1. Nonlinear Odderon evolution in the Color Glass Condensate Nonlinear Odderon evolution in the Color Glass Condensate Kazunori Itakura (SPhT, CEA/Saclay) in collaboration with E. Iancu (Saclay), L. McLerran and Y. Hatta (BNL)

2. Motivation Motivation Increasing interest in physics beyond the Balitsky- Kovchegov equation (2 point function) - dipole correlations, fluctuation - evolutions of multipoint functions - effects of non-dipole operators (quartic or sextupole operators) Does the Color Glass Condensate (JIMWLK equation) contain correct n-Reggeon dynamics described by the BKP equation? (Bartels, Kwiecinski- Praszalowicz) - linearization (weak field limit) of n-point evolution equations - extension of BKP to nonlinear regime The first step: to find a description of 3 gluon exchange in CGC  Odderon in CGC !

3. Perturbative QCD Odderon In QCD, the odderon is a three Reggeized gluon exchange which is odd under the charge conjugation cf) BFKL Pomeron = 2 gluon exchange, C-even What is the relevant operator for the odderon? - Pomeron = tr(V x + V y ) with strong field (saturation)  2 gluon operator {  (x)-  (y)} 2 in weak field limit (  (x) is the minus component of the gauge field) - Gauge invariant combination of 3 gluons? How to construct them?

4. C-odd operators Charge conjugation Fermions mesonic baryonic Gauge fields any combination of 3 gluons with d-symbol is C-odd.  (+ even, -- odd) 

5. Intuitive construction of S-matrix Dipole-CGC scattering in eikonal approximation scattering of a dipole in one gauge configuration average over the random gauge field should be taken 3 quarks-CGC scattering can be formulated in a similar way with B operator stay at the same transverse positions

6. C-odd S-matrix C-odd S-matrix (dipole-CGC scattering) Transition from C-even to C-odd dipole states Relevant operator O dipole (x,y) = tr(V x + V y ) – tr(V y + V x ) = 2i Im tr(V x + V y ) - constructed from gauge fields, but has the same symmetry as for the fermionic dipole operator M(x,y)-M(y,x)  anti-symmetric under the exchange of x and y O dipole (x,y) = - O dipole (y,x) - imaginary part of the dipole operator tr(V x + V y ). Real part of the scattering amplitude T (S = 1 + iT) Weak field expansion  leading order is 3 gluons - should be gauge invariant combination

7. C-odd S-matrix C-odd S-matrix (3q-CGC scattering) Transition from C-even to C-odd 3 quark states “baryonic Wilson lines” Relevant operator difference between proton and anti-proton cross sections Weak field expansion 3 gluons with d-symbol, gauge invariant, symmetric under the exchange of coordinates (x,y,z)

8. Evolution of the dipole odderon (I) Non-linear evolution eq. for the odderon operator can be easily obtained from the Balitsky eq. for tr(V + x V y ). - N(x,y) = 1- 1/Nc Re tr(V + x V y ) is the usual “scatt. amplitude” (real) - the whole equation is consistent with the symmetry O dipole (x,y) = - O dipole (y,x) and N(x,y) = N(y,x) - becomes equivalent to Kovchegov-Symanowsky-Wallon (2004) if one assumes factorization . - linear part = the BFKL eq. (but with different initial condition)  reproduces the BKP solution with the largest intercept found by Bartels, Lipatov and Vacca (KSW,04) - intercept reduces due to saturation As N(x,y)  1, O dipole (x,y) becomes decreasing ! BFKL

9. 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! - One Pomeron can split into two Odderons! N=1, O=0 is the stable fixed point.

10. Comments on the proton-odderon Baryonic Wilson line operator multiplying the identity One can rewrite the proton-odderon operator as - subtracted dipole terms which do not contribute to 3 gluons - can compute the evolution equation for O proton (x,y,z) (but the result is very complicated) - the equivalence is valid only for Nc=3 - reproduces the same 3 gluon operator in the weak field limit - partly contains the impact factor  doesn ’ t satisfy the BKP eq. itself

11. Summary We have identified the C-odd Odderon operator for dipole-CGC scattering. It is still given by the two point dipole operator, but its imaginary part O dipole (x, y) = tr(V x + V y ) – tr(V y + V x ). The nonlinear evolution equation for the dipole-odderon is easily obtained from the Balitsky equation. The nonlinear terms represent the coupling to the Pomeron. The evolution equation for the Pomeron (real part of the S matrix) should be modified due to Odderon. One Pomeron can split into two Odderons. The similar analysis can be done for 3 quarks-CGC scattering with the baryonic Wilson line operator, which is a three point function and generates gauge invariant 3 gluon combination in the weak field limit.