Itakura

Perturbative Odderon in the Color Glass Condensate: 

Perturbative Odderon in the Color Glass Condensate in collaboration with E. Iancu (Saclay), L. McLerran & Y. Hatta (BNL) Kazunori Itakura (SPhT, CEA/Saclay  KEK in two weeks) based on hep-ph/0501171

Outline: 

Outline Introduction Odderon in Regge theory and in perturbative QCD Why Odderon in CGC?? C-odd operators in CGC Relevant operators for dipole-CGC & 3quark-CGC scatterings Odderon evolutions dipole-CGC scattering  decomposition of the Balitsky equation, BFKL equation in weak-field regime 3quark-CGC scattering  new equation, reduces to the BKP eq. in weak-field regime Summary

Odderon / Introduction (I): 

Odderon = Leading “C-odd” exchange in hadron scatt. at high energies. “C-odd” counterpart of the Pomeron (see Ewerz’s talk) Odderon / Introduction (I)

Perturbative Odderon / Introduction (II): 

The BKP equation for 3 gluons [Bartels, Kwiecinski-Praszalowicz ‘80] F: amplitude for exchange of three reggeized gluons in a color singlet C-odd state Pair-wise interaction between two gluons among three  BFKL evolution HBFKL The physical amplitude is obtained after convoluting the impact factor of the projectile Two solutions for BKP eq. with 3 gluons: Janik-Wosiek (‘99) , Bartels, Lipatov & Vacca (‘00) Perturbative Odderon / Introduction (II) * * * *

Why Odderon in CGC? / Introduction (III): 

Why Odderon in CGC? / Introduction (III) Perturbative Pomeron in the Color Glass Condensate dipole-CGC scattering ( dipole operator + JIMWLK equation) The relevant operator for the Pomeron (see talks by Venugopalan, Iancu) Two reggeized gluon exchange in linear regime two Wilson lines in nonlinear regime  BFKL equation But n-reggeon dynamics (BKP) is also important at high energy Need to investigate n-reggeon dynamics in the CGC which is in principle applicable for n-reggeons. The first step: 3 gluon exchange in linear regime  Odderon ! What is the relevant operator for the Odderon exchange??? Can we reproduce the BKP equation in the CGC???

General strategies in CGC: 

General strategies in CGC

C-odd operator in dipole-CGC scatt.: 

Transition from C-odd to C-even dipole states 　　　　　　　　　　　　　　　　　　　　　 Relevant operator - anti-symmetric under the exchange of x and y: O(x,y) = - O(y,x) - imaginary part of the dipole operator. Weak field expansion  leading order is 3 gluons gauge invariant combination! (a  a + c) C-odd operator in dipole-CGC scatt.

C-odd operator in 3-quark--CGC scatt.: 

C-odd operator in 3-quark--CGC scatt. Consider the scattering of a color singlet “3-quark state” and transition from C-even to C-odd 3 quark states Relevant operator “baryonic Wilson lines” Weak field expansion 3 gluons with d-symbol, gauge invariant ________ ________ all the possible ways of attaching

Evolution of the dipole Odderon: 

Evolution of the dipole Odderon Evolution eq. for the dipole Odderon  “imaginary part” of the Balitsky eq. couple to the Pomeron N(x,y) = 1- 1/Nc Re tr(V+xVy) becomes equivalent to Kovchegov-Szymanowski-Wallon (‘04) if one assumes factorization <NO>  <N><O>. initial condition computable with a classical gauge field + color averaging or in an extended McLerran-Venugopalan model (Jeon-Venugopalan ‘05) linear part = the BFKL eq. (but with C-odd initial condition)  reproduces the BKP solution with the largest intercept found by Bartels, Lipatov & Vacca (KSW,04) intercept reduces due to saturation: <O(x,y)> decreasing as <N(x,y)>  1 Evolution of N(x,y) is also modified due to Odderon: 2 Odderons 1 Pomeron BFKL * * * * * *

Evolution of the dipole Odderon (II): 

Evolution of the dipole Odderon (II) The presence of imaginary part (odderon) affects the evolution equation for the scattering amplitude N(x,y). Balitsky equation new contribution! - Two Odderons can merge into one Pomeron! N=1, O=0 is the stable fixed point.

3-quark--Odderon operator: 

3-quark--Odderon operator Baryonic Wilson line operator multiplying the identity One can rewrite 3quark-Odderon operator as manifestly gauge invariant reduces to dipole-Odderon operator when two coordinates are the same Oproton(x,z,z) = O(x,z) diquark ~ antiquark can compute nonlinear evolution equation for Oproton(x,y,z)  complicated

Evolution of 3quark-Odderon operatorin the weak-field limit: 

: weak field limit of The BKP equation appears as the equation for 3 point Green function with infra-red singularities removed Evolution of 3quark-Odderon operator in the weak-field limit

Relation to the traditional approach: 

Relation to the traditional approach Traditional description CGC formalism Our operator partly contains the information of the impact factor  Gauge invariant impact factor gauge invariance BKP equation LC wavefunction

Summary: 

Summary Identified the relevant operator for C-odd Odderon exchange in dipole-CGC scattering  imaginary part of the dipole operator (2pt fnc), O(x,y) = [ tr(Vx+ Vy) – tr(Vy+ Vx) ] / 2iNc. in 3-quark--CGC scattering  a 3 point fnc constructed from baryonic Wilson line operator Both reduce to 3 gluons with d-symbol in the weak-field limit Evolution equations for these operators  JIMWLK eq. dipole--CGC scattering Imaginary part of the Balitsky eq. Nonlinear terms represent coupling to the Pomeron. 3-quark--CGC scattering Complicated in the nonlinear (strong field) regime Reproduce the BKP equation in the weak-field limit