View source for Yang–Mills theory

{{Short description|Quantum field theory}}
{{Use American English|date = March 2019}}
{{unsolved|physics|Yang–Mills theory and the mass gap. Quantum particles described by the theory have mass but the classical waves of the field travel at the speed of light.<ref>{{Cite web |title=Yang-Mills & The Mass Gap |url=https://www.claymath.org/millennium/yang-mills-the-maths-gap/ |access-date=2024-04-09 |website=Clay Mathematics Institute |language=en-US}}</ref>}}
{{Quantum field theory}}
'''Yang–Mills theory''' is a quantum field theory for nuclear binding devised by [[Chen Ning Yang]] and [[Robert Mills (physicist)|Robert Mills]] in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a [[gauge theory]] based on a [[special unitary group]] {{math|SU(''n'')}}, or more generally any [[compact Lie group]]. A Yang–Mills theory seeks to describe the behavior of elementary particles using these [[non-abelian group|non-abelian]] [[Lie group]]s and is at the core of the unification of the [[electromagnetic force]] and [[weak force]]s (i.e. {{math|U(1) × SU(2)}}) as well as [[quantum chromodynamics]], the theory of the [[strong force]] (based on {{math|SU(3)}}). Thus it forms the basis of the understanding of the [[Standard Model]] of particle physics.

== History and qualitative description ==
=== Gauge theory in electrodynamics ===
All known fundamental interactions can be described in terms of gauge theories, but working this out took decades.<ref name=oraifearthagh>{{Cite journal |last1=O’Raifeartaigh |first1=Lochlainn |last2=Straumann |first2=Norbert |date=2000-01-01 |title=Gauge theory: Historical origins and some modern developments |url=https://link.aps.org/doi/10.1103/RevModPhys.72.1 |journal=Reviews of Modern Physics |language=en |volume=72 |issue=1 |pages=1–23 |doi=10.1103/RevModPhys.72.1 |bibcode=2000RvMP...72....1O |issn=0034-6861|url-access=subscription }}</ref> [[Hermann Weyl]]'s pioneering work on this project started in 1915 when his colleague [[Emmy Noether]] proved that every conserved physical quantity has a matching symmetry, and culminated in 1928 when he published his book applying the geometrical theory of symmetry ([[group theory]]) to quantum mechanics.<ref name=Baggott40/>{{rp|194}} Weyl named the relevant symmetry in [[Noether's theorem]] the "gauge symmetry", by analogy to distance standardization in [[Track gauge|railroad gauges]].

[[Erwin Schrödinger]] in 1922, three years before working on his equation, connected Weyl's group concept to electron charge. Schrödinger showed that the group <math>U(1)</math> produced a phase shift <math>e^{i\theta}</math> in electromagnetic fields that matched the conservation of electric charge.<ref name=Baggott40/>{{rp|198}} As the theory of [[quantum electrodynamics]] developed in the 1930's and 1940's the <math>U(1)</math> group transformations played a central role. Many physicists thought there must be an analog for the dynamics of nucleons. 
[[Chen Ning Yang]] in particular was obsessed with this possibility.

=== Yang and Mills find the nuclear force gauge theory ===
Yang's core idea was to look for a conserved quantity in nuclear physics comparable to electric charge and use it to develop a corresponding gauge theory comparable to electrodynamics. He settled on conservation of [[isospin]], a quantum number that distinguishes a neutron from a proton, but he made no progress on a theory.<ref name=Baggott40/>{{rp|200}} Taking a break from Princeton in the summer of 1953, Yang met a collaborator who could help: [[Robert Mills (physicist)|Robert Mills]]. As Mills himself describes:<blockquote>"During the academic year 1953–1954, Yang was a visitor to [[Brookhaven National Laboratory]] ... I was at Brookhaven also ... and was assigned to the same office as Yang. Yang, who has demonstrated on a number of occasions his generosity to physicists beginning their careers, told me about his idea of generalizing gauge invariance and we discussed it at some length ... I was able to contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however, the key ideas were Yang's."<ref>
{{cite book
 |last1=Gray |first1=Jeremy
 |last2=Wilson |first2=Robin
 |date=2012-12-06
 |title=Mathematical Conversations: Selections from the ''Mathematical Intelligencer''
 |publisher=Springer Science & Business Media
 |isbn=9781461301950
 |page=63
 |url=https://books.google.com/books?id=e0TTBwAAQBAJ&q=during+the+academic+year+1953-1954+yang+was+a+visitor+to+the+brookhaven+national+laboratory...+I+was+at+brookhaven+also...+and+was+assigned&pg=PA63
 |via=Google Books
}}
</ref>
</blockquote>

