Production of double P-wave heavy quarkonia at a super Z factory

Abstract

The production of double P-wave heavy quarkonia is studied systematically within the framework of non-relativistic quantum chromodynamics at the exclusive process of the electron–positron collision annihilation at the CM energy s=91.1876\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{s}=91.1876$$\\end{document} GeV of the Z0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z^0$$\\end{document} pole, i.e., double P-wave charmonia, double P-wave Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c$$\\end{document} mesons, and double P-wave bottomonia in e+e-→γ∗/Z0→|(QQ′¯)[n]⟩+|(Q′Q¯)[n′]⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^+e^-\\rightarrow \\gamma ^*/Z^0 \\rightarrow |(Q\\bar{Q'})[n]\\rangle +|(Q'\\bar{Q})[n^\\prime ]\\rangle $$\\end{document} (Q/Q′=c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q/Q'=c$$\\end{document}- or b-quarks) at a future super Z factory, where [n] or [n′]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[n^\\prime ]$$\\end{document} stands for the color-singlet [1P1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[^1P_1]$$\\end{document} and [3PJ]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[^3P_J]$$\\end{document} (J=0,1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J=0,1,2$$\\end{document}) Fock states. As an improved trace technology is useful for calculating the complicated double P-wave channels, the analytical result can be obtained at the amplitude level. According to our study, the generation rates for the double P-wave heavy quarkonia at the future super Z factory are considerable. The values obtained for the total cross sections of the double P-wave charmonium for σ(|(cc¯)[1]⟩+|(cc¯)[1]⟩)=4.191-0.498+0.489×10-3fb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma {(|(c\\bar{c})[1]\\rangle +|(c\\bar{c})[1]\\rangle )}=4.191^{+0.489}_{-0.498}\ imes 10^{-3}~fb$$\\end{document}, the total cross sections of the double P-wave Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c$$\\end{document} mesons for σ(|(cb¯)[1]⟩+|(bc¯)[1]⟩)=0.2045-0.0080+0.0084fb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma {(|(c\\bar{b})[1]\\rangle +|(b\\bar{c})[1]\\rangle )}=0.2045^{+0.0084}_{-0.0080}~fb$$\\end{document}, and the total cross sections of the double P-wave bottomonium for σ(|(bb¯)[1]⟩+|(bb¯)[1]⟩)=5.244-0.136+0.134×10-3fb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma {(|(b\\bar{b})[1]\\rangle +|(b\\bar{b})[1]\\rangle )}=5.244^{+0.134}_{-0.136}\ imes 10^{-3}~fb$$\\end{document}. The main uncertainties come from the mass of heavy quarkonia and the corresponding first-derivative radial wave functions at the origin under the Buchmüller–Tye (BT) potential. The number of events in the production of double P-wave charmonium, Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c$$\\end{document}-meson, and bottomonium states via e-e+→γ∗/Z0→|(QQ¯′)[n]⟩+|(Q′Q¯)[n′]⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^-e^+\\rightarrow \\gamma ^*/Z^0 \\rightarrow |(Q\\bar{Q}')[n]\\rangle + |(Q'\\bar{Q})[n^\\prime ]\\rangle $$\\end{document} at the integrated luminosity L≈1034cm-2s-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{L}\\approx 10^{34}cm^{-2}s^{-1}$$\\end{document} at the super Z factory is almost unobservable.