Using the field-theoretic renormalization group technique in the two-loop approximation, the influence of helicity (spatial parity violation) on the turbulent vector Prandtl number is investigated in the model of a passive vector field advected by the turbulent helical environment driven by the stochastic Navier-Stokes equation. It is shown that the presence of helicity in the turbulent environment can significantly decrease the value of the turbulent vector Prandtl number by up to 15% of its nonhelical value. This result is compared to the corresponding results obtained recently for the turbulent Prandtl number of a passively advected scalar quantity as well as for the turbulent magnetic Prandtl number of a weak magnetic field in the framework of the kinematic magnetohydrodynamic turbulence. It is shown that the behavior of the turbulent vector Prandtl number as function of the helicity parameter is much closer to the corresponding behavior of the turbulent Prandtl number of the scalar quantity than to the behavior of the turbulent magnetic Prandtl number.
A two-particle self-consistency is rarely part of mean-field theories. It is, however, essential for avoiding spurious critical transitions and unphysical behavior. We present a general scheme for constructing analytically controllable approximations with self-consistent equations for the twoparticle vertices based on the parquet equations. We explain in details how to reduce the full set of parquet equations not to miss quantum criticality in strong coupling. We further introduce a decoupling of convolutions of the dynamical variables in the Bethe-Salpeter equations to make them analytically solvable. We connect the self-energy with the two-particle vertices to satisfy the Ward identity and the Schwinger-Dyson equation. We discuss the role of the one-particle self-consistency in making the approximations reliable in the whole spectrum of the input parameters. We demonstrate the general construction on the simplest static approximation that we apply to the Kondo behavior of the single-impurity Anderson model.
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