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Short-range Interaction and Nonrelativistic φ 4 Theory in Various Dimensions

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 φ 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 renormal-ization group flow of higher-dimensional operators. There is one ancient claim that triviality of the 4-dimensional φ 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 16/3π times the Compton wavelength.