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The axial-vector mesons $f_1(1285)$ and $f_1(1420)$ are particularly viewed as the mixtures of flavor states $f_n$ and $f_s$ with mixing angle $\varphi$. In order to determine this angle, we study the $B_{d,s}^0\to J/\psi f_1(1285,1420)$ and $B_{d,s}^0\to \eta_c f_1(1285,1420)$ decays in perturbative QCD (PQCD) approach, including the effects of vertex corrections, nonfactorizable diagrams and penguin operators. Not only the branching fractions, but also the direct $CP$ asymmetries and the polarization fractions are calculated. It is found that the branching fractions of these decays are large enough to be measured in the running LHCb and Belle-II experiments. Moreover, in comparison with the observed ${\cal B}(B_{d,s}^0 \to J/\psi f_1(1285))$, $B_s^0 \to (J/\psi, \eta_c) f_1(1420)$ decays have large branching fractions, which could be measured promisingly through $f_1(1420) \to K_S^0 K^\pm \pi^\mp$ in experiments. We also propose several ratios that could be used to further constrain the absolute value of the mixing angle $\varphi$, but its sign cannot be determined yet in these decays. The direct CP asymmetries of these decays indicate the penguin pollution in the $B_d^0 \to (J/\psi, \eta_c) f_1$ decays cannot be neglected. We acknowledge that there are large theoretical uncertainties arising from the distribution amplitudes of axial-vector mesons and charmonium states, and more precise nonperturbative parameters are called. The comparisons between our results and future experimental data would help us to understand the nature of $f_1$ states and to test the PQCD approach., Comment: 18 pages, 3 figures, journal version, published in EPJC

Three-body $B$ decays not only significantly broaden the study of $B$ meson decay mechanisms, but also provide information of resonant particles. Because of complicate dynamics, it is very hard for us to study the whole phase space in a specific approach. In this review, we take $B\to K({\cal R}\to) K^+K^-$ decays as examples and show the application of the perturbative QCD (PQCD) approach in studying the quasi-two-body $B$ decays, where two particles move collinearly with large energy and the bachelor one recoils back. To describe the dynamics of two collinear particles, the ($S$, $P$ and $D$)-wave functions of kaon-pair with different waves are introduced. By keeping the transverse momenta, all possible diagrams including the hard spectator diagrams and annihilation ones can be calculated in PQCD approach. Most results are well consistent with the current measurements from BaBar, Belle and LHCb experiments. Moreover, under the narrow-width approximation we can extract the branching fractions of the two-body decays involving the resonant states, and also predict the branching fractions of the corresponding quasi-two-body decays $B\to K(\cal{R}\to )\pi^+\pi^-$. All prediction are expected to be tested in the ongoing LHCb and Belle-II experiments., Comment: 12 pages,4 figures, invited mini-review for the special issue "Heavy Flavor Physics and CP Violation" of Advances in High Energy Physics

In this work, we investigate the $B_s\to V S$ decays in the perturbative QCD approach, where $V$ and $S$ denote the vector meson and scalar meson respectively. Based on the two-quark structure, considering two different scenarios for describing the scalar mesons, we calculate the branching fractions and the direct $CP$ asymmetries of all $B_s\to VS$ decays. Most branching fractions are predicted to be at $10^{-7}$ to $10^{-5}$, which could be measured in the LHCb and Belle-II experiments, especially for these color-allowed $B_s\to \kappa(800)(K_0^*(1430))K^*$ decays. It is found that the branching fractions of $B_s\to K_0^{*0}(1430)\bar{K}^{*0}$ and $B_s\to K_0^{*+}(1430)\bar{K}^{*-}$ are very sensitive to the scenarios, which can be used to determine whether $K_0^{*0}(1430)$ belongs to the ground state or the first excited state, if the data were available. We also note that some decays have large direct $CP$ asymmetries, some of which are also sensitive to the scenarios, such as the $B_s \to a_0^+(1450)K^{*-}$ and the $B_s\to f_0(1500) K^{*0}$ decays. Since the experimental measurements of $B_s\to VS$ decays are on the way, combined with the available data in the future, we expect the theoretical predictions will shed light on the structure of the scalar mesons., Comment: 18 Pages, 3 Figures

