On perturbative limits of quadrupole evolution in QCD at high energy

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.