αs(mZ) from τ decays with matching conditions at three loops

Using the recent four-loop calculations of the QCD β-function and the three-loop matching coefficients we study the induced errors in αs(mZ) obtained from αs(mτ ) due to the evolution procedure. We show that, when consistent matching and running is used at this order, these errors are pushed below 0.0005 in αs(mZ). The beta function and the quark mass anomalous dimension govern the evolution of the strong coupling constant and the quark masses through the renormalization group (RG) equations, da d logμ2 = β(a) = −a ( β0 + β1a+ β2a 2 + β3a 3 ) +O(a) , d log m̄q d logμ2 = γm(a) = −a ( γ0 + γ1a + γ2a 2 + γ3a 3 )