78 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

in the space P ((0, ∞)ε) ⊂

C∞((0,

∞)ε) of periods, such that for all l ∈ Z≥0,

there is a small ε asymptotic expansion

∂l

∂lL

wγ(P (ε, L),I)

∂l

∂lL

r≥0

gr(ε)Φr.

This means that for each l and each x ∈ X, there exists a non-decreasing

sequence dR ∈ Z, tending to ∞, such that

lim

ε→0

ε−dR

∂l

∂lL

⎛

⎝wγ(P

(ε, L),I) −

r≥0

gr(ε)Φr⎠

⎞

x

= 0

in the topological vector space Hom(E

⊗T (γ),

R) ⊗ Ax. Here, Hom(E

⊗T (γ),

R)

is given the strong topology (i.e. the topology of uniform convergence on

bounded subsets).

Also, each gr has a finite order pole at ε = 0. This means that the limit

limε→0

εkgr(ε)

is 0 for some k 0.

Further, each

Φr(L, e) ∈ Hom(E

⊗T (γ),C∞((0,

∞)L))

has a small L asymptotic expansion

Φr hs(L)Ψr,s

where each Ψr,s ∈ Oloc(E , A ). The definition of small L asymptotic expan-

sion is in the same sense as before: there exists a non-decreasing sequence

dS ∈ Z≥0, tending to ∞, such that, for all x ∈ X, and all S ∈ Z≥0,

lim

L→0

L−dS

Φr(e) −

S

s=0

hs(L)Ψr,s(e)

x

= 0

in the topological vector space Hom(E

⊗T (γ),

R) ⊗ Ax.

It follows from this theorem that it makes sense to define the next coun-

terterm I(I,K)

CT

simply by

I(I,K)(ε,

CT

L) = Singε

⎛

⎝W(I,K)

⎛

⎝P

(ε, L),I −

(r,s) (I,K)

rI(r,s)(ε)⎠⎠

CT

⎞⎞

.

We would like to show the following properties of I(I,K)(ε,

CT

L).

(1) I(I,K)(ε,

CT

L) is independent of L.

(2) I(I,K)(ε)

CT

is local, that is, it is an element of

Oloc(E , A ) ⊗alg

C∞((0,

∞)ε).