Taming the long distance effects in the $D^{+}_{s}\rightarrow \pi^{+}\ell^{-}\ell^{+}$ decay
The semileptonic decay $D^{+}_{s}\rightarrow \pi^{+}\ell^{-}\ell^{+}$, where $\ell=e, \ \mu$, is used as a reference channel when looking for suppressed or forbidden processes of the standard model of the form $D_{s}\rightarrow P\ell^{\pm}\ell^{\prime+}$. The process is dominated by hadronic resonances from the $\eta$ to the $\phi$ meson region, usually excluded due to the large uncertainties associated. In this work, we explore the role of the parameters involved in the resonance region. We focus in the relative strong phase between the $\rho$ and the $\phi$ meson ($\delta_{\rho\phi}$). We argue that this parameter can be bound from LHCb data of the di-muon invariant mass in the $D^{+}_{s}\rightarrow \pi^{+}\mu^{-}\mu^{+}$ decay and we perform a fit to provide a first approach to it. We obtain $\delta_{\rho\phi}=(0.44 \pm 0.24) \pi$. In contradistinction, it would be useful to consider observables insensitive to this phase. We first compute the invariant mass of the lepton pair at a given angle of one of the leptons emission with respect to the pion, and then we compute the forward-backward distribution at such angle. We show that for the latter observable the resonance region is smeared, exhibiting no dependence on the $\delta_{\rho\phi}$ phase, although global effects are observed by comparing with the pure phase space approach. In order to consider this observable in more general scenarios, we analyse the behavior for the resonant background process $D^{+}_{s}\rightarrow \pi\pi\pi$, the short distance contribution in the standard model, and the charged Higgs mediated process as given by the two Higgs doublet model, which exhibit distinguishable features among themselves.