Measurement of the matrix elements for the decays $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $\eta /{\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$

Based on a sample of $1.31\times 10^9$ $J/\psi$ events collected with the BESIII detector at the BEPCII collider, Dalitz plot analyses of selected 79,625 $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ events, 33,908 $\eta \to {\pi }^0{\pi }^0{\pi }^0$ events, and 1,888 ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$ events are performed. The measured matrix elements of $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ are in reasonable agreement with previous measurements. The Dalitz plot slope parameters of $\eta \to {\pi }^0{\pi }^0{\pi }^0$ and ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$ are determined to be $-0.055\pm 0.014\pm 0.004$ and $-0.640\pm 0.046\pm 0.047$, respectively, where the first uncertainties are statistical and the second systematic. Both values are consistent with previous measurements, while the precision of the latter one is improved by a factor of 3. Final state interactions are found to have an important role in those decays.

Figure 1

Invariant mass spectrum of ${\pi }^{+}{\pi }^{-}{\pi }^0$ obtained after the kinematic fit without the $\eta$ mass constraint applied. The dots with error bars are for data, and the shaded histogram is for background events estimated from the inclusive MC sample.

Figure 2

Dalitz plot for $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ in the data sample.

Figure 3

Projections of the Dalitz plot (a) $X$ and (b) $Y$ for $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ obtained from data (dots with error bars) and phase space distributed MC events (dashed line). The result of the fit described in the text (solid line) is also plotted.

Figure 4

(a) Invariant mass spectrum of ${\pi }^0{\pi }^0{\pi }^0$ for $\eta \to {\pi }^0{\pi }^0{\pi }^0$ obtained from the data (dots with error bars) and estimated from the inclusive MC sample (shaded histogram). (b) Distribution of the kinematic variable $Z$ for $\eta \to {\pi }^0{\pi }^0{\pi }^0$ obtained from the data (dots with error bars) and phase space distributed MC events, where the $Z$ distribution is flat from $Z=0$ to $Z\sim 0.76$ and then drops to zero at $Z=1$ (dashed line). The inset shows the $Z$ range which is used for the fit to extract the slope parameter $\alpha$. Overlaid on the data is the result of the fit (solid line in the inset).

Figure 5

(a) Invariant mass spectrum of ${\pi }^0{\pi }^0{\pi }^0$ for ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$ obtained from the data (dots with error bars), estimated from the inclusive MC sample (shaded), and ${\eta }^{\prime }\to \eta {\pi }^0{\pi }^0$ MC events (hatched). (b) Distribution of the kinematic variable $Z$ for ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$ obtained from the data (dots with error bars), phase space distributed MC events (dashed line), ${\eta }^{\prime }$ sideband regions (shaded), and ${\eta }^{\prime }\to \eta {\pi }^0{\pi }^0$ MC events (hatched). The result of the fit (solid line) is overlaid on the data in the insert.

Figure 6

Comparison of experimental measurements and theoretical predictions of the matrix elements for $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$. *Theoretical predictions without error. **BSE denotes the Bethe–Salpeter equation. [29, 30, 31, 32] are the corresponding references.

Figure 7

Comparison of experimental measurements and theoretical predictions of the matrix elements for (a) $\eta \to {\pi }^0{\pi }^0{\pi }^0$ and (b) ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$. *Theoretical predictions without error. **BSE denotes the Bethe–Salpeter equation. [33, 34, 35] are the corresponding references.