Goldstone-boson equivalence theorem ($\mathcal{M}(V_L\cdots )=\mathcal{M}(\chi \cdots )+O(m_V/E)$)

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.

The Goldstone-boson equivalence theorem reduces high-energy electroweak longitudinal scattering to scalar scattering. Without the radial Higgs boson, a nonlinear sigma-model description gives an amplitude of order $s/v^2$. In the corresponding linear scalar model, the contact and Higgs-exchange terms give, for the charged-to-neutral Goldstone channel,The growing terms cancel, leaving $-m_H^2/v^2$ at high energy. Thus the Higgs mechanism supplies both vector masses and the interactions required for controlled longitudinal scattering. The tree expression is not valid as an unresummed approximation arbitrarily close to the Higgs pole.