Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 1 / b

For a real field, ${\varphi }_{-\mathbf{k}}={\varphi }_{\mathbf{k}}^{\ast }$. The stated Fourier transform givesUsingin both quadratic terms yields

Write $A(k,T)=\gamma k^2+{\mu }^2(T)$ and use units with $k_B=1$. The Gaussian functional integral gives the fluctuation free-energy density, up to terms linear in $T$ that do not affect the heat capacity,Since the heat capacity per volume is $c=-T{d}^{2}f/d{T}^2$,where a dot denotes $d/dT$. ThusFor ${\mu }^2=a(T-T_c)$, these reduce to $g(T)=2aT$ and $h(T)=a^2{T}^2$.

The most singular contribution as ${\mu }^2\downarrow 0$ isRescaling $\mathbf{k}=(\mu /\sqrt{\gamma })\mathbf{q}$ shows that its singular part is proportional toBecause ${\mu }^2\propto T-T_c$, the heat-capacity critical exponent isfor $d<4$. The term with one propagator is less singular. At $d=4$ the power is replaced by a logarithmic singularity, identifying four as the upper critical dimension of this Gaussian heat-capacity correction.