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

Based on a sample of $(1.0087\pm 0.0044)\times {10}^{10}$ $J\mathrm{/}\psi$ events collected at BESIII, the transition form factor of the $\eta$ meson is extracted by analyzing $J\mathrm{/}\psi \to \gamma {\eta }^{\mathrm{\prime }},{\eta }^{\mathrm{\prime }}\to {\pi }^{+}{\pi }^{-}\eta ,\eta \to \gamma l^{+}{l}^{-}$ ($l$=$e$, $\mu$) events. The measured slope of the transition form factor is ${\mathrm{\Lambda }}^{-2}=1.645\pm {0.093}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.024}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{.}}$ (GeV/$c^2$)${}^{-2}$ for the di-electron channel and ${\mathrm{\Lambda }}^{-2}=1.645\pm {0.343}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.017}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{.}}$ (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 $\mathcal{B}(\eta \to \gamma e^{+}{e}^{-})=(6.79\pm {0.04}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.36}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{.}})\times {10}^{-3}$ and $\mathcal{B}(\eta \to \gamma {\mu }^{+}{\mu }^{-})=(2.97\pm {0.11}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.07}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{.}})\times {10}^{-4}$. By combining with the results based on the $J\mathrm{/}\psi \to \gamma \eta ,\eta \to \gamma e^{+}{e}^{-}$ events from the previous BESIII measurement, we determine ${\mathrm{\Lambda }}^{-2}=1.707\pm {0.076}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}}\pm {0.029}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{.}}$ (GeV/$c^2$)${}^{-2}$ and $\mathcal{B}(\eta \to \gamma e^{+}{e}^{-})=(6.93\pm {0.28}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{.}})\times {10}^{-3}$. In addition, we search for the dark photon ($A^{\mathrm{\prime }}$) using the combined events. No significant signal is observed, and the upper limits on $\mathcal{B}(\eta \to \gamma A^{\mathrm{\prime }},A^{\mathrm{\prime }}\to e^{+}{e}^{-})$ are set at 90% confidence level for different $A^{\mathrm{\prime }}$ mass hypotheses.