Math2

Higgs sector

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

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

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:

$${\mathcal {L}}_{\mathrm {H} }=\varphi ^{\dagger }\left({\partial ^{\mu }}-{i \over 2}\left(g'Y_{\mathrm {W} }B^{\mu }+g{\vec {\tau }}{\vec {W}}^{\mu }\right)\right)\left(\partial _{\mu }+{i \over 2}\left(g'Y_{\mathrm {W} }B_{\mu }+g{\vec {\tau }}{\vec {W}}_{\mu }\right)\right)\varphi \ -\ {\lambda ^{2} \over 4}\left(\varphi ^{\dagger }\varphi -v^{2}\right)^{2}\;,$$

which can also be written as:

${\mathcal {L}}_{\mathrm {H} }=\left|\left(\partial _{\mu }+{i \over 2}\left(g'Y_{\mathrm {W} }B_{\mu }+g{\vec {\tau }}{\vec {W}}_{\mu }\right)\right)\varphi \right|^{2}\ -\ {\lambda ^{2} \over 4}\left(\varphi ^{\dagger }\varphi -v^{2}\right)^{2}\;.$