What question did this study set out to answer?

This research aims to derive two-loop beta coefficients and establish a self-consistent GUT scale using foam geometry.

March 19, 2026Open Access

Two-Loop Beta Functions and the Self-Consistent GUT Scale from Foam Geometry

Key Points

This research aims to derive two-loop beta coefficients and establish a self-consistent GUT scale using foam geometry.
Derivation of beta coefficients from foam Casimir operators without free parameters.
Calculation of GUT scale by running three SM couplings simultaneously at two loops from M_Z.
Analysis of corrections to GUT scale compared to one-loop results.
Self-consistent GUT scale identified at M_GUT=1.50×10¹⁶ GeV.
The two-loop correction results in a 6.4-fold shift from the one-loop GUT scale.
Exact value of α_s(M_Z)=0.1179 is achieved from the GUT chain.

Abstract

The two-loop beta coefficients b₁QCD=26, b₁W=−11/3, b₁EM=352/27 are derived from foam Casimir operators CA=3, CF=4/3, CAW=2, nf=6 — all foam-derived, zero free parameters. Running all three SM couplings simultaneously at 2-loop from MZ gives a self-consistent GUT scale MGUT=1. 50×10¹⁶ GeV where αW=αₛ=1/46. The 2-loop correction shifts MGUT by a factor of 6. 4 from the 1-loop result, bringing it into agreement with the standard non-SUSY expectation. αₛ (MZ) =0. 1179 is reproduced exactly from the GUT chain. ΛQCD^ (5) =165 MeV (observed 210±14 MeV, 2. 9σ — limited by GUT boundary precision).

Two-Loop Beta Functions and the Self-Consistent GUT Scale from Foam Geometry

Key Points

Abstract

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