Renormalization of the Three-Body System with Short-Range Interactions

TLDR

The three‑body problem with short‑range forces becomes nonperturbative at momenta comparable to the inverse two‑body scattering length, requiring summation of infinitely many diagrams. The authors aim to renormalize this nonrelativistic three‑body system by absorbing cutoff dependence into a single counterterm and determining its running. They absorb the cutoff dependence into a single three‑body counterterm and compute its running with the cutoff. The summed diagrams produce a cutoff dependence absent in perturbation theory, and the authors note its significance for effective field theory in nuclear and molecular physics.

Abstract

We discuss renormalization of the nonrelativistic three-body problem with short-range forces. The problem becomes nonperturbative at momenta of the order of the inverse of the two-body scattering length, and an infinite number of graphs must be summed. This summation leads to a cutoff dependence that does not appear in any order in perturbation theory. We argue that this cutoff dependence can be absorbed in a single three-body counterterm and compute the running of the three-body force with the cutoff. We comment on the relevance of this result for the effective field theory program in nuclear and molecular physics.