Asymmetry in flavour changing electromagnetic transitions of vector-like quarks

Vector-like quarks have been of interest for a plethora of experimentally motivated reasons and have come under increased investigation recently due to their inclusion to the Standard Model (SM) appreciably improving global fits to several flavour physics and precision electroweak measurements. The addition of vector-like quark singlets breaks the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and enables tree-level flavour changing neutral (FCN) vertices. The resulting radiative flavour changing decays of these particles through the electromagnetic transition dipole moment are a key means to study their properties and search for them at experiments like the Large Hadron Collider (LHC). Despite these radiative decays providing such a clean experimental signature, an explicit analytical study of the branching ratios and $CP$ violation resulting from these loop level processes has thus far evaded attention. We provide the formulation for the decay rates and $CP$ asymmetry resulting from a combination of the imaginary components of the loop integrals and complex phases in the quark mixing matrix. We then apply our analytical results to study phenomenology of these states for several global fits pertaining to vector-like isosinglets $t'$ and $b'$. We find that clean collider signatures and polarisation observables can be generated for both $t'$ and $b'$.