Some Problems about the Source of Mass in the Electroweak Theory

We review the electroweak theory to find out some noteworthy issues. In this theory, the Higgs mechanism makes the gauge bosons obtain their mass. If the vacuum states of the Higgs fields are the sources of mass for the massive gauge bosons W and Z even electron e-, then this lowest energy v of the Higgs field must be smaller than the Higgs boson of 125 GeV even smaller than the electron’s rest mass of 0.511 MeV. It shall be like the zero-point energy of a linearly harmonic oscillator and those massive gauge bosons consist of many such lowest-energy quanta. However, substituting the weak coupling constant g=0.77 into the mass equation of the W boson, it directly gives v equal to 208 GeV much heavier than the Higgs boson and the similar results have been revealed in some textbooks about twenty years ago. If it means v lower than the Higgs boson, then the lowest-energy of the Higgs field is negative! Furthermore, the scalar Higgs boson is a charge-zero (q=0) and spin-zero (S=0) massive particle so the vacuum states of the Higgs fields have the same characteristics if they were treated as the lowest-energy quanta. However, the massive gauge bosons W and Z are all spin-1 (S=1) particles and moreover, W bosons are charged. Therefore, how to constitute those massive gauges bosons from the vacuum states of the Higgs fields becomes a questionable concept. On the other hand, due to the local gauge invariance, all mass terms have to be removed for fermions and the Yukawa coupling can provide their mass through the Higgs mechanism. It is also a similar problem that the fermion like electron is a spin-1/2 (S=1/2) massive particle and how to constitute the mass of electron from the vacuum states like the spin-zero Higgs bosons is another serious problem. Those considerations cause seriously ponder whether the Yukawa coupling is the way to provide the mass of fermion? Especially, the electron-positron pair production from two photons directly tells us that the mass of electron and positron is much easier from the photon fields through the coupling above 1.02 MeV. Even for the scalar Higgs boson H0, it can come from different parent particles. We also mathematically discuss the symmetry of the scalar field Φ under the gauge transformation and find the Lagrangian still holding its symmetry even Φ ⟶ -Φ.