Editing Electroweak interaction (section)

=== 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>