Radiative decays of charged leptons as constraints of unitarity polygons for active-sterile neutrino mixing and CP violation

We calculate the rates of radiative beta--> alpha-+gamma decays for (alpha,beta)=(e,mu), (e,tau) and (mu,tau) by taking the unitary gauge in the (3+n) active-sterile neutrino mixing scheme, and make it clear that constraints on the unitarity of the 3x3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U extracted from beta--> alpha-+gamma decays in the minimal unitarity violation scheme differ from those obtained in the canonical seesaw mechanism with n heavy Majorana neutrinos by a factor 5/3. In such a natural seesaw case we show that the rates of beta--> alpha-+gamma can be used to cleanly and strongly constrain the effective apex of a unitarity polygon, and compare its geometry with the geometry of its three sub-triangles formed by two vectors U alpha iU beta i and U alpha jU beta j (for i not equal j) in the complex plane. We find that the areas of such sub-triangles can be described in terms of the Jarlskog-like invariants of CP violation J alpha beta ij, and their small differences signify slight unitarity violation of the PMNS matrix U.