Measurement of branching fractions of $\psi \left(3686\right)\to \varphi {\eta }^{\prime }$, $\varphi {f}_{1}\left(1285\right)$ and $\varphi \eta \left(1405\right)$

Using a sample of $448.1×{10}^{6}\text{}\text{}\psi \left(3686\right)$ events collected with the BESIII detector, we perform a study of the decay $\psi \left(3686\right)\to \varphi {\pi }^{+}{\pi }^{-}\eta $. The branching fraction of $\psi \left(3686\right)\to \varphi {\eta }^{\prime }$ is determined to be $\left(1.51±0.16±0.12\right)×{10}^{-5}$, which is consistent with the previous measurement but with significantly improved precision. The resonances ${f}_{1}\left(1285\right)$ and $\eta \left(1405\right)$ are clearly observed in the ${\pi }^{+}{\pi }^{-}\eta $ mass spectrum with statistical significances of $18\sigma $ and $9.7\sigma $, respectively. The corresponding product branching fractions are measured to be $B\left(\psi \left(3686\right)\to \varphi {f}_{1}\left(1285\right)$, ${f}_{1}\left(1285\right)\to {\pi }^{+}{\pi }^{-}\eta \right)=\left(1.03±0.10±0.09\right)×{10}^{-5}$ and $B\left(\psi \left(3686\right)\to \varphi \eta \left(1405\right)$, $\eta \left(1405\right)\to {\pi }^{+}{\pi }^{-}\eta \right)=\left(8.46±1.37±0.92\right)×{10}^{-6}$. These results are used to test the perturbative QCD "12% rule."