Lattice Gauge Theories

Abstract

Lattice Gauge Theories reflects the number of colours carried by the quarks. Since there are eight generators of SU(3), there are eight massless "gluons" carrying a colour charge which mediate the strong interactions between the fundamental constituents of matter. By the emission or absorption of a gluon, a quark can change its colour.QCD is an asymptotically free theory ('t Hooft, 1972;Politzer, 1973;Gross and Wilczek, 1973). Asymptotic freedom tells us that the forces between quarks become weak for small quark separations. Because of this asymptotic freedom property it was possible for the first time to carry out quantitative perturbative calculations of observables in strong interaction physics which are sensitive to the short distance structure of QCD. * In particular it allowed one to study the Bjorken scaling violations observed in deep inelastic lepton nucleon scattering at SLAC. QCD is the only theory we know that can account for these scaling violations.The asymptotic freedom property of QCD is intimately connected with the fact that it is based on a non-abelian gauge group. As a consequence of this non-abelian structure the coloured gluons, which mediate the interactions between quarks, can couple to themselves. These self couplings, one believes, are responsible for quark confinement. Since the coupling strength becomes small for small separations of the quarks, one can speculate that the forces may become strong for large separations. This could explain why these fundamental constituents of matter have never been seen free in nature, and why only colour neutral hadrons are observed. A confirmation that QCD accounts for quark confinement can however only come from a non-perturbative treatment of this theory, since confinement is a consequence of the dynamics at large distances where perturbation theory breaks down.Until 1974 all predictions of QCD were restricted to the perturbative regime. The breakthrough came with the lattice formulation of QCD by Kenneth Wilson (1974), which opened the way to the study of non-perturbative phenomena using numerical methods. By now lattice gauge theories have become a branch of particle physics in its own right, and their intimate connection to statistical mechanics make them of interest to elementary particle physicists as well as to physicists working in the latter mentioned field. Hence also those readers who are not acquainted with quantum field theory, but are working in statistical mechanics, can profit from a study of lattice gauge theories. Conversely, elementary particle physicists have profited enormously from the computational methods used in statistical mechanics, such as the high temperature expansion, cluster expansion, mean field approximation, renormalization group methods, and numerical methods. * For an early review see Politzer (1974). * For a comprehensive discussion of the PI-method in quantum mechanics in the real-time formulation, the reader should confer the book by Feynman and Hibbs (1965).

Hybrid static potential flux tubes from SU(2) and SU(3) lattice gauge theory

“…ref. [53]). Since we are considering purely gluonic observables, we expect that there is little difference between our pure gauge theory results and corresponding results in full QCD.…”

Perfect Abelian dominance of quark confinement in SU(3) QCD

“…Although its analytical proof is not yet known from QCD, lattice QCD Monte Carlo simulations show that the static quark-antiquark (QQ) potential is well reproduced by a sum of the Coulomb and linear confinement terms as [1][2][3] …”

“…The linear confinement term σr is considered to be caused by "one-dimensional squeezing" of the interquark colorelectric flux, which is shown by lattice QCD studies [1]. Historically, such a one-dimensional property of hadrons leads to several interesting theoretical frameworks, such as string theory, the flux-tube picture [4], and the Lund model [5] for hadron reactions based on the Schwinger mechanism.…”

Kibble-Zurek mechanism in a quantum link model

“…Although the freezing of the evolution in the impulse region has been shown to be an oversimplified assumption, the KZM prediction of Eq. (5) has been verified in various systems [24,25,57].…”