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Furthermore if one considers loop corrections, so that the coupling ##g## acquires an RG flow, then this will no longer be a symmetry of the theory and under ##g\rightarrow \sigma\beta(g)## the current satisfies ##\partial_{\mu}D^{\mu} = \sum_i \beta(g_i)\partial_{g_i} \mathcal{L}## as an operator equation, where ##\mathcal{L}## is the Lagrangian and ##\beta(g) = \Lambda \frac{d g}{d\Lambda}## is the usual beta function. However this claim is not proven and I have been trying for hours to prove this relation starting from the partition function ##\mathcal{Z} = \int \mathcal{D}\varphi e^{S}## and trying a calculation similar to the proof of the Ward identity, with no luck. I suspect my issue is I do not know how exactly the beta function is defined in terms of ##\mathcal{Z}## as opposed to its definition in terms of the RG flow.

Does anyone know of or have a reference for a calculation proving this claim? Thanks in advance!