stylistic:tone
Technical aspects and mathematical formulation

In the Standard Model, the Higgs field is a four-component scalar field that forms a complex doublet of the weak isospin SU(2) symmetry:

while the field has charge + 1 / 2 under the weak hypercharge U(1) symmetry.

Note: This article uses the scaling convention where the electric charge, Q, the weak isospin, T, and the weak hypercharge, Y, are related by Q = T + Y. A different convention used in most other Wikipedia articles is Q = T + 1 / 2 Y.

The Higgs part of the Lagrangian is

where W μ a {\displaystyle W_{\mu \,a}} and B μ {\displaystyle B_{\mu }} are the gauge bosons of the SU(2) and U(1) symmetries, g {\displaystyle g} and g ′ {\displaystyle g'} their respective coupling constants, σ a {\displaystyle \sigma ^{a}} are the Pauli matrices (a complete set of generators of the SU(2) symmetry), and λ > 0 {\displaystyle \lambda >0} and μ H 2 > 0 {\displaystyle \mu _{\text{H}}^{2}>0}, so that the ground state breaks the SU(2) symmetry (see figure).

The ground state of the Higgs field (the bottom of the potential) is degenerate with different ground states related to each other by a SU(2) gauge transformation. It is always possible to pick a gauge such that in the ground state ϕ 1 = ϕ 2 = ϕ 3 = 0 {\displaystyle \phi ^{1}=\phi ^{2}=\phi ^{3}=0}. The expectation value of ϕ 0 {\displaystyle \phi ^{0}} in the ground state (the vacuum expectation value or VEV) is then ⟨ ϕ 0 ⟩ = 1 2 v {\displaystyle \left\langle \phi ^{0}\right\rangle ={\tfrac {1}{\sqrt {2\,}}}v}, where v = 1 λ | μ H | {\displaystyle v={\tfrac {1}{\sqrt {\lambda \,}}}\left|\mu _{\text{H}}\right|}. The measured value of this parameter is ~ 246 GeV/ c. It has units of mass, and is the only free parameter of the Standard Model that is not a dimensionless number. Quadratic terms in W μ {\displaystyle W_{\mu }} and B μ {\displaystyle B_{\mu }} arise, which give masses to the W and Z bosons:

with their ratio determining the Weinberg angle, cos θ W = m W m Z = | g | g 2 + g ′ 2 {\textstyle \cos \theta _{\text{W}}={\frac {m_{\text{W}}}{\ m_{\text{Z}}\ }}={\frac {\left|\,g\,\right|}{\ {\sqrt {g^{2}+{g'}^{2}\ }}\ }}}, and leave a massless U(1) photon, γ {\displaystyle \gamma }. The mass of the Higgs boson itself is given by

The quarks and the leptons interact with the Higgs field through Yukawa interaction terms:

where (d, u, e, ν) L,R i {\displaystyle (d,u,e,\nu)_{\text{L,R}}^{i}} are left-handed and right-handed quarks and leptons of the i th generation, λ u,d,e i j {\displaystyle \lambda _{\text{u,d,e}}^{i\,j}} are matrices of Yukawa couplings where h.c. denotes the hermitian conjugate of all the preceding terms. In the symmetry breaking ground state, only the terms containing ϕ 0 {\displaystyle \phi ^{0}} remain, giving rise to mass terms for the fermions. Rotating the quark and lepton fields to the basis where the matrices of Yukawa couplings are diagonal, one gets

where the masses of the fermions are m u,d,e i = 1 2 λ u,d,e i v {\displaystyle m_{\text{u,d,e}}^{i}={\tfrac {1}{\sqrt {2\ }}}\lambda _{\text{u,d,e}}^{i}v}, and λ u,d,e i {\displaystyle \lambda _{\text{u,d,e}}^{i}} denote the eigenvalues of the Yukawa matrices.