Partial wave analysis of $J/\psi \to \gamma \eta {\eta }^{\prime }$

Based on a sample of $(10.09\pm 0.04)\times {10}^9$ $J/\psi$ events collected with the BESIII detector operating at the BEPCII storage ring, a partial wave analysis of the decay $J/\psi \to \gamma \eta {\eta }^{\prime }$ is performed. An isoscalar state with exotic quantum numbers $J^{PC}=1^{-+}$, denoted as ${\eta }_1(1855)$, has been observed for the first time with statistical significance larger than $19\sigma$. Its mass and width are measured to be $(1855\pm 9_{-1}^{+6})\mathrm{MeV}/c^2$ and $(188\pm {18}_{-8}^{+3})\mathrm{MeV}$, respectively. The first uncertainties are statistical and the second are systematic. The product branching fraction $\mathcal{B}(J/\psi \to \gamma {\eta }_1(1855))\mathcal{B}({\eta }_1(1855)\to \eta {\eta }^{\prime })$ is measured to be $(2.70\pm {0.41}_{-0.35}^{+0.16})\times {10}^{-6}$. In addition, an upper limit on the ratio of branching fractions $\mathcal{B}(f_0(1710)\to \eta {\eta }^{\prime })/\mathcal{B}(f_0(1710)\to \pi \pi )$ is determined to be $1.61\times {10}^{-3}$ at 90% confidence level, which lends support to the hypothesis that the $f_0(1710)$ has a large glueball component.

Figure 1

Invariant mass distributions for (a,c,e) $J/\psi \to \gamma \eta {\eta }^{\prime }$, ${\eta }^{\prime }\to \gamma {\pi }^{+}{\pi }^{-}$ and (b,d,f) $J/\psi \to \gamma \eta {\eta }^{\prime }$, ${\eta }^{\prime }\to \eta {\pi }^{+}{\pi }^{-}$. (a,b) Invariant mass distributions for ${\eta }^{\prime }$ candidates. The events between the blue arrows are the selected signal events, and events between the red arrows on the same side are selected sideband events. The black points with error bars are data, the red solid line is the fit result, the blue dashed line is the ${\eta }^{\prime }$ signal shape from signal MC, and the green dashed line is the 2nd degree polynomial function for the background. (c,d) Invariant mass distributions of the $\eta {\eta }^{\prime }$ for the selected $\gamma \eta {\eta }^{\prime }$ candidates. The points with error bars are data and the red line shows the background events estimated from the ${\eta }^{\prime }$ sideband. (e,f) The corresponding Dalitz plots for the selected $\gamma \eta {\eta }^{\prime }$ candidates.

Figure 2

Background-subtracted data (black points) and the PWA fit projections (lines) for (a,b,c) the invariant mass distributions of (a) $\eta {\eta }^{\prime }$, (b) $\gamma \eta$, and (c) $\gamma {\eta }^{\prime }$, and (d) the distribution of $\cos {\theta }_{\eta }$, where ${\theta }_{\eta }$ is the angle of the $\eta$ momentum in the $\eta {\eta }^{\prime }$ helicity coordinate system. The red lines are the total fit projections from the baseline PWA. The blue lines are the total fit projections from a fit excluding the ${\eta }_1$ component. The dashed lines for the $1^{-+},0^{++},2^{++},4^{++}$ and $1^{+-}$ contributions are the coherent sums of amplitudes for each $J^{PC}$.

Figure 3

Dalitz plots for (a) the baseline PWA, (b) the selected data, and (c) background estimated from the ${\eta }^{\prime }$ sideband.

Figure 4

The change in negative log-likelihood values for a range of ${\eta }_1$ resonance parameters in the baseline set of amplitudes. The red dashed line indicates that the statistical significances of the hypotheses under the red dashed line are higher than $5\sigma$.

Figure 5

The distributions of the unnormalized moments $⟨Y_L^0⟩$ ($L=0$, 1, 2, 3 and 4) for $J/\psi \to \gamma \eta {\eta }^{\prime }$ as functions of the $\eta {\eta }^{\prime }$ mass. Black dots with error bars represent the background-subtracted data weighted with angular moments; the red solid lines represent the baseline fit projections; and the blue dotted lines represent the projections from a fit excluding the ${\eta }_1$ component.

Figure 6

Background-subtracted data (black points) and the PWA fit projections (lines) for the $\cos {\theta }_{\eta }$ distribution when the $\eta {\eta }^{\prime }$ mass is restricted to the regions: (a) [1.5,1.7], (b) [1.7,2.0], and (c) $[2.0,3.2]\mathrm{GeV}/c^2$. The red lines are the total fit projections from the baseline PWA. The blue lines are the total fit projections from a fit excluding the ${\eta }_1$ component.