Goldstone current matrix element in a phase-only scalar theory ($⟨0|J^{\mu }(x)|\pi (p)⟩=-if{p}^{\mu }{e}^{-ipx}$)

For a canonically normalized massless phase scalar field with shift $\pi \mapsto \pi +f\alpha$, the Noether current is $J^{\mu }=f{\partial }^{\mu }\pi$. With invariant one-particle normalization, the displayed matrix element follows from $⟨0|\pi (x)|\pi (p)⟩=e^{-ipx}$. Its divergence is $-f{p}^2{e}^{-ipx}=0$ on the massless shell, despite a nonzero current matrix element. This is the elementary current coupling of a Goldstone boson; conservation constrains its momentum contraction, not every component separately.