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

Abstract

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 $U_{\alpha i}^{}{U}_{\beta i}^{\ast }$ and $U_{\alpha j}^{}{U}_{\beta j}^{\ast }$ (for $i\ne j$) 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 ${\mathcal{J}}_{\alpha \beta }^{ij}$, and their small differences signify slight unitarity violation of the PMNS matrix $U$.

Comments: 26 pages, 5 figures, and the version accepted for publication in EPJC