The effect of decay cascade via an intermediate resonance in the $\gamma p \to \pi^0 \eta p $ reaction

The $\gamma p \to \pi^0 \eta p$ reaction has been investigated by the CBELSA/TAPS Collaboration, revealing a narrow structure in the $\eta p$ invariant mass distributions at a mass of $1700$ MeV. In this study, we explore the possibility that the narrow structure is caused by a decay cascade via an intermediate nucleon resonance decaying to $\eta p$ final states. The candidates for the intermediate nucleon resonances are $N(1700)3/2^{-}$ and $N(1710)1/2^{+}$, with masses near the observed structure. We consider the $t$-channel $\rho$- and $\omega$-exchange diagrams, the $u$-channel nucleon-pole exchange diagram, the contact term, and the $s$-channel pole diagrams of nucleon, $\Delta$, and nucleon resonances when constructing the reaction amplitudes to reproduce the stripped individual contribution of the narrow structure. Our analysis indicates that the signature strength of the decay cascade $\gamma p \to \pi^{0}N(1700)3/2^{-} \to \pi^{0}\eta p$ is too weak to reach the experimental curve of the narrow structure due to the small decay branching ratio of $N(1700)3/2^{-}$ to $\eta p$. Although the decay cascade $\gamma p \to \pi^{0}N(1710)1/2^{+} \to \pi^{0}\eta p$ can qualitatively reproduce the experimental curve of the invariant mass distributions, its cross-section width is much larger than that of the corresponding experimental curve. Therefore, we conclude that the decay cascade via an intermediate nucleon resonance could not be the reason leading to the narrow structure in the $\eta p$ invariant mass distributions of the $\gamma p \to \pi^0 \eta p$ reaction.