Anomalous scaling of the magnetic field in the Kazantsev-Kraichnan model

Using the field theoretic renormalization group technique and the operator-product expansion, the problem of anomalous scaling in magnetohydrodynamic turbulence is investigated in the framework of the Kazantsev-Kraichnan model, where the magnetic field is passively advected by the δ-correlated in time Gaussian velocity field, up to the second order of the perturbation theory (two-loop approximation). The model is studied in the isotropic case as well as in the case of large-scale anisotropy. It is shown that the two-loop corrections to the critical dimensions of relevant composite operators are much more important in the model of the passively advected vector field studied in the present paper, i.e. they lead to a more pronounced anomalous scaling behavior than in the corresponding problem of a passively advected scalar field in the framework of the Kraichnan model. Explicit two-loop expressions for the anomalous exponents of the single-time correlation functions of the magnetic field of arbitrary order are present. Their hierarchies with respect to the degree of anisotropy are investigated and it is shown that the hierarchies which are valid at the one-loop level of approximation are not destroyed by two-loop corrections. Thus, the leading contributions to the even correlation functions are given by the exponents from the isotropic shell in accordance with the predictions given at the one-loop level of approximation. The persistence of anisotropy in the inertial interval is studied by the behavior of the odd correlation functions. It is shown that the two-loop corrections to the scaling behavior of the odd dimensionless ratios of the correlation functions confirm the persistence of anisotropy in the inertial interval.