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At tree level and energies much greater than a gauge-vector mass, an amplitude with longitudinal external gauge bosons equals the corresponding amplitude with their would-be Goldstone bosons, up to conventional external-state phases and relative $O(m_V/E)$ corrections for a nonzero leading amplitude. In a renormalizable gauge, the broken-symmetry identities relate $k_{\mu }{\mathcal{M}}^{\mu }$ to $m_V{\mathcal{M}}_{\chi }$; inserting ${\epsilon }_L^{\mu }=k^{\mu }/m_V+O(m_V/E)$ explains the result. Loop-level versions include field-normalization factors. This is different from an equivalence theorem for field redefinitions.

On shell, the contact vertex is $4i{\lambda }^2{m}^2$. At a cubic vertex with two external on-shell legs and internal momentum $k$, the rule reduces to $2i\lambda (k^2-m^2)$. The propagator cancels one such factor, so the three exchange diagrams sum toFor equal-mass two-to-two scattering, $s+t+u=4m^2$, and the exchange sum is $-4i{\lambda }^2{m}^2$. It cancels the contact diagram exactly. Thus the connected on-shell amplitude vanishes, as required by the equivalence theorem for field redefinitions: an invertible local field redefinition cannot turn a free theory into a physically interacting one.