arXiv subjects

Zhen-Jun Xiao

Publications and source records attributed to Zhen-Jun Xiao.

At least 37 records · Page 2Linked to original sources

arXiv2022

Next-to-next-to-leading order matching of beauty-charmed meson $B_{c}$ and $B^*_{c}$ decay constants

We present the next-to-next-to-leading order (NNLO) QCD corrections to the decay constants for both the pseudoscalar and vector beauty-charmed mesons $B_{c}$ and $B^*_{c}$ in nonrelativistic QCD effective theory. Explicit NNLO calculation verified that the $B_c$ decay constant from pseudoscalar current is identical with the $B_c$ decay constant from axial-vector current. The NNLO result for the vector decay constant of $B^*_{c}$ meson is novel. Combined with the latest extraction of nonrelativistic QCD long-distance matrix elements of $B_c$ meson, we give the branching ratios of leptonic decays of $B_{c}$ and $B^*_{c}$ mesons. In addition, the novel anomalous dimension for the flavor-changing heavy quark vector current in nonrelativistic QCD effective theory are helpful to investigate the threshold behaviours of two different heavy quarks.

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arXiv2022

$Λ_b\to p$ transition form factors in perturbative QCD

We reanalyze the $Λ_b\to p$ transition form factors in the perturbative QCD (PQCD) approach by including higher-twist light-cone distribution amplitudes (LCDAs) of a $Λ_b$ baryon and a proton. The previous PQCD evaluation performed decades ago with only the leading-twist $Λ_b$ baryon and proton LCDAs gave the form factors, which are two orders of magnitude smaller than indicated by experimental data. We find that the twist-4 $Λ_b$ baryon LCDAs and the twist-4 and -5 proton LCDAs contribute dominantly, and the enhanced form factors become consistent with those from lattice QCD and other nonperturbative methods. The estimated branching ratios of the semileptonic decays $Λ_b\to p\ell\barν_\ell$ and the hadronic decay $Λ_b\to pπ$ are also close to the data. It implies that the $b$ quark mass is not really heavy enough, and higher-power contributions play a crucial role, similar to the observation made in analyses of $B$ meson transition form factors. With the formalism established in this work, we are ready to study various exclusive heavy baryon decays systematically in the PQCD approach.

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arXiv2022

Next-to-leading order QCD calculation of $B_c$ to charmonium tensor form factors

We present a next-to-leading order (NLO) QCD corrections to $B_c\to η_c$ and $B_c\to J/ψ$ tensor form factors within nonrelativistic QCD (NRQCD) framework. The full analytical results for $B_c$ to S-wave charmonium tensor form factors are obtained. We also studied the asymptotic behaviours of tensor form factors in hierarchy heavy quark limit, i.e. $m_b\to\infty,~ m_c\to\infty, ~\mathrm{and }~m_c/m_b\to0$. A compact expression for tensor form factors are given analytically in the hierarchy heavy quark limit. The relation among different form factors is also analyzed especially at large momentum recoil point. The numerical results for the $B_c$ to charmonium tensor form factors in all the physical region are given in the end.

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arXiv2022

Study of $B_{(s)}^0 \to ϕϕ\to (K^+K^-)(K^+K^-)$ decays in the perturbative QCD approach

In this work, we make a detailed analysis on the penguin-dominant processes $B_{(s)}^0 \to ϕϕ\to (K^+K^-)(K^+K^-)$ in the perturbative QCD (PQCD) approach. In addition to the dominant $P$-wave resonance, the scalar background $f_0(980) \to K^+K^-$ is also accounted for. We improve the Gegenbauer moments in $KK$ two-meson distribution amplitudes by fitting the PQCD factorization formulas to measured branching ratios of three-body and four-body $B$ decays. We extract the branching ratios of two-body $B_{(s)}^0 \to ϕϕ$ decays from the corresponding four-body decay modes and calculate the relevant polarization fractions together with two relative phases $ϕ_{\parallel,\perp}$, which are consistent with the previous theoretical predictions. The PQCD predictions for the "true" triple product asymmetries (TPAs) are zero which are expected in the standard model due to the vanishing weak phase difference, and support the current data reported by the CDF and LHCb Collaborations. A large "fake" TPA $\mathcal{A}_\text{T-fake}^1=30.4\%$ of the decay $B^0_s \to ϕϕ\to (K^+K^-)(K^+K^-)$ is predicted for the first time, which indicates the presence of the significant final-state interactions. The TPAs of the rare decay channel $B^0 \to ϕϕ\to (K^+K^-)(K^+K^-)$ are also predicted and can be tested in the near future.

