High-statistics measurement of the $\eta \to 3{\pi }^0$ decay at the Mainz Microtron

The largest, at the moment, statistics of $7\times {10}^6\eta \to 3{\pi }^0$ decays, based on $6.2\times {10}^7\eta$ mesons produced in the $\gamma p\to \eta p$ reaction, has been accumulated by the A2 Collaboration at the Mainz Microtron, MAMI. It allowed a detailed study of the $\eta \to 3{\pi }^0$ dynamics beyond its conventional parametrization with just the quadratic slope parameter $\alpha$ and enabled, for the first time, a measurement of the second-order term and a better understanding of the cusp structure in the neutral decay. The present data are also compared to recent theoretical calculations that predict a nonlinear dependence along the quadratic distance from the Dalitz-plot center.

Figure 1

Dalitz plot and its $z=X^2+Y^2={\rho }^2/{\rho }_{\max }^2$ distribution for the phase-space $\eta \to 3{\pi }^0$ decay.

Figure 2

Comparison of the experimental data [9, 10, 11, 24, 25, 26, 27, 28, 29, 30] (plotted by black points), which were used in the RPP [23] to obtain the averaged value (shown by the vertical lines) for the $\eta \to 3{\pi }^0$ quadratic slope parameter $\alpha$, to each other and to various calculations [2, 3, 4, 16, 17, 19, 21, 22, 31] (colored points). Ref. [2] gives the magnitude of $\alpha$ for the analysis made in Ref. [1], in which its value was not given.

Figure 3

$m\left(3{\pi }^0\right)$ invariant-mass distributions for events selected at $\mathrm{CL}>1\%$ by testing the $\gamma p\to 3{\pi }^{0}p\to 6\gamma p$ hypothesis for the data of Run I (left) and Run II (right).

Figure 4

Comparison of the experimental $\eta \to 3\pi$ Dalitz plots: (a) the full $\eta \to 3{\pi }^0$ plot (six entries per decay) from the present analysis of $\sim 7\times {10}^6$ decays; (b) one sextant of the $\eta \to 3{\pi }^0$ plot (one entry per decay) for the angle range ${30}^{\circ }<\varphi <{90}^{\circ }$ in Eq. (3); (c) the $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ plot (without boundary bins) from the KLOE analysis of $\sim 4.7\times {10}^6$ decays [14].

Figure 5

Experimental $z$ distributions obtained (a) individually from Run I (blue circles) and Run II (red triangles), and (b) the combined results (black triangles). The earlier A2 data from Ref. [11] are depicted in (a) by green open squares. The NREFT calculation by the Bonn group [16] is shown in (a) by the black long-dash-dotted line. The prediction from the dispersive analysis by the Bern group [21] is shown in (a) by the magenta long-dashed line with an error band. The prediction based on the extended chiral Khuri-Treiman formalism [22] is shown in (a) by the black dotted line. The fit of the combined $z$ distribution with the leading-order term (fit no. 2 in Table 1) is shown in (b) by the cyan long-dashed line. The fits of the Dalitz-plot sextant with Eq. (4), namely nos. 4 and 6 from Table 1, are shown in (b) by the yellow solid and the green dashed lines, respectively. The isospin-limit results from fitting both the present $\eta \to 3{\pi }^0$ and KLOE's $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ [14] data within the NREFT framework from Ref. [35] are shown by the blue dash-dotted line. The isospin-breaking results from fitting solely the $\eta \to 3{\pi }^0$ data within the same NREFT framework are shown by the red dotted line.

Figure 6

Ratios of the experimental $m\left({\pi }^0{\pi }^0\right)$ invariant-mass distributions to phase space obtained (a) individually from Run I (blue circles) and Run II (red triangles), and (b) the combined results (black triangles). The earlier A2 data from Ref. [11] are depicted in (a) by green open squares. The NREFT calculation by the Bonn group [16] is shown in (a) by the black long-dash-dotted line. The prediction from the dispersive analysis by the Bern group [21, 33] is shown in (a) by the magenta long-dashed line. The prediction based on the extended chiral Khuri-Treiman formalism [22] is shown in (a) by the black dotted line. The combined $m\left({\pi }^0{\pi }^0\right)$ distribution is compared in (b) to the results of fitting a sextant (${30}^{\circ }<\varphi <{90}^{\circ }$) of the acceptance-corrected $\eta \to 3{\pi }^0$ Dalitz plot with the density function of Eq. (4): fits no. 1 (cyan long-dashed line), no. 4 (yellow solid line), and no. 6 (green dashed line) in Table 1. The isospin-limit results from fitting both the present $\eta \to 3{\pi }^0$ and KLOE's $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ [14] data within the NREFT framework from Ref. [35] are shown by the blue dash-dotted line. The isospin-breaking results from fitting solely the $\eta \to 3{\pi }^0$ data within the same NREFT framework are shown by the red dotted line.