In the summer of 1953, Yang and Mills extended the concept of gauge theory for [[abelian group]]s, e.g. [[quantum electrodynamics]], to non-abelian groups, selecting the group {{math| [[SU(2)]] }} to provide an explanation for isospin conservation in collisions involving the strong interactions. Yang's presentation of the work at Princeton in February&nbsp;1954 was challenged by Pauli, asking about the mass in the field developed with the gauge invariance idea.<ref name=Baggott40>{{Cite book |last=Baggott |first=J.E. |year=2013 |title=The Quantum Story: A history in 40&nbsp;moments |edition=Impression&nbsp;3 |place=Oxford, UK |publisher=Oxford University Press |isbn=978-0-19-956684-6 }}</ref>{{rp|202}} Pauli knew that this might be an issue as he had worked on applying gauge invariance but chose not to publish it, viewing the massless excitations of the theory to be "unphysical 'shadow particles'".<ref name=oraifearthagh/>{{rp|13}} Yang and Mills published in October&nbsp;1954; near the end of the paper, they admit:
{{blockquote| We next come to the question of the mass of the <math>b</math> quantum, to which we do not have a satisfactory answer.<ref name=YM>
{{cite journal
 |first1=C.N. |last1=Yang   |author-link1=Chen-Ning Yang
 |first2=R.   |last2=Mills  |author-link2=Robert Mills (physicist)
 |year=1954
 |title=Conservation of isotopic spin and isotopic gauge invariance
 |journal=[[Physical Review]]
 |volume=96 |issue=1 |pages=191–195
 |doi=10.1103/PhysRev.96.191 |doi-access=free
 |bibcode = 1954PhRv...96..191Y
}}
</ref>}}
This problem of unphysical massless excitation blocked further progress.<ref name=Baggott40/>

The idea was set aside until 1960, when the concept of particles acquiring mass through [[symmetry breaking]] in massless theories was put forward, initially by [[Jeffrey Goldstone]], [[Yoichiro Nambu]], and [[Giovanni Jona-Lasinio]]. This prompted a significant restart of Yang–Mills theory studies that proved successful in the formulation of both [[electroweak interaction|electroweak unification]] and [[quantum chromodynamics]] (QCD). The electroweak interaction is described by the gauge group {{math|SU(2) × U(1)}}, while QCD is an {{math| [[SU(3)]] }} Yang–Mills theory. The massless gauge bosons of the electroweak {{math|SU(2) × U(1)}} mix after [[spontaneous symmetry breaking]] to produce [[W and Z bosons|the three massive bosons]] of the weak interaction ({{math|{{SubatomicParticle|W boson+}}}}, {{math|{{SubatomicParticle|W boson-}}}}, and {{math|{{SubatomicParticle|Z boson0}}}}) as well as the still-massless [[photon]] field. The dynamics of the photon field and its interactions with matter are, in turn, governed by the {{math|U(1)}} gauge theory of quantum electrodynamics. The [[Standard Model]] combines the [[strong interaction]] with the unified electroweak interaction (unifying the [[weak interaction|weak]] and [[electromagnetic interaction]]) through the symmetry group {{math|SU(3) × SU(2) × U(1)}}. In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observed [[Running coupling|running of the coupling]] constants it is believed{{citation needed|reason=By whom?|date=January 2016}} they all converge to a single value at very high energies.

