CPasymmetry in top-quark radiative decays

We analyze the $\mathrm{CP}$ asymmetry in the top-quark radiative decay $t\to q\gamma$, where $q=c, u \mathrm{and} \gamma$ is an on-shell photon, due to the one-loop penguin diagram in the three-generation standard model. This penguin-dominated $\mathrm{CP}$ asymmetry is a new phenomenon of heavy-quark physics, and corresponds to the onset in the amplitude of the physical thresholds $t\to \mathrm{real}\mathrm{}\left\{\mathrm{Wl}\right\}\mathrm{}$, where $l=b, s, d$ is a quark in the loop. A $\mathrm{CP}$ asymmetry in a radiative decay to an on-shell photon in the light-quark sector (e.g. $B$-meson decays) cannot be generated by the one-loop penguin. The origin of this asymmetry is thus formally of some interest, although the phenomenological application turns out to be rather limited, due to extremely small branching ratios for $t\to q\gamma$, at least to one-loop order (however, our calculations raise an interesting possibility with respect to the role of QCD corrections to this asymmetry). This calculation should also serve as a useful benchmark for the study of possible $\mathrm{CP}$ asymmetries in the top-quark sector, in the context of new physics.