Observation of the decay $J/\psi \to {e}^{+}{e}^{-}\eta \left(1405\right)$ with $\eta \left(1405\right)\to {\pi }^{0}{f}_{0}\left(980\right)$

Using a data sample of $\left(10087±44\right)×{10}^{6}\text{}\text{}J/\psi $ events collected by the BESIII detector in 2009, 2012, 2018 and 2019, the electromagnetic Dalitz process $J/\psi \to {e}^{+}{e}^{-}\eta \left(1405\right)$ is observed via the decay $\eta \left(1405\right)\to {\pi }^{0}{f}_{0}\left(980\right)$, ${f}_{0}\left(980\right)\to {\pi }^{+}{\pi }^{-}$, with a significance of about $9.8\sigma $. The branching fraction of this decay is measured to be $B\left(J/\psi \to {e}^{+}{e}^{-}\eta \left(1405\right)\to {e}^{+}{e}^{-}{\pi }^{0}{f}_{0}\left(980\right)\to {e}^{+}{e}^{-}{\pi }^{0}{\pi }^{+}{\pi }^{-}\right)=\left(2.04±0.20\left(\mathrm{stat}\right)±\phantom{\rule{0ex}{0ex}}0.08\left(\mathrm{syst}\right)\right)×{10}^{-7}$. The branching fraction ratio $B\left(J/\psi \to {e}^{+}{e}^{-}\eta \left(1405\right)\right)/B\left(J/\psi \to \gamma \eta \left(1405\right)\right)$ is determined to be $\left(1.36±0.17\left(\mathrm{stat}\right)±0.06\left(\mathrm{syst}\right)\right)×{10}^{-2}$. Furthermore, an ${e}^{+}{e}^{-}$ invariant mass dependent transition form factor of $J/\psi \to {e}^{+}{e}^{-}\eta \left(1405\right)$ is presented for the first time. The obtained result provides input for different theoretical models and is valuable for the improved understanding the intrinsic structure of the $\eta \left(1405\right)$ meson.