Explicit rephasing to Kobayashi-Maskawa representation and fundamental phase structure of CP violation

In this letter, we construct an explicit rephasing transformation that converts an arbitrary unitary matrix into the Kobayashi–Maskawa (KM) parameterization and identify all independent CP phases in the mixing matrix as the arguments of its matrix elements. Furthermore, by applying this rephasing transformation to the fermion diagonalization matrices $U^{f}$, we show that the Majorana phases are represented by fermion-specific phases $δ^{ν, e}{\rm KM}$ and their relative phases. In particular, by neglecting the 3-1 elements $U{31}^{ν,e}$ of the diagonalization matrices for the two fermions, the KM phase $δ_{\rm KM}$ is concisely expressed by fermion-specific rephasing invariants involving two relative phases $δ_{\rm KM} = \arg \left [1 + ({U^{e * }{21} U^ν{21} / U^{e * }{11} U^ν{11} }) \right ] + \arg \left [ - { U_{32}^{e *} U_{32}^ν / U^{e * }{22} U^ν{22} } \right] $.

💡 Research Summary

In this paper the author presents a fully explicit construction of a rephasing transformation that maps any arbitrary unitary matrix onto the original Kobayashi‑Maskawa (KM) parametrization. While the existence of such a transformation has long been assumed, the actual procedure—i.e. the determination of the left‑hand and right‑hand phase matrices that bring a generic matrix into the KM form—has never been written out in detail. The author first defines the KM matrix (U_{1}) together with a diagonal Majorana‑phase matrix (P) (Eq. 1) and imposes six phase‑fixing conditions (Eq. 2) that uniquely specify the KM convention (all mixing angles positive, the overall phase of the determinant fixed to (\delta_{\rm KM}+\pi)).

A generic unitary matrix (U) is then factorised as (U = L,U_{1},R) where (L=\mathrm{diag}(e^{i\gamma_{L1}},e^{i\gamma_{L2}},e^{i\gamma_{L3}})) and (R=\mathrm{diag}(e^{-i\gamma_{R1}},e^{-i\gamma_{R2}},e^{-i\gamma_{R3}})) (Eq. 3). By separating absolute values and arguments of the matrix elements, the five independent equations (Eq. 7) determine the differences (\gamma_{Li}-\gamma_{Rj}) in terms of the arguments of the original matrix entries. One overall phase remains undetermined; the author chooses (\gamma_{R1}) as the free parameter so that the Majorana phases are left untouched. The final explicit rephasing is given in Eq. (8), which shows that the transformation merely removes five phases from the original matrix, leaving the KM phase (\delta_{\rm KM}) expressed as a single argument of a product of matrix elements (Eq. 4). This expression is equivalent to the well‑known Jarlskog invariant and to the sum rule (\delta_{\rm KM}= \pi-\delta_{\rm PDG}-\alpha+\gamma).

Having established the KM rephasing, the author applies the same machinery to the diagonalisation matrices of neutrinos ((U^{\nu})) and charged leptons ((U^{e})). Each of these matrices is written as (U^{f}= \Phi^{f}{L},U^{f}{1},\Phi^{f}{R}) with (f=\nu,e) (Eq. 16). The “fermion‑specific” KM phases (\delta^{f}{\rm KM}) are defined analogously to Eq. (17). The left‑hand phase matrices combine into a diagonal matrix (\Phi_{L}= \mathrm{diag}(e^{i\rho_{1}},e^{i\rho_{2}},e^{i\rho_{3}})) where (\rho_{i}= \arg!\big(U^{e*}{i1}U^{\nu}{i1}\big)) (Eq. 18). Only the differences (\rho_{i}-\rho_{j}) are physical; they are invariant under a global rephasing of the lepton mixing matrix.

The Majorana phases (\alpha_{2,3}) are then expressed as functions of the fermion‑specific KM phases, the relative phases (\rho_{i}-\rho_{j}), and the absolute values of the matrix elements (Eqs. 22–25). In the limit where the 3‑1 elements of both diagonalisation matrices are set to zero, the fermion‑specific KM phases drop out, and the observable KM phase (\delta_{\rm KM}) depends only on two relative phases. The author derives a compact formula (Eq. 32)
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