The CSS Hamiltonian: high energy evolution of rapidity dependent observables

TL;DR

This paper derives a new evolution Hamiltonian for rapidity-dependent observables at high energy, which differs from the traditional JIMWLK equation and aligns with Collins-Soper-Sterman evolution in certain regimes.

Contribution

The paper introduces the CSS Hamiltonian, a novel evolution operator for small-x observables at high rapidity, extending beyond the JIMWLK framework.

Findings

The CSS Hamiltonian is valid for both dilute and dense regimes.

It reproduces the Collins-Soper-Sterman evolution for gluon TMDs.

The derived evolution differs from the JIMWLK equation at high rapidity.

Abstract

We consider evolution of observables which depend on a small but fixed value of longitudinal momentum fraction $x$, to high rapidity, such that $\eta>\ln 1/x$. We show that this evolution is not given by the JIMWLK (or BK) equation. We derive the evolution Hamiltonian - $H_{CSS-JIMWLK}$ which generates this evolution in the cases of dilute and dense projectile wave function. The two limits yield identical results for $H_{CSS-JIMWLK}$. We show that the resulting evolution for the gluon TMD is identical to the (double logarithmic) perturbative Collins-Soper-Sterman evolution equation in the longitudinal resolution parameter at a fixed and very large transverse resolution.