Transcript Perturbative Odderon in the Color Glass Condensate Yoshitaka Hatta (RIKEN BNL) in collaboration with E.

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)
Outline

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
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
CAC 1   AT
a
b
c
dabc A A A
Contributes to hadronic cross section difference between
the direct and crossed channel processes at very high energy
p
p
s
p
p
 odd 1
Odderon: Where to look for ?
Diffractive pseudoscalar meson production in DIS

 0 ,c ,...

p
Dip region in the pp and pp
differential cross section
CNI effect in the double transverse-spin asymmetry
ANN
Leader & Trueman (2000)
Theoretical status of perturbative odderon

BKP equation
Bartels (1980), Kwiecinski & Praszalowicz (1980)
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Mapping onto exactly solvable 1D Heisenberg spin chains
Lipatov (1993), Faddeev & Korchemsky (1994)

Two exact solutions
Janik & Wosiek (1999)
Bartels, Lipatov & Vacca (2000)
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odd  1
odd  1
Formulation in Mueller’s dipole model
Kovchegov, Szymanowski & Wallon (2004)

Formulation in the CGC
The Bartels-Kwiecinski-Praszalowicz equation
  ln s
BFKL
: rapidity
BFKL
cf. the BFKL equation
BFKL
BFKL
1
2
3
BFKL
1
2
Exchange the BFKL kernel between all possible pairs in one step of evolution
d
1
f (k1 , k2,k3 )  ( H12BFKL  H 23BFKL  H 31BFKL ){k }{k '} f ( k '1 , k '2 , k '3 )
d
2
 H BKP f (k1, k2 , k3 )
Generalized Leading Log Approximation
(GLLA)
Bartels (1980)
s
BFKL
4 s ln 2
2 ImTif    TinT fn*
violation of
unitarity bound
n
Need to consider more than two reggeized gluon exchanges
BFKL
BFKL

T  T ( N )
BFKL
BFKL
BFKL
N 1
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
f (k1, k2 , k3 ,...)  f ( x , y , z ,...)
Introduce the complex coordinate in 2D
z1  x1  ix2 , z1  x1  ix2 ,
etc.
..
BKP (BFKL) Hamiltonian is invariant under the Mobius transformation
(2D conformal symmetry)
azi  b
zi 
czi  d
a b 

  SL(2,C)
c
d


Large-Nc BKP equation
s0
Heisenberg spin chain
…..
N
1
……
N
4
4
2
3
1
3
2
H BKP  H  H
+ terms containing
 ( zi  z j ) ( zi  z j )
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   A  1 / g distributed according to a weight function W [ ] .
Hadron-CGC scattering amplitudes T are first calculated with fixed
background field
, then averaged over
.


 T    [ D ]T ( )W [ ]
W [ ]
satisfies the JIMWLK equation
small-x evolution equation for
 T 
Jalilian-Marian, Iancu, McLerran,
Weigert, Leonidov, Kovner
Example:
Dipole-CGC scattering
x
x  t  z

S-matrix with a fixed background field
x
1
S xy [ ] 
T r[Vx ( )Vy ( )]  1  N xy [ ]
NC
y
Vx ( )  P exp{ig  dx ( x  , x )}
JIMWLK
Balitsky equation
Weak field approximation
JIMWLK
N xy [ ]  g 2 (x   y )2
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
x
Dipole-CGC scattering
y
CGC
Sxy [ ]  1  N xy [ ]  iOxy [ ]
Odderon amplitude
Weak field approximation
O( x , y )  g 3d abc {3(xa yb yc  xaxb yc )  xaxbxc   ya yb yc }
“CGC Green’s function”
(dipole) JIMWLK
odd  1
Non-linear evolution equation for
the dipole-odderon amplitude
Decomposition of the Balitsky equation into real and imaginary parts
Merging of two odderons
 
 O   exp{cs2 2}
Levin-Tuchin law
Saturation effect suppresses the odderon amplitude
x
y
3-quark-CGC scattering
z
CGC
Odderon amplitude
The 3-quark amplitude and the dipole amplitude are related.
Evolution equation for the 3-quark
odderon amplitude in the linear regime
 B( x , y , z ) 
 B( x , y , z ) 
Closed equation for the gauge invariant amplitude  B( x , y , z ) 
IR and UV safe
Relation to the BKP equation ?
Equivalence to the BKP equation
Impact factor
x
y
z
 B( x , y , z ) 
f (k1, k2 , k3 )
JIMWLK equation
BKP equation
Identify
d
abc
      f ( x , y , z )
a
x
b
y
c
z
They satisfy the same equation provided one uses
the dipole JIMWLK equation for d abc   a b c 
The Janik-Wosiek (JW) solution (1999)
Lipatov’s ansatz
f
JW
h
3
 z12 z23 z31 
( x , y , z )   2 2 2  g (  )
 z14 z24 z34 

z12 z34
z14 z32
: anharmonic ratio
Obtained within the formalism of LFK
The odderon intercept
Vanishes at equal points
H BKP  H  H
odd  1  0.2472
s NC

fJW ( x , x , z )  BJW ( x , x , z )   0
+ terms containing
 ( 2) ( x  y )
..
Vanishing for solutions in the Mobius representation
The Bartels-Lipatov-Vacca (BLV) solution (2000)
fBLV (k1 , k2 , k3 )  
ijk
(ki  k j )2
ki2k 2j
E ( ,n ) (ki  k j , kk )
BFKL eigenfunction
Constructed directly in the momentum space,
The odderon intercept
odd  1
(n  1,  0)
..
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