Scattering amplitudes versus potentials in nuclear effective field theory: is there a potential compromise?

In effective field theory physical quantities, in particular observables, are expressed as a power series in terms of a small expansion parameter. For non-perturbative systems, for instance nuclear physics, this requires the non-perturbative treatment of at least a part of the interaction (or the potential, if we are dealing with a non-relativistic system). This is not entirely trivial and as a consequence different interpretations on how to treat these systems have appeared. A practical approach is to expand the effective potential, where this potential is later fully iterated in the Schrödinger equation for obtaining amplitudes and observables. The expectation is that this will lead to observables that will have an implicit power counting expansion. Here we explicitly check whether the amplitudes are actually following the same power counting as the potential. It happens that reality does not necessarily conform to expectations and the amplitudes will often violate the power counting with which the potential has been expanded. A more formal approach is to formulate the expansion directly in terms of amplitudes and observables, which is the original aim of the effective field theory idea. Yet this second approach is technically complicated. We explore here the possibility of constructing potentials that when fully iterated will make sure that amplitudes indeed are expansible in terms of a small expansion parameter.