section_path: Theoretical aspects/Construction of the Standard Model Lagrangian/Higgs sector
section_name: Higgs sector
paragraph_text: Higgs sector
  
  In the Standard Model, the Higgs field is an SU (2) L {\displaystyle \operatorname {SU} (2)_{\text{L}}} doublet of complex scalar fields with four degrees of freedom:
  
  φ = (φ + φ 0) = 1 2 (φ 1 + i φ 2 φ 3 + i φ 4), {\displaystyle \varphi ={\begin{pmatrix}\varphi ^{+}\\\varphi ^{0}\end{pmatrix}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\varphi _{1}+i\varphi _{2}\\\varphi _{3}+i\varphi _{4}\end{pmatrix}},} where the superscripts + and 0 indicate the electric charge Q {\displaystyle Q} of the components. The weak hypercharge Y W {\displaystyle Y_{\text{W}}} of both components is 1. Before symmetry breaking, the Higgs Lagrangian is L H = (D μ φ) † (D μ φ) − V (φ), {\displaystyle {\mathcal {L}}_{\text{H}}=\left(D_{\mu }\varphi \right)^{\dagger }\left(D^{\mu }\varphi \right)-V(\varphi),} where D μ {\displaystyle D_{\mu }} is the electroweak gauge covariant derivative defined above and V (φ) {\displaystyle V(\varphi)} is the potential of the Higgs field. The square of the covariant derivative leads to three and four point interactions between the electroweak gauge fields W μ a {\displaystyle W_{\mu }^{a}} and B μ {\displaystyle B_{\mu }} and the scalar field φ {\displaystyle \varphi }. The scalar potential is given by V (φ) = − μ 2 φ † φ + λ (φ † φ) 2, {\displaystyle V(\varphi)=-\mu ^{2}\varphi ^{\dagger }\varphi +\lambda \left(\varphi ^{\dagger }\varphi \right)^{2},} where μ 2 > 0 {\displaystyle \mu ^{2}>0}, so that φ {\displaystyle \varphi } acquires a non-zero Vacuum expectation value, which generates masses for the Electroweak gauge fields (the Higgs' mechanism), and λ > 0 {\displaystyle \lambda >0}, so that the potential is bounded from below. The quartic term describes self-interactions of the scalar field φ {\displaystyle \varphi }.
  
  The minimum of the potential is degenerate with an infinite number of equivalent ground state solutions, which occurs when φ † φ = μ 2 2 λ {\displaystyle \varphi ^{\dagger }\varphi ={\tfrac {\mu ^{2}}{2\lambda }}}. It is possible to perform a gauge transformation on φ {\displaystyle \varphi } such that the ground state is transformed to a basis where φ 1 = φ 2 = φ 4 = 0 {\displaystyle \varphi _{1}=\varphi _{2}=\varphi _{4}=0} and φ 3 = μ λ ≡ v {\displaystyle \varphi _{3}={\tfrac {\mu }{\sqrt {\lambda }}}\equiv v}. This breaks the symmetry of the ground state. The expectation value of φ {\displaystyle \varphi } now becomes ⟨ φ ⟩ = 1 2 (0 v), {\displaystyle \langle \varphi \rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0\\v\end{pmatrix}},} where v {\displaystyle v} has units of mass and sets the scale of electroweak physics. This is the only dimensional parameter of the Standard Model and has a measured value of ~ 246 GeV/ c.
  
  After symmetry breaking, the masses of the W {\displaystyle {\text{W}}} and Z {\displaystyle {\text{Z}}} are given by m W = 1 2 g v {\displaystyle m_{\text{W}}={\frac {1}{2}}gv} and m Z = 1 2 g 2 + g ′ 2 v {\displaystyle m_{\text{Z}}={\frac {1}{2}}{\sqrt {g^{2}+g'^{2}}}v}, which can be viewed as predictions of the theory. The photon remains massless. The mass of the Higgs Boson is m H = 2 μ 2 = 2 λ v {\displaystyle m_{\text{H}}={\sqrt {2\mu ^{2}}}={\sqrt {2\lambda }}v}. Since μ {\displaystyle \mu } and λ {\displaystyle \lambda } are free parameters, the Higgs' mass could not be predicted beforehand and had to be determined experimentally.