Hidden $U(1)_Y$ Ward-Takahashi identities in the spontaneously brokenAbelian Higgs model and the decoupling of certain heavy particles in its simple extensions

Abstract: This work is dedicated to the memory of R. Stora. The spontaneously broken (SSB) $U(1)Y$ Abelian Higgs model (AHM) (the gauge theory of a scalar $\phi \propto (H+i\pi)= {\tilde H}e^{i{\tilde \pi}/<H>}$ and a transverse vector A) has a massless pseudo-scalar $\pi$ in Lorenz gauge. Physical states have a conserved global current and Goldstone theorem (GT). $\tilde \pi$ becomes a Nambu-Goldstone boson (NGB). Slavnov-Taylor identities keep on-shell T-matrix elements of physical states independent of anomaly-free gauge, and global, transformations, yielding towers of $\phi$-sector Ward-Takahashi Identities (WTI), and constraining external $\phi$ dynamics. Ultraviolet quadratic divergences (UVQD) contribute only to $m\pi^2$, forced by the GT to 0, so all UVQD vanish. Weak-scale renormalized gauge-independent Higgs pole-mass and VEV are therefore not fine-tuned. The NGB is "eaten" and decouples, hiding the $U(1)Y$ WTI from observable particle physics. Our regularization-scheme-independent results are unchanged by the addition of certain heavy fields as the extended WTI and GT cause all relevant operators to vanish. We prove 5 SSB decoupling theorems, illustrating them with two examples: a heavy $>>m{Weak}$ $Z_2$-symmetric singlet real scalar field with 0 VEV; and a heavy singlet right-handed type 1 see-saw Majorana neutrino. Including all loops we prove that certain heavy degrees of freedom decouple from the low-energy effective Lagrangian, contributing only irrelevant operators after renormalization. The $\nu_R^{M}$ cannot completely decouple, but becomes invisible in practice. The NGB decouples, but our hidden SSB $U(1)Y$ WTI, and a ${\tilde \pi}$ shift symmetry, protect the low-energy SSB AHM physics from loop contributions of heavy particles! Gauge-independent observable weak-scale $m{H;Pole}$ and $<H>$ are Goldstone Exceptionally Natural, not fine-tuned.