66 2. THEORIES, LAGRANGIANS AND COUNTERTERMS exists. Let us suppose, by induction, that these counterterms are local and independent of L. Then, we define the next counterterm by Ii,k CT (L, ε) = Singε Wi,k ⎛ ⎝P (ε, L),I − (r,s) (i,k) r Ir,s CT (ε)⎠ ⎞ . The identity Wi,k ⎛ ⎝P (ε, L),I − (r,s) (i,k) r Ir,s CT (ε) − i ICT i,k (L, ε)⎠ ⎞ = Wi,k ⎛ ⎝P (ε, L),I − (r,s) (i,k) r ICT r,s (ε)⎠ ⎞ − ICT i,k (L, ε) shows that the limit lim ε→0 W≤(i.k) ⎛ ⎝P (ε, L),I − (r,s) (i,k) r ICT r,s (ε) − i ICT i,k (L, ε)⎠ ⎞ exists. To show locality of the counterterm ICT i,k (L, ε), it suffices, as before, to show that it is independent of L. If L L, we have ICT i,k (L , ε) = Sing ε Wi,k ⎛ ⎝P (ε, L ),I − (r,s) (i,k) r ICT r,s (ε)⎠ ⎞ = Sing ε Wi,k ⎛ ⎝P (L, L ),W ⎛ ⎝P (ε, L),I − (r,s) (i,k) r ICT r,s (ε)⎠⎠ ⎞⎞ = Sing ε Wi,k ⎛ ⎝P (L, L ),W (i,k) ⎛ ⎝P (ε, L),I − (r,s) (i,k) r ICT r,s (ε)⎠ ⎞ + i Wi,k ⎛ ⎝P (ε, L),I − (r,s) (i,k) r ICT r,s (ε)⎠⎠ ⎞⎞ = Sing ε Wi,k ⎛ ⎝P (L, L ),W (i,k) ⎛ ⎝P (ε, L),I − (r,s) (i,k) r ICT r,s (ε)⎠⎠ ⎞⎞ + Singε Wi,k ⎛ ⎝P (ε, L),I − (r,s) (i,k) r Ir,s CT (ε)⎠ ⎞
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