Data determination of HQET parameters in inclusive charm decays

This work delves into the phenomenology of electronic inclusive decays of $D$ mesons, encompassing $D^0,D^{+},D_s^{+}\to X{e}^{+}\nu$. The theoretical formulas for the decay widths and electron energy moments of these decays are presented as expansions with powers of ${\alpha }_s$ and ${\mathrm{\Lambda }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}\mathrm{/}{m}_c$. 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} μ^2_π(D^{0,+}) &= (0.09\pm 0.05) $\mathrm{G}\mathrm{e}\mathrm{V}$^2, \qquad \qquad μ^2_π(D^{+}_s) = (0.11\pm 0.05) $\mathrm{G}\mathrm{e}\mathrm{V}$^2, \nonumber \ μ^2_G(D^{0,+}) &= (0.32\pm 0.02) $\mathrm{G}\mathrm{e}\mathrm{V}$^2, \qquad \qquad μ^2_G(D^{+}s) = (0.43\pm 0.02) $\mathrm{G}\mathrm{e}\mathrm{V}$^2, \nonumber \ ρ_D^3(D^{0,+}) &= (-0.003\pm 0.002) $\mathrm{G}\mathrm{e}\mathrm{V}$^3, \qquad\ ρ_D^3(D^{+}s) = (-0.004\pm 0.002) $\mathrm{G}\mathrm{e}\mathrm{V}$^3, \nonumber \ ρ{LS}^3(D^{0,+}) &= (0.004\pm 0.002) $\mathrm{G}\mathrm{e}\mathrm{V}$^3, \qquad \ \ \ ρ{LS}^3(D^{+}_s) = (0.005\pm 0.002) $\mathrm{G}\mathrm{e}\mathrm{V}$^3 . \nonumber \end{align} These determined parameters will play a crucial role as inputs in various physical quantities, including $D$ meson lifetimes.