Probing the center-vortex area law ind=3: The role of inert vortices

Abstract

In center-vortex theory, beyond the simplest picture of confinement several conceptual problems arise that are the subject of this paper. Recall that confinement arises through configuration averaging of phase factors associated with the gauge center group, raised to powers depending on the total Gauss link number of a vortex ensemble with a given Wilson loop. The simplest approach to confinement counts this link number by counting the number of vortices, considered in $d=3$ as infinitely long closed self-avoiding random walks of fixed step length, piercing any surface spanning the Wilson loop. Problems arise because a given vortex may pierce a given spanning surface several times without being linked or without contributing a nontrivial phase factor, or it may contribute a nontrivial phase factor appropriate to a smaller number of pierce points. We estimate the dilution factor $\ensuremath{\alpha}$, due to these inert or partially inert vortices, that reduces the ratio of fundamental string tension ${K}_{F}$ to the areal density $\ensuremath{\rho}$ of vortices from the ratio given by elementary approaches and find $\ensuremath{\alpha}=0.6\ifmmode\pm\else\textpm\fi{}0.1$. Then we show how inert vortices resolve the problem that the link number of a given vortex-Wilson-loop configuration is the same for any spanning surface of whatever area, yet a unique area (of a minimal surface) appears in the area law. Third, we discuss semiquantitatively a configuration of two distinct Wilson loops separated by a variable distance, and show how inert vortices govern the transition between two possible forms of the area law (one at small loop separation, the other at large), and point out the different behaviors in $SU(2)$ and higher groups, notably $SU(3)$. The result is a finite-range van der Waals force between the two loops. Finally, in a problem related to the double-loop problem, we argue that the analogs of inert vortices do not affect the fact that, in the $SU(3)$ baryonic area law, the mesonic string tension appears.