# Weak Interactions

Although weak interactions can be described by a non–Abelian $SU(2)$ group too, the phenomenology of Nature complicates the form of the corresponding Lagrangian.
The CP violation can be formalized assuming that weak interactions should act only on left–chirality states. Left–handed fermions therefore form a weak isospin doublet

where the left–handed states are

The electron neutrino, known to be nearly massless, is idealized to be exactly massless. As a consequence its right–handed state

$\nu_R=\frac{1}{2}(1+\gamma_5)\nu = 0$

would not exist and this leads to have only one right–handed fermion, which constitute the weak–isospin singlet

$R\equiv e_R=\frac{1}{2}(1+\gamma_5)e.$

The group that describes the symmetry of weak isospin is therefore called $SU(2)_L$, where $L$ stands for “left”. Anyway, neutral currents can couple even right–handed fermions, although in a different way.

The construction of a model which takes account of the peculiarities of the weak interaction was achieved by Glashow, Weinberg and Salam at the end of the 1960s. Weak and electromagnetic interactions were unified in the single $SU(2)_L \otimes U(1)_Y$ symmetry group where $Y$, the weak hypercharge, is the generator for the group and it is obtained from the Gell-Mann — Nishijima relation $Q=\frac{Y}{2}+I_3$.
The request of local gauge invariance leads to the introduction of four vector bosons.

Weak and electromagnetic interactions are considered unified in the single $U(1)_Y\otimes SU(2)_L$ symmetry group where $Y$, the weak hypercharge, is the generator of the group that is given by the Gell-Mann–Nishijima relation between electrical charge and isospin $Q=\frac{Y}{2}+I_3$.
The subscript $L$ is used to denote left-handed spinors recording the vector-axial nature of the charged currents and the subscript $C$ refers to the charge of strong interactions given by the colour.

The gauge fields associated with $SU(2)_L$ are $W^1$, $W^2$ and $W^0$, while $B$ is associated with the $U(1)_Y$ group. Charged bosons $W^\pm$ come from

$W^\pm_\mu=\frac{W_\mu^1\mp iW_\mu^2}{\sqrt 2}.$

The photon and the neutral $Z^0$ boson can be obtained as a combination of the neutral fields

$W^0_\mu= Z_\mu cos\theta_W + A_\mu sin\theta_W$$B^0_\mu=-Z_\mu sin\theta_W + A_\mu cos\theta_W$

where the rotation angle $\theta_W$ takes the name of electroweak mixing angle$A_\mu$ is the photon field, $Z_\mu$ is the field associated to the $Z^0$ boson.