Measurement of the Branching Fraction For the Semi-Leptonic Decays $D^{0(+)}\to ��^{-(0)}��^+��_��$ and Test of Lepton Flavor Universality

Using a data sample corresponding to an integrated luminosity of $2.93\,\rm fb^{-1}$ taken at a center-of-mass energy of 3.773\,GeV with the BESIII detector operated at the BEPCII collider, we perform an analysis of the semi-leptonic decays $D^{0(+)}\to ��^{-(0)}��^+��_��$. The branching fractions of $D^0\to ��^-��^+��_��$ and $D^+\to ��^0��^+��_��$ are measured to be $(0.272 \pm 0.008_{\rm stat.} \pm 0.006_{\rm syst.})\%$ and $(0.350 \pm 0.011_{\rm stat.} \pm 0.010_{\rm syst.})\%$, respectively, where the former is of much improved precision compared to previous results and the latter is determined for the first time. Using these results along with previous BESIII measurements of $D^{0(+)}\to ��^{-(0)}e^+��_e$, we calculate the branching fraction ratios to be ${\mathcal R}^0\equiv {\mathcal B}_{D^{0}\to ��^{-}��^+��_��}/{\mathcal B}_{D^{0}\to ��^{-}e^+��_e}=0.922\pm 0.030_{\rm stat.}\pm0.022_{\rm syst.}$ and ${\mathcal R}^+\equiv {\mathcal B}_{D^{+}\to ��^{0}��^+��_��}/{\mathcal B}_{D^{+}\to ��^{0}e^+��_e}=0.964\pm 0.037_{\rm stat.}\pm0.026_{\rm syst.}$, which are compatible with the theoretical expectation of lepton flavor universality within $1.7��$ and $0.5��$, respectively. We also examine the branching fraction ratios in different four-momentum transfer square regions, and find no significant deviations from the standard model predictions.