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More than four decades ago, Lee and Symanzik proved that, in the GellMann-Levy (GML) model with partially conserved axial-vector currents (PCAC),tadpole renormalization (a Higgs Vacuum Stability Condition) forces allS-matrix ultra-violet quadratic divergences (UVQD) to be absorbed into thephysical renormalized pseudo-scalar pion (pole) mass squared. We show that thisincludes "new" UVQD (widely unfamiliar to modern audiences). We also show thattadpole renormalization is an automatic consequence of Ward-Takahashiidentities.  We prove that all UVQD therefore vanish identically in the Goldstone-modelimit, where pions are Nambu-Goldstone Bosons (NGB), and where Lee andSymanzik's Goldstone Symmetry Restoration Condition (a renormalizationprescription) enforces spontaneous symmetry breaking and the massless-ness ofNGB. Axial-vector current conservation is restored as is SU(2)(L-R) chiralsymmetry: the vanishing of UVQD is therefore achieved in the Goldstone-mode byrestoration of an exact symmetry, and therefore (by definition) withoutfine-tuning!  A weak-scale Higgs mass is therefore not UVQD fine-tuned in the spontaneouslybroken GML LSM. That is simply another (albeit unfamiliar) consequence of theGoldstone Theorem. Hence Goldstone-mode O(4) LSM symmetries are sufficient toensure that the theory does not suffer from the Higgs Fine Tuning Problem. Thisis contrary to the widely accepted belief that UVQD in the Higgs mass lead tosuch problems in the O(4) LSM, which are then presumed to be inherited by theStandard Model (SM). The key observation is to regard the spontaneously brokenO(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 GMLmodel. We then break the O(4) symmetry to SU(2)L with SM Yukawa couplings, andshow that the above remains true.

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