Exploring rapidity regularization schemes at low $x$ with the DIS longitudinal structure function

TL;DR

This work introduces three rapidity regulators—$\eta^+$, $\eta^-$, and pure rapidity—for handling rapidity divergences in low-$x$ QCD with gluon saturation, enabling the evolution variable (either $k^+$, $k^-$, or rapidity) to be chosen at the outset. The authors apply these regulators to the NLO calculation of the dipole factorization of the longitudinal DIS structure function $F_L$, detailing the evaluation of $\gamma^*_L \to q\bar{q}$ and $q\bar{q}g$ Fock-state corrections, including self-energy and vertex diagrams, and performing Fourier transforms to mixed space. They perform rapidity subtraction to obtain rapidity-renormalized dipole operators and show that the dipole and $qqg$ contributions reproduce known results (Beuf 2017) with consistent evolution ranges for the BK/JIMWLK equations, while highlighting regulator-dependent scheme effects on collinear/anticollinear logs. The study demonstrates that rapidity regulators can resolve ambiguities in evolution-variable choices and disentangle soft from rapidity divergences, paving the way for improved NLO predictions in DIS at low $x$ and informing future calculations of other observables such as $F_T$ and dijet production. These regulators thus offer a systematic framework to connect high-energy resummation with DGLAP-like dynamics in a unified, regulator-aware manner.

Abstract

We propose three possible rapidity regulators for higher-order calculations in low $x$ QCD with gluon saturation, as alternatives to the usual lower cut-off for the integrals over the light-cone momentum $k^+$. These rapidity regulators are closely related to the $η$ regulator and to the pure rapidity regulator, which have been used primarily in studies of transverse-momentum-dependent (TMD) factorization within the soft-collinear effective theory (SCET). By choosing one of the three rapidity regulators that we propose, formulated in terms of $k^+$, $k^-$ or rapidity respectively, one can set from the start of the calculation in which of these three variables one wishes to formulate the low $x$ evolution equations, which is one of the main advantages of our approach. As a test of the viability of these rapidity regulators and of their practical implementation in higher order calculations with gluon saturation effects, we use them to revisit the calculation of the NLO corrections to the dipole factorization of the $F_L$ structure function in inclusive DIS at low $x$.

Figures (2)

• Figure 1: Light-front perturbation theory diagrams contributing to NLO corrections to the $\gamma_L^* \to q \bar{q}$ light-front wave-function. Vertical dotted lines indicate the energy denominators associated with an intermediate Fock state.
• Figure 2: Diagrams contributing to the $F_L$ structure function at NLO with a $q\bar{q} g$ intermediate Fock state scattering on the target.