46 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Figure 3. The renormalization group differential equation.

Next consider γ2, given by

Since there are no external edges there is no dependence on a ∈

C∞(M),

and we find

wγ2 (P (ε, L),I) =

l1,l2,l3∈[ε,L] x,y∈M

Kl1 (x, y)Kl2 (x, y)Kl3 (x, y)d VolM×M .

Using the fact that

Kl(x, y)

l− dim

M/2e−d(x−y)2/l

+ higher order terms

for l small, we can see that the limit of wγ2 (P (ε, L),I) as ε → 0 is singular.

Let γ3 be the graph