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Using continuum extrapolated lattice data we trace a family of runningcouplings in three-flavour QCD over a large range of scales from about 4 to 128GeV. The scale is set by the finite space time volume so that recursive finitesize techniques can be applied, and Schr\"odinger functional (SF) boundaryconditions enable direct simulations in the chiral limit. Compared to earlierstudies we have improved on both statistical and systematic errors. Using theSF coupling to implicitly define a reference scale $1/L_0\approx 4$ GeV through$\bar{g}^2(L_0) =2.012$, we quote $L_0 \Lambda^{N_{\rm f}=3}_{\overline{\rmMS}} =0.0791(21)$. This error is dominated by statistics; in particular, theremnant perturbative uncertainty is negligible and very well controlled, byconnecting to infinite renormalization scale from different scales $2^n/L_0$for $n=0,1,\ldots,5$. An intermediate step in this connection may involve anymember of a one-parameter family of SF couplings. This provides an excellentopportunity for tests of perturbation theory some of which have been publishedin a letter [1]. The results indicate that for our target precision of 3 percent in $L_0 \Lambda^{N_{\rm f}=3}_{\overline{\rm MS}}$, a reliable estimate ofthe truncation error requires non-perturbative data for a sufficiently largerange of values of $\alpha_s=\bar{g}^2/(4\pi)$. In the present work we reachthis precision by studying scales that vary by a factor $2^5= 32$, reachingdown to $\alpha_s\approx 0.1$. We here provide the details of our analysis andan extended discussion.

A non-perturbative exploration of the high energy regime in $N_\text{f}=3$ QCD