The "Goldstone Exception" II: Absence of a Higgs Fine-Tuning Problem in the Spontaneously Broken Limit of the Gell Mann Levy Linear Sigma Model: O(4) with PCAC and SU(2)_L with PCAC and Standard Model Quarks and Leptons

More than four decades ago, Lee and Symanzik proved that, in the Gell Mann-Levy (GML) model with partially conserved axial-vector currents (PCAC), tadpole renormalization (a Higgs Vacuum Stability Condition) forces all S-matrix ultra-violet quadratic divergences (UVQD) to be absorbed into the physical renormalized pseudo-scalar pion (pole) mass squared. We show that this includes "new" UVQD (widely unfamiliar to modern audiences). We also show that tadpole renormalization is an automatic consequence of Ward-Takahashi identities. We prove that all UVQD therefore vanish identically in the Goldstone-mode limit, where pions are Nambu-Goldstone Bosons (NGB), and where Lee and Symanzik's Goldstone Symmetry Restoration Condition (a renormalization prescription) enforces spontaneous symmetry breaking and the massless-ness of NGB. Axial-vector current conservation is restored as is SU(2)(L-R) chiral symmetry: the vanishing of UVQD is therefore achieved in the Goldstone-mode by restoration of an exact symmetry, and therefore (by definition) without fine-tuning! A weak-scale Higgs mass is therefore not UVQD fine-tuned in the spontaneously broken GML LSM. That is simply another (albeit unfamiliar) consequence of the Goldstone Theorem. Hence Goldstone-mode O(4) LSM symmetries are sufficient to ensure that the theory does not suffer from the Higgs Fine Tuning Problem. This is contrary to the widely accepted belief that UVQD in the Higgs mass lead to such problems in the O(4) LSM, which are then presumed to be inherited by the Standard Model (SM). The key observation is to regard the spontaneously broken O(4) LSM as the Goldstone-mode limit of the GML LSM. We prove this first at 1-loop then at all loop orders for the pure scalar GML model. We then break the O(4) symmetry to SU(2)L with SM Yukawa couplings, and show that the above remains true.