QCD perturbative expansions in large orders

Abstract

We consider implications of divergencies of QCD perturbative expansions in large orders for the total cross sections of e +e − annihilation into hadrons at high energies. The divergencies bring in uncertainties which are suppressed as powers of the total energy √ s and we present numerical estimates of these uncertainties. The leading uncertainty is associated with the so-called ultraviolet renormalon. It scales with energy like s −1 ln −2 s and at the energies of order Z-boson mass does not exceed 10 −6. The next-to-leading uncertainty is due to the infrared renormalon and scales like s −2 ln −2 s. This contribution is not Borel summable and its treatment calls for some input on non-perturbative physics. We consider then the status of the instanton singularity. Although it produces only a subleading, Borel non-summable contribution it reflects a specific high-energy phenomenon, that is the growth of multi-jet production with the multiplicity. Phenomenologically, the corresponding events would look as anomalous isotropic multi-jet events. The mechanisms of the termination of this growth lie outside the perturbation theory itself. We observe that even if such events dominate the total cross section asymptotically, the onset of this asymptotics could be delayed until the (1–10) TeV region.