Short-distance tachyonic gluon mass and 1/Q² corrections

We consider the assumption that a tachyonic gluon mass imitates short-distance nonperturbative physics of QCD. The phenomenological implications include modifications of the QCD sum rules for correlators of currents with various quantum numbers. The new 1/Q^2 terms allow to resolve in a natural way old puzzles in the pion and scalar-gluonium channels. They lead to a slight reduction of the values of the running light quark masses from the (pseudo)scalar sum rules and of alpha_s(M_\tau) from tau decay data. Further tests can be provided by precision measurements of the correlators on the lattice and by the e^+e^- --> hadrons data.