1. Blois workshopK. Itakura (CEA/Saclay)1 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

2. Blois workshopK. Itakura (CEA/Saclay)2 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

3. Blois workshopK. Itakura (CEA/Saclay)3 Odderon Odderon = Leading “C-odd” exchange in hadron scatt. at high energies. “C-odd” counterpart of the Pomeron (see Ewerz’s talk) Odderon / Introduction (I) Regge theory  “soft” Odderon [Lukaszuk-Nicolescu ’73] Elastic amplitude odd under “crossing” (a+b  a+b vs “crossed” a+b  a+b) A -- : “particle-particle scatt” – “particle-antiparticle scatt” ODD under charge conjugation p  p -- - * Perturbative QCD  “hard” Odderon three reggeized gluon exchange in C-odd state (exists only for ) C-odd three gluon operator * * Experimental status  not conclusive so far…

4. Blois workshopK. Itakura (CEA/Saclay)4 The BKP equation for 3 gluons 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 H BFKL 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) * * * *

5. Blois workshopK. Itakura (CEA/Saclay)5 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??? Can we reproduce the BKP equation in the CGC???

6. Blois workshopK. Itakura (CEA/Saclay)6 Determine the relevant operators for scatt. btw a projectile and the CGC A projectile traverses a strong “random” gauge field created by the CGC. - the eikonal approximation - The operator is evaluated with averaging over the color field W  [  ]: weight function  randomness General strategies in CGC ex)Dipole-CGC scattering: the relevant operator leads to the Balitsky eq. Compute the evolution equations from the JIMWLK equation JIMWLK eq. = evolution equation for the weight function in the target. easily converted into the equations for operators. can be made simple for gauge invariant operators IR finiteness manifest

7. Blois workshopK. Itakura (CEA/Saclay)7 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! (    + c) C-odd operatorin dipole-CGC scatt. C-odd operator in dipole-CGC scatt.

8. Blois workshopK. Itakura (CEA/Saclay)8 C-odd operator in3-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

9. Blois workshopK. Itakura (CEA/Saclay)9 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 + x V y ) becomes equivalent to Kovchegov-Szymanowski-Wallon (‘04) if one assumes factorization . 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: decreasing as  1 Evolution of N(x,y) is also modified due to Odderon: 2 Odderons  1 Pomeron BFKL * * * * * *

10. Blois workshopK. Itakura (CEA/Saclay)10 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.

11. Blois workshopK. Itakura (CEA/Saclay)11 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 O proton (x,z,z) = O(x,z) diquark ~ antiquark can compute nonlinear evolution equation for O proton (x,y,z)  complicated

12. Blois workshopK. Itakura (CEA/Saclay)12 : 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

13. Blois workshopK. Itakura (CEA/Saclay)13 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

14. Blois workshopK. Itakura (CEA/Saclay)14 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(V x + V y ) – tr(V y + V x ) ] / 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