\Gamma(\frac{1}{2(1-\eta)}) }{\Gamma(\frac{\eta}{2(1-\eta)}) }  \right)^{4(1-\eta)}.\end{aligned}$$

This expression agrees with the one obtained in \[\].

We note that one cannot use the effective representation of single-spin operators [@GNT] to obtain the short-distance correlation function for the XXZ spin-$\frac{1}{2}$ chain and in particular to obtain exact amplitudes of the correlation functions of bilinear spin operators with the fusion rules of underlying conformal theory. This is so, because conformal symmetry is only an effective property of the model and at short distances the XXZ chain is not conformally invariant, because of irrelevant, in infrared limit, corrections (the leading ones can be found in \[\]). Due to this reason the constant $B_2$ is not related to coefficients $A^x_0$ and $A^x_1$ appearing in Eq. (3) of the main text in any simple way and also we cannot determine the exact numerical prefactor in front of the staggered part of the bilinear spin correlation function \[the $B_1$ coefficient in Eq. (7) of the main text cannot be fixed with currently known methods\].

Calculating anomalous dimension $\gamma_{B{_2}}$
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In this appendix we show how to calculate the anomalous dimension of the field $\cos {\sqrt{8 \pi } \Theta}$ picked up upon renormalization due to marginally irrelevant perturbations of the 