Renormalization group, operator product expansion, and anomalous scaling in a model of advected passive scalar

Field theoretical renormalization group methods are applied to the Obukhov-Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance [v(t,x)v(t',x)] - [v(t,x)v(t',x')] proportional to delta(t -t')\x-x'\(epsilon). Inertial range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain essential or ''dangerous'' composite operators [powers of the local dissipation rate], whose negative critical dimensions determine anomalous exponents. The main technical result is the calculation of the anomalous exponents in the order epsilon(2) Of the epsilon expansion. Generalization of the results obtained to the case of a "slow" velocity field is also presented.