Earth mover's distance as a measure of CP violation

We introduce a new unbinned two sample test statistic sensitive to CP violation utilizing the optimal transport plan associated with the Wasserstein (earth mover's) distance. The efficacy of the test statistic is shown via two examples of CP asymmetric distributions with varying sample sizes: the Dalitz distributions of $B^0 \rightarrow K^+\pi^-\pi^0$ and of $D^0 \rightarrow \pi^+\pi^-\pi^0$ decays. The windowed version of the Wasserstein distance test statistic is shown to have comparable sensitivity to CP violation as the commonly used energy test statistic, but also retains information about the localized distributions of CP asymmetry over the Dalitz plot. For large statistic datasets we introduce two modified Wasserstein distance based test statistics -- the binned and the sliced Wasserstein distance statistics, which show comparable sensitivity to CP violation, but improved computing time and memory scalings. Finally, general extensions and applications of the introduced statistics are discussed.