The Renormalization Group Limit Cycle for the 1/r^2 Potential

TL;DR

This paper investigates the renormalization group limit cycle behavior in the 1/r^2 potential when regularized with different short-distance potentials, revealing that a delta-shell regularization inevitably leads to a limit cycle.

Contribution

It demonstrates that using a delta-shell potential for regularization enforces an unavoidable RG limit cycle in the 1/r^2 potential, unlike other regularizations.

Findings

Delta-shell regularization yields a log-periodic coupling constant with infinite discontinuities.

RG limit cycle behavior is inherent in the delta-shell regularization of the 1/r^2 potential.

Proper renormalization reproduces physical observables at low energies despite the limit cycle.

Abstract

Previous work has shown that if an attractive 1/r^2 potential is regularized at short distances by a spherical square-well potential, renormalization allows multiple solutions for the depth of the square well. The depth can be chosen to be a continuous function of the short-distance cutoff R, but it can also be a log-periodic function of R with finite discontinuities, corresponding to a renormalization group (RG) limit cycle. We consider the regularization with a delta-shell potential. In this case, the coupling constant is uniquely determined to be a log-periodic function of R with infinite discontinuities, and an RG limit cycle is unavoidable. In general, a regularization with an RG limit cycle is selected as the correct renormalization of the 1/r^2 potential by the conditions that the cutoff radius R can be made arbitrarily small and that physical observables are reproduced accurately at all energies much less than hbar^2/mR^2.