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arXiv2021

Study of $B_{(s)} \to (ππ)(Kπ)$ decays in the perturbative QCD approach

In this work, we provide estimates of the branching ratios, direct $CP$ asymmetries and triple product asymmetries in $B_{(s)} \to (ππ)(Kπ)$ decays in the perturbative QCD approach, where the $ππ$ and $Kπ$ invariant mass spectra are dominated by the vector resonances $ρ(770)$ and $K^*(892)$, respectively. Some scalar backgrounds, such as $f_0(500,980) \to ππ$ and $K^*_0(1430) \to Kπ$ are also accounted for. The $ρ(700)$ is parametrized by the Gounaris-Sakurai function. The relativistic Breit-Wigner formula for the $f_0(500)$ and Flatté model for the $f_0(980)$ are adopted to parameterize the time-like scalar form factors $F_S(ω^2)$. We also use the D.V. Bugg model to parameterize the $f_0(500)$ and compare the relevant theoretical predictions from different models. While in the region of $Kπ$ invariant mass, the $K^*_0(1430)$ is described with the LASS lineshape and the $K^*(892)$ is modeled by the Breit-Wigner function. We find that the decay rates for the considered decay modes agree with currently available data within errors. As a by-product, we extract the branching ratios of two-body decays $B_{(s)} \to ρ(770)K^*(892)$ from the corresponding four-body decay modes and calculate the relevant polarization fractions. Our prediction of longitudinal polarization fraction for $B^0\to ρ(770)^0 K^*(892)^0$ decay deviates a lot from the recent LHCb measurement, which should be resolved. It is shown that the direct $CP$ asymmetries are large due to the sizable interference between the tree and penguin contributions, but they are small for the tree-dominant or penguin-dominant processes. The PQCD predictions for the triple product asymmetries are small which are expected in the standard model, and consistent with the current data reported by the LHCb Collaboration.Our results can be tested by the future precise data from the LHCb and Belle II experiments.

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arXiv2021

Study of $B_s\to ϕ\ell^+ \ell^-$ decays in the PQCD factorization approach with lattice QCD input

In this paper, we studied systematically the semileptonic decays $B_s \to ϕl^+ l^-$ with $l^-=(e^-,μ^-,τ^-)$ by using the perturbative QCD (PQCD) and the "PQCD+Lattice" factorization approach, respectively. We first evaluated all relevant form factors $F_i(q^2)$ in the low $q^2$ region using the PQCD approach, and we also took the available lattice QCD results at the high-$q^2$ region as additional input to improve the extrapolation of $F_i(q^2)$ from the low-$q^2$ region to the endpoint $q^2_{max}$. We then calculated the branching ratios and many other physical observables: $A_{FB}^{l}$, $F_L^ϕ$, $S_{3,4,7}$, $A_{5,6,8,9}$ and the clean angular observables $P_{1,2,3} $ and $P^{\prime}_{4,5,6,8}$.From our studies, we find the following points: (a) the PQCD and "PQCD+Lattice" predictions of ${\cal B}( B_s \to ϕμ^+ μ^-)$ are about $7\times 10^{-7}$, which agree well with the LHCb measured values and the QCD sum rule prediction within still large errors; (b) we defined and calculated the ratios of the branching ratios $R_ϕ^{eμ}$ and $R_ϕ^{μτ}$; (c) the PQCD and "PQCD+Lattice" predictions of the longitudinal polarization $F_L$, the CP averaged angular coefficients $S_{3,4,7} $ and the CP asymmetry angular coefficients $A_{5,6,8,9}$, agree with the LHCb measurements in all considered bins within the still large experimental errors; and (d) for those currently still unknown observables $R_ϕ^{eμ}, R_ϕ^{μτ}, A_{FB}^{l}, P_{1,2,3}$ and $P^{\prime}_{4,5,6,8}$, we suggest LHCb and Belle-II Collaboration to measure them in their experiments.

