On perturbative limits of quadrupole evolution in QCD at high energy

TL;DR

This paper investigates the perturbative limits of quadrupole evolution in high-energy QCD, showing that linearized equations relate to known BFKL and BJKP equations, and discusses deriving reggeized gluon evolution from JIMWLK.

Contribution

It demonstrates that the linearized quadrupole evolution reduces to BFKL and BJKP equations, linking small x QCD evolution to established formalisms.

Findings

Quadratic terms satisfy the BFKL equation.

Quartic terms relate to the BJKP equation.

Linearized evolution connects to reggeized gluon amplitudes.

Abstract

We consider the perturbative (weak field) limit of the small $x$ QCD evolution equation for quadrupole, the normalized trace of four Wilson lines in the fundamental representation, which appears in di-hadron angular correlation in high energy collisions. We linearize the quadrupole evolution equation and then expand the Wilson lines in powers of $g\, A_{\mu}$ where $A_{\mu}$ is the gauge field. The quadratic terms in the expansion ($\sim g^2\, A^2$) satisfy the BFKL equation as has been recently shown. We then consider the quartic terms ($\sim g^4\, A^4$) in the expansion and show that the linearized quadrupole evolution equation, written in terms of color charge density $\rho$, reduces to the well-known BJKP equation for the imaginary part of four-reggeized gluon exchange amplitude. We comment on the possibility that the BJKP equation for the evolution of a $n$-reggeized gluon state can be obtained from the JIMWLK evolution equation for the normalized trace of $n$ fundamental Wilson lines when non-linear (recombination) terms are neglected.