SQUARE45

The Weinberg angle

\theta_W

defines the mixing between the neutral gauge bosons

B_{\mu}

(hypercharge) and

W^3_{\mu}

(weak isospin) to form the physical photon

A_{\mu}

and the

Z

boson

Z_{\mu}

. The transformation is given by the rotation matrix: \n

\begin{pmatrix} A_{\mu} \ Z_{\mu} \end{pmatrix} = \begin{pmatrix} \cos \theta_W & \sin \theta_W \end{pmatrix} \begin{pmatrix} B_{\mu} \ W^3_{\mu} \end{pmatrix}

\nThis angle relates the coupling constants and masses:

\cos \theta_W = \frac{M_Z}{M_Z} \frac{M_W}{M_W} \frac{1}{\sqrt{1 - M_W^2/M_Z^2}}

(or simply

\cos \theta_W = \sqrt{M_W^2/M_Z^2}

in the limit of small mixing) and determines the electric charge

e = g \sin \theta_W

Weinberg Angle