Perturbative Odderon in the Color Glass Condensate Yoshitaka Hatta (RIKEN BNL) in collaboration with E. Iancu, K. Itakura L. McLerran pppp Ref. hep-ph/0501171 (to appear in Nucl.Phys.A)

Outline What is an odderon ? Theoretical status of perturbative odderon Color Glass Condensate formalism Odderon exchange in the CGC Remarks on the odderon solutions

Odderon in QCD C-odd compound state of gluons To lowest order in pQCD, the odderon is a three-gluon exchange Contributes to hadronic cross section difference between the direct and crossed channel processes at very high energy

Odderon: Where to look for ? Diffractive pseudoscalar meson production in DIS Dip region in the and differential cross section CNI effect in the double transverse-spin asymmetry Leader & Trueman (2000)

Theoretical status of perturbative odderon BKP equation Bartels (1980), Kwiecinski & Praszalowicz (1980) Mapping onto exactly solvable 1D Heisenberg spin chains Lipatov (1993), Faddeev & Korchemsky (1994) Two exact solutions Janik & Wosiek (1999) Bartels, Lipatov & Vacca (2000) Formulation in Mueller’s dipole model Kovchegov, Szymanowski & Wallon (2004) Formulation in the CGC

The Bartels-Kwiecinski-Praszalowicz equation : rapidity BFKL BFKL cf. the BFKL equation BFKL BFKL BFKL 1 2 3 1 2 Exchange the BFKL kernel between all possible pairs in one step of evolution

Generalized Leading Log Approximation (GLLA) Bartels (1980) BFKL violation of unitarity bound Need to consider more than two reggeized gluon exchanges BFKL BFKL BFKL BFKL BFKL BFKL 1 2 3 4 …. N

High energy QCD in large Nc limit is completely integrable Lipatov, Faddeev & Korchemsky Fourier transform to impact parameter space Introduce the complex coordinate in 2D etc. .. BKP (BFKL) Hamiltonian is invariant under the Mobius transformation (2D conformal symmetry) SL(2,C)

Large-Nc BKP equation Heisenberg spin chain ….. 4 N …… 4 N 1 3 1 2 3 2 + terms containing Hamiltonian exhibits holomorphic separability, identical to that of a spin chain Solution to the Yang-Baxter equation known  N integrals of motion Eigenvalue (odderon intercept) obtained by the Bethe ansatz

Color Glass Condensate formalism McLerran & Venugopalan (1994) An effective theory for the gluon saturation at high energy A high energy hadron (nucleus) is replaced with static, strong classical color fields distributed according to a weight function . Hadron-CGC scattering amplitudes are first calculated with fixed background field , then averaged over . satisfies the JIMWLK equation small-x evolution equation for Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner

Dipole-CGC scattering Example: Dipole-CGC scattering S-matrix with a fixed background field JIMWLK Balitsky equation Weak field approximation JIMWLK BFKL equation

Simplifying the JIMWLK equation Evolution equation for a scattering amplitude JIMWLK kernel For a color singlet projectile, the JIMWLK equation can equivalently be written in a manifestly IR finite form. “Dipole”- JIMWLK

Construction of the odderon exchange amplitude in CGC Dipole-CGC scattering CGC Odderon amplitude Weak field approximation “CGC Green’s function” (dipole) JIMWLK

Non-linear evolution equation for the dipole-odderon amplitude Decomposition of the Balitsky equation into real and imaginary parts Merging of two odderons Levin-Tuchin law Saturation effect suppresses the odderon amplitude

3-quark-CGC scattering Odderon amplitude The 3-quark amplitude and the dipole amplitude are related.

Evolution equation for the 3-quark odderon amplitude in the linear regime Closed equation for the gauge invariant amplitude IR and UV safe Relation to the BKP equation ?

Equivalence to the BKP equation Impact factor JIMWLK equation BKP equation Identify They satisfy the same equation provided one uses the dipole JIMWLK equation for

The Janik-Wosiek (JW) solution (1999) Lipatov’s ansatz : anharmonic ratio Obtained within the formalism of LFK The odderon intercept Vanishes at equal points + terms containing .. Vanishing for solutions in the Mobius representation

The Bartels-Lipatov-Vacca (BLV) solution (2000) BFKL eigenfunction Constructed directly in the momentum space, The odderon intercept Does not vanish at equal point (lies outside the Mobius representaion) The only solution for the dipole scattering (Kovchegov, Szymanowski & Wallon) Largest intercept solution also for the 3-quark scattering ! ..

Summary The first study of C-odd amplitude (odderon) in the CGC framework We derived evolution equations for the dipole-CGC and 3-quark (“proton”)-CGC scatterings, with the help of the dipole-JIMWLK equation. Equivalence to the BKP equation established. Gauge invariant evolution equations in coordinate space explicitly stay outside the Mobius space. Starting point for a study of arbitrary N-point amplitudes in CGC ..