Confinement by Monopole Loops in Inhomogeneous Magnetic Field

Abstract

We show that a generalized Polyakov mechanism can lead to confinement at weak coupling in $3+1$ dimensions when the theory is placed in a non-trivial, spatially varying magnetic field background. Depending on the magnitude of the field and the length scale of its spatial variation, the "dual" Schwinger mechanism for monopole-antimonopole pair creation may or may not be operative. At the threshold, monopole loops in the Euclidean description develop an almost flat direction. In this regime, confinement arises in a way similar to the $2+1$ dimensional Polyakov mechanism and the monopoles and antimonopoles are effectively replaced by deconfined "bits" of a monopole loop.

Figures (6)

• Figure 1: The Euclidean action $S_E(l_{\tau},l_{z})$ above, equal to, and below the critical value of $B$. For $B>B_{cr}$ there is a saddle point solution (black dot in the left plot). At the critical value $B=B_{cr}$ there is a flat direction (black line in the middle plot). Here we used $S_{M}=d=1$.
• Figure 2: A finite-energy background made with staggered currents can create a pattern of oscillating magnetic field in two directions.
• Figure 3: The instanton–monopole–loop solution (\ref{['instmonloop']}) for the oscillatory background for different $C$. Approaching the critical value the loop becomes infinitely elongated in the Euclidean time direction while remaining fixed at the scale of the spatial fluctuation. This effect is generic for any type of oscillation, not only the simple trigonometric one. We used $m_{M}=1$ and $k_z=1$.
• Figure 4: The deconfined bits of the monopole loop (\ref{['taucrittrig']}) at the critical value $B_{cr}$. We used $m_{M}=1$ and $k_z=1$. The position in $\tau$ can be arbitrary (the constant in (\ref{['taucrittrig']}) which here in the plot is set to $0$).
• Figure 5: Critical line in the $B$-$d$ plane and the other conditions of Eq. (\ref{['conditions']}).