Broken gauge symmetry

This is a broken gauge theory at low energy, which can be expressed as in Eq. (686) as a gauge theory accompanied by a broken gauge symmetry. Assume a simple Lagrangian that couples the left-handed fields , to the right-handed boson and the right-handed fields v ir to the left-handed boson ... 

Lee, B.W. and Zinn-Justin, J. (1972). Spontaneously broken gauge symmetries. 

From the foregoing, it becomes clear that fields and potentials are freely intermingled in the symmetry-broken Lagrangians of the Higgs mechanism. To close this section, we address the question of whether potentials are physical (Faraday and Maxwell) or mathematical (Heaviside) using the non-Abelian Stokes theorem for any gauge symmetry ... 

The most telling example is the way in which the, often sterile, laws based on the symmetry of special relativity acquire physical significance within the broken symmetry of general relativity. It removes the major anomaly of time-reversible laws of microphysics underpinning reversible macro effects, and shows how local, rather than global gauge-symmetry breaking may cause the creation of massive particles. 

The second Higgs field acts in such a way that if the vacuum expectation value is zero, ( ) = 0, then the symmetry breaking mechanism effectively collapses to the Higgs mechanism of the standard SU(2) x U(l) electroweak theory. The result is a vector electromagnetic gauge theory 0(3)/> and a broken chiral SU(2) weak interaction theory. The mass of the vector boson sector is in the A(3) boson plus the W and Z° particles. 

Englert, F. and Brout, R. (1964). Broken symmetry and the mass of gauge vector mesons, Phys. Rev. Lett. 13, 321-323. 

The many-electron Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian is neither gauge invariant — it does not even contain vector potentials — nor Lorentz covariant as these symmetries have explicitly been broken in section 8.1. Moreover, the first-quantized relativistic many-particle Hamiltonians suffer from serious conceptual problems [217], which are solely related to the unbounded spectrum of the one-electron Dirac Hamiltonian h. ... 

What has happened is that in the spontaneously broken symmetry the gauge boson has acquired mass at the expense of the would-be Goldstone boson, which simply disappears. For each vector gauge field that gets massive we need one complex scalar field, one piece of which becomes unphysical and disappears (it reappears as the longitudinal mode of the vector field) leaving one real scalar physical field, the Higgs boson. 

The minimal standard model, a gauge theory with broken symmetry, constructed according to the rules studied in the previous chapters, provides a... 

P. W. Higgs, Broken symmetries, massless particles and gauge fields, Phys. Lett. 12,132 (1964). 

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