Prospective analysis of CKM element $\mathrm{\mid }{V}_{cd}\mathrm{\mid }$ and $D^{+}$-meson decay constant from leptonic decays $D^{+}\to {\mathrm{\ell }}^{+}\nu$

The leptonic decay of $D^{+}$-meson has attracted significant interest due to its unique characteristics. In this paper, we carry out an investigation into the $D^{+}$-meson leptonic decays $D^{+}\to {\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ with $\mathrm{\ell }=(e,\mu ,\tau )$ by employing the QCD sum rules approach. In which the $D^{+}$-meson decay constant $f_{D^{+}}$ is an important input parameter in the process. To enhance the accuracy of our calculations for $f_{D^{+}}$, we consider the quark propagator and vertex up to dimension-six within the framework of background field theory. Consequently, we obtain the QCD sum rule expression for $f_{D^{+}}$ up to dimension-six condensates, yielding $f_{D^{+}}=203.0\pm 1.5\mathrm{M}\mathrm{e}\mathrm{V}$. Our results are in good agreement with BESIII measurements and theoretical predictions. We also present the integrated decay widths for the $D^{+}$-meson in three channels $\mathrm{\Gamma }(D^{+}\to e^{+}{\nu }_e)=(5.263_{-0.075}^{+0.076})\times {10}^{-21}\mathrm{G}\mathrm{e}\mathrm{V}$, $\mathrm{\Gamma }(D^{+}\to {\mu }^{+}{\nu }_{\mu })=(2.236_{-0.032}^{+0.032})\times {10}^{-16}\mathrm{G}\mathrm{e}\mathrm{V}$ and $\mathrm{\Gamma }(D^{+}\to {\tau }^{+}{\nu }_{\tau })=(5.958_{-0.085}^{+0.086})\times {10}^{-16}\mathrm{G}\mathrm{e}\mathrm{V}$. Accordingly, we compute the branching fraction $\mathcal{B}(D^{+}\to {\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }})$ with the electron, muon and tau channels, which are $\mathcal{B}(D^{+}\to e^{+}{\nu }_e)=(8.260_{-0.118}^{+0.119})\times {10}^{-9}$, $\mathcal{B}(D^{+}\to {\mu }^{+}{\nu }_{\mu })=(3.508_{-0.050}^{+0.051})\times {10}^{-4}$ and $\mathcal{B}(D^{+}\to {\tau }^{+}{\nu }_{\tau })=(0.935_{-0.013}^{+0.013})\times {10}^{-3}$. Furthermore, we present our prediction for the CKM matrix element $\mathrm{\mid }{V}_{cd}\mathrm{\mid }$ using the branching fraction $\mathcal{B}(D^{+}\to {\mu }^{+}{\nu }_{\mu })$ obtained from BESIII Collaboration, yielding $\mathrm{\mid }{V}_{cd}\mathrm{\mid }=0.227_{-0.001}^{+0.002}$.