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, Left: We show the ratio of the obtained cut-o given matching at N versus N ? 1 order using the RGEs at N -loop order. The coloured(grey) contours use the two(one)-loop RGEs, therefore showing the ratio of the matching at two(one)-loop versus one-loop(tree-level), respectively. Right: Ratio of the calculation performed using both matching and RGEs at two-loop order versus the leading order, Dierence in the predicted cut-o scale depending on the matching performed as a function of the singlet VEV v S and the heavy CP-even Higgs mass m H

. Rges-;-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·, Here we have xed the physical parameters such that m h = 125 GeV, tan ? = 0.2, while the remaining parameters are chosen as ? 1 = 0 GeV and ? 2 = 1000 GeV, p.178

, The value of ?(m t ) is obtained by requiring m h = 125.15 GeV, with dierent orders of matchings depending on the curve. The solid line corresponds to the use of two-loop matching and two-loop RGE running, the dashed line to tree-level matching and two-loop RGEs, and the dotted line to treelevel matching and one-loop RGEs. Note that because of the cancellation that occurs in the one-loop correction for small ? SH (discussed in the main text), the curves we would have obtained using one-loop matching would have been very similar to those with tree-level matching, Higgs quartic coupling as a function of the renormalisation scale Q, having taken ? SH = 0.28, ? S = 0.1 and m S = 500 GeV

. ·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·, The orange shaded region of parameter space corresponds to UV-complete theories, i.e. none of the three quartic couplings (?, ? SH , ? S ) become non-perturbative and the constraints from unitarity are not violated before the Planck scale; the black shaded region corresponds to theories with stable vacua. The thin blue lines give ? as a function of ? SH when imposing m h = 125.15 GeV with a matching condition at respectively tree-level (dotted curve), one-loop order (dashed curve) and two-loop order (solid line). The other parameters of the scalar sector are ? S = 0.1, and M S = 500 GeV, Dierent phases in the Z 2 SSM shown in the ? SH ? ? plane, where the couplings are taken at scale Q = m t, p.182

, Simplied comparison between the running of ? in the SM with and without vector-like states. Here, we used full one-loop (dashed lines) and two-loop RGEs (full lines) in both models and as starting point the SM best-t values from ref, p.184

. ·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·, Black lines correspond to ? 0 = 10 18 GeV, blue lines to 10 11 GeV. The background shows the two-loop shift in ?(m t ) in percent, dened as (? (2) ? ? (1) )/? (1), Contours of the scale ? 0 at which ? runs negative with full two-loop, p.185

. ·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·-·, The stale of metastability ? 0 in the case of large Y t using two-loop running with two-loop matching for ?. The black contours show the size of ? 0 with respect to using one-loop matching, p.186