3. WILSONIAN LOW ENERGY THEORIES 11
Figure 1. The first few expressions in the renormalization
group flow from scale ε to scale L. The dotted lines indicate
incoming particles. The blobs indicate interactions between
these particles. The symbol Ii,k
eff
[L] indicates the
i
term in
the contribution to the interaction of k particles at length-
scale L. The solid lines indicate the propagation of a particle
between two interactions; particles are allowed to propagate
with proper time between ε and L.
We should think of Ii,k
eff
[L] as being a contribution to the interaction of k
particles which come together in a region of size around L.
Figure 1 shows how to express, graphically, the world-line version of the
renormalization group flow.
3.6. So far in this section, we have sketched the definition of two versions
of the renormalization group flow: one based on energy, and one based on
length. There is a more general definition of the renormalization group flow
which includes these two as special cases. This more general version is based
on the concept of parametrix.
