Simultaneous influence of helicity and compressibility on anomalous scaling of the magnetic field in the Kazantsev-Kraichnan model

Using the field theoretic renormalization group technique and the operator product expansion, the systematic investigation of the influence of the spatial parity violation on the anomalous scaling behavior of correlation functions of the weak passive magnetic field in the framework of the compressible Kazantsev-Kraichnan model with the presence of a large-scale anisotropy is performed up to the second order of the perturbation theory (two-loop approximation). The renormalization group analysis of the model is done and the two-loop explicit expressions for the anomalous and critical dimensions of the leading composite operators are found as functions of the helicity and compressibility parameters and their anisotropic hierarchies are discussed. It is shown that for arbitrary values of the helicity parameter and for physically acceptable (small enough) values of the compressibility parameter, the main role is played by the composite operators near the isotropic shell in accordance with the Kolmogorov's local isotropy restoration hypothesis. The anomalous dimensions of the relevant composite operators are then compared with the anomalous dimensions of the corresponding leading composite operators in the Kraichnan model of passively advected scalar field. The significant difference between these two sets of anomalous dimensions is discussed. The two-loop inertial-range scaling exponents of the single-time two-point correlation functions of the magnetic field are found and their dependence on the helicity and compressibility parameters is studied in detail. It is shown that while the presence of the helicity leads to more pronounced anomalous scaling for correlation functions of arbitrary order, the compressibility, in general, makes the anomalous scaling more pronounced in comparison to the incompressible case only for low-order correlation functions. The persistence of the anisotropy deep inside the inertial interval is investigated using the appropriate odd ratios of the correlation functions. It is shown that, in general, the persistence of the anisotropy is much more pronounced in the helical systems, while in the compressible turbulent environments this is true only for low-order odd ratios of the correlation functions.

Figure 1

Graphical representation of the important propagators of the model. The end with a slash in the propagator ${\langle b_i^{\prime }{b}_j\rangle }_0$ corresponds to the field ${\mathbf{b}}^{\prime }$ and the end without a slash corresponds to the field $\mathbf{b}$.

Figure 2

The graphical representation of the interaction vertex of the model.

Figure 3

The only self-energy Feynman diagram which contributes to the UV renormalization of the model.

Figure 4

The Feynman diagrams for the function ${\mathrm{\Gamma }}_{N,p}(x;\mathbf{b})$ in the two-loop approximation. The Feynman rules are the same as in Sec. 2. The black circle denotes the vertex of the composite operator $F_{N,p}$.

Figure 5

Dependence of the two-loop correction ${\gamma }_{2,0}^{*\left(2\right)}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$.

Figure 6

Dependence of the total two-loop anomalous dimension ${\gamma }_{2,0}^{*}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$ and $\varepsilon =1$.

Figure 7

Dependence of the two-loop correction ${\gamma }_{3,1}^{*\left(2\right)}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$.

Figure 8

Dependence of the total two-loop anomalous dimension ${\gamma }_{3,1}^{*}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$ and $\varepsilon =1$.

Figure 9

Dependence of the two-loop correction ${\gamma }_{4,0}^{*\left(2\right)}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$.

Figure 10

Dependence of the total two-loop anomalous dimension ${\gamma }_{4,0}^{*}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$ and $\varepsilon =1$.

Figure 11

Dependence of the two-loop correction ${\gamma }_{5,1}^{*\left(2\right)}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$.

Figure 12

Dependence of the total two-loop anomalous dimension ${\gamma }_{5,1}^{*}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$ and $\varepsilon =1$.

Figure 13

Dependence of the two-loop correction ${\gamma }_{6,0}^{*\left(2\right)}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$.

Figure 14

Dependence of the total two-loop anomalous dimension ${\gamma }_{6,0}^{*}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$ and $\varepsilon =1$.

Figure 15

Dependence of the two-loop correction ${\gamma }_{7,1}^{*\left(2\right)}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$.

Figure 16

Dependence of the total two-loop anomalous dimension ${\gamma }_{7,1}^{*}$ on the parameters of compressibility $\alpha$ and helicity $\rho$ for $d=3$ and $\varepsilon =1$.

Figure 17

Dependence of the total two-loop anomalous dimension ${\gamma }_{2,0}^{*}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\rho =0$ and $\varepsilon =1$.

Figure 18

Dependence of the total two-loop anomalous dimension ${\gamma }_{3,1}^{*}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\rho =0$ and $\varepsilon =1$.

Figure 19

Dependence of the total two-loop anomalous dimension ${\gamma }_{4,0}^{*}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\rho =0$ and $\varepsilon =1$.

Figure 20

Dependence of the total two-loop anomalous dimension ${\gamma }_{5,1}^{*}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\rho =0$ and $\varepsilon =1$.

