Editing Electroweak interaction (section)

== Lagrangian ==

=== Before electroweak symmetry breaking ===
The [[Lagrangian (field theory)|Lagrangian]] for the electroweak interactions is divided into four parts before [[electroweak symmetry breaking]] manifests,
: <math>\mathcal{L}_{\mathrm{EW}} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y~.</math>

The <math>\mathcal{L}_g</math> term describes the interaction between the three {{mvar|W}}  vector bosons and the {{mvar|B}} [[vector boson]], 
: <math>\mathcal{L}_g = -\tfrac{1}{4} W_{a}^{\mu\nu}W_{\mu\nu}^a - \tfrac{1}{4} B^{\mu\nu}B_{\mu\nu},</math>
where <math>W_{a}^{\mu\nu}</math> (<math>a=1,2,3</math>) and <math>B^{\mu\nu}</math> are the [[field strength tensor]]s for the weak isospin and weak hypercharge gauge fields.

<math>\mathcal{L}_f</math> is the [[kinetic term]] for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the [[gauge covariant derivative]],
: <math>\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,</math>
where the subscript {{mvar|j}} sums over the three generations of fermions; {{mvar|Q}}, {{mvar|u}}, and {{mvar|d}} are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields; and {{mvar|L}} and {{mvar|e}} are the left-handed doublet and right-handed singlet electron fields.
The [[Feynman slash notation|Feynman slash]] <math>D\!\!\!\!/</math> means the contraction of the 4-gradient with the [[Dirac matrices]], defined as
: <math>D\!\!\!\!/ \equiv \gamma^\mu\ D_\mu,</math>
and the covariant derivative (excluding the [[gluon]] gauge field for the [[strong interaction]]) is defined as
: <math>\ D_\mu \equiv \partial_\mu - i\ \frac{g'}{2}\ Y\ B_\mu - i\ \frac{g}{2}\ T_j\ W_\mu^j.</math>
Here <math>\ Y\ </math> is the weak hypercharge and the <math>\ T_j\ </math> are the components of the weak isospin.

The <math>\mathcal{L}_h</math> term describes the [[Higgs field]] <math>h</math> and its interactions with itself and the gauge bosons,
: <math>\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\ ,</math>
where <math>v</math> is the [[vacuum expectation value]].

The <math>\ \mathcal{L}_y\ </math> term describes the [[Yukawa interaction]] with the fermions, 
: <math>\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.} ~,</math>
and generates their masses,  manifest when the Higgs field acquires a nonzero vacuum expectation value, discussed next. The <math>\ y_k^{ij}\ ,</math> for <math>\ k \in \{ \mathrm{u, d, e} \}\ ,</math> 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 {{val|159.5|1.5|ul=GeV}}<ref>
{{cite journal
  |author1   = D'Onofrio, Michela
  |author2   = Rummukainen, Kari
  |year      = 2016
  |title     = Standard model cross-over on the lattice
  |journal   = Phys. Rev. D
  |volume    = 93  |number = 2  |page = 025003
  |doi       = 10.1103/PhysRevD.93.025003  |s2cid  = 119261776
  |bibcode   = 2016PhRvD..93b5003D  |hdl = 10138/159845 |hdl-access= free
  |arxiv     = 1508.07161
}}
</ref>
(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.
: <math>\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} ~.</math>

The kinetic term <math>\mathcal{L}_K</math> 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)
: <math> 
\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}
</math>
where the sum runs over all the fermions of the theory (quarks and leptons), and the fields <math>\ A_{\mu\nu}\ ,</math> <math>\ Z_{\mu\nu}\ ,</math> <math>\ W^-_{\mu\nu}\ ,</math> and <math>\ W^+_{\mu\nu} \equiv (W^-_{\mu\nu})^\dagger\ </math> are given as
: <math>X^{a}_{\mu\nu} = \partial_\mu X^{a}_\nu - \partial_\nu X^{a}_\mu + g f^{abc}X^{b}_{\mu}X^{c}_{\nu} ~,</math>
with <math>X</math> to be replaced by the relevant field (<math>A,</math> <math>Z,</math> <math>W^\pm</math>) and  {{mvar|f&thinsp;{{sup|abc}} }} by the structure constants of the appropriate gauge group.

The neutral current <math>\ \mathcal{L}_\mathrm{N}\ </math> and charged current <math>\ \mathcal{L}_\mathrm{C}\ </math> components of the Lagrangian contain the interactions between the fermions and gauge bosons,
: <math>\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 ~,</math>
where <math>~e = g\ \sin \theta_\mathrm{W} = g'\ \cos \theta_\mathrm{W} ~.</math> The electromagnetic current <math>\; J_\mu^{\mathrm{em}} \;</math> is
: <math>J_\mu^\mathrm{em} = \sum_f \ q_f\ \overline{f}\ \gamma_\mu\ f ~,</math>
where <math>\ q_f\ </math> is the fermions' electric charges. 
The neutral weak current <math>\ J_\mu^3\ </math> is
: <math>J_\mu^3 = \sum_f\ T^3_f\ \overline{f}\ \gamma_\mu\ \frac{\ 1-\gamma^5\ }{2}\ f ~,</math>
where <math>T^3_f</math> is the fermions' weak isospin.{{efn|name=note_chiral_factors| Note the factors <math>~\tfrac{1}{2}\ (1-\gamma^5)~</math> in the weak coupling formulas: These factors are deliberately inserted to expunge any left-[[chirality (physics)|chiral]] components of the spinor fields. This is why electroweak theory is said to be a '<nowiki/>''[[chiral theory]]''<nowiki/>'.}}

The charged current part of the Lagrangian is given by
: <math>\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.} ~,</math>
where <math>\ \nu\ </math> is the right-handed singlet neutrino field, and the [[CKM matrix]] <math>M_{ij}^\mathrm{CKM}</math> determines the mixing between mass and weak eigenstates of the quarks.{{efn|name=note_chiral_factors}}

<math>\mathcal{L}_\mathrm{H}</math> contains the Higgs three-point and four-point self interaction terms,
: <math>\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 ~.</math>

<math>\mathcal{L}_{\mathrm{HV}}</math> contains the Higgs interactions with gauge vector bosons,
: <math>\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) ~.</math>

<math>\mathcal{L}_{\mathrm{WWV}}</math> contains the gauge three-point self interactions,
: <math>\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] ~.</math>

<math>\mathcal{L}_{\mathrm{WWVV}}</math> contains the gauge four-point self interactions,
: <math>
\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}
</math>

<math>\ \mathcal{L}_\mathrm{Y}\ </math> contains the Yukawa interactions between the fermions and the Higgs field,
: <math>\mathcal{L}_\mathrm{Y} = -\sum_f\ \frac{\ g\ m_f\ }{2\ m_\mathrm{W}} \; \overline{f}\ f\ H ~.</math>