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Kraichnan's model of passive scalar advection in which the driving velocityfield has fast temporal decorrelation is studied as a case model forunderstanding the appearance of anomalous scaling in turbulent systems. Wedemonstrate how the techniques of renormalized perturbation theory lead (afterexact resummations) to equations for the statistical quantities that revealalso non perturbative effects. It is shown that ultraviolet divergences in thediagrammatic expansion translate into anomalous scaling with the inner lengthacting as the renormalization scale. In this paper we compute analytically theinfinite set of anomalous exponents that stem from the ultraviolet divergences.Notwithstanding, non-perturbative effects furnish a possibility of anomalousscaling based on the outer renormalization scale. The mechanism for thisintricate behavior is examined and explained in detail. We show that in thelanguage of L'vov, Procaccia and Fairhall [Phys. Rev. E {\bf 50}, 4684 (1994)]the problem is ``critical" i.e. the anomalous exponent of the scalar primaryfield $\Delta=\Delta_c$. This is precisely the condition that allows foranomalous scaling in the structure functions as well, and we prove that thisanomaly must be based on the outer renormalization scale. Finally, we derivethe scaling laws that were proposed by Kraichnan for this problem, and showthat his scaling exponents are consistent with our theory.Comment: 43 pages, revtex

Anomalous scaling in a model of passive scalar advection: Exact results