[[Phenomenology (particle physics)|Phenomenology]] at lower energies in quantum chromodynamics is not completely understood due to the difficulties of managing such a theory with a strong coupling. This may be the reason why [[color confinement|confinement]] has not been theoretically proven, though it is a consistent experimental observation.  This shows why QCD confinement at low energy is a mathematical problem of great relevance, and why the [[Yang–Mills existence and mass gap]] problem is a [[Millennium Prize Problems|Millennium Prize Problem]].

=== Parallel work on non-Abelian gauge theories ===
In 1953, in a private correspondence, [[Wolfgang Pauli]] formulated a six-dimensional theory of [[Einstein's field equations]] of [[general relativity]], extending the five-dimensional theory of [[Kaluza–Klein theory|Kaluza, Klein]], [[Vladimir Fock|Fock]], and others to a higher-dimensional internal space.<ref name = Straumann>{{cite arXiv |last=Straumann |first=N. |year=2000 |title=On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953 |eprint=gr-qc/0012054}}</ref> However, there is no evidence that Pauli developed the [[Lagrangian (field theory)|Lagrangian]] of a [[gauge field]] or the quantization of it. Because Pauli found that his theory "leads to some rather unphysical shadow particles", he refrained from publishing his results formally.<ref name=Straumann/> Although Pauli did not publish his six-dimensional theory, he gave two seminar lectures about it in Zürich in November 1953.<ref name = Straumann />

In January&nbsp;1954 [[Ronald Shaw (physicist)|Ronald Shaw]], a graduate student at the [[University of Cambridge]] also developed a non-Abelian gauge theory for nuclear forces.<ref name=shaw>
{{cite journal
 |last1=Atiyah |first1=M. |authorlink1=Michael Atiyah
 |date=2017
 |title=Ronald Shaw 1929–2016 by Michael Atiyah (1954)
 |journal=Trinity College Annual Record
 |volume=2017 |issue= |pages=137–146
 |type=memorial
 |url=https://issuu.com/trinityalumni/docs/trinity_ar_2017_web
}}
</ref>
However, the theory needed massless particles in order to maintain [[gauge invariance]].  Since no such massless particles were known at the time, Shaw and his supervisor [[Abdus Salam]] chose not to publish their work.<ref name=shaw/>
Shortly after Yang and Mills published their paper in October&nbsp;1954, Salam encouraged Shaw to publish his work to mark his contribution. Shaw declined, and instead it only forms a chapter of his PhD thesis published in 1956.<ref>
{{cite thesis
 |last=Shaw |first=Ronald
 |date=September 1956
 |title=The problem of particle types and other contributions to the theory of elementary particles
 |degree=Ph.D.
 |at=ch.&nbsp;3, pp.&nbsp;34–46
 |publisher=[[University of Cambridge]]
}}
</ref><ref>
{{cite book
 |last=Fraser |first=Gordon
 |date=2008
 |title=Cosmic Anger: Abdus Salam – the first Muslim Nobel scientist
 |place=Oxford, UK
 |publisher=Oxford University Press
 |isbn=978-0199208463
 |page=117
}}
</ref>

