Editing Lattice gauge theory (section)

==Quantum triviality==
Lattice gauge theory is also important for the study of [[quantum triviality]] by the real-space [[renormalization group]].<ref>{{cite journal | last=Wilson | first=Kenneth G. |author-link=Kenneth G. Wilson| title=The renormalization group: Critical phenomena and the Kondo problem | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=47 | issue=4 | date=1975-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.47.773 | pages=773–840| bibcode=1975RvMP...47..773W }}</ref> The most important information in the RG flow are what's called the ''fixed points''.

The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be ''trivial'' or noninteracting.  Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question.<ref>{{cite journal
 | author=[[David J E Callaway|D. J. E. Callaway]]
 | year=1988
 | title=Triviality Pursuit: Can Elementary Scalar Particles Exist?
 | journal=[[Physics Reports]]
 | volume=167
 | issue=5 | pages=241–320
 | doi=10.1016/0370-1573(88)90008-7
|bibcode = 1988PhR...167..241C }}</ref>

Triviality has yet to be proven rigorously, but lattice computations 
have provided strong evidence for this.<ref>[[Kenneth G. Wilson|K.G. Wilson]](1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. '''47''', 4, 773.</ref><ref>{{Cite journal|last1=Callaway|first1=D.J.E.|last2=Petronzio|first2=R.|doi=10.1016/0550-3213(87)90657-2|title=Is the standard model Higgs mass predictable?|journal=[[Nuclear Physics B]]|volume=292|pages=497–526|year=1987|bibcode=1987NuPhB.292..497C|url=https://cds.cern.ch/record/172532}}{{cite journal| last=Heller| first=Urs| author2=Markus Klomfass |author3=Herbert Neuberger |author4=Pavols Vranas | s2cid=7146602|date=1993-09-20|journal=[[Nuclear Physics B]]| volume=405| doi=10.1016/0550-3213(93)90559-8| pages=555–573|arxiv = hep-ph/9303215 |bibcode = 1993NuPhB.405..555H|title=Numerical analysis of the Higgs mass triviality bound|issue=2–3 }}, which suggests {{math|''M<sub>H</sub>'' < 710 GeV}}.</ref><ref>{{Cite journal | doi = 10.1016/0550-3213(84)90246-3 | title = Monte Carlo renormalization group study of {{math|''φ''<sup>4</sup>}} field theory| journal = Nuclear Physics B| volume = 240| issue = 4| pages = 577| year = 1984| last1 = Callaway | first1 = D. J. E. | last2 = Petronzio | first2 = R. |bibcode = 1984NuPhB.240..577C | url = https://cds.cern.ch/record/150964}}</ref><ref>{{cite journal| last1=Gies| first1=Holger|last2=Jaeckel| first2=Joerg| s2cid=222197| title=Renormalization Flow of QED| journal=[[Physical Review Letters]]| date=2004-09-09| volume=93| doi=10.1103/PhysRevLett.93.110405 | page=110405| bibcode=2004PhRvL..93k0405G|arxiv = hep-ph/0405183|issue=11| pmid=15447325}}</ref><ref>{{cite journal | doi = 10.1016/0550-3213(86)90431-1 | title =Can elementary scalar particles exist?: (II). Scalar electrodynamics | journal = Nuclear Physics B| volume = 277| issue = 1| pages = 50–66| year = 1986| last1 = Callaway | first1 = D. J. E. | last2 = Petronzio | first2 = R. |bibcode = 1986NuPhB.277...50C | url =https://cds.cern.ch/record/167168 }}</ref><ref>{{cite journal |last1=Göckeler|first1=M. |last2=Horsley|first2=R. |last3=Linke|first3=V. |last4=Rakow|first4=P. |last5=Schierholz|first5=G. |last6=Stüben|first6=H. |s2cid=119494925 |year=1998|title=Is There a Landau Pole Problem in QED?|journal=[[Physical Review Letters]]|volume=80|doi=10.1103/PhysRevLett.80.4119| pages=4119–4122| bibcode=1998PhRvL..80.4119G|arxiv = hep-th/9712244|issue=19 }}</ref>   This fact is important as quantum triviality can be used to bound or even predict parameters such as the mass of [[Higgs boson]].  Lattice calculations have been useful in this context.<ref>For example, {{Cite journal|last1=Callaway|first1=D.J.E.|last2=Petronzio|first2=R.|doi=10.1016/0550-3213(87)90657-2|title=Is the standard model Higgs mass predictable?|journal=[[Nuclear Physics B]]|volume=292|pages=497–526|year=1987|bibcode=1987NuPhB.292..497C|url=https://cds.cern.ch/record/172532}}{{cite journal| last=Heller| first=Urs| author2=Markus Klomfass |author3=Herbert Neuberger |author4=Pavols Vranas | s2cid=7146602|date=1993-09-20|journal=[[Nuclear Physics B]]| volume=405| doi=10.1016/0550-3213(93)90559-8| pages=555–573|arxiv = hep-ph/9303215 |bibcode = 1993NuPhB.405..555H|title=Numerical analysis of the Higgs mass triviality bound|issue=2–3 }}, which suggests {{math|''M<sub>H</sub>'' < 710 GeV}}.</ref>