Figure 21

Dependence of the total two-loop anomalous dimension ${\gamma }_{6,0}^{*}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\rho =0$ and $\varepsilon =1$.

Figure 22

Dependence of the total two-loop anomalous dimension ${\gamma }_{7,1}^{*}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\rho =0$ and $\varepsilon =1$.

Figure 23

Dependence of the one-loop anomalous dimensions ${\gamma }_{2,0}^{*\left(1\right)}\varepsilon$ and ${\gamma }_{2,0}^{\prime *\left(1\right)}\varepsilon$ and total two-loop anomalous dimensions ${\gamma }_{2,0}^{*}$ and ${\gamma }_{2,0}^{\prime *}$ on the compressibility parameter $\alpha$ for $d=3,\rho =0,$ and $\varepsilon =1$, i.e., here ${\gamma }_{2,0}^{*\left(1\right)}\varepsilon \equiv {\gamma }_{2,0}^{*\left(1\right)}$ and ${\gamma }_{2,0}^{\prime *\left(1\right)}\varepsilon \equiv {\gamma }_{2,0}^{\prime *\left(1\right)}$.

Figure 24

Dependence of the one-loop anomalous dimensions ${\gamma }_{3,1}^{*\left(1\right)}\varepsilon$ and ${\gamma }_{3,1}^{\prime *\left(1\right)}\varepsilon$ and total two-loop anomalous dimensions ${\gamma }_{3,1}^{*}$ and ${\gamma }_{3,1}^{\prime *}$ on the compressibility parameter $\alpha$ for $d=3,\rho =0,$ and $\varepsilon =1$, i.e., here ${\gamma }_{3,1}^{*\left(1\right)}\varepsilon \equiv {\gamma }_{3,1}^{*\left(1\right)}$ and ${\gamma }_{3,1}^{\prime *\left(1\right)}\varepsilon \equiv {\gamma }_{3,1}^{\prime *\left(1\right)}$.

Figure 25

Dependence of the one-loop anomalous dimensions ${\gamma }_{4,0}^{*\left(1\right)}\varepsilon$ and ${\gamma }_{4,0}^{\prime *\left(1\right)}\varepsilon$ and total two-loop anomalous dimensions ${\gamma }_{4,0}^{*}$ and ${\gamma }_{4,0}^{\prime *}$ on the compressibility parameter $\alpha$ for $d=3,\rho =0,$ and $\varepsilon =1$, i.e., here ${\gamma }_{4,0}^{*\left(1\right)}\varepsilon \equiv {\gamma }_{4,0}^{*\left(1\right)}$ and ${\gamma }_{4,0}^{\prime *\left(1\right)}\varepsilon \equiv {\gamma }_{4,0}^{\prime *\left(1\right)}$.

Figure 26

Dependence of the one-loop anomalous dimensions ${\gamma }_{5,1}^{*\left(1\right)}\varepsilon$ and ${\gamma }_{5,1}^{\prime *\left(1\right)}\varepsilon$ and total two-loop anomalous dimensions ${\gamma }_{5,1}^{*}$ and ${\gamma }_{5,1}^{\prime *}$ on the compressibility parameter $\alpha$ for $d=3,\rho =0,$ and $\varepsilon =1$, i.e., here ${\gamma }_{5,1}^{*\left(1\right)}\varepsilon \equiv {\gamma }_{5,1}^{*\left(1\right)}$ and ${\gamma }_{5,1}^{\prime *\left(1\right)}\varepsilon \equiv {\gamma }_{5,1}^{\prime *\left(1\right)}$.

Figure 27

Dependence of the one-loop anomalous dimensions ${\gamma }_{6,0}^{*\left(1\right)}\varepsilon$ and ${\gamma }_{6,0}^{\prime *\left(1\right)}\varepsilon$ and total two-loop anomalous dimensions ${\gamma }_{6,0}^{*}$ and ${\gamma }_{6,0}^{\prime *}$ on the compressibility parameter $\alpha$ for $d=3,\rho =0,$ and $\varepsilon =1$, i.e., here ${\gamma }_{6,0}^{*\left(1\right)}\varepsilon \equiv {\gamma }_{6,0}^{*\left(1\right)}$ and ${\gamma }_{6,0}^{\prime *\left(1\right)}\varepsilon \equiv {\gamma }_{6,0}^{\prime *\left(1\right)}$.

Figure 28

Dependence of the one-loop anomalous dimensions ${\gamma }_{7,1}^{*\left(1\right)}\varepsilon$ and ${\gamma }_{7,1}^{\prime *\left(1\right)}\varepsilon$ and total two-loop anomalous dimensions ${\gamma }_{7,1}^{*}$ and ${\gamma }_{7,1}^{\prime *}$ on the compressibility parameter $\alpha$ for $d=3,\rho =0,$ and $\varepsilon =1$, i.e., here ${\gamma }_{7,1}^{*\left(1\right)}\varepsilon \equiv {\gamma }_{7,1}^{*\left(1\right)}$ and ${\gamma }_{7,1}^{\prime *\left(1\right)}\varepsilon \equiv {\gamma }_{7,1}^{\prime *\left(1\right)}$.

