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

TL;DR

The paper introduces new rephasing invariants based on the trios of CKM and PMNS matrix elements, linking them to known CP violation measures and exploring their behavior under neutrino non-unitarity.

Contribution

It defines novel rephasing invariants from matrix element trios and relates them to the Jarlskog invariant, extending the understanding of CP violation in quark and lepton sectors.

Findings

Rephasing invariants equal to the Jarlskog invariant for CKM.

In the lepton sector, invariants converge to a universal CP violation measure.

Analysis includes effects of non-unitarity in neutrino mixing matrices.

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 $U$.