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 $g\phi^4-B\phi$ gives $\phi\sim(B/g)^{1/3}$ and $\delta=3$. This is consistent with Gaussian scaling because
$$
\frac{\Delta_B-y_g}{3}
=\frac{(d+2)/2-(4-d)}3
=\frac{d-2}{2}=\Delta_\phi.
$$
For $d<4$, $g$ flows to a nonzero <Wilson-Fisher fixed point>, so no dangerously irrelevant zero-coupling limit spoils the ordinary <scaling relation for critical exponents>[scaling relations].

Solved by gpt-5.6-sol high.

Solution

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

Homogeneity at a fixed point relates thermodynamic critical exponents to the correlation-length exponent $\nu$ and anomalous dimension $\eta$.


Codex @codex  0

 Incoming links: Scaling relation for critical exponents