62 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Theorem 10.1.1. There exists a unique series of local counterterms

Ii,k

CT

(ε) ∈

Oloc(C∞(M

)) ⊗alg P((0, ∞)) 0,

for all i 0,k ≥ 0, with Ii,k

CT

homogeneous of degree k as a function of

a ∈

C∞(M),

such that, for all L ∈ (0, ∞], the limit

lim

ε→0

W

⎛

⎝P

(ε, L),I −

i,k

iIi,k CT

(ε)⎠

⎞

exists.

Here the symbol ⊗alg denotes the algebraic tensor product, so only finite

sums are allowed.

10.2. We will construct our counterterms using induction on the genus

and number of external edges of the Feynman graphs. Later, we will see a

very short (though unilluminating) construction of the counterterms, which

does not use Feynman graphs. For reasons of exposition, we will introduce

the Feynman graph picture first.

Let Γi,k denote the set of all stable graphs of genus i with k external

edges. Let

Wi,k (P, I) =

γ∈Γi,k

wγ(P (ε, L),I).

Thus,

W (P, I) =

iWi,k

(P, I) .

If the graph γ is of genus zero, and so is a tree, then limε→0 wγ(P (ε, L),I)

converges. Thus, the first counterterms we need to construct are those from

graphs with one loop and one external edge. Let us define

I1,1

CT

(ε, L) = Singε W1,1 (P (ε, L),I) .

Section 9 explains the meaning of the singular part Singε of W1,1 (P (ε, L),I).

We need to check that this has the desired properties. It is immediate

from the definition that

W1,1

(

P (ε, L),I − I1,1

CT

(ε, L)

)

= W1,1 (P (ε, L),I) − I1,1

CT

(ε, L)

and so the limit limε→0 W1,1

(

P (ε, L),I − I1,1

CT

(ε, L)

)

exists.

Next, we need to check that

Lemma. I1,1

CT

(ε, L) is local.

First we will show that

Lemma. I1,1

CT

(ε, L) is independent of L.

Figure 5 illustrates

d

dL

W1,1 (P (ε, L),I). This expression is non-singular,

as it is obtained by contracting the distribution I0,3 on

C∞(M 3)

with the

smooth function KL ∈

C∞(M 2).

Therefore I1,1

CT

(ε, L), which we defined to

be the singular part of W1,1 (P (ε, L),I), is independent of L.