== Mathematical overview ==
{{See also|Yang–Mills equations}}
{{multiple image
 |perrow = 2 |total_width=280 |left
 | image1 = -y-(x^2+y^2+1) plot; BPST instanton.png
 | image2 = X-(x^2+y^2+1) plot; BPST instanton.png
 | image3 = Curvature of BPST Instanton.png
 | image4 = BPST on sphere.png
 | footer = The {{math|d''x''<sup>1</sup>⊗''σ''<sub>3</sub>}} coefficient of a [[BPST instanton]] on the {{math|(''x''<sup>1</sup>,''x''<sup>2</sup>)}}-slice of {{math|ℝ<sup>4</sup>}} where {{math|''σ''<sub>3</sub>}} is the third [[Pauli matrix]] (top left). The {{math|d''x''<sup>2</sup>⊗''σ''<sub>3</sub>}} coefficient (top right). These coefficients determine the restriction of the BPST instanton {{mvar|A}} with {{nobr|{{math|''g''{{=}}2, ''ρ''{{=}}1, ''z''{{=}}0}}}} to this slice. The corresponding field strength centered around {{math|''z''{{=}}0}} (bottom left). A visual representation of the field strength of a BPST instanton with center {{mvar|z}} on the [[compactification (mathematics)|compactification]] {{math|''S''<sup>4</sup>}} of {{math|ℝ<sup>4</sup>}} (bottom right). The BPST instanton is a classical [[instanton]] solution to the [[Yang–Mills equations]] on {{math|ℝ<sup>4</sup>}}.
}}

Yang–Mills theories are special examples of gauge theories with a non-abelian symmetry group given by the [[Lagrangian (field theory)|Lagrangian]]

:<math>\ \mathcal{L}_\mathrm{gf} = -\tfrac{1}{2}\operatorname{tr}(F^2) = - \tfrac{1}{4} F^{a\mu \nu} F_{\mu \nu}^a\ </math>

with the generators <math>\ T^a\ </math> of the [[Lie algebra]], indexed by {{mvar|a}}, corresponding to the {{mvar|F}}-quantities (the [[curvature]] or field-strength form) satisfying
:<math>\ \operatorname{tr}\left( T^a\ T^b \right) = \tfrac{1}{2} \delta^{ab}\ , \qquad \left[ T^a,\ T^b \right] = i\ f^{abc}\ T^c ~.</math>

Here, the {{mvar|f&thinsp;<sup>abc</sup>}} are [[structure constant]]s of the Lie algebra (totally antisymmetric if the generators of the Lie algebra are normalised such that <math>\ \operatorname{tr}(T^a\ T^b)\ </math> is proportional to <math>\ \delta^{ab}\ </math>), the [[covariant derivative]] is defined as

:<math>\ D_\mu = I\ \partial_\mu - i\ g\ T^a\ A^a_\mu\ ,</math>

{{mvar|I}} is the [[identity matrix]] (matching the size of the generators), <math>\ A^a_\mu\ </math> is the [[Four-vector|vector]] potential, and {{mvar|g}} is the [[coupling constant]]. In four dimensions, the coupling constant {{mvar|g}} is a pure number and for a {{math|SU(''n'')}} group one has <math>\ a, b, c = 1 \ldots n^2-1 ~.</math>

The relation

:<math>\ F_{\mu \nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g\ f^{abc}\ A_\mu^b\ A_\nu^c\ </math>

can be derived by the [[commutator]]

:<math>\ \left[ D_\mu, D_\nu \right] = -i\ g\ T^a\ F_{\mu\nu}^a ~.</math>

The field has the property of being self-interacting and the equations of motion that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this theory only by [[perturbation theory]] with small nonlinearities.{{Citation needed|reason=The use of the word "only" needs clarification and a proper citation will give contextual meaning that is evidently missing.|date=June 2023}}

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for ''a'' indices (e.g. <math>\ f^{abc} = f_{abc}\ </math>), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature, <math>\ \eta_{\mu \nu } = {\rm diag}(+---) ~.</math>

From the given Lagrangian one can derive the equations of motion given by

:<math>\ \partial^\mu F_{\mu\nu}^a + g\ f^{abc}\ A^{\mu b}\ F_{\mu\nu}^c = 0 ~.</math>

