Past exam of the gauge field theory course of the University of Cambridge / 2016 / Paper 1 / 2

Chirality distinguishes the two inequivalent Weyl spinor representations of the proper Lorentz group. For a Dirac spinor, define ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the chiral projectorsThey satisfy $P_L{P}_R=0$ and $P_L+P_R=I$. Proper Lorentz transformations preserve each chiral representation, whereas parity exchanges them. Helicity instead measures the projection of spin angular momentum along momentum. For a massive particle it can change under a boost that overtakes the particle, so it must not be identified with chirality. For massless positive-energy particles the left-chiral field annihilates negative-helicity particles and creates positive-helicity antiparticles.

The Standard Model is a chiral gauge theory: left- and right-chiral fields have different gauge group representations. In the convention $Q=T_3+Y$, one generation containsThe minimal field content has no right-handed neutrino. A sterile right-handed field is a possible extension, not a required member of that minimal list. The table displays weak isospin, hypercharge and color charge assignments, not assignments of massive-particle helicity.

The weak charged current contains $\bar{\nu }{\gamma }^{\mu }{P}_{L}e{W}_{\mu }^{+}$ and its Hermitian conjugate, with coupling $g/\sqrt{2}$. Only the left-chiral lepton doublet participates, giving parity violation. Both chiralities can couple to the Z boson, but their coefficients differ:Here $T_3$ is that of the left-chiral component. For the electron, it is $-1/2$. The photon and gluon couplings of a charged massive fermion are vectorlike; this does not make the full electroweak interaction vectorlike.

A bare Dirac mass term $m\bar{\psi }\psi =m({\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L)$ joins the two chiral representations. Their different electroweak quantum numbers forbid that term before electroweak symmetry breaking. A Yukawa interaction, such as $-y_e{\bar{L}}_{L}H{e}_R+h.c.$, is gauge invariant; the Higgs doublet expectation value then gives $m_e=y_{e}v/\sqrt{2}$. Up-type quarks use the conjugate doublet. Thus chirality explains the weak charge assignments, their parity violation, and why masses arise through the Higgs mechanism. The consistency of these chiral assignments also requires Standard Model anomaly cancellation.

Gauge boson self-interactions follow from noncommuting generators, rather than a separate assumption that the force carriers should attract one another. With $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }^a{T}^a$ and $[T^a,T^b]=i{f}^{abc}{T}^c$, the Yang-Mills field strength isThe last term is required so that $F_{\mu \nu }$ transforms covariantly under the gauge group. Let $G_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a$. Expanding $\mathcal{L}=-F_{\mu \nu }^a{F}^{a\mu \nu }/4$ gives a quadratic kinetic term and the cubic and quartic Yang-Mills self-interactionsThe structure constants and one gauge coupling fix both types of vertex. For an Abelian group the structure constants vanish: the renormalizable pure electromagnetic field action has no elementary three- or four-photon vertex. Photon scattering through charged-particle loops is a different statement, not a contradiction.

In Quantum chromodynamics, the non-Abelian $SU(3{)}_c$ structure produces three- and four-gluon vertices; the gluons themselves carry color charge. These vertices influence multi-jet production and the renormalization group running of the strong coupling. In the electroweak sector the $SU(2{)}_L$ fields mix with the Abelian field after electroweak symmetry breaking. The resulting charged-vector vertices include $W^{+}{W}^{-}\gamma$ and $W^{+}{W}^{-}Z$, with couplings $e$ and $g\cos {\theta }_W$, together with the related four-boson vertices. Neutral-only triple vertices such as $ZZ\gamma$ are absent at tree level in the Standard Model.

The electroweak triple gauge coupling can be measured using $e^{+}{e}^{-}\to W^{+}{W}^{-}$. The neutrino exchange and neutral-vector exchange diagrams interfere: the gauge-related coefficients cancel the most rapidly growing longitudinal-polarization terms. Their relative strengths and angular dependence therefore test the theory more sharply than merely observing a pair of massive vector bosons. The electron-mass-suppressed Higgs contribution completes the corresponding mass-dependent cancellation. Adding arbitrary independent three- and four-vector couplings would generally destroy these relations.

At the Large Electron–Positron Collider, the L3 experiment fitted charged triple-gauge parameters using hadronic and semileptonic W-pair events, complemented by single-W and single-photon channels. These measurements agreed with the predicted self-couplings. Their interpretation uses reconstructed decay products and differential distributions, rather than direct pictures of a three-boson vertex.^[ref]

The evidence tests several linked claims: a weak neutral current, massive charged and neutral mediators, their chiral couplings, and the non-Abelian interactions among those mediators. A theory must describe these with a common set of electroweak parameters.

The Gargamelle neutral-current experiment supplied a particularly direct early test. A neutrino beam crossed a heavy-liquid bubble chamber; charged reaction products made visible tracks, while incoming and outgoing neutrinos did not. Neutral-current candidates had a hadronic vertex without an accompanying charged lepton, or an isolated recoiling electron. The visible final state differs from a muon-neutrino charged-current event with an outgoing muon. Neutral hadrons entering from surrounding material could imitate a hadronic vertex, so background rejection was essential. The 1973 observations supported an interaction carried by a neutral weak mediator; they did not directly reconstruct an on-shell Z resonance.^[ref]

The UA1 experiment and UA2 experiment collider discoveries supplied complementary evidence in 1983. A proton–antiproton collider provided enough energy to produce the mediators. Charged W candidates used a high-transverse-momentum charged lepton and missing transverse momentum from an escaping neutrino; because its longitudinal momentum was not measured, a transverse-mass distribution was useful. Neutral Z candidates instead formed a charged-lepton pair with an invariant-mass peak. Observing both kinds of events and their masses tested the predicted mediator structure.^[ref]

The Weinberg angle connects independent measurements. At tree level, a Higgs doublet givesso $m_W=m_Z\cos {\theta }_W$. Neutral-current scattering, the mediator masses and electromagnetic measurements must be consistent with one mixing angle after a common renormalization prescription and radiative corrections are included. Precise Z-resonance widths and asymmetries test the left- and right-chiral couplings; a mass peak alone would not establish those coupling assignments.

At higher collider energies, W-pair production tested the electroweak triple gauge couplings. Reconstructing hadronic jets or a charged lepton plus missing momentum identifies W decays; total rates and production/decay angles constrain the $WW\gamma$ and $WWZ$ vertices. Their non-Abelian relations are required for the characteristic interference with neutrino exchange. L3's 2004 analysis used these channels and found agreement with the Standard Model expectation.^[ref]

Finally, the 2012 Higgs boson discovery and subsequent coupling tests address the symmetry-breaking sector, beyond the existence of gauge mediators. The initial observation was a new resonance in several decay channels; establishing all its properties requires further measurements. These are mutually supporting tests of the electroweak interaction, with distinct apparatus, observables and background controls, rather than a claim that any single event proves the entire theory.