K$ (or $K \pi \bar K$) channel alone. In the next section we address the production of the $\pi a_0(980)$ channel.

![The $K\pi\bar K$ production cross section from Eq. showing detail of the dashed line in Fig \[Fig:4\].[]{data-label="Fig:4bis"}](fig4bis.eps "fig:"){width="50.00000%"}\

The $\pi a_0(980)$ decay mode of the $f_1(1285)$
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This problem was also addressed in Ref. [@Aceti:2015zva] but at the peak of the $f_1(1285)$. Now we extend it to higher energies according to Eq. (\[Eq:5\]), but replacing the width $\Gamma_{R,23}$ by $\Gamma_{f_1(1285),\pi a_0(980)}$. Following Ref. [@Aceti:2015zva] and sticking to the simplified $K^* \bar K$ decay of the former section, we have to look at the diagram of Fig. \[Fig:5\], where we consider the $a_0(980)$ decay into the $\pi^0 \eta$ channel.

![Triangle diagram leading to the production of $\pi a_0(980)$, the latter is observed in $\pi^0 \eta$. In brackets the momenta of the particles.[]{data-label="Fig:5"}](fig5.eps "fig:"){width="40.00000%"}\

The amplitude for the diagram of Fig. \[Fig:5\] for the $f_1(1285)$ at rest ($\vec{P}=0$) is given by $$\label{Eq:14}
  t_{f_1, \pi \pi^0 \eta} = g_{f_1, K^* \bar K} \, g_{K^*, K \pi} \, t_T \, t_{K \bar K \to \pi^0 \eta} (\widetilde{m}_{\rm inv}),$$ with $\widetilde{m}_{\rm inv}$ the $\pi^0 \eta$ invariant mass, and where $t_T$ stands for the triangle 