The hyperbolic rotation group of neutral meson mixing and $CP$ violation

TL;DR

This paper introduces a geometric interpretation of neutral meson mixing and CP violation using hyperbolic rotations in a complex Lie group, providing new insights into meson decay processes and CKM angle measurements.

Contribution

It presents a novel mathematical framework modeling meson mixing and CP violation as hyperbolic rotations within the Lie group SO(1,1,C), enabling geometric visualization and analysis.

Findings

Derived charm mixing correction on CPV parameters.

Analyzed effects of charm mixing on GLW, ADS, GGSZ methods.

Described combined charm and strange mixing with CPV.

Abstract

Neutral meson mixing and $CP$ violation are very well known weak processes that involve decays to meson states that are, in general, a superposition of flavor eigenstates. This paper describes a mathematical interpretation of the time-dependent mixing amplitudes as a complex hyperbolic rotation of the time evolution of those amplitudes without mixing, which involves a Lie group $SO(1,1,\mathbb{C})$. This allows a geometric interpretation of mixing as a curve into the $SO(1,1,\mathbb{C})$ manifold, parameterized with the proper decay time, where $CP$ violation is the image of this curve at $t = 0$. To show the power of this new interpretation, it is applied to several aspects of the measurement of the CKM angle $\gamma$ in $B$ decays to neutral $D$ mesons. On one hand, the charm mixing correction on the $CPV$ parameters is derived. On the other hand, it is shown how the expressions used in GLW, ADS and GGSZ methods are affected by charm mixing. Finally, the complete example with both charm and strange mixing and $CPV$ is described.