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arXiv2021

Weak decays of bottom-charm baryons: $\mathcal{B}_{bc}\to\mathcal{B}_bP$

After the discovery of the double-charm baryon $Ξ_{cc}^{++}$ by LHCb, one of the most important topics is to search for the bottom-charm baryons which contain a $b$ quark, a $c$ quark and a light quark. In this work, we study the two-body non-leptonic weak decays of a bottom-charm baryon into a spin-$1/2$ bottomed baryon and a light pseudoscalar meson with the short-distance contributions calculated under the factorization hypothesis and the long-distance contributions considering the final-state-interaction effects. The branching fractions of all fifty-seven decay channels are estimated. The results indicate that $Ξ_{bc}^+\toΞ_b^0π^+$, $Ξ_{bc}^{0}\toΞ_{b}^{-}π^+$ and $Ω_{bc}^0\toΩ_b^-π^+$ decay modes have relatively large decay rates and thus could be used to experimentally search for the bottom-charm baryons. The topological diagrams and the SU(3) symmetry of bottom-charm baryon decays are discussed.

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arXiv2021

$B_{d,s}^0 \to f_1 f_1$ decays with $f_1(1285)-f_1(1420)$ mixing in the perturbative QCD approach

We investigate the $B_{d,s}^0 \to f_1 f_1$ decays in the framework of perturbative QCD(PQCD) approach with a referenced value $ϕ_{f_1} \sim 24^\circ$. Here, $f_1$ denotes the axial-vector mesons $f_1(1285)$ and $f_1(1420)$ with mixing angle $ϕ_{f_1}$ in the quark-flavor basis. The observables such as branching ratios, direct CP violations, and polarization fractions of the $B_s^0 \to f_1 f_1$ decays are predicted for the first time. We find that: (i) the almost pure penguin modes $B_s^0 \to f_1 f_1$ have large branching ratios in the order of $10^{-6} \sim 10^{-5}$ due to the Cabibbo-Kobayashi-Maskawa enhancement and generally constructive interferences between the amplitudes of $B_s^0 \to f_n f_s$ and $B_s^0 \to f_s f_s$ with $f_n$ and $f_s$ being the quark-flavor states of $f_1$ mesons; (ii) The observables receive important contributions from the weak annihilation diagrams in the PQCD approach. In particular, without the annihilation contributions, the $B_s^0 \to f_1(1420) f_1(1420)$ branching ratio will decrease about 81% and its longitudinal polarization fraction will reduce around 43%. And (iii) the dependence of the $B_{d,s}^0 \to f_1 f_1$ decay rates on $ϕ_{f_1}$ exhibits some interesting line shapes, whose confirmations would be helpful to constrain the determination of $ϕ_{f_1}$ inversely. All the PQCD predictions await for the (near) future examinations at Large Hadron Collider beauty and/or Belle-II experiments to further understand the properties of the axial-vector mesons and the perturbative dynamics released from the considered decay modes.

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arXiv2021

Rescattering mechanism of weak decays of double-charm baryons

The doubly charmed baryon $Ξ_{cc}^{++}$ was recently observed by LHCb via the decay processes of $Ξ_{cc}^{++}\to Λ_c^+ K^-π^+π^+$ and $Ξ_c^+π^+$. These discovery channels were successfully predicted in the framework that the short-distance contributions are calculated under the factorization hypothesis and the long-distance contributions are estimated using the rescattering mechanism for the final-state-interaction effects. In this paper, we illustrate the above framework in details by systematic studies on the two-body baryonic decays $\mathcal{B}_{cc}\to\mathcal{B}_{c}P$ involving the doubly charmed baryons $\mathcal{B}_{cc}=(Ξ_{cc}^{++} , Ξ_{cc}^+,Ω_{cc}^+)$, the singly charmed baryons $\mathcal{B}_{c}=(\mathcal{B}_{\bar{3}}, \mathcal{B}_{6})$ and the light pseudoscalar mesons $P=(π,K,η_{1,8})$.

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arXiv2020

Next-to-leading-logarithm $k_T$ resummation for $B_c\to J/ψ$ decays

We derive the $k_T$ resummation for a transverse-momentum-dependent charmonium wave function, which involves the bottom quark mass $m_b$, the charm quark mass $m_c$, and the charm quark transverse momentum $k_T$, up to the next-to-leading-logarithm (NLL) accuracy under the hierarchy $m_b\gg m_c \gg k_T$. The resultant Sudakov factor is employed in the perutrbative QCD (PQCD) approach to the $B_c\to J/ψ$ transition form factor $A_0^{B_c \to J/ψ}(0)$ and the $B_c^+ \to J/ψπ^+$ decay. We compare the NLL resummation effect on these processes with the leading-logarithm one in the literature, and find a $(5-10)\%$ enhancement to the form factor $A_0^{B_c \to J/ψ}(0)$ and a $(10-20)\%$ enhancement to the decay rate ${\rm BR}(B_c^+ \to J/ψπ^+)$. The improved $k_T$ resummation formalism is applicable to the PQCD analysis of heavy meson decays to other charmonia.

