The real photon structure functions in massive parton model in NLO

We investigate the one-gluon-exchange ($\alpha \alpha_s$) corrections to the real photon structure functions $W_{TT} $, $W_{LT}$, $W_{TT}^{a} $ and $W_{TT}^\tau$ in the massive parton model. We employ a technique based on the Cutkosky rules and the reduction of Feynman integrals to master integrals. We show that a positivity constraint, which is derived from the Cauchy-Schwarz inequality, is satisfied among the unpolarized and polarized structure functions $W_{TT}$, $W_{TT}^a$ and $W_{TT}^\tau$ calculated up to the next-to-leading order in QCD.