Color-magnetic correlations in SU(2) and SU(3) lattice QCD

We study the two-point field-strength correlation g2⟨Gμνa(s)Gαβb(s′)⟩ in the Landau gauge in SU(2) and SU(3) quenched lattice QCD, as well as the gluon propagator g2⟨Aμa(s)Aνb(s′)⟩. The Landau-gauge gluon propagator g2⟨Aμa(s)Aμa(s′)⟩ is well described by the Yukawa-type function e−mr/r with r≡|s−s′| for r=0.1–1.0 fm in both SU(2) and SU(3) QCD. Next, motivated by color-magnetic instabilities in the QCD vacuum, we investigate the perpendicular-type color-magnetic correlation, C⊥(r)≡g2⟨Hza(s)Hza(s+r⊥^))⟩ (⊥^: unit vector on the xy plane), and the parallel-type one, C∥(r)≡g2⟨Hza(s)Hza(s+r∥^)⟩ (∥^: unit vector on the tz plane). These two quantities reproduce all the correlation of g2⟨Gμνa(s)Gαβb(s′)⟩, due to the Lorentz and global SU(Nc) color symmetries in the Landau gauge. Curiously, the perpendicular-type color-magnetic correlation C⊥(r) is found to be always negative for arbitrary r, except for the same-point correlation at r=0. In contrast, the parallel-type color-magnetic correlation C∥(r) is always positive. In the infrared region of r≳0.4 fm, C⊥(r) and C∥(r) strongly cancel each other, which leads to a significant cancelation in the sum of the field-strength correlations as ∑μ,νg2⟨Gμνa(s)Gμνa(s′)⟩∝C⊥(|s−s′|)+C∥(|s−s′|)≃0. Finally, we decompose the field-strength correlation into quadratic, cubic, and quartic terms of the gluon field Aμ in the Landau gauge. For the perpendicular-type color-magnetic correlation C⊥(r), the quadratic term is always negative, which is explained by the Yukawa-type gluon propagator. The quartic term gives a relatively small contribution. In the infrared region, the cubic term is positive and tends to cancel with the quadratic term, resulting in a small value of C⊥(r).