Lattice QCD

In physics, lattice quantum chromodynamics (lattice QCD) is a theory of quarks and gluons formulated on a space-time lattice. That is, it is a lattice model of quantum chromodynamics, a special case of a lattice gauge theory or lattice field theory. At the moment, this is a quite well established non-perturbative approach to solving the theory of Quantum Chromodynamics.

Analytic or perturbative solutions in QCD are hard or impossible due to the highly nonlinear nature of the strong force. The formulation of QCD on a discrete rather than continuous space-time naturally introduces a momentum cut off at the order 1/a, where a is the lattice spacing, which regularizes the theory. As a result lattice QCD is mathematically well-defined. Most importantly, lattice QCD provides the framework for investigation of non-perturbative phenomena such as confinement and quark-gluon plasma formation, which are intractable by means of analytic field theories.

In lattice QCD spacetime is represented not as continuous but as a crystalline lattice, vertices connected by lines. Quarks may reside only on vertices and gluons can only travel along lines. While this is understood to be a fiction, as the spacing between vertices is reduced to zero, the theory will approach continuum QCD. As this is computationally impossible, lattice QCD calculations often involve analysis at different lattice spacings to determine the lattice-spacing dependence, which can then be extrapolated to the continuum. “Lattice QCD has always been plagued by the fact that towards realistic quark masses the simulation costs increase drastically due to the large correlation lengths of the light states and the large lattice volumes that one employs in order to avoid finite-size (FS) effects. This difficulty has, in fact, prevented us until this day from carrying out simulations with truly physical parameters. Instead one resorts to computations with several un-physically large quark masses in FS-effect-free volumes, followed by an extrapolation to the physical quark mass.”

This technique is only applicable in the domain of low density and high temperature. At higher densities, the region of greatest interest, the fermion sign problem renders the results useless. Lattice QCD predicts that confined quarks will become released to quark-gluon plasma around energies of 170 MeV. Lattice QCD's limitation to low density does not allow investigation of the color flavor locked states (CFL) at higher densities.

Lattice QCD has already made contact with experiments at various fields with good results. For example the mass of the proton could be determined theoretically with an error of less than 2 percent. A particular important tool of the theory showing the confinement of the underlying fields is the Wilson loop variable, described in a separate article.

See also

• QCD matter

References

• Creutz, Michael "Quarks, Gluons and Lattices" (Cambridge, 1983)
• Degrand and De Tar "Lattice Methods for Quantum Chromodynamics" (World Scientific, 2006)
• Montvay and Münster "Quantum Fields on a Lattice" (Cambridge 1997)
• Weingarten, Donald H. "Quarks by Computer", Scientific American, 274:2, 116-120 (February 1996)
• Wilson, K.G., and Kogut, J., "Lattice Gauge Theory ...", Rev. Mod. Phys. 55 , 775 (1983)

External links

• Gupta - Introduction to Lattice QCD
• Lombardo - Lattice QCD at Finite Temperature and Density
• Chandrasekharan, Wiese - An Introduction to Chiral Symmetry on the Lattice