Lattice Gauge Theory

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LATTICE GAUGE THEORY

Asymptotic freedom has defined the history of quantum chromodynamics (QCD), the gauge theory of quarks and gluons that describes subnuclear physics. Asymptotic freedom means that the interaction energies between quarks weaken relative to their kinetic energies at short distances, less than about 1/3 fm. This weakening allows us to analyze short-distance interactions using an expansion in powers of the interaction energy divided by the kinetic energy. This expansion is called perturbation theory, and perturbation theory was well developed when QCD's asymptotic freedom was discovered in1974. Consequently most tests of QCD during the next two decades focused exclusively on its short-distance behavior.

Unfortunately hadrons, such as the proton and the neutron, are several times larger than 1/3 fm. Thus the physics of hadronic structure is highly nonperturbative and, in 1974, was impossible to compute. Within months of the discovery of asymptotic freedom, however, Kenneth Wilson introduced a new formulation of QCD, called lattice QCD, that facilitated nonperturbative, numerical simulations of QCD. An early triumph of lattice QCD was Wilson's demonstration that quarks are confined within hadrons in the strong-coupling limit of the theory, but further progress was very slow until the 1990s. Today Wilson's theory—the first lattice gauge theory studied by particle physicists—provides the only rigorous approach for computing long-distance properties of QCD, including such things as the masses and structure of hadrons.

The Lattice Approximation

Wilson's innovation was to replace continuous space and time by a rectangular lattice of discrete points or sites in space and time, separated from each other by a fixed lattice spacing in each direction. In the lattice approximation, the fields that describe quarks and gluons are specified only at the lattice sites. Thus the fields within a hadron can be specified by a finite number of numbers—the field values at each lattice site inside the hadron—and the problem becomes tractable on a computer.

The QCD path integral, which defines the quantum theory, becomes an ordinary multidimensional integral in the lattice approximation. In principle any property of the theory can be computed using this integral. The integration variables are the values of the eight gauge fields at each of the lattice sites. Since a typical lattice today has a lattice spacing between 0.1 and 0.2 fm, and covers a volume of about 3 fm, lattices with 20⁴ sites or more are common. Consequently the path integral involves the evaluation of millions of nested integrals. Special numerical techniques, called Monte Carlo simulations, are used to evaluate these integrals. The simplest simulations, using very coarse lattices and severe approximations, can be completed on a laptop within an hour. High-quality simulations, however, require months of running on clusters composed of hundreds of PCs.

Quantum fields, unlike most classical fields, have important structure at all length scales. This suggests that a grid approximation of the sort used in lattice QCD should fail because it omits all structure at distances smaller than the lattice spacing. In fact, the effects of this missing structure can be mimicked by modifying the integrand of the path integral. The modifications are computed using perturbation theory because they come from short distances, where asymptotic freedom renders QCD perturbative. Thus lattice QCD is actually a hybrid of perturbative techniques for physics at scales smaller than the lattice spacing and with numerical, nonperturbative techniques for physics at scales of order the lattice spacing or larger.

Application in QCD and Beyond

Lattice QCD simulations are commonly used to compute the properties of single hadrons. They are particularly effective for hadrons that are stable or nearly stable with respect to strong interactions. Particle masses, radii, magnetic moments, and other aspects of a particle's structure are all readily computed. In addition, QCD simulations are used to calculate electroweak form factors, structure functions, and decay amplitudes that couple photons, W bosons, or Z bosons to a hadron. This last application is particularly important for heavy-quark physics, with its focus on heavy-quark decays mediated by weak interactions.

Lattice QCD simulations are also used to study the behavior of QCD at high temperatures. Such simulations provide insights into the behavior of matter in extreme conditions, such as might be found in stellar interiors or in the very early universe.

After slow progress in the 1970s and 1980s, the 1990s saw rapid improvements in lattice QCD techniques and in the computer hardware needed for the simulations. As a result, simulation errors were reduced from 100 percent to between 10 and 20 percent for a wide variety of nonperturbative quantities. The first decade of the twenty-first century will see these errors fall by another order of magnitude, and lattice QCD will play an increasingly important role in high-precision studies of the weak interactions of heavy quarks.

Lattice methods are applicable to other field theories as well. They have been used to explore the Higgs sector of the Standard Model in the limit of large Higgs mass, where the interactions become strong. These techniques could well be important for studying physics beyond the Standard Model. Most realistic quantum field theories have strong interactions either at low energies (e.g., QCD) or at high energies (e.g., gravity). The only exceptions are theories in which symmetries are spontaneously broken(e.g., electroweak interactions). But even in these theories, the most natural mechanism for spontaneous symmetry breaking is dynamic in origin and again involves strong coupling. Lattice methods, which must be extended to cover such models, offer the best hope of dealing with all such strong-interaction phenomena.