Final answer:
The factor of half in the Lagrangian for the Z boson's mass arises because the Z boson is neutral. Lagrangians for real fields include a 1/2 factor to avoid overcounting self-interactions. This supports the prediction of the electroweak theory, confirmed by CERN experiments.
Step-by-step explanation:
The student asked why there is a factor of half in the Lagrangian mass term for the Z boson compared to the W boson. The factor of 1/2 for the Z boson is due to the fact that the W bosons (W+ and W-) are charged and can be treated as distinct particles, while the Z boson is neutral and only one such particle exists.
In the Lagrangian, terms including mass are quadratic in the fields, and for real (uncharged) fields, this quadratic term comes with a factor of 1/2 to avoid double-counting in the field's self-interaction, especially when considering the expansion of the fields around the vacuum expectation value (VEV) after symmetry breaking. This is consistent with the foundations of quantum field theory, where the Lagrangian should accurately reflect the physical degrees of freedom and their interactions.
When discussing the unification of forces, the electroweak theory describes how at energies above approximately 100 GeV, corresponding to extremely small distances, the electromagnetic and weak nuclear forces become the same, and carrier particles like the W and Z bosons can be produced. These predictions were spectacularly confirmed at CERN with the detection of the W and Z bosons at predicted masses, and later with the discovery of the Higgs boson, providing further evidence for the theory.