Logarithmic enhancements in conformal perturbation theory and their real time interpretation

TL;DR

This paper investigates the divergence structures in conformal perturbation theory, revealing how logarithmic enhancements relate to resonant phenomena in time-dependent perturbations, with implications for both planar and cylindrical geometries.

Contribution

It demonstrates the connection between logarithmic divergences in conformal perturbation theory and resonant behavior in time-dependent quantum systems.

Findings

Logarithmic divergences correspond to resonant transitions in time-dependent perturbation theory.

Divergence structures are similar on the plane and cylinder when perturbations have position-dependent profiles.

Resonant behavior explains the logarithmic enhancements observed in correlation function corrections.

Abstract

We study various corrections of correlation functions to leading order in conformal perturbation theory, both on the cylinder and on the plane. Many problems on the cylinder are mathematically equivalent to those in the plane if we give the perturbations a position dependent scaling profile. The integrals to be done are then similar to the study of correlation functions with one additional insertion at the center of the profile. We will be primarily interested in the divergence structure of these corrections when computed in dimensional regularization. In particular, we show that the logarithmic divergences (enhancements) that show up in the plane under these circumstances can be understood in terms of resonant behavior in time dependent perturbation theory, for a transition between states that is induced by an oscillatory perturbation on the cylinder.