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The weak-scale U(1) Abelian Higgs Model (AHM) is the simplest spontaneoussymmetry breaking (SSB) gauge theory. The extended AHM (E-AHM) adds certainheavy scalars $\Phi$ and fermions $\psi$. In Lorenz gauge, these theories havea global U(1) conserved physical current, but no conserved charge. As shown byKibble, the Goldstone theorem applies, there is a massless derivatively coupledNambu-Goldstone boson (NGB). Proof of all-loop-orders renormalizability andunitarity is tricky because the BRST-invariant Lagrangian is not U(1)symmetric. Nevertheless, Slavnov-Taylor identities guarantee that on-shellT-matrix elements of physical states are independent of anomaly-free gaugetransformations. We observe that they are therefore also independent of theusual anomaly-free U(1) global transformations. It follows that the associatedglobal current, is exactly conserved for amplitudes of physical states. Weidentify corresponding Ward-Takahashi identities (WTI). In Lorenz gauge, twotowers of "1-soft-pion" global WTI govern the scalar-sector, and represent anew global U(1)xBRST symmetry not of the Lagrangian but of the physics. Thefirst gives relations among off-shell Green's functions, the second governson-shell T-matrix elements, replacing the Adler self-consistency conditions.These WTI constrain the all-loop-orders scalar-sector low-energy effectiveLagrangian. Consequently, certain heavy CP-conserving heavy matterrepresentations decouple completely in the $M_{Heavy}^2/m_{Weak}^2 \to \infty$limit. SSB (E-)AHM physics therefore has more symmetry than does itsBRST-invariant Lagrantian. The NGB decouples from the observable particlespectrum in the usual way, when the observable vector absorbs it, as if it werea gauge transformation, hiding both towers of WTI from observable particlephysics.

physical review. d/physical review. d.

Global $U(1)_Y$xBRST symmetry and the LSS theorem: Ward-Takahashi  identities governing Green's functions, on-shell T-Matrix elements, and the  effective potential, in the spontaneously broken extended Abelian Higgs model