10. CONSTRUCTING LOCAL COUNTERTERMS 65

As before, it is immediate that

Wi,k

⎛

⎝P

(ε, L),I −

(j,l) (i,k)

jIj,l CT

(ε) −

iIi,k CT

(ε,

L)⎠

⎞

= Wi,k

⎛

⎝P

(ε, L),I −

(j,l) (i,k)

jIj,l CT (ε)⎠

⎞

− Ii,k

CT

(ε, L)

is non-singular.

What we need to show is that

(1) Ii,k

CT

(ε, L) is independent of L.

(2) Ii,k

CT

(ε, L) is local.

As before, the second statement follows from the first one. To show

independence of L it suﬃces to show that

(†)

d

dL

Wi,k

⎛

⎝P

(ε, L),I −

(j,l) (i,k)

jIj,l CT

(ε)⎠

⎞

is non-singular, that is, the limit of this expression as ε → 0 exists. A proof

of this is illustrated in figure 6. In this diagram, an expression for (†) is

given as a sum over graphs whose vertices are labeled by

Wj,l

⎛

⎝P

(ε, L),I −

(r,s)≤(j,l)

rIr,s CT

(ε)⎠

⎞

for various (j, l) (i, k). We know by induction that these are non-singular.

On the unique edge of these graphs we put the heat kernel KL, which is

smooth. Thus, the expression resulting from each graph is non-singular.

10.4. In fact, the use of Feynman graphs is not at all necessary in this

proof; I first came up with the argument by thinking of W (P, I) as

W (P, I) = log (exp( ∂P ) exp(I/ )) .

From this expression, one can see that

W

(

P (L, L ),W (P (ε, L),I)

)

= W

(

P (ε, L ),I

)

Wi,k (P (ε, L),I) = Wi,k

(

P (ε, L),I

(i,k)

)

+ Ii,k.

The first identity is obvious, and the second identity can be seen (for in-

stance) using the expression of W (P, I) in terms of Feynman graphs.

These two identities are all that is really needed for the argument. In-

deed, suppose we have constructed counterterms Ir,s

CT

(ε) for all (r, s) (i, k),

such that, for all L, the limit

lim

ε→0

W

(i.k)

⎛

⎝P

(ε, L),I −

(r,s) (i,k)

rIr,s CT

(ε)⎠

⎞