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)
rIr,s CT
(ε)⎠

.
The identity
Wi,k

⎝P
(ε, L),I
(r,s) (i,k)
rIr,s CT
(ε)
iIi,k CT
(L,
ε)⎠

=
Wi,k

⎝P
(ε, L),I
(r,s) (i,k)
rIr,s CT
(ε)⎠

Ii,k
CT
(L, ε)
shows that the limit
lim
ε→0
W≤(i.k)

⎝P
(ε, L),I
(r,s) (i,k)
rIr,s CT
(ε)
iIi,k CT
(L,
ε)⎠

exists.
To show locality of the counterterm Ii,k
CT
(L, ε), it suffices, as before, to
show that it is independent of L. If L L, we have
Ii,k
CT
(L , ε) = Singε Wi,k

⎝P
(ε, L ),I
(r,s) (i,k)
rIr,s CT (ε)⎠

= Singε Wi,k

⎝P
(L, L ),W

⎝P
(ε, L),I
(r,s) (i,k)
rIr,s CT
(ε)⎠⎠
⎞⎞
= Singε Wi,k

⎝P
(L, L ),W
(i,k)

⎝P
(ε, L),I
(r,s) (i,k)
rIr,s CT (ε)⎠

+
iWi,k

⎝P
(ε, L),I
(r,s) (i,k)
rIr,s CT
(ε)⎠⎠
⎞⎞
= Singε Wi,k

⎝P
(L, L ),W
(i,k)

⎝P
(ε, L),I
(r,s) (i,k)
rIr,s CT (ε)⎠⎠
⎞⎞
+ Singε Wi,k

⎝P
(ε, L),I
(r,s) (i,k)
rIr,s CT
(ε)⎠
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