Global U ( 1 ) Y ⊗ BRST symmetry and the LSS theorem: Ward-Takahashi identities governing Green’s functions, on-shell T -matrix elements, and the effective potential in the scalar sector of the spontaneously broken extended Abelian Higgs model

The weak-scale U(1)Y Abelian Higgs model (AHM) is the simplest spontaneous symmetry breaking (SSB) gauge theory: a scalar ϕ=12(H+iπ)≡12H˜eiπ˜/⟨H⟩ and a vector Aμ. The extended AHM (E-AHM) adds certain heavy (MΦ2,Mψ2∼MHeavy2≫⟨H⟩2∼mWeak2) spin S=0 scalars Φ and S=12 fermions ψ. In Lorenz gauge, ∂μAμ=0, the SSB AHM (and E-AHM) has a global U(1)Y conserved physical current, but no conserved charge. As shown by T. W. B. Kibble, the Goldstone theorem applies, so π˜ is a massless derivatively coupled Nambu-Goldstone boson (NGB). Proof of all-loop-orders renormalizability and unitarity for the SSB case is tricky because the Becchi-Rouet-Stora-Tyutin (BRST)-invariant Lagrangian is not U(1)Y symmetric. Nevertheless, Slavnov-Taylor identities guarantee that on-shell T-matrix elements of physical states Aμ,ϕ, Φ, ψ (but not ghosts ω, η¯) are independent of anomaly-free local U(1)Y gauge transformations. We observe here that they are therefore also independent of the usual anomaly-free U(1)Y global/rigid transformations. It follows that the associated global current, which is classically conserved only up to gauge-fixing terms, is exactly conserved for amplitudes of physical states in the AHM and E-AHM. We identify corresponding “undeformed” [i.e. with full global U(1)Y symmetry] Ward-Takahashi identities (WTI). The proof of renormalizability and unitarity, which relies on BRST invariance, is undisturbed. In Lorenz gauge, two towers of “1-soft-pion” SSB global WTI govern the ϕ-sector, and represent a new global U(1)Y⊗BRST symmetry not of the Lagrangian but of the physics. The first gives relations among off-shell Green’s functions, yielding powerful constraints on the all-loop-orders ϕ-sector SSB E-AHM low-energy effective Lagrangian and an additional global shift symmetry for the NGB: π˜→π˜+⟨H⟩θ. A second tower, governing on-shell T-matrix elements, replaces the old Adler self-consistency conditions with those for gauge theories, further severely constrains the effective potential, and guarantees infrared finiteness for zero NGB (π˜) mass. The on-shell WTI include a Lee-Stora-Symanzik theorem, also for gauge theories. This enforces the strong condition mπ2=0 on the pseudoscalar π (not just the much weaker condition mπ˜2=0 on the NGB π˜), and causes all relevant-operator contributions to the effective Lagrangian to vanish exactly. In consequence, certain heavy CP-conserving Φ, ψ matter decouple completely in the mHeavy2/mweak2→∞ limit. We prove four new low-energy heavy