Conformal phase diagram of complete asymptotically free theories

We investigate the ultraviolet and infrared fixed point structure of gauge-Yukawa theories featuring a single gauge coupling, Yukawa coupling and scalar self-coupling. Our investigations are performed using the two loop gauge beta function, one loop Yukawa beta function, and one loop scalar beta function. We provide the general conditions that the beta function coefficients must abide for the theory to be complete asymptotically free while simultaneously possessing an infrared stable fixed point. We also uncover special trajectories in coupling space along which some couplings are both asymptotically safe and infrared conformal.