Editing Wilson loop (section)

===Renormalization===

Since Wilson loops are operators of the gauge fields, the [[regularization (physics)|regularization]] and [[renormalization]] of the underlying Yang–Mills theory fields and couplings does not prevent the Wilson loops from requiring additional renormalization corrections. In a renormalized Yang–Mills theory, the particular way that the Wilson loops get renormalized depends on the geometry of the loop under consideration. The main features are<ref>{{cite journal|last1=Korchemskaya|first1=I.A.|authorlink1=|last2=Korchemsky|first2=G.P.|authorlink2=|date=1992|title=On light-like Wilson loops|url=https://dx.doi.org/10.1016/0370-2693%2892%2991895-G|journal=Physics Letters B|volume=287|issue=1|pages=169–175|doi=10.1016/0370-2693(92)91895-G|pmid=|arxiv=|bibcode=1992PhLB..287..169K |s2cid=|access-date=|url-access=subscription}}</ref><ref>{{cite journal|last1=Polyakov|first1=A.M.|authorlink1=Alexander Markovich Polyakov|date=1980|title=Gauge fields as rings of glue|url=https://dx.doi.org/10.1016/0550-3213%2880%2990507-6|journal=Nuclear Physics B|volume=164|issue=|pages=171–188|doi=10.1016/0550-3213(80)90507-6|pmid=|arxiv=|bibcode=1980NuPhB.164..171P |s2cid=|access-date=|url-access=subscription}}</ref><ref>{{cite journal|last1=Brandt|first1=R.A.|authorlink1=|last2=Neri|first2=F.|authorlink2=|last3=Sato|first3=M.|authorlink3=|date=1981|title=Renormalization of loop functions for all loops|url=https://link.aps.org/doi/10.1103/PhysRevD.24.879|journal=Phys. Rev. D|volume=24|issue=4|pages=879–902|doi=10.1103/PhysRevD.24.879|pmid=|arxiv=|bibcode=1981PhRvD..24..879B |s2cid=|access-date=|url-access=subscription}}</ref><ref>{{cite journal|last1=Korchemsky|first1=G.P.|author-link2=Anatoly Radyushkin|last2=Radyushkin|first2=A.V.|date=1987|title=Renormalization of the Wilson loops beyond the leading order|url=https://dx.doi.org/10.1016/0550-3213%2887%2990277-X|journal=Nuclear Physics B|volume=283|issue=|pages=342–364|doi=10.1016/0550-3213(87)90277-X|pmid=|arxiv=|bibcode=1987NuPhB.283..342K |s2cid=|access-date=|url-access=subscription}}</ref>
* Smooth non-intersecting curve: This can only have linear divergences proportional to the contour which can be removed through multiplicative renormalization.
* Non-intersecting curve with [[cusp (singularity)|cusps]]: Each cusp results in an additional local multiplicative renormalization factor <math>Z[\phi]</math> that depends on the cusp angle <math>\phi</math>.
* Self-intersections: This leads to operator mixing between the Wilson loops associated with the full loop and the subloops.
* Lightlike segments: These give rise to additional logarithmic divergences.