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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 angulardistribution which can include arbitrary partial waves of $\pi\pi$. Retainingthe S-wave and P-wave contributions we study the branching ratios,forward--backward asymmetries and a few other observables. The P-wavecontribution is dominated by $\rho^0$ resonance, and the S-wave contribution isanalyzed using the unitarized chiral perturbation theory. The obtainedbranching fraction for $D\to \rho\ell\nu$, at the order $10^{-3}$, isconsistent with the available experimental data. The S-wave contribution has abranching ratio at the order of $10^{-4}$, and this prediction can be tested byexperiments like BESIII and LHCb. Future measurements can also be used toexamine the $\pi$--$\pi$ scattering phase shift.

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