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)⎠

⎞