Math2

Higgs sector

In the Standard Model, the Higgs field is a complex scalar of the group SU(2)_L:

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

where the indices + and 0 indicate the electric charge (Q) of the components. The weak isospin ($Y_W$) of both components is 1.

Before symmetry breaking, the Higgs Lagrangian is: $${{ℒ}}_{\mathrm{H}}={\phi }^{†}({\partial }^{\mu }-\frac{i}{2}(g^{\prime }{Y}_{\mathrm{W}}{B}^{\mu }+g\overrightarrow{\tau }{\overrightarrow{W}}^{\mu }))({\partial }_{\mu }+\frac{i}{2}(g^{\prime }{Y}_{\mathrm{W}}{B}_{\mu }+g\overrightarrow{\tau }{\overrightarrow{W}}_{\mu }))\phi -\frac{{\lambda }^2}{4}{({\phi }^{†}\phi -v^2)}^2,$$

which can also be written as:

${{ℒ}}_{\mathrm{H}}={\left|({\partial }_{\mu }+\frac{i}{2}(g^{\prime }{Y}_{\mathrm{W}}{B}_{\mu }+g\overrightarrow{\tau }{\overrightarrow{W}}_{\mu }))\phi \right|}^2-\frac{{\lambda }^2}{4}{({\phi }^{†}\phi -v^2)}^2.$