next section we will discuss analyticity of correlators, introduce the contour integral giving the infra-red limit and show that the contribution from the cut vanishes for large $\rho$. We also show that the contour integral is the Borel transform of $F(s)$. In section 3 we define $F(s)$ for the central charge, show its analyticity and discuss an approximation method for calculating the central charge. An exact bound on $I(\rho)$ is given. In section 4 we use this approximation on the free massive bosonic and fermionic theory together with the minimal models perturbed by $\phi_{(1,3)}$. In section 5 we discuss the $\varphi^4$ theory and introduce a function giving the critical exponents $\nu, \eta$ in the infra-red limit, and obtain estimates of $\nu$ and $\eta$ from the limit of the Borel transform.

Defining the contour integral
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The physical quantities of a quantum field theory can be written in terms of Green functions which have well known analyticity properties. As an example consider the two point function of a scalar field $A(p)$. The Källen-Lehmann spectral representation writes this as a sum over intermediate states $$\label{kallenlehmann}
  \langle A(p)A(0)\rangle = \int_0^\infty\! d\mu^2\ \tilde{c}(\mu^2)\frac{1}{p^2+\mu^2}
\nonumber$$ where $d\mu^2 \tilde{c}(\mu^2)$ is the spectral density. Introducing the complex 