Branching ratios and CP asymmetries of the quasi-two-body decays Bc→K0∗(1430,1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow \\ K^{*}_0(1430,1950) D_{(s)} \\rightarrow K \\pi D_{(s)} $$\\end{document} in the PQCD approach

Abstract

In this paper, we investigate the quasi-two-body decays Bc→K0∗(1430,1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430,1950) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} within the perturbative QCD (PQCD) framework. The S-wave two-meson distribution amplitudes (DAs) are introduced to describe the final state interactions of the Kπ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\pi $$\\end{document} pair, which involve the time-like form factors and the Gegenbauer polynomials. In the calculations, we adopt two kinds of parameterization schemes to describe the time-like form factors: one is the relativistic Breit–Wigner (RBW) formula, which is usually more suitable for the narrow resonances, and the other is the LASS line shape proposed by the LASS Collaboration, which includes both the resonant and nonresonant components. We find that the branching ratios and the direct CP violations for the decays Bc→K0∗(1430)D(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430) D_{(s)}$$\\end{document} obtained from those of the quasi-two-body decays Bc→K0∗(1430)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} under the narrow width approximation (NWA) can be consistent well with the previous PQCD results calculated in the two-body framework by assuming that K0∗(1430)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1430)$$\\end{document} is the lowest lying q¯s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{q} s$$\\end{document} state, which is the so-called scenario II (SII). We conclude that the LASS parameterization is more suitable to describe the K0∗(1430)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_0^{*}(1430)$$\\end{document} than the RBW formula, and the nonresonant components play an important role in the branching ratios of the decays Bc→K0∗(1430)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document}. In view of the large difference between the decay width measurements for the K0∗(1950)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_0^{*}(1950)$$\\end{document} given by BaBar and LASS collaborations, we calculate the branching ratios and the CP violations for the quasi-two-body decays Bc→K0∗(1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1950) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} by using two values, ΓK0∗(1950)=0.100±0.04\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{K^*_0(1950)}=0.100\\pm 0.04$$\\end{document} GeV and ΓK0∗(1950)=0.201±0.034\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{K^*_0(1950)}=0.201\\pm 0.034$$\\end{document} GeV, besides the two kinds of parameterizations for the resonance K0∗(1950)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1950)$$\\end{document}. We find that the branching ratios and the direct CP violations for the decays Bc→K0∗(1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1950) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} have not as large difference between the two parameterizations as the case of decays involving the K0∗(1430)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1430)$$\\end{document}, especially for the results with ΓK0∗(1950)=0.201±0.034\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{K^*_0(1950)}=0.201\\pm 0.034$$\\end{document} GeV. The effect of the nonresonant component in the K0∗(1950)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1950)$$\\end{document} may be not so serious as that in the K0∗(1430)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*_0(1430)$$\\end{document}. The quasi-two-body decays Bc+→K0∗+(1430)D0→K0π+D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^+_c \\rightarrow K^{*+}_0(1430) D^{0} \\rightarrow K^0 \\pi ^+ D^{0}$$\\end{document} and Bc+→K0∗0(1430)D+→K+π-D+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^+_c \\rightarrow K^{*0}_0(1430) D^{+} \\rightarrow K^+ \\pi ^- D^{+}$$\\end{document} have large branching ratios, which can reach to the order of 10-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-4}$$\\end{document} and are most likely to be observed in the future LHCb experiments. Furthermore, the branching ratios of the quasi-two-body decays Bc→K0∗(1950)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1950) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document} are about one order smaller than those of the corresponding decays Bc→K0∗(1430)D(s)→KπD(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c \\rightarrow K_0^{*}(1430) D_{(s)} \\rightarrow K \\pi D_{(s)}$$\\end{document}.