Building light nuclei from neutrons, protons, and pions

An effective theory is a systematic approximation to some underlying dynamics (which may be known or unknown) that is valid in some specified regime. An effective theory is not a model, since its systematic character means that, in principle, predictions of arbitrary accuracy may be made. However, if this is to be true then a small parameter, such as the α of quantum electrodynamics, must govern the systematic approximation scheme. As we shall see here, in many modern effective theories the expansion parameter is a ratio of two physical scales. For instance, in effective theories of supersymmetric physics “beyond the standard model” the ratio would be of p or m, the momentum or mass of a standard model particle, to MSUSY. In an effective theory for finite-proton-size effects in the hydrogen atom the small parameter would be rp, the proton size, divided by rb, the Bohr radius. The smallness of this parameter is then indicative of the domain of validity of the effective theory (ET). In this sense effective theories, like revolutions, carry the seeds of their own destruction, since the failure of the expansion to converge is a signal to the user that he or she is pushing the theory beyond its limits. Within the radius of convergence of the ET the ET “works” because of the following fundamental tenet: