Power counting with one-pion exchange

Abstract

Techniques developed for handing inverse-power-law potentials in atomic physics are applied to the tensor one-pion exchange potential to determine the regions in which it can be treated perturbatively. In $S,P$, and $D$ waves, the critical values of the relative momentum are less than or of the order of 400 MeV. The Wilsonian renormalization group (RG) is then used to determine the power counting for short-range interaction in the presence of this potential. In the $P$ and $D$ waves, where there are no low-energy bound or virtual states, these interactions have half-integer RG eigenvalues and are substantially promoted relative to naive expectations. These results are independent of whether the tensor force is attractive or repulsive. In the ${}^{3}{S}_{1}$ channel, the leading term is relevant, but it is demoted by half an order compared to the counting for the effective-range expansion with only a short-range potential. The tensor force can be treated perturbatively in those $F$ waves and above that do not couple to $P$ or $D$ waves. The corresponding power counting is the usual one given by naive dimensional analysis.