# Presentation: 087942

## Perturbative Odderon in the Dipole Model

- Yuri Kovchegov
- The Ohio State University
- based on work done in collaboration with
-  L. Szymanowski and S. Wallon, hep-ph/0309281

## The BKP Equation

- The BFKL equation describes
- evolution of the 2-reggeon system
- shown below  (a C-even exchange)

- The BKP equation describes evolution
- of any n-reggeon system. For
- 3-reggeons it gives a C-odd amplitude

- Pomeron

- Odderon

## The Odderon Problem

- While a complete set of eigenfunctions
- for the BFKL equation has been
- found long time ago, it has not been
- found for the BKP equation with
- 3 reggeons. Instead there are several
- solutions of the 3-reggeon BKP
- equation on the market, some giving
- different leading odderon intercepts.
- Since a complete set of eigenfunctions
- has not been found, one can not prove
- that any particular solution is dominant
- at high energy, since there is always a
- chance to find more.

## Dipole Approach

- Our approach is somewhat pragmatic: for us an odderon
- exchange is any diffractive process that changes the C-parity
- of the quark-antiquark light cone wave function (e.g. a C-odd
- virtual photon can become a C-even pion, etc.). Corresponding
- diffractive dipole amplitude has to satisfy the following conditon:

- C: x0 ↔ x1
-      z ↔ 1-z

## Lowest Order

- At the lowest order the C-odd odderon amplitude is given by
- the three-gluon exchange in the color-singlet state:

-     with c0 the color factor. The target
- is located at _0_ transverse coordinate.

- The above amplitude obviously changes sign under x0 ↔ x1 !
- (All eikonal amplitudes are even under z ↔ 1-z interchange.)

## Mueller’s Dipole Model

- To include the quantum evolution in a dipole amplitude one
- has to use the approach developed by A. H. Mueller in ’93-’94.
- The goal is to resum leading logs of energy, log s, just like
- for the BFKL equation.
- Emission of a small-x
- gluon taken in the
- large-NC limit would
- split the original color
- dipole in two:

- 0

- 1

- 2

- Emission of a single gluon with longitudinal momentum fraction z2
- and transverse coordinate x2 would bring in a factor

- z

- 1-z

- z2

## Mueller’s Dipole Model

- For the dipole scattering amplitude N one can write down an
- equation resumming all successive gluon emissions:

- A. H. Mueller, ‘93

- This equation is equivalent to the BFKL equation (more later)!

## One Rung of Odderon Evolution

- Let us include one rung of the small-x evolution in the framework
- of A. H. Mueller’s dipole model. We are working in the large NC limit, so the graphs are planar and the gluon is given by double line:
- One rung of evolution gives

- Substituting the amplitude O0 from the previous slide we get

- which is non-zero and C-odd (changes sign under x0 ↔ x1).

-  “C-oddness” survives dipole evolution!

## Odderon Evolution Equation

- Now we can easily
- write down an
- evolution equation
- for the odderon
- amplitude in the dipole
- model.

- This is the same evolution equation as for the BFKL Pomeron
- in the dipole model! How is the Odderon equation different
- from the Pomeron equation?

## Odderon Evolution Equation

-  Odderon and Pomeron are described by the _same evolution _
- _equation_ in Mueller’s dipole model!
-  The only difference is in the _initial conditions_: Odderon has
- a C-odd initial condition,
- while the Pomeron
- has a C-even initial condition
- given by 2-gluon exchange:

-  Initial Conditions single out either the Odderon or the
-    Pomeron contributions depending on their C-parity!

## Solution of the Dipole Odderon Equation

- Since the Odderon evolution equation is equivalent to the
- BFKL Pomeron equation the solution can be easily constructed.
- First one shows that the eigenfunctions of conformal algebra
- Casimirs

- are also the eigenfunctions of the equation’s kernel

- with the eigenvalues being the same as for the BFKL equation

## Solution of the Odderon Equation

- Since

- the C-odd initial condition would pick only the eigenfunctions
-                         with odd n. These are the eigenfunctions of the
- odderon evolution equation.
- The eigenvalues would then be given by
- with odd n.

- The odderon solution would then look like

## Solution of the Odderon Equation

- The leading high energy odderon intercept would then be
- given by
- in agreement with Bartels, Lipatov and Vacca (BLV).

- To find the leading high energy term in

- one notes that the saddle point of -integrals is given by
- =0 for any n. Analyzing the saddle point values (n,for
- odd n one can see that n=1 term is the largest.

- Odderon amplitude is at most constant with energy!

## Connection to BLV Solution

- Bartels, Lipatov and Vacca have found the following
- eigenfunctions of the 3-reggeon BKP equation in momentum space:

- where the sum goes over all permutations of indices 1,2,3.

## Connection to BLV Solution

- Let’s see shat the BLV solution translates into for the dipole
- amplitude in transverse coordinate space. To do that let’s connect the BLV amplitude to an incoming dipole 01:

- A straightforward calculation shows that BLV eigenfunctions
- give the dipole amplitude

- which is also given by function                           up to a constant!

## Connection to BLV Solution

-  BLV eigenfunctions, when connected to a dipole, give the same eigenfunctions                           (with odd n) as found from the dipole evolution equation.
-  The corresponding eigenvalues are identical in the BLV and in our cases
-  This leads to the dominant Odderon intercept
-  Our solution is identical to BLV!
-  Since we have a complete set of solutions to the dipole equation, we have proven that there is no other odderon solution giving larger intercept than the BLV solution as long as dipole-anything scattering is concerned!

## Are we solving the same equation?

- Is there an equivalence between the BKP equation for 3
- reggeons and the dipole equation?

- Here we are going to show that the equivalence exists, as long
- as dipole scattering is concerned.

