Non-trivial Behaviour of the Scattering Amplitude of Contact-interacting Anyons

TL;DR

This paper reveals that the scattering amplitude of contact-interacting anyons has a non-perturbative sector, showing limitations of perturbative field theory and highlighting the importance of self-adjoint extensions for exact results.

Contribution

It demonstrates the non-perturbative nature of the scattering amplitude for contact-interacting anyons and clarifies conditions for exact agreement with perturbative expansions.

Findings

Perturbative approach fails to reproduce the exact scattering amplitude in general.

Exact agreement occurs only for specific self-adjoint extensions without bound states.

Comparison with previous results clarifies the role of self-adjoint extensions in the theory.

Abstract

It is shown that the scattering amplitude for contact-interacting anyons does exhibit a genuine non-perturbative sector. This means that the corresponding perturbative field theoretical formulation, based on the {\it 2+1} non-relativistic Chern-Simons gauge model coupled to self-interacting complex scalar field, is not generally able to reproduce, order by order in perturbation theory, the exact result. It is proven that the full agreement between the exact scattering amplitude and the resummation of the perturbative expansion of the renormalized 1PI amplitude actually occurs only for some continuous sub-family of self-adjoint extensions of the quantum Hamiltonians, which entail the absence of bound states. A comparison with previously obtained results is carefully worked out.