Quark loop contribution to BFKL evolution: Running coupling and leading-N_f NLO intercept

TL;DR

This work analyzes quark-loop (sea quark) effects on BFKL evolution within Mueller's dipole formalism, deriving an all-orders running-coupling BFKL equation and the leading-$N_f$ contribution to the NLO BFKL intercept in momentum space. It demonstrates that the running-coupling structure forms a triumvirate of couplings and verifies the bootstrap conjecture of Levin and Braun for an unintegrated gluon observable, while providing an independent cross-check against the classic Fadin–Lipatov and Camici–Ciafaloni results. The intercept’s value is shown to be observable-dependent, with a consistent translation between momentum-space unintegrated gluon evolution and coordinate-space dipole evolution (Balitsky's framework). The paper also generalizes to non-forward kinematics, clarifying the relation between different observables and consolidating the link between BK/JIMWLK running-coupling corrections and the NLO BFKL sector.

Abstract

We study the sea quark contribution to the BFKL kernel in the framework of Mueller's dipole model using the results of our earlier calculation. We first obtain the BFKL equation with the running coupling constant. We observe that the ``triumvirate'' structure of the running coupling found previously for non-linear evolution equations is preserved for the BFKL equation. In fact, we rederive the equation conjectured by Levin and by Braun, albeit for the unintegrated gluon distribution with a slightly unconventional normalization. We obtain the leading-N_f contribution to the NLO BFKL kernel in transverse momentum space and use it to calculate the leading-N_f contribution to the NLO BFKL pomeron intercept for the unintegrated gluon distribution. Our result agrees with the well-known results of Camici and Ciafaloni and of Fadin and Lipatov. We show how to translate this intercept to the case of the quark dipole scattering amplitude and find that it maps onto the expression found by Balitsky.

Figures (1)

• Figure 1: Diagrams contributing a factor $N_f$ at order $\alpha_s^2 \ln(1/x)$. The line from bottom left to top right indicates the target field crossed by the fast moving constituents of the projectile. Their transverse positions are fixed. The ovals indicate insertions of $N({\bm x}_0,{\bm z}_1,Y)$, $N({\bm x}_0,{\bm z},Y)$ and $N({\bm x}_0,{\bm x}_1,Y)$ respectively.