Probing $B^{+}\to K^{+}$ semi-leptonic FCNC decay with new physics effects under PQCD approach

Recently, the Belle II Collaboration reported the branching fraction $\mathcal{B}(B^{+}\to K^{+}\nu \bar{\nu })=(2.3\pm 0.7)\times {10}^{-5}$ with a significance of $3.5\sigma$, which is $2.7\sigma$ above the SM expectation. Motivated by this measurement, we calculate this decay channel at the NLO and twist-3 level using the PQCD approach. By combining the lattice QCD data and our results, we obtain form factors with improved accuracy. Using these form factors, we precisely estimate the branching ratios of $B^{+}\to K^{+}$ semileptonic FCNC decays, including $B^{+}\to K^{+}\nu \bar{\nu }$ and $B^{+}\to K^{+}{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$. To explain the anomalies in these two processes, we introduce a leptoquark model as a new physics scenario. Using five possible types of leptoquarks, we successfully explain the latest experimental measurements, and further constraints on the leptoquarks are derived.