What the Gribov copy tells about confinement and the theory of dynamical chiral symmetry breaking

Abstract

We performed lattice Landau gauge QCD simulation on $\ensuremath{\beta}=6.0,{16}^{4},{24}^{4},{32}^{4}$ and $\ensuremath{\beta}=6.4,{32}^{4},{48}^{4}$ and ${56}^{4}$ by adopting the gauge fixing that minimizes the norm of the gauge field, and measured the running coupling by using the gluon propagator and the ghost propagator. In view of ambiguity in the vertex renormalization factor ${\stackrel{\texttildelow{}}{Z}}_{1}$ in the lattice, we adjust the normalization of the running coupling by the perturbative QCD results near the highest momentum point. It has a maximum ${\ensuremath{\alpha}}_{s}(q)\ensuremath{\simeq}2.1(3)$ at around $q=0.5\text{ }\text{ }\mathrm{G}\mathrm{e}\mathrm{V}$ and decreases as $q$ approaches 0, and the Kugo-Ojima parameter reached $\ensuremath{-}0.83(2)$. The infrared exponent of the ghost propagator at $0.4\text{ }\text{ }\mathrm{G}\mathrm{e}\mathrm{V}$ region is ${\ensuremath{\alpha}}_{G}=0.20$ but there is an exceptional Gribov copy with ${\ensuremath{\alpha}}_{G}=0.27$. The features of the exceptional Gribov copy are investigated by measuring four one-dimensional Fourier transform (1-d FT) of the gluon propagator transverse to each lattice axis. We observe, in general, correlation between absolute value of the Kugo-Ojima parameter and the degree of reflection positivity violation in the 1-d FT of the gluon propagator. The 1-d FT of the exceptional Gribov copy has an axis whose samplewise gluon propagator manifestly violates reflection positivity, and the average of the Cartan subalgebra components of the Kugo-Ojima parameter along this axis is consistent to $\ensuremath{-}1$. The running coupling of the ensemble average shows a suppression at 0 momentum, but when the ghost propagator of the exceptional Gribov copy is adopted, the suppression disappears and the data implies presence of the infrared fixed point ${\ensuremath{\alpha}}_{s}(0)\ensuremath{\sim}2.5(5)$ and $\ensuremath{\kappa}=0.5$ suggested by the Dyson-Schwinger approach in the multiplicative renormalizable scheme. Comparison with the SU(2) QCD and ${N}_{f}=2$ unquenched SU(3) QCD are also made.