The possible $K{\bar{K}}^{\ast }$ and $D{\bar{D}}^{\ast }$ bound and resonance states by solving Schrodinger equation

Abstract

The Schrodinger equation with a Yukawa type of potential is solved analytically. When different boundary conditions are taken into account, a series of solutions are indicated as Bessel function, the first kind of Hankel function and the second kind of Hankel function, respectively. Subsequently, the scattering processes of $K{\bar{K}}^{\ast }$ and $D{\bar{D}}^{\ast }$ are investigated. In the $K{\bar{K}}^{\ast }$ sector, the $f_1(1285)$ particle is treated as a $K{\bar{K}}^{\ast }$ bound state, therefore, the coupling constant in the $K{\bar{K}}^{\ast }$ Yukawa potential can be fixed according to the binding energy of the $f_1(1285)$ particle. Consequently, a $K{\bar{K}}^{\ast }$ resonance state is generated by solving the Schrodinger equation with the outgoing wave condition, which lie at $1417-i18$MeV on the complex energy plane. It is reasonable to assume that the $K{\bar{K}}^{\ast }$ resonance state at $1417-i18$MeV might correspond to the $f_1(1420)$ particle in the review of Particle Data Group(PDG).In the $D{\bar{D}}^{\ast }$ sector, since the $X(3872)$ particle is almost located at the $D{\bar{D}}^{\ast }$ threshold, the binding energy of it equals to zero approximately. Therefore, the coupling constant in the $D{\bar{D}}^{\ast }$ Yukawa potential is determined, which is related to the first zero point of the zero order Bessel function. Similarly to the $K{\bar{K}}^{\ast }$ case, four resonance states are produced as solutions of the Schrodinger equation with the outgoing wave condition. It is assumed that the resonance states at $3885-i1$MeV, $4029-i108$ MeV, $4328-i191$MeV and $4772-i267$MeV might be associated with the $Zc(3900)$, the $X(3940)$, the ${\chi }_{c1}(4274)$ and ${\chi }_{c1}(4685)$ particles, respectively. It is noted that all solutions are isospin degenerate.

9 pages, 2 tables, 4 figures, to be published in Communications in Theoretical Physics

References in corpus (14)

• Pentaquark and Tetraquark states
• An updated review of the new hadron states
• Chiral perturbation theory for heavy hadrons and chiral effective field theory for heavy hadronic molecules
• X(3872) and Other Possible Heavy Molecular States
• Is X(3872) {\sl Really} a Molecular State?
• Is X(3872) a molecule?
• Reconciling the X(3872) with the near-threshold enhancement in the D^0\bar{D}^{*0} final state
• Hadron Loops: General Theorems and Application to Charmonium
• Isospin breaking, coupled-channel effects, and X(3872)
• Interplay of quark and meson degrees of freedom in a near-threshold resonance
• Ambiversion of X(3872)
• Charmonium spectroscopy above thresholds
• Revising the $f_1(1420)$ resonance
• X(3872) as a 1D2 charmonium state