New rephasing invariants and CP violation built from the trios of the CKM or PMNS matrix elements

Abstract

Given the $3\times 3$ Cabibbo-Kobayashi-Maskawa (CKM) quark flavor mixing matrix $V$, we define a new set of rephasing invariants in terms of the "trios" of its nine elements: ${◊}_{\hat{I}\pm {\hat{I}}^2{\hat{I}}^3}^{ijk}\equiv (V_{\hat{I}\pm i}^{}{V}_{{\hat{I}}^{2}j}^{}{V}_{{\hat{I}}^{3}k}^{})/\det V$ with $\hat{I}\pm \ne {\hat{I}}^2\ne {\hat{I}}^3$ and $i\ne j\ne k$ running respectively over $(u,c,t)$ and $(d,s,b)$. We find that $\mathrm{Im}{◊}_{\hat{I}\pm {\hat{I}}^2{\hat{I}}^3}^{ijk}=-\mathcal{J}$ holds, where $\mathcal{J}$ is the well-known Jarlskog invariant of weak CP violation. Analogous rephasing invariants ${⧫}_{\hat{I}\pm {\hat{I}}^2{\hat{I}}^3}^{ijk}\equiv (U_{\hat{I}\pm I}^{}{U}_{{\hat{I}}^{2}j}^{}{U}_{{\hat{I}}^{3}k}^{})/\det U$ can be defined for the $3\times 3$ Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix $U$, where $\hat{I}\pm \ne {\hat{I}}^2\ne {\hat{I}}^3$ and $i\ne j\ne k$ run respectively over $(e,\hat{I}¼,\ddot{I}„)$ and $(1,2,3)$. Taking into account small non-unitarity of $U$ based on the canonical seesaw mechanism for neutrino mass generation, we calculate $\mathrm{Im}{⧫}_{\hat{I}\pm {\hat{I}}^2{\hat{I}}^3}^{ijk}$ with the help of a full Euler-like block parametrization of the seesaw flavor structure and demonstrate that their leading terms converge to a universal invariant ${\mathcal{J}}_{\hat{I}}^{}½$ in the unitarity limit of $U$.

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