Abstract
Considering the fact that some excited states of the heavy quarkonia (charmonium and bottomonium) are still missing in experimental observations and potential applications of the relevant wave functions of the bound states, we re-analyze the spectrum and the relevant wave functions of the heavy quarkonia within the framework of Bethe-Salpeter (B.S.) equation with a proper QCD-inspired kernel. Such a kernel for the heavy quarkonia, relating to potential of the non-relativistic quark model, is instantaneous, so we call the corresponding B.S. equation as BS-In equation throughout the paper. Particularly, a new way to solve the B.S. equation, which is different from the traditional ones, is proposed here, and with it not only the known spectrum for the heavy quarkonia is re-generated, but also an important issue is brought in, i.e., the obtained solutions of the equation ‘automatically’ include the ‘fine’, ‘hyperfine’ splittings and the wave function mixture, such as S-D wave mixing in J PC = 1−− states, P-F wave mixing in J PC = 2++ states for charmonium, bottomonium etc. It is pointed out that the best place to test the wave mixture probably is at Z-factory (e + e − collider running at Z-boson pole with extremely high luminosity).
Similar content being viewed by others
References
Eichten E, Gottfried K, Kinoshita T, et al. The spectrum of charmonium. Phys Rev Lett, 1975, 34: 369–372 [Erratum-ibid. 1976, 36: 1276]; Eichten E, Gottfried K, Kinoshita T, et al. Charmonium: The model. Phys Rev D, 1978, 17: 3090–3117 [Erratum-ibid. 1980, 21: 313]
Eichten E, Gottfried K, Kinoshita T, et al. Charmonium: Comparison with experiment. Phys Rev D, 1980, 21: 203–233
Godfrey S, Isgur N. Mesons in a relativized quark model with chromodynamics. Phys Rev D, 1985, 32: 189–231; Godfrey S, Kokoski R. Properties of P-wave mesons with one heavy quark. Phys Rev D, 1991, 43: 1679–1687; Isgur N, Scora D, Grinstein B, et al. Semileptonic B and D decays in the quark model. Phys Rev D, 1989, 39: 799–818
Chen Y Q, Kuang Y P. Improved QCD-motivated heavy-quark potentials with explicit \( \Lambda _{\overline {MS} } \) dependence. Phys Rev D, 1992, 46: 1165–1171 [Erratum-ibid. 1993, 47: 350]
Lou J Z, Qin D H, Ding Y B, et al. c anti-b System: Mass Spectra and Decays. Commun Theor Phys, 1998, 30: 93–100; Qiao C F, Huang H W, Chao H T. Possible retardation effects of quark confinement on the meson spectrum. Phys Rev D, 1996, 54: 2273–2278
Yang C B, Cai X. Heavy-light mesons in a relativistic potential model. Phys Rev D, 1995, 51: 6332–6337
Zeng J, Van Orden J W, Roberts C D. Heavy mesons in a relativistic model. Phys Rev D, 1995, 52: 5229–5241
Salpeter E E, Bethe H A. A relativistic equation for bound state problems. Phys Rev, 1951, 84: 1232–1242
Salpeter E E. Mass corrections to the fine structure of hydrogen-like atoms. Phys Rev, 1952, 87: 328–343
Chang C H, Chen J K. The instantaneous Bethe-Salpeter equation and its analog: The Breit-like equation. Commun Theor Phys, 2005, 44: 646–650
Kim C S, Wang G L. Average kinetic energy of heavy quark (µ 2π ) inside heavy meson in 0− state by Bethe C Salpeter method. Phys Lett B, 2004, 584: 285–293
Chang C H, Chen J K, Li X Q, et al. Instantaneous Bethe-Salpeter equation and its exact solution. Commun Theor Phys, 2005, 43: 113–118
Rosner J. P-wave mesons with one heavy quark. Comm Nucl Part Phys, 1986, 16: 109–137
Isgur N, Wise M B. Excited charm mesons in semileptonic b decay and their contributions to a bjorken sum rule. Phys Rev D, 1991, 43: 819–828
Wang G L. Decay constants of heavy vector mesons in relativistic Bethe-Salpeter method. Phys Lett B, 2006, 633: 492–496
Wang G L. Decay constants of P-wave mesons. Phys Lett B, 2007, 650: 15–21
Amsler C, Doser M, Antonelli M, et al. (Particle Data Group). Review of particle physics. Phys Lett B, 2008, 667: 1–6
Aubert B, Bona M, Karyotakis Y, et al. (BABAR Collaboration). Observation of the bottomonium ground state in the decay ϒ(3S) → ϒηb . Phys Rev Lett, 2008, 101: 071801
Resag J, Münz C R. Heavy quarkonia in the instantaneous Bethe-Salpeter model. Nucl Phys A, 1995, 590: 735–749
Chang C H, Wang J X, Wu X G. Production of a heavy quarkonium and a photon and ISR at Z peak in e + e − collider. Sci China Phys Mech Astron, 2010, 53
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Chang, C., Wang, G. Spectrum for heavy quarkonia and mixture of the relevant wave functions within the framework of Bethe-Salpeter equation. Sci. China Phys. Mech. Astron. 53, 2005–2018 (2010). https://doi.org/10.1007/s11433-010-4156-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11433-010-4156-1