The Matrix of Unitarity Triangle Angles for Quarks

TL;DR

This paper introduces a matrix of unitarity triangle angles for quark mixing, which is basis- and phase-independent, directly observable, and provides a new way to represent and analyze quark mixing data.

Contribution

It defines the angle matrix $\

Findings

The angle matrix $\\Phi$ is basis- and phase-convention independent.

Several entries of the $\\Phi$ matrix are already measurable in experiments.

Provides translation formulas between the mixing matrix and the angle matrix.

Abstract

In the context of quark (as for lepton) mixing, we introduce the concept of the matrix of unitarity triangle angles $\Phi$, emphasising that it carries equivalent information to the complex mixing matrix $V$ itself. The angle matrix $\Phi$ has the added advantage, with respect to $V$, of being both basis-and phase-convention independent and consequently observable (indeed several $\Phi$-matrix entries, eg. $\Phi_{cs}=\alpha$, $\Phi_{us}=\beta$ etc. are already long-studied as directly measurable/measured in $B$-physics experiments). We give complete translation formulae between the mixing-matrix and angle-matrix representations. We go on to consider briefly the present state of the experimental data on the full angle matrix and some of the prospects for the future, with reference to both the quark and lepton cases.