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.
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