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64 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
Figure 6. This diagram, which is just the infinitesimal ver-
sion of the renormalization group equation, explains why
Ii,k
CT
(ε, L) is independent of L. In this diagram, W∗,∗ is short-
hand for
W∗,∗
⎛
⎝P
(ε, L),I −
(a,b) (i,k)
Ia,b
CT (ε)⎠
⎞
.
(1) For all L, the limit
lim
ε→0
W
(i,k)
⎛
⎝P
(ε, L),I −
(j,l) (i,k)
jIj,l CT
(ε)⎠
⎞
exists.
(2) The counterterms Ij,l
CT
(ε) are local for (j, k) (i, k).
Then, we define the counterterm
Ii,k
CT
(ε, L) = Singε Wi,k
⎛
⎝P
(ε, L),I −
(j,l) (i,k)
jIj,l CT
(ε)⎠
⎞
.

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