Coupled-channel analysis of the near-threshold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretch

The possible existence of nucleon-antinucleon bound states has been studied for decades. We investigate the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>e</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>e</a:mi><a:mo>−</a:mo></a:msup><a:mo stretchy="false">→</a:mo><a:mi>p</a:mi><a:mover accent="true"><a:mi>p</a:mi><a:mo stretchy="false">¯</a:mo></a:mover></a:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msup><f:mi>e</f:mi><f:mo>+</f:mo></f:msup><f:msup><f:mi>e</f:mi><f:mo>−</f:mo></f:msup><f:mo stretchy="false">→</f:mo><f:mi>n</f:mi><f:mover accent="true"><f:mi>n</f:mi><f:mo stretchy="false">¯</f:mo></f:mover></f:math> 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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>p</k:mi></k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mover accent="true"><m:mi>p</m:mi><m:mo stretchy="false">¯</m:mo></m:mover></m:math> are considered, and the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>N</q:mi><q:mover accent="true"><q:mi>N</q:mi><q:mo stretchy="false">¯</q:mo></q:mover></q:math> 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 <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msup><u:mi>e</u:mi><u:mo>+</u:mo></u:msup><u:msup><u:mi>e</u:mi><u:mo>−</u:mo></u:msup><u:mo stretchy="false">→</u:mo><u:mi>N</u:mi><u:mover accent="true"><u:mi>N</u:mi><u:mo stretchy="false">¯</u:mo></u:mover></u:math> cross sections from the BESIII and SND collaborations, a <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>N</z:mi><z:mover accent="true"><z:mi>N</z:mi><z:mo stretchy="false">¯</z:mo></z:mover></z:math> quasibound state is found just above the <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>p</db:mi><db:mover accent="true"><db:mi>p</db:mi><db:mo stretchy="false">¯</db:mo></db:mover></db:math> threshold, and another <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mi>N</hb:mi><hb:mover accent="true"><hb:mi>N</hb:mi><hb:mo stretchy="false">¯</hb:mo></hb:mover></hb:math> 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 <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:mi>N</lb:mi><lb:mover accent="true"><lb:mi>N</lb:mi><lb:mo stretchy="false">¯</lb:mo></lb:mover></lb:math> thresholds.