High orders perturbation theory and dual models for Yang-Mills theories

We start with the QCD sum rules which are originally based on the idea that it is power-like corrections to the parton model which are related to the confinement. Moreover, the naive use of the Operator Product Expansion ensures that there is a 'gap' in the powers of ΛQCD which miss the quadratic terms and start with the quartic term, proportional to the gluon condensate, <(Gμνa)2>. We review how this hypothesis stood against various checks through the last three decades and how it was modified and developed. We conclude that it was modified rather unexpectedly, through inclusion of the missing link, that is quadratic corrections which are dual to long perturbative series. Moreover, the quadratic corrections are conveniently parameterized in terms of extra dimensions. The emerging phenomenology of the quadratic corrections seems most interesting at this moment. In particular, the dual models do not incorporate the so called infrared renormalon.