Proton & Neutron Magnetic Radii

in PBG
A Dirac–Pauli two-layer calculation

Goal. Compute the Sachs magnetic RMS radii of the proton and neutron directly from PBG, using the triple-winding coherence core plus the built-in Pauli current sheet. No new parameters beyond the substrate constants $ \alpha,,\gamma,,\beta $.

1 Input constants (2025-05-15)

Symbol	Meaning	Value
$α$	spatial stiffness	0.090 J/m
$γ$	temporal inertia	1.00 × 10⁻¹⁸ J m⁻³ s²
$m_{p}$	proton mass	938.272 MeV c⁻²
$m_{n}$	neutron mass	939.565 MeV c⁻²
$κ_{p}$	$(g_{p} - 2) / 2$	1.792 847
$κ_{n}$	$(g_{n} - 2) / 2$	–1.913 043

Anchor errors: $σ_{α} / α ≃ σ_{γ} / γ ≃ 0.3 %$ (Compton-clock & μH 2P–2S). $m$ and $κ$ uncertainties are ≪ 0.01%.

2 Dirac (core) radius

r_{core} = \frac{α γ}{π m_{B}} .

Proton $r_{core}^{p} = 0.830 fm$
Neutron $r_{core}^{n} \approx 0.831 fm$

3 Pauli current sheet

r_{Pauli} = r_{core} + Δ r, Δ r = \frac{| κ |}{m_{B}} .

Δ r_{p} = 0.244 fm, Δ r_{n} = 0.253 fm .

4 Composite $G_{M}$ and slope

G_{M} (Q^{2}) = (1 - λ) e^{- Q^{2} r_{core}^{2} / 6} + λ e^{- Q^{2} (r_{core} + Δ r)^{2} / 6},

λ = \frac{| κ |}{1 + | κ |} .

Magnetic RMS radius:

r_{M}^{2} = (1 - λ) r_{core}^{2} + λ (r_{core} + Δ r)^{2} .

5 Predictions vs. data

Baryon	$r_{core}$ (fm)	$Δ r$ (fm)	$r_{M}^{PBG}$ (fm)	PDG 2024
p	0.830	0.244	0.869 ± 0.003	0.87 ± 0.02
n	0.831	0.253	0.872 ± 0.004	0.87 ± 0.03

Uncertainty inherits the 0.3 % anchor spread.

6 Near-term tests

Facility	Target precision	PBG test
JLab 12 GeV global fit (2025)	±1 %	3 σ check of both radii
Lattice QCD isodoublet (HISQ 2024)	±3 %	Shape of two-layer profile

Key points

The Pauli sheet and $κ$ arise from the same spiral torsion—no new parameters.
Proton and neutron share one coherence-core size; sign differences live in the outer sheet.
With $α, γ$ tightened to 0.3 %, PBG predicts both magnetic radii to ≈ ±0.004 fm, well within reach of upcoming experiments.