Gauge Coupling Unification, Vacuum Stability, and Proton Decay in a Trinification-Based E₆ Grand Unified Theory

This work presents a complete theoretical investigation of gauge coupling unification, Higgs vacuum stability, and proton decay within a non-supersymmetric grand unified theory based on the exceptional Lie group E₆ with an intermediate trinification symmetry structure SU(3)₍C₎ × SU(3)₍L₎ × SU(3)₍R₎.

The Standard Model fermions are embedded into the fundamental 27-dimensional representation of E₆, providing a unified description of quarks, leptons, right-handed neutrinos, and exotic vectorlike fermions. The symmetry breaking chain proceeds through an intermediate trinification stage, improving gauge coupling convergence and allowing consistent unification near 10¹⁶ GeV.

Renormalization group evolution is computed explicitly using one- and two-loop beta functions, including threshold corrections from heavy scalar and fermion multiplets. The Higgs quartic coupling is evolved up to the Planck scale, demonstrating metastable vacuum consistency with current experimental measurements of the Higgs and top quark masses.

Proton decay is analyzed using dimension-6 gauge boson mediated operators. The predicted proton lifetime is approximately 10³⁴–10³⁶ years, placing the model within experimental reach of next-generation detectors such as Hyper-Kamiokande, DUNE, and JUNO.

The framework remains perturbatively consistent up to the Planck scale and provides a complete, anomaly-free, and experimentally testable realization of exceptional group grand unification.

This repository includes:

• Full theoretical derivations

• Complete renormalization group equations

• Particle spectrum specification

• Vacuum stability analysis

• Proton decay rate calculations

This work contributes to ongoing research into exceptional group unification, vacuum criticality, and experimentally testable extensions of the Standard Model.