Color-magnetic correlations in SU(N) lattice QCD

Abstract

Motivated by color-magnetic instabilities in QCD, we investigate field-strength correlations in both SU(2) and SU(3) lattice QCD. In the Euclidean Landau gauge, we numerically calculate the perpendicular-type color-magnetic correlation, $C_{\perp }(r)\equiv g^2⟨H_z^a(s)H_z^a(s+r\hat{\perp }))⟩$ with $\perp \equiv x,y$, and the parallel-type one, $C_{\parallel }(r)\equiv g^2⟨H_z^a(s)H_z^a(s+r\hat{\parallel })⟩$ with $\parallel \equiv z,t$. In the Landau gauge, all two-point field-strength correlations $g^2⟨G_{\mu \nu }^a(s)G_{\alpha \beta }^b(s^{\prime })⟩$ are described by these two quantities, due to the Lorentz and global SU($N_c$) color symmetries. Curiously, the perpendicular-type color-magnetic correlation $C_{\perp }(r)$ is found to be always negative for arbitrary $r$, except for the same point of $r=0$. The parallel-type color-magnetic correlation $C_{\parallel }(r)$ is always positive. In the infrared region, $C_{\perp }(r)$ and $C_{\parallel }(r)$ strongly cancel each other, which leads to an approximate cancellation for the sum of the field-strength correlations as ${\sum }_{\mu ,\nu }⟨G_{\mu \nu }^a(s)G_{\mu \nu }^a(s^{\prime })⟩\propto C_{\perp }(|s-s^{\prime }|)+C_{\parallel }(|s-s^{\prime }|)\simeq 0$. Next, we decompose the perpendicular-type color-magnetic correlation $C_{\perp }(r)$ into quadratic, cubic and quartic terms of the gluon field $A_{\mu }$. The quadratic term is always negative, which is explained by the Yukawa-type gluon propagator $⟨A_{\mu }^a(s)A_{\mu }^a(s^{\prime })⟩\propto e^{-mr}/r$ with $r\equiv |s-s^{\prime }|$ in the Landau gauge. 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 $C_{\perp }(r)$.

Comments: 8 pages, 5 figures