Resonances in $D^0\to\pi^+\pi^-\ell^+\ell^-$ and sensitivity to New Physics

The study of processes involving the weak decays of the charm quark offers unique possibilities to test the Standard Model from the up-type sector and to probe different New Physics scenarios. In here we focus on the rare decays $D^0\to\pi^+\pi^-\ell^+\ell^-$. The effectiveness of the Glashow-Iliopoulos-Maiani mechanism in charm results in a series of observables that serve as null tests of the Standard Model. Simultaneously, the dominant Standard Model contribution to such decays comes from long-distance dynamics. Following previous literature we describe this with the mediation of resonances. Motivated by recent experimental results on analogous semileptonic decays we investigate the effect of the $S$-wave of the pion pair by considering the scalar resonance $f_0(500)$. The experimental mass distributions and angular observables are described well with our model and there is a significant improvement on the agreement when accounting for $f_0(500)$, which amounts to around 20\% of the total branching ratio. Furthermore, we propose and estimate the size of a series of additional observables to be measured, which can help probe the $S$-wave and act as complementary null tests of the Standard Model.