- (Many thanks to Al Mueller for discussions which led us to
- come up with this proof.)

## Are we solving the same equation?

- Let us start with the same three-gluon initial conditions and
- see how BKP and dipole equation kernels act on it.

- The 3-gluon onium-onium scattering
- amplitude is equal to

- which can be rewritten as

- with coefficients

## One Rung of BKP

- One rung of BKP evolution is
- shown here (horizontal line is
- the full BFKL kernel):

- BLV showed that BFKL kernel
- acting on their eigenfunction

- gives

- To go to coordinate space we recall
- that there

## One Rung of BKP

- Therefore

- Since

- than

## One Rung of Dipole Evolution

- Since, as we showed above

- than

- In the rest frame of one of the onia
- a rung of dipole evolution looks like

-  Higher order iterations will be the same by induction.

-  BKP and dipole equations are identical! (for dipole scattering)

## Linear Evolution Summary

-  We have shown that for dipole-anything scattering 3-reggeon
- BKP equation is equivalent to the dipole evolution equation.
-  The dipole evolution equation has a well-known complete
- set of eigenfunctions                           and eigenvalues.
- C-odd initial conditions single out the ones
- with odd n.
-  We have therefore constructed a _complete set of solutions_
- for the odderon exchange amplitude, which turned out to be
- equivalent to the BLV solution.
-  The leading Odderon intercept is

## Saturation/Color Glass Effects

- At very high energies the gluonic fields become strong
- leading to parton saturation.
- Saturation is crucial for unitarization of total DIS cross
- sections (at fixed impact parameter) mediated by the BFKL
- pomeron exchange. Saturation effects slow down the
- corresponding growth of the cross sections from
- power-of-energy (~s) to a constant.
- With zero intercept odderon exchange amplitude is in no
- danger of violating unitarity bound. Nevertheless, the
- saturation effects can give an order 1 correction to
- the odderon exchange amplitude.

## Saturation effects for C-even Amplitude

- In the rest frame of the
- target the DIS process
- is shown on the right,
- and it factorizes into

- The dipole-nucleus amplitude in
- the classical approximation is

- A.H. Mueller, ‘90

- with _rapidity_ Y=ln(1/x)

**1/Q****S**

## Quantum Evolution

- As energy increases
- the higher Fock states
- including gluons on top
- of the quark-antiquark
- pair become important.
- They generate a
- cascade of gluons.

- These extra gluons bring in powers of  S ln s, such that
- when S << 1 and  ln s >>1  this parameter is S ln s ~ 1.

## Nonlinear Evolution Equation

- 

- Defining rapidity Y=ln s we can resum the dipole cascade

- Yu. K., ’99, large NC QCD
- I. Balitsky, ’96, HE effective lagrangian

-  Linear part is BFKL, quadratic term brings in damping

- initial condition

## Nonlinear Equation

- 

- In the* *traditional
- language this
- corresponds to summing up
- “fan” diagrams like this one

- Gluon _recombination_ in the nuclear wave function corresponds
- to _splitting_ in the projectile wave function.

## Nonlinear Equation: Saturation

- 

- Gluon recombination tries to reduce the number of gluons in the wave function. At very high energy recombination begins to compensate gluon splitting. Gluon density reaches a limit and does not grow anymore. So do
- total DIS cross sections. Unitarity is restored!

## Including Saturation Effects in the Odderon Amplitude

- To include saturation effects in odderon evolution one has to first include multiple rescatterings a la Glauber-Mueller / McLerran-Venugopalan:

- LO Odderon

- For a nuclear target, two-gluon multiple rescatterings bring in
- powers of                        , so that single odderon exchange
- makes the amplitude to be of the order

## Including Saturation Effects

- Resummation of
- multiple rescatterings
- gives initial conditions
- for the evolution which
- we will construct next:

- (cf. Jeon and Venugopalan, ’05)

## Odderon Evolution in Color Glass

- Now one can write down a nonlinear evolution equation for the
- odderon amplitude, including the effects of multiple pomeron
- exchanges in the color glass condensate:

- (cf. evolution equation for reggeon by K. Itakura, Yu. K., L. McLerran, D. Teaney, ’03
-                  and generalization of this equation by Hatta, Iancu, Itakura, McLerran, ‘05)

- Here N is the C-even (forward) dipole amplitude on the target.
- N varies from ~0 at low energies to 1 at high energies.

-  Saturation effects only _decrease_ the Odderon intercept!

## Feynman Diagram Analogy

- In the traditional Feynman diagram language we are
- resumming graphs like

## Saturation Summary

-  We have constructed an evolution equation for the odderon
- amplitude in the saturation / Color Glass regime.
-  The equation appears to lead to a decrease of the odderon
- intercept at very high energy. That means that the intercept
- will go from 0 to negative as energy increases, making the
- odderon amplitude a _decreasing_ function of energy.
-  Saturation has (probably) been observed at HERA. Does it
- mean that we should expect the odderon exchange amplitude
- to be a decreasing function of energy at low Q2 in HERA
- kinematics? Any other phenomenological predictions? Can
- saturation be the reason why they have not found odderon yet?

## Conclusions

-  We have found the most general solution for the Odderon
- exchange amplitude for a dipole scattering on any target and
- showed that it is equivalent to the BLV solution. We have shown
- that the leading high energy odderon intercept is given by
-                                  in agreement with the results of
-                                  Bartels, Lipatov and Vacca.
-  We have shown that in Mueller’s dipole model both the BFKL
- Pomeron and the Odderon are described by the same evolution
- equation with C-even and C-odd init. conditions correspondingly.
-  We have written an evolution equation for the Odderon in the
- Color Glass regime and showed that saturation can only reduce
- the Odderon intercept at high energies.