Hidden $U(1)_Y$ Ward-Takahashi identities, absence of Brout-Englert-Higgs fine-tuning and decoupling of certain heavy particles due to spontaneous symmetry breaking I: The extended Abelian Higgs model

This work is dedicated to the memory of R. Stora who co-concieved its seminal ideas, and expressed his intention to join as a co-author. The weak-scale $U(1)_Y$ Abelian Higgs model (AHM) is the SSB gauge theory of a complex scalar $\phi=(H+i\pi)/\sqrt{2}={\tilde H}/\sqrt{2}exp[i{\tilde \pi}/ ]$ and a vector $A^\mu$. The extended AHM (E-AHM) adds heavy ($M>>m_{Weak}$) $U(1)_Y$ scalar $\Phi$ and fermion $\psi$ particles. In Lorenz gauge, the SSB E-AHM has a massless pseudo-scalar $\pi$, a conserved $U(1)_Y$ global current and a Goldstone Theorem. $\tilde\pi$ becomes a Nambu-Goldstone boson (NGB). Since Slavnov-Taylor IDs guarantee that on-shell T-matrix elements of physical states are independent of anomaly-free $U(1)_Y$ gauge transformations, we observe that they are therefore also independent of anomaly-free $U(1)_Y$ rigid/global transformations. Two towers of resulting $U(1)_Y$ Ward-Takahashi Identities (WTI) give relations among Green's functions and among T-matrix elements, and constraints on the $\phi$-sector dynamics. All UV-divergent and finite relevant operators originating in virtual loops are absorbed into a pseudo-Nambu-Goldstone Boson (NGB) mass-squared $m_\pi^2$ that appears in intermediate calculational steps. The Goldstone Theorem enforces $m_\pi^2=0$, so all relevant operators vanish! After quartic-coupling renormalization, $\Phi$ and $\psi$ contribute only irrelevant operators and decouple. The renormalized gauge-independent H pole-mass and VEV are therefore not fine-tuned. The NGB decouples from the observable particle spectrum as usual, when the physical vector particle absorbs it, as if it were a gauge transformation. Our WTI are then "hidden" from observable particle physics, but those Hidden WTI and the embedded shift symmetry for ${\tilde\pi}$, have protected the physics of the low-energy SSB AHM weak-scale theory from loop contributions of heavy particles!