Amplitude Analysis of the Isospin-Violating Decay $J\mathrm{/}\psi \to \gamma \eta {\pi }^0$

Using $(10087\pm 44)\times {10}^6$ $J\mathrm{/}\psi$ events collected with the BESIII detector, we perform the first amplitude analysis of the process $J\mathrm{/}\psi \to \gamma \eta {\pi }^0$. The decay is dominated by the intermediate processes $J\mathrm{/}\psi \to {\pi }^0{b}_1(1235{)}^0\to \gamma \eta {\pi }^0$, $J\mathrm{/}\psi \to {\pi }^0\rho (1450{)}^0\to \gamma \eta {\pi }^0$ and $J\mathrm{/}\psi \to \eta h_1(1170)\to \gamma \eta {\pi }^0$. Contributions from $J\mathrm{/}\psi \to \gamma a_0(980{)}^0\to \gamma \eta {\pi }^0$, $J\mathrm{/}\psi \to \gamma a_2(1320{)}^0\to \gamma \eta {\pi }^0$ and $J\mathrm{/}\psi \to \gamma a_2(1700{)}^0\to \gamma \eta {\pi }^0$ are observed with a statistical significance exceeding $5\sigma$, constituting the first observation of radiative transitions of $J\mathrm{/}\psi$ to isospin-triplet scalar mesons. The total branching fraction of $J\mathrm{/}\psi \to \gamma \eta {\pi }^0$ is measured to be $(25.7\pm 0.3\pm 1.5)\times {10}^{-6}$, where the first uncertainty is statistical and the second systematic. This result is consistent with the previous measurement, with the precision improved by more than a factor of two.