Renormalizing Chiral Nuclear Forces: A Case Study of 3P0
  We discuss in this paper the subleading contact interactions, or counterterms, in the 3P0 channel of nucleon-nucleon scattering up to O(Q^3), where, already at leading order, Weinberg's original power counting (WPC) scheme fails to fulfill renormalization group invariance due to the singular attraction of one-pion exchange. Treating the subleading interactions as perturbations and using renormalization group invariance as the criterion, we investigate whether WPC, although missing the leading order, could prescribe correct subleading counterterms. We find that the answer is negative and, instead, that the structure of counterterms agrees with a modified version of naive dimensional analysis. Using 3P0 as an example, we also study the cutoffs where the subleading potential can be iterated together with the leading one. 
Bingwei Long and C.-J. Yang
nucl-th