Before electroweak symmetry breaking
The Lagrangian for the electroweak interactions is divided into four parts before electroweak symmetry breaking becomes manifest,
The term describes the interaction between the three W vector bosons and the B vector boson,
where () and are the field strength tensors for the weak isospin and weak hypercharge gauge fields.
is the kinetic term for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the gauge covariant derivative,
where the subscript j sums over the three generations of fermions; Q, u, and d are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields; and L and e are the left-handed doublet and right-handed singlet electron fields.
The Feynman slash means the contraction of the 4-gradient with the Dirac matrices, defined as
and the covariant derivative (excluding the gluon gauge field for the strong interaction) is defined as
Here is the weak hypercharge and the are the components of the weak isospin.
The term describes the Higgs field and its interactions with itself and the gauge bosons,
where is the vacuum expectation value.
The term describes the Yukawa interaction with the fermions,
and generates their masses, manifest when the Higgs field acquires a nonzero vacuum expectation value, discussed next. The for are matrices of Yukawa couplings.
After electroweak symmetry breaking
The Lagrangian reorganizes itself as the Higgs boson acquires a non-vanishing vacuum expectation value dictated by the potential of the previous section. As a result of this rewriting, the symmetry breaking becomes manifest. In the history of the universe, this is believed to have happened shortly after the hot big bang, when the universe was at a temperature 159.5±1.5 GeV[12]
(assuming the Standard Model of particle physics).
Due to its complexity, this Lagrangian is best described by breaking it up into several parts as follows.
The kinetic term contains all the quadratic terms of the Lagrangian, which include the dynamic terms (the partial derivatives) and the mass terms (conspicuously absent from the Lagrangian before symmetry breaking)
where the sum runs over all the fermions of the theory (quarks and leptons), and the fields and are given as
with to be replaced by the relevant field ( ) and f abc by the structure constants of the appropriate gauge group.
The neutral current and charged current components of the Lagrangian contain the interactions between the fermions and gauge bosons,
where The electromagnetic current is
where is the fermions' electric charges.
The neutral weak current is
where is the fermions' weak isospin.[lower-alpha 4]
The charged current part of the Lagrangian is given by
where is the right-handed singlet neutrino field, and the CKM matrix determines the mixing between mass and weak eigenstates of the quarks.[lower-alpha 4]
contains the Higgs three-point and four-point self interaction terms,
contains the Higgs interactions with gauge vector bosons,
contains the gauge three-point self interactions,
contains the gauge four-point self interactions,
contains the Yukawa interactions between the fermions and the Higgs field,