Polarized Virtual Photon Structure Function $g_2^{\gamma }$ and Twist-3 Effects in QCD

We investigate the twist-3 effects in the polarized virtual photon structure. The structure functions $g_1^{\gamma }$ and $g_2^{\gamma }$ of polarized photon could be experimentally studied in the future polarized $ep$ or $e^{+}{e}^{-}$ colliders. The leading contributions to $g_1^{\gamma }$ are the twist-2 effects, while another structure function $g_2^{\gamma }$, which only exists for the virtual photon target, receives not only the twist-2 but also twist-3 contributions. We first show that the twist-3 effects actually exist in the box-diagram contributions and we extract the twist-3 part, which can also be reproduced by the pure QED operator product expansion. We then calculate the non-trivial lowest moment ($n=3$) of the twist-3 contribution to $g_2^{\gamma }$ in QCD. For large $N_c$ (the number of colors), the QCD analysis of the twist-3 effects in the flavor nonsinglet part of $g_2^{\gamma }$ becomes tractable and we can obtain its moments in a compact form for all $n$.