Study of the decay ${D}^{0}\to \rho \left(770{\right)}^{-}{e}^{+}{\nu }_{e}$

We present a study of the semileptonic decay ${D}^{0}\to {\pi }^{-}{\pi }^{0}{e}^{+}{\nu }_{e}$ using an ${e}^{+}{e}^{-}$ annihilation data sample of $7.93\text{}\text{}{\mathrm{fb}}^{-1}$ collected at the center-of-mass energy of 3.773 GeV with the BESIII detector. The branching fraction of ${D}^{0}\to \rho \left(770{\right)}^{-}{e}^{+}{\nu }_{e}$ is measured to be $\left(1.439±0.033\left(\mathrm{stat}\right)±0.027\left(\mathrm{syst}\right)\right)×{10}^{-3}$, which is a factor 1.6 more precise than previous measurements. By performing an amplitude analysis, we measure the hadronic form-factor ratios of ${D}^{0}\to \rho \left(770{\right)}^{-}{e}^{+}{\nu }_{e}$ at ${q}^{2}=0$ assuming the single-pole-dominance parametrization: ${r}_{V}=\phantom{\rule{0ex}{0ex}}V\left(0\right)/{A}_{1}\left(0\right)=1.548±0.079\left(\mathrm{stat}\right)±0.041\left(\mathrm{syst}\right)$ and ${r}_{2}={A}_{2}\left(0\right)/{A}_{1}\left(0\right)=0.823±0.056\left(\mathrm{stat}\right)±0.026\left(\mathrm{syst}\right)$.