  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.
