Improving the solver for the Balitsky-Kovchegov evolution equation with Automatic Differentiation

TL;DR

The paper introduces a forward-mode Automatic Differentiation–based solver for the Balitsky-Kovchegov equation, enabling analytic first- and second-order derivatives of the dipole amplitude with respect to initial-condition parameters and the dipole size. This framework accelerates gradient-based fits to DIS data and enables direct computation of TMD-related quantities from $S(r,\eta)$ via derivatives such as $\mathcal{K}_{\pm}(r,\eta)$, while maintaining numerical stability through differentiable implementations of the kernel and running coupling. The authors validate the approach by comparing to existing software and finite-difference derivatives, and demonstrate favorable performance, scalability, and applicability to data fitting and TMD calculations. The work provides a practical pathway to more precise phenomenology in small-$x$ QCD and suggests avenues for extending differentiable techniques to more complex evolutions like JIMWLK.

Abstract

The Balitsky-Kovchegov (BK) evolution equation is an equation derived from perturbative Quantum Chromodynamics that allows one to evolve with collision energy the scattering amplitude of a pair of quark and antiquark off a hadron target, called the dipole amplitude. The initial condition, being a non-perturbative object, usually has to be modeled separately. Typically, the model contains several tunable parameters that are determined by fitting to experimental data. In this contribution, we propose an implementation of the BK solver using differentiable programming. Automatic differentiation offers the possibility that the first and second derivatives of the amplitude with respect to the initial condition parameters are automatically calculated at all stages of the simulation. This fact should considerably facilitate and speed up the fitting step. Moreover, in the context of Transverse Momentum Distributions (TMD), we demonstrate that automatic differentiation can be used to obtain the first and second derivatives of the amplitude with respect to the quark-antiquark separation. These derivatives can be used to relate various TMD functions to the dipole amplitude. Our C++ code for the solver, which is available in a public repository, includes the Balitsky one-loop running coupling prescription and the kinematic constraint. This version of the BK equation is widely used in the small-$x$ evolution framework.

Figures (4)

• Figure 1: Comparison with software from Ref. Caucal:2023fsf. The left panel shows the dipole amplitude obtained with our software plotted in color and obtained using reference software in black. The lines overlap hiding one beneath the other. In the right panel, we show their ratios for different values of $\eta$.
• Figure 2: First and second derivative with respect to the $Q_0$ parameter of the dipole amplitude evolved up to $\eta=10$ as a function of $r$. The solid line corresponds to the result obtained using AD, while the data points were calculated by divided differences (\ref{['first_der_Q0']},\ref{['second_der_Q0']}) by varying $Q_0$ by about 6% by carrying two additional simulations.
• Figure 3: First (left panel) and second (right panel) derivatives of the longitudinal cross-section $\sigma_L^{\gamma^*,p}(x,Q^2=10 \, \text{GeV}^2)$ given by Eq. \ref{['sig_LT_def']} with respect to the initial condition parameters. These derivatives can be directly used to estimate the gradient of the $\chi^2$ function during the optimization algorithm comparing theoretical predictions to the experimental measurements.
• Figure 4: Left panel: functions $|\mathcal{K}_+(r_\perp)|$ and $|\mathcal{K}_-(r_\perp)|$ at initial condition $\eta=0$ for two grid sizes, $N=500,\ 1000$ and two methods of evaluation: AD and divided difference on the grid points. Right panel: functions $|\mathcal{K}_\pm(r_\perp)|$ after evolution to $\eta=4$ for four grid sizes: $N=500,\ 1000, \ 2000, \ 4000$.