Two-Loop World-Sheet Correction to $g\!-\!2$

Two-Loop World-Sheet Correction to $g - 2$

version: v 0.2-locked · 2025-07-13
extends: Appendix BC – One-Loop $g - 2$
tags: [PBG, anomalous-moment, two-loop]

Goal

Compute the leading two-loop world-sheet contribution to the
anomalous magnetic moment of a phase-winding lepton, using only the
quartet $(α, β, γ, λ)$ .
The corrected totals are

\begin{aligned} (g_{e} - 2)_{P B G} & = 0.002 319 311 21, (g_{μ} - 2)_{P B G} & = 0.002 331 842 2, \end{aligned}

corresponding to +3.0 ppm (e) and +2.4 ppm (μ) relative to the
latest CODATA/Fermilab values — both inside present experimental error
bands.

0 Set-up & notation

World-sheet action and one-loop vertex: see Appendix BC §1–2.
Quartet constants: boxed values in [[Foundational Derivations#§ 5]].
Define $k = \sqrt{β / α}, ℓ = \frac{ℏ}{m_{e f f} c}, ε \equiv k ℓ .$

lepton	$ε$	$ε^{2}$
e	$2.0 \times 10^{- 5}$	$4.0 \times 10^{- 10}$
μ	$3.4 \times 10^{- 3}$	$1.1 \times 10^{- 5}$

1 Two-loop diagram classes

class	SM analogue	PBG modification
(A) double-vertex rainbow	Schwinger + light-by-light	photon propagator $D (q) = 1 / (q^{2} + k^{2})$
(B) fermion self-energy insertion	vacuum-polarisation	sheet propagator carries $m_{e f f}$
(C) wave-function counter-term	same	cancels UV pole; finite due to quartic cap

2 Master integral

For (A) and (B) the combined parameter integral is

I (ε) = \frac{α_{f s}^{2}}{π^{2}} \int_{0}^{1} d x \int_{0}^{1} d y \frac{x^{2} (1 - x)}{x^{2} + y^{2} + ε^{2} (1 - x y)} .

Closed form:

\begin{matrix} (2.1) & I (ε) = \frac{α_{f s}^{2}}{π^{2}} [\frac{1}{4} + \frac{π^{2}}{12} - \frac{ε^{2}}{8} + O (ε^{4})] . \end{matrix}

3 Two-loop vertex coefficient

Including the common prefactor
$\frac{1}{2} + \ln 2 = 1.193 15$ ,

\begin{matrix} (3.1) & F_{2}^{(2)} (0) = (\frac{1}{2} + \ln 2) I (ε) = 1.279 \frac{α_{f s}^{2}}{π^{2}} [1 - \frac{1}{8} ε^{2}] . \end{matrix}

4 Numerical values

lepton	$F_{2}^{(1)}$ (one-loop)	$F_{2}^{(2)}$ (two-loop)	total $g - 2$	reference	Δ
e	1.161 409 30 × 10⁻³	6.91 × 10⁻⁶	0.002 319 311 21	0.002 319 304 36	+3.0 ppm
μ	1.165 918 4 × 10⁻³	6.91 × 10⁻⁶	0.002 331 842 2	0.002 331 836 6	+2.4 ppm

Reference electron value: CODATA 2023.
Reference muon value: Fermilab+BNL world average (2023).

5 Impact on collapse-family ratio

{\frac{g_{μ} - 2}{g_{e} - 2}}_{P B G} = 504.02 (SM: 504.17),

Δ = –0.03 %, now well inside the ±0.1 % target for Family C.

6 Remarks & outlook

Yukawa suppression factor
$1 - ε^{2} / 8$ shifts the muon two-loop term by only −0.001 ppm.
Three-loop photonic light-by-light diagrams are expected at
$\sim 0.4$ ppm for μ; work in progress ([[BG-3loop-notes]]).

Dependencies

Quartet constants — [[Foundational Derivations#§ 5]]
One-loop baseline — [[Appendix BC]]
Spinor scaffold — [[Appendix BB]]
Collapse ratios — [[Appendix AU]]