How do we end up with the renormalization group equations in the Wilsonian perspective?

We start with a Lagrangian $$L$$, which is valid up to some scale $$\Lambda$$ and includes couplings $$g,m$$.

In the Wilsonian perspective, we note that the contributions from fluctuations at scales between some arbitrary scale $$\lambda$$ and the cutoff $$\Lambda$$ are $$\propto \ln(\lambda /\Lambda)$$. The key idea is then that we can use the cutoff $$\lambda$$ instead of $$\Lambda$$ if we include counter terms $$\delta L(\lambda,\Lambda)$$. In particular, we end up with a renormalized effective Lagrangian $$L_R = L + \delta L(\lambda,\Lambda) ,$$ which is valid up to the scale $$\lambda$$. One way to understand this effective Lagrangian is by defining new parameters $$g_R, m_R$$ that are scale dependent. Specifically, we have $$g_R (\lambda) := g +\frac{g^2}{12 \pi^2} \ln(\lambda /\Lambda)$$, where $$g$$ is the "bare" coupling that appears in the original Lagrangian which is valid for cutoff $$\Lambda$$.

It is then usually argued (see, for example, page 84 here) that this implies that if we measure $$g_R$$ at some scale $$\mu_0$$, we can predict its value at any other scale $$\mu$$ by using the analogous evolution equation $$g_R (\mu) = g(\mu_0) +\frac{g(\mu_0)^2}{12 \pi^2} \ln(\mu /\mu_0)$$

This seems to make sense at first glance, but raises many questions upon closer inspection. So how can this actually be derived? My main issue is that the scales $$\mu$$ and $$\mu_0$$ are physical scales relevant to the energy in a given process, whereas $$\lambda$$ and $$\Lambda$$ are cutoffs. Moreover, $$g$$ has no scale dependence. It's the charge that appears in the original Lagrangian $$L$$ which is valid up to the cutoff scale $$\Lambda$$. But there is no reason why this $$g$$ should depend on the energy scale in a given process.

• The coupling $g$ is energy scale independent at the tree/classical level. Taking into account the quantum/loop/renormalization corrections, $g$ is inevitably incoming-momentum/energy dependent for a particle-particle scattering process. – MadMax Jul 8 at 15:31