Study of the process $e^+e^-\to\pi^0\pi^0 J/\psi$ and neutral charmonium-like state $Z_c(3900)^0$

Cross sections of the process $e^+e^-\to\pi^0\pi^0 J/\psi$ at center-of-mass energies between 3.808 and 4.600 GeV are measured with high precision by using 12.4 $fb^{-1}$ of data samples collected with the BESIII detector operating at the BEPCII collider facility. A fit to the measured energy-dependent cross sections confirms the existence of the charmonium-like state $Y(4220)$. The mass and width of the $Y(4220)$ are determined to be $(4220.4\pm2.4\pm2.3)$ ${\rm MeV}/c^{2}$ and $(46.2\pm4.7\pm2.1)$ MeV, respectively, where the first uncertainties are statistical and the second systematic. The mass and width are consistent with those measured in the process $e^+e^-\to\pi^+\pi^- J/\psi$. The neutral charmonium-like state $Z_c(3900)^0$ is observed prominently in the $\pi^0 J/\psi$ invariant-mass spectrum, and, for the first time, an amplitude analysis is performed to study its properties. The spin-parity of $Z_c(3900)^0$ is determined to be $J^{P}=1^{+}$, and the pole position is $(3893.1\pm2.2\pm3.0)-i(22.2\pm2.6\pm7.0)$ ${\rm MeV}/c^2$, which is consistent with previous studies of electrically charged $Z_c(3900)^\pm$. In addition, cross sections of $e^+e^-\to\pi^0 Z_c(3900)^0\to \pi^0\pi^0 J/\psi$ are extracted, and the corresponding lineshape is found to agree with that of the $Y(4220)$.