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

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