Perturbative%20Odderon%20in%20the%20%20%20Color%20Glass%20Condensate

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Perturbative Odderon in the Color Glass
Condensate

• Yoshitaka Hatta
• (RIKEN BNL)
• in collaboration with
• E. Iancu, K. Itakura L. McLerran

Ref. hep-ph/0501171 (to appear in Nucl.Phys.A)
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Outline

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

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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
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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)
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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 Muellers dipole model
• Kovchegov,
  Szymanowski Wallon (2004)
• Formulation in the CGC

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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
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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
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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)
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Large-Nc BKP equation
Heisenberg spin chain
..
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N

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N
1
3
1
2
3
2
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
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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
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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
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Example Dipole-CGC scattering
S-matrix with a fixed background field
JIMWLK
Balitsky equation
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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
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Construction of the odderon exchange amplitude in
CGC
Dipole-CGC scattering
CGC
Odderon amplitude
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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
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3-quark-CGC scattering
CGC
Odderon amplitude
The 3-quark amplitude and the dipole amplitude
are related.
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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 ?
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Equivalence to the BKP equation
Impact factor
BKP equation
JIMWLK equation
Identify
They satisfy the same equation provided one uses
the dipole JIMWLK equation for
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The Janik-Wosiek (JW) solution (1999)
Lipatovs ansatz
anharmonic ratio
Obtained within the formalism of LFK The odderon
intercept
Vanishes at equal points



..
Vanishing for solutions in the Mobius
representation
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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 !
..
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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

..