Does the background shift affects the renormalization group equations?

Question

In Section 21 of "Quantum Field theory" by Mark Srednicki, it is shown that there are two equivalent ways to get the quantum action of the shifted field $\phi'= \phi-\tilde{\phi}$, where $\phi$ is the original field and $\tilde{\phi}$ is the background. (See Eq. (21.27)) One way is to first perform the shift at the classical level and then derive the quantum action of $\phi'$ by treating $\tilde{\phi}$ as a new parameter. The other is to first derive the quantum action of $\phi$ and then perform the background shift.

Similarly, the renormalization group (RG) equations of all the parameters in the shifted action $S'$ can be derived in those two ways. The first way is to directly derive the RG equations for $S'$. The second way is to first derive the RG equations for the original action, and then get the RG equations for $S'$ using the parameter relations given by the background shift. My question is: do these two ways lead to the same RG equations?

Please let me know if I'm not clear enough.

This post imported from StackExchange Physics at 2019-03-12 18:52 (UTC), posted by SE-user Karl