Lattice QCD

Quantum field theory
Feynman diagram
History
Background
Symmetries
Tools
Equations
Standard Model
Incomplete theories
Scientists

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. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered.[1][2]

Analytic or perturbative solutions in low-energy QCD are hard or impossible to obtain due to the highly nonlinear nature of the strong force and the large coupling constant at low energies. 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 increases as the lattice spacing decreases, results must be extrapolated to a = 0 (the continuum limit) by repeated calculations at different lattice spacings a.

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.[3] These simulations typically utilize algorithms based upon molecular dynamics or microcanonical ensemble algorithms.[4][5]

At present, lattice QCD is primarily applicable at low densities where the numerical sign problem does not interfere with calculations. 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 successfully agreed with many experiments. For example, the mass of the proton has been determined theoretically with an error of less than 2 percent.[6] Lattice QCD predicts that the transition from confined quarks to quark–gluon plasma occurs around a temperature of 150 MeV (1.7×1012 K), within the range of experimental measurements.[7][8]

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.[9]

  1. ^ Wilson, K. (1974). "Confinement of quarks". Physical Review D. 10 (8): 2445. Bibcode:1974PhRvD..10.2445W. doi:10.1103/PhysRevD.10.2445.
  2. ^ Davies, C. T. H.; Follana, E.; Gray, A.; Lepage, G. P.; Mason, Q.; Nobes, M.; Shigemitsu, J.; Trottier, H. D.; Wingate, M.; Aubin, C.; Bernard, C.; et al. (2004). "High-Precision Lattice QCD Confronts Experiment". Physical Review Letters. 92 (2): 022001. arXiv:hep-lat/0304004. Bibcode:2004PhRvL..92b2001D. doi:10.1103/PhysRevLett.92.022001. ISSN 0031-9007. PMID 14753930. S2CID 16205350.
  3. ^ 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. S2CID 119259340.
  4. ^ 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.
  5. ^ David J. E. Callaway and Aneesur Rahman (1983). "Lattice gauge theory in the microcanonical ensemble" (PDF). Physical Review. D28 (6): 1506–1514. Bibcode:1983PhRvD..28.1506C. doi:10.1103/PhysRevD.28.1506.
  6. ^ S. Dürr; Z. Fodor; J. Frison; et al. (2008). "Ab Initio Determination of Light Hadron Masses". Science. 322 (5905): 1224–7. arXiv:0906.3599. Bibcode:2008Sci...322.1224D. doi:10.1126/science.1163233. PMID 19023076. S2CID 14225402.
  7. ^ P. Petreczky (2012). "Lattice QCD at non-zero temperature". J. Phys. G. 39 (9): 093002. arXiv:1203.5320. Bibcode:2012JPhG...39i3002P. doi:10.1088/0954-3899/39/9/093002. S2CID 119193093.
  8. ^ Rafelski, Johann (September 2015). "Melting hadrons, boiling quarks". The European Physical Journal A. 51 (9): 114. arXiv:1508.03260. Bibcode:2015EPJA...51..114R. doi:10.1140/epja/i2015-15114-0.
  9. ^ Bennett, Ed; Lucini, Biagio; Del Debbio, Luigi; Jordan, Kirk; Patella, Agostino; Pica, Claudio; Rago, Antonio; Trottier, H. D.; Wingate, M.; Aubin, C.; Bernard, C.; Burch, T.; DeTar, C.; Gottlieb, Steven; Gregory, E. B.; Heller, U. M.; Hetrick, J. E.; Osborn, J.; Sugar, R.; Toussaint, D.; Di Pierro, M.; El-Khadra, A.; Kronfeld, A. S.; Mackenzie, P. B.; Menscher, D.; Simone, J. (2016). "BSMBench: A flexible and scalable HPC benchmark from beyond the standard model physics". 2016 International Conference on High Performance Computing & Simulation (HPCS). pp. 834–839. arXiv:1401.3733. doi:10.1109/HPCSim.2016.7568421. ISBN 978-1-5090-2088-1. S2CID 115229961.