Renormalization group functional equations

Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling $\sigma$ functions and lead to exact functional relations for the local flow $\beta$ functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of $\sigma$ are sometimes not true fixed points under continuous changes in scale and zeroes of $\beta$ do not necessarily signal fixed points of the flow but instead may only indicate turning points of the trajectories.

Figure 1

A 2-loop trajectory with $g(0)=1/2$.

Figure 2

Two-loop IR fixed point exhibited by $\sigma (y)$ (solid red) and ${\sigma }^{-1}(y)$ (dashed red) versus $y$. Light gray curves are the functional square roots of $\sigma$ and ${\sigma }^{-1}$.

Figure 3

One-loop “geometrostatic” IR fixed point exhibited by $\sigma (\mathfrak{S})$ (solid red) and ${\sigma }^{-1}(\mathfrak{S})$ (dashed red) versus $\mathfrak{S}$, and a few other fixed $t$ slices of the ${\sigma }_t(\mathfrak{S})$ surface (thin gray curves).

Figure 4

Two branches of ${\Psi }^{-1}(z)$, in green, and various approximations (orange dashes).

Figure 5

$\sigma (u)$ (solid red) and ${\sigma }^{-1}(u)$ (dashed red) versus $u$, and a few other fixed $t$ slices of the ${\sigma }_t(u)$ surface (thin gray curves).

Figure 6

Lattice (solid blue) and 2-loop (dashed blue) $g^2(t)$ trajectories, with $g^2(0)=3$.

Figure 7

Various $\beta (g^2)$ functions: 2-loop (black), 3-loop MS (green), 3-loop SFR (orange), 4-loop MS (blue), lattice (red).

Figure 8

Two-loop (black) and lattice (red) $\beta$’s, the latter to both cubic (dashed) and quartic order.

Figure 9

Step-scaling function $\sigma (u)=\tan (\frac{\pi }{4}+\arctan u)$, in blue, compared to the identity map.

Figure 10

Toy model $\sigma (u)=2u(1-u)$ (solid red) and ${\sigma }^{-1}(u)$ (dashed red) versus $u$.

Figure 11

The toy $\beta$ function, $\beta (u)=\frac{1}{2}(\ln 2)(2u-1)\ln (1-2u)$.

Figure 12

A trajectory for the $\sigma (u)=2u(1-u)$ model, with $u(0)=1/4$.

Figure 13

Toy model $\sigma (u)=\frac{11}{4}u(1-u)$ (solid red) and ${\sigma }^{-1}(u)$ (dashed red) versus $u$.

Figure 14

Six branches of the multivalued $\beta$ function for the model with $\sigma (u)=\frac{11}{4}u(1-u)$.

Figure 15

A trajectory with $u(0)=1/4$ for the $s=11/4$ multivalued $\beta$ function.

Figure 16

A trajectory with $u(0)=1/4$ for the $s=10/3$ multivalued $\beta$ function with a 2-cycle limit.

Figure 17

A trajectory with $u(0)=1/4$ for the $s=4$ multivalued $\beta$ function leading to chaotic evolution.