Renormalizing chiral nuclear forces: Triplet channels

Abstract

We discuss the subleading contact interactions, or counterterms, of the triplet channels of nucleon-nucleon scattering in the framework of chiral effective field theory, with $S$ and $P$ waves as the examples. The triplet channels are special in that they allow the singular attraction of one-pion exchange to modify Weinberg's original power-counting (WPC) scheme. With renormalization group invariance as the constraint, our power counting for the triplet channels can be summarized as a modified version of naive dimensional analysis in which, when compared with WPC, all of the counterterms in a given partial wave (leading or subleading) are enhanced by the same amount. More specifically, this means that WPC needs no modification in ${}^{3}\phantom{\rule{-0.16em}{0ex}}{S}_{1}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{D}_{1}$ and ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}$, whereas a two-order enhancement is necessary in both ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{0}$ and ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{2}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{F}_{2}$.