Spectral dimension as a tool for analyzing non-perturbative propagators

TL;DR

This paper investigates how the spectral dimension of non-perturbative gauge boson propagators varies across scales in Yang-Mills theories, revealing scale-dependent geometric properties and their relation to propagator features.

Contribution

It introduces a method to analyze the scale-dependent spectral dimension of gauge boson propagators derived from various non-perturbative approaches in Yang-Mills theories.

Findings

Spectral dimension decreases slightly in the ultraviolet due to anomalous dimensions.

Spectral dimension approaches one at large times for propagators with positivity violation.

Longest time intervals relate to the momentum scale of the propagator's maximum.

Abstract

We derive general properties of the scale-dependent effective spectral dimensions of non-perturbative gauge boson propagators as they appear as solutions from different methods in Yang-Mills theories. In the ultraviolet and for short time scales the anomalous dimensions of the propagators lead to a slight decrease of the spectral dimension as compared to the one of a free propagator. Lowering the momentum scale, the spectral dimension decreases further. The class of propagators which display a maximum at Euclidean momenta, and thus violate positivity, always approaches a spectral dimension of one for large times. We also show that the longest time intervals are not related to the deep infrared but to the momentum scale defined by the position of the maximum.