P -wave coupled channel effects in electron-positron annihilation

Abstract

$P$-wave coupled channel effects arising from the $D\overline{D}$, $D{\overline{D}}^{*}+\mathrm{c}.\mathrm{c}.$, and ${D}^{*}{\overline{D}}^{*}$ thresholds in ${e}^{+}{e}^{\ensuremath{-}}$ 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 ${e}^{+}{e}^{\ensuremath{-}}$ 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 ${J}^{\mathrm{PC}}={1}^{\ensuremath{-}+}$. In the mass region of the $D\overline{D}$, $D{\overline{D}}^{*}+\mathrm{c}.\mathrm{c}.$, and ${D}^{*}{\overline{D}}^{*}$ thresholds, there are two quark model bare states, i.e. the $\ensuremath{\psi}(3770)$ and $\ensuremath{\psi}(4040)$, which are assigned as $({1}^{3}{D}_{1})$ and $({3}^{1}{S}_{1})$ states, respectively. By an overall fit of the cross sections of ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}D\overline{D}$, $D{\overline{D}}^{*}+\mathrm{c}.\mathrm{c}.$, ${D}^{*}{\overline{D}}^{*}$, we determine the physical coupling constants to each channel and extract the pole positions of the $\ensuremath{\psi}(3770)$ and $\ensuremath{\psi}(4040)$. 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 ${D}^{*}$. Besides the two poles, we also find a pole a few MeV above the $D{\overline{D}}^{*}+\mathrm{c}.\mathrm{c}.$ 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 ${D}^{*}{\overline{D}}^{*}$ channel with high luminosity experiments.