Editing Wilson loop (section)

==Properties==

===Makeenko–Migdal loop equation===

Similarly to the [[functional derivative]] which acts on [[functional (mathematics)|functions of functions]], functions of loops admit two types of [[derivative]]s called the area derivative and the perimeter derivative. To define the former, consider a contour <math>\gamma</math> and another contour <math>\gamma_{\delta \sigma_{\mu\nu}}</math> which is the same contour but with an extra small loop at <math>x</math> in the <math>\mu</math>-<math>\nu</math> plane with area <math>\delta \sigma_{\mu\nu}=dx_\mu \wedge dx_\nu</math>. Then the area derivative of the loop functional <math>F[\gamma]</math> is defined through the same idea as the usual derivative, as the normalized difference between the functional of the two loops<ref>{{cite journal|last1=Migdal|first1=A.A.|authorlink1=Alexander Arkadyevich Migdal|date=1983|title=Loop Equations and 1/N Expansion|url=|journal=Phys. Rep.|volume=102|issue=4|pages=199–290|doi=10.1016/0370-1573(83)90076-5|pmid=|arxiv=|s2cid=|access-date=}}</ref>

:<math>
\frac{\delta F[\gamma]}{\delta \sigma_{\mu\nu}(x)} = \frac{1}{\delta \sigma_{\mu\nu}(x)}[F[\gamma_{\delta \sigma_{\mu\nu}}]-F[\gamma]].
</math>

The perimeter derivative is similarly defined whereby now <math>\gamma_{\delta x_\mu}</math> is a slight deformation of the contour <math>\gamma</math> which at position <math>x</math> has a small extruding loop of length <math>\delta x_\mu</math> in the <math>\mu</math> direction and of zero area. The perimeter derivative <math>\partial_\mu^x</math> of the loop functional is then defined as

:<math>
\partial_\mu^x F[\gamma] = \frac{1}{\delta x_\mu}[F[\gamma_{\delta x_\mu}]-F[\gamma]].
</math>

In the [[1/N expansion|large N-limit]], the Wilson loop vacuum expectation value satisfies a closed functional form equation called the Makeenko–Migdal equation<ref>{{cite journal|last1=Makeenko|first1=Y.M.|authorlink1=|last2=Migdal|first2=A.A.|authorlink2=Alexander Arkadyevich Migdal|date=1979|title=Exact Equation for the Loop Average in Multicolor QCD|url=|journal=Phys. Lett. B|volume=88|issue=1–2|pages=135–137|doi=10.1016/0370-2693(79)90131-X|pmid=|arxiv=|bibcode=1979PhLB...88..135M |s2cid=|access-date=}}</ref>

:<math>
\partial^x_\mu \frac{\delta}{\delta \sigma_{\mu\nu}(x)}\langle W[\gamma]\rangle = g^2 N \oint_\gamma dy_\nu \delta^{(D)}(x-y) \langle W[\gamma_{yx}]\rangle \langle W[\gamma_{xy}]\rangle.
</math>

Here <math>\gamma = \gamma_{xy}\cup \gamma_{yx}</math> with <math>\gamma_{xy}</math> being a line that does not close from <math>x</math> to <math>y</math>, with the two points however close to each other. The equation can also be written for finite <math>N</math>, but in this case it does not factorize and instead leads to expectation values of products of Wilson loops, rather than the product of their expectation values.<ref>{{cite book|last=Năstase|first=H.|author-link=Horațiu Năstase|date=2019|title=Introduction to Quantum Field Theory|url=|doi=|location=|publisher=Cambridge University Press|chapter=50|pages=469–472|isbn=978-1108493994}}</ref> This gives rise to an infinite chain of coupled equations for different Wilson loop expectation values, analogous to the [[Schwinger–Dyson equation]]s. The Makeenko–Migdal equation has been solved exactly in two dimensional <math>\text{U}(\infty)</math> theory.<ref>{{cite journal|last1=Kazakov|first1=V.A.|authorlink1=|last2=Kostov|first2=I.K.|authorlink2=|date=1980|title=Non-linear strings in two-dimensional U(∞) gauge theory|url=https://dx.doi.org/10.1016/0550-3213%2880%2990072-3|journal=Nuclear Physics B|volume=176|issue=1|pages=199–215|doi=10.1016/0550-3213(80)90072-3|pmid=|arxiv=|bibcode=1980NuPhB.176..199K |s2cid=|access-date=|url-access=subscription}}</ref>

