Renormalising NN scattering: is power counting powerless?

Abstract

The renormalisation of NN scattering in theories with zero-range interactions is examined using a cut-off regularisation where the cut-off is taken to infinity, dimensional regularisation (DR) with minimal subtraction, and DR with power-divergence subtraction. In the infinite cut-off limit power counting breaks down: terms of different orders in the potential contribute to the scattering amplitude at the same order. Minimal subtraction does yield a systematic expansion, but with a very limited range of validity for systems that have unnaturally large scattering lengths. For a finite cut-off, the behaviour of the couplings as the cut-off is lowered shows that a theory with a natural scattering length approaches an IR fixed point. In the corresponding effective theory, loop corrections can be treated perturbatively. In contrast, if there is an IR fixed point for systems with an infinite scattering length it must be a nonperturbative one, with no power counting. For such systems, power-divergence subtraction appears to yield a systematic expansion, but with a different power counting from Weinberg's. However the scheme omits IR divergent terms that would otherwise lead to nonperturbative behaviour and so the interpretation of the fixed point remains unclear.