Quark model, large order behavior and nonperturbative wave functions in QCD

We discuss a few, apparently different (but actually, tightly related) problems: 1. 1. the relation between QCD and valence quark model, 2. 2. the evaluation of the nonlocal condensate 〈q(x)q(0)〉, its relation to the heavy-light qQ quark system and to constituent quark mass, 3. 3. the asymptotic behavior of the nonperturbative pion wave function ψ(k⊥2,x) at x → 0, 1, k⊥2 → ∞ and 4. 4. the large order behavior of perturbative series. The analysis is based on such general methods as dispersion relations, duality and PCAC. We use the steepest descent method (also known as semiclassical, or instanton calculus), introduced by Lipatov to calculate the n-th moment of the ψ(k⊥2,x) with result 〈k⊥2n〉 ∼ n!.. This information fixes the asymptotic behavior of wf at large k⊥2. This behavior turned out to be Gaussian, as commonly used in the phenomenological analyses. The same method determines the asymptotic behavior of the mixed local vacuum condensates 〈qGμνnq〉 ∼ n! at large n as well as the nonlocal vacuum condensate 〈q(x)q(0)〉 which naturally arises in the description of the heavy-light qQ quark system. The relation between the nonlocal condensate and the constituent quark mass is also discussed.