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

At the Gaussian fixed point, set $a_d=12{\Omega }_{d-1}{\Lambda }^{d-2}/(2\pi {)}^d$. The linearized flow isThe mass direction, corresponding to ${\varphi }^2$, has coupling dimension $y_t=2$. For $d\ne 2$, the second eigendirection isand has $y_g=4-d$; it is the tadpole-subtracted mixture of ${\varphi }^4$ and ${\varphi }^2$. The corresponding operator dimensions are $d-2$ and $2d-4$.

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For $d<4$, $y_g=4-d>0$, so the quartic coupling is a relevant direction of a fixed point. Generic Ising-like systems therefore flow away from the Gaussian point toward the interacting Wilson-Fisher fixed point, which controls the Ising universality class below four dimensions.

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Put $r={\mu }^2/{\Lambda }^2$ and $\tilde{g}={\Lambda }^{d-4}g$. At $d=4-ϵ$, ${\Omega }_3/(2\pi {)}^4=1/(8{\pi }^2)$ givesThe nonzero fixed point iswhich is the stated Wilson-Fisher fixed point.

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Linearization at the fixed point givesThe relevant eigendirection is $(\delta r,\delta \tilde{g})=(1,O({ϵ}^2))$. An irrelevant eigendirection isThus the mass-like combination is relevant and the quartic-like combination is irrelevant; neither is marginal for $ϵ>0$.

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