Editing Wilson loop (section)

==Lattice formulation==

In [[lattice field theory]], Wilson lines and loops play a fundamental role in formulating gauge fields on the [[lattice (group)|lattice]]. The smallest Wilson lines on the lattice, those between two adjacent lattice points, are known as links, with a single link starting from a lattice point <math>n</math> going in the <math>\mu</math> direction denoted by <math>U_\mu(n)</math>. Four links around a single square are known as a plaquette, with their trace forming the smallest Wilson loop.<ref>{{cite book|last1=Baulieu|first1=L.|author-link1=|last2=Iliopoulos|first2=J.|author-link2=John Iliopoulos|last3=Sénéor|first3=R.|author-link3=:fr:Roland Sénéor|date=2017|title=From Classical to Quantum Fields|url=|doi=|location=|publisher=Oxford University Press|chapter=25|page=720|isbn=978-0198788409}}</ref> It is these plaquettes that are used to construct the lattice gauge action known as the [[Wilson action]]. Larger Wilson loops are expressed as products of link variables along some loop <math>\gamma</math>, denoted by<ref>{{cite book|last1=Montvay|first1=I.|last2=Munster|first2=G.|date=1994|title=Quantum Fields on a Lattice|series=Cambridge Monographs on Mathematical Physics|url=|doi=10.1017/CBO9780511470783|location=Cambridge|publisher=Cambridge University Press|chapter=43|page=105|isbn=9780511470783|s2cid=118339104 }}</ref>

:<math>
L[U] = \text{tr} \bigg[\prod_{n \in \gamma} U_\mu(n)\bigg].
</math>

These Wilson loops are used to study confinement and quark potentials [[computational physics|numerically]]. [[Linear combination]]s of Wilson loops are also used as interpolating operators that give rise to [[glueball|glueball states]].<ref>{{cite book|last1=DeGrand|first1=T.|last2=DeTar|first2=C.|date=2006|title=Lattice Methods for Quantum Chromodynamics|series=|url=|doi=10.1142/6065|location=|publisher=World Scientific Publishing|chapter=11|pages=232–233|bibcode=2006lmqc.book.....D |isbn=978-9812567277}}</ref> The glueball masses can then be extracted from the [[correlation function (quantum field theory)|correlation function]] between these interpolators.<ref>{{cite journal|last1=Chen|first1=Y.|authorlink1=|display-authors=etal|date=2006|title=Glueball spectrum and matrix elements on anisotropic lattices|url=|journal=Phys. Rev. D|volume=73|issue=1|page=014516|doi=10.1103/PhysRevD.73.014516|pmid=|arxiv=hep-lat/0510074|bibcode=2006PhRvD..73a4516C |s2cid=15741174|access-date=}}</ref>

The lattice formulation of the Wilson loops also allows for an analytic demonstration of confinement in the [[Coupling constant#Weak and strong coupling|strongly coupled]] phase, assuming the [[quenched approximation]] where quark loops are neglected.<ref>{{cite book|last=Yndurain|first=F.J.|author-link=Francisco José Ynduráin|date=2006|title=The Theory of Quark and Gluon Interactions|url=|doi=|location=|edition=4|publisher=Springer|chapter=9|page=383|isbn=978-3540332091}}</ref> This is done by expanding out the Wilson action as a [[power series]] of traces of plaquettes, where the first non-vanishing term in the expectation value of the Wilson loop in an <math>\text{SU}(3)</math> gauge theory gives rise to an area law with a string tension of the form<ref>{{cite book|last1=Gattringer|first1=C.|last2=Lang|first2=C.B.|date=2009|title=Quantum Chromodynamics on the Lattice: An Introductory Presentation|series=Lecture Notes in Physics 788|url=|doi=10.1007/978-3-642-01850-3|location=|publisher=Springer|chapter=3|pages=58–62|isbn=978-3642018497}}</ref><ref>{{cite journal|last1=Drouffe|first1=J.M.|authorlink1=|last2=Zuber|first2=J.B.|authorlink2=Jean-Bernard Zuber|date=1983|title=Strong coupling and mean field methods in lattice gauge theories|url=https://dx.doi.org/10.1016/0370-1573%2883%2990034-0|journal=Physics Reports|volume=102|issue=1|pages=1–119|doi=10.1016/0370-1573(83)90034-0|pmid=|arxiv=|bibcode=1983PhR...102....1D |s2cid=|access-date=|url-access=subscription}}</ref>

:<math>
\sigma = - \frac{1}{a^2}\ln \bigg(\frac{\beta}{18}\bigg)(1+\mathcal O(\beta)),
</math>

where <math>\beta =6/g^2</math> is the inverse coupling constant and <math>a</math> is the lattice spacing. While this argument holds for both the abelian and non-abelian case, compact [[electrodynamics]] only exhibits confinement at strong coupling, with there being a [[phase transition]] to the Coulomb phase at <math>\beta \sim 1.01</math>, leaving the theory deconfined at weak coupling.<ref>{{cite journal|last1=Lautrup|first1=B.E.|authorlink1=Benny Lautrup|last2=Nauenberg|first2=M.|authorlink2=Michael Nauenberg|date=1980|title=Phase Transition in Four-Dimensional Compact QED|url=https://cds.cern.ch/record/133835|journal=Phys. Lett. B|volume=95|issue=1|pages=63–66|doi=10.1016/0370-2693(80)90400-1|pmid=|arxiv=|bibcode=1980PhLB...95...63L |s2cid=|access-date=}}</ref><ref>{{cite journal|last1=Guth|first1=A.H.|authorlink1=Alan Guth|date=1980|title=Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory|url=https://link.aps.org/doi/10.1103/PhysRevD.21.2291|journal=Phys. Rev. D|volume=21|issue=8|pages=2291–2307|doi=10.1103/PhysRevD.21.2291|pmid=|arxiv=|bibcode=1980PhRvD..21.2291G |s2cid=|access-date=|url-access=subscription}}</ref> Such a phase transition is not believed to exist for <math>\text{SU}(N)</math> gauge theories at [[absolute zero|zero temperature]], instead they exhibit confinement at all values of the coupling constant.