The possible KK¯* and DD¯* bound and resonance states by solving the 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 a Bessel function, the first kind of Hankel function and the second kind of Hankel function, respectively. Subsequently, the scattering processes of KK¯* and DD¯* are investigated. In the KK¯* sector, the f ₁(1285) particle is treated as a KK¯* bound state, therefore, the coupling constant in the KK¯* Yukawa potential can be fixed according to the binding energy of the f ₁(1285) particle. Consequently, a KK¯* 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 KK¯* resonance state at 1417 − i18 MeV might correspond to the f ₁(1420) particle in the review of the Particle Data Group. In the DD¯* sector, since the X(3872) particle is almost located at the DD¯* threshold, its binding energy is approximately equal to zero. Therefore, the coupling constant in the DD¯* Yukawa potential is determined, which is related to the first zero point of the zero-order Bessel function. Similarly to the KK¯* 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.