Color-magnetic correlations in SU(N) lattice QCD

TL;DR

The paper addresses color-magnetic instabilities in QCD by evaluating two-point color-magnetic field-strength correlations in SU(2) and SU(3) lattice QCD within the Euclidean Landau gauge. It focuses on perpendicular-type and parallel-type correlations, $C_{ ext{perp}}(r)$ and $C_{ ext{parallel}}(r)$, finding $C_{ ext{perp}}(r)<0$ for all $r>0$ and $C_{ ext{parallel}}(r)>0$, with strong infrared cancellations between them in the total field-strength correlator. By decomposing $C_{ ext{perp}}(r)$ into quadratic, cubic, and quartic gluon-field contributions, the authors show the quadratic term is negative due to a Yukawa-type propagator $igrace{ ext{with } igrace{D(r) o A m e^{-mr}/r}ig}$, while the cubic term is positive and cancels part of the quadratic in the infrared, leaving a small net. The findings support a highly stochastic color-magnetic vacuum in QCD, consistent with stochastic vacuum models and the Savvidy/Copenhagen vacuum picture, and suggest that simple Abelian flux-conservation arguments do not explain the observed correlations.

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, $C_{\perp}(r) \equiv g^2 \langle H_z^a(s)H_z^a(s + r\hat \perp)) \rangle$ with $\perp \equiv x, y$, and the parallel-type one, $C_{\parallel}(r) \equiv g^2 \langle H_z^a(s)H_z^a(s + r\hat \parallel) \rangle$ with $\parallel~\equiv z, t$. In the Landau gauge, all two-point field-strength correlations $g^2 \langle G^a_{μν}(s)G^b_{αβ}(s')\rangle$ 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 $\sum_{μ, ν} \langle G^a_{μν}(s)G^a_{μν}(s')\rangle \propto C_{\perp}(|s-s'|)+ C_{\parallel}(|s-s'|) \simeq 0$. Next, we decompose the perpendicular-type color-magnetic correlation $C_{\perp}(r)$ into quadratic, cubic and quartic terms of the gluon field $A_μ$. The quadratic term is always negative, which is explained by the Yukawa-type gluon propagator $\langle A^a_μ(s)A^a_μ(s')\rangle \propto e^{-mr}/r$ with $r\equiv |s-s'|$ 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 $C_{\perp}(r)$.

Figures (5)

• Figure 1: Landau-gauge gluon propagator $D(r)\equiv g^2 \langle A_\mu^a(s)A_\mu^a(s')\rangle$ plotted against $r\equiv |s-s'|$ in SU(3) (left) and SU(2) (right) lattice QCD. The curve is the best-fit Yukawa function.
• Figure 2: The perpendicular-type color-magnetic correlation $C_{\perp}(r) \equiv g^2 \langle H^a_z(s)H^a_z(s+r\hat{\perp})\rangle$ ($\perp~\equiv x,y$) in the Landau gauge in SU(3) (left) and SU(2) (right) lattice QCD.
• Figure 3: An example of the plaquette correlator to extract the gauge-invariant field-strength correlation in lattice QCD.
• Figure 4: The parallel-type color-magnetic correlation $C_{\parallel}(r) \equiv g^2 \langle H^a_z(s)H^a_z(s+r\hat{\parallel})\rangle$ ($\parallel~\equiv z,t$) in the Landau gauge in SU(3) (left) and SU(2) (right) lattice QCD.
• Figure 5: Each contribution of the quadratic (red), cubic (blue) and quartic (green) terms of the perpendicular-type color-magnetic correlation $C_{\perp}(r)$ (black) in the Landau gauge in SU(3) (left) and SU(2) (right) lattice QCD. The red dotted line denotes the curve of Eq. (\ref{['eq:Yukawa-perp']}) derived from the Yukawa-type propagator $D_{\rm Yukawa}(r)$.