The possible K K ¯ * and D D ¯ * bound and resonance states by solving the Schrodinger equation

Abstract

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 a Bessel function, the first kind of Hankel function and the second kind of Hankel function, respectively. Subsequently, the scattering processes of K K ¯ * and D D ¯ * are investigated. In the K K ¯ * sector, the f 1(1285) particle is treated as a K K ¯ * bound state, therefore, the coupling constant in the K K ¯ * Yukawa potential can be fixed according to the binding energy of the f 1(1285) particle. Consequently, a K K ¯ * resonance state is generated by solving the Schrodinger equation with the outgoing wave condition, which lies at 1417 − i18 MeV on the complex energy plane. It is reasonable to assume that the K K ¯ * resonance state at 1417 − i18 MeV might correspond to the f 1(1420) particle in the review of the Particle Data Group. In the D D ¯ * sector, since the X(3872) particle is almost located at the D D ¯ * threshold, its binding energy is approximately equal to zero. Therefore, the coupling constant in the D D ¯ * Yukawa potential is determined, which is related to the first zero point of the zero-order Bessel function. Similarly to the K K ¯ * 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 χ c1(4274) and χ c1(4685) particles, respectively. It is noted that all solutions are isospin degenerate.