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

Given the 3 × 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: ◊ α β γ i j k ≡ ( V α i V β j V γ k ) / det V with α ≠ β ≠ γ and i ≠ j ≠ k running respectively over ( u, c, t ) and ( d, s, b ). We find that I m ◊ α β γ i j k = − J holds, where J is the well-known Jarlskog invariant of weak CP violation. Analogous rephasing invariants ⧫ α β γ i j k ≡ ( U α i U β j U γ k ) / det U can be defined for the 3 × 3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix U , where α ≠ β ≠ γ and i ≠ j ≠ k run respectively over ( e, μ, τ ) and (1, 2, 3). Taking into account small non-unitarity of U based on the canonical seesaw mechanism for neutrino mass generation, we calculate I m ⧫ α β γ i j k 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 J ν in the unitarity limit of U .