From confining fields on the lattice to higher dimensions in the continuum

Abstract

We discuss the relation between lattice phenomenology of confining fields in the vacuum state of Yang-Mills theories (mostly SU(2) case) and continuum theories. In the continuum, understanding of the confinement is most straightforward in the dual formulation which involves higher dimensions. We try to bridge these two approaches to the confinement, let it be on a rudimentary level. We review lattice data on low-dimensional vacuum defects, like monopoles, center vortices. There is certain resemblance to dual strings, domain walls considered in the continuum version of Yang-Mills theories.

Magnetic monopoles in the high temperature phase of Yang–Mills theories

“…As we have emphasized above, very similar problems are met when studying center dominance in the maximal center gauge (MCG) [22][23][24][25] and some possible interpretations have been discussed [24,26]. One should also study how the problem is related to the choice of a particular starting point on the gauge orbit (belonging to Landau gauge in our case) and perform more extensive studies in which the global maximum of the MAG functional is searched for via better algorithms, like for instance simulated annealing [27].…”

Center vortices as rigid strings

“…In order to describe condensation one should use the concepts of Euclidean interacting quantum field theory instead of random walks which describe the states of only one particle. Condensate then corresponds to nonzero background field, as in the Higgs model (for a related discussion see, e.g, [22,23]). It is not clear whether this picture remains valid for the theory of random surfaces, because required nonperturbative apparatus of string field theory is almost not developed.…”

“…Quadratic divergence in the effective action (1) corresponds to vacuum oscillations of these two-dimensional fields [22,23]. Localization on center vortices was also directly observed in lattice simulations -for instance, it was found that in the maximal Abelian gauge almost all trajectories of Abelian monopoles belong to center vortices [20].…”

“…Up to now the only way to see directly the content of the conjectured two-dimensional field theory is to impose maximal Abelian gauge and to look at the trajectories of Abelian monopoles [19,22,23,28], which populate densely the surfaces of center vortices. In real simulations about 95% of monopole trajectories belong to center vortices [19,20].…”

“…Besides quadratic divergence, this also yields a −1 ∼ Λ UV divergence in the bare string tension in physical units: a −2 σ 0 (a) = a −2 A + a −1 B, where A is given by (2) and B = 2.2 ± 0.1f m −1 . Such divergence usually corresponds to self-energy of one-dimensional objects [4,8,22,23]. If the density of one-dimensional objects per unit of vortex area ρ 1D scales in physical units and is constant in the continuum limit, while the bare mass of these objects is UV divergent and is close to the critical value m bare = a [28].…”