How to read renormalization group flow plots?

Question

From what I know, the renormalization group tells us how the coupling constants of a theory change as the energy scale is varied. Thus as you vary the energy scale you trace out a path or flow in the space of all possible coupling constants.

With this picture in mind, how does one interpret the plots often associated with renormalization group flows (for example, Figure 12.1 or 12.2 in Peskin & Schroeder on pages 403 and 405) where the coupling constants are on the various axes.

Is the energy scale what is implicitly being used to parameterize the curves in the plot? If so, isn't it ambiguous what direction corresponds to increased energy?

Answer 1

Your text uses the parameter b<1, an effective energy sort of thing, whose shrinking leads to decreasing effective coupling, $$ \lambda ' = \lambda \left (1-\frac{3\lambda^2}{16\pi} \ln (1/b)\right ), \tag {12.28} $$ "infrared freedom", where the arrows in Fig (20.2) (b) indicate its shrinking.

But these are parametric slopes where b or the particular monotonic logarithm function of it have been eliminated in the ($m^2$, λ) plot provided. There is no sense of how fast a point in these RG trajectories flows on the plot, only the direction with decreasing b, towards the origin, absent a video of the flows. Any such monotonic flow function, eliminated, would give you the same trajectories.