Color-magnetic correlations in $SU(N)$ lattice QCD

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 $g^2 \langle G^a_{\mu\nu}(s)G^b_{\alpha\beta}(s')\rangle$ 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 $\sum_{\mu, \nu} \langle G^a_{\mu\nu}(s)G^a_{\mu\nu}(s')\rangle \propto C_{\perp}(|s-s'|)+ C_{\parallel}(|s-s'|) \simeq 0$. Next, we decompose the perpendicular-type color-magnetic correlation $C_{\perp}(r)$ into quadratic, cubic and quartic terms of the gluon field $A_\mu$. The quadratic term is always negative, which is explained by the Yukawa-type gluon propagator $\langle A^a_\mu(s)A^a_\mu(s')\rangle \propto e^{-mr}/r$ 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 $C_{\perp}(r)$.