Charmonium States in QCD-Inspired Quark Potential Model Using Gaussian Expansion Method

Abstract

We study the mass spectrum and electromagnetic processes of charmonium system with the spin-dependent potentials fully taking into account in the solution of the Schroedinger equation and the results for the pure scalar and scalar-vector mixing linear confining potentials are compared. It is revealed that the scalar-vector mixing confinement is important for reproducing the mass spectrum and decay widths and the vector component is found to be around 22%, the long-standing discrepancy in M1 radiative transitions of J/ψ and ψ ′ is alleviated by means of the state wave functions obtained via the Hamiltonian with the full spin-dependent potential. This work also intends to identify few of the copious higher charmonium-like states as the cc ones. Particularly, the newly observed X(4160) and X(4350) are assigned as M (2 1 D 2 ) = 4164.9 MeV and M (3 3 P 2 ) = 4352.4 MeV, which strongly favor the J P C = 2 −+ , 2 ++ assignments respectively. The corresponding radiative transitions, leptonic and two-photon decay widths have been also calculated for the further experimental study.Keywords charmonium · potential model · Gaussian expansion method · XYZ mesons 1 IntroductionDue to the impressive increase of experimental results, charmonium (cc) spectroscopy has renewed great interest recently, coming along with the striking disagreement with theoretical expectations [1,2,3]. The unexpected and still-fascinating X(3872) has been joined by more than a dozen other charmonium-like states, while the series of vacancy have been left on the cc list. It is urgent to identify the possible new members of charmonium family from the abundant observations. The QCD inspired potential models have been playing an important role in investigating heavy quarkonium, owning to the presence of large nonperturbative effects in this energy region. Most quark potential models [4,5,6,7,8,9,10,11,12,13,14,15,16] have common ingredients under the nonrelativistic limit, despite some differences in the detailed corrections for relativistic and coupled channel effects, which typically are the Coulomb-like term induced by one-gluon exchange plus the long-range confining potential expected from nonperturbative QCD. Anyway, the nature of confining mechanism has been veiled so far. In the original Cornell model [17,18], it was assumed as Lorentz scalar, which gives a vanishing long-range magnetic contribution and agrees with the flux tube picture of quark

Bottomonium states versus recent experimental observations in the QCD-inspired potential model

“…Fitting with the mass of the ten well-established bb states, the vector scale parameter ε is determined and implies the 18.51% vector component of confining interaction. Combining our previous work on charmonium systems [23], we found that the scalar-vector mixing linear confinement of heavy quarkonium seems to be important and nearly about onefifth; explicitly, it is slightly lower than 4% for the bottomonium mesons.…”

“…R nS (0) is the radial S wave function at the origin, and R ′′ nD (0) is the second derivative of the radial D-wave function at the origin. They are explicitly analytic in the Gaussian basis space [23]. Table 2 presents the numerical results.…”

“…The variationally Gaussian expanded wave functions give rise to the analytic formulas for the overlap integral and the transition matrix elements [23]. The angular matrix element C f i is…”

A relativistic wave equation with a local kinetic operator and an energy-dependent effective interaction for the study of hadronic systems

“…By using, for the interaction, only central potentials, we shall not take into account the effects related to the spin-spin, spin-orbit and tensor interaction. These corrective contributions, that are extremely relevant for a detailed study of charmonium spectroscopy [5][6][7][8], can be introduced perturbatively carefully considering the Lorentz transformation properties of the interaction operators. The aim of the following analysis is only to demonstrate that our relativistic equation, with a local kinetic operator, can adequately reproduce the main structure of the charmonium spectrum for the low-lying resonances.…”

“…(1), is exactly consistent only for the zero component of a vector field. If a scalar (effective) field is considered, as it is usually done, in particular, for the study ¯ and ¯ spectra [5][6][7][8], one should add the corresponding scalar interaction operators to the constituent masses by means of the substitution that will be discussed in Section 3. However, these scalar interaction operators would appear in the square roots of the relativistic energies, giving rise to very serious difficulties for the calculations, unless an approximate Taylor expansion of the square roots is performed.…”