Observation of ${D}^{+}\to {f}_{0}\left(500\right){\mu }^{+}{\nu }_{\mu }$ and study of ${D}^{+}\to {\pi }^{+}{\pi }^{-}{\ell }^{+}{\nu }_{\ell }$ decay dynamics

Using $2.93\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ collision data collected with the BESIII detector at the center-of-mass energy of 3.773 GeV, we investigate the semileptonic decays ${D}^{+}\to {\pi }^{+}{\pi }^{-}{\ell }^{+}{\nu }_{\ell }$ ($\ell =e$ and $\mu $). The ${D}^{+}\to {f}_{0}\left(500\right){\mu }^{+}{\nu }_{\mu }$ decay is observed for the first time. By analyzing simultaneously the differential decay rates of ${D}^{+}\to {f}_{0}\left(500\right){\mu }^{+}{\nu }_{\mu }$ and ${D}^{+}\to {f}_{0}\left(500\right){e}^{+}{\nu }_{e}$ in different ${\ell }^{+}{\nu }_{\ell }$ four-momentum transfer intervals, the product of the relevant hadronic form factor ${f}_{+}^{{f}_{0}}\left(0\right)$ and the magnitude of the $c\to d$ Cabibbo-Kobayashi-Maskawa matrix element $|{V}_{cd}|$ is determined to be ${f}_{+}^{{f}_{0}}\left(0\right)|{V}_{cd}|=0.143±0.01{4}_{\mathrm{stat}}±0.01{1}_{\mathrm{syst}}$ for the first time. With the input of $|{V}_{cd}|$ from the global fit in the standard model, we determine ${f}_{+}^{{f}_{0}}\left(0\right)=0.63±0.0{6}_{\mathrm{stat}}±0.0{5}_{\mathrm{syst}}$. The absolute branching fractions of ${D}^{+}\to {f}_{0}\left(500{\right)}_{\left({\pi }^{+}{\pi }^{-}\right)}{\mu }^{+}{\nu }_{\mu }$ and ${D}^{+}\to {\rho }_{\left({\pi }^{+}{\pi }^{-}\right)}^{0}{\mu }^{+}{\nu }_{\mu }$ are determined as $\left(0.72±0.1{3}_{\mathrm{stat}}±0.0{8}_{\mathrm{syst}}\right)×{10}^{-3}$ and $\left(1.64±0.1{3}_{\mathrm{stat}}±\phantom{\rule{0ex}{0ex}}0.1{0}_{\mathrm{syst}}\right)×{10}^{-3}$. Combining these results with those of previous BESIII measurements on their semielectronic counterparts from the same data sample, we test lepton flavor universality by measuring the branching fraction ratios ${B}_{{D}^{+}\to {\rho }^{0}{\mu }^{+}{\nu }_{\mu }}/{B}_{{D}^{+}\to {\rho }^{0}{e}^{+}{\nu }_{e}}=0.88±0.10$ and ${B}_{{D}^{+}\to {f}_{0}\left(500\right){\mu }^{+}{\nu }_{\mu }}/{B}_{{D}^{+}\to {f}_{0}\left(500\right){e}^{+}{\nu }_{e}}=\phantom{\rule{0ex}{0ex}}1.14±0.26$, which are compatible with the standard model expectation.