We consider two stochastic equations that describe the turbulent transfer of a passive scalar field θ(x) ≡ θ(t,x) and generalize the known Obukhov-Kraichnan model to the case of a possible compressibility and large-scale anisotropy. The pair correlation function of the field θ(x) is characterized by an infinite collection of anomalous indices, which have previously been found exactly using the zero-mode method. In the quantum field formulation, these indices are identified with the critical dimensions of an infinite family of tensor composite operators that are quadratic in the field θ(x), which allows obtaining exact values for the latter (the values not restricted to the ε-expansion) and then using them to find the corresponding renormalization constants. The identification of the correlation function indices with the composite-operator dimensions itself is supported by a direct calculation of the critical dimensions in the one-loop approximation.