coupling constants (which is the case for conserved quantities) there will be no multiplicative logarithmic corrections in the corresponding correlation function.

Moreover, apart from the logarithmic correction we can even determine the precise numerical prefactor, by comparing with the exact results for $\eta<1$ [@LT]. Since in the infrared limit the running coupling constant $g_{\perp}$ flows to zero, one can estimate the (renormalized) correlation function at large scale, from perturbative expansion in $g_{\perp}$.

We note that $G^{x,x}_{B_2}$ obeys the following Callan-Symanzik (CS) RG equation, $$\label{RGEQ}
\!\Big(\frac{\partial}  {\partial \ln \tilde r} +\!\!\!\!\sum_{j=\{||,\bot \}}\!\!\!\!\! \beta_j\frac{ \partial}{ \partial g_j}-2\gamma_{B_2}({\bm{g}})\Big) G^{x,x}_{B_2}(r,\tilde r, {\bm{g}}(\tilde r)) =0,$$ where ${\bm{g}}(\tilde r)=(g_{||}(\tilde r), g_{\bot}(\tilde r))$, $\beta_j$ are their beta functions presented in Eq. (5) of the main text and $\gamma_{B_2}({\bm{g}})=2-4\pi g_{||}/\sqrt{3}$ is the anomalous dimension of $\hat O^{x}_2$ calculated in the previous section. The CS RG equation is equivalent to the one presented in the main text Eq. (13), up to the sign in front of the anomalous dimension $\gamma_{B_2}$ (due to the fact that increase of the short-distance cutoff is equivalent to decreasing the distances measured in units of the new cutoff).

CS Eq. (\[RGEQ\]) defines the evolution of the two-point correlation function $G$ under variation of the 