At the <Gaussian fixed point>, set $a_d=12\Omega_{d-1}\Lambda^{d-2}/(2\pi)^d$. The linearized flow is
$$
\frac d{d\log\zeta}
\begin{pmatrix}\mu^2\\g\end{pmatrix}
=\begin{pmatrix}2&a_d\\0&4-d\end{pmatrix}
\begin{pmatrix}\mu^2\\g\end{pmatrix}.
$$
The mass direction, corresponding to $\phi^2$, has coupling dimension $y_t=2$. For $d\ne2$, the second eigendirection is
$$
(\delta\mu^2,\delta g)=\left(-\frac{a_d}{d-2},1\right)\delta g
$$
and has $y_g=4-d$; it is the tadpole-subtracted mixture of $\phi^4$ and $\phi^2$. The corresponding operator dimensions are $d-2$ and $2d-4$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 303 / 2 / b / i / Solution

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At the Gaussian fixed point all interaction couplings vanish and scaling dimensions follow from the quadratic theory.


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