Prospective analysis of CKM element |Vcd| and D+-meson decay constant from leptonic decays D+ → ℓ+ν

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 $D^{+}\to {\ell }^{+}{\nu }_{\ell }$ with $\ell =(e,\mu ,\tau )$ 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 $f_{D^{+}}=203.0\pm 1.5\mathrm{MeV}$. Our result is in good agreement with BESIII measurements and theoretical predictions. We also present the integrated decay widths for the $D^{+}$-meson in three channels $\Gamma (D^{+}\to e^{+}{\nu }_e)=\left({5.263}_{-0.075}^{+0.076}\right)\times {10}^{-21}\mathrm{GeV}$, $\Gamma (D^{+}\to {\mu }^{+}{\nu }_{\mu })=\left({2.236}_{-0.032}^{+0.032}\right)\times {10}^{-16}\mathrm{GeV}$ and $\Gamma (D^{+}\to {\tau }^{+}{\nu }_{\tau })=\left({5.958}_{-0.085}^{+0.086}\right)\times {10}^{-16}\mathrm{GeV}$. Accordingly, we compute the branching fractions $B(D^{+}\to {\ell }^{+}{\nu }_{\ell })$ with the electron, muon and tau channels, which are $B(D^{+}\to e^{+}{\nu }_e)=\left({8.260}_{-0.118}^{+0.119}\right)\times {10}^{-9}$, $B(D^{+}\to {\mu }^{+}{\nu }_{\mu })=\left({3.508}_{-0.050}^{+0.051}\right)\times {10}^{-4}$ and $B(D^{+}\to {\tau }^{+}{\nu }_{\tau })=\left({0.935}_{-0.013}^{+0.013}\right)\times {10}^{-3}$. Furthermore, we present our prediction for the CKM matrix element $\left|V_{cd}\right|$ using the branching fraction $B(D^{+}\to {\mu }^{+}{\nu }_{\mu })$ obtained from BESIII Collaboration, yielding $\left|V_{cd}\right|={0.227}_{-0.001}^{+0.002}$.