Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 44 / 2 / d

Using functional derivatives of the interaction functional,Dividing the flow by $e^{-S_1}$ therefore givesIn the first term, remove one leg from each of two interaction vertices and join them with a line weighted by ${\dot{C}}_{\Lambda }(p)$. This produces the tree joining of two vertices. In the second, remove two legs from one vertex and contract them with that line, raising the loop order by one, with tadpole diagrams as the simplest example. The factor one half accounts for interchanging the contracted ends; the displayed signs are the signs in the interaction-action flow.

The varying cutoff replaces an internal propagator by its cutoff derivative. Repeated tree joins and loop closures express how eliminated high-momentum fluctuations generate the vertices of the Wilsonian effective action. The action is not restricted to one-particle-irreducible Feynman diagrams: connected tree joins also occur. Field-independent vacuum contributions can again be absorbed into normalization.