Small-$x$ evolution of dipole amplitude in momentum space: forward--off-forward correspondence

TL;DR

This paper demonstrates that the small-$x$ evolution of off-forward dipole amplitudes in momentum space can be derived from forward amplitudes, linking high-energy dipole GTMDs evolution to small-$x$ dipole TMDs.

Contribution

It establishes a correspondence between forward and off-forward dipole amplitudes in momentum space, simplifying the study of their evolution at small-$x$.

Findings

Evolution of off-forward amplitudes determined by forward amplitudes with rescaled momenta.

Translation symmetry in position space implies dependence only on quark-antiquark separation.

High energy evolution of dipole GTMDs can be studied via dipole TMDs at small-$x$.

Abstract

We have shown that the small-$x$ evolution of the off-forward leading-log dipole scattering amplitudes, both pomeron and odderon, in the momentum space can be completely determined by the evolution of the respective forward amplitudes, with rescaled momenta. In position space, if there is translation symmetry (assumption of a large nucleus), the dipole cross section depends on the positions of quarks and anti-quarks only through their separation. The present study is an equivalent proposition in the momentum space -- where translation symmetry in momentum bifurcates the amplitudes into two translationally symmetric functions along the ${\bf k}$ line in the ${\bf k}-{\bf \Delta}$ plane. It also shows that high energy evolutions of dipole GTMDs can be achieved only by studying the evolution of dipole TMDs at small-$x$.