Lattice calculation of coordinate-space vector and axial-vector current correlators in QCD

We study the vector and axial-vector current correlators in perturbative and non-perturbative regimes of QCD. The correlators in Euclidean coordinate space are calculated on the lattice using the M\"obius domain-wall fermion formulation at three lattice spacings covering 0.044--0.080~fm. The dynamical quark effects of $2+1$ light flavors are included. The sum $V+A$ and the difference $V-A$ of the vector ($V$) and axial-vector ($A$) current correlators calculated on the lattice after extrapolating to the physical point agree with those converted from the ALEPH experimental data of hadronic $\tau$ decays. The level of the agreement in the $V+A$ channel is about $1.3\sigma$ or smaller in the region of $|x|\ge0.4$~fm, while that in the $V-A$ channel is about $1.8\sigma$ at $|x|=0.74$~fm and smaller at other distances. We also extract the hiral condensate from the short-distance correlators on the lattice using the PCAC relation. Its result extrapolated to the chiral and continuum limit is compatible with other estimates at low energies.