Reconstruction of the Dipole Amplitude in the Dipole Picture as a mathematical Inverse Problem

TL;DR

This work reframes the inference of the dipole amplitude $N(r,x)$ from inclusive DIS data in the dipole picture as a discrete linear inverse problem, analogous to tomographic reconstruction. By expressing the reduced cross section $\sigma_r$ as a linear integral transform of $N(r,x)$ and discretizing the problem, standard regularized inversion algorithms can recover $N$ without a fixed parametrization, enabling uncertainty quantification. Closure tests using ground-truth dipole parametrizations demonstrate robust recovery of the intermediate $r$-regime and the method’s ability to distinguish different parametrizations, even with heavy-quark effects included in the forward operator. The approach offers a parametrization-bias-free route to small-$x$ phenomenology, with potential applications to HERA and EIC data and future comparisons to BK/JIMWLK evolution, while highlighting avenues for improving forward-theory order and discretization in subsequent work.

Abstract

We show that the inference problem of constraining the dipole amplitude with inclusive deep inelastic scattering data can be written into a discrete linear inverse problem, in an analogous manner as can be done for computed tomography. To this formulation of the problem, we apply standard inverse problems methods and algorithms to reconstruct known dipole amplitudes from simulated reduced cross section data with realistic precision. The main difference of this approach to previous works is that this implementation does not require any fit parametrization of the dipole amplitude. The freedom from parametrization also enables us for the first time to quantify the uncertainties of the inferred dipole amplitude in a novel more general framework. This mathematical approach to small-$x$ phenomenology opens a path to parametrization bias free inference of the dipole amplitude from HERA and Electron--Ion Collider data.

Figures (10)

• Figure 1: Depiction of the virtual photon--proton deep inelastic scattering in the dipole picture. At leading order in light-cone perturbation theory, the incoming virtual photon quantum state fluctuates into a quark--antiquark state, which is able to probe the strong nuclear force field of the proton. In this figure, time proceeds left to right, and vertical axis is the separation between the probe $\gamma^*$ and target proton, which are primarily traveling in opposite directions in the collider experiment. Key quantities are the transverse size of the quark--antiquark dipole-state $r$, the transverse separation of the target and the incoming quantum state $b$, and the fractional momentum of the quarks in the state $z$. The dipole amplitude---the quantity which describes the scattering process of the dipole-state off the target shown in the center---is to be inferred from the collider experiment data.
• Figure 2: Comparison of considered reconstruction methods. Altogether nine algorithms were compared, and the best performing six are shown here. The 1 order priorconditioned Tikhonov--Phillips regularization algorithm was chosen as the main method to be used for the rest of the paper for its consistent good performance, and less pronounced bad behavior in challenging regimes of the reconstruction.
• Figure 3: Reconstructions of the dipole amplitude at various fixed Bjorken-$x$ from generated reduced cross section data compared to the ground-truth dipole amplitude. The ground-truth dipole amplitude is the 4-parameter Bayesian fit from Ref. Casuga:2023dcf.
• Figure 4: Reconstructions of the dipole amplitude shown with linear vertical axis to show the growth of uncertainty at large dipole sizes, where the reconstruction becomes unsensitive to large changes in the dipole, due to the asymptotic vanishing of the forward operator at large $r$. Nevertheless, the reconstructions manage to recover the ground-truth fairly accurately even at large $r$.
• Figure 5: Ratio of the reconstructions with respect to the ground-truth fit dipole amplitude to show the relative precision of the reconstructions. In ideal conditions the "noiseless" reconstruction recovers the ground-truth quite accurately, however with the statistical fluctuations introduced to the data, the reconstruction becomes less accurate, and the relative uncertainties grow especially at small and large $r$.