Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 3 / a / vi / Solution

For $d=4-ϵ$, define the dimensionless couplings $r={\mu }^2/{\Lambda }^2$ and $u=\tilde{g}={\Lambda }^{-ϵ}g$. Since $K_4=1/(8{\pi }^2)$, the leading epsilon expansion of the flow isThere is a Gaussian fixed point $(r_{\ast },u_{\ast })=(0,0)$. Its thermal eigenvalue is $y_t=2$, so its correlation-length critical exponent is $\boxed{{\nu }_{\mathrm{G}}=1/2}$.

The interacting Wilson-Fisher fixed point isLinearizing the renormalization-group flow gives the thermal eigenvalueTaking its reciprocal gives