Gauge Theories in Particle Physics 2026, 45 - Local non-Abelian symmetries in Lagrangian QFT

Write locally, because of sign convention,

(Eqs. 13.64-65). This reflects in the field operators. The Feynman rule for the fermion-W uses the W-propagator

(Eq. 13.69), which is initially unsuitable for describing weak interactions, since weak interactions are associated with several symmetry violations. The locally SU(3)_c-invariant Lagrangian for a quark triplet q_f (13.3.1).

The electromagnetic interactions of the charged massless spin-1 particles carries the charge e, so the standard wave equation for this interaction is the same as for the gauge A field A^μ, which implies [D^μ, D^ν] = ieF^μν (Eq. 13.78), and fails to produce a unique wave equation. The ambiguity is made possible by introducing an arbitrary parameter δ in the wave equation so that D²X^μ - D^νD^μX_ν + ieδF^μνX_ν = 0 (Eq. 13.79). The additional term is the ambiguous magnetic moment, and contributes to the magnetic moment coupling of the X-particle to the electromagnetic field. The full term is fully U(1)-gauge covariant. The theory is renormalizable if the photon, the X-particle and its antiparticle are the members of an SU(2) gauge triplet. Written as □X^μ - ∂^ν∂^μX_ν = VX^μ (Eq. 13.80). With V = -ie{[∂^ν(A_νX^μ) + A^ν∂_νX^μ] - (1+δ)[∂^ν(A^μX_ν) + A^ν∂^μX_ν] + δ[∂^μ(A^νX_ν) + A^μ∂^νX_ν]} (Eq. 13.81) with dropped terms of O(e²). At lowest order perturbation theory the transformation under the potential V is -i∫X_μ*(f)VX^μ(i)d⁴x (Eq. 13.83) from which the vertices are determined under evaluation (13.3.2). From the YM-Lagrangian follows the definiteness via SU(2) gauge theory with massless gauge fields. It introduces an auxiliary field B, for which (D^μ)_ki∂_μB_i = 0 (Eq. 13.115). B then is not a free field in the non-Abelian case, which extends to ∂_μW^μ.