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Suppose I have a Lagrangian $\mathcal{L}[\phi]$ with $\phi$ a cyclic variable, which means that the Lagrangian is symmetric under shift of $\phi\rightarrow\phi+c\quad$.

The equation of motion will be simply the conservation of the canonical momentum:$$\partial_{\mu}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}[\phi]:=\partial_{\mu}J^{\mu}=0$$

My question is: can in general the value of $J^{\mu}$ enter any observable?

More precisely, will any operator containing $J^{\mu}$ be generated at loop level?

If so, how will such operator be made generally?

Or somewhat equivalently: is the scalar $J_{\mu}J^{\mu}$ relevant in any way?

Edit: I realised that since $\mathcal{L}\supset J^{\mu}\partial_{\mu}\phi\;,\;\;$ the combination $J^{\mu}\partial_{\mu}\phi$ will enter the stress energy tensor. Is this the only way $J^{\mu}$ enters any observable?

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Generally I can observe $J^{\mu}$ whenever I add to the Lagrangian a coupling of the field $\phi$ to some other field, say $\psi$, in such a way that there is a vertex containing one $(\partial^{\mu})\phi$ line and only a vector combination of $\psi$ lines, say $b_{\mu}(\psi)$.

This vertex generates an operator $\langle \,J^{\mu}\,\,b_{\mu}(\psi)\,\rangle$ which depends on $J^{\mu}$.

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