Renormalization and Non-perturbative Dynamics in Conformal Quantum Mechanics

TL;DR

This paper investigates conformal quantum mechanics, analyzing ultraviolet divergences, beta functions, and non-perturbative effects in models with inverse square potentials across different dimensions.

Contribution

It provides a detailed perturbative and non-perturbative analysis of the beta function and divergences in conformal quantum mechanics with inverse square potentials.

Findings

Computed the beta function to all orders in both sectors.

Derived explicit series results for non-perturbative effects.

Analyzed ultraviolet divergences in models with two couplings.

Abstract

We study conformal quantum mechanics by first considering the perturbative $S$-matrix in various dimensions. The model has two couplings and we study perturbatively the degree of ultraviolet divergences arising in the interplay between the two couplings. We then focus on the inverse square potential in one spatial dimension and compute the beta function to arbitrarily perturbative and non-perturbative orders. This we do in both the bound state sector and scattering sector. We provide explicit, exact and infinite series results of the first few non-perturbative orders.