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

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.