Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 303 / 3 / a / v / Solution

For $d>4$, the quartic coupling has $y_g=4-d<0$, but setting it to zero removes the stabilizing term. It is a dangerously irrelevant coupling. At $t=0$, minimizing $g{\varphi }^4-B\varphi$ gives $\varphi \sim (B/g{)}^{1/3}$ and $\delta =3$. This is consistent with Gaussian scaling becauseFor $d<4$, $g$ flows to a nonzero Wilson-Fisher fixed point, so no dangerously irrelevant zero-coupling limit spoils the ordinary scaling relations.