Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to orderɛ3

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the Gaussian velocity field with zero mean and correlation function approximately equal to delta(t-t('))/k(d + epsilon). Inertial-range anomalous exponents, identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients, are calculated within the epsilon expansion to order epsilon(3) (three-loop approximation), including the exponents in anisotropic sectors. The main goal of the paper is to give the complete derivation of this third-order result, and to present and explain in detail the corresponding calculational techniques. The character and convergence properties of the epsilon expansion are discussed, the improved "inverse" epsilon expansion is proposed, and the comparison with the existing nonperturbative results is given.