Continuing our effort to build a consistent power counting for chiral nuclear effective field theory (EFT), we discuss the subleading contact interactions, or counterterms, in the singlet channels of nucleon-nucleon scattering, with renormalization group invariance as the constraint. We argue that the rather large cutoff error of the leading amplitude requires O(Q) of the EFT expansion to be non-vanishing, contrary to Weinberg's original power counting. This, together with the ultraviolet divergences of two-pion exchanges in distortedwave expansion, leads to enhancement of the 1 S 0 counterterms and results in a pionless theory-like power counting for the singlet channels. * Electronic address: bingwei@jlab.org † Electronic address:
We discuss the subleading contact interactions, or counterterms, of the triplet channels of nucleonnucleon 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 that, when compared with WPC, all 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 S1 − 3 D1 and 3 P1 whereas a two-order enhancement is necessary in both 3 P0 and 3 P2 − 3 F2.
The leading-order nucleon-nucleon (NN) potential derived from chiral perturbation theory consists of one-pion exchange plus short-distance contact interactions. We show that in the 1 S0 and 3 S1-3 D1 channels renormalization of the Lippmann-Schwinger equation for this potential can be achieved by performing one subtraction. This subtraction requires as its only input knowledge of the NN scattering lengths. This procedure leads to a set of integral equations for the partial-wave NN t-matrix which give cutoff-independent results for the corresponding NN phase shifts. This reformulation of the NN scattering equation offers practical advantages, because only observable quantities appear in the integral equation. The scattering equation may then be analytically continued to negative energies, where information on bound-state energies and wave functions can be extracted.
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