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Since the discovery of a Higgs particle the effective Higgs potential of theStandard Model or extensions and the stability of the ground statecorresponding to its minimum at the electroweak scale have been subject to alot of investigation. The vacuum expectation value of the scalar SU(2) doubletfield in the Standard Model, which is responsible for the masses of elementaryparticles, may in fact not be at the global minimum of the effective Higgspotential. The question whether there is a deeper minimum at some large scaleis closely linked to the behaviour of the running quartic Higgsself-interaction $\lambda(\mu)$. In this talk an update on the analysis of theevolution of this coupling is given. We use three-loop beta-functions for theStandard Model couplings, two-loop matching between on-shell and MSbarquantities and compare the theoretical precision achieved in this way to theprecision in the latest experimental values for the key parameters.

Standard Model beta-functions to three-loop order and vacuum stability