The $c \to u \gamma$ Contribution to Weak Radiative Charm Decay

The $c \to u \gamma$ transition does not occur at the tree level in the Standard Model and is strongly GIM-suppressed at one loop. The leading QCD logarithms are known to enhance the amplitude by two orders of magnitude. We point out that the amplitude increases further by two orders of magnitude after including the complete 2-loop QCD contributions. Nevertheless, $\Delta S =0$ radiative decays of charmed hadrons remain dominated by the $c \to d \bar{d} u \gamma$ and $c \to s \bar{s} u \gamma$ quark subprocesses.