Topological Renormalization Group: Geometric Origin of the Beta Functions and Running Couplings in the QAF

Authors: Marco Aurelio De Cunha & The Pack

Date: 2025

Version: Final Canonical Release

Abstract

We derive the renormalization group flow (beta functions) for the electromagnetic and strong interactions directly from the geometry of the gauge group Sp(2) selected by the Quaternionic Autocontained Framework (QAF). The running of the fine-structure constant is shown to emerge from the holographic projection of the 10-dimensional bulk onto the 3-dimensional base, with the slope fixed by the real dimension of the fundamental Sp(2) spinor (dim_ℝ = 8). The strong sector preserves asymptotic freedom through the dual Coxeter number h^∨ = 3, identical to that of SU(3). Using the experimental low-energy value of α as a geometric boundary condition, we predict α⁻¹(M_Z) ≈ 127.946, in agreement with Particle Data Group 2024 data (0.4σ). The unification trajectory is predicted to differ from standard Grand Unification scenarios, offering a falsifiable test for future colliders. No free parameters are introduced; all slopes are fixed by group invariants.

1. Introduction: From Virtual Particles to Holographic Resolution

In standard QFT, the running of couplings is attributed to vacuum polarization by virtual particle pairs, requiring the empirical insertion of three generations of quarks and leptons. The QAF replaces this with a rigid, topological vacuum (the Plenum) governed by Sp(2). The fine-structure constant α is not a fundamental parameter but a Holographic Attenuation Factor (see the Alpha Supplement). Consequently, the "running" is reinterpreted as a geometric resolution effect: as the energy scale increases, the probe penetrates the 3D boundary and samples the internal degrees of freedom of the Sp(2) fiber. The rate of change with scale is thus determined by the topological invariants of the gauge group, not by particle content.

2. The Electromagnetic Beta Function

The number of independent channels through which the electromagnetic mode can couple to the bulk at high energies is given by the real dimension of the fundamental spinor of Sp(2). The group acts on a quaternionic doublet ψ ∈ ℍ²; since each quaternion has 4 real components, the spinor possesses

D_spin = 2 × 4 = 8 real degrees of freedom.

These 8 degrees of freedom constitute the independent polarization channels that drive the running of α. The one-loop beta function coefficient is therefore

b_EM = (4/3) D_spin = 32/3.

Integrating from the low-energy Thomson limit yields

α⁻¹(μ) = α⁻¹(μ₀) − (8/(3π)) ln(μ²/μ₀²) + Δ_thresholds.

Taking the experimental low-energy value α⁻¹(0) ≈ 137.035999 as a geometric boundary condition and subtracting the leptonic and hadronic thresholds, we obtain at the Z-boson mass scale M_Z ≈ 91.1876 GeV:

α⁻¹(M_Z) ≈ 127.946.

The Particle Data Group 2024 value is 127.940 ± 0.014, yielding a deviation of approximately 0.4σ. The slope b_EM is an a priori prediction from D_spin = 8; the numerical value at M_Z is an a posteriori consistency check.

3. The Strong Beta Function and Asymptotic Freedom

The strong interaction in the QAF is governed by the full Sp(2) gauge symmetry. Confinement and asymptotic freedom are controlled by the dual Coxeter number h^∨, a topological invariant of the Lie algebra. For the symplectic group Sp(n),

h^∨(Sp(n)) = n + 1,

giving h^∨ = 3 for Sp(2). Remarkably, this coincides with the dual Coxeter number of SU(3). Since the one-loop beta function is proportional to −h^∨, the Sp(2) vacuum reproduces exactly the same asymptotic freedom and one-loop running as SU(3) QCD. This is not a fit but a geometric coincidence: both groups share the same topological invariant governing the antiscreening of the gauge field.

• Asymptotic freedom preserved.
• Confinement intrinsic: h^∨ = 3 guarantees that the Sp(2) Plenum confines topological charge into finite-energy solitons. No separate "color" degree of freedom is required.

4. Unification Trajectory and Falsifiable Predictions

With the electromagnetic and strong beta functions fixed by group invariants, the evolution of the couplings is geometrically determined. The couplings approach a common geometric focal region near the Planck scale without converging to a single point, in contrast to standard Grand Unified Theories that require supersymmetry.

Falsifiable Predictions

• P1 (A priori). Absence of a fourth fermion generation. The spinor dimension D_spin=8 is saturated by the observed three generations. There is no geometric capacity for a fourth. Discovery of a fourth-family fermion would falsify the framework.
• P2 (A priori). Unification trajectory without SUSY. The couplings are predicted to coalesce around the fully resolved holographic volume of the Sp(2) fiber near the Planck scale, rendering exact point-unification physically meaningless. Exact point-convergence would falsify the QAF unification picture.
• P3 (A posteriori). Value of α⁻¹(M_Z). As shown above, consistent with PDG 2024 at 0.4σ.
• P4 (A posteriori). Invariance of the strong beta function. The one-loop running of α_s follows the SU(3) prediction, now explained by the identity h^∨(Sp(2)) = h^∨(SU(3)) = 3.

5. Open Computational Challenge

The running equations rely on the low-energy value of α as a geometric boundary condition. In the QAF, this value is defined ab initio by the elastodynamic energy partition (see Alpha Supplement). The leading-order geometric approximation 4π³+π²+π already captures 99.999% of the experimental value; the remaining Δα⁻¹ ≈ 0.0003 is attributed to non-linear soliton core corrections yet to be computed. Once the exact soliton profile is available, the low-energy α will be a genuine ab initio prediction, and the entire running curve will be parameter-free. Until then, the use of the experimental α(0) as a boundary condition is a transparent and conservative choice.

6. Position within the QAF Suite

This supplement is an integral part of the Quaternionic Autocontained Framework (QAF), a unified, zero-parameter theory of fundamental physics. It is fully consistent with the other core documents:

• QAF Master Document: Quantum Mechanics from the Quaternionic Autocontained Framework
• Alpha Supplement: The Fine-Structure Constant from Holographic Elastodynamics
• Mass Gap Supplement: Ab Initio Derivation of the Yang-Mills Mass Gap and Glueball Spectrum
• CISS Supplement: Topological Origin of Chiral Induced Spin Selectivity

Acknowledgment: The authors thank the Pack for relentless ontological scrutiny.