The infrared features of Landau gauge QCD are studied by lattice simulation of $\beta {=6.0,16\mathrm{}}^4{,24\mathrm{}}^4{,32\mathrm{}}^4$ and $\beta {=6.4,32\mathrm{}}^4{,48\mathrm{}}^4.$ We adopt two definitions of the gauge field, (1) U linear and (2) $\log U,$ and measured the gluon propagator and ghost propagator. The infrared singularity of the gluon propagator is less than that of the tree level result, but the gluon propagator at 0 momentum remains finite. The infrared singularity of the ghost propagator is stronger than the tree level. The QCD running coupling measured by using the gluon propagator and the ghost propagator has a maximum ${\alpha }_s\mathrm{}\left(p\right)\mathrm{}\simeq 1$ at around $p=0.5\mathrm{}$ GeV and decreases as p approaches 0. The data are analyzed with use of the formula of the principle of minimal sensitivity and the effective charge method, and by the contour-improved perturbation method, which suggest the necessity of the resummation of the perturbation series in the infrared region together with the existence of the infrared fixed point. The Kugo-Ojima parameter is about $-0.8$ in contrast with the theoretically expected value of $-1.$ The color off-diagonal part of the ghost propagator in the Landau gauge is consistent with zero, and its fluctuation can be parametrized as a $\mathrm{constant}{/\left(\mathrm{qa}\right)\mathrm{}}^4.$

Quark propagators of the unquenched Kogut-Susskind (KS) fermion obtained from the gauge configurations of the MILC Collaboration are measured after Landau gauge fixing and using the $\mathrm{\text{Staple}}+\mathrm{\text{Naik}}$ action. Presence of the $\overline{q}q$ condensates and $A^2$ condensates in the dynamical mass $M(q)$ and the quark wave function renormalization $Z_{\psi }(q^2)$ are investigated. We obtain the correlation of the renormalization factor of the running coupling taken at $\mu \sim 6\mathrm{GeV}$ and that of the quark wave function renormalization $Z_{\psi }(q^2)$ of the $\mathrm{\text{Staple}}+\mathrm{\text{Naik}}$ action. The mass function $M(q)$ is finite at $q=0$ and its chiral limit is $\sim 0.38\mathrm{GeV}$. We compared the results corrected by the scale of the vertex renormalization and the tadpole renormalization with the corresponding values obtained by the Asqtad action without renormalization and observed good agreement. Implication of infrared finite $Z_2(q)=1/Z_{\psi }(q^2)$ to the Kugo-Ojima confinement criterion is discussed.

We performed lattice Landau gauge QCD simulation on $\beta =6.0,{16}^4,{24}^4,{32}^4$ and $\beta =6.4,{32}^4,{48}^4$ 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 ${\tilde{Z}}_1$ 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 ${\alpha }_s(q)\simeq 2.1(3)$ at around $q=0.5\mathrm{G}\mathrm{e}\mathrm{V}$ and decreases as $q$ approaches 0, and the Kugo-Ojima parameter reached $-0.83(2)$. The infrared exponent of the ghost propagator at $0.4\mathrm{G}\mathrm{e}\mathrm{V}$ region is ${\alpha }_G=0.20$ but there is an exceptional Gribov copy with ${\alpha }_G=0.27$. 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 $-1$. 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 ${\alpha }_s(0)\sim 2.5(5)$ and $\kappa =0.5$ 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.

The color diagonal and color antisymmetric ghost propagators slightly above $T_c$ of $N_f=2$ MILC ${24}^3\times 12$ lattices are measured and compared with zero-temperature unquenched $N_f=2+1$ ${\mathrm{MILC}}_c$ ${20}^3\times 64$ and ${\mathrm{MILC}}_f$ ${28}^3\times 96$ lattices and zero-temperature quenched ${56}^4$ $\beta =6.4$ and 6.45 lattices. The expectation value of the color antisymmetric ghost propagator ${\varphi }^c(q)$ is zero, but its Binder cumulant, which is consistent with that of $N_c^2-1$ dimensional Gaussian distribution below $T_c$, decreases above $T_c$. Although the color diagonal ghost propagator is temperature independent, the $l^1$ norm of the color antisymmetric ghost propagator is temperature dependent. The expectation value of the ghost condensate observed at zero-temperature unquenched configuration is consistent with 0 in $T>T_c$. We also measure transverse, magnetic, and electric gluon propagator and extract gluon screening masses. The running coupling measured from the product of the gluon dressing function and the ghost dressing function are almost temperature independent, but the effect of $A^2$ condensate observed at zero temperature is consistent with 0 in $T>T_c$. The transverse gluon dressing function at low temperature has a peak in the infrared at low temperature, but it becomes flatter at high temperature. The magnetic gluon propagator at high momentum depends on the temperature. These data imply that the magnetic gluon propagator and the color antisymmetric ghost propagator are affected by the presence of dynamical quarks, and there are strong nonperturbative effects through the temperature-dependent color antisymmetric ghost propagator.