By analyzing an $e^+e^-$ annihilation data sample corresponding to an integrated luminosity of $2.93~mathrm{fb}^{-1}$ collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, we measure the branching fraction of the $D^0to rho^- mu^+ u_mu$ decay for the first time. We obtain ${mathcal B}_{D^0to rho^- mu^+ u_mu}=(1.35pm0.09_{rm stat}pm0.09_{rm syst})times 10^{-3}$. Combining with theoretical predictions, we extract the CKM matrix element $|V_{cd}|=0.204pm0.007_{rm stat}pm0.007_{rm syst}pm0.014_{rm theory}$. Using the world average of ${mathcal B}_{D^0to rho^- e^+ u_e}$, we find a branching fraction ratio of ${mathcal B}_{D^0to rho^- mu^+ u_mu}/{mathcal B}_{D^0to rho^- e^+ u_e}=0.90pm0.11$, which agrees with the theoretical expectation of lepton flavor universality within the uncertainty. Combining the world average of ${mathcal B}_{D^+to rho^0 mu^+ u_mu}$ and the lifetimes of $D^{0(+)}$, we obtain a partial decay width ratio of ${Gamma}_{D^0to rho^- mu^+ u_{mu}}/(2{Gamma}_{D^+to rho^0 mu^+ u_{mu}}) = 0.71pm0.14$, which is consistent with the isospin symmetry expectation of one within $2.1sigma$. For the reported values of ${mathcal B}_{D^0to rho^- mu^+ u_mu}/{mathcal B}_{D^0to rho^- e^+ u_e}$ and ${Gamma}_{D^0to rho^- mu^+ u_{mu}}/2{Gamma}_{D^+to rho^0 mu^+ u_{mu}}$, the uncertainty is the quadratic sum of the statistical and systematic uncertainties.
تحميل البحث