Determination of the initial condition for the Balitsky-Kovchegov equation with transformers

TL;DR

This work tackles the computational bottleneck of fitting the BK initial condition for small-$x$ QCD by replacing repeated BK solves with a transformer-based emulator of the dipole amplitude $N(r,x)$. The authors generate a large BK-solution library across initial-condition parameters, train a transformer to interpolate $N(r,x)$, and build a second emulator to predict the DIS reduced cross section $\sigma_r$ efficiently. They perform global fits to HERA $e^+p$ DIS data for two starting points $x_0$ and both fixed and free $\gamma$, finding that a smaller $x_0$ yields better agreement and that the LO BK framework with a MV-like initial condition can describe the data within uncertainties. The approach drastically reduces computation time and provides a path to LO+ and beyond global analyses in small-$x$ physics.

Abstract

In the high-energy limit of QCD, scattering off nucleons and nuclei can be described in terms of Wilson-line correlators whose energy dependence is perturbative. The energy dependence of the two-point correlator, called the dipole amplitude, is governed by the Balitsky-Kovchegov (BK) equation. The initial condition for the BK equation can be fitted to the experimental data, which requires evolving the dipole amplitude for a large set of different parameter values. In this work, we train a transformer model to learn the energy dependence of the dipole amplitude, skipping the time-consuming numerical evaluation of the BK equation. The transformer predicts the learned dipole amplitude and the leading order inclusive deep inelastic scattering cross section very accurately, allowing for efficient fitting of the initial condition to the experimental data. Using this setup, we fit the initial condition of the BK equation to the inclusive deep inelastic scattering data from HERA and consider two different starting points $x_0$ for the evolution. We find better agreement with the experimental data for a smaller $x_0$. This work paves the way for future studies involving global fits of the dipole amplitude at leading order and beyond.

Figures (7)

• Figure 1: Left: Two dimensional histogram comparing the exact and transformer emulated values of $\ln N(r,x)$ for the BK dipole amplitude. The color scale indicates the density of samples, and the dashed line represents exact agreement. The mean and median relative errors on the validation set are $0.09\%$ and $0.05\%$, respectively. Right: Distribution of the signed relative errors, zoomed to $\pm1\%$. This highlights that the vast majority of predictions fall within a few per mille of the true values, demonstrating the emulator's high accuracy.
• Figure 2: The emulated dipole amplitude $N(r,x)$ (lines) is compared against the exact BK solution (points) for a representative out-of-training-sample parameter set at three different rapidities, $x = 10^{-3}$, $10^{-5}$, and $10^{-7}$. The excellent agreement in both log-linear (left) and log-log (right) scales across the entire range of $r$ demonstrates the model's robust extrapolation capability. The chosen parameters are $Q_{s0}^2\simeq0.07$ GeV$^2$, $\gamma=1.01$, $e_c=24.68$, $C^2\simeq4.65$, and $x_0=0.01$, corresponding roughly to physical values found in previous fits Lappi:2013zmaCasuga:2023dcf.
• Figure 3: Left: Parity plot comparing the transformer emulator predictions for the DIS reduced cross section $\sigma_r/(\sigma_0/2)$ (y-axis) against the exact dipole model values (x-axis). The dashed line indicates perfect agreement, with mean and median relative errors of $0.115\%$ and $0.073\%$, respectively. Right: Distribution of the signed relative errors, shown within $\pm 1\%$, demonstrating that the overwhelming majority of predictions lie well below the per-mille level. Together these demonstrate the high accuracy of the DIS surrogate model.
• Figure 4: Comparison of the model prediction with the theory cross sections for $e^+p$ DIS at three center-of-mass energies, $\sqrt{s}=318.1$, $251.5$, and $224.9~\mathrm{GeV}$. The green circles denote the theory values, while the red curves show the model prediction evaluated with a representative parameter set outside the training replicas. Each panel is drawn at fixed $Q^2$ ($2$, $15$, $20~\mathrm{GeV}^2$ respectively).
• Figure 5: Reduced cross section $\sigma_r(x,Q^2)$ compared with the combined HERA $e^+p$ data at four center-of-mass energies, $\sqrt{s}=318.1$, $300.3$, $251.5$, and $224.9$ GeV. The curves show the result of the five-parameter fit (with $\gamma$ free) performed with the evolution starting point $x_0=0.05$. The shaded bands represent the $2\sigma$ uncertainty of the fitted prediction.