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We present analytical results for the leading top-Yukawa and QCD contributionto the $\beta$-function for the Higgs self-coupling $\lambda$ of the StandardModel at four-loop level, namely the part $\propto y_t^4 g_s^6$ independentlyconfirming a result given in [1]. We also give the contribution $\propto y_t^2g_s^6$ of the anomalous dimension of the Higgs field as well as the terms$\propto y_t g_s^8$ to the top-Yukawa $\beta$-function which can also bederived from the anomalous dimension of the top quark mass. We compare theresults with the RG functions of the correlators of two and four scalarcurrents in pure QCD and find a new relation between the anomalous dimension$\gamma_0$ of the QCD vacuum energy and the anomalous dimension $\gamma_m^{SS}$appearing in the RG equation of the correlator of two scalar currents. Togetherwith the recently computed top-Yukawa and QCD contributions to $\beta_{g_s}$[2,3] the $\beta$-functions presented here constitute the leading four-loopcontributions to the evolution of the Higgs self-coupling. A numerical estimateof these terms at the scale of the top-quark mass is presented as well as ananalysis of the impact on the evolution of $\lambda$ up to the Planck scale andthe vacuum stability problem.

Leading QCD-induced four-loop contributions to the $\beta$-function of the Higgs self-coupling in the SM and vacuum stability