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: ${◊}_{\alpha \beta \gamma }^{ijk}\equiv (V_{\alpha i}^{}{V}_{\beta j}^{}{V}_{\gamma k}^{})/\det V$ with $\alpha \ne \beta \ne \gamma$ and $i\ne j\ne k$ running respectively over $(u,c,t)$ and $(d,s,b)$. We find that $\mathrm{Im}{◊}_{\alpha \beta \gamma }^{ijk}=-\mathcal{J}$ holds, where $\mathcal{J}$ is the well-known Jarlskog invariant of weak CP violation. Analogous rephasing invariants ${⧫}_{\alpha \beta \gamma }^{ijk}\equiv (U_{\alpha I}^{}{U}_{\beta j}^{}{U}_{\gamma k}^{})/\det U$ can be defined for the $3\times 3$ Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix $U$, where $\alpha \ne \beta \ne \gamma$ and $i\ne j\ne k$ run respectively over $(e,\mu ,\tau )$ and $(1,2,3)$. Taking into account small non-unitarity of $U$ based on the canonical seesaw mechanism for neutrino mass generation, we calculate $\mathrm{Im}{⧫}_{\alpha \beta \gamma }^{ijk}$ 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 ${\mathcal{J}}_{\nu }^{}$ in the unitarity limit of $U$.

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