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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    $$F^{\gamma }_{2} (x,Q^{2})$$F  2  γ    (  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    $$F^{\gamma }_{2} (x,Q^{2})$$F  2  γ    (  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.

european physical journal. c, particles and fields

Analytic derivation of the photon structure function $$F^{\gamma }_{2} (x,Q^{2})$$ at the leading order and the next-to-leading order approximations based on the Laplace transform method