Observation of $J/psito gammaetapi^{0}$

We present the first study of the process $J/psi rightarrow gammaetapi^{0}$ using $(223.7pm1.4)times10^{6}$ $J/psi$ events accumulated with the BESIII detector at the BEPCII facility. The branching fraction for $J/psi rightarrow gammaetapi^{0}$ is measured to be $mathcal{B}(J/psi rightarrow gammaetapi^{0}) =(2.14pm0.18(stat)pm0.25(syst))times10^{-5}$. With a Bayesian approach, the upper limits of the branching fractions $mathcal{B}(J/psi rightarrow gamma a_{0}(980),a_0(980)rightarrowetapi^0)$ and $mathcal{B}(J/psi rightarrow gamma a_{2}(1320),a_2(1320)rightarrowetapi^0)$ are determined to be $2.5times10^{-6}$ and $6.6times10^{-6}$ at the 95% confidence level, respectively. All of these measurements are given for the first time.