38 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Proof. We will prove this by first verifying the result for P = 0, and then checking that both sides satisfy the same differential equation as a function of P . When P = 0, we have seen that W (0,I) = I which of course is the same as log exp( ∂P ) exp(I/ ). Now let us turn to proving the general case. It is easier to consider the exponentiated version: so we will actually verify that exp ( −1 W (P, I) ) = exp( ∂P ) exp(I/ ). We will do this by verifying that, if ε is a parameter of square zero, and P Sym2 U, exp ( −1 W ( P + εP , I )) = (1 + ε∂P ) exp ( −1 W (P, I) ) . Let a1,...,ak U, and let us consider ∂a1 · · · ∂ak exp ( −1 W (P, I) ) (0). It will suffice to prove a similar differential equation for this expression. It follows immediately from the definition of the weight function wγ(P, I) of a graph that ∂a1 · · · ∂ak eW (P,I)/ (0) = γ,φ g(γ) |Aut(γ, φ)| wγ,φ(P, I)(a1,...,ak) where the sum is over all possibly disconnected stable graphs γ with an isomorphism φ : {1,...,k} T (γ). The automorphism group Aut(γ, φ) consists of those automorphisms preserving the ordering φ of the set of tails of γ. Let ε be a parameter of square zero, and let P Sym2 U. Let us consider varying P to P + εP . We find that d ∂a1 · · · ∂ak exp ( −1 W ( P + εP , I )) (0) = γ,e,φ g(γ) |Aut(γ, e, φ)| wγ,e,φ(P, I)(a1,...,ak). Here, the sum is over possibly disconnected stable graphs γ with a distin- guished edge e E(γ). The weight wγ,e,φ is defined in the same way as wγ,φ except that the distinguished edge e is labelled by P , whereas all other edges are labelled by P . The automorphism group considered must preserve the edge e as well as φ. Given any graph γ with a distinguished edge e, we can cut along this edge to get another graph γ with two more tails. These tails can be ordered
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