Abstract
In this paper, we present the analytical solutions, based on the Laplace transform method, for the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi evolution equations of the photon at the leading order (LO) and next-to-leading order (NLO) approximations in perturbative QCD. Using these solutions, we derive the singlet, non-singlet, gluon, and photon distribution functions of the photon and also the photon structure function F^{gamma }_{2} (x,Q^{2}) at the LO and NLO approximations. We show that the resulting distribution functions are in agreement with the results of the parameterization model formulated by Gluck, Reya, and Vogt (GRV) (Phys Rev D 46(5):1973, 1992). Moreover, our numerical results of F^{gamma }_{2} (x,Q^{2}) are comparable with the results achieved by Aurenche, Fontannaz, and Guillet (AFG) (Eur Phys J C Part Fields 44(3):395–409, 2005) and also with the experimental data released by the L3, DELPHI, OPAL, ALEPH and PLUTO collaborations.
Highlights
Photons in the interactions: the direct photon and the resolved photon
The Dokshitzer– Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations are a set of coupled integro-differential equations that give the parton distribution functions in perturbative QCD
The QCD DGLAP evolution equations of the proton at leading order (LO) and next-to-leading order (NLO) have been solved by using the Laplace transform [20,21,22,23] and the Mellin transform methods [24,25] and have yielded the proton structure function and singlet, non-singlet, and gluon distribution functions inside the proton
Summary
After observing the photon structure function in different interactions, theorists have been interested in studying such a structure function. The QCD DGLAP evolution equations of the proton at LO and NLO have been solved by using the Laplace transform [20,21,22,23] and the Mellin transform methods [24,25] and have yielded the proton structure function and singlet, non-singlet, and gluon distribution functions inside the proton. We apply the Laplace transform method to solve the QCD⊗QED evolution equations of the photon at LO and NLO QCD and at LO QED On this basis, we extract the singlet, gluon, and photon distribution functions of the photon at the scale Q2 as follows: Fsγ (x, Q2) = F (Fsγ0(x), Gγ0 (x), M0γ (x), Fnγs0(x)), (1). In Appendix B, we present the functions of a set of coefficients in the Laplace s space, which are dependent on the splitting functions
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