Short-range Interaction and Nonrelativistic Phi**4 Theory in Various Dimensions

Abstract

We employ the effective field theory method to systematically study the short-range interaction in two-body sector in 2, 3 and 4 spacetime dimensions, respectively. The phi**4 theory is taken as a specific example and matched onto the nonrelativistic effective theory to one loop level. An exact, Lorentz-invariant expression for the S-wave amplitude is presented, from which the nonperturbative information can be easily extracted. We pay particular attention to the renormalization group analysis in the 3 dimensions, and show that relativistic effects qualitatively change the renormalization group flow of higher-dimensional operators. There is one ancient claim that triviality of the 4-dimensional phi**4 theory can be substantiated in the nonrelativistic limit. We illustrate that this assertion arises from treating the interaction between two nonrelativistic particles as literally zero-range, which is incompatible with the Uncertainty Principle. The S-wave effective range in this theory is identified to be approximately 16/3pi times the Compton wavelength.