Not yet reviewed by Pith; the record is open.

The possible existence of nucleon-antinucleon bound states has been studied for decades. We investigate the $e^+e^-\to p\bar{p}$ and $e^+e^-\to n\bar{n}$ cross sections in the nonrelativistic effective field theory framework. The proton-antiproton and neutron-antineutron coupled-channel final state interactions are considered and found responsible for near-threshold enhancements. Both the proton-neutron mass difference and the Coulomb interaction between $p$ and $\bar{p}$ are considered, and the $N\bar{N}$ strong interactions are taken into account through a short-distance optical potential. By fitting the low energy constants in the amplitudes to the data for the near-threshold $e^+e^-\to N\bar{N}$ cross sections from the BESIII and SND Collaborations, a $N\bar{N}$ quasi-bound state is found just above the $p\bar{p}$ threshold, and another $N\bar{N}$ pole is found on the unphysical Riemann sheet, farther away from the threshold. The constructed coupled-channel amplitude with Coulomb effects also offers a framework that can be used directly in experimental analyses on fine structures near the $N\bar{N}$ thresholds.