Fig. 11

Running couplings of \({N_{\mathrm{f}}}= 3\) QCD from integrating the non-perturbative \(\beta \)-functions in the SF- and GF-schemes [31, 50]. They are matched non-perturbatively at the scale \(\mu _0\) defined by \({\bar{g}}^2_{\mathrm{SF_{\nu =0}}}(\mu _0) = 2.012\) by computing \({\bar{g}}^2_{\mathrm{GF}}(\mu _0/2)=2.6723(64)\) [50]. The scales \(\mu _{\mathrm{PT}}=16\mu _0\) and \(\mu _{\mathrm{had}}\) defined by \({\bar{g}}^2_{\mathrm{GF}}(\mu _{\mathrm{had}})=11.31\) are also shown, as well as the perturbative prediction for the \(\mathrm{SF}_{\nu =0}\)-coupling for \(\mu >\mu _{\mathrm{PT}}\) using the 3-loop \(\beta \)-function. The red curves correspond to the results for \(\alpha ^{(3)}_{\overline{\mathrm{MS}}}(\mu )\) obtained from \(\varLambda _{\overline{\mathrm{MS}}}^{(3)} =341(12)\,\mathrm{MeV}\) [13], considering different perturbative orders for the \(\beta \)-function in the \(\overline{\mathrm{MS}}\)-scheme