Renormalization group and spectra of the generalized P\"oschl-Teller potential

TL;DR

This paper investigates the spectral properties and renormalization group flow of the generalized P"oschl-Teller potential, revealing phenomena like anomalous length scales, symmetry breaking, and complex spectra across various parameter regimes.

Contribution

It introduces a renormalization procedure for the potential, analyzes the RG flow, and explores symmetry breaking and spectral features for all parameter ranges, including non-self-adjoint cases.

Findings

Discovery of an anomalous length scale via dimensional transmutation.

Identification of spontaneous breaking of conformal and supersymmetry.

Comprehensive classification of bound, anti-bound, and metastable states.

Abstract

We study the P\"oschl-Teller potential $V(x) = \alpha^2 g_s \sinh^{-2}(\alpha x) + \alpha^2 g_c \cosh^{-2}(\alpha x)$, for every value of the dimensionless parameters $g_s$ and $g_c$, including the less usual ranges for which the regular singularity at the origin prevents the Hamiltonian from being self-adjoint. We apply a renormalization procedure to obtain a family of well-defined energy eigenfunctions, and study the associated renormalization group (RG) flow. We find an anomalous length scale that appears by dimensional transmutation, and spontaneously breaks the asymptotic conformal symmetry near the singularity, which is also explicitly broken by the dimensionful parameter $\alpha$ in the potential. These two competing ways of breaking conformal symmetry give the RG flow a rich structure, with phenomena such as a possible region of walking coupling, massive phases, and non-trivial limits even when the anomalous dimension is absent. We show that supersymmetry of the potential, when present, is also spontaneously broken, along with asymptotic conformal symmetry. We use the family of eigenfunctions to compute the S-matrix in all regions of parameter space, for any value of anomalous scale, and systematically study the poles of the S-matrix to classify all bound, anti-bound and metastable states, including quasi-normal modes. The anomalous scale, as expected, changes the spectra in non-trivial ways.