Consistency between renormalization group running of the chiral operator and the counting rule—case of the chiral pion production operator

Abstract

In nuclear chiral perturbation theory (χPT), an operator is defined in a space with a cutoff which may be varied within a certain range. The operator runs as a result of the variation of the cutoff (renormalization group (RG) running). In order for χPT to be useful, the operator should run in a way consistent with the counting rule, that is, the running of chiral counter terms has to be of natural size. We vary the cutoff using the Wilsonian renormalization group (WRG) equation and examine this consistency. As an example, we study the s-wave pion production operator for NN → dπ, derived in χPT. We demonstrate that the WRG running does not generate any chiral-symmetry-violating (CSV) interaction, provided that we start with an operator which does not contain a CSV term. We analytically show how the counter terms are generated in the WRG running in the case of the infinitesimal cutoff reduction. Based on the analytic result, we argue a range of the cutoff variation for which the running of the counter terms is of natural size. Then, we numerically confirm this.