Current matrix element in HAL QCD's wave function equivalent potential method

We give a formula to calculate a matrix element of a conserved current in the effective quantum mechanics defined by the wave function equivalent potentials proposed by HAL QCD collaboration. As a first step, a non-relativistic field theory with two channel coupling is considered as the original theory, with which a wave function equivalent HAL QCD potential is obtained in a closed analytic form. The external field method is used to derive the formula by demanding that the result should agree with the original theory. With this formula, the matrix element is obtained by sandwiching the effective current operator between the left and the right eigen functions of the effective Hamiltonian associated with the HAL QCD potential. In addition to the naive one-body current, the effective current operator contains an additional two-body term emerging from the degrees of freedom which has been integrated out.