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