The rare decay $B^+ \to K^+\ell^+\ell^-(\nu\bar{\nu})$ under the QCD sum rules approach

In the paper, we conduct a detailed investigation of the rare decay processes of charged meson, specifically $B^+ \to K^+\ell^+\ell^-$ with $\ell=(e,\mu,\tau)$ and $B^+ \to K^+\nu\bar{\nu}$. These processes involve flavor-changing-neutral-current (FCNC) transitions, namely $b\to s\ell^+\ell^-$ and $b\to s\nu\bar{\nu}$. The essential components $B\to K$ scalar, vector and tensor transition form factors (TFFs) are calculated by using the QCD light-cone sum rules approach up to next-to-leading order QCD corrections. In which, the kaon twist-2 and twist-3 light-cone distribution amplitudes are calculated from both the QCD sum rules within the framework of background field theory and the light-cone harmonic oscillator model. The TFFs at large recoil point are $f_+^{BK}(0)=f_0^{BK}(0) =0.328_{-0.028}^{+0.032}$ and $f_{\rm T}^{BK}(0)=0.277_{-0.024}^{+0.028}$, respectively. To achieve the behavior of those TFFs in the whole $q^2$-region, we extrapolate them by utilizing the simplified $z(q^2)$-series expansion. Furthermore, we compute the differential branching fractions with respect to the squared dilepton invariant mass for the two different decay channels and present the corresponding curves. Our predictions of total branching fraction are ${\cal B}(B^+\to K^+ e^+ e^-)=6.633_{-1.070}^{+1.341}\times 10^{-7}$, ${\cal B}(B^+\to K^+ \mu^+ \mu^-)=6.620_{-1.056}^{+1.323}\times 10^{-7}$, ${\cal B}(B^+\to K^+ \tau^+ \tau^-)=1.760_{-0.197}^{+0.241}\times 10^{-7}$, and ${\cal B}(B^+\to K^+ \nu\bar{\nu})=4.135_{-0.655}^{+0.820}\times 10^{-6}$, respectively. Lastly, the observables such as the lepton universality $\mathcal{R}_{K}$ and the angular distribution `flat term' $F_{\rm H}^\ell$ are given, which show good agreement with the theoretical and experimental predictions.