Analytic Derivation of the Longitudinal Proton Structure Function FL(x, Q2) and the Reduced Cross Section σr(x, Q2) at the Leading Order and the Next-to-leading Order Approximations

• Published in: International journal of theoretical physics Vol. 60; no. 10; pp. 3822 - 3849
• Main Authors: Zarrin, S., Dadfar, S.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer Nature B.V
• Subjects: Approximation; Cross-sections; Dipoles; Distribution functions; Elementary Particles; Inelastic scattering; Laplace transforms; Large Hadron Collider; Mathematical and Computational Physics; Partons; Physics; Physics and Astronomy; Protons; Quantum chromodynamics; Quantum Field Theory; Quantum Physics; Theoretical; Proton; Structure function; Longitudinal; Parton distribution functions
• ISSN: 0020-7748; 1572-9575
• DOI: 10.1007/s10773-021-04947-1

We present a set of formulas to extract the longitudinal proton structure function F L ( x , Q 2 ) and the reduced cross section σ r ( x , Q 2 ) by using Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations and Altarelli-Martinelli equation at the leading order (LO) and the next-to-leading order (NLO) approximations in perturbative quantum chromodynamics (QCD). These formulas are obtained by the Laplace transform method at the virtualities higher than Q 0 2 . We show that, the obtained equations for F L and σ r depend only on the parton distribution functions (PDF’s) at the initial scale Q 0 2 . We obtain the corresponding numerical results in a range of the virtuality Q 0 2 ≤ Q 2 ≤ 800 G e V 2 and Bjorken scale 10 − 4 ≤ x ≤ 1 and compare them with the results achieved by MSTW2008 (Martin et al. Eur. Phys. J 63 (2), 189–285, 2009 ), WT (White and Thorne Phy. Rev. D 75 (3), 034005, 2007 ) and Dipole (b-Sat) model (H. Kowalski, L. Motyka, G. Watt, Phys. Rev. D 74 (7), 074016, 2006 ) predictions and also with data released by the Hadron Electron Ring Accelerator (HERA). Our numerical results show an acceptable agreement with the deep inelastic scattering (DIS) experimental data throughout over a wide range of the Bjorken scale x and the virtuality Q 2 , and then can be applied in analyses of the Large Hadron Collider and Future Circular Collider projects.