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Direct determination of the structure functions $F_{L}$ , $F_{S}$ , and $G$ from $F_{2}$ and $d F_{2} / d Q^{2}$ to $O (α_{s}^{2})$

G. R. Boroun ^*

Loyal Durand ^†,§

Phuoc Ha ^‡

Department of Physics, Razi University, Kermanshah 67149, Iran

Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA

Department of Physics, Astronomy, and Geosciences, Towson University, Towson, Maryland 21252, USA

^*Contact author: boroun@razi.ac.ir
^†Contact author: ldurand@hep.wisc.edu
^‡Contact author: pdha@towson.edu
^§Present address: 415 Pearl Court, Aspen, Colorado 81611, USA.

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Phys. Rev. D 114, 034029 – Published 13 August, 2026

DOI: https://doi.org/10.1103/psnk-s9bg

Abstract

We extend the results of Lappi et al. [Eur. Phys. J. C 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 $d F_{2} / \ln (Q^{2})$ modulo nonsinglet 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 $O (α_{s}^{2})$ , correcting and extending the mixed-order results of Lappi et al..

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References (28)

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Phys. Rev. D 114, 034029– Published 13 August, 2026

Vol. 114, Iss. 3 — 1 August 2026

Received 9 May 2026
Accepted 21 July 2026

DOI: https://doi.org/10.1103/psnk-s9bg

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Direct determination of the structure functions $F_{L}$ , $F_{S}$ , and $G$ from $F_{2}$ and $d F_{2} / d Q^{2}$ to $O (α_{s}^{2})$

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Direct determination of the structure functions FL, FS, and G from F2 and dF2/dQ2 to O(αs2)

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Direct determination of the structure functions $F_{L}$ , $F_{S}$ , and $G$ from $F_{2}$ and $d F_{2} / d Q^{2}$ to $O (α_{s}^{2})$