Implications of the evidence for direct $\mathbf{CP}$ 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 $10\%$ 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.