= -{\tilde N}_q^2~\sqrt{\frac{\pi}{\xi_0}}  
\left(f_q(2\xi_0) + f'_q(2\xi_0)\right)~(\frac{\alpha}{\xi_0})^{q-2}
+ {\cal O}(\alpha^{q-3})~, \nonumber \\
&~& \epsilon^3~q^2(q-1)~\langle {\cal D}^{(q-1)}(z) {\cal D}^{(q-2)}(0)\rangle
= {\cal O}(\alpha^{q-3})~, 
\nonumber \\
&~& \epsilon^4~q^2(q-1)^2~\langle {\cal D}^{(q-2)}(z) {\cal D}^{(q-2)}(0)\rangle
= {\cal O}(\alpha^{q-4})~\end{aligned}$$ where ${\tilde N}_q = \frac{\Gamma (1+q)}{2\pi}$, and $\xi_0$ has been defined in (\[xi0\]).

Notice that the above algebra is plagued by world-sheet ultraviolet divergences as $\epsilon^2 \to 0^+$, thereby making the approach to the fixed (conformal) point subtle. As becomes obvious from (\[xi0\]), the non-trivial fixed point $\epsilon \to 0^+$ corresponds to $L/a \to +\infty$, i.e. it is an infrared world-sheet fixed point. In order to understand the approach to the infrared fixed point, it is important to make a few remarks first, motivated by physical considerations.

From the integral expression of the regularized Heaviside function [@kmw] (\[rep\]) it becomes obvious that a scale $1/\epsilon$ for the target time is introduced. This, together with the fact that the scale $\epsilon$ is connected (\[xi0\]) to the world renormalization-group scales $L/a$, implies naturally the introduction of a ‘renormalized’ $\sigma$-model coupling/velocity $v_{R,i}(\tfrac{1}{\epsilon}) $ at the scale $\tfrac{1}{\epsilon}$: $$v_{R,i}(\frac{1}{\epsilon}) \sim \left(\frac{1}{\epsilon}\right)^{q-1}
\label{velrenorm}$$ for a trajectory $y_i (t) \sim t^q$. This normalization would imply the following rescaling of the operators $${\cal D}^{(q-n)} \rightarrow \epsilon^{q-1}{\cal 