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# Lattice QCD

Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time.

Analytic or perturbative solutions in low-energy QCD are hard or impossible due to the highly nonlinear nature of the strong force. This formulation of QCD in discrete rather than continuous spacetime 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 a 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, fields representing quarks are defined at lattice sites (which leads to fermion doubling), while the gluon fields are defined on the links connecting neighboring sites. This approximation approaches continuum QCD as the spacing between lattice sites is reduced to zero. Because the computational cost of numerical simulations can increase dramatically as the lattice spacing decreases, results are often extrapolated to a = 0 by repeated calculations at different lattice spacings a that are large enough to be tractable.

Numerical lattice QCD calculations using Monte Carlo methods can be extremely computationally intensive, requiring the use of the largest available supercomputers. To reduce the computational burden, the so-called quenched approximation can be used, in which the quark fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard.[1] These simulations typically utilize algorithms based upon molecular dynamics or microcanonical ensemble algorithms.[2][3]

At present, lattice QCD is primarily applicable at low densities where the numerical sign problem does not interfere with calculations. Lattice QCD predicts that confined quarks will become released to quark-gluon plasma around energies of 170 MeV. Monte Carlo methods are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC2D).

Lattice QCD has already made successful contact with many experiments. For example the mass of the proton has been determined theoretically with an error of less than 2 percent.[4]

Lattice QCD has also been used as a benchmark for high-performance computing, an approach originally developed in the context of the IBM Blue Gene supercomputer.

Techniques

Monte-Carlo simulations

Monte-Carlo is a method to pseudo-randomly sample a large space of variables. The importance sampling technique used to select the gauge configurations in the Monte-Carlo simulation imposes the use of Euclidean time, by a Wick rotation of space-time.

In lattice Monte-Carlo simulations the aim is to calculate correlation functions. This is done by explicitly calculating the action, using field configurations which are chosen according to the distribution function, which depends on the action and the fields. Usually one starts with the gauge bosons part and gauge-fermion interaction part of the action to calculate the gauge configurations, and then uses the simulated gauge configurations to calculate hadronic propagators and correlation functions.

Fermions on the lattice

Lattice QCD is a way to solve the theory exactly from first principles, without any assumptions, to the desired precision. However, in practice the calculation power is limited, which requires a smart use of the available resources. One needs to choose an action which gives the best physical description of the system, with minimum errors, using the available computational power. The limited computer resources force one to use physical constants which are different from their true physical values:

The lattice discretization means a finite lattice spacing and size, which do not exist in the continuous and infinite space-time. In addition to the automatic error introduced by this, the limited resources force the use of smaller physical lattices and larger lattice spacing than wanted in order to minimize errors.

Another unphysical quantity is the quark masses. Quark masses are steadily going down, but to-date (2010) they are typically too high with respect to the real value.

In order to compensate for the errors one improves the lattice action in various ways, to minimize mainly finite spacing errors.

Lattice perturbation theory

The lattice was initially introduced by Wilson as a framework for studying strongly coupled theories, such as QCD, non-perturbatively. it was found to be a regularization also suitable for perturbative calculations. Perturbation theory involves an expansion in the coupling constant, and is well-justified in high-energy QCD where the coupling constant is small, while it fails completely when the coupling is large and higher order corrections are larger than lower orders in the perturbative series. In this region non-perturbative methods, such as Monte-Carlo sampling of the correlation function, are necessary.

Lattice perturbation theory can also provide results for condensed matter theory. One can use the lattice to represent the real atomic crystal. In this case the lattice spacing is a real physical value, and not an artifact of the calculation which has to be removed, and a quantum field theory can be formulated and solved on the physical lattice.

See also

Lattice field theory

Lattice gauge theory

QCD matter

QCD sum rules

Notes

^ A. Bazavov et al. (2010). "Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks". Reviews of Modern Physics 82 (2): 1349–1417. arXiv:0903.3598. Bibcode 2010RvMP...82.1349B. doi:10.1103/RevModPhys.82.1349.

^ David J. E. Callaway and Aneesur Rahman (1982). "Microcanonical Ensemble Formulation of Lattice Gauge Theory". Physical Review Letters 49 (9): 613–616. Bibcode 1982PhRvL..49..613C. doi:10.1103/PhysRevLett.49.613.

^ David J. E. Callaway and Aneesur Rahman (1983). "Lattice gauge theory in the microcanonical ensemble". Physical Review D28 (6): 1506–1514. Bibcode 1983PhRvD..28.1506C. doi:10.1103/PhysRevD.28.1506.

^ S. Dürr, Z. Fodor, J. Frison et al. (2008). "Ab Initio Determination of Light Hadron Masses". Science 322 (5905): 1224. arXiv:0906.3599. Bibcode 2008Sci...322.1224D. doi:10.1126/science.1163233. PMID 19023076.

Further reading

M. Creutz, Quarks, gluons and lattices, Cambridge University Press 1985.

I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge University Press 1997.

J. Smit, Introduction to Quantum Fields on a Lattice, Cambridge University Press 2002.

H. Rothe, Lattice Gauge Theories, An Introduction, World Scientific 2005.

T. DeGrand and C. DeTar, Lattice Methods for Quantum Chromodynamics, World Scientific 2006.

C. Gattringer and C. B. Lang, Quantum Chromodynamics on the Lattice, Springer 2010.

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

Kuti, Julius - Lattice QCD and String Theory

The FermiQCD Library for Lattice Field theory

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