Using a data sample corresponding to an integrated luminosity of 2.93~fb$^{ -1}$ recorded by the BESIII detector at a center-of-mass energy of $3.773$ GeV, we present an analysis of the decays $\bar{ D}^0\to\pi^+\pi^0 e^-\bar{ \nu}_e$ and $D^+\to\pi^-\pi^+ e^+\nu_e$. By performing a partial wave analysis, the $\pi^+\pi^-$ $S$-wave contribution to $D^+\to\pi^-\pi^+ e^+\nu_e$ is observed to be $(25.7\pm1.6\pm1.1)$% with a statistical significance greater than 10$\sigma$, besides the dominant $P$-wave contribution. This is the first observation of the $S$-wave contribution. We measure the branching fractions $\mathcal{ B}(D^{ 0} \to \rho^- e^+ \nu_e) = (1.445\pm 0.058 \pm 0.039) \times10^{ -3}$, $\mathcal{ B}(D^{ +} \to \rho^0 e^+ \nu_e) = (1.860\pm 0.070 \pm 0.061) \times10^{ -3}$, and $\mathcal{ B}(D^{ +} \to f_0(500) e^+ \nu_e, f_0(500)\to\pi^+\pi^-) = (6.30\pm 0.43 \pm 0.32) \times10^{ -4}$. An upper limit of $\mathcal{ B}(D^{ +} \to f_0(980) e^+ \nu_e, f_0(980)\to\pi^+\pi^-) < 2.8 \times10^{ -5}$ is set at the 90% confidence level. We also obtain the hadronic form factor ratios of $D\to \rho e^+\nu_e$ at $q^{ 2}=0$ assuming the single-pole dominance parameterization: $r_{ V}=\frac{ V(0)}{ A_{ 1}(0)}=1.695\pm0.083\pm0.051$, $r_{ 2}=\frac{ A_{ 2}(0)}{ A_{ 1}(0)}=0.845\pm0.056\pm0.039$.
Categories: hep-ex hep-lat