for $g_{||}$, $$\label{inteq}
g_{||}(\tilde r)={\sqrt{3}(1-\eta)} \coth{\left(2(1-\eta)\ln{\tilde r}\right)}/({2\pi}).$$

Using Eq. (\[inteq\]) and Eq. (\[GFPA\]) we get from Eq. (\[BR\]) $$\begin{aligned}
\label{ppl}
G^{x,x}_{B_2}(r)&=&Const.   \left (  \frac{1}{r}  \right )^{4\eta} \left (  \frac{1-(Ar)^{-4(1-\eta)} }{1-r^{-4(1-\eta)}_0}  \right )^{2} .\end{aligned}$$ In obtaining Eq. (\[ppl\]) we used the following table integral $\int \mathrm {d}x \coth{\alpha x} =\ln( \sinh(\alpha x))/\alpha$ and put $r_1=Ar$. As an artifact of the finite order perturbation theory approximation (for correlation function, beta functions and anomalous dimension), $G^{x,x}_{B_2}(r)$ in Eq. (\[ppl\]) contains some arbitrary number $A$.

Choosing the ($\eta$ dependent) normalization constant in such a way that the leading behavior of $G^{x,x}_{B_2}(r)$, for distances $ \ln{r}\gg 1/(1-\eta)$, becomes identical to that shown in Eq. (11) of the main text, we obtain for $\eta\to 1$ over an intermediate range of distances $1\ll \ln{r}\ll1/(1-\eta)$, $$\label{pl}
G^{x,x}_{B_2}(r)\simeq {B_2 (4(1-\eta)\ln{(Ar)})^2}/{r^4},$$ where for $\eta \to 1$ $$B_2\simeq \frac{1-2(1-\eta)\ln{\frac{(1-\eta)}{6\pi}}+O((1-\eta)^2) }{ (1-\eta)^2 2^7 \pi^4 }.$$ For obtaining Eq. (\[pl\]) from Eq. (\[ppl\]) we have used the following equation, $$\lim_{\eta \to 1}(r^{2(1-\eta)}- r^{-2(1-\eta)})=4(1-\eta) \ln{r}+O((1-\eta)^3).$$

Number $A$ in Eq. (\[pl\]) can be estimated by going to higher order approximation in perturbation theory [@LT; @Lukyanov98]; however it will contain some fitting constant depending on the order of truncation of the perturbative series. If the complete 