Before electroweak symmetry breaking The
Lagrangian for the electroweak interactions is divided into four parts before
electroweak symmetry breaking manifests, : \mathcal{L}_{\mathrm{EW}} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y~. The \mathcal{L}_g term describes the interaction between the three vector bosons and the
vector boson, : \mathcal{L}_g = -\tfrac{1}{4} W_{a}^{\mu\nu}W_{\mu\nu}^a - \tfrac{1}{4} B^{\mu\nu}B_{\mu\nu}, where W_{a}^{\mu\nu} (a=1,2,3) and B^{\mu\nu} are the
field strength tensors for the weak isospin and weak hypercharge gauge fields. \mathcal{L}_f is the
kinetic term for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the
gauge covariant derivative, : \mathcal{L}_f = \overline{Q}_j iD\!\!\!\!/\; Q_j+ \overline{u}_j iD\!\!\!\!/\; u_j+ \overline{d}_j iD\!\!\!\!/\; d_j + \overline{L}_j iD\!\!\!\!/\; L_j + \overline{e}_j iD\!\!\!\!/\; e_j, where the subscript sums over the three generations of fermions; , , and are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields; and and are the left-handed doublet and right-handed singlet electron fields. The
Feynman slash D\!\!\!\!/ means the contraction of the 4-gradient with the
Dirac matrices, defined as : D\!\!\!\!/ \equiv \gamma^\mu\ D_\mu, and the covariant derivative (excluding the
gluon gauge field for the
strong interaction) is defined as : \ D_\mu \equiv \partial_\mu - i\ \frac{g'}{2}\ Y\ B_\mu - i\ \frac{g}{2}\ T_j\ W_\mu^j. Here \ Y\ is the weak hypercharge and the \ T_j\ are the components of the weak isospin. The \mathcal{L}_h term describes the
Higgs field h and its interactions with itself and the gauge bosons, : \mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\ , where v is the
vacuum expectation value. The \ \mathcal{L}_y\ term describes the
Yukawa interaction with the fermions, : \mathcal{L}_y = - y_{u}^{ij}\epsilon^{ab}\ h_b^\dagger\ \overline{Q}_{ia} u_j^c - y_{d}^{ij}\ h\ \overline{Q}_i d^c_j - y_{e}^{ij}\ h\ \overline{L}_i e^c_j + \mathrm{h.c.} ~, and generates their masses, manifest when the Higgs field acquires a nonzero vacuum expectation value, discussed next. The \ y_k^{ij}\ , for \ k \in \{ \mathrm{u, d, e} \}\ , are matrices of Yukawa couplings.
After electroweak symmetry breaking The Lagrangian reorganizes itself as the Higgs field 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 (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. : \mathcal{L}_{\mathrm{EW}} = \mathcal{L}_\mathrm{K} + \mathcal{L}_\mathrm{N} + \mathcal{L}_\mathrm{C} + \mathcal{L}_\mathrm{H} + \mathcal{L}_{\mathrm{HV}} + \mathcal{L}_{\mathrm{WWV}} + \mathcal{L}_{\mathrm{WWVV}} + \mathcal{L}_\mathrm{Y} ~. The kinetic term \mathcal{L}_K 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) : \begin{align} \mathcal{L}_\mathrm{K} = \sum_f \overline{f}(i\partial\!\!\!/\!\;-m_f)\ f - \frac{1}{4}\ A_{\mu\nu}\ A^{\mu\nu} - \frac{1}{2}\ W^+_{\mu\nu}\ W^{-\mu\nu} + m_W^2\ W^+_\mu\ W^{-\mu} \\ \qquad -\frac{1}{4}\ Z_{\mu\nu}Z^{\mu\nu} + \frac{1}{2}\ m_Z^2\ Z_\mu\ Z^\mu + \frac{1}{2}\ (\partial^\mu\ H)(\partial_\mu\ H) - \frac{1}{2}\ m_H^2\ H^2 ~, \end{align} where the sum runs over all the fermions of the theory (quarks and leptons), and the fields \ A_{\mu\nu}\ , \ Z_{\mu\nu}\ , \ W^-_{\mu\nu}\ , and \ W^+_{\mu\nu} \equiv (W^-_{\mu\nu})^\dagger\ are given as : X^{a}_{\mu\nu} = \partial_\mu