The Electroweak Theory

All the forces described above are formulated in the Standard Model as gauge symmetric quantum field theories. The strong force is described by quantum chromodynamics (QCD), whereas the electromagnetic and weak forces are both described by the electroweak theory.

Any system can be described by its Lagrangian. The Lagrangian contains creation and annihilation operators that act at a particular position in space, and thus they are field operators, hence the name ``quantum field theory.''

The theory contains fermionic fields representing the fermions and when local symmetry is imposed, gauge fields arise that form the interaction terms with the fermionic fields. These gauge fields are then identified as the gauge bosons discussed earlier.

The Electroweak theory is an ${\rm SU}(2)_{L}\times {\rm U}(1)_{Y}$ gauge symmetric field theory. It has four gauge fields, three associated with weak isospin, these being the ${\bf W}^{\mu}$ fields. The subscript on the SU(2)$_{L}$ indicates that these fields only couple to left-handed fermions. There is then the $B^{\mu}$ field that couples to the weak hypercharge (Y) of particles. All fermions have non-zero weak hypercharge.

The four fields cannot be directly associated with the four gauge bosons discussed earlier, for a start they represent massless particles. These fields may be connected to the massive gauge bosons via the Higgs mechanism [9]. This causes the gauge fields to mix. The ${W}^{+\mu}$ and ${W}^{-\mu}$ fields gain mass from the vacuum expectation value. These two fields can then be directly related with the ${\rm W}^{+}$ and ${\rm W}^{-}$ bosons. The ${W}^{0\mu}$ and $B^{\mu}$ fields mix to form two new fields, the $Z^{\mu}$ and $A^{\mu}$ fields, that can be identified as the ${\rm Z}^{0}$ and $\gamma$ bosons respectively. The form of the mixed fields is shown below.

$$Z^{\mu} = \cos\theta_{w}{W}^{0\mu} - \sin\theta_{w}B^{\mu}$$ (1.7)

$$A^{\mu} = \sin\theta_{w}{W}^{0\mu} + \cos\theta_{w}B^{\mu}$$ (1.8)