\int \mathrm{d^2} {{\bm{x}}} \mathcal{H'}_1({\bm{x}})  \cos {\sqrt{8 \pi } \Theta (r)} \rangle_G \nonumber\\
=\frac{8\pi g^0_{||}}{\sqrt{3}} \ln{(r/r_0)}r^{-4}.\end{aligned}$$

Combining this correction with Eq. (\[cw\]) we obtain, up to the first order in coupling constants, $$\label{cw1}
G^{x,x}_{B_2}(r) \sim r^{-4}( 1+\frac{8\pi g^0_{||}}{\sqrt{3}} \ln{r/r_0}).$$ Hence, the effective expansion parameter of perturbation series increases logarithmically at large distances, $g^0_{||}\to  g^0_{||}  \ln{(r/r_0)}$.

RG is a way to resum the leading logarithmic divergences of the infinite perturbation series occurring in the $r\to \infty$ limit. One can obtain from Eq. (\[cw1\]) directly at the $SU(2)$ antiferromagnetic point the double-logarithmic correction of the correlation function as follows. At $\eta=1$ we have $g_{\bot}=g_{||}=g$ and considering it as a small perturbation the following connection between the bare and renormalized couplings exists from the one-loop beta function, $$\label{KTSU2}
g(r)=\frac{g^0}{1+4g^0 \ln(r/r_0)/\sqrt{3}}.$$ Hence to the lowest order in coupling constant we can make a substitution, $$\label{subst}
 1+\frac{8\pi g^0}{\sqrt{3}} \ln{\frac{r}{r_0}}= (g^0/g(r))^2+\cdots$$ and represent Eq. (\[cw1\]) in the following form, $$\label{cw2}
G^{x,x}_{B_2}(r)  \sim {r}^{-4}(g^0/g(r))^2+\cdots.$$ From Eq. (\[KTSU2\]), at large distances, $g(r) \simeq \sqrt{3}/(4 \ln(r/r_0))$, and plugging this into Eq. (\[cw2\]) produces multiplicative double-logarithmic correction of the algebraic $1/r^4$ decay of the correlation function $G^{x,x}_{B_2}(r)$. Note that if the anomalous dimension of the operator does not depend on the 