N-loop running should be combined with N-loop matching
We investigate the high-scale behaviour of Higgs sectors beyond the Standard Model, pointing out that the proper matching of the quartic couplings before applying the renormalisation group equations (RGEs) is of crucial importance for reliable predictions at larger energy scales. In particular, the common practice of leading-order parameters in the RGE evolution is insufficient to make precise statements on a given model's UV behaviour, typically resulting in uncertainties of many orders of magnitude. We argue that, before applying N-loop RGEs, a matching should even be performed at N-loop order in contrast to common lore. We show both analytical and numerical results where the impact is sizeable for three minimal extensions of the Standard Model: a singlet extension, a second Higgs doublet and finally vector-like quarks. We highlight that the known two-loop RGEs tend to moderate the running of their one-loop counterparts, typically delaying the appearance of Landau poles. For the addition of vector-like quarks we show that the complete two-loop matching and RGE evolution hints at a stabilisation of the electroweak vacuum at high energies, in contrast to results in the literature.