Infinite-cutoff renormalization of the chiral nucleon-nucleon interaction at N3LO

TL;DR

This paper investigates the application of infinite-cutoff renormalization to chiral nucleon-nucleon interactions at N3LO, revealing issues with convergence and systematic improvement, and concluding that the method is inappropriate for such interactions.

Contribution

It demonstrates that infinite-cutoff renormalization fails to produce consistent, systematic results for chiral NN interactions, highlighting the importance of the physical cutoff scale.

Findings

No convergence in lower partial waves with increasing order.

Convergence to empirical phase shifts in higher partial waves (except 3G5).

Inconsistent number of counterterms per partial wave, indicating erratic power counting.

Abstract

Naively, the "best" method of renormalization is the one where a momentum cutoff is taken to infinity while maintaining stable results due to a cutoff-dependent adjustment of counterterms. We have applied this renormalization method in the non-perturbative calculation of phase-shifts for nucleon-nucleon (NN) scattering using chiral NN potentials up to next-to-next-to-next-to-leading order (N3LO). For lower partial waves, we find that there is either no convergence with increasing order or, if convergence occurs, the results do not always converge to the empirical values. For higher partial waves, we always observe convergence to the empirical phase shifts (except for the 3G5 state). Furthermore, no matter what the order is, one can use only one or no counterterm per partial wave, creating a rather erratic scheme of power counting that does not allow for a systematic order-by-order improvement of the predictions. The conclusion is that infinite-cutoff renormalization is inappropriate for chiral NN interactions, which should not come as a surprise, since the chiral effective field theory, these interactions are based upon, is designed for momenta below the chiral-symmetry breaking scale of about 1 GeV. Therefore, this value for the hard scale should also be perceived as the appropriate upper limit for the momentum cutoff.