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

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

3. 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

4. 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)

5. 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

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

7. 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
…. N

8. 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)

9. 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

10. 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

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

12. 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

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

14. 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

15. 3-quark-CGC scattering
Odderon amplitude
The 3-quark amplitude and the dipole amplitude are related.

16. 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 ?

17. 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

18. 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

19. 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 ! ..

20. 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 ..