4.1. Weinberg-Salam Model

Electroweak interaction is determined by local gauge invariance of Lagrangian. Specifically, we have two types of interactions

\[\begin{split}- i g \mathbf J_\mu \cdot \mathbf W^\mu = & - i g \bar \xi_L \gamma_\mu \mathbf T\cdot \mathbf W^\mu \xi_L \\ - i \frac{g'}{2} j_\mu ^Y B^\mu = & - i g' \bar \psi \gamma_\mu \frac{Y}{2} \psi B^\mu,\end{split}\]

where \(\mathbf T\) and \(Y\) are generators of \(SU(2)_L\) and \(U(1)\). The doublet \(\xi_L\) and singlet \(\psi\) fermions and bosons.

For a group multiplication \(G = SU(2)_L \times U(1)\), the generators are related

\[Q = T^3 + \frac{Y}{2},\]

where \(Q\) is the generator of group \(G\).

Then we can write down the currents

\[j^{\mathrm{em}}_\mu = j^3_{\mu} + \frac{1}{2}j^{Y}_\mu.\]

Since we know the neutral currents \(j^3_{\mu}\) and \(\frac{1}{2}j^{Y}_\mu\), we can calculate the EM current by adding them up. Hence we acutally can relate \(g\) and \(g'\) by looking at the coefficient of \(A^\mu\) field. More specific proof of this is to use the Higgs field and find the actual electromagnetic field \(A\)

Why does the Weinberg-Salam model work

The choice of vacuum in Weinberg-Salam model is quite unique.