The semileptonic decays $B\mathrm{/}{B}_s\to (\pi ,K)(l^{+}{l}^{-},l\nu ,\nu \bar{\nu })$ in the perturbative QCD approach beyond the leading-order

In this paper we first calculate the form factors of $B\to (\pi ,K)$ and $B_s \to K $ transitions by employing the perturbative QCD (pQCD) factorization approach with the inclusion of the next-to-leading-order(NLO) corrections, and then we calculate the branching ratios of the corresponding semileptonic decays $B\mathrm{/}{B}_s\to (\pi ,K)(l^{+}{l}^{-},l\nu ,\nu \bar{\nu })$ (here $l$ denotes $e,\mu$ and $\tau$). Based on the numerical calculations and phenomenological analysis, we found the following results: (a) For $B\to (\pi ,K)$ and $B_s\to K$ transition form factors $F_{0,+,\mathrm{t}}(q^2)$, the NLO pQCD predictions for the values of $F_{0,+,\mathrm{t}}(0)$ and their $q^2$-dependence agree well with those obtained from other methods; (b) For $\bar{B}^0 \to \pi^+l^-\bar{\nu}l, \bar{K}^0 l^+ l^- $and$B^- \to \pi^0 l^-\bar{\nu}l, K^- l^+ l^-$ decay modes, the NLO pQCD predictions for their branching ratios agree very well with the measured values; © By comparing the pQCD predictions for $Br(\bar{B}^0 \to \pi^+l^-\bar{\nu}l)$ with the measured decay rate we extract out the magnitude of $V_{ub}$: $|V{ub}|= (3.80^{+0.56}{-0.50}(theor.)) \times 10^{-3}$;(d)Wealsodefinedseveralratiosofthebranchingratios,$R\nu, R_C$and$R_{N1,N2,N3}$, and presented the corresponding pQCD predictions, which will be tested by LHCb and the forthcoming Super-B experiments.