On the Lepton Mass Spectrum as Radial Breathing Modes of an Extended Dirac-Type Electron

In the Standard Model of particle physics, the pronounced mass hierarchyamong the three charged lepton generations (e; ; ) is accommodated viaarbitrary empirical Yukawa couplings without an underlying dynamical explanation.In this work, we provide a quantitative realization of Paul Dirac’s1962 conjecture, which posited that heavy leptons correspond to excited radialbreathing modes of an extended electron. Overcoming the mechanicalshortcomings of Dirac’s original hollow membrane model, we formulate a continuous,non-singular field configuration inspired by Mie’s non-linear electrodynamics,where the potential (r) = 􀀀(e=) exp(􀀀=r) eliminates the ultravioletdivergence at the origin and is locally stabilized against Coulombrepulsion by an exact internal Poincaré-type cohesive field. Hydrostatic equilibriumstrictly fixes the absolute ground-state radius at r0 = = ln 2, where = ~=(mec) is the classical electron radius.By quantizing the radial volume pulsations via non-linear recurrence relationsand semiclassical phase-space WKB techniques, the system yields adiscrete mass eigenvalue spectrum with zero free parameters: the groundstate (N = 0) defines the electron rest mass (0:511 MeV); the first excitedbreathing mode (N = 1) analytically yields the muon mass m me(1+ 32 ) 105:55 MeV (99:9% agreement with experimental data); and the second mode (N = 2), regulated by the asymptotic flattening of the well, sets the tau leptonmass at 1777:3 MeV (99:97% agreement). A candidate third excited mode(N = 3) is predicted at 29:6 GeV, whose ultra-short lifetime ( 10􀀀26 s) andcontinuum dissolution explain its absence in tracking detectors and reconcileit with the LEP invisible Z-width constraint (N = 3). This framework providesa dynamical derivation of Nambu’s empirical mass quantum (me=2 ) andidentifies E0 = mec2= 70:03 MeV as a fundamental mass-generating scale,presenting a causal, non-perturbative foundation that complements standardquantum field theory.