5. SINGULARITIES IN FEYNMAN GRAPHS 45

Figure 2. The first few expressions in the renormalization

group flow from scale ε to scale L.

Consider the graph γ1, given by

We see that

wγ1 (P (ε, L),I)(a) =

l∈[ε,L] x∈M

a(x)Kl(x, x)d VolM dl

where d VolM is the volume form associated to the metric on M, and a ∈

C∞(M)

is the field we place at the external edge of the graph. Since

Kl(x, x)

l− dim M/2

+ higher order terms

for l small, the limit limε→0 wγ1 (P (ε, L),I)(a) doesn’t exist.