Study of $ {f}_{1}(1420)$ and $ \eta (1405)$ in the decay $ J/\psi \to \gamma {\pi }^{0}{\pi }^{0}{\pi }^{0}$

A partial-wave analysis is performed on the decay $ J/\psi \to \gamma {\pi }^{0}{\pi }^{0}{\pi }^{0}$ within the $ {\pi }^{0}{\pi }^{0}{\pi }^{0}$ invariant-mass region below $ 1.6\text{}\text{}\mathrm{GeV}/{c}^{2}$, using $ (10.09\pm 0.04)\times {10}^{9}$ $ J/\psi $ events collected with the BESIII detector. Significant isospin-violating decays of $ \eta (1405)$ and $ {f}_{1}(1420)$ into $ {f}_{0}(980){\pi }^{0}$ are observed. For the first time, three axial vectors, $ {f}_{1}(1285)$, $ {f}_{1}(1420)$, and $ {f}_{1}(1510)$, are observed to decay into $ {\pi }^{0}{\pi }^{0}{\pi }^{0}$. The product branching fractions of these resonances are reported.