Consistency between renormalization group running of chiral operator and counting rule -- Case of chiral pion production operator --

摘要

In nuclear chiral perturbation theory (ChPT), an operator is defined in aspace with a cutoff which may be varied within a certain range. The operatorruns as a result of the variation of the cutoff [renormalization group (RG)running]. In order for ChPT to be useful, the operator should run in a wayconsistent with the counting rule; that is, the running of chiral counter termshave to be of natural size. We vary the cutoff using the Wilsonianrenormalization group (WRG) equation, and examine this consistency. As anexample, we study the s-wave pion production operator for NN\to d pi, derivedin ChPT. We demonstrate that the WRG running does not generate anychiral-symmetry-violating (CSV) interaction, provided that we start with anoperator which does not contain a CSV term. We analytically show how thecounter terms are generated in the WRG running in case of the infinitesimalcutoff reduction. Based on the analytic result, we argue a range of the cutoffvariation for which the running of the counter terms is of natural size. Then,we numerically confirm this.