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arXiv2020

Resonant contributions to three-body $B_{(s)} \to [ D^{()}, \bar{D}^{()} ] K^+K^-$ decays in the perturbative QCD approach

In this work, we study the $S$, $P$ and $D$ wave resonance contributions to three-body decays $B_{(s)} \to [ D^{(*)}, \bar{D}^{(*)} ] K^+K^-$ by employing the perturbative QCD (PQCD) approach, where the kaon-kaon invariant mass spectra are dominated by the $f_0(980),f_0(1370),$ $ϕ(1020),ϕ(1680),f_2(1270),f^{\prime}_2(1525),f_2(1750)$ and $f_2(1950)$ resonances. The $KK$ $S$-wave component $f_0(980)$ is modeled with the Flatté formalism, while other resonances are described by the relativistic Breit-Wigner (BW) line shape. The corresponding decay channels are studied by constructing the kaon-kaon distribution amplitude $Φ_{KK}$, which captures important final state interactions between the kaon pair in the resonant region. We found that the PQCD predictions for the branching ratios for most considered decays agree with currently available data within errors. The associated polarization fractions of those vector-vector and vector-tensor decay modes are also predicted, which are expected to be tested in the near future experiments. The invariant mass spectra for the corresponding resonances in the $B_{(s)} \to [D^{(*)}, \bar{D}^{(*)} ] K^+K^-$ decays are well established, which can be confronted with the precise data from the LHCb and Belle II experiments.

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arXiv2020

Study of $\bar{B}_s\to K^{(*)}\ell^+ \ell^-$ decays in the PQCD factorization approach with the lattice QCD input

In this paper, we studied systematically the semileptonic decays $\bar{B}_s \to K^{(*)} \ell^+ \ell^-$ with $l^-=(e^-,μ^-,τ^-)$ by using the perturbative QCD (PQCD) and the "PQCD+Lattice" factorization approach, respectively. We first evaluated all relevant form factors $F_i(q^2)$ in the low $q^2$ region using the PQCD approach, and we also take the available Lattice QCD results at the end point $q^2_{max}$ as additional inputs to improve the extrapolation of the form factors to the high $q^2$ region. We calculated the branching ratios and other twelve kinds of physical observables. From our studies, we find the following points: (a) for $ \bar{B}_s \to K l^+ l^-$ decays, the PQCD and "PQCD+Lattice" predictions for branching ratios (BRs) ${\cal B}(\bar{B}_s \to K l^+ l^-)$, the ratios of the BRs $R_K^{eμ}$ and $R_K^{μτ}$, and the longitudinal polarization asymmetry of the leptons $P_L$ agree well within errors; (b) the PQCD and "PQCD+Lattice" predictions for the CP averaged branching ratio ${\cal B}(\bar{B}_s \to K^* μ^+ μ^-)$ are $(3.17^{+0.95}_{-0.78})\times 10^{-8}$ and $(2.48^{+0.56}_{-0.50})\times 10^{-8}$ respectively, which agree well with the LHCb measured value $(2.9\pm 1.1) \times 10^{-8}$ and the light-cone sum rule (LCSR) prediction; (c) for the ratios $R_{K^\ast}^{eμ}$ and $R_{K^\ast}^{μτ}$, the PQCD and "PQCD+Lattice" predictions agree well with each other and have a small error less than $10\%$; (d) for the direct CP asymmetries ${\cal A}_{CP}$ of all considered decay modes, they are always very small as expected: less than $5\%$ in magnitude; and (e) for the angular observables $P_{1,2,3}$ and $P^\prime_{4,5,6,8}$ , our theoretical predictions for each kind of lepton are consistent within errors. We believe that above predictions could be tested by future LHCb and Belle-II experiments.

