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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 ofinterest for decades as they offer a key probe for studying top quarkproperties. However an explicit analytical study of the branching ratios and$CP$ asymmetries resulting from these loop level processes has thus far evadedattention. In this work, we provide the formulation for the $CP$ asymmetryresulting from the total kinetic contribution of the loop integrals and theirimaginary parts, as well as an updated numerical computation of the predictedStandard Model (SM) branching fractions. These rare processes are suppressed inthe SM by the Glashow-Iliopoulos-Maiani (GIM) mechanism. However, the resultspresented here can easily be exported for use in minimal extensions of the SMincluding vector-like quarks or in Two-Higgs-Doublet models where radiativefermionic decay processes can be enhanced relative to the SM by several ordersof magnitude. Such processes provide an experimentally clean signature for newfundamental physics and can potentially be tested by current colliderexperiments. These topical beyond the SM theories are an elegant means toprovide improved global fits to the latest results emerging from flavourphysics, CKM and precision electroweak measurements.

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