Lattice-motivated QCD coupling and hadronic contribution to muon $g-2$

We present an updated version of a QCD coupling which fulfills various physically motivated conditions: at high momenta it practically coincides with the perturbative QCD (pQCD) coupling; at intermediate momenta it reproduces correctly the physics of the semihadronic tau decay; and at very low momenta it is suppressed as suggested by large-volume lattice calculations. An earlier presented analysis is updated here in the sense that the Adler function, in the regime $|Q^2| \lesssim 1 \ {\rm GeV}^2$, is evaluated by a renormalon-motivated resummation method. This Adler function is then used here in the evaluation of the quantities related with the semihadronic (strangeless) $\tau$-decay spectral functions, including Borel-Laplace sum rules in the (V+A)-channel. The analysis is then extended to the evaluation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, $a_{\mu}^{\rm had(1)}$, where we include in the Adler function the V-channel higher-twist OPE terms which are regulated in the infrared (IR) by mass parameters which are expected to be $\lesssim 1$ GeV. The correct value of $a_{\mu}^{\rm had(1)}$ can be reproduced with the mentioned IR-regulating mass parameters if the value of the condensate $\langle O_4 \rangle_{\rm V+A}$ is positive (and thus the gluon condensate value is positive). This restriction and the requirement of the acceptable quality of the fits to the various mentioned sum rules then lead us to the restriction $0.1171 < \alpha_s(M_Z^2;{\overline{MS}}) < 0.1180$.