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

The little group of a massive four-momentum is $SO(3)$, so a massive spin-one one-particle state has $2s+1=3$ polarizations. For $p^{\mu }=(E,0,0,p)$ and metric signature $(+---)$, one convenient basis isAll three obey $p_{\mu }{ϵ}^{\mu }=0$ and ${ϵ}^{\ast }\cdot ϵ=-1$. Their completeness relation isFor a massless momentum, the finite-helicity representations of the $ISO(2)$ little group carry only the two helicity states $\lambda =\pm 1$. Gauge equivalence removes the timelike and longitudinal polarizations; correspondingly, ${ϵ}_L$ has no finite $M_X\to 0$ limit.

The quadratic Proca action isIts momentum-space kinetic operator isInverting it on transverse and longitudinal projectors gives the Proca propagator

For static sources, the spatial Fourier transform of a massive scalar propagator isThus massive-vector exchange likewise produces a Yukawa potential, $V(r)\propto e^{-M_{X}r}/r$, with range $M_X^{-1}$ instead of the infinite range of the Coulomb potential.

Quantum chromodynamics supplies the requested massless-vector counterexample. Its gluons are massless, but confinement and the QCD mass gap prevent a long-range color force between color-singlet asymptotic states. The short range is generated by strong dynamics rather than by a vector-boson mass.

A gauge boson couples to the conserved current of a charged fermion throughwith the appropriate chiral projector when the gauge representation is chiral. At momentum transfer $|q^2|\ll M_X^2$, its propagator can be expanded aswhere the longitudinal term drops for a conserved current. The operation of integrating out a field, applied to $X_{\mu }$, gives the local four-fermion interactionup to the normalization used for $J^{\mu }$. Hence the effective Fermi coupling scales as $G\sim g_X^2/M_X^2$. For Standard Model charged currents, the conventional normalization is

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.