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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 305

With $T^a=\tau^a/2$, hypercharge $Y_H=1/2$, and
$$
D_\mu H=\left(\partial_\mu-igW_\mu^aT^a-\frac{i g'}2B_\mu\right)H,
$$
the most general local renormalizable bosonic Lagrangian is
$$
\mathcal L_{\mathrm{bos}}
=-\frac14W_{\mu\nu}^aW^{a\mu\nu}
-\frac14B_{\mu\nu}B^{\mu\nu}
+(D_\mu H)^\dagger D^\mu H
-m^2H^\dagger H-\lambda(H^\dagger H)^2.
$$
Canonical normalization leaves four parameters:
$$
g,\qquad g',\qquad m^2,\qquad\lambda.
$$
Stability requires $\lambda>0$. If $m^2>0$, the minimum is $H=0$ and the electroweak symmetry is unbroken. If $m^2<0$,
$$
H^\dagger H=\frac{v^2}{2},
\qquad
v^2=-\frac{m^2}{\lambda},
$$
and the <electroweak symmetry breaking> pattern is
$$
SU(2)_L\times U(1)_Y\longrightarrow U(1)_{\mathrm{EM}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 2 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 2 / a / Solution

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