I apologize, but QFT is not my domain. What I ask is connected with the question Do the fields exist without electric charges?Do the fields exist without electric charges? .
By analogy with the electron and proton, that carry the electric charges of the electrostatic field, could a neutron and anti-neutron represent the positive, respectively negative, charges of some field? I am sure that, if the answer is positive, the supposed field is not the gravitational one. In the gravitational field all the charges (masses) are positive.