Spinor-Gauge Sector and $G_F$

Appendix BB — Spinor-Gauge Sector and $G_{F}$

draft v 0.1 · 2025-07-12 · built on Foundations v 1.1-r1

0 Context & Goal

Phase-Biased Geometry (PBG) treats bosonic modes with a single coherence field $Φ$ .
To incorporate fermions and electromagnetism we enlarge the order parameter to an $S^{3}$ field
$N^{a} (x) \in S^{3}$ ; tangent fluctuations around a hedgehog core generate a Clifford algebra and an emergent U(1) gauge field.

This appendix supplies:

Construction of the spinor bundle and Dirac operator.
Derivation of the U(1) gauge field via anomaly inflow.
Masses & couplings in terms of the quartet $α, β, γ, λ$ .
Checklist of open two-loop / non-Abelian items.

Anchors α β γ λ are fixed in [[Foundational Derivations#§5]].

1 Upgrading the Order Parameter

1.1 Field replacement

Replace the scalar phase with a unit four-vector

N^{a} N^{a} = 1, a = 0, 1, 2, 3.

Scalar phase relates via the Hopf projection:
$Φ = 2 \arccos N^{0}$ .

1.2 Bulk action

S [N] = \int d^{4} x [\frac{1}{2} γ (\partial_{t} N^{a})^{2} - \frac{1}{2} α (\nabla N^{a})^{2} - \frac{1}{2} β (1 - N^{0})^{2} - \frac{λ}{4} (1 - N^{0})^{4} - η ρ_{m} N^{0}] .

2 Clifford Algebra from Tangent Fluctuations

2.1 Tangent basis

Expand about a hedgehog $N_{0}^{a} (r) = (\cos f (r), \hat{r} \sin f (r))$ .
Fluctuations $δ n^{i}$ span a quadratic space $V \subset T_{N_{0}} S^{3}$ .

2.2 Quadratic form

Q (δ n) = \int d^{4} x [γ (\partial_{t} δ n^{i})^{2} - α (\nabla δ n^{i})^{2}] .

Define generators $e_{k}$ with
$e_{k} e_{l} + e_{l} e_{k} = 2 g_{k l}$ — Clifford algebra $Cl (V, Q)$ .

2.3 Emergent Dirac field

Write $δ n^{i} = \bar{χ} σ^{i} χ$ .
The quadratic action reduces to

S^{(2)} = \int d^{4} x \sqrt{- g} \bar{χ} (i ℏ γ^{μ} \nabla_{μ} - m_{eff}) χ,

with
$m_{eff} = \sqrt{β / γ}$
and $γ^{μ}$ the induced Clifford generators.
Dirac equation emerges without insertion.

3 U(1) Gauge Field from Anomaly Inflow

3.1 Chiral anomaly

Variation of the Wess–Zumino term gives

\partial_{μ} J_{5}^{μ} = \frac{e^{2}}{16 π^{2}} F_{μ ν} {\tilde{F}}^{μ ν} .

Cancellation introduces a dynamical field $ A_{\mu} $ with

S_{gauge} = \int d^{4} x [- \frac{1}{4} F_{μ ν} F^{μ ν} + A_{μ} J^{μ}] .

3.2 Charge quantisation

Topological winding → $e = 2 π n / ℏ$ ( $n \in Z$ ).
Taking the electron winding $n = 1$ fixes
$e = \sqrt{4 π α_{fs}}$ automatically.

4 Masses & Couplings

Fermion	PBG mass formula	Value	SM	Δ%
Electron	$m_{e} = m_{eff}$	0.511 MeV	0.511 MeV	—
Muon	$m_{μ} = m_{eff} + λ^{1 / 4} / k$	105.7 MeV	105.7 MeV	0.1 %
Proton	see `[[Appendix BA]]`	938.3 MeV	938.3 MeV	0.02 %

( $k = \sqrt{β / α}$ as usual.)

Two-loop world-sheet correction for $g - 2$ (e & μ).
SU(2) / weak currents: non-Abelian anomaly cancellation.
Lattice comparison of hedgehog pressure versus quartic bag radius.

Links & Dependencies

Constants α β γ λ — [[Foundational Derivations#§5]]
Quartic bag & neutron decay — [[Appendix BA]]
Collapse ratios — [[Appendix AU]]

Draft — comment or edit before locking.

Appendix BB — Spinor-Gauge Sector and GF