Comment on Reparametrization Invariance of Quark-Lepton Complementarity

TL;DR

This paper investigates the quark-lepton complementarity (QLC) by deriving reparametrization invariant expressions for the mixing angles, revealing that the observed relations are accidental and dependent on parametrization.

Contribution

It introduces reparametrization invariant formulas for QLC relations, clarifying their dependence on parametrization and providing exact and approximate expressions.

Findings

Exact QLC relations expressed in terms of mixing matrix elements.

Reparametrization invariance shows QLC relations are accidental.

Derived formulas include deviations from exact complementarity.

Abstract

We study the complementarity between quark and lepton mixing angles (QLC), the sum of an angle in quark mixing and the corresponding angle in lepton mixing is $\pi/4$. Experimentally in the standard PDG parametrization, two such relations exist approximately. These QLC relations are accidental which only manifest themselves in the PDG parametrization. We propose reparametrization invariant expressions for the complementarity relations in terms of the magnitude of the elements in the quark and lepton mixing matrices. In the exact QLC limit, it is found that $|V_{us}/V_{ud}| + |V_{e2}/V_{e1}| + |V_{us}/V_{ud}| |V_{e2}/V_{e1}| =1$ and $|V_{cb}/V_{tb}| + |V_{\mu 3}/V_{\tau 3}| +|V_{cb}/V_{tb}|| {V_{\mu 3}}/V_{\tau 3}| =1$. Expressions with deviations from exact complementarity are obtained. Implications of these relations are also discussed.