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)⎠

⎞

.