has a non-trivial infra-red fixed point. We obtain the renormalisation group improved perturbative calculation of $c_{IR}$ and compare with the approximation .

The free boson
--------------

We take the action for the free bosonic theory in 2 dimensions with a mass $m$ to be $$\label{bosonvirk}
  S=\int d^2x\, \left(\frac{1}{2}\p_\mu\varphi(x)\p^\mu
  \varphi(x)+\frac{1}{2} m^2\varphi^2(x)\right) .$$ The perturbation away from the conformal field theory is thus given by the purely massive term $\frac{1}{2}m^2\varphi^2$. The theory is still a free theory off criticality and the correlator $\langle TT \rangle$ can be calculated exactly in the whole scaling region from the ultra-violet to the infra-red. With this normalisation the energy-momentum tensor becomes $$\label{Tbose}
  T(z,\bar{z})=
T_{zz}(z,\bar{z})=-2\pi :\p\varphi (z,\bar{z})\p \varphi(z,\bar{z}):$$ with the correlator $$\begin{aligned}
  \label{schwingerbose}
  \langle T(z,\bar{z})T(w,\bar{w})\rangle &=& (2\pi)^2\langle :\p
  \varphi(z,\bar{z})\p \varphi(z,\bar{z}):\, :\p
  \varphi(w,\bar{w})\p \varphi(w,\bar{w}): \rangle \nonumber \\ 
&=&  2 (2\pi)^2
\langle \p \varphi(z,\bar{z})\p \varphi(w,\bar{w})\rangle ^2,  \end{aligned}$$ as only the double contractions survive. Using the form (\[boseG\]) of the free propagator then (\[schwingerbose\]) is $2(\p_z \p_z K_0(m|z|))^2$ where we have set $w=0$ using translation invariance. We now use the identities $K_n'=-\frac{1}{2}(K_{n-1}+K_{n+1})$, $K_2(x)=K_0(x)+\frac{2}{x}K_1(x)$, $K_{-1}(x)=K_1(x)$ and (\[schwingerbose\]) becomes $$\label{schwinger2}
 \langle T(z,\bar{z})T(0,0)\rangle =
  \frac{m^2 |z|^2}{8z^4}\left(
  4K_1^2(m|z|)\!+\! m^2 |z|^2 K_0^2(m|z|)\!+\! 4m|z| K_1(m|z|)K_0(m|z|)\right),
\nonumber$$ hence $F(s)$ with $k=4$ becomes $$\label{Fbose}
  F(s)=\left(\frac{ms^{1/2}}{2}\right)^2\left(\frac{4}{s^{1/2}}K_1^2(ms^{1/4}) +
       m^2K_0^2(m s^{1/4})+\frac{4m}{s^{1/4}}K_0(ms^{1/4})K_1(ms^{1/4})
       