Figure 29

Dependence of the total two-loop anomalous dimensions ${\gamma }_{N,p}^{*}$ (solid curves) and ${\gamma }_{N,p}^{\prime *}$ (dashed curves) for $N=2$ and $p=0,2$ on the compressibility parameter $\alpha$ for $d=3, \rho =0$, and $\varepsilon =1$.

Figure 30

Dependence of the total two-loop anomalous dimensions ${\gamma }_{N,p}^{*}$ (solid curves) and ${\gamma }_{N,p}^{\prime *}$ (dashed curves) for $N=3$ and $p=1,3$ on the compressibility parameter $\alpha$ for $d=3, \rho =0$, and $\varepsilon =1$.

Figure 31

Dependence of the total two-loop anomalous dimensions ${\gamma }_{N,p}^{*}$ (solid curves) and ${\gamma }_{N,p}^{\prime *}$ (dashed curves) for $N=4$ and $p=0,2,4$ on the compressibility parameter $\alpha$ for $d=3, \rho =0$, and $\varepsilon =1$.

Figure 32

Dependence of the total two-loop anomalous dimensions ${\gamma }_{N,p}^{*}$ (solid curves) and ${\gamma }_{N,p}^{\prime *}$ (dashed curves) for $N=5$ and $p=1,3,5$ on the compressibility parameter $\alpha$ for $d=3, \rho =0$, and $\varepsilon =1$.

Figure 33

Dependence of the total two-loop anomalous dimensions ${\gamma }_{N,p}^{*}$ (solid curves) and ${\gamma }_{N,p}^{\prime *}$ (dashed curves) for $N=6$ and $p=0,2,4,6$ on the compressibility parameter $\alpha$ for $d=3, \rho =0$, and $\varepsilon =1$.

Figure 34

Dependence of the total two-loop anomalous dimensions ${\gamma }_{N,p}^{*}$ (solid curves) and ${\gamma }_{N,p}^{\prime *}$ (dashed curves) for $N=7$ and $p=1,3,5,7$ on the compressibility parameter $\alpha$ for $d=3, \rho =0$, and $\varepsilon =1$.

Figure 35

Dependence of the total two-loop anomalous dimension ${\gamma }_{2,0}^{\prime *}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\varepsilon =1$.

Figure 36

Dependence of the total two-loop anomalous dimension ${\gamma }_{3,1}^{\prime *}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\varepsilon =1$.

Figure 37

Dependence of the total two-loop anomalous dimension ${\gamma }_{4,0}^{\prime *}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\varepsilon =1$.

Figure 38

Dependence of the total two-loop anomalous dimension ${\gamma }_{5,1}^{\prime *}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\varepsilon =1$.

Figure 39

Dependence of the total two-loop anomalous dimension ${\gamma }_{6,0}^{\prime *}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\varepsilon =1$.

Figure 40

Dependence of the total two-loop anomalous dimension ${\gamma }_{7,1}^{\prime *}$ on the parameter of compressibility $\alpha$ and on the spatial dimension $d$ for $\varepsilon =1$.

Figure 41

Dependence of the total two-loop scaling exponent ${\zeta }_{2,1}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 42

Dependence of the total two-loop scaling exponent ${\zeta }_{3,1}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 43

Dependence of the total two-loop scaling exponent ${\zeta }_{4,1}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 44

Dependence of the total two-loop scaling exponent ${\zeta }_{4,2}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 45

Dependence of the total two-loop scaling exponent ${\zeta }_{5,1}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 46

Dependence of the total two-loop scaling exponent ${\zeta }_{5,3}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 47

Dependence of the total two-loop scaling exponent ${\zeta }_{6,1}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 48

Dependence of the total two-loop scaling exponent ${\zeta }_{6,2}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 49

Dependence of the total two-loop scaling exponent ${\zeta }_{6,3}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 50

Dependence of the total two-loop scaling exponent ${\zeta }_{7,1}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 51

Dependence of the total two-loop scaling exponent ${\zeta }_{7,3}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 52

Dependence of the total two-loop scaling exponent ${\zeta }_{7,5}$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 53

Dependence of the total two-loop exponent ${\xi }_3$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 54

Dependence of the total two-loop exponent ${\xi }_4$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 55

Dependence of the total two-loop exponent ${\xi }_5$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 56

Dependence of the total two-loop exponent ${\xi }_6$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.

Figure 57

Dependence of the total two-loop exponent ${\xi }_7$ on the parameters $\alpha$ and $\rho$ for spatial dimension $d=3$ and for $\varepsilon =1$.