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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 305

Use <unitary gauge> to set
$$
H=\frac1{\sqrt2}\begin{pmatrix}0\\v+h\end{pmatrix}.
$$
Then
$$
(D_\mu H)^\dagger D^\mu H
=\frac12(\partial_\mu h)^2
+\frac{(v+h)^2}{8}
\left[g^2\{(W_\mu^1)^2+(W_\mu^2)^2\}
+(gW_\mu^3-g'B_\mu)^2\right].
$$
Define
$$
W_\mu^\pm=\frac{W_\mu^1\mp iW_\mu^2}{\sqrt2},
\quad
Z_\mu=\frac{gW_\mu^3-g'B_\mu}{\sqrt{g^2+g'^2}},
\quad
A_\mu=\frac{g'W_\mu^3+gB_\mu}{\sqrt{g^2+g'^2}}.
$$
The propagating fields are $W^\pm_\mu,Z_\mu,A_\mu$, and the <Higgs boson> $h$. Expanding
$$
V(H)=\lambda\left(H^\dagger H-\frac{v^2}{2}\right)^2
=\lambda v^2h^2+\lambda vh^3+\frac\lambda4h^4
$$
and reading the quadratic terms gives
$$
m_W=\frac{gv}{2},
\qquad
m_Z=\frac{v}{2}\sqrt{g^2+g'^2},
\qquad
m_A=0,
\qquad
m_h=\sqrt{2\lambda}\,v.
$$
The factors $(v+h)^2$ in the kinetic term also give the corresponding Higgs-vector interactions.

Solved by gpt-5.6-sol high.

Solution

Paper 305 / 4 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 4 / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 4 / i / Solution

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