Editing Lattice QCD (section)

{{short description|Quantum chromodynamics on a lattice}}
{{Quantum field theory}}
'''Lattice QCD''' is a well-established non-[[Perturbation theory (quantum mechanics)|perturbative]] approach to solving the [[quantum chromodynamics]] (QCD) theory of [[quark]]s and [[gluon]]s. It is a [[lattice gauge theory]] formulated on a grid or [[lattice (group)|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.<ref name="wilson">{{cite journal | authorlink=Kenneth G. Wilson | first=K. | last= Wilson | journal=[[Physical Review D]]| volume=10 | issue=8 | page=2445 | title=Confinement of quarks | year= 1974 | doi=10.1103/PhysRevD.10.2445|bibcode = 1974PhRvD..10.2445W }}</ref><ref name="DaviesFollana2004">{{cite journal|last1=Davies|first1=C. T. H.|authorlink1=Christine Davies|last2=Follana|first2=E.|last3=Gray|first3=A.|last4=Lepage|first4=G. P.|last5=Mason|first5=Q.|last6=Nobes|first6=M.|last7=Shigemitsu|first7=J.|author7-link= Junko Shigemitsu |last8=Trottier|first8=H. D.|last9=Wingate|first9=M.|last10=Aubin|first10=C.|last11=Bernard|first11=C.|last12=Burch|first12=T.|last13=DeTar|first13=C.|last14=Gottlieb|first14=Steven|last15=Gregory|first15=E. B.|last16=Heller|first16=U. M.|last17=Hetrick|first17=J. E.|last18=Osborn|first18=J.|last19=Sugar|first19=R.|last20=Toussaint|first20=D.|last21=Pierro|first21=M. Di|last22=El-Khadra|first22=A.|last23=Kronfeld|first23=A. S.|last24=Mackenzie|first24=P. B.|last25=Menscher|first25=D.|last26=Simone|first26=J.|title=High-Precision Lattice QCD Confronts Experiment|display-authors=11|journal=[[Physical Review Letters]]|volume=92|issue=2|pages=022001|year=2004|issn=0031-9007|doi=10.1103/PhysRevLett.92.022001|pmid=14753930|arxiv=hep-lat/0304004|bibcode=2004PhRvL..92b2001D|s2cid=16205350}}</ref>
 
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#QCD and asymptotic freedom|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 [[colour confinement|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 [[extrapolation|extrapolated]] to ''a = 0'' (the [[continuum limit]]) by repeated calculations at different lattice spacings ''a''.

Numerical lattice QCD calculations using [[Monte Carlo method]]s can be extremely computationally intensive, requiring the use of the largest available [[supercomputer]]s.  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.<ref name="Bazavov">{{cite journal | author=A. Bazavov| title=Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks | journal=Reviews of Modern Physics | volume=82 | issue=2 | year=2010 | pages=1349–1417 | doi=10.1103/RevModPhys.82.1349 | arxiv=0903.3598 | bibcode=2010RvMP...82.1349B| s2cid=119259340 |display-authors=etal}}</ref>  These simulations typically utilize algorithms based upon [[molecular dynamics]] or [[microcanonical ensemble]] algorithms.<ref>{{cite journal | author=[[David Callaway|David J. E. Callaway]] and [[Aneesur Rahman]] | title=Microcanonical Ensemble Formulation of Lattice Gauge Theory | journal=Physical Review Letters | volume=49 | year=1982 | issue=9 |pages=613–616 | doi=10.1103/PhysRevLett.49.613 | bibcode=1982PhRvL..49..613C}}</ref><ref>{{cite journal | author=[[David Callaway|David J. E. Callaway]] and [[Aneesur Rahman]] | title=Lattice gauge theory in the microcanonical ensemble | journal=Physical Review | volume=D28 |year=1983 | issue=6 | pages=1506–1514 | doi=10.1103/PhysRevD.28.1506|bibcode = 1983PhRvD..28.1506C | url=https://cds.cern.ch/record/144746/files/PhysRevD.28.1506.pdf }}</ref>

At present, lattice QCD is primarily applicable at low densities where the [[numerical sign problem]] does not interfere with calculations. [[Monte Carlo method]]s are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC<sub>2</sub>D).

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.<ref>{{cite journal | journal=Science | volume=322 | issue=5905 | pages=1224–7 |author1=S. Dürr |author2=Z. Fodor |author3=J. Frison | title=Ab Initio Determination of Light Hadron Masses | year=2008 | doi=10.1126/science.1163233 | pmid=19023076 | arxiv=0906.3599|bibcode = 2008Sci...322.1224D | s2cid=14225402 |display-authors=etal}}</ref> Lattice QCD predicts that the transition from confined quarks to [[quark–gluon plasma]] occurs around a temperature of {{val|150|ul=MeV}} ({{val|1.7e12|ul=K}}), within the range of experimental measurements.<ref>{{cite journal | author=P. Petreczky | title=Lattice QCD at non-zero temperature | journal=J. Phys. G | volume=39  | issue=9 | year=2012 | pages= 093002 | doi=10.1088/0954-3899/39/9/093002 | arxiv=1203.5320 |bibcode = 2012JPhG...39i3002P | s2cid=119193093 }}</ref><ref>{{cite journal |last1=Rafelski |first1=Johann |title=Melting hadrons, boiling quarks |journal=The European Physical Journal A |date=September 2015 |volume=51 |issue=9 |pages=114 |doi=10.1140/epja/i2015-15114-0 |arxiv=1508.03260 |bibcode=2015EPJA...51..114R |doi-access=free }}</ref>

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.<ref>{{Cite book |arxiv = 1401.3733|doi = 10.1109/HPCSim.2016.7568421|chapter = BSMBench: A flexible and scalable HPC benchmark from beyond the standard model physics|title = 2016 International Conference on High Performance Computing & Simulation (HPCS)|pages = 834–839|year = 2016|last1 = Bennett|first1 = Ed|last2 = Lucini|first2 = Biagio|last3 = Del Debbio|first3 = Luigi|last4 = Jordan|first4 = Kirk|last5 = Patella|first5 = Agostino|last6 = Pica|first6 = Claudio|last7 = Rago|first7 = Antonio|last8 = Trottier|first8 = H. D.|last9 = Wingate|first9 = M.|last10 = Aubin|first10 = C.|last11 = Bernard|first11 = C.|last12 = Burch|first12 = T.|last13 = DeTar|first13 = C.|last14 = Gottlieb|first14 = Steven|last15 = Gregory|first15 = E. B.|last16 = Heller|first16 = U. M.|last17 = Hetrick|first17 = J. E.|last18 = Osborn|first18 = J.|last19 = Sugar|first19 = R.|last20 = Toussaint|first20 = D.|last21 = Di Pierro|first21 = M.|last22 = El-Khadra|first22 = A.|last23 = Kronfeld|first23 = A. S.|last24 = Mackenzie|first24 = P. B.|last25 = Menscher|first25 = D.|last26 = Simone|first26 = J.|isbn = 978-1-5090-2088-1|s2cid = 115229961}}</ref>