Spectral Functions of Gauge Theories with Banks-Zaks Fixed Points

TL;DR

This paper analytically studies the spectral functions of matter-gauge theories with Banks-Zaks fixed points, revealing the behavior of propagators across different coupling regimes and providing insights into their analytic structure.

Contribution

It introduces a comprehensive analytical framework for spectral functions in gauge theories with Banks-Zaks fixed points, including higher-loop spectral functions and bounds on the conformal window.

Findings

At weak coupling, propagators have a K"allén-Lehmann spectral representation.

At strong coupling, complex branch cuts prevent a causal spectral representation.

Derived relations for scaling exponents and provided an algorithm for higher-loop running coupling.

Abstract

We investigate spectral functions of matter-gauge theories that are asymptotically free in the ultraviolet and display a Banks-Zaks conformal fixed point in the infrared. Using perturbation theory, Callan-Symanzik resummations, and UV-IR connecting renormalisation group trajectories, we analytically determine the gluon, quark, and ghost propagators in the entire complex momentum plane. At weak coupling, we find that a K\"all\'en-Lehmann spectral representation of propagators is achieved for all fields, and determine suitable ranges for gauge-fixing parameters. At strong coupling, a proliferation of complex conjugated branch cuts renders a causal representation impossible. We also derive relations for scaling exponents that determine the presence or absence of propagator non-analyticities. Further results include spectral functions for all fields up to five loop order, bounds on the conformal window, and an algorithm to find running gauge coupling analytically at higher loops. Implications of our findings and extensions to other theories are discussed.