通过求解薛定谔方程研究可能的 $K \ar{K}^*$ 和 $D \ar{D}^*$ 束缚态与共振态
The possible $K \bar{K}^*$ and $D \bar{D}^*$ bound and resonance states by solving Schrodinger equation
AI总结：
本文通过解析求解 Yukawa 势薛定谔方程，研究了 $K \ar{K}^*$ 和 $D \ar{D}^*$ 的束缚态与共振态，并利用 $f_1(1285)$ 和 $X(3872)$ 确定耦合常数，预言了多个可能对应已知粒子的共振态。
摘要：
本文解析求解了具有 Yukawa 型势的薛定谔方程。当考虑不同边界条件时，分别得到一系列由 Bessel 函数、第一类 Hankel 函数和第二类 Hankel 函数表示的解。随后，研究了 $K \ar{K}^*$ 和 $D \ar{D}^*$ 的散射过程。在 $K \ar{K}^*$ 扇区中，将 $f_1(1285)$ 粒子视为 $K \ar{K}^*$ 束缚态，因此可以根据 $f_1(1285)$ 粒子的结合能确定 $K \ar{K}^*$ Yukawa 势中的耦合常数。进而，通过求解具有出射波条件的薛定谔方程，得到一个位于复能量平面上 $1417-i18$MeV 处的 $K \ar{K}^*$ 共振态。可以合理地假设 $1417-i18$MeV 处的 $K \ar{K}^*$ 共振态可能对应于 Particle Data Group（PDG）综述中的 $f_1(1420)$ 粒子。在 $D \ar{D}^*$ 扇区中，由于 $X(3872)$ 粒子几乎位于 $D \ar{D}^*$ 阈值处，其结合能近似为零。因此，$D \ar{D}^*$ Yukawa 势中的耦合常数被确定，它与零阶 Bessel 函数的第一个零点相关。与 $K \ar{K}^*$ 情形类似，通过求解具有出射波条件的薛定谔方程，得到四个共振态作为解。假设位于 $3885-i1$MeV、$4029-i108$MeV、$4328-i191$MeV 和 $4772-i267$MeV 的共振态可能分别与 $Zc(3900)$、$X(3940)$、$χ_{c1}(4274)$ 和 $χ_{c1}(4685)$ 粒子相关联。需要指出的是，所有解都是同位旋简并的。
英文摘要：
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 $K \bar{K}^*$ and $D \bar{ D}^*$ are investigated. In the $K \bar{K}^*$ sector, the $f_1(1285)$ particle is treated as a $K \bar{K}^*$ bound state, therefore, the coupling constant in the $K \bar{K}^*$ Yukawa potential can be fixed according to the binding energy of the $f_1(1285)$ particle. Consequently, a $K \bar{K}^*$ 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 $K \bar{K}^*$ resonance state at $1417-i18$MeV might correspond to the $f_1(1420)$ particle in the review of Particle Data Group(PDG).In the $D \bar{D}^*$ sector, since the $X(3872)$ particle is almost located at the $D \bar{ D}^*$ threshold, the binding energy of it equals to zero approximately. Therefore, the coupling constant in the $D \bar{ D}^*$ Yukawa potential is determined, which is related to the first zero point of the zero order Bessel function. Similarly to the $K \bar{K}^*$ 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.
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