SUSY shields the scaling symmetry of conformal quantum mechanics

TL;DR

This paper demonstrates that supersymmetry can unify different regimes of the inverse square potential, restoring continuous scaling symmetry by eliminating the anomalous scale introduced through renormalization.

Contribution

It reveals a dynamical supersymmetry linking strongly repulsive and weak-medium coupling regimes, restoring conformal invariance in quantum mechanics.

Findings

Supersymmetry relates different coupling regimes of the inverse square potential.

Imposing SUSY invariance removes the anomalous length scale.

Continuous scaling symmetry is restored through supersymmetric constraints.

Abstract

Renormalization of the inverse square potential usually breaks its classical conformal invariance. In a strongly attractive potential, the scaling symmetry is broken to a discrete subgroup while, in a strongly repulsive potential, it is preserved at quantum level. In the intermediate, weak-medium range of the coupling, an anomalous length scale appears due to a flow of the renormalization group away from a critical point. We show that potentials with couplings in the strongly-repulsive and in the weak-medium ranges can be related by a dynamical supersymmetry. Imposing SUSY invariance unifies these two ranges, and fixes the anomalous scale to zero, thus restoring the continuous scaling symmetry.