Probing the nature of $Y(4260)$ in the isospin violating process $Y(4260)\to J/\psi \eta {\pi }^0$

The isospin violation process $Y(4260)\to J/\psi \eta {\pi }^0$ is studied assuming that $Y(4260)$ is a $D_1\overline{D}+c.c$. hadronic molecule. In association with the production of the $Z_c(3900)$, which is treated as a $D{\overline{D}}^{*}+c.c$. hadronic molecule, this process can help us distinguish the molecular nature of $Y(4260)$ from other scenarios since the incomplete cancellation between the charged and neutral-meson loops, which are prominent in the molecular picture only, produces a peak in the $e^{+}{e}^{-}\to Y(4260)\to J/\psi \eta {\pi }^0$ cross section at the $D_1\overline{D}+c.c$. threshold and a very prominent peak in the $J/\psi \eta$ invariant mass spectrum in between the $D{\overline{D}}^{*}+c.c$. thresholds, the latter being much narrower than the corresponding one in the isospin conserving channel, i.e. $J/\psi {\pi }^{+}{\pi }^{-}$. The partial width of $Y(4260)\to J/\psi \eta {\pi }^0$ is about $4\times {10}^{-4}$ times that of $Y(4260)\to J/\psi {\pi }^{+}{\pi }^{-}$. The cross section of $e^{+}{e}^{-}\to Y(4260)\to J/\psi \eta {\pi }^0$ at the $D_1\overline{D}+c.c$. threshold is about 0.05 pb, which is much larger than that produced by the nearby resonances. These features are the direct consequence of the assumed nature of these two states, especially for the $Y(4260)$, which might be accessible at high-statistics experiments such as BESIII and LHCb.

Figure 1

Feynman diagrams for $Y(4260)\to J/\psi \eta {\pi }^0$, which are classified as: (a) Triangle diagrams with the explicit $Z_c(3900)$ pole; (b–d) Box diagrams without the pole.

Figure 2

The schematic diagram for the ${\pi }^0$-$\eta$ mixing. The square denotes that all the allowed diagrams in Fig. 1 contributing to $Y(4260)\to J/\psi {\pi }^0{\pi }^0$ are included, and the round dot represents the mixing between ${\pi }^0$ and $\eta$.

Figure 3

The invariant mass spectra for the final state $J/\psi \eta$ (left column), $J/\psi {\pi }^0$ (middle column), and $\eta {\pi }^0$ (right column) in $Y(4260)\to J/\psi \eta {\pi }^0$ evaluated for a total energy of 4260 MeV. Figures (a–c) in the first row are the contributions from the triangle diagrams, Figures (d–f) in the second row are for the box diagrams, Figures (g–i) in the third row are for the ${\pi }^0$-$\eta$ mixing through triangle diagrams, and Figures (j–l) in the fourth row are for the ${\pi }^0$-$\eta$ mixing through box diagrams. The solid (red) and dashed (blue) lines stand for the contributions with and without considering the width of $D_1$. The three vertical lines in the left column denote the $D^0{\overline{D}}^{*0}$ and $D^{+}{\overline{D}}^{*-}$ threshold, and the $Z_c(3900)$ mass, respectively. All the differential partial widths have been normalized to the partial width of the isospin conserving process, i.e. ${\mathrm{\Gamma }}_{Y(4260)\to J/\psi {\pi }^{+}{\pi }^{-}}$.

Figure 4

The predicted cross section for $e^{+}{e}^{-}\to Y(4260)\to J\psi \eta {\pi }^0$. The two vertical short-dashed lines denote the charged and neutral $D_1\overline{D}$ thresholds, respectively, while the long-dashed line denotes the location of the $Y(4260)$ pole as it emerged from the fit to the isospin conserving data. The grey band shows the variation of our prediction, when the parameters are allowed to vary within the statistical uncertainty allowed by the fit to the isospin conserving data.