Possible $K \bar{K}^*$ and $D \bar{D}^*$ resonances by solving Schrödinger equation

TL;DR

The paper investigates whether a simple one-pion-exchange potential can generate near-threshold hadronic molecules in the $K \bar{K}^*$ and $D \bar{D}^*$ sectors. It solves the radial Schrödinger equation in the S-wave with a Yukawa-type potential $V(r)=-g^2\frac{e^{-mr}}{d}$, fixes the coupling $g$ from bound-state energies via $\alpha=2g\sqrt{2\mu d}$, and then obtains resonances from the outgoing-wave condition $H_{\rho}^{(2)}(\alpha)=0$, yielding complex energies $E=M-i\frac{\Gamma}{2}$. The results produce bound states consistent with $f_1(1285)$ and near-threshold states $f_1(1378)$; resonances near $M\sim 1400$ MeV identified with $f_1(1420)$, and several $D\bar{D}^*$-related resonances matching $\chi_{c1}$ and $X$ states, supporting a molecular interpretation. The work demonstrates a coherent bound–resonance relation in these hadronic systems and extends to charged channels, offering a unified framework for near-threshold hadron spectroscopy.

Abstract

The one-pion exchange interaction between the kaon and the vector antikaon is investigated by solving the Schrödinger equation in the S-wave approximation. In addition to the particle $f_1(1285)$, another bound state of $K \bar{K}^*$ is obtained, which is approximately 9 MeV below the threshold of $K \bar{K}^*$ and labeled $f_1(1378)$ for convenience in this manuscript. Under the outgoing wave condition, two resonance states of $K \bar{K}^*$ are produced with different coupling constants fixed with the binding energies of $f_1(1285)$ and $f_1(1378)$, respectively. Both of the resonance states are located in the vicinity of 1400 MeV, and thus it is reasonable to assume that these two resonance states correspond to the $f_1(1420)$ particle in the review of the Particle Data Group simultaneously. This method is extended to study the $D \bar{D}^*$ system analogously. When the particle $χ_{c1}(3872)$ is treated as a bound state of $D \bar{D}^*$, the particles $T_{c\bar{c}1}(3900)$, $T_{c \bar{c}}(4020)$ and $X(3940)$ can be obtained as solutions of the Schrödinger equation under the outgoing wave condition, which implies the spin and parity of these particles are all $J^P=1^+$.

Figures (2)

• Figure 1: Bessel functions $J_\rho(\alpha)$ with $\rho=3.703$(Solid line) and $\rho=1.106$(Dash line) are depicted, respectively.
• Figure 2: $1/|H_\rho^{(2)}(\alpha)|^2$ .vs. the complex energy $E$. $D^0 \bar{D}^{*0}$ channel with the zero point at $\alpha=2.405$ (Left panel) and $D^{-} D^{*+}$ channel with the zero point at $\alpha=4.8583$ (Right panel). The pole of $1/|H_\rho^{(2)}(\alpha)|^2$ corresponds to a zero-point of the second kind of Hankel function $H_\rho^{(2)}(\alpha)$, which represents a $D \bar{D}^{*}$ resonance state, as labeled in the figure.