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

We calculate the rates of radiative β${}^{-}$ → α${}^{-}$ + γ decays for (α ,β) = (e, μ), (e, τ) and (μ, τ) 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 3×3 Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix U extracted from β${}^{-}$ → α${}^{-}$ + γ 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 β${}^{-}$ → α${}^{-}$ + γ 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${}_{\alpha i}$U${}_{\beta i}^{\ast }$ and U${}_{\alpha j}$U${}_{\beta j}^{\ast }$ (for i ≠ 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.