Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 2

The Wigner classification labels a massive one-particle state by $|p;j,m_s⟩$, where $p^2=m^2>0$, $p^0>0$, $j$ is the spin representation of the $SO(3)$ little group, and $m_s=-j,\dots ,j$. A finite-spin massless state is $|p;\lambda ⟩$, where $p^2=0$, $p^0>0$, and the helicity $\lambda$ labels a one-dimensional representation of the rotational part of the $ISO(2)$ little group. A parity-invariant theory pairs nonzero helicities $\lambda$ and $-\lambda$.

The physical polarization counts areA scalar obeys the Klein-Gordon equation. A spinor obeys the Dirac equation, and a massless irreducible spinor additionally has a fixed chirality. A massive vector obeys the Proca equation, whose divergence gives ${\partial }_{\mu }{A}^{\mu }=0$ and leaves three polarizations. A massless vector instead has the gauge redundancy $A_{\mu }\sim A_{\mu }+{\partial }_{\mu }\alpha$, leaving two transverse polarizations. A massive symmetric tensor obeys the Fierz-Pauli equations: symmetry, transversality, and tracelessness leave five components. A massless tensor has linearized diffeomorphism redundancy $h_{\mu \nu }\sim h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, leaving helicities $\pm 2$. A redundancy identifies field configurations representing the same physical state; gauge invariance is invariance under that identification.

Write an amplitude with an external massless vector of momentum $k$ asChanging the polarization representative by ${ϵ}_{\mu }\mapsto {ϵ}_{\mu }+c,k_{\mu }$ is a gauge transformation. The physical amplitude must be unchanged, so the Ward identity isThus a longitudinal pure-gauge polarization decouples from the S-matrix. This on-shell statement is the amplitude counterpart of the off-shell Ward-Takahashi identity.

The leading Soft photon theorem for a photon of momentum $q$ iswhere ${\eta }_i=+1$ for outgoing and $-1$ for incoming particles. Replacing $ϵ$ by $q$ and applying the Ward identity gives ${\sum }_i{\eta }_i{Q}_i=0$, which is electric charge conservation.

The analogous Weinberg soft theorem for helicity two forces every particle to couple to a massless spin-two field with the same strength; its Ward identity reduces to momentum conservation and yields the equivalence principle. For helicity strictly greater than two, the corresponding polynomial constraints have no nontrivial solution for a generic interacting S-matrix under the usual assumptions of Lorentz invariance, locality, and finitely many particle species. This is the soft-theorem obstruction to ordinary long-range interactions mediated by massless higher-spin particles.

The longitudinal polarization of a massive vector behaves as ${ϵ}_L^{\mu }(p)\sim p^{\mu }/m$ at high energy. Generic self-interactions therefore make amplitudes grow as powers of $E/m$, eventually violating partial-wave unitarity and destroying perturbative predictivity. Simply adding a Proca mass also spoils the gauge cancellations that control ultraviolet behavior.

The Standard Model obtains the $W$ and $Z$ masses through the Higgs mechanism inside a renormalizable gauge theory. Goldstone bosons supply the longitudinal polarizations, and relations among gauge and Higgs couplings enforce cancellations of the growing terms. In particular, Higgs exchange cancels the remaining $E^2$ growth in longitudinal vector-boson scattering.