Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 304 / 3

The inverse Fourier transform is linear:Splitting the integration domain into $p^2<b^2{\Lambda }^2$ and $b^2{\Lambda }^2<p^2<{\Lambda }^2$, then inserting the definitions of $\tilde{\varphi }$ and $\tilde{\chi }$, gives

The disjoint Fourier supports make the quadratic cross term $\int \partial \varphi \cdot \partial \chi$ vanish. ThereforewhereDefine the high-mode free generating functional with source conventionThen inserting $\chi =-\delta /\delta J$ reproduces every high-field factor, andExpanding the interaction exponential and applying Wick theorem evaluates the Wilsonian effective action as a sum of connected diagrams whose internal lines are restricted to the high-momentum shell.

The original cubic vertex gives $g$. At order $g^3$, the one-particle-irreducible correction to the cubic interaction is the triangle diagram: one slow external leg leaves each of three cubic vertices, and three high-mode propagators join the vertices cyclically. Diagrams with a high-momentum bridge carrying only a sum of vanishing external momenta do not contribute to the local zero-momentum coupling; external self-energy diagrams belong to the neglected field rescaling.

The relevant interaction at each vertex is $(g/2)\varphi {\chi }^2$. The third connected cumulant contributesWick's theorem gives $8G_{>}(x-y)G_{>}(y-z)G_{>}(z-x)$. Projecting onto a local interaction therefore givesFor $d=6$, the area of the unit five-sphere is ${\Omega }_5={\pi }^3$, and henceThus