On the renormalization-group analysis of the SM: loops, uncertainties, and vacuum stability

Renormalization-group equations (RGE) is one of the key tools in studying high-energy behavior of the Standard Model (SM). We begin by reviewing one-loop RGE for the dimensionless couplings of the SM and proceed to the state-of-the-art results. Our study focuses on the RGE solutions at different loop orders. We compare not only the standard (``diagonal'') loop counting when one considers gauge, Yukawa, and scalar self-coupling beta functions at the same order but also ``non-diagonal'' ones, inspired by the so-called Weyl consistency conditions. We discuss the initial conditions for RGE (``matching'') for different loop configurations and study the uncertainties of running couplings both related to the limited precision of the experimental input (``parametric'') and the missing high-order corrections (``theoretical''). As an application of our analysis we also estimate the electroweak vacuum decay probability and study how the uncertainties in the running parameters affect the latter. We argue that ``non-diagonal'' beta functions, if coupled with a more consistent ``non-diagonal'' matching, lead to larger theoretical uncertainty than ``diagonal'' ones.