A reanalysis of the BESIII experimental data [@BESIII:2012aa] from this new perspective would be most interesting.

The singularity of the triangle diagram
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The shoulder for the $f_1(1285)$ decay to $\pi\pi^0\eta$ around 1400 MeV has its origin in the singularity that the triangle diagram of Fig. \[Fig:5\] develops at 1420 MeV. Triangular singularities were studied by Landau in Ref. [@Landau:1959fi] and stem from triangle diagrams when the intermediate particles are all placed on-shell and the momenta are parallel ($-\vec{q}$ and $\vec{k}$ of Fig. \[Fig:5\] go in the same direction). There is also a further constraint which is that the mechanism can lead to a classical process with $R\to K^*\bar K$; $K^*\to K\pi$; $K\bar K\to a_0(980)$, which is stated by the Coleman-Norton theorem [@Coleman:1965xm]. A pedagogical and practically easy way to visualize it, and see where the triangular singularity appears, is shown in Ref. [@Bayar:2016ftu] (see Eq. (18) of that paper).

In the present case we can see the origin of the singularities by looking at the expression of Eq. (\[Eq:17\]). The two factors in the denominator that develop singularities are $$\label{Eq:22}
D_1=P^0-\omega^{*}(\vec{q}\,)-\omega(\vec{q}\,)+i\epsilon$$ and $$\label{Eq:23}
D_2=P^0-k^0-\omega(\vec{q}\,)-\omega(\vec{q}+\vec{k}\,)+i\epsilon\, .$$

When they are zero with $\vec q$ and $\vec k$ in opposite directions, one 