Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 2

A momentum-shell renormalization group step has three parts:

1. Split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ into slow modes $|k|<\Lambda /\zeta$ and fast modes $\Lambda /\zeta <|k|<\Lambda$, then integrate out ${\varphi }_{>}$ using a cumulant expansion to obtain a Wilsonian effective action for ${\varphi }_{<}$.
2. Rescale momenta by $k^{\prime }=\zeta k$, equivalently coordinates by $x^{\prime }=x/\zeta$, restoring the cutoff from $\Lambda /\zeta$ to $\Lambda$.
3. Rescale the field by ${\varphi }^{\prime }(x^{\prime })={\zeta }^{(d-2)/2}{\varphi }_{<}(x)$ at the Gaussian fixed point, restoring the normalized kinetic term.

The resulting functional has the original form and cutoff but new masses and couplings. Iteration produces a renormalization-group flow.

Requiring $\int d^{d}x(\nabla \varphi {)}^2$ to be dimensionless gives the engineering dimensionsandThus ${\mu }_0$ is relevant. The cubic coupling is relevant, marginal, or irrelevant for $d<6$, $d=6$, or $d>6$; the quartic coupling has the same classification around $d=4$; and the sextic coupling has it around $d=3$.

LetAt order $g_0^2$, the connected six-point topology consists of two quartic vertices joined by one fast line. The second cumulant givesFor a sharp shell and external slow momenta approaching zero, the connecting momentum cannot lie in the fast shell, so this nonlocal tree topology gives no local zero-momentum ${\varphi }_{<}^6$ coupling.

At order $g_0^3$, the local connected contribution is the triangle of three quartic vertices, with two external slow legs at each vertex and one fast propagator along each edge. The factor isConsequently, at zero external momentum,Ignoring contributions involving ${\lambda }_0,{\gamma }_0$ and anomalous field rescaling, the running sextic coupling is thereforeThe diagrams are respectively the one-line two-vertex six-point tree and the three-vertex triangular one-loop six-point diagram; only the latter contributes to the local shell coupling at vanishing external momenta.