in question is well known, namely the central charge of the unitary minimal models perturbed by the relevant field $\phi_{(1,3)}$, and the critical exponents of $\varphi^4$ theory in three dimensions.

We also considered the perturbed minimal models in [@lkpm] where this method was shown to improve upon renormalisation group improved perturbation theory by a factor 2. $F(s)$ is given as the two point function of the energy momentum tensor, and we will here address some of the details left out of our discussion in [@lkpm]. We first show the analytic properties of $F(s)$, and we then obtain a bound on the exact non perturbative expression for $I(\rho)$, and this allows us to obtain an approximate value for calculating the central charge.

We employ this approximation method to calculate the central charge for the free fermionic and bosonic theories perturbed by a mass term and the unitary minimal models perturbed by the relevant operator $\phi_{(1,3)}$. The free massive theories can be solved exactly and these then demonstrate how $I(\rho)$ behaves in an exact case. For the perturbed minimal models we first calculate $F(s)$ as a renormalisation group improvement of a one loop perturbative calculation and show that this expression has the 