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The Goldstone theorem protects naturalness, and the absence of Brout-Englert-Higgs fine-tuning, in spontaneously broken SO(2)

The Gell-Mann-Levy (GML), Schwinger and Standard Models were previously shown to lack a Brout-Englert-Higgs (BEH) fine-tuning problem due to quadratic divergences, with finite Euclidean cut-off Λ, because of the symmetries obeyed by all O(Λ^2) contributions. We extend those results to finite contributions from certain M_{Heavy}^2>> m_{BEH}^2 particles in SO(2) versions of GML and Schwinger. We demonstrate explicit 1-loop physical naturalness for two SO(2) singlet examples: a heavy real scalar S and a right-handed Type 1 see-saw Majorana neutrino. We prove that for low |q^2| the heavy degrees of freedom contribute, at worst, marginal operators in spontaneously broken SO(2) Schwinger. The key GML lesson from these examples is that the pseudo Nambu-Goldstone boson (NGB) mass-squared must be properly renormalized. A true NGB value, m_3^2 = 0, is then protected by the Goldstone theorem. For the Schwinger model, two crucial observations emerge: global Ward-Takahashi identities (WTI) force all relevant operators into the pseudo-NGB mass-squared; and WTI enforce the Goldstone theorem by forbidding all relevant operator contributions in the spontaneously broken Goldstone mode, π_3 is a massless NGB there. Goldstone mode, with weak-scale m_{BEH}^2 \& <H>^2, is not-fine-tuned even as a low-energy effective theory with certain high-mass-scale extensions. Its "Goldstone Exceptional Naturalness (GEN)," where all relevant operators vanish, a powerful suppression of fine-tuning, is simply another (albeit un-familiar) consequence of WTI, spontaneous symmetry breaking and the Goldstone theorem. If GEN can somehow be extended to the Standard Model (SM), there should be no expectation that LHC will discover any Beyond the SM physics unrelated to neutrino mixing, i.e. the only known experimentally necessary modification of the Standard Model plus General Relativity paradigm.