Higgs Boson - Technical Aspects and Mathematical Formulation

Technical Aspects and Mathematical Formulation

See also: Standard Model (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:

\phi=\frac{1}{\sqrt{2}} \left( \begin{array}{c} \phi^1 + i\phi^2 \\ \phi^0+i\phi^3 \end{array} \right)\;,

(1)

while the field has charge +1/2 under the weak hypercharge U(1) symmetry (in the convention where the electric charge, Q, the weak isospin, I3, and the weak hypercharge, Y, are related by Q = I3 + Y).

The Higgs part of the Lagrangian is

(2)

where and are the gauge bosons of the SU(2) and U(1) symmetries, and their respective coupling constants, (where are the Pauli matrices) a complete set generators of the SU(2) symmetry, and and, 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 . The expectation value of in the ground state (the vacuum expectation value or vev) is then, where . The measured value of this parameter is ~246 GeV/c2. It has units of mass, and is the only free parameter of the Standard Model that is not a dimensionless number. Quadratic terms in and arise, which give masses to the W and Z bosons:

(3)

(4)

with their ratio determining the Weinberg angle, and leave a massless U(1) photon, .

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

\begin{align}\mathcal{L}_{Y} = &-\lambda_u^{ij}\frac{\phi^0-i\phi^3}{\sqrt{2}}\overline u_L^i u_R^j +\lambda_u^{ij}\frac{\phi^1-i\phi^2}{\sqrt{2}}\overline d_L^i u_R^j\\ &-\lambda_d^{ij}\frac{\phi^0+i\phi^3}{\sqrt{2}}\overline d_L^i d_R^j -\lambda_d^{ij}\frac{\phi^1+i\phi^2}{\sqrt{2}}\overline u_L^i d_R^j\\ &-\lambda_e^{ij}\frac{\phi^0+i\phi^3}{\sqrt{2}}\overline e_L^i e_R^j -\lambda_e^{ij}\frac{\phi^1+i\phi^2}{\sqrt{2}}\overline \nu_L^i e_R^j + \textrm{h.c.},\end{align}

(5)

where are left-handed and right-handed quarks and leptons of the ith generation, are matrices of Yukawa couplings where h.c. denotes the hermitian conjugate terms. In the symmetry breaking ground state, only the terms containing 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

(6)

where the masses of the fermions are, and denote the eigenvalues of the Yukawa matrices.

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