scale factor $s\in \mathbb{C}$ $$\label{scalefac}
  \langle A(sp)A(0)\rangle = \int_0^\infty\! d\mu^2\ \tilde{c}(\mu^2)\frac{1}{s^2p^2+\mu^2}
\nonumber$$ then shows that $\langle A(sp)A(0)\rangle$ is analytic in the positive half-plane $\Re (s)>0$.

Generally physical quantities will not be analytic in the positive half-plane but in some sector $S(\alpha)$ of the complex plane, defined as $$\label{sector}
S(\alpha)=\ \{z=re^{i\phi}\ |\
0<r<\infty,\ -\tfrac{\alpha}{2}
<\phi<\tfrac{\alpha}{2} \}.$$ Note that $0\notin S(\alpha)$. We want to calculate the infra-red limit $F_{IR}$ of a physical quantity $f(x)$, where $F_{IR}=\lim_{|x|\rightarrow \infty}f(x)$, and we will denote the UV limit $F_{UV}=\lim_{|x|\rightarrow 0}f(x)$. We are considering functions which have well defined limits in $S(\alpha)$, i.e.  $\lim_{|s|\rightarrow \infty}F(s)=F_{IR}$ and $\lim_{s\rightarrow 0}F(s)=F_{UV}$ for $s\in S(\alpha)$.

In the two examples we will consider the physical quantities $f(x)$ are given as functionals of a two point correlator resulting in analytic functions $F(s)$. A large class of physical quantities will in this way preserve the analytic structure of the Green functions in the theory.

The scale parameter $s$ can of course be introduced into the physical quantity in a number of ways. We will define by $\tilde{F}(s)$ the analytical continuation of $f(x)$ where $s$ is introduced as $\tilde{F}(s)\equiv
f(sx)|_{x=1}$ so that $\tilde{F}(s)$ has the expansion $\tilde{F}(s)=\sum
\tilde{F}_n s^n$ around the origin, and $\tilde{F}(s)$ is 