Exploring Photoproduced $\eta^{(}{'^{)}} \pi^{0}$ Systems in the Search for Exotic Hadrons at GlueX

Physicists have long been captivated by the spectrum of hadrons, aiming to better comprehend the fundamental building blocks of matter. While various experiments have laid the foundation for this spectrum, Lattice Quantum Chromodynamics has revealed new states with forbidden $J^{PC}$ values. This has challenged the constituent quark model, suggesting spin-exotic hybrid mesons that could reshape our understanding of hadronic structure and quark-gluon interactions through gluonic excitations. The GlueX experiment at Jefferson Lab has played a pivotal role in this pursuit, with its efforts centered on analyzing photoproduction data. In order to investigate the lightest predicted exotic state with $J^{PC}=1^{-+}$, known as the $\pi_1(1600)$ meson, significant attention has been directed towards both $\eta\pi^{0}$ and $\eta' \pi^{0}$ systems. Detailed ongoing amplitude analysis studies of $\gamma p \rightarrow \eta^{(}{'^{)}}\pi^{0} p$, leveraging the polarization of the photon beam at the GlueX experiment will be highlighted. Furthermore, first studies on the moments of angular distributions for these channels will also be shown, along with the differential cross section results for the $a_2(1320)$ meson. These results aid in identifying the dominant production mechanisms and offer valuable insights into complex hadronic interactions. Ultimately, this contributes to the ongoing search for and future identification of exotic hybrid meson candidates.