Disentangling QCD and New Physics in $D^+\to\pi^+\ell^+\ell^-$

In this paper we consider the decay $D^+\to\pi^+\ell^+\ell^-$, addressing in particular the resonance contributions as well as the relatively large contributions from the weak annihilation diagrams. For the weak annihilation diagrams we include known results from QCD factorisation at low $q^2$ and at high $q^2$, adapting the existing calculation for $B$ decays in the Operator Product Expansion. The hadronic resonance contributions are obtained through a dispersion relation, modelling the spectral functions as towers of Regge-like resonances in each channel, as suggested by Shifman, imposing the partonic behaviour in the deep Euclidean. The parameters of the model are extracted using $e^+e^-\to{\rm (hadrons)}$ and $\tau\to {\rm (hadrons)}+\nu_\tau$ data as well as the branching ratios for the resonant decays $D^+\to\pi^+\,R\,(R\to\ell^+\ell^-)$, with $R=\rho$, $\omega$, and $\phi$. We perform a thorough error analysis, and present our results for the Standard Model differential branching ratio as a function of $q^2$. Focusing then on the observables $F_H$ and $A_{\rm FB}$, we consider the sensitivity of this channel to effects of physics beyond the Standard Model, both in a model independent way and for the case of leptoquarks.