Anomalous Scaling in the Kinematic Magnetohydrodynamic Turbulence

The problem of the anomalous scaling in the kinematic magnetohydrodynamic turbulence is investigated using the field theoretic renormalization group method and the operator product expansion technique. The anomalous dimensions of all leading composite operators, which drive the anomalous scaling of the correlation functions of a weak passive magnetic field, are determined up to the second order of the perturbation theory (i.e., in the two-loop approximation in the field theoretic terminology) in the presence of a large scale anisotropy for physically the most interesting three-dimensional case. It is shown that the leading role in the anomalous scaling properties of the model is played by the anomalous dimensions of the composite operators near the isotropic shell, in accordance with the Kolmogorov’s local isotropy restoration hypothesis. The importance of the two-loop corrections to the anomalous dimensions of the leading composite operators is demonstrated.