Observation of $D^+ to f_0(500) e^+ u_e$ and Improved Measurements of $D torho e^+ u_e$

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 $bar{D}^0topi^+pi^0 e^-bar{ u}_e$ and $D^+topi^-pi^+ e^+ u_e$. By performing a partial wave analysis, the $pi^+pi^-$ $S$-wave contribution to $D^+topi^-pi^+ e^+ u_e$ is observed to be $(25.7pm1.6pm1.1)$% 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 $mathcal{B}(D^{0} to rho^- e^+ u_e) = (1.445pm 0.058 pm 0.039) times10^{-3}$, $mathcal{B}(D^{+} to rho^0 e^+ u_e) = (1.860pm 0.070 pm 0.061) times10^{-3}$, and $mathcal{B}(D^{+} to f_0(500) e^+ u_e, f_0(500)topi^+pi^-) = (6.30pm 0.43 pm 0.32) times10^{-4}$. An upper limit of $mathcal{B}(D^{+} to f_0(980) e^+ u_e, f_0(980)topi^+pi^-) < 2.8 times10^{-5}$ is set at the 90% confidence level. We also obtain the hadronic form factor ratios of $Dto rho e^+ u_e$ at $q^{2}=0$ assuming the single-pole dominance parameterization: $r_{V}=frac{V(0)}{A_{1}(0)}=1.695pm0.083pm0.051$, $r_{2}=frac{A_{2}(0)}{A_{1}(0)}=0.845pm0.056pm0.039$.