= Higgs cancellation in longitudinal vector scattering
{c}
{title2=$\mathcal M\simeq s/v^2-s^2/[v^2(s-m_H^2)]$}
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,
$$
\mathcal M=-\frac{m_H^2}{v^2}-\frac{m_H^4}{v^2(s-m_H^2)}
=\frac{s}{v^2}-\frac{s^2}{v^2(s-m_H^2)}.
$$
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.
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