The UD Derivation of the Muon-Electron Mass Ratio

ABSTRACT The muon-electron mass ratio m_μ/mₑ ≈ 207 is an important constant in particle physics. The Standard Model cannot explain this ratio; it must be taken as a free input parameter. This paper derives the muon-electron mass ratio from first principles within the UD theory framework. The electron is identified as the bound state of DU (quantum fluctuations) with the scalar background field DD (matter condensation). The muon is identified as the bound state of DU with the vector background field UD (dark matter). The background field ratio UD/DD = e^π follows rigorously from the mutual containment principle and the total normalization axiom. The factor 2π√2 emerges from the spherical wavefunction normalization and spin-1/2 normalization. The vector coupling renormalization gives 1/ (1-α). Combining all factors yields m_μ/mₑ = 2π√2 e^π/ (1-α) = 207. 13, which differs from the experimental value 206. 768 by only 0. 18%. Including the estimated scalar vertex correction (~0. 23%) would bring the prediction to ≈206. 6, even closer to experiment. This is the second piece of the three-generation fermion mass puzzle in UD theory. KEY FINDINGS 1. The electron is the bound state of DU with the scalar background DD; the muon is the bound state of DU with the vector background UD. 2. The background field ratio UD/DD = e^π is rigorously derived from the UD axioms (total normalization + mutual containment). 3. The mass of each bound state is proportional to its background field value, justified by first-order perturbation theory. 4. The quantum characteristic factor 2π√2 emerges from spherical wavefunction normalization and spin-1/2 normalization. 5. The vector coupling renormalization factor 1/ (1-α) is adopted from standard QED, which UD theory reduces to in the appropriate limit. 6. The theoretical prediction 207. 13 differs from the experimental value 206. 768 by only 0. 18%. 7. Including the estimated scalar vertex correction (~0. 23%) would bring the prediction to ≈206. 6. 8. The muon is heavier but rarer than the electron because the UD background is stronger but more diffuse and less stable. 9. Internal consistency with the proton-electron mass ratio is verified: mₚ/m_μ ≈ 8. 88. ELIMINATED ASSUMPTIONS - Muon mass as free parameter → Derived from UD axioms- Lepton generations as independent → Bound states with different backgrounds- Mass hierarchy as unexplained → Consequence of background field asymmetry Keywords: UD theory, muon mass, electron mass, lepton generations, bound states, mass ratio