Euclidean Approach to the Bethe-Salpeter Equation for Scattering

It is shown that the mass-shell scattering amplitude in the physical scattering region (two-particle branch cut) can be directly obtained from the resolvent of the Bethe-Salpeter kernel transformed according to Wick (rotation of the energy integration paths to the imaginary axis). As an illustration and under the assumption that the kernel is approximated by any finite set of irreducible graphs, the scattering amplitude in ${\phi }^3$ theory is given in terms of a Fredholm formula whose convergence is explicitly demonstrated.