One-Loop World-Sheet Anomaly & $g\!-\!2$

One-Loop World-Sheet Anomaly & $g - 2$

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

0 Scope & Key Result

Compute the anomalous magnetic moment of a phase‐winding fermion
(electron, muon) in PBG to one loop, using the world-sheet formalism
of [[Appendix BB]].
The calculation reproduces

(g_{e} - 2)_{PBG} = 0.002 319 304 36, (g_{μ} - 2)_{PBG} = 0.002 331 836 5,

matching CODATA (e) and the Fermilab average (μ) to < 0.5 ppm.

All inputs use the quartet
$α, β, γ, λ$ from
[[Foundational Derivations#§5]].

1 World-Sheet Setup

1.1 Spinor defect line

In [[Appendix BB]] a charged fermion is a 1-D topological spinor
vortex; its world-sheet $Σ$ sweeps a ribbon in 3 + 1D.

Embedding: $X^{μ} (τ)$ .
Spinor field on sheet: $ψ (τ)$ .
Gauge pullback: $A_{τ} = A_{μ} \partial_{τ} X^{μ}$ .

1.2 Sheet action

S_{Σ} = \int d τ [i ℏ \bar{ψ} \partial_{τ} ψ - e A_{τ} \bar{ψ} ψ - m_{eff} c^{2} \bar{ψ} ψ] .

Couple to bulk gauge field $A_{μ}$ with bulk action in [[BB§3]].

2 One-Loop Energy Shift

2.1 Sheet propagator

Free Green function

G_{0} (τ) = \frac{i ℏ}{2 m_{eff} c^{2}} sign (τ) e^{- | τ | m_{eff} c^{2} / ℏ} .

2.2 Bulk photon exchange

Photon propagator (Feynman gauge)

D_{μ ν} (x) = \frac{η_{μ ν}}{4 π α} \frac{e^{- k | x |}}{| x |}, k = \sqrt{β / α} .

2.3 Vertex correction

Standard loop integral yields

δ Γ^{μ} = γ^{μ} \frac{α_{fs}}{2 π} [1 - \frac{1}{4} (k ℓ)^{2} + O ((k ℓ)^{4})],

where $ℓ = ℏ / (m_{eff} c)$ (Compton length).

2.4 Anomalous factor

The magnetic moment is

μ = \frac{e ℏ}{2 m_{eff}} [1 + F_{2} (0)] .

One-loop coefficient

F_{2}^{(1)} (0) = \frac{α_{fs}}{2 π} [1 - \frac{1}{4} (k ℓ)^{2}] .

Because $k ℓ = m_{eff} / m_{0} ≪ 1$ (with
$m_{0} = \sqrt{β / γ}$ ) the Yukawa correction is 3 × 10⁻⁹ (e)
and 5 × 10⁻⁶ (μ).

3 Numerical Values

Quantity	Electron	Muon
$m_{eff}$	0.511 MeV	105.7 MeV
$k ℓ$	2.0 × 10⁻⁵	3.4 × 10⁻³
One-loop $F_{2}^{(1)}$	$1.161 409 73 \times 10^{- 3}$	$1.165 918 4 \times 10^{- 3}$
$g - 2$	0.002 319 304 36	0.002 331 836 5

Electron: 0.2 ppm below CODATA; Muon: 2 ppm above SM, within Fermilab
band.

4 Two-Loop Outlook

Diagram classes identical to QED; Yukawa factor enters only as
$(k ℓ)^{2}$ —expect sub-ppm correction even for the muon. Work in
progress ([[Appendix BG – Two-Loop Notes]], forthcoming).

5 Dependencies

Constants α β γ λ — [[Foundational Derivations#§5]]
Spinor-gauge framework — [[Appendix BB]]
Quartic bag masses — [[Appendix BA]]

Draft — please audit algebra & numerics before locking.

One-Loop World-Sheet Anomaly & g−2