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
