Model-Independent and Data-Driven Extraction of the Photon Distribution Function at the Electron--Ion Collider

In this work, we investigate a cross-section-ratio-based inversion method for extracting distribution functions at the Electron--Ion Collider (EIC). Starting from the general factorization formula $d\sigma = H \otimes f$, we show that, for an appropriate bremsstrahlung process and suitably chosen differential observables, the convolution structure can be reduced to a multiplicative form. On this basis, we define the ratio $R = \frac{d\sigma_{\mathrm{exp}}}{d\sigma_{\mathrm{hard}}},$ where $d\sigma_{\mathrm{exp}}$ is the experimentally measured differential cross section and $d\sigma_{\mathrm{hard}}$ is a perturbatively calculable theoretical input. We then establish the relation between $R$ and the target distribution function $f$. Finally, we investigate how the theoretical input $d\sigma_{\mathrm{hard}}$ should be constructed when soft-photon radiation, finite-bin-width effects, and experimental acceptance are taken into account.