Past exam of the particle physics course of the University of Oxford / 2022 / 5 / b / Solution

Set $G=\sqrt{g^2+g^{\prime 2}}$ and $s_W=\sin {\theta }_W$. First expand the charged term using the ladder operators:Substituting the neutral gauge fields as well, the covariant derivative becomesMatching $W^3$ to $g{\tau }_3$ requiresMatching $B$ to $g^{\prime }(Q-{\tau }_3)=g^{\prime }Y/2$ requiresFor $g,g^{\prime }>0$, elimination yieldsEquivalently, $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$ and $1/e^2=1/g^2+1/g^{\prime 2}$. The conventional Weinberg angle lies in the first quadrant.