D_{} = + iqA_{}.
| (14) |

L = D^{ }D_{} - V() - F_{}F^{ }.
| (15) |

with any differentiable function. Continuing in exactly the same way as for the Goldstone model with a negative and expressing the Lagrangian in terms of the variables and as defined in (2.15) the result is

The Lagrangian clearly has a massive vector boson field

(x) = [v + (x)].
| (16) |

In summary, it is seen that a complex scalar field and a massless vector field, both with two degrees of freedom, in (2.20) as a result of the Higgs mechanism was transformed in (2.24) into one real scalar field with one degree of freedom and a massive vector boson field with 3 degrees of freedom. A massless spin 1 particle has two transverse polarised states while a massive spin 1 particle has an additional longitudinal polarised state. It should be noted that the field only disappears if the bosons are massless. As shown in section 2.1.1 this requires the vacuum state to be degenerate ie. the Higgs mechanism will only work with a degenerate vacuum.

The Higgs mechanism was demonstrated here for a U(1) gauge invariant
Lagrangian. To extend it to the SU(2) `x` U(1) gauge invariant
Lagrangian of the electroweak theory is relatively simple. The
starting point is a Lagrangian with a complex scalar doublet and four
massless vector bosons. Counting degrees of freedom gives four from
the scalars and eight from the vector bosons.

Through the Higgs mechanism the Lagrangian is transformed into one real scalar, three massive vector and one massless vector boson. The massless vector boson is of course to be identified with the photon and the single remaining scalar with the Higgs boson. Counting degrees of freedom again gives one from the Higgs, two from the photon and nine from the massive vector bosons, again adding up to twelve.

Introducing the masses of the vector bosons with one doublet of complex scalars is the simplest scenario, in principle an infinite number of scalar fields can be introduced. The simplest supersymmetric models instead have five scalar fields left after the Higgs mechanism: a doublet of charged scalars, two neutral scalars and one neutral pseudoscalar.

The masses of the particles in the standard model are given as

m_{H} = v, m_{W} = vg, m_{Z} = ,
| (17) |

v^{ 2} = , = ,
| (18) |

The vacuum expectation value is easily extracted as

v = = 246GeV ,
| (19) |

For reference the full electroweak and Higgs sector part of the standard model Lagrangian is given. It takes the form

L = L _{0} + L _{FB} + L _{FH} + L _{BB} + L _{BH} + L _{HH}
| (20) |

and the remaining terms the interaction between fermions and bosons, fermions and the Higgs, bosons and bosons, bosons and the Higgs and finally Higgs self interactions. Only the fermion Higgs interaction term

L _{FH} = - m_{f}
| (21) |

A remarkable feature of introducing massive fermions in the theory is that their interaction terms in the Lagrangian become proportional to their mass as seen in (2.30). While this makes it impossible to put constraints on the Higgs mass and the fermion masses, it has the consequence that the Higgs particle will couple to the fermions in proportion to their mass. The supersymmetric extensions of the Standard Model have different predictions for the coupling constants and hence a measurement of several coupling constants can be used to look into the theory behind the Higgs mechanism.