Naturalness made easy: two-loop naturalness bounds on minimal SM extensions

Abstract: The main result of this paper is a collection of conservative naturalness bounds on minimal extensions of the standard model by (vector-like) fermionic or scalar gauge multiplets. Within, we advocate for an intuitive and physical concept of naturalness built upon the renormalisation group equations. In the effective field theory of the standard model plus a gauge multiplet with mass $M$, the low scale Higgs mass parameter is a calculable function of $\overline{\rm MS}$ input parameters defined at some high scale $\Lambda_h > M$. If the Higgs mass is very sensitive to these input parameters, then this signifies a naturalness problem. To sensibly capture the sensitivity, it is shown how a sensitivity measure can be rigorously derived as a Bayesian model comparison, which reduces in a relevant limit to a Barbieri--Giudice-like fine-tuning measure. This measure is fully generalisable to any perturbative EFT. The interesting results of our two-loop renormalisation group study are as follows: for $\Lambda_h=\Lambda_{Pl}$ we find "$10\%$ fine-tuning" bounds on the masses of various gauge multiplets of $M<\mathcal{O}(1$--$10)$ TeV, with bounds on fermionic gauge multiplets significantly weaker than for scalars; these bounds remain finite in the limit $\Lambda_h\to M^+$, weakening to $M<\mathcal{O}(10$--$100)$ TeV; and bounds on coloured multiplets are no more severe than for electroweak multiplets, since they only directly correct the Higgs mass at three-loop.