a minimum before approaching its limiting value of $c^*_{IR}$. Since this minimum occurs at the largest value of $\rho$ for which $I_n(\rho)$ is a reasonable approximation to $I(\rho)$ and the true value of $c_{IR}$ is given by $I(\infty)$, it is this minimum of $I_n(\rho)$ that we will use to provide a better estimate of $c_{IR}$. The approximation then becomes $$\label{approxcir}
  c_{IR}=I_n(\rho_m)$$ where $\rho_m$ is the value where $I_n(\rho)$ attains its minimum. Below we consider the quantity $\Delta c_{exact}=c_{UV}-c_{IR}$ and denote the approximation to it $\Delta c_{approx}=c_{UV}-I_{n}(\rho_m)$, we call the perturbative value $\Delta c_{pert}=c_{UV}-\lim_{\rho\rightarrow
  \infty}I_n(\rho)=c_{UV}-c_{IR}^*$. This approximation rests upon the assumption that the exact function is monotonically decreasing from $c_{UV}$ to $c_{IR}$, or at least that its minimum value is close to $c_{IR}$. Below we will show that this is indeed the case for $k=4$, at least to a very good approximation.

Exact bound on $\Delta c$
-------------------------

As discussed above can the exact value $I(\rho)$ possibly be smaller than the asymptotic value $c_{IR}$. We will denote by ${\rho'}_m$ the value where $I(\rho)$ attains its minimum and $\Delta c_{est}=c_{UV}-I({\rho'}_m)$, then $\Delta c_{exact}-\Delta
c_{est}$ measures the undershoot of the exact function $I(\rho)$ compared with its asymptotic value $c_{IR}$ (see figure 3). From 