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

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 $2d-4$.