Observation of ${D}^{+}\to {f}_{0}\left(500\right){e}^{+}{\nu }_{e}$ and Improved Measurements of $D\to \rho {e}^{+}{\nu }_{e}$

Using a data sample corresponding to an integrated luminosity of $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ recorded by the BESIII detector at a center-of-mass energy of 3.773 GeV, we present an analysis of the decays ${D}^{0}\to {\pi }^{-}{\pi }^{0}{e}^{+}{\nu }_{e}$ 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 $\left(25.7±1.6±1.1\right)%$ 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 $B\left({D}^{0}\to {\rho }^{-}{e}^{+}{\nu }_{e}\right)=\left(1.445±0.058±0.039\right)×1{0}^{-3}$, $B\left({D}^{+}\to {\rho }^{0}{e}^{+}{\nu }_{e}\right)=\phantom{\rule{0ex}{0ex}}\left(1.860±0.070±0.061\right)×1{0}^{-3}$, and $B\left({D}^{+}\to {f}_{0}\left(500\right){e}^{+}{\nu }_{e},{f}_{0}\left(500\right)\to {\pi }^{+}{\pi }^{-}\right)=\left(6.30±0.43±0.32\right)×1{0}^{-4}$. An upper limit of $B\left({D}^{+}\to {f}_{0}\left(980\right){e}^{+}{\nu }_{e},{f}_{0}\left(980\right)\to {\pi }^{+}{\pi }^{-}\right)<2.8×1{0}^{-5}$ 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 parametrization: ${r}_{V}=\left\{\left[V\left(0\right)\right]/\left[{A}_{1}\left(0\right)\right]\right\}=1.695±0.083±0.051$, ${r}_{2}=\left\{\left[{A}_{2}\left(0\right)\right]/\left[{A}_{1}\left(0\right)\right]\right\}=\phantom{\rule{0ex}{0ex}}0.845±0.056±0.039$.