Measurement of the $η$ transition form factor through $η' \rightarrow π^+π^-η$ decay
Based on a sample of $(1.0087\pm0.0044)\times10^{10}$ $J/ψ$ events collected at BESIII, the transition form factor of the $η$ meson is extracted by analyzing $J/ψ\toγη',~η'\toπ^+π^-η,~η\toγl^+l^-$ ($l$=$e$, $μ$) events. The measured slope of the transition form factor is $Λ^{-2}=1.645\pm0.093_{\rm stat.}\pm {0.024_{\rm sys.}}$ (GeV/$c^2$)$^{-2}$ for the di-electron channel and $Λ^{-2}=1.645\pm0.343_{\rm stat.}\pm0.017_{\rm sys.}$ (GeV/$c^2$)$^{-2}$ for the di-muon channel. The branching fractions for $η\rightarrowγe^+e^-$ and $η\rightarrowγμ^+μ^-$ are measured to be $\mathcal{B}(η\toγe^+e^-)=(6.79\pm0.04_{\rm stat.}\pm0.36_{\rm sys.})\times 10^{-3}$ and $\mathcal{B}(η\toγμ^+μ^-)=(2.97\pm0.11_{\rm stat.}\pm0.07_{\rm sys.})\times 10^{-4}$. By combining with the results based on the $J/ψ\toγη,~η\toγe^+e^-$ events from the previous BESIII measurement, we determine $Λ^{-2}=1.707\pm0.076_{\rm stat.}\pm0.029_{\rm sys.}$ (GeV/$c^2$)$^{-2}$ and $\mathcal{B}(η\toγe^+e^-)=(6.93\pm0.28_{\rm tot.})\times 10^{-3}$. In addition, we search for the dark photon ($A'$) using the combined events. No significant signal is observed, and the upper limits on $\mathcal{B}(η\toγA',~A'\to e^+e^-)$ are set at 90\% confidence level for different $A'$ mass hypotheses.