Using the perturbative QCD approach, we studied the effects from hadronic structure of photon on the pure annihilation rediative decays $B\to\phi\gamma$ and $B_s\to(\rho^0,\omega)\gamma$. These decays have small branching fractions due to the power suppression by the $\Lambda/m_B$, which make them very sensitive to the next-leading power corrections. The quark components and the related two-particle distribution amplitudes of a final state photon are introduced. The branching fractions can be enhanced remarkably by the factorizable and nonfactorizable emission diagrams. The branching fraction of $B\to \phi\gamma$ even increases by about 40 times, and those of $B_s \to \rho^0\gamma$ and $B_s \to \omega\gamma$ are at the order of ${\cal O}(10^{-10})$. We also note that the ratio of branching fractions of $B_s \to \rho^0\gamma$ and $B_s \to \omega\gamma$ is very sensitive to the effects from hadronic structure of photon. All above results could be tested in future., Comment: 9 pages, 1 figure, to be published in Chinese Physics C

Motivated by the measurements of branching fractions of the quasi-two-body decays $B^0\to K^{*0}(f_0(980)\to )\pi^+\pi^-$ and $B^0\to \rho^0(f_0(980)\to )\pi^+\pi^-$, we study the charmless $B\to V (f_0(980)\to ) \pi^+\pi^-$ decays in the perturbative QCD approach. Supposing that $f_0(980)$ is a two-quark state and mixture of $n\bar n=(u\bar u+d\bar d)/\sqrt 2$ and $s\bar s$ with the mixing angle $\theta$, we calculate the branching fractions of these decays with the new introduced $s$-wave $\pi^+\pi^-$-pair wave function. When the mixing angle $\theta$ lies in the range $[135^{\circ}, 155^{\circ}]$, the calculated branching fractions of the $B^0\to K^{*0}f_0(980)\to K^{*0}\pi^+\pi^-$ and $B^0\to \rho^0f_0(980)\to \rho^0\pi^+\pi^-$ decays are in agreement with the experimental data. The branching fractions of other decays could be measured in the current LHCb and Belle II experiments. Considering the isospin symmetry, we also estimate the branching fractions of the quasi-two-body decays $B\to Vf_0(980)\to V\pi^0\pi^0$, which are half of those the corresponding decays $B\to Vf_0(980)\to V\pi^+\pi^-$. Moreover, the direct $CP$ asymmetries of these decays are also calculated, and some of them can be tested in the current experiments., Comment: 10 pages, 2 figures

In 2017, LHCb collaboration reported their first observation of the rare decays $B_s \to \phi (f_0(980)$ $/f_2(1270) \to ) \pi^+\pi^-$ and the evidence of $B^0 \to \phi(f_0(980)/f_2(1270)\to)\pi^+\pi^-$. Motivated by this, we study these quasi-two-body decays in the perturbative QCD approach. The branching fractions, $CP$ asymmetries and the polarization fractions are calculated. We find that within the appropriate two-meson wave functions, the calculated branching fractions are in agreement with the measurements of LHCb. Based on the narrow-width approximation, We also calculate the branching fractions of the quasi-two-body $B_{d,s}\to \phi(f_0(980)/f_2(1270)\to) \pi^0\pi^0$ and $B_{d,s}\to \phi(f_2(1270)\to) K^+K^-$, and hope the predictions to be tested in the ongoing LHCb and Belle II experiments. Moreover, the processes $B_{d,s}\to \phi f_2(1270)$ are also analyzed under this approximation. We note that the $CP$ asymmetries of these decays are very small, because these decays are either penguin dominant or pure penguin processes., Comment: 21 Pages, 1 figure

