Physics – High Energy Physics – High Energy Physics - Phenomenology
Scientific paper
2011-12-09
Physics
High Energy Physics
High Energy Physics - Phenomenology
17 pages, 10 figures; version 2 -- typos corrected, figures improved, minor wording changes
Scientific paper
More than four decades ago, B.W. Lee and K. Symanzik proved that, in a generic set of O(4) linear sigma models (LSM) in the f-pi vs. mpi-squared-by-lambda-squared half-plane, Ward-Takahashi identities, along with tadpole renormalization, i.e. a Higgs Vacuum Stability Condition, force all S-matrix ultra-violet quadratic divergences (UV-QD) to be absorbed into the physical renormalized pseudo-scalar pion mass squared. We show that all such UV-QD, together with any finite remnants, therefore vanish identically in the "Goldstone-mode" limit, where pions are Nambu-Goldstone Bosons (NGB) and axial-vector current conservation is restored (i.e. when Lee and Symanzik's Goldstone Symmetry Restoration Condition is enforced, as required by Goldstone's theorem for the spontaneously broken theory). The scalar "Higgs" mass is therefore not "quadratically fine-tuned" in the spontaneously broken theory. Hence Goldstone-mode O(4) LSM symmetries are sufficient to ensure that any finite remnant, after UV-QD cancellation, does not suffer from the Higgs Fine Tuning Problem or Naturalness Problem. This is contrary to the lore that quadratic divergences in the Higgs mass, observable already at 1-loop, lead to such problems in the O(4) LSM, and are then directly inherited by the Standard Model. It was recently shown that, including all-orders perturbative electro-weak and QCD loops, and independent of 1-loop ultraviolet regularization scheme, all finite remnants of UV-QD in the Standard Model (SM) S-Matrix are absorbed into NGB mass-squareds and vanish identically in the spontaneously broken SM. The SM Higgs mass is therefore not fine-tuned. This paper's results on the O(4) LSM lead us to a simple and intuitive understanding of that SM result, arising from the embedding of O(4) LSM into the SM.
Freese Katherine
Lynn Bryan W.
Podolsky Dmitry I.
Starkman Glenn D.
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