On the other hand, in the difference $G^{a,a}_r- G^{a,b}_r$, $b\neq a$, the leading oscillatory terms cancel [@comment0] and from this quantity and Eq. (\[ee1\]) we can obtain the desired uniform part of the correlation function $G^{a,a}_{u}(r)$. In Fig. 2 we plot numerical data for $r^4(G^{a,a}_r-G^{a,b}_r)$, which includes both uniform and staggered components. For the uniform component our analytical result is $\ln^2r/(4\pi^4)$, following from Eq. (\[mainresult\]) and the relation $G^{a,s}_r-G^{a,b}_r=\frac{3}{2}G^{a,a}_{u}(r)+\cdots$, where dots indicate sub-leading contribution. We will calculate the leading oscillatory contribution in $$\!\!2(G^{x,x}_r\!\!-\!G^{x,y}_r)=\! \langle \! { (S^x_0S^x_1-\!S^y_0S^y_1 )(S^x_rS^x_{r+1}-\!S^y_rS^y_{r+1} )} \! \rangle.$$

![Difference of bilinear spin correlation functions of the Heisenberg spin-$\frac{1}{2}$ AFM chain $\rho^4(G^{a,a}_r-G^{a,b}_r)$ for $a\neq  b$ and for $L=96$ sites chain with periodic boundary conditions (PBC). Bullets indicate numerical data and continuous lines are analytical curves, $\ln^2(c_0\rho)/(4\pi^4)\pm \tilde c{\ln^{\frac{3}{2}}(c_0 \rho)}/\rho$, where $+$ sign corresponds to even $r$ data and $-$ sign to odd $r$ data. Constant $c_0\simeq 22$ and $\tilde c$ is fitted to $\tilde c\simeq 0.0023$. Dashed line is analytical result without taking into account leading oscillatory contribution. Inset shows $L=144$ sites case with $c_0\simeq 23$ and the same value of $\tilde c\simeq 0.0023$. []{data-label="fig:pd"}](Osc48New.eps){width="8.5cm"}

In bosonization $ (-1)^r(S^x_rS^x_{r+1}-S^y_rS^y_{r+1})\to \hat D(r)$, where $\hat D(r) \sim e^{i{\sqrt{8\pi}\Theta(r) }}\cos{\sqrt{2 \pi} 