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

Abstract

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.

Comments: 19 pages, 9 figures, 2 tables. To match the published version