= Weinberg angle
{c}
{title2=$\theta_W$}
{wiki}
= Weak mixing angle
{synonym}
The <Weinberg angle> rotates the neutral <gauge fields> into the <photon> and <Z boson>:
$$
\begin{pmatrix}A_\mu\\ Z_\mu\end{pmatrix}
=\begin{pmatrix}\sin\theta_W&\cos\theta_W\\ \cos\theta_W&-\sin\theta_W\end{pmatrix}
\begin{pmatrix}W^3_\mu\\ B_\mu\end{pmatrix},\qquad
\tan\theta_W=\frac{g'}{g},\qquad
e=g\sin\theta_W=g'\cos\theta_W.
$$
$g$ belongs to $SU(2)_L$, $g'$ to $U(1)_Y$, and $e$ is the positive magnitude of the <electron> <electric charge>. These definitions use $Q=T_3+Y$; doubling the definition of <hypercharge> requires halving its coefficient in the <covariant derivative>. The <electromagnetic coupling of a weak multiplet> follows by substituting this rotation.
For a <fermion> component, the corresponding neutral coefficient is
$$
g\cos\theta_W T_3-g'\sin\theta_WY
=\frac{g}{\cos\theta_W}\left(T_3-Q\sin^2\theta_W\right).
$$
For a left-handed charged <lepton>, $T_3=-1/2$ and $Q=-1$, giving $g_L=-1/2+\sin^2\theta_W$. Its right-handed component has $T_3=0$, giving $g_R=\sin^2\theta_W$. This algebra links the <neutral current> to the electromagnetic charge assignment. Together with the <Higgs doublet>, it also gives the <tree-level electroweak gauge-boson masses> with $m_W/m_Z=\cos\theta_W$; precise comparisons require a defined renormalization convention and <radiative corrections>.
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