Two separate renormalization group equations?

[geoduck]

Are there two separate renormalization group equations?

One for how the physical coupling constants change with time, and one for how the bare parameters/coupling constants change with cutoff?

Is there a relationship between the two?

It just seems that textbooks use the term renormalization group loosely, and I can't tell sometimes which one are they referring to, the bare or physical coupling constants.

Also, who really cares how the bare coupling constants change with cutoff? Isn't the prize how the physical coupling constants change with energy? Isn't that the only thing that can be measured experimentally?

 

[geoduck]

Not sure how to edit my post, but I meant physical coupling constants changing with energy, not time, while the bare coupling constants change with cutoff (or dimension if regularizing dimensionally).

If you have:

$$\lambda_P (\mu)=\lambda(\Lambda)+C\lambda^2(\Lambda) \log(\frac{\Lambda}{\mu})$$

then it seems that they obey the same RG equation:

$$\mu \frac{d\lambda_P}{ d\mu}=-C\lambda_{P}^2$$

and

$$\Lambda \frac{d\lambda}{ d\Lambda}=-C\lambda^2$$