Spontaneous Symmetry Breakdown and theμ−e−γInteraction

Spontaneous breakdowns of symmetries have been examined for a system of two charged fields of zero bare mass (the "muon" and "electron" fields) interacting minimally with the electromagnetic field. Upon arranging the two fields into an "isotopic" doublet, the Lagrangian is seen to posses $\mathrm{SU}\left(2\right)\mathrm{}$ symmetry. Three possibilities are available: (a) no spontaneous breakdown of the $\mathrm{SU}\left(2\right)\mathrm{}$ symmetry is allowed and the muon-electron system remains a degenerate doublet; (b) a partial breakdown occurs in which a mass splitting develops but the heavier muon remains stable; (c) a complete breakdown occurs in which the muon decays into an electron plus a photon. Using the high-energy scheme of Baker, Johnson, and Willey, approximate solutions for the one-fermion Green's function and vertex function are examined. (The approximation scheme has the advantage that no ad hoc cutoffs need be invoked.) The solutions obtained permit case (b) to occur but not case (c), provided improper Lorentz invariance is imposed. It is shown, at least for the one-fermion Green's function, that no solutions breaking $P$, $C$, or $T$ invariance can arise.