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

Abstract

We study the two-point field-strength correlation $g^2⟨G_{\mu \nu }^a(s)G_{\alpha \beta }^b(s^{\prime })⟩$ in the Landau gauge in SU(2) and SU(3) quenched lattice QCD, as well as the gluon propagator $g^2⟨A_{\mu }^a(s)A_{\nu }^b(s^{\prime })⟩$. The Landau-gauge gluon propagator $g^2⟨A_{\mu }^a(s)A_{\mu }^a(s^{\prime })⟩$ is well described by the Yukawa-type function $e^{-mr}/r$ with $r\equiv |s-s^{\prime }|$ for $r=0.1-1.0\mathrm{fm}$ 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, $C_{\perp }(r)\equiv g^2⟨H_z^a(s)H_z^a(s+r\hat{\perp }))⟩$ ($\hat{\perp }$: unit vector on the $xy$-plane), and the parallel-type one, $C_{\parallel }(r)\equiv g^2⟨H_z^a(s)H_z^a(s+r\hat{\parallel })⟩$ ($\hat{\parallel }$: unit vector on the $tz$-plane). These two quantities reproduce all the correlation of $g^2⟨G_{\mu \nu }^a(s)G_{\alpha \beta }^b(s^{\prime })⟩$, due to the Lorentz and global SU($N_c$) color symmetries in the Landau gauge. Curiously, the perpendicular-type color-magnetic correlation $C_{\perp }(r)$ is found to be always negative for arbitrary $r$, except for the same-point correlation. In contrast, the parallel-type color-magnetic correlation $C_{\parallel }(r)$ is always positive. In the infrared region of $r\gtrsim 0.4\mathrm{fm}$, $C_{\perp }(r)$ and $C_{\parallel }(r)$ strongly cancel each other, which leads to a significant cancellation in the sum of the field-strength correlations as ${\sum }_{\mu ,\nu }{g}^2⟨G_{\mu \nu }^a(s)G_{\mu \nu }^a(s^{\prime })⟩\propto C_{\perp }(|s-s^{\prime }|)+C_{\parallel }(|s-s^{\prime }|)\simeq 0$. Finally, we decompose the field-strength correlation into quadratic, cubic and quartic terms of the gluon field $A_{\mu }$ in the Landau gauge.

arXiv comment16 pages, 12 figures

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