One-loop structure of parton distribution for the gluon condensate and "zero modes"

We present results for one-loop corrections to the recently introduced "gluon condensate" PDF $F(x)$. In particular, we give expression for the $gg$-part of its evolution kernel. To enforce strict compliance with the gauge invariance requirements, we have used on-shell states for external gluons, and have obtained identical results both in Feynman and light-cone gauges. No "zero mode" $\delta (x)$ terms were found for the twist-4 gluon PDF $F(x)$. However a $q^2 \delta (x)$ term was found for the $\xi=0$ GPD $F(x,q^2)$ at nonzero momentum transfer $q$. Overall, our results do not agree with the original attempt of one-loop calculations of $F(x)$ for gluon states, which sets alarm warning for calculations that use matrix elements with virtual external gluons and for lattice renormalization procedures based on their results.