ln}^2|z/L|^2
+ \dots  \end{aligned}$$ where $\dots $ denote terms that vanish as $\epsilon \to 0^+$.

To avoid the divergences coming from the $\epsilon^{-2}$ factors, the following condition must be satisfied: there must be a solution $\xi_0=\xi_0(q)$ of the equation: ${\cal H}(\xi_0) \equiv \frac{\sqrt{\pi}}{\sqrt{\xi_0}}
f_q(2\xi_0) - h^2_q(\xi_0)=0$. The existence of such a solution can be verified numerically (see figure 2). Analytically this can be confirmed by looking at the asymptotic behaviour of the function ${\cal H}(x)$ as $x \to \infty$, which yields a negative value: ${\cal H}(x\to \infty) \sim 
-\frac{\pi^3x^{2q}e^{2x}}{\Gamma^2 (\frac{1+q}{2})  
\Gamma^2(1 + \frac{q}{2})} < 0$. This behaviour comes entirely from the term $h_q^2(x)$, given that $f_q(x\to \infty) \to 0^+$.

\[sol\]

As we shall show below, for various values of $q$, near the fixed point $\epsilon \to 0^+$, one can construct higher order logarithmic algebras, whose highest power is determined by the dominant terms in the operator algebra of correlators (\[offshellalg\]), (\[tdope\]). To this end, we first remark that in the above analysis we have dealt with a small but otherwise arbitrary parameter $\epsilon$, which allows us to keep as many powers as required by (\[offshellalg\]) in conjunction with the value of $q$. The value of $\epsilon$ determines the distance from 