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Using a data sample of $(10087\pm44)\times 10^6$ $J/\psi$ events collected bythe BESIII detector in 2009, 2012, 2018 and 2019, the electromagnetic Dalitzprocess $J/\psi \to e^+ e^- \eta(1405)$ is observed via the decay $\eta(1405)\to \pi^0 f_0(980)$, $f_0(980) \to \pi^+ \pi^-$, with a significance of about$9.6\sigma$. The branching fraction of this decay is measured to be ${\mathcalB}(J/\psi \to e^+ e^- \pi^0 \eta(1405) \to e^+ e^- \pi^0 f_0(980) \to e^+ e^-\pi^0 \pi^+ \pi^-)=(2.02\pm0.24(\rm{stat.})\pm0.09(\rm{syst.}))\times 10^{-7}$.The branching-fraction ratio ${\mathcal B}(J/\psi \to e^+ e^-\eta(1405))$/${\mathcal B}(J/\psi \to \gamma \eta(1405))$ is determined to be$(1.35\pm0.19(\rm{stat.})\pm0.06(\rm{syst.}))\times10^{-2}$. Furthermore, an$e^+e^-$ invariant-mass dependent transition form factor of $J/\psi \to e^+e^-\eta(1405)$ is presented for the first time. The obtained result providesinput for different theoretical models, and is valuable for the improvedunderstanding the intrinsic structure of the $\eta(1405)$ meson.

Observation of the decay $J/\psi \to e^+ e^- \eta(1405)$ with $\eta(1405) \to \pi^0 f_0(980)$