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 wγ 1 (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 wγ 1 (P (ε, L),I)(a) doesn’t exist.
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