Putting <math>\ F_{\mu\nu}=T^aF^a_{\mu\nu}\ ,</math> these can be rewritten as

:<math>\ \left( D^\mu F_{\mu\nu} \right)^a = 0 ~.</math>

A [[Bianchi identity]] holds

:<math>\ \left( D_\mu\ F_{\nu \kappa} \right)^a + \left( D_\kappa\ F_{\mu \nu} \right)^a + \left( D_\nu\ F_{\kappa \mu} \right)^a = 0\ </math>

which is equivalent to the [[Jacobi identity]]

:<math>\ \left[ D_{\mu}, \left[ D_{\nu},D_{\kappa} \right] \right] + \left[ D_{\kappa}, \left[ D_{\mu},D_{\nu} \right] \right] + \left[ D_{\nu}, \left[ D_{\kappa},D_{\mu} \right] \right] = 0\ </math>

since <math>\ \left[ D_{\mu},F^a_{\nu\kappa} \right] = D_{\mu}\ F^a_{\nu\kappa} ~.</math> Define the [[Hodge star operator|dual]] strength tensor
<math>\ \tilde{F}^{\mu\nu} = \tfrac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}\ ,</math> then the Bianchi identity can be rewritten as

:<math>\ D_{\mu}\tilde{F}^{\mu\nu} = 0 ~.</math>

A source <math>\ J_\mu^a\ </math> enters into the equations of motion as

:<math>\ \partial^\mu F_{\mu\nu}^a + g\ f^{abc}\ A^{b\mu}\ F_{\mu\nu}^c = -J_\nu^a ~.</math>

Note that the currents must properly change under gauge group transformations.

We give here some comments about the physical dimensions of the coupling. In {{mvar|D}} dimensions, the field scales as <math>\ \left[A\right]=\left[ L^{\left(\tfrac{2-D}{2}\right)} \right]\ </math> and so the coupling must scale as <math>\ \left[g^2\right] = \left[L^{\left(D-4\right)}\right] ~.</math> This implies that Yang–Mills theory is not [[renormalization|renormalizable]] for dimensions greater than four. Furthermore, for {{nobr|{{math|''D'' {{=}} 4}} ,}} the coupling is dimensionless and both the field and the square of the coupling have the same dimensions of the field and the coupling of a massless quartic [[scalar field theory]]. So, these theories share the [[scale invariance]] at the classical level.

== Quantization ==
A method of quantizing the Yang–Mills theory is by functional methods, i.e. [[Path integral formulation|path integrals]]. One introduces a generating functional for {{mvar|n}}-point functions as

:<math>\ Z[j] = \int [\mathrm{d}A]\ \exp\left[- \tfrac{i}{2} \int \mathrm{d}^4x\ \operatorname{tr}\left( F^{\mu \nu}\ F_{\mu \nu}\right) + i\ \int \mathrm{d}^4x\ j^a_\mu(x)\ A^{a\mu}(x) \right]\ ,</math>

but this integral has no meaning as it is because the potential vector can be arbitrarily chosen due to the [[gauge freedom]]. This problem was already known for quantum electrodynamics but here becomes more severe due to non-abelian properties of the gauge group. A way out has been given by [[Ludvig Faddeev]] and [[Victor Popov]] with the introduction of a ''ghost field'' (see [[Faddeev–Popov ghost]]) that has the property of being unphysical since, although it agrees with [[Fermi–Dirac statistics]], it is a complex scalar field, which violates the [[spin–statistics theorem]]. So, we can write the generating functional as

