D}^{(q-n)}$$ As a consequence, the factors $\epsilon^{2(1-q)}$ in (\[defj\]), (\[offshellalg\]) are removed. In the context of the world-sheet field theory this renormalization can be interpreted as a subtraction of the ultraviolet divergences by the addition of appropriate counterterms in the $\sigma$ model.

The approach to the infrared fixed point $\epsilon \to 0^+$ can now be made by looking at the [*connected*]{} two point correlators between the operators ${\cal D}^{(q)}$ defined by $$\label{connected}  
\langle {\cal A}{\cal B} \rangle_c = \langle {\cal A}{\cal B} \rangle - 
\langle {\cal A} \rangle~\langle {\cal B} \rangle~,$$ where the one-point functions are given by: $$\begin{aligned}
&~&\langle {\cal D}^{(s)}\rangle = N_s\int 
\frac{d\omega}{(\omega -i\epsilon)^{s+1}}\langle e^{i\omega t} \rangle 
=N_s\int \frac{d\omega}{(\omega -i\epsilon)^{s+1}} e^{-\omega^2\alpha} = {\tilde N}_s~\epsilon^{-s}~h_s(\epsilon^2\alpha)~, \nonumber \\
&~&h_s(x) 
= -\frac{x^{s/2}}{2}\left(\frac{4\pi}{\Gamma (\frac{1+s}{2})}~
\sqrt{\pi}~F(1+\frac{s}{2}~, \frac{3}{2}~, x) - \frac{2\pi}{\Gamma (1 + 
\frac{s}{2})}~F(\frac{1 + s}{2}~, \frac{1}{2}~, x)\right).\end{aligned}$$

For the two-point function of the ${\cal D}^{(q)}$ operator the result is: $$\begin{aligned}
&~& \langle {\cal D}^{(q)}(z){\cal D}^{(q)}(0) \rangle_c =  
{\tilde N}_q \epsilon^{-2}\left(\frac{\sqrt{\pi}}{\xi_0}
f_q(2\xi_0+2\epsilon^2{\rm ln}|z/L|^2) - h^2_q(\xi_0)\right)~.\end{aligned}$$

Expanding in powers of $\epsilon$, we obtain $$\begin{aligned}
 \label{dd}
&~& \langle {\cal D}^{(q)}(z) {\cal D}^{(q)}(0) \rangle_c =  
{\tilde N}_q \epsilon^{-2}\left(\frac{\sqrt{\pi}}{\sqrt{\xi_0}}
f_q(2\xi_0) - h^2_q(\xi_0)\right) + 
{\tilde N}^2_q~\frac{\sqrt{\pi}}{\sqrt{\xi_0}}~f'_q(2\xi_0)2
{\rm ln}|z/L|^2 + \nonumber \\
&~& +~ \epsilon^2 {\tilde N}_q^2 \sqrt{\frac{\pi}{\xi_0}}
\frac{1}{2}f''_q(2\xi_0)4 {\rm 