Recent anomalies in decays induced by $b\to c \ell^- \bar\nu_\ell$ transitions raise the question about such phenomena in the $D$ decays induced by $c\to (s,d)\ell^+\nu_\ell$ transitions. In the experimental side, current measurements on the pure leptonic and semileptonic $D$ decays agree with the standard model predictions, such agreements can be used to constrain the new physics (NP) contributions. In this work, we extend the standard model by assuming general effective Hamiltonians describing the $c\to (s,d)\ell^+\nu_\ell$ transitions including the full set of the four-fermion operators. Within the latest experimental data, we perform a minimum $\chi^2$ fit of the Wilson coefficient corresponding to each operator. The results show that the Wilson coefficients of scalar operators in muon sector are at the order of ${\cal O}(10^{-2})$, and others are at the order of ${\cal O}(10^{-3})$. The lepton flavor universality could be violated by the scalar operators. We also calculate the branching fractions, the forward-backward asymmetries and polarizations of final vector mesons and leptons with the fitted Wilson coefficients of scalar and tensor operators. It is found that the pure leptonic decays are very sensitive to the scalar operators. The effects of NP on the semileptonic decays with electron are negligible, while for the semileptonic decays with muon the effects of scalar operators will show up in the forward-backward asymmetries and polarizations of muon of $D \to P\mu^+ \nu_\mu$. The future measurements in BESIII and Belle II experiments will help us to test effects of NP and to further test new physics models., Comment: 28 Pages,9 figures

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 $\phi_{f_1} \sim 24^\circ$. Here, $f_1$ denotes the axial-vector mesons $f_1(1285)$ and $f_1(1420)$ with mixing angle $\phi_{f_1}$ 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 $\phi_{f_1}$ exhibits some interesting line shapes, whose confirmations would be helpful to constrain the determination of $\phi_{f_1}$ 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., Comment: 19 pages, 3 figures, and 6 tables. the values in (A12) corrected, the numerical results and conclusions remain unchanged

We suggest that the uncertain state $f_X(1500)$ observed by Belle and BaBar more than a decade ago, which has been viewed as a single scalar or a combination of several even spin resonances, is the vector $\rho^0(1450)$ reported recently by LHCb. Adopting the perturbative QCD approach, we determine the di-kaon distribution amplitudes with the $\rho^0(1450)$ resonance from the LHCb data for the quasi-two-body decays $B^{\pm}\to \pi^{\pm}\rho^0(1450)\to\pi^{\pm}K^+K^-$. It is then shown that the $B^+ \to K^+K^+K^-$ decay spectrum around the invariant mass $M(K^+K^-)\sim 1.5~\rm GeV$ measured by BaBar can be well described by the resonant contribution from $\rho^0(1450)$. The broad structure in the $B^{+}\to K^{+} K_SK_S$ spectrum around the invariant mass $1.5~\rm GeV$ of a $K_SK_S$ pair, which $\rho^0(1450)$ cannot decay into because of Bose-Einstein statistics, can be accounted for by a nonresonant $S$-wave contribution alone. The branching fractions and/or the direct $CP$ asymmetries of the $B^{\pm}\to \pi^{\pm}\rho^0(1450)\to\pi^{\pm}K^+K^-$, $B^{+}\to K^{+}\rho^0(1450)\to K^+K^+K^-$ and $B^{0}\to K^{0}\rho^0(1450)\to K^0K^+K^-$ modes are predicted, which can be tested at the ongoing LHCb and Belle-II experiments. We encourage experimental colleagues to scrutinize our postulation by analyzing relevant data with higher precision., Comment: 7 Pages, 3 Figures. Accepted by PRD

Motivated by the first untagged decay-time-integrated amplitude analysis of $B_s \to K_SK^{\mp}\pi^{\pm}$ decays performed by LHCb collaboration, where the decay amplitudes are modeled to contain the resonant contributions from intermediate resonances $K^*(892)$, $K_0^*(1430)$ and $K_2^*(1430)$, we comprehensively investigate the quasi-two-body $B_{s} \to K^0/\overline{\kern -0.2em K}^0 K^{\pm}\pi^{\mp}$ decays, and calculate the branching fractions and the time-dependent $CP$ asymmetries within the perturbative QCD approach based on the $k_T$ factorization. In the quasi-two-body space region the calculated branching fractions with the considered intermediate resonances are in good agreement with the experimental results of LHCb by adopting proper $K\pi$ pair wave function, describing the interaction between the kaon and pion in the $K\pi$ pair. Furthermore,within the obtained branching fractions of the quasi-two-body decays, we also calculate the branching fractions of corresponding two-body decays, and the results consist with the LHCb measurements and the earlier studies with errors. For these considered decays, since the final states are not flavour-specific, the time-dependent $CP$ could be measured. We calculate six $CP$-violation observables, which can be tested in the ongoing LHCb experiment., Comment: 20 pages

