Chiral dynamics, S-wave contributions and angular analysis in $D\to \pi\pi \ell\bar\nu$

We present a theoretical analysis of the $D^-\to \pi^+\pi^- \ell\bar\nu$ and $\bar D^0\to \pi^+\pi^0 \ell\bar\nu$ decays. We construct a general angular distribution which can include arbitrary partial waves of $\pi\pi$. Retaining the S-wave and P-wave contributions we study the branching ratios, forward--backward asymmetries and a few other observables. The P-wave contribution is dominated by $\rho^0$ resonance, and the S-wave contribution is analyzed using the unitarized chiral perturbation theory. The obtained branching fraction for $D\to \rho\ell\nu$, at the order $10^{-3}$, is consistent with the available experimental data. The S-wave contribution has a branching ratio at the order of $10^{-4}$, and this prediction can be tested by experiments like BESIII and LHCb. Future measurements can also be used to examine the $\pi$--$\pi$ scattering phase shift.