### Speakers

### Description

The product of the gluon dressing function and the square of the ghost dressing function in the Landau gauge can be regarded to represent, apart from the inverse power corrections $1/Q^{2n}$, a nonperturbative generalization $A(Q^2)$ of the perturbative QCD running coupling $a(Q^2)$ ($\equiv \alpha_s(Q^2)/\pi$). Recent large volume lattice calculations for these dressing functions strongly indicate that such a generalized coupling goes to zero as $A(Q^2) \sim Q^2$ when the squared momenta $Q^2$ go to zero ($Q^2 \ll 1 \ {\rm GeV}^2$). We construct such a QCD coupling $A(Q^2)$ which fulfills also various other physically motivated conditions. At high momenta it becomes the underlying perturbative coupling $a(Q^2)$ to a very high precision. And at intermediately low momenta $Q^2 \sim 1 \ {\rm GeV}^2$ it gives results consistent with the data of the semihadronic $\tau$ lepton decays as measured by OPAL and ALEPH. The coupling is constructed in a dispersive way, ensuring as a byproduct the holomorphic behavior of $A(Q^2)$ in the complex $Q^2$-plane which reflects the holomorphic behavior of the spacelike QCD observables. Application of the Borel sum rules to $\tau$-decay $V+A$ spectral functions allows us to obtain values for the gluon (dimension-4) condensate and the dimension-6 condensate, which reproduce the measured OPAL and ALEPH data to a significantly better precision than the perturbative ${\overline {\rm MS}}$ coupling (+OPE) approach. The comparison with the experimental $V$-channel Adler function, related with the $e^+ e^- \to$ hadrons ratio, at low $Q^2 \sim 1 \ {\rm GeV}^2$, also gives results considerably better than with the usual ${\overline {\rm MS}}$ pQCD+OPE approach.