Measurement of the $ \eta $ transition form factor through the $ {\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\eta $ decay

Based on a sample of $ (1.0087\pm 0.0044)\times {10}^{10}\text{}\text{}J/\psi $ events collected at BESIII, the transition form factor of the $ \eta $ meson is extracted by analyzing $ J/\psi \to \gamma {\eta }^{\prime },{\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\eta $ and $ \eta \to \gamma {l}^{+}{l}^{-}$ ($ l=e$, $ \mu $) events. The measured slope of the transition form factor is $ {\mathrm{\Lambda }}^{-2}=1.668\pm 0.09{3}_{\mathrm{stat}}\pm 0.02{4}_{\mathrm{sys}}\text{}\text{}\text{}(\mathrm{GeV}/{c}^{2}{)}^{-2}$ for the di-electron channel and $ {\mathrm{\Lambda }}^{-2}=1.645\pm 0.34{3}_{\mathrm{stat}}\pm 0.01{7}_{\mathrm{sys}}\text{}\text{}(\mathrm{GeV}/{c}^{2}{)}^{-2}$ for the di-muon channel. The branching fractions for $ \eta \to \gamma {e}^{+}{e}^{-}$ and $ \eta \to \gamma {\mu }^{+}{\mu }^{-}$ are measured to be $ \mathcal{B}(\eta \to \gamma {e}^{+}{e}^{-})=(6.79\pm 0.0{5}_{\mathrm{stat}}\pm 0.3{6}_{\mathrm{sys}})\times {10}^{-3}$ and $ \mathcal{B}(\eta \to \gamma {\mu }^{+}{\mu }^{-})=(2.97\pm 0.1{2}_{\mathrm{stat}}\pm 0.0{7}_{\mathrm{sys}})\times {10}^{-4}$. By combining with the results based on the $ J/\psi \to \gamma \eta $ and $ \eta \to \gamma {e}^{+}{e}^{-}$ events from the previous BESIII measurement, we determine $ {\mathrm{\Lambda }}^{-2}=1.707\pm 0.07{6}_{\mathrm{stat}}\pm 0.02{9}_{\mathrm{sys}}\text{}\text{}(\mathrm{GeV}/{c}^{2}{)}^{-2}$ and $ \mathcal{B}(\eta \to \gamma {e}^{+}{e}^{-})=(6.93\pm 0.2{8}_{\mathrm{tot}})\times {10}^{-3}$. In addition, we search for the dark photon ($ {A}^{\prime }$) using the combined events. No significant signal is observed, and the upper limits on $ \mathcal{B}(\eta \to \gamma {A}^{\prime },{A}^{\prime }\to {e}^{+}{e}^{-})$ are set at the 90% confidence level for different $ {A}^{\prime }$ mass hypotheses.