Interacting fixed points in four-dimensional gauge theories coupled to matter are investigated using perturbation theory up to three loop order. It is shown how fixed points, scaling exponents, and anomalous dimensions are obtained as a systematic power series in a small parameter. The underlying ordering principle is explained and contrasted with conventional perturbation theory and Weyl consistency conditions. We then determine the conformal window with asymptotic safety from the complete nextto-next-to-leading order in perturbation theory. Limits for the conformal window arise due to fixed point mergers, the onset of strong coupling, or vacuum instability. A consistent picture is uncovered by comparing various levels of approximation. The theory remains perturbative in the entire conformal window, with vacuum stability dictating the tightest constraints. We also speculate about a secondary conformal window at strong coupling and estimate its lower limit. Implications for model building and cosmology are indicated.

We study four-dimensional gauge theories coupled to fermions in the fundamental and meson-like scalars. All requisite beta functions are provided for general gauge group and fermion representation. In the regime where asymptotic freedom is absent, we determine all interacting fixed points using perturbation theory up to three loop in the gauge and two loop in the Yukawa and quartic couplings. We find that the conformal window of ultraviolet fixed points is narrowed-down by finite-N corrections beyond the Veneziano limit. We also find a new infrared fixed point whose main features such as scaling exponents, UV-IR connecting trajectories, and phase diagram are provided. Both fixed points collide upon varying the number of fermion flavours N f , and conformality is lost through a saddle-node bifurcation. We further revisit the prospect for ultraviolet fixed points in the large N f limit where matter field fluctuations dominate. Unlike at weak coupling, we do not find clear evidence for new scaling solutions even in the presence of scalar and Yukawa couplings.Contents *