1$, the expression for $B_2$ is only valid for evaluating correlations Eq.(\[bl\]) at exponentially large distances, $r\gg e^{1/(1-\eta)}$. In the limit of the $SU(2)$ AFM point we apply RG improved perturbation theory [@Affleck98]. We note that $G^{x,x}_{B_2}$ obeys the following RG equation, $$\label{RGEQ}
\Big(\frac{\partial}  {\partial \ln r} +\sum_{j=\{||,\bot \}} \beta_j\frac{ \partial}{ \partial g_j}+2\gamma_{B_2}({\bm{g}})\Big) G^{x,x}_{B_2}(r,{\bm{g}}) =0,$$ where $\beta_j$ are beta functions presented in Eq. (\[KTRG\]) and $\gamma_{B_2}({\bm{g}})=2-4\pi g_{||}/\sqrt{3}$ is the anomalous dimension of the $\hat O^x_2$ operator, calculated in Appendix B. This allows us to follow the approach [@Affleck98; @Lukyanov98] that led to the exact expression of the single-spin correlation function Eq. (\[su2\]). Solving the RG Eq. (\[RGEQ\]) and integrating over $\gamma_{B_2}(\ln{r})$ in the solution, as shown explicitly in Appendix C, gives for $1-\eta \ll 1$ the following behavior over an intermediate range $1\ll \ln{r}\ll1/(1-\eta)$: $G^{x,x}_{B_2}(r)\simeq {B_2 (4(1-\eta)\ln{r})^2}/{r^4}$. Then taking the limit $\eta\to 1$ and using the limiting expression of the amplitude in Eq. (\[LT\]), $B_2\to {1}/({2^7\pi^4(1-\eta)^2})$, we obtain the following exact asymptotic expression for $\ln r \gg 1$, $$\label{ee}
G^{x,x}_{B_2}(r)= \frac{1}{8\pi^4}  \frac{\ln^2r}{ r^4}.$$

Let us consider now the mixed correlation function of bilinear spin operators at the Heisenberg AFM point, $$\langle  \langle S^{a}_0S^{a}_{1}S^{b}_{r}S^{b}_{r+1} \rangle \rangle=G^{a,b}_r=(-1)^r G^{a,b}_{s}(r)  +G^{a,b}_{u}(r),$$ for $a \neq 