loop integral with three propagators,

$$\label{Eq:15}
  t_T = i \int \frac{\mathrm{d}^4q}{(2\pi)^4}\, \vec{\epsilon}_{f_1}  \cdot  \vec{\epsilon}_{K^*}\;
  \vec{\epsilon}_{K^*} \cdot (2\vec{k} + \vec{q}) \,
  \frac{1}{q^2 - m_K^2 +i\epsilon}\,
  \frac{1}{(P-q)^2 - m_{K^*}^2 +i m_{K^*} \Gamma_{K^*}}\,
  \frac{1}{(P-q-k)^2 - m_K^2 +i\epsilon} \, ,$$

and $t_{K \bar K \to \pi^0 \eta}$ is evaluated using the chiral unitary approach of Ref. [@Oller:1997ti], with the input used in the study of weak decays of $B$ and $D$ mesons in Refs. [@Liang:2014tia; @Xie:2014tma].

The integral of $q^0$ in Eq. (\[Eq:15\]) can be done analytically and it is done in Ref. [@Aceti:2015zva] with the result

$$\label{Eq:16}
  t_T = \widetilde{t}_T \, \vec{\epsilon}_{f_1} \cdot \vec{k} ,$$

and

$$\begin{aligned}
\label{Eq:17}
 \nonumber \widetilde{t}_T &= \int \frac{\mathrm{d}^3q}{(2\pi)^3}
 \left(2 + \frac{\vec{k} \cdot \vec{q}}{\vec{k}^2} \right)\,
 \frac{1}{8\, \omega(q) \omega^{\prime}(q) \omega^*(q)}\,
  \frac{1}{k^0 -\omega^{\prime}(q) -\omega^{*}(q) +i\epsilon}\,
  \frac{1}{P^0 -\omega^*(q) -\omega(q) +i\epsilon}\\
&\times
\frac{2P^0 \omega(q) +2k^0\omega^{\prime}(q)
-2(\omega(q) +\omega^{\prime}(q))
(\omega(q) +\omega^{\prime}(q) +\omega^{*}(q))}{(P^0 -\omega(q) -\omega^{\prime}(q) -k^0+i\epsilon) (P^0+\omega(q)+ \omega^{\prime}(q) -k^0 -i\epsilon)}\, ,\end{aligned}$$

with $\omega(q) = \sqrt{\vec{q}^{\,2} +m_K^2}$, $\omega^{\prime}(q) = \sqrt{(\vec{q}+\vec{k})^2 +m_K^2}$, $\omega^{*}(q) = \sqrt{\vec{q}^{\,2} +m_{K^*}^2}$ , (or $\sqrt{\vec{q}^{\,2} +m_{K^*}^2} - i \Gamma_{K^*}/2$ if the $K^*$ width is considered as in Eq. (\[Eq:15\])). The width $\Gamma_{f_1(1285),\pi\pi^0\eta}$ is given by $$\label{Eq:18}
\Gamma_{f_1,\pi\pi^0\eta}=\int{\frac{\mathrm{d}\Gamma_{f_1,\pi\pi^0\eta}}{\mathrm{d}\tilde{m}_{\rm inv}}\mathrm{d}\tilde{m}_{\rm inv}}\, ,$$ with $\mathrm{d}\Gamma_{f_1,\pi\pi^0\eta}/\mathrm{d}\tilde{m}_{\rm inv}$ obtained using Eq. (\[Eq:10\]) with $m_{\rm inv}\to \tilde{m}_{\rm inv}$ (of $\pi^0\eta$), 