Figure 1

The Belle and Crystal Ball data for the $\gamma \gamma \to {\pi }^0\eta$ cross section. $\theta$ denotes the output angle of the ${\pi }^0$ (or $\eta$) in the $\gamma \gamma$ center-of-mass system. The average statistical error of the Belle data is approximately $\pm 0.4\mathrm{nb}$, the shaded band shows the size of their systematic error. The solid, dashed, and dotted lines correspond to the total, helicity 0, and $S$ wave $\gamma \gamma \to {\pi }^0\eta$ cross sections caused by the elementary $\rho$ and $\omega$ exchanges for $|\cos \theta |\le 0.8$.Reuse & Permissions

Figure 2

Diagrammatical representation for the helicity amplitudes $\gamma \gamma \to {\pi }^0\eta$.Reuse & Permissions

Figure 3

The Born $\rho$, $\omega$, $K^{*}$, $K$ exchange diagrams for $\gamma \gamma \to {\pi }^0\eta$, $\gamma \gamma \to {\pi }^0{\eta }^{\prime }$, and $\gamma \gamma \to K\overline{K}$.Reuse & Permissions

Figure 4

Fits to the Belle data. The curves in (a), (b), and (c) correspond to variant 1. (a) The total cross section $\gamma \gamma \to {\pi }^0\eta$ (solid line 1), its helicity 0 (solid line 2), and helicity 2 (dotted line) parts, the contribution from the $\gamma \gamma \to K^{+}{K}^{-}\to {\pi }^0\eta$ rescattering with intermediate $K^{+}{K}^{-}$ produced via the Born $K$ exchange (solid line 3), the contribution from the $\gamma \gamma \to K\overline{K}\to {\pi }^0\eta$ rescattering with intermediate $K\overline{K}$ produced via the Born $K$ and $K^{*}$ exchanges (dashed line), the contribution from the Born $\rho$ and $\omega$ exchange amplitude with $\lambda =0$ (dot-dashed line), and the combined contribution from the Born one and the $S$ wave $\gamma \gamma \to ({\pi }^0\eta +{\pi }^0{\eta }^{\prime })\to {\pi }^0\eta$ rescattering (solid line 4); see Figs. 2, 3. (b) The solid lines 1 and 2 are the same as in (a), the short-dashed line corresponds to the contribution from the $\gamma \gamma \to {\pi }^0\eta$ direct transition amplitude for the $a_0-a_0^{\prime }$ resonance complex [see Eq. (10)], the dotted line shows the total contribution from the $a_0-a_0^{\prime }$ resonance one, and the long-dashed line corresponds to the helicity 0 cross section without the contribution from the direct transition amplitude. The resulting solid line in (c) corresponds to the solid line 1 in (a) [or (b)], folded, in the region $0.84\mathrm{GeV}<\sqrt{s}<1.15\mathrm{GeV}$, with a Gaussian with $\sigma =10\mathrm{MeV}$ mass resolution; the shaded band shows the size of the systematic error of the data. (d), (e), and (f) The same as (c), (a), and (b), respectively, for variant 2.Reuse & Permissions

Figure 5

Modulus of $T_0^1(s)$, inelasticity ${\eta }_0^1(s)$ (a), and phase shifts (b) of the $S$ wave amplitude ${\pi }^0\eta \to {\pi }^0\eta$ for variant 1 (here $a_0^1=0.0098$); (c) and (d) the same for variant 2 (here $a_0^1=0.0066$).Reuse & Permissions