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arXiv2020

$P$-wave contributions to $B_{(s)}\toψKπ$ decays in perturbative QCD approach

In this work, we studied the quasi-two-body decays $B_{(s)} \to ψ[K^*(892), K^*(1410),$ $K^*(1680)] \to ψKπ$ by employing the perturbative QCD (PQCD) factorization approach, where the charmonia $ψ$ represents $J/ψ$ and $ψ(2S)$. The corresponding decay channels are studied by constructing the kaon-pion distribution amplitude (DA) $Φ_{K π}^{\rm P}$, which contained the important final state interactions between the kaon and pion in the resonant region. The relativistic Breit-Wigner formulas are adopted to parameterize the time-like form factor $F_{Kπ}$ appeared in the kaon-pion DAs. The SU(3) flavor symmetry breaking effect resulting from the mass difference between kaon and pion is taken into account, which makes significant contributions to the longitudinal polarizations. We accommodate well the observed branching ratios and the polarization fractions of the $B_{(s)} \to ψK^*(892) \to ψKπ$ by tuning the hadronic parameters for the kaon-pion DAs. The PQCD predictions for the $B_{(s)} \to ψ[K^*(1410), K^*(1680)] \to ψKπ$ modes from the same set of parameters can be tested by the future precise data from the LHCb and the Belle II experiments.

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arXiv2020

$S$, $P$ and $D$-wave resonance contributions to $B_{(s)} \to η_c(1S,2S) Kπ$ decays in the perturbative QCD approach

In this work, we analyze the three-body $B_{(s)} \to η_c(1S,2S) K π$ decays within the framework of the perturbative QCD approach (PQCD) under the quasi-two-body approximation, where the kaon-pion invariant mass spectra are dominated by the $K_0^*(1430)^0,K_0^*(1950)^0,K^*(892)^0,K^*(1410)^0,K^*(1680)^0$ and $K_2^*(1430)^0$ resonances. The time-like form factors are adopted to parametrize the corresponding $S$, $P$, $D$-wave kaon-pion distribution amplitudes for the concerned decay modes, which describe the final-state interactions between the kaon and pion in the resonant region. The $Kπ$ $S$-wave component at low $Kπ$ mass is described by the LASS line shape, while the time-like form factors of other resonances are modeled by the relativistic Breit-Wigner function. We find the following main points: (a) the PQCD predictions of the branching ratios for most considered $B \to η_c(1S)(K^{*0}\to )K^+π^-$ decays agree well with the currently available data within errors; (b) for ${\cal B}(B^0 \to η_c (K_0^*(1430)\to )K^+π^-)$ and ${\cal B}(B^0 \to η_c K^+π^-({\rm NR}))$ (here NR means nonresonant), our predictions of the branching ratios are a bit smaller than the measured ones; and (c) the PQCD results for the $D$-wave contributions considered in this work can be tested once the precise data from the future LHCb and Belle-II experiments are available.

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arXiv2020

$B_{(s)} \to η_c(P,V)$ decays and effects of the next-to-leading order contributions in the perturbative QCD approach

By employing the perturbative QCD (PQCD) factorization approach, we studied the sixteen $B/B_s \to η_c (π, K, η^{(\prime)},ρ,K^*,ω,ϕ)$ decays with the inclusion of the currently known next-to-leading order (NLO) contributions. We found the following main points: (a) for the five measured $B \to η_c (K,K^*)$ and $B_s\to η_cϕ$ decays, the NLO contributions can provide $ (80-180)\%$ enhancements to the leading order (LO) PQCD predictions of their branching ratios, which play an important role to help us to interpret the data; (b) for the seven ratios $R_{1,\cdots,7}$ of the branching ratios defined among the properly selected pair of the considered decay modes, the PQCD predictions for the values of $R_{3,4,5}$ agree well with those currently available measurements from BaBar and Belle Collaboration; (c) for $B^0 \to η_c K_S^0$ decay, the PQCD predictions for both the direct and mixing induced CP asymmetries do agree very well with the measured values within errors; and (d) the PQCD predictions for ratios $R_{1,2}$ and $R_{6,7}$ also agree with the general expectations and will be tested by the future experiments.

