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: $\lozenge^{ijk}_{\alpha\beta\gamma} \equiv (V^{}_{\alpha i} V^{}_{\beta j} V^{}_{\gamma k})/\det V$ with $\alpha \neq \beta \neq \gamma$ and $i \neq j \neq k$ running respectively over $(u, c, t)$ and $(d, s, b)$. We find that ${\rm Im} \lozenge^{ijk}_{\alpha\beta\gamma} = - {\cal J}$ holds, where ${\cal J}$ is the well-known Jarlskog invariant of weak CP violation. Analogous rephasing invariants $\blacklozenge^{ijk}_{\alpha\beta\gamma} \equiv (U^{}_{\alpha I} U^{}_{\beta j} U^{}_{\gamma k})/\det U$ can be defined for the $3\times 3$ Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix $U$, where $\alpha \neq \beta \neq \gamma$ and $i \neq j \neq k$ 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 ${\rm Im} \blacklozenge^{ijk}_{\alpha\beta\gamma}$ 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 ${\cal J}^{}_\nu$ in the unitarity limit of $U$.