In this work, we study the ($S$, $P$ and $D$)-wave $K^+K^-$ contributions to $B\to KKK$ decays in the perturbative QCD approach at leading order. Within the two-meson wave functions describing the nonperturbative dynamics in the kaon-pair for different waves, we calculate the branching fractions and the direct $CP$ asymmetries of these decay modes in the corresponding resonance regions. Most of our numerical results are well consistent with the current measurements. We note that the narrow-width approximation is invalid in the quasi-two-body decays $B\to Kf_0(980)\to KKK$. For other decays, under the narrow-width approximation we can extract the branching fractions of the corresponding two-body decays involving the intermediate resonant states, and the related branching fractions agree with the current experimental data well. Furthermore, we also predict the corresponding quasi-two-body decays $B\to K\pi^+\pi^-$, which are expected to be measured in the ongoing LHCb and Belle-II experiments., Comment: 28 Pages, and 3 figures

Winograd's minimal filtering algorithm has been widely used in Convolutional Neural Networks (CNNs) to reduce the number of multiplications for faster processing. However, it is only effective on convolutions with kernel size as 3x3 and stride as 1, because it suffers from significantly increased FLOPs and numerical accuracy problem for kernel size larger than 3x3 and fails on convolution with stride larger than 1. In this paper, we propose a novel Decomposable Winograd Method (DWM), which breaks through the limitation of original Winograd's minimal filtering algorithm to a wide and general convolutions. DWM decomposes kernels with large size or large stride to several small kernels with stride as 1 for further applying Winograd method, so that DWM can reduce the number of multiplications while keeping the numerical accuracy. It enables the fast exploring of larger kernel size and larger stride value in CNNs for high performance and accuracy and even the potential for new CNNs. Comparing against the original Winograd, the proposed DWM is able to support all kinds of convolutions with a speedup of ~2, without affecting the numerical accuracy., Comment: Accepted by AAAI 2020

Motivated by the observations of the decays $B^0 \to K_0^{*}(1430)^0 f_0(980)$ and $ B^0 \to K_2^{*}(1430)^0 f_0(980)$ from BaBar collaboration, we study the $B^{0(+)} \to K_{0,2}^{*}(1430)^{0(+)} f_0(980)/\sigma$ decays in the perturbative QCD approach for the first time. In the absence of reliable nonperturbative wave functions we only assume the scalar meson $f_0(980)$ and $\sigma$ are two-quark ground states. In our calculations, these decays are all dominated by the hard-scattering emission and annihilation diagrams, while the factorizable emission diagrams are forbidden or suppressed heavily by the vector decay constants. Furthermore, the branching fractions are sensitive to the mixing between $f_0(980)$ and $\sigma$. Comparing our results with the experimental data, a large mixing angle $\theta$ is favored. Taking $\theta=145^\circ$, the orders of branching fractions of $B \to K_0^{*}(1430)^0 \sigma$, $B \to K_{2}^{*}(1430)^0 \sigma$ and $B \to K_{0,2}^{*}(1430)^0 f_0(980)$ are predicted to be $10^{-4}$, $10^{-5}$ and $10^{-6}$, respectively, which can be measured in the current experiments such as LHCb and Belle-2. In addition, although these decays are penguin dominant, the mixing also leads to large direct $CP$ asymmetries in these decays. With the precise data in future, our results could shed light on the inner structure of the scalar mesons and can be used to determine the mixing angle of the $\sigma-f_0(980)$ system., Comment: 21 pages, 5 figures

