Perturbative Odderon in the Dipole Model

TL;DR

The paper demonstrates that perturbative QCD Odderon dynamics in Mueller's dipole model are governed by the dipole-BFKL evolution with a $C$-odd initial condition, yielding a leading intercept $α_{odd}-1 = 0$ and reproducing the BLV Odderon. The Odderon solution is expressed through odd $n$ eigenfunctions $E^{n,ν}$ with eigenvalues $χ(n,ν)$, and the general solution is a BLV-equivalent superposition ${ m O}( ho_1, ho_2; ho_{1'}, ho_{2'};Y) = rac{}{} obreak \sum_{ ext{odd }n} obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak obreak $n$, ν} e^{2 ar{α}_s χ(n,ν) Y} E^{n,ν}( ho_{10}, ho_{20}) E^{n,ν*}( ho_{1'0}, ho_{2'0})$. Including saturation via a nonlinear equation with the $C$-even amplitude $N$ shows that saturation introduces negative nonlinear terms, likely driving the Odderon amplitude down with energy. The work also clarifies the connection to BLV/BKP and discusses the limitations of the dipole model in bounding the Odderon intercept for $pp$ scattering due to diagrams not captured by the dipole framework. Overall, the results affirm that the dipole model provides a consistent perturbative Odderon description and highlights saturation as a mechanism for its suppression at high energy.

Abstract

We show that, in the framework of Mueller's dipole model, the perturbative QCD odderon is described by the dipole model equivalent of the BFKL equation with a $C$-odd initial condition. The eigenfunctions and eigenvalues of the odderon solution are the same as for the dipole BFKL equation and are given by the functions $E^{n,ν}$ and $χ(n,ν)$ correspondingly, where the $C$-odd initial condition allows only for odd values of $n$. The leading high-energy odderon intercept is given by $α_{odd} - 1 = \frac{2 \as N_c}π χ(n=1 ,ν=0) = 0$ in agreement with the solution found by Bartels, Lipatov and Vacca. We proceed by writing down an evolution equation for the odderon including the effects of parton saturation. We argue that saturation makes the odderon solution a decreasing function of energy.

Figures (6)

• Figure 1: Lowest order odderon exchange diagram which will be used as initial condition for small-$x$ evolution. The gluon lines connect to both the quark and the anti-quark in the onium above.
• Figure 2: One step of dipole evolution with the three-gluon exchange initial conditions.
• Figure 3: Dipole evolution equation for the odderon exchange amplitude.
• Figure 4: Onium-onium scattering.
• Figure 5: Dipole's impact factor.