improved perturbation theory [@Cardy86; @Affleck+89], $$\label{su2}
\langle S^{a}_{0}S^{b}_r \rangle=(-1)^r\delta^{ab}\frac{\sqrt{\ln r}}{(2\pi)^{3/2}r}-\frac{\delta^{ab}}{4\pi^2 r^2},$$ where $a,b=x,y,z$.

Prior to the analytical works [@Affleck98; @Lukyanov98], the numerical prefactor of the staggered term in Eq. (\[su2\]) has been estimated by numerical simulations [@KomaMizukoshi] as $0.065$, which is close to the exact value $1/(2\pi)^{3/2}\simeq 0.0635$.

Bilinear spin correlation functions
===================================

We will generalize the approach leading to the exact assymptotic expressions of single-spin correlation functions at the Heisenberg AFM point [@Affleck98; @Lukyanov98], Eq. (\[su2\]), for the calculation of the uniform part of the correlation function of bilinear spin operators, $S_{r}^aS_{r+1}^a$.

Let us first address the correlation function of the bilinear in $S^x$ operator in the gapless region in the vicinity of (but not directly at) the $SU(2)$ AFM point. Up to the subleading corrections we have $$\begin{aligned}
\label{ccor}
\!\!\!\! \langle  S_{0}^xS_{1}^x S_{r}^xS_{r+1}^x \rangle\!=\!B_0+ \frac{(-1)^r\! B_1}{r^{1/\eta}}  +  \frac{B_2}{r^{4\eta}}+   \frac{B_3}{r^{4/\eta}}+\frac{B_4}{r^4}.\end{aligned}$$ The $B_l$’s, for $l\ge 2$, are amplitudes of the correlation functions of the following $\hat O^{x}_l$ operators, $$\begin{aligned}
\label{prop}
 \hat O^{x}_2 &\sim& \cos \sqrt{8 \pi \eta} \Theta, \quad \hat O^{x}_3 \sim \cos \sqrt{8 \pi/ \eta} \Phi \nonumber \\
\hat O^{x}_4 &\sim & (\partial_r \Phi)^2 +  \beta^{x}_{\eta} (\partial_r \Theta)^2, \quad \partial_r\Theta= \Pi\end{aligned}$$ and $\beta^{x}_{\eta}$ is, similarly to other proportionality coefficients 