Improved measurement of the branching fraction of $h_{c}\rightarrowγη^\prime/η$ and search for $h_{c}\rightarrowγπ^0$

The processes $h_c\toγP(P = η^\prime,~η,~π^0)$ are studied with a sample of $(27.12\pm0.14)\times10^{8}$ $ψ(3686)$ events collected by the BESIII detector at the BEPCII collider. The decay $h_{c}\rightarrowγη$ is observed for the first time with the significance of $9.0\,σ$, and the branching fraction is determined to be $(3.77\pm0.55\pm0.13\pm0.26)\times10^{-4}$, while $\mathscr{B}(h_{c}\rightarrowγη^\prime)$ is measured to be $(1.40\pm0.11\pm0.04\pm0.10)\times10^{-3}$, where the first uncertainties are statistical, the second systematic, and the third from the branching fraction of $ψ(3686)\rightarrowπ^{0}h_c$. The combination of these results allows for a precise determination of $R_{h_c}=\frac{\mathscr{B}(h_c\rightarrowγη)}{\mathscr{B}(h_c\rightarrowγη^\prime)}$, which is calculated to be $(27.0\pm4.4\pm1.0)\%$. The results are valuable for gaining a deeper understanding of $η-η^\prime$ mixing, and its manifestation within quantum chromodynamics. No significant signal is found for the decay $h_c\rightarrowγπ^{0}$, and an upper limit is placed on its branching fraction of $\mathscr{B}(h_c\rightarrowγπ^{0})<5.0\times10^{-5}$, at the 90$\%$ confidence level.