Recent experimental results of ${\cal R}(D^{(*)})$ deviate from the standard model (SM) by $3.1\sigma$, suggesting a new physics (NP) that affects the $b\to c \tau \bar\nu_\tau$ transition. Motivated by this, we investigate the possible NP effects in the $\Lambda_b\to\Lambda_c \tau\bar\nu_\tau$ decay. For this purpose, assuming the neutrinos are left-handed, we calculate in detail the helicity amplitudes of $\Lambda_b\to\Lambda_c \ell\bar\nu_\ell$ decays with all possible four-fermion operators. Within the latest results of $\Lambda_b\to\Lambda_c$ form factors from lattice QCD calculations, we study these decays in a model-independent manner. The differential and total branching fractions and other observables are calculated. In SM, we obtain the ratio ${\cal R}(\Lambda_c)=0.33\pm0.01$. Supposing that NP only affects the third generation fermions, we present the correlations among ${\cal R}(D)$, ${\cal R}(D^*)$ and ${\cal R}(\Lambda_c)$. We perform a minimum $\chi^2$ fit of the wilson coefficient of each operator to the latest experimental data of different observables. It is found that the left-handed scalar operator ${\cal O}_{SL}$ affects the branching fraction remarkably, and the ratio ${\cal R}(\Lambda_c)$ can be enhanced by $30\%$. For other operators, the ratio amounts to $0.38\pm0.02$, which is larger than prediction of SM by $20\%$. Using the fitted values of the wilson coefficients of the single NP operators, we also give a prognosis for the physical observables of $\Lambda_b\to\Lambda_c \tau\bar\nu_\tau$. Furthermore, we also study the effects of three typical NP models on $\Lambda_b\to\Lambda_c \tau\bar\nu_\tau$. We hope our results can be tested in the current LHCb experiment and the future high energy experiments., Comment: 24 pages,8 figures

Encouraged by the global agreement between theoretical predictions and experimental measurements for $B \to J/\psi V$ decays, we extend that perturbative QCD formalism to $B_{d,s}^0 \to J/\psi f_0(500) [f_0(980)]$ decays at the presently known next-to-leading order in the quark-antiquark description of $f_0(500)$ and $f_0(980)$. With the angle $\phi_f \approx 25^\circ$ of the $f_0(500)-f_0(980)$ mixing in the quark-flavor basis, we find that the branching ratios of the $B_d^0 \to J/\psi f_0(500) (\to \pi^+ \pi^-)$ and $B_{d,s}^0 \to J/\psi f_0(980) (\to \pi^+ \pi^-)$ modes generally agree with the current data or the upper limits within uncertainties, except for the seemingly challenging $B_s^0 \to J/\psi f_0(500) (\to \pi^+\pi^-)$ one. Then, we further explore the relevant observables of the $B_{d,s}^0 \to J/\psi f_0(500) [f_0(980)]$ decays, which could provide further constraints on the mixing angle $\phi_f$ and/or SU(3) flavor symmetry breaking effects. As a byproduct, we predict ${\rm BR}(B_{d}^0 \to J/\psi f_0(980)(\to K^+ K^-))=5.8^{+3.1}_{-2.9} \times 10^{-7}$ and ${\rm BR}(B_{s}^0 \to J/\psi f_0(980)(\to K^+ K^-)) =4.6^{+2.6}_{-2.3} \times 10^{-5}$. All theoretical predictions await the future examinations with high precision., Comment: RevTex file, 15 pages, 5 figures; final version to appear in PRD

In this paper we calculate the power corrections to the pion transition form factor within the framework of perturbative QCD approach on the basis of $k_T$ factorization. The power suppressed contributions from higher twist pion wave functions and the hadronic structure of photon are investigated. We find that there exists strong cancellation between the two kinds contributions, thus the total power corrections considered currently are very small, and the prediction of the leading power contribution with joint resummation improved perturbative QCD approach is almost unchanged. This result confirms that the pion transition form factor is a good platform to constrain the nonperturbative parameters in pion wave functions. Moreover, our result can accommodate the anomalous data from BaBar, or agrees with results from Belle according to the choice of Gegebauer moment in the pion wave function, and the more precise experimental data from Belle-II is expected., Comment: 5 figures, 1 table