$P$-wave coupled channel effects in electron-positron annihilation

Abstract

$P$-wave coupled channel effects arising from the $D\bar{D}$, $D{\bar{D}}^{\ast }+c.c.$, and $D^{\ast }{\bar{D}}^{\ast }$ thresholds in $e^{+}{e}^{-}$ annihilations are systematically studied. We provide an exploratory study by solving the Lippmann-Schwinger equation with short-ranged contact potentials obtained in the heavy quark limit. These contact potentials can be extracted from the $P$-wave interactions in the $e^{+}{e}^{-}$ annihilations, and then be employed to investigate possible isosinglet $P$-wave hadronic molecules. In particular, such an investigation may provide information about exotic candidates with quantum numbers $J^{PC}=1^{-+}$. In the mass region of the $D\bar{D}$, $D{\bar{D}}^{\ast }+c.c.$, and $D^{\ast }{\bar{D}}^{\ast }$ thresholds, there are two quark model bare states, i.e. the $\psi (3770)$ and $\psi (4040)$, which are assigned as $(1^3{D}_1)$ and $(3^1{S}_1)$ states, respectively. By an overall fit of the cross sections of $e^{+}{e}^{-}\to D\bar{D}$, $D{\bar{D}}^{\ast }+c.c.$, $D^{\ast }{\bar{D}}^{\ast }$, we determine the physical coupling constants to each channel and extract the pole positions of the $\psi (3770)$ and $\psi (4040)$. The deviation of the ratios from that in the heavy quark spin symmetry (HQSS) limit reflects the HQSS breaking effect due to the mass splitting between the $D$ and the $D^{\ast }$. Besides the two poles, we also find a pole a few MeV above the $D{\bar{D}}^{\ast }+c.c.$ threshold which can be related to the so-called $G(3900)$ observed earlier by BABAR and Belle. This scenario can be further scrutinized by measuring the angular distribution in the $D^{\ast }{\bar{D}}^{\ast }$ channel with high luminosity experiments.

Comments: 15 pages, 2 figures