Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 1 / e

At energy $E\gg M_X$, the longitudinal polarization behaves as ${ϵ}_L^{\mu }\sim p^{\mu }/M_X$. A Proca theory with arbitrary interactions can therefore produce longitudinal-vector scattering amplitudes growing like powers of $E/M_X$, eventually violating partial-wave unitarity and invalidating perturbation theory.

In a gauge-theory completion, the Higgs mechanism identifies the longitudinal mode with a would-be Goldstone boson. Gauge relations fix the vector, Goldstone, and Higgs couplings. Higgs-exchange diagrams then cancel the leading energy growth of the gauge diagrams, as in longitudinal $W$-boson scattering, leaving amplitudes compatible with perturbative unitarity up to the scale where the full renormalizable theory itself becomes strongly coupled.