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

We calculate the rates of radiative $\beta^- \to \alpha^- + \gamma$ decays for $(\alpha, \beta) = (e, \mu)$, $(e, \tau)$ and $(\mu, \tau)$ by taking the {\it unitary} gauge in the $(3+n)$ active-sterile neutrino mixing scheme, and make it clear that constraints on the unitarity of the $3\times 3$ Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix $U$ extracted from $\beta^- \to \alpha^- + \gamma$ decays in the {\it 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 $\beta^- \to \alpha^- + \gamma$ 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 $U^{}_{\alpha i} U^*_{\beta i}$ and $U^{}_{\alpha j} U^*_{\beta j}$ (for $i \neq j$) 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 ${\cal J}^{ij}_{\alpha\beta}$, and their small differences signify slight unitarity violation of the PMNS matrix $U$.