Direct determination of the structure functions $F_L$, $F_S$ and $G$ from $F_2$ and $dF_2/dQ^2$ to $O(\alpha_s^2)$

TL;DR

This paper derives direct expressions for key structure functions in deep inelastic scattering using measured data, extending previous results to include higher-order corrections.

Contribution

It provides a method to determine $F_L$, $F_S$, and $G$ directly from $F_2$ and its derivative, including corrections up to order $\alpha_s^2$.

Findings

Expressions for structure functions are valid to $O(\alpha_s^2)$.

Results correct and extend previous mixed-order calculations.

Non-singlet corrections are small at very small $x$ and manageable.

Abstract

We extend the results of Lappi {\em et al.}, Eur.~Phys.~J.~C {\bf 84}, 84 (2024), to show that it is possible to obtain expressions for the longitudinal, singlet and gluon structure functions $F_L$, $F_S$ and $G$ in deep inelastic scattering directly in terms of the measured functions $F_2$ and $dF_2/\ln(Q^2)$ {\em modulo} non-singlet corrections expected to be small at very small $x$. The latter can be treated at low $x$ using existing quark distributions. Our results are presented consistently to $O(\alpha_s^2)$, correcting and extending the mixed-order results of Lappi {\em et al.}.