Study of four-body decays B(s)→(ππ)(ππ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{(s)} \\rightarrow (\\pi \\pi )(\\pi \\pi )$$\\end{document} in the perturbative QCD approach

Abstract

In this work, we make a systematical study on the four-body B(s)→(ππ)(ππ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{(s)} \\rightarrow (\\pi \\pi )(\\pi \\pi )$$\\end{document} decays in the perturbative QCD (PQCD) approach, where the ππ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi \\pi $$\\end{document} invariant mass spectra are dominated by the vector resonance ρ(770)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho (770)$$\\end{document} and the scalar resonance f0(980)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_0(980)$$\\end{document}. We improve the Gengenbauer moments for the longitudinal P-wave two-pion distribution amplitudes (DAs) by fitting the PQCD factorization formulas to measured branching ratios of three-body and four-body B decays. With the fitted Gegenbauer moments, we make predictions for the branching ratios and direct CP asymmetries of four-body B(s)→(ππ)(ππ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{(s)} \\rightarrow (\\pi \\pi )(\\pi \\pi )$$\\end{document} decays. As a by-product, we extract the branching ratios of two-body B(s)→ρρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{(s)} \\rightarrow \\rho \\rho $$\\end{document} from the corresponding four-body decay modes and calculate the relevant polarization fractions. We find that the B(B0→ρ+ρ-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{B}(B^0 \\rightarrow \\rho ^+\\rho ^-)$$\\end{document} is consistent with the previous theoretical predictions and data. The leading-order PQCD calculations of the B(B+→ρ+ρ0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{B}(B^+\\rightarrow \\rho ^+\\rho ^0)$$\\end{document}, B(B0→ρ0ρ0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{B}(B^0\\rightarrow \\rho ^0\\rho ^0)$$\\end{document} and the f0(B0→ρ0ρ0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_0(B^0\\rightarrow \\rho ^0\\rho ^0)$$\\end{document} are a bit lower than the experimental measurements, which should be further examined. It is shown that the direct CP asymmetries are large when both tree and penguin contributions are comparable to each other, but small for the tree-dominant or penguin-dominant processes. In addition to the direct CP asymmetries, the “true” and “fake” triple-product asymmetries (TPAs) originating from the interference among various helicity amplitudes in the B(s)→(ππ)(ππ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{(s)}\\rightarrow (\\pi \\pi )(\\pi \\pi )$$\\end{document} decays are also analyzed. The sizable averaged TPA AT-true1,ave=25.26%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{A}_{\ ext {T-true}}^{1, \ ext {ave}}=25.26\\%$$\\end{document} of the color-suppressed decay B0→ρ0ρ0→(π+π-)(π+π-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^0\\rightarrow \\rho ^0\\rho ^0 \\rightarrow (\\pi ^+\\pi ^-)(\\pi ^+\\pi ^-)$$\\end{document} is predicted for the first time, which deviates a lot from the so-called “true” TPA AT-true1=7.92%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}_\ ext {T-true}^1=7.92\\%$$\\end{document} due to the large direct CP violation. A large “fake” TPA AT-fake1=24.96%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}_\ ext {T-fake}^1=24.96\\%$$\\end{document} of the decay B0→ρ0ρ0→(π+π-)(π+π-)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^0\\rightarrow \\rho ^0\\rho ^0 \\rightarrow (\\pi ^+\\pi ^-)(\\pi ^+\\pi ^-)$$\\end{document} is also found, which indicates the significance of the final-state interactions. The predictions in this work can be tested by LHCb and Belle-II experiments in the near future.