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

TL;DR

This work analyzes two-point color-magnetic and field-strength correlators in SU(2) and SU(3) lattice QCD within the Landau gauge. The gluon propagator is well fit by a Yukawa-type form $D_{ ext{Yukawa}}(r)=A m\,e^{-mr}/r$ with $m\approx 0.66$ GeV, indicating an infrared mass scale. The perpendicular-type color-magnetic correlation $C_{\perp}(r)$ is negative while the parallel-type $C_{\parallel}(r)$ is positive, with their sum nearly vanishing for $r\gtrsim 0.4$ fm, implying near-zero total field-strength correlations in the infrared. A decomposition into quadratic, cubic, and quartic gluon-field contributions explains the signs and cancellations, highlighting a highly stochastic QCD vacuum and potential links to dimensional-reduction ideas and stochastic vacuum models. These results deepen understanding of the QCD vacuum structure and its nonperturbative magnetic fluctuations in both SU(2) and SU(3) gauge theories.

Abstract

We study the two-point field-strength correlation $g^2 \langle G_{μν}^a(s)G^b_{αβ}(s') \rangle$ in the Landau gauge in SU(2) and SU(3) quenched lattice QCD, as well as the gluon propagator $g^2 \langle A_μ^a (s)A_ν^b(s') \rangle$. The Landau-gauge gluon propagator $g^2 \langle A_μ^a (s)A_μ^a(s') \rangle$ is well described by the Yukawa-type function $e^{-mr}/r$ with $r\equiv |s-s'|$ for $r=0.1-1.0~{\rm fm}$ 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, $C_{\perp}(r) \equiv g^2\langle H_z^a(s)H_z^a(s + r \hat \perp)) \rangle$ ($\hat \perp$: unit vector on the $xy$-plane), and the parallel-type one, $C_{\parallel}(r) \equiv g^2 \langle H_z^a(s)H_z^a(s + r \hat \parallel) \rangle$ ($\hat \parallel$: unit vector on the $tz$-plane). These two quantities reproduce all the correlation of $g^2\langle G^a_{μν}(s)G^b_{αβ}(s')\rangle$, 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 $r \gtrsim 0.4~{\rm fm}$, $C_{\perp}(r)$ and $C_{\parallel}(r)$ strongly cancel each other, which leads to a significant cancellation in the sum of the field-strength correlations as $\sum_{μ, ν} g^2\langle G^a_{μν}(s)G^a_{μν}(s')\rangle \propto C_{\perp}(|s-s'|)+ C_{\parallel}(|s-s'|) \simeq 0$. Finally, we decompose the field-strength correlation into quadratic, cubic and quartic terms of the gluon field $A_μ$ in the Landau gauge.

Figures (12)

• Figure 1: Plaquette correlations corresponding to the spatial color-magnetic correlations at the same Euclidean time, $C_{\perp}(r)$ and $C_{\parallel}(r)$, in lattice QCD. The upper plaquette correlation corresponds to $C_{\perp}(r) = g^2\langle H_z^a(s) H_z^a(s+r\hat{x}) \rangle$, and the lower one $C_{\parallel}(r) = g^2\langle H_z^a(s) H_z^a(s+r\hat{z}) \rangle$.
• Figure 2: Landau-gauge gluon propagator $g^2\langle A_\mu^a(s)A_\mu^a(s')\rangle$ plotted against $r\equiv |s-s'|$ in SU(3) (upper) and SU(2) (lower) lattice QCD. The curve is the best-fit Yukawa-type function $D_{\rm Yukawa}(r)$ in Eq. (\ref{['eq:Yukawa-type']}).
• Figure 3: Perpendicular-type color-magnetic correlation $C_{\perp}(r) \equiv g^2\langle H^a_z(s)H^a_z(s+r \hat{\perp})\rangle$ in the Landau gauge in SU(3) (upper) and SU(2) (lower) lattice QCD. $\hat{\perp}$ denotes a unit vector on the $xy$-plane.
• Figure 4: Example of the plaquette correlator to extract the gauge-invariant field-strength correlation in lattice QCD.
• Figure 5: Parallel-type color-magnetic correlation $C_{\parallel}(r) \equiv g^2\langle H^a_z(s)H^a_z(s+r \hat{\parallel})\rangle$ in the Landau gauge in SU(3) (upper) and SU(2) (lower) lattice QCD. $\hat{\parallel}$ denotes a unit vector on the $zt$-plane.