:<math>\begin{align}
Z[j,\bar\varepsilon,\varepsilon] & = \int [\mathrm{d}\ A] [\mathrm{d}\ \bar c] [\mathrm{d}\ c]\ \exp\Bigl\{ i\ S_F\ \left[\partial A, A\right] + i\ S_{gf}\left[\partial A\right] + i\ S_g\left[\partial c, \partial\bar c, c,\bar c, A \right] \Bigr\} \\
&\exp\left\{i\int \mathrm{d}^4x\ j^a_\mu(x)A^{a\mu}(x)+i\int \mathrm{d}^4x\ \left[\bar c^a(x)\ \varepsilon^a(x) + \bar\varepsilon^a(x)\ c^a(x)\right]\right\}
\end{align}</math>

being

:<math>S_F=- \tfrac{1}{2} \int \mathrm{d}^4 x\ \operatorname{tr}\left( F^{\mu \nu}\ F_{\mu \nu} \right)\ </math>

for the field,

:<math>S_{gf} = -\frac{1}{2\xi} \int \mathrm{d}^4 x\ (\partial\cdot A)^2\ </math>

for the gauge fixing and

:<math>\ S_g = -\int \mathrm{d}^4 x\ \left(\bar c^a\ \partial_\mu\partial^\mu c^a + g\ \bar c^a\ f^{abc}\ \partial_\mu\ A^{b\mu}\ c^c \right)\ </math>

for the ghost. This is the expression commonly used to derive Feynman's rules (see [[Feynman diagram]]). Here we have {{mvar|c<sup>a</sup>}} for the ghost field while {{mvar|ξ}} fixes the gauge's choice for the quantization. Feynman's rules obtained from this functional are the following

[[File:FeynRulesEN.jpg|center|488px]]

These rules for Feynman's diagrams can be obtained when the generating functional given above is rewritten as

:<math>\begin{align}
Z[j,\bar\varepsilon,\varepsilon] &= \exp\left(-i\ g\int \mathrm{d}^4x\ \frac{\delta}{i\ \delta\ \bar\varepsilon^a(x)}\ f^{abc}\partial_\mu\ \frac{i\ \delta}{\delta\ j^b_\mu(x)}\ \frac{i\ \delta}{\delta\ \varepsilon^c(x)} \right)\\
& \qquad \times \exp\left(-i\ g\int \mathrm{d}^4x\ f^{abc}\partial_\mu\frac{i\ \delta}{\delta\ j^a_\nu(x)}\frac{i\ \delta}{\delta\ j^b_\mu(x)}\ \frac{i\ \delta}{\delta\ j^{c\nu}(x)}\right)\\
& \qquad \qquad \times \exp\left(-i\ \frac{g^2}{4}\int \mathrm{d}^4x\ f^{abc}\ f^{ars}\frac{i\ \delta}{\delta\ j^b_\mu(x)}\ \frac{i\ \delta}{\delta\ j^c_\nu(x)}\ \frac{\ i\delta}{\delta\ j^{r\mu}(x)} \frac{i\ \delta}{\delta\ j^{s\nu}(x)} \right) \\
& \qquad \qquad \qquad \times Z_0[j,\bar\varepsilon,\varepsilon]
\end{align}</math>

with

:<math> Z_0[j,\bar\varepsilon,\varepsilon] = \exp \left( -\int \mathrm{d}^4x\ \mathrm{d}^4y\ \bar\varepsilon^a(x)\ C^{ab}(x-y)\ \varepsilon^b(y) \right)\exp \left( \tfrac{1}{2} \int \mathrm{d}^4x\ \mathrm{d}^4y\ j^a_\mu(x)\ D^{ab\mu\nu}(x-y)\ j^b_\nu(y) \right)\ </math>

being the generating functional of the free theory. Expanding in {{mvar|g}} and computing the [[functional derivative]]s, we are able to obtain all the {{mvar|n}}-point functions with perturbation theory. Using [[LSZ reduction formula]] we get from the {{mvar|n}}-point functions the corresponding process amplitudes, [[Cross section (physics)|cross sections]] and [[decay rate]]s. The theory is renormalizable and corrections are finite at any order of perturbation theory.

For quantum electrodynamics the ghost field decouples because the gauge group is abelian. This can be seen from the coupling between the gauge field and the ghost field that is <math>\ \bar c^a\ f^{abc}\ \partial_\mu A^{b\mu}\ c^c ~.</math> For the abelian case, all the structure constants <math>\ f^{abc}\ </math> are zero and so there is no coupling. In the non-abelian case, the ghost field appears as a useful way to rewrite the quantum field theory without physical consequences on the observables of the theory such as cross sections or decay rates.

