$CP$ asymmetries in the rare top decays $t\to c\gamma$ and $t\to c g$

The rare radiative flavour changing top decays $t\to c\gamma$ and $t\to cg$ (and the even rarer $t\to u\gamma$ and $t\to ug$) have been processes of interest for decades as they offer a key probe for studying top quark properties. However an explicit analytical study of the branching ratios and $CP$ asymmetries resulting from these loop level processes has thus far evaded attention. In this work, we provide the formulation for the $CP$ asymmetry resulting from the total kinetic contribution of the loop integrals and their imaginary parts, as well as an updated numerical computation of the predicted Standard Model (SM) branching fractions. These rare processes are suppressed in the SM by the Glashow-Iliopoulos-Maiani (GIM) mechanism. However, the results presented here can easily be exported for use in minimal extensions of the SM including vector-like quarks or in Two-Higgs-Doublet models where radiative fermionic decay processes can be enhanced relative to the SM by several orders of magnitude. Such processes provide an experimentally clean signature for new fundamental physics and can potentially be tested by current collider experiments. These topical beyond the SM theories are an elegant means to provide improved global fits to the latest results emerging from flavour physics, CKM and precision electroweak measurements.