Paper XXXV: Electroweak Symmetry Breaking from Hopf Vacuum Structure

Title: Paper XXXV: Electroweak Symmetry Breaking from Hopf Vacuum Structure

Author: Alexander Novickis (alex.novickis@gmail.com)

The Standard Model's electroweak sector — SU(2)_L × U(1)_Y gauge symmetry, spontaneous breaking to U(1)_em, three massive and one massless gauge boson, and a scalar Higgs field — is conventionally built from an ad hoc scalar doublet with two free parameters (μ², λ). We show that this entire structure emerges from the topology of the Hopf fibration S³ → S² (with S¹ fiber). The S³ isometry group SO(4) ≅ (SU(2)_L × SU(2)_R)/Z₂ provides the gauge symmetry; squashing the Hopf fiber reduces SU(2)_R to U(1)_Y, carrying the Z₂ through to give the electroweak group in its global form U(2) = (SU(2)_L × U(1)_Y)/Z₂ — forced independently by the matter content (the center acts trivially on every Standard-Model fermion), i.e. the electroweak factor of (SU(3)×SU(2)×U(1))/Z₆. When the soliton vacuum condenses — selecting a point on S² — the pattern SU(2)_L × U(1)_Y → U(1)_em follows without any scalar potential. The W^± mass arises from the Skyrme term's energy cost for fiber reorientation; the Z⁰ mass from electroweak mixing; the photon remains massless as the unbroken U(1)_em generator. The Weinberg angle sin²θ_W = 3/13 is derived geometrically with zero free parameters. The Higgs mass is obtained structurally as m_H = √(2λ)·v, with the electroweak scale v = √2 m_e α^−44/17 ≈ 245 GeV derived to 0.4% (the exponent 44/17 = (27+17)/17 with the charm count 27 forced by collective-sector character content and the generation step a single α-quantum, so y_t ≈ 1 is a consistency check rather than an input; the chain's sole remaining structural condition is the central-step form) and the quartic λ near-critical, so m_H is structural and metastable, in the near-critical window [123, 130] GeV containing the measured 125.25. All precision electroweak observables are reproduced at the per-mille level. The hierarchy problem is addressed by the topological determination of the electroweak scale together with the near-criticality of the Higgs self-coupling.

Keywords: physics, electroweak, higgs, soliton, topology, symmetrybreaking, hopf, weinberg

DOI: 10.5281/zenodo.19109199

Series: Paper XXXV in the Hopf Soliton Programme