Open issues in confinement, for the lattice and for center vortices

Abstract

Center vortices have been around for more than thirty years, well-confirmed on the lattice, and very successful in explaining the basics of confinement, yet there are still open questions unstudied either on the lattice or in theory. The first is that basic confinement in the center vortex picture is topological, coming from gluonic solitons where the gluons have no direct coupling to the Wilson loop, and makes no reference to any particular surface (whose area would appear in the area law) or fluctuation dynamics of this surface. Only in d = 2 (flat Wilson loops) is it obvious what surface must be involved, and in this dimension there is no room for fluctuations. This makes it hard to understand the Lüscher term and other properties of the fluctuating confinement surface for d > 2. I make the obvious, but unconfirmed to date, conjecture that in topological confinement for non-planar Wilson loops the area law is the exponential of a string tension times the area of a minimal surface spanning the Wilson loop. Less obvious is whether, in this purely topological picture, this minimal surface shows the correct Lüscher term, or whether this term must come from gluons propagating between points on the Wilson loop (as possibly described by fishnet graphs and their relative the gluon chain model). Closely-related issues are the structure of the area law for two coaxial Wilson loops, as the distance between them along the axis grows; the resulting Casimir force between hadrons; and the behavior of k-string tensions for SU(N) with N > 3. I suggest a program of both lattice and theoretical studies, focused on center vortices and the pinch technique, to explore these and other issues: 1) Calculate the area law and its fluctuations for non-planar Wilson loops, or for pairs of flat Wilson loops, in a center-vortex-like ground state with a gas of vortices, but with no gluon-Wilson loop coupling. 2) Study more closely a picture I outline here of reconciling center vortices and minimal surfaces with fishnet graphs and the gluon-chain model, with the key ingredient of dynamically-massive gluons. 3) Extend beyond perturbation theory the old lattice work of Dashen and Gross on background-field Feynman gauge fixing to extract the gauge-invariant off-shell Green's functions of the pinch technique. © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.