section_path: Theoretical aspects/Construction of the Standard Model Lagrangian/Electroweak sector
section_name: Electroweak sector
paragraph_text: Electroweak sector
  
  The electroweak sector is a Yang–Mills gauge theory with the symmetry group U(1) × SU(2),
  
  L EW = Q ¯ L j i γ μ D μ Q L j + u ¯ R j i γ μ D μ u R j + d ¯ R j i γ μ D μ d R j + ℓ ¯ L j i γ μ D μ ℓ L j + e ¯ R j i γ μ D μ e R j − 1 4 W a μ ν W μ ν a − 1 4 B μ ν B μ ν, {\displaystyle {\mathcal {L}}_{\text{EW}}={\overline {Q}}_{Lj}i\gamma ^{\mu }D_{\mu }Q_{Lj}+{\overline {u}}_{Rj}i\gamma ^{\mu }D_{\mu }u_{Rj}+{\overline {d}}_{Rj}i\gamma ^{\mu }D_{\mu }d_{Rj}+{\overline {\ell }}_{Lj}i\gamma ^{\mu }D_{\mu }\ell _{Lj}+{\overline {e}}_{Rj}i\gamma ^{\mu }D_{\mu }e_{Rj}-{\tfrac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\tfrac {1}{4}}B^{\mu \nu }B_{\mu \nu },}
  
  where the subscript j {\displaystyle j} sums over the three generations of fermions; Q L, u R {\displaystyle Q_{L},u_{R}}, and d R {\displaystyle d_{R}} are the left-handed doublet, right-handed singlet up type, and right handed singlet down type quark fields; and ℓ L {\displaystyle \ell _{L}} and e R {\displaystyle e_{R}} are the left-handed doublet and right-handed singlet lepton fields.
  
  The electroweak gauge covariant derivative is defined as D μ ≡ ∂ μ − i g ′ 1 2 Y W B μ − i g 1 2 τ → L W → μ {\displaystyle D_{\mu }\equiv \partial _{\mu }-ig'{\tfrac {1}{2}}Y_{\text{W}}B_{\mu }-ig{\tfrac {1}{2}}{\vec {\tau }}_{\text{L}}{\vec {W}}_{\mu }}, where
  
  Notice that the addition of fermion mass terms into the electroweak Lagrangian is forbidden, since terms of the form m ψ ¯ ψ {\displaystyle m{\overline {\psi }}\psi } do not respect U(1) × SU(2) gauge invariance. Neither is it possible to add explicit mass terms for the U(1) and SU(2) gauge fields. The Higgs mechanism is responsible for the generation of the gauge boson masses, and the fermion masses result from Yukawa-type interactions with the Higgs field.