 Solution

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.