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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 the power-like corrections to the parton model which are related to the confinement. The naive use of the Operator Product Expansion ensures that there is a 'gap' in the powers of $Λ_{QCD}$ which miss the quadratic terms and start with the quartic term, proportional to the gluon condensate, $<(G_{μν}^a)^2>$. We review how this hypothesis stood against various checks through the last three decades and how it was modified through inclusion of the missing link, that is quadratic corrections. In field theoretic language the quadratic corrections are dual to long perturbative series. In the dual description, the quadratic corrections are conveniently parameterized in terms of the metric in extra dimensions. We emphasize that the dual models do not incorporate the so called infrared renormalon.