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We present analytical results for the QCD $\beta$-function extended to thegaugeless limit of the unbroken phase of the Standard Model at four-loop level.Apart from the strong coupling itself we include the top-Yukawa contributionand the Higgs self-coupling. We observe a non-naive $\gamma_5$ contribution atorder $y_t^4 g_s^4$, a feature not encountered in lower loop orders.

Top-Yukawa effects on the $\beta$-function of the strong coupling in the SM at four-loop level