b$. Using bosonization [@GNT] one can show that the leading staggered part of the mixed correlation function $G^{a,b}_{s}(r)$ behaves identically to $G^{a,a}_{s}(r)$ [@comment0]. To study the long-distance asymptotics of the uniform part of the mixed bilinear correlation function, $G^{a,b}_{u}(r)$ for $\ln{r}\gg 1$, it is useful to look at the correlation function of the Hamiltonian density, $$\langle \langle   ( {\bf S}_0{\bf S}_1 )( {\bf S}_r{\bf S}_{r+1})  \rangle \rangle=(-1)^rG^E_s(r)+G^E_u(r),$$ where $ G^E_s(r)=9 G^{a,a}_{s}(r)+\cdots$, and we will use the important property that the uniform part $G^E_u(r)$ can not contain multiplicative logarithmic corrections due to energy conservation [@comment]. Since energy density does not pick up anomalous dimension due to marginally irrelevant perturbations, the correlation function of energy density behaves similarly to the correlation function of the energy-momentum tensor of the unperturbed conformally invariant Gaussian or Wess-Zumino model [@comment01], thus $G^E_u(r)\sim 1/r^4$. This means that logarithmic contributions, such as in Eq. (\[ee\]), all must be canceled by the mixed terms. Hence, at the $SU(2)$ AFM point, for the leading behavior of the uniform parts of the bilinear spin correlation functions, we obtain, $$\label{ee1}
2G^{a,b\neq a}_{u}(r)= - G^{a, a}_{u}(r).$$ Let us rewrite the relation (\[correctidenx\]) in the following way: $\sum_r S^{x}_{r}S^{x}_{r+1} \to  \int \mathrm{d} r  \{ 