Observing the conserved canonical momenta

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?

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 $$J^{\mu}$$.