In local quantum field theories, conserved current operators are obtained by Neother’s theorem. However, this is not the case with effective theories with non-local interactions which are obtained after some of the degrees of freedom are integrated out. Consider for a instance, a system made of a proton and a neutron interacting through one pion exchange potential. The electromagnetic charge is carried by the proton and the pion exchanged between proton and neutron. Since the charged pion exists only in the instantaneous potential, pionic contribution to the conserved current (exchange current) is missing . Since HAL QCD potentials are obtained by integrating out degrees of freedom of quarks and gluons and leaving only two particular hadrons, similar problems exists. In this talk, as a first attempt, we consider a secondary quantized non-relativistic two-channel coupling model to give an explicit example to solve this problem. The HAL QCD potential is obtained by integrating out the closed channel in the elastic two-particle scattering region. We derive the formula to calculate the current matrix elements by demanding the effective quantum mechanics defined by the HAL QCD potential responds to the external field in the same way as the original two-channel coupling model.

Hadron Structure