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 re
duction 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.
We investigate the one-gluon-exchange ($alpha alpha_s$) corrections to the polarized real photon structure function $g_1^gamma(x,Q^2)$ in the massive parton model. We employ a technique based on the Cutkosky rules and the reduction of Feynman integra
ls to master integrals. The NLO contribution is noticeable at large $x$ and does not vanish at the threshold of the massive quark pair production due to the Coulomb singularity. It is found that the first moment sum rule of $g_1^gamma$ is satisfied up to the NLO.
We consider a different power counting in potential NRQCD by incorporating the static potential exactly in the leading order Hamiltonian. We compute the leading relativistic corrections to the inclusive electromagnetic decay ratios in this new scheme
. The effect of this new power counting is found to be large (even for top). We produce an updated value for the $eta_b$ decay to two photons. This scheme also brings consistency between the weak coupling computation and the experimental value of the charmonium decay ratio.
