Three-loop beta function for the Higgs self-coupling

In the last two years the renormalization group functions for the couplings and fields of the Standard Model have been computed at three-loop level. The evolution of the self-coupling $\lambda$ of the Standard Model Higgs boson is of particular importance due to its close connection with the stability of the Standard Model vacuum state. In this talk the three-loop corrections to the $\beta$-function for this crucial coupling are discussed. The calculation of three-loop $\beta$-functions and anomalous dimensions poses special technical challenges, such as the huge number of diagrams and the proper treatment of $\gamma_5$ in dimensional regularization. In order to avoid infrared divergences resulting from setting external momenta to zero in the case of the Higgs self-coupling an auxiliary mass is used to compute the ultraviolet divergences needed for the renormalization constants. This method is explained in some detail. Finally, an update for the status of the vacuum stability problem in the Standard Model up to the Planck scale is presented.