Towards the renormalisation of the Standard Model effective field theory to dimension eight: Bosonic interactions I

We compute the one-loop renormalisation group running of the bosonic Standard Model effective operators to order $v^4/Lambda^4$, with $vsim 246$ GeV being the electroweak scale and $Lambda$ the unknown new physics threshold. We concentrate on the effects triggered by pairs of the leading dimension-six interactions, namely those that can arise at tree level in weakly-coupled ultraviolet completions of the Standard Model. We highlight some interesting consequences, including the interplay between positivity bounds and the form of the anomalous dimensions; the non renormalisation of the $S$ and $U$ parameters; or the importance of radiative corrections to the Higgs potential for the electroweak phase transition. As a byproduct of this work, we provide a complete Green basis of operators involving only the Higgs and derivatives at dimension-eight, comprising 13 redundant interactions.