Role of a triangle singularity in the $\pi N\left(1535\right)$ contribution to $\gamma p\to p{\pi }^0\eta$

We have studied the $\gamma p\to p{\pi }^0\eta$ reaction paying attention to the two main mechanisms at low energies, the $\gamma p\to \mathrm{\Delta }\left(1700\right)\to \eta \mathrm{\Delta }\left(1232\right)$ and the $\gamma p\to \mathrm{\Delta }\left(1700\right)\to \pi N\left(1535\right)$. Both are driven by the photoexcitation of the $\mathrm{\Delta }\left(1700\right)$ and the second one involves a mechanism that leads to a triangle singularity. We are able to evaluate quantitatively the cross section for this process and show that it agrees with the experimental determination. Yet there are some differences with the standard partial wave analysis which does not include explicitly the triangle singularity. The exercise also shows the convenience of exploring possible triangle singularities in other reactions and how a standard partial wave analysis can be extended to accommodate them.

Figure 1

Cross section for $\gamma p\to {\pi }^0\eta p$. ${\left|a\right|}^2$ stands for the mechanism of Fig. 2. ${\left|b\right|}^2$ stands for the triangle mechanism of Fig. 3. ${|a+b|}^2$ stands for the coherent sum of the two former mechanisms and ${\left|a\right|}^2+{\left|b\right|}^2$ stands for the incoherent sum.

Figure 2

Mechanism for $\gamma p\to \mathrm{\Delta }\left(1700\right)\to \eta \mathrm{\Delta }\left(1232\right)\to \eta {\pi }^{0}p$ driven by $\mathrm{\Delta }\left(1700\right)(3/2^{-})$ photoproduction.

Figure 3

Triangle diagram leading to the production of $\pi N\left(1535\right)$ ($\eta p$). In parenthesis, the momenta of the particles.

Figure 4

Standard mechanism for $\gamma p\to \pi N\left(1535\right)$ production.

Figure 5

$|t_T{|}^2$, $\mathrm{Re}\left(t_T\right)$, $\mathrm{Im}\left(t_T\right)$, and $|t_T|$ as a function of the $\gamma p$ energy $\sqrt{s}$. The mass of the $\eta p$ system is taken at the $N\left(1535\right)$ mass of 1543 MeV determined in Ref. [41].

Figure 6

Cross section for the $\gamma p\to \pi N\left(1535\right)$ with the triangle mechanism and the standard mechanism of Fig. 4.