Data determination of HQET parameters in inclusive charm decays

This work delves into the phenomenology of electronic inclusive decays of $D$ mesons, encompassing $D^0, D^+, D^+_s\to Xe^{+}\nu$. The theoretical formulas for the decay widths and electron energy moments of these decays are presented as expansions with powers of $\alpha_s$ and $\Lambda_{\rm QCD}/m_c$. Remarkably, the expansion exhibits excellent convergence properties when we choose the 1S mass scheme for charm. The formulas are subsequently fitted to experimental data, and the $D$ meson matrix elements of operators in the heavy quark effective theory are hence determined by data for the first time, including \begin{align} \mu^2_\pi(D^{0,+}) &= (0.09\pm 0.05) \mathrm{GeV}^2, \qquad \qquad \mu^2_\pi(D^{+}_s) = (0.11\pm 0.05) \mathrm{GeV}^2, \nonumber \\ \mu^2_G(D^{0,+}) &= (0.32\pm 0.02) \mathrm{GeV}^2, \qquad \qquad \mu^2_G(D^{+}_s) = (0.43\pm 0.02) \mathrm{GeV}^2, \nonumber \\ \rho_D^3(D^{0,+}) &= (-0.003\pm 0.002) \mathrm{GeV}^3, \qquad\ \rho_D^3(D^{+}_s) = (-0.004\pm 0.002) \mathrm{GeV}^3, \nonumber \\ \rho_{LS}^3(D^{0,+}) &= (0.004\pm 0.002) \mathrm{GeV}^3, \qquad \ \ \ \rho_{LS}^3(D^{+}_s) = (0.005\pm 0.002) \mathrm{GeV}^3 . \nonumber \end{align} These determined parameters will play a crucial role as inputs in various physical quantities, including $D$ meson lifetimes.