Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 303 / 3 / iv / Solution

Besides $g=0$, the coupling beta function vanishes atThe mass fixed-point equation isso the root continuously connected to the origin isPerturbation theory requires $b/c\ll 1$, ensuring that the fixed-point coupling is small and the omitted higher powers are suppressed.

For increasing infrared scale $s$, positive $g<g_{\ast }$ flows upward toward $g_{\ast }$ and $g>g_{\ast }$ flows downward toward it. The mass direction remains relevant, so only the tuned critical trajectory reaches the interacting fixed point; trajectories on either side leave toward the two phases.

The relevant thermal eigenvalue is obtained from the stability matrix of a renormalization-group fixed point by differentiating the mass beta function:It follows that