Renormalization in the Three-body Problem with Resonant P-wave Interactions

TL;DR

This paper investigates the renormalization properties of a minimal zero-range P-wave interaction model in the three-body problem, revealing ultraviolet limit cycles and the conditions under which Efimov physics occurs.

Contribution

It introduces a renormalizable zero-range P-wave interaction model and analyzes its behavior in two- and three-body sectors, highlighting the role of negative-probability states and limit cycles.

Findings

Renormalizable in the two-atom sector but exhibits unphysical high-energy behavior.

Presence of ultraviolet limit cycles indicating discrete scale invariance.

Efimov effect appears in the unphysical limit where parameters go to zero.

Abstract

Resonant P-wave interactions can be described by a minimal zero-range model defined by a truncated effective range expansion, so that the only 2-body interaction parameters are the inverse scattering volume 1/a_P and the P-wave effective range r_P. This minimal model can be formulated as a local quantum field theory with a P-wave interaction between atom fields and a molecular field. In the two-atom sector, the model is renormalizable, but it has unphysical behavior at high energies, because there are negative-probability states with momentum scale r_P. In the sector with three atoms, two of which are identical, renormalization in some parity and angular-momentum channels involves an ultraviolet limit cycle, indicating asymptotic discrete scale invariance. The Efimov effect occurs in the unitary limit a_P^(-1/3), r_P \to 0, but this limit is unphysical because there are low-energy states with negative probability. The minimal model can be of physical relevance only at energies small compared to the energy scale set by r_P, where the effects of negative-probability states are suppressed.