Implications of the evidence for direct $\mathbf{C}\mathbf{P}$ violation in $D\to {\pi }^{+}{\pi }^{-}$ decays

The observation of $CP$ violation in the difference of $CP$ asymmetries between $D\to K^{+}{K}^{-}$ and $D\to {\pi }^{+}{\pi }^{-}$ has raised a debate whether the observed asymmetries can be regarded as a signal of physics beyond the standard model (SM). In this paper we obtain all the topological amplitudes and isospin amplitudes directly from measured observables for $D\to \pi \pi$. These results unambiguously imply a very large penguin contribution, having a central value $4.74$ times the magnitude of the amplitude for $D^0\to {\pi }^{+}{\pi }^{-}$. This fitted central value differs from a reasonable SM estimate of $10\mathrm{\%}$ with a significance greater than $3.3\sigma$. In contrast to previous studies, we present model-independent arguments based only on unitarity of re-scattering amplitudes to show that large penguins cannot arise from re-scattering alone and likely indicate physics beyond the SM. In a model-independent approach we show how a very small contribution from physics beyond the SM with a large weak phase alleviates the problem.