Solving the Inverse Source Problem in Femtoscopy with a Toy Model

TL;DR

The paper tackles the inverse problem of extracting emission source functions from momentum correlation functions in femtoscopy. It employs the Koonin-Pratt formulation and a toy setup with a square-well interaction and Gaussian-type sources, solving the ill-posed inversion via Tikhonov regularization with the L-curve criterion for parameter choice. The results demonstrate that Gaussian and mixed Gaussian sources can be accurately reconstructed from perturbed CFs (1%–10% noise) when the wave functions are known, validating the method's potential for realistic hadron–hadron systems. This approach offers a mathematically principled pathway to retrieve realistic source profiles, which in turn improves the extraction of hadron–hadron interactions from experimental CF data.

Abstract

Hadron-hadron interactions, as a non-perturbative effect, play a significant role in understanding phenomenological problems in particle physics. Femtoscopy is a powerful tool in heavy-ion collision experiments, enabling the extraction of hadron-hadron interactions via momentum-correlation functions (CFs). These CFs are generally factorized into a convolution of source functions and hadron-hadron wave functions, with the latter encoding information about hadron-hadron interactions. However, source functions remain ambiguous and are commonly approximated by a Gaussian form. Reconstructing source functions from experimental correlation data constitutes an ``inverse problem." To address it, we propose a toy model based on the Tikhonov regularization. Employing a square potential well of four distinct potential strengths, we calculate the CFs for inputs of a Gaussian source function and its hybrid form. The obtained CFs are subsequently used to reconstruct the source functions via the Tikhonov regularization. Our results demonstrate that the Gaussian source function can be successfully reconstructed, indicating the potential of this approach for extracting realistic source functions of hadron pairs of interest in the future.

Figures (4)

• Figure 1: Unregularized solutions (red curves) for the four sources, exhibiting unstable reconstructions that deviate from the benchmark solutions (black curves) by 17 to 18 orders of magnitude.
• Figure 2: Reconstructed solutions $S_\alpha^\delta$ (red lines) and the associated error bands (pink shaded areas) for $V_0=-10\,\mathrm{MeV}$ with $1\%$ uncertainty, compared to the benchmarks $S_t$ (black line).
• Figure 3: Reconstructed solutions $S_\alpha^\delta$ (red lines) and the associated error bands (pink shaded areas) for $V_0=-10\,\mathrm{MeV}$ with $10\%$ uncertainty, compared to the benchmarks $S_t$ (black line).
• Figure 4: Reconstruction of the source $S_3$ under three potential $25,-25$ and $-75\,\mathrm{MeV}$ with $10\%$ (top row) and $1\%$ (bottom row) uncertainty. Reconstructed solutions $S_\alpha^\delta$ (red lines) and the associated error bands (pink shaded areas), compared to the benchmarks $S_t$ (black line).