X^{a}_\nu - \partial_\nu X^{a}_\mu + g f^{abc}X^{b}_{\mu}X^{c}_{\nu} ~, with X to be replaced by the relevant field (A, Z, W^\pm) and by the structure constants of the appropriate gauge group. The neutral current \ \mathcal{L}_\mathrm{N}\ and charged current \ \mathcal{L}_\mathrm{C}\ components of the Lagrangian contain the interactions between the fermions and gauge bosons, : \mathcal{L}_\mathrm{N} = e\ J_\mu^\mathrm{em}\ A^\mu + \frac{g}{\ \cos\theta_W\ }\ (\ J_\mu^3 - \sin^2\theta_W\ J_\mu^\mathrm{em}\ )\ Z^\mu ~, where ~e = g\ \sin \theta_\mathrm{W} = g'\ \cos \theta_\mathrm{W} ~. The electromagnetic current \; J_\mu^{\mathrm{em}} \; is : J_\mu^\mathrm{em} = \sum_f \ q_f\ \overline{f}\ \gamma_\mu\ f ~, where \ q_f\ is the fermions' electric charges. The neutral weak current \ J_\mu^3\ is : J_\mu^3 = \sum_f\ T^3_f\ \overline{f}\ \gamma_\mu\ \frac{\ 1-\gamma^5\ }{2}\ f ~, where T^3_f is the fermions' weak isospin.{{efn|name=note_chiral_factors| Note the factors ~\tfrac{1}{2}\ (1-\gamma^5)~ in the weak coupling formulas: These factors are deliberately inserted to expunge any left-
chiral components of the spinor fields. This is why electroweak theory is said to be a
chiral theory.}} The charged current part of the Lagrangian is given by : \mathcal{L}_\mathrm{C} = -\frac{g}{\ \sqrt{2 \;}\ }\ \left[\ \overline{u}_i\ \gamma^\mu\ \frac{\ 1 - \gamma^5\ }{2} \; M^{\mathrm{CKM}}_{ij}\ d_j + \overline{\nu}_i\ \gamma^\mu\;\frac{\ 1-\gamma^5\ }{2} \; e_i\ \right]\ W_\mu^{+} + \mathrm{h.c.} ~, where \ \nu\ is the right-handed singlet neutrino field, and the
CKM matrix M_{ij}^\mathrm{CKM} determines the mixing between mass and weak eigenstates of the quarks. \mathcal{L}_\mathrm{H} contains the Higgs three-point and four-point self interaction terms, : \mathcal{L}_\mathrm{H} = -\frac{\ g\ m_\mathrm{H}^2\,}{\ 4\ m_\mathrm{W}\ }\;H^3 - \frac{\ g^2\ m_\mathrm{H}^2\ }{32\ m_\mathrm{W}^2}\;H^4 ~. \mathcal{L}_{\mathrm{HV}} contains the Higgs interactions with gauge vector bosons, : \mathcal{L}_\mathrm{HV} =\left(\ g\ m_\mathrm{HV} + \frac{\ g^2\ }{4}\;H^2\ \right)\left(\ W^{+}_\mu\ W^{-\mu} + \frac{1}{\ 2\ \cos^2\ \theta_\mathrm{W}\ }\;Z_\mu\ Z^\mu\ \right) ~. \mathcal{L}_{\mathrm{WWV}} contains the gauge three-point self interactions, : \mathcal{L}_{\mathrm{WWV}} = -i\ g\ \left[\; \left(\ W_{\mu\nu}^{+}\ W^{-\mu} - W^{+\mu}\ W^{-}_{\mu\nu}\ \right)\left(\ A^\nu\ \sin \theta_\mathrm{W} - Z^\nu\ \cos\theta_\mathrm{W}\ \right) + W^{-}_\nu\ W^{+}_\mu\ \left(\ A^{\mu\nu}\ \sin \theta_\mathrm{W} - Z^{\mu\nu}\ \cos \theta_\mathrm{W}\ \right) \;\right] ~. \mathcal{L}_{\mathrm{WWVV}} contains the gauge four-point self interactions, : \begin{align} \mathcal{L}_{\mathrm{WWVV}} = -\frac{\ g^2\ }{4}\ \Biggl\{\ &\Bigl[\ 2\ W^{+}_\mu\ W^{-\mu} + (\ A_\mu\ \sin \theta_\mathrm{W} - Z_\mu\ \cos \theta_\mathrm{W} \ )^2\ \Bigr]^2 \\ &- \Bigl[\ W_\mu^{+}\ W_\nu^{-} + W^{+}_\nu\ W^{-}_\mu + \left(\ A_\mu\ \sin \theta_\mathrm{W} - Z_\mu\ \cos \theta_\mathrm{W}\ \right)\left(\ A_\nu\ \sin \theta_\mathrm{W} - Z_\nu\ \cos \theta_\mathrm{W}\ \right)\ \Bigr]^2\,\Biggr\} ~. \end{align} \ \mathcal{L}_\mathrm{Y}\ contains the Yukawa interactions between the fermions and the Higgs field, : \mathcal{L}_\mathrm{Y} = -\sum_f\ \frac{\ g\ m_f\ }{2\ m_\mathrm{W}} \; \overline{f}\ f\ H ~. == See also ==