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 $\phi \propto (H+i\pi)= {\tilde H}e^{i{\tilde \pi}/ }$ 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 $ $ are Goldstone Exceptionally Natural, not fine-tuned.