Higher spatial derivative field theories

In this work, we employ renormalization group methods to study the general behavior of field theories possessing anisotropic scaling in the spacetime variables. The Lorentz symmetry breaking that accompanies these models are either soft, if no higher spatial derivative is present, or it may have a more complex structure if higher spatial derivatives are also included. Both situations are discussed in models with only scalar fields and also in models with fermions as a Yukawa-like model.

Figure 1

One-loop corrections to the two-point vertex functions of the $\phi$ (first row) and $\varphi$ (second row) fields.

Figure 2

Cutkosky rule at one-loop order.

Figure 3

Two-point function of order $\lambda$.

Figure 4

Interacting vertices for the Yukawa model.

Figure 5

Lowest-order contributions to the self-energy of the scalar field.

Figure 6

Two-point fermionic function.

Figure 7

Three- point vertex function.

Figure 8

Four-point scalar functions.

Figure 9

The ultraviolet behaviors of the parameters $a_{\phi }$ and $a_{\psi }$.

Figure 10

The ultraviolet behaviors of the parameters $b_{\phi }$ and $b_{\psi }$.

Figure 11

The general behavior of the coupling constant $g$. The singularity in the UV region is similar to the Landau pole.

Figure 12

The infrared behaviors of the parameters $a_{\phi }$ and $a_{\psi }$.

Figure 13

The infrared behaviors of the parameters $b_{\phi }$ and $b_{\psi }$.