Improved measurement of the branching fraction of $h_c\to \gamma {\eta }^{\mathrm{\prime }}\mathrm{/}\eta$ and search for $h_c\to \gamma {\pi }^0$

The processes $h_c\to \gamma P(P={\eta }^{\mathrm{\prime }},\eta ,{\pi }^0)$ are studied with a sample of $(27.12\pm 0.14)\times {10}^8$ $\psi (3686)$ events collected by the BESIII detector at the BEPCII collider. The decay $h_c\to \gamma \eta$ is observed for the first time with the significance of $9.0\sigma$, and the branching fraction is determined to be $(3.77\pm 0.55\pm 0.13\pm 0.26)\times {10}^{-4}$, while $\mathcal{B}(h_c\to \gamma {\eta }^{\mathrm{\prime }})$ is measured to be $(1.40\pm 0.11\pm 0.04\pm 0.10)\times {10}^{-3}$, where the first uncertainties are statistical, the second systematic, and the third from the branching fraction of $\psi (3686)\to {\pi }^0{h}_c$. The combination of these results allows for a precise determination of $R_{h_c}=\frac{\mathcal{B}(h_c\to \gamma \eta )}{\mathcal{B}(h_c\to \gamma {\eta }^{\mathrm{\prime }})}$, which is calculated to be $(27.0\pm 4.4\pm 1.0)\mathrm{\%}$. The results are valuable for gaining a deeper understanding of $\eta -{\eta }^{\mathrm{\prime }}$ mixing, and its manifestation within quantum chromodynamics. No significant signal is found for the decay $h_c\to \gamma {\pi }^0$, and an upper limit is placed on its branching fraction of $\mathcal{B}(h_c\to \gamma {\pi }^0)<5.0\times {10}^{-5}$, at the 90$\mathrm{\%}$ confidence level.