XXZ chain reduces to the Heisenberg antiferromagnetic (AFM) chain.

Despite being exactly solvable, calculating correlation functions (objects that provide direct connection between theoretical calculations and experimental observations) from the microscopic XXZ model for $\lambda>-1$ is a formidable task, due to the complicated form of the wave functions [@Godin]. On the other hand, effective approaches, have allowed asymptotically exact calculation of the spin correlation functions in the gapless regime $-1<\lambda\le 1$ [@LZ; @Affleck98; @Lukyanov99; @LT].

Effective theory, describing the low-energy properties of the XXZ spin-1/2 chain for $-1<\lambda<1$, is given by the Gaussian model [@GNT], $$\label{ham-LL}
 H_G= \int  \mathcal{H}_G(r)\mathrm{d} r=\frac{v}{2}\int \mathrm{d}r \Big\{  (\partial_{r}\Phi(r))^{2}+\Pi^{2}(r)\Big\},$$ where $\Phi$ is a real bosonic field with the compactification radius $R$, $\Phi=\Phi+2\pi R$, and $\Pi$ is its conjugate momentum, $[\Phi(r),\Pi(r')]=i\delta(r-r')$. Spin-wave velocity $v$ and $R$ are known analytically as functions of $\lambda$ from the exact solution of the model (\[ham-XXZ0\]) [@JohnsonKrinskyMcCoy73].

In this work, using an effective approach, we determine exact asymptotic expressions of the uniform parts of two-point correlation functions of bilinear spin operators in the Heisenberg AFM spin-$\frac{1}{2}$ chain. Apart from the algebraic decay, we identify the logarithmic contribution and determine the exact numerical prefactor. Our calculations are similar to the ones that were performed 