the end. This will also allow us to go below the threshold of $K^* \bar K$ production.

To account for the decay, we look into the diagram of Fig. \[Fig:3\].

![Decay diagram for $f_1(1285) \to K^* \bar K$ considering the $K \pi$ decay channel of the $K^*$.[]{data-label="Fig:3"}](fig3.eps "fig:"){width="40.00000%"}\

The decay width of the $f_1(1285)$ for Fig. \[Fig:3\] is given by $$\label{Eq:9}
   \Gamma_{f_1, K^* \bar K \to K \pi \bar K } \equiv
   \int \frac{\mathrm{d} \Gamma_{f_1, K \pi \bar K}}{\mathrm{d}m_{\rm inv}}\mathrm{d}m_{\rm inv},$$ where $m_{\rm inv}$ stands for the $K \pi$ invariant mass, and $$\label{Eq:10}
   \frac{\mathrm{d} \Gamma_{f_1, K \pi \bar K}}{\mathrm{d}m_{\rm inv}} =
   \frac{1}{(2\pi)^3}\frac{1}{4M_{\rm inv}^2} \, p_{\bar K} \, \widetilde{p}_\pi \, \overline{\sum} \, \sum \, |t'|^2,$$ where $p_{\bar K}$ is the $\bar K$ momentum in the $f_1(1285)$ rest frame and $\widetilde{p}_\pi$ the $\pi$ momentum in the $K \pi$ rest frame, and $t'$ contains now the $K^*$ propagator, the $f_1(1285) \to K^* K$ and $K^* \to K \pi$ couplings. There is no need to evaluate $t'$ explicitly since we can use the same step that led us from Eq. (\[Eq:1\]) to Eq. (\[Eq:5\]) and we can write $$\begin{aligned}
\label{Eq:11}
   \nonumber \frac{\mathrm{d} \Gamma_{f_1, K \pi \bar K}}{\mathrm{d}m_{\rm inv}} &=
   \frac{1}{(2\pi)^3}\frac{1}{4M_{\rm inv}^2} \, g_{f_1,K^* \bar 