# Does the background shift affects the renormalization group equations?

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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
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