===Mandelstam identities===

Gauge groups that admit fundamental representations in terms of <math>N\times N</math> matrices have Wilson loops that satisfy a set of identities called the Mandelstam identities, with these identities reflecting the particular properties of the underlying gauge group.<ref>{{cite journal|last1=Mandelstam|first1=S.|authorlink1=Stanley Mandelstam|date=1968|title=Feynman Rules for Electromagnetic and Yang–Mills Fields from the Gauge-Independent Field-Theoretic Formalism|url=https://link.aps.org/doi/10.1103/PhysRev.175.1580|journal=Phys. Rev.|volume=175|issue=5|pages=1580–1603|doi=10.1103/PhysRev.175.1580|pmid=|arxiv=|bibcode=1968PhRv..175.1580M |s2cid=|access-date=|url-access=subscription}}</ref> The identities apply to loops formed from two or more subloops, with <math>\gamma = \gamma_2 \circ \gamma_1</math> being a loop formed by first going around <math>\gamma_1</math> and then going around <math>\gamma_2</math>.

The Mandelstam identity of the first kind states that <math>W[\gamma_1\circ \gamma_2] = W[\gamma_2 \circ \gamma_1]</math>, with this holding for any gauge group in any dimension. Mandelstam identities of the second kind are acquired by noting that in <math>N</math> dimensions, any object with <math>N+1</math> [[antisymmetric tensor|totally antisymmetric]] indices vanishes, meaning that <math>\delta^{a_1}_{[b_1}\delta^{a_2}_{b_2}\cdots \delta^{a_{N+1}}_{b_{N+1}]} = 0</math>. In the fundamental representation, the holonomies used to form the Wilson loops are <math>N\times N</math> [[matrix representation]]s of the gauge groups. Contracting <math>N+1</math> holonomies with the [[Kronecker delta|delta functions]] yields a set of identities between Wilson loops. These can be written in terms the objects <math>M_K</math> defined iteratively so that <math>M_1[\gamma] = W[\gamma]</math> and

:<math>
(K+1)M_{K+1}[\gamma_1, \dots, \gamma_{K+1}] = W[\gamma_{K+1}]M_K[\gamma_1,\dots, \gamma_K] - M_K[\gamma_1 \circ \gamma_{K+1},\gamma_2, \dots, \gamma_K] -\cdots - M_K[\gamma_1, \gamma_2, \dots, \gamma_K\circ \gamma_{K+1}].
</math>

In this notation the Mandelstam identities of the second kind are<ref>{{cite book|last=Gambini|first=R.|author-link=|date=2008|title=Loops, Knots, Gauge Theories|url=|doi=|location=|publisher=|chapter=3|pages=63–67|isbn=978-0521654753}}</ref>

:<math>
M_{N+1}[\gamma_1, \dots, \gamma_{N+1}] = 0.
</math>

For example, for a <math>\text{U}(1)</math> gauge group this gives <math>W[\gamma_1]W[\gamma_2] = W[\gamma_1\circ \gamma_2]</math>.

If the fundamental representation are matrices of unit [[determinant]], then it also holds that <math>M_N(\gamma, \dots, \gamma)=1</math>. For example, applying this identity to <math>\text{SU}(2)</math> gives

:<math>
W[\gamma_1]W[\gamma_2] = W[\gamma_1\circ \gamma_2^{-1}]+W[\gamma_1\circ \gamma_2].
</math>

Fundamental representations consisting of [[unitary matrix|unitary matrices]] satisfy <math>W[\gamma] = W^*[\gamma^{-1}]</math>. Furthermore, while the equality <math>W[I] = N</math> holds for all gauge groups in the fundamental representations, for unitary groups it moreover holds that <math>|W[\gamma]|\leq N</math>.

===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.