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.
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