saturation scale. **
**For A >> 1, coupling runs as a function of this scale**

**McLerran, RV**
**Kovchegov**

## Random walk of SU(3) Color charges

**Denote SU(3) representations by (m, n) :**

**Recursion relation for SU(3) :**

**=**

**+**

**+**

## =  multiplicity of (m, n) state in the kth iteration

**=  ****multiplicity of (m, n) state in the kth iteration**

**Trinomial coefficients:**

**Solution:**

**Again, use Stirling’s formula...**

## Quadratic Casimir

**Quadratic Casimir**

**Cubic Casimir**

**Probability:**

**Dimension of **
**representation**

**Note:**** **

## Prove: 

**Classical color charge:**

**Proof:**

**For any SU(3) representation,**

**Canonically conjugate “Darboux” variables**

## Canonical phase space volume of SU(3):

**Johnson; **
**Marinov;**
**Alexeev,Fadeev,**
**Shatashvilli**

**Hence,**** **

**Measure of probability integral has identical argument **
**in m & n to RHS- hence can express in terms of LHS.**
** ****End of Proof.**

## Path Integrals:

**As in SU(2), with**

**MV Path integral measure for SU(3):**

**Generates Odderon **
**excitations!**

**Path integral approach reproduces diagrammatic **
**computations of  dipole and baryon C= -1 operators**

##  Representations of order             dominate for k >> 1 

** ****These representations are classical - can be represented **
**    ****by an SU(3) classical path integral **
** ****Can repeat analysis for gluon and quark-anti-quark pairs**
**    ****- only quadratic Casimir contributes...**
**  ****Can add glue representations to quarks-result as for **
**     ****valence quarks - with larger weight.**

**To summarize...**

## On to the Odderon...

## DIS:

**DIS:**

**IN CGC:**

## Dipole Odderon operator:

**Hatta et al.**

**To lowest order**

## Can compute < O > with “SU(3) measure”

**To lowest order,**** **

**To all orders (in the parton density)**

**Kovchegov, Szymanowski, Wallon**

## 3-quark Baryon state scattering off CGC

**Dosch, Ewerz, Schatz**

**To lowest order,**** **

**All order result feasible, but tedious...**

## Conclusions

** ****The ground state of a large nucleus contains configurations **
**     ****that generate Odderon excitations - these can be traced **
**     ****to the random walk of valence partons in color space.**
**  ****These results represent a “rigorous” proof  in the large **
**     ****A asymptotics of QCD at high energies**
**   ****Dipole Odderon and Baryon Odderon operators **
**      ****are computed**
**   ****Phenomenological consequences - to be investigated **
**      ****further**

## Why are higher dim. Reps. classical?

**Invariance group                  or  higher dim. rep.****   **

**Yaffe**

**For a system prepared in this state, uncertainity in mom. **
**& position vanishes in this limit -> Coherent States**

**Gitman, Shelepin**

## SU(N)

**Can follow same recursion procedure...**
**Quadratic Casimir dominates: **
**successive N-2 Casimirs parametrically **
**suppressed by **