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arXiv2020

Semileptonic decays $B/B_s \to (D^{()},D_s^{()}) l ν_l$ in the PQCD approach with the lattice QCD input

In this paper, we studied the semileptonic $B/B_s \to (D^{(*)},D_s^{(*)}) lν_l$ decays in the framework of the standard model (SM) , by employing the perturbative QCD (PQCD) factorization formalism combining with the lattice QCD inputs of the relevant transition form factors. We calculated the branching ratios ${\cal B}(B_{(s)} \to D_{(s)}^{(*)} l ν_l )$ with $l=(e,μ.τ)$ , the ratios of the branching ratios $R(D^{(*)})$ and $R(D_s^{(*)} )$ and other physical observables $P_τ(D_{(s)}^{(*)})$, $F_L(D^*_{(s)})$ and $A_{FB}(τ)$. The "PQCD+Lattice" predictions for ${\cal B}(B \to D^{(*)} lν_l)$ and $R(D^{(*)})$ do agree well with those currently available experimental measurements within the errors. For the ratios $R(D_s)$ and $R(D_s^*)$, the "PQCD+Lattice" predictions agree well with other known predictions. For both $P_τ(D^*)$ and $F_L(D^*)$, our theoretical predictions agree well with the measured values within errors. Our theoretical predictions about the considered semileptonic $B/B_s$ decays could be tested in the near future LHCb and the Belle II experiments.

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arXiv2019

Semileptonic decays $B_c \to (η_c,J/ψ) l \barν_l $ in the "PQCD + Lattice" approach

In this paper, we studied the semileptonic decays $B_c^- \to (η_c, J/ψ) l ^- \barν_l$ by employing the PQCD factorization approach, using the newly defined distribution amplitudes of the $B_c$ meson and the new kind of parametrization for extrapolation of the form factors , and also taking into account the lattice QCD results about the relevant form factors at several points. We found the following main results: (a) the PQCD predictions for the branching ratios of $B_c \to (η_c,J/ψ) l \barν$ decays will become a little smaller by about $(5-16)\%$ when the lattice input are taken into account in the extrapolation of the relevant form factors; (b) the PQCD predictions for the ratio $R_{η_c, J/ψ}$ and the longitudinal polarization $P_τ$ are $R_{η_c}=0.34\pm 0.01 , R_{J/ψ}=0.28\pm 0.01$, $P_τ(η_c) = 0.37\pm 0.01$ and $P_τ(J/ψ) = -0.55 \pm 0.01$ ; and (c) after the inclusion of the lattice input the theoretical predictions changed slightly: $R_{η_c}=0.31\pm 0.01$, $ R_{ J/ψ}=0.27\pm 0.01$, $P_τ( η_c) = 0.36 \pm 0.01$ and $P_τ( J/ψ) = -0.53\pm 0.01$. The theoretical predictions for $R_{ J/ψ}$ agree with the measured one within errors, and other predictions could be tested in the near future LHCb experiments.

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arXiv2019

Anatomy of $B_s \to PP $ decays and effects of the next-to-leading order contributions in the perturbative QCD approach

By employing the perturbative QCD (PQCD) factorization approach, we made a systematic investigation for the CP-averaged branching ratios and the CP-violating asymmetries of the thirteen $\bar{B}^0_s \to PP $ decays ( here $P=(π,K, η, η^{\prime})$) with the inclusion of all currently known next-to-leading order (NLO) contributions, and compared our results with the measured values or the theoretical predictions from other different approaches. We focused on the examination of the effects of the NLO contributions and found the following points: (a) for $\bar{B}_s^0 \to (K^0 \bar{K}^0, K^+ K^-, η^\prime η^\prime,π^- K^+)$ decays, the NLO contributions can provide a $(40-150) \%$ enhancement or a $20\%$ reduction to the corresponding leading order (LO) PQCD predictions for their decay rates and result in a much better agreement between the PQCD predictions and the experimental measurements; (b) for the pure annihilation decay $\bar{B}_s^0 \to π^+π^-$, the PQCD prediction still remain consistent with the data after the inclusion of the small NLO reduction; (c) the PQCD predictions for the ratio $(f_s/f_d)\cdot {\cal B}(B_s^0\to π^+π^-)/{\cal B}(B^0\to K^+π^-)$ and $(f_s/f_d)\cdot {\cal B}(B_s^0\to π^+ K^-)/{\cal B}(B^0\to K^+π^-)$ become agree very well with the measured ones after the inclusion of a $40\%$ NLO reduction; (d) for $\bar{B}_s^0 \to K^+ K^-$ and $\bar{B}_s^0 \to π^- K^+$ decays, the NLO PQCD predictions for their CP-violating asymmetries do agree very well with the measured values in both the sign and the magnitude; and (e) for all $\bar{B}_s^0 \to P P$ decays, we also compared our results with those obtained in the QCD factorization approach and soft-collinear effective theory and discussed their similarities and differences.

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