One of the most important results obtained for Yang–Mills theory is [[asymptotic freedom]]. This result can be obtained by assuming that the [[coupling constant]] {{mvar|g}} is small (so small nonlinearities), as for high energies, and applying [[perturbation theory]]. The relevance of this result is due to the fact that a Yang–Mills theory that describes strong interaction and asymptotic freedom permits proper treatment of experimental results coming from [[deep inelastic scattering]].

To obtain the behavior of the Yang–Mills theory at high energies, and so to prove asymptotic freedom, one applies perturbation theory assuming a small coupling. This is verified [[a posteriori]] in the [[Ultraviolet divergence|ultraviolet limit]]. In the opposite limit, the infrared limit, the situation is the opposite, as the coupling is too large for perturbation theory to be reliable. Most of the difficulties that research meets is just managing the theory at low energies. That is the interesting case, being inherent to the description of hadronic matter and, more generally, to all the observed bound states of gluons and quarks and their confinement (see [[hadrons]]). The most used method to study the theory in this limit is to try to solve it on computers (see [[lattice gauge theory]]). In this case, large computational resources are needed to be sure the correct limit of infinite volume (smaller lattice spacing) is obtained. This is the limit the results must be compared with. Smaller spacing and larger coupling are not independent of each other, and larger computational resources are needed for each. As of today, the situation appears somewhat satisfactory for the hadronic spectrum and the computation of the gluon and ghost propagators, but the [[glueball]] and [[exotic meson|hybrids]] spectra are yet a questioned matter in view of the experimental observation of such exotic states. Indeed, the {{mvar|σ}} resonance<ref name=ccl>
{{cite journal
 |first1=I. |last1=Caprini
 |first2=G. |last2=Colangelo
 |first3=H. |last3=Leutwyler
 |year=2006
 |title=Mass and width of the lowest resonance in QCD
 |journal=[[Physical Review Letters]]
 |volume=96 |issue=13 |page=132001
 |doi=10.1103/PhysRevLett.96.132001  |bibcode=2006PhRvL..96m2001C
 |arxiv = hep-ph/0512364  |pmid=16711979  |s2cid=42504317
}}
</ref><ref name=ygp>
{{cite journal
 |first1=F.J. |last1=Yndurain
 |first2=R.   |last2=Garcia-Martin
 |first3=J.R. |last3=Pelaez
 |year=2007
 |title=Experimental status of the ππ isoscalar S wave at low energy: {{mvar|f}}<sub>0</sub>(600) pole and scattering length
 |journal=[[Physical Review D]]
 |volume=76 |issue=7 |page=074034
 |doi=10.1103/PhysRevD.76.074034 |arxiv = hep-ph/0701025
 |bibcode = 2007PhRvD..76g4034G |s2cid=119434312
}}
</ref>
is not seen in any of such lattice computations and contrasting interpretations have been put forward. This is a hotly debated issue.

== Open problems ==
Yang–Mills theories met with general acceptance in the physics community after [[Gerard 't Hooft]], in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor [[Martinus Veltman]].<ref>
{{cite journal
 | last1 = 't&nbsp;Hooft | first1 = G.  |author1-link = Gerard 't Hooft
 | last2 = Veltman       | first2 = M.  |author2-link = Martinus Veltman
 | doi = 10.1016/0550-3213(72)90279-9
 | title = Regularization and renormalization of gauge fields
 | journal = Nuclear Physics B
 | volume = 44 | issue = 1 | pages = 189–213
 | year = 1972
 | bibcode = 1972NuPhB..44..189T
 | hdl = 1874/4845 | url = https://repositorio.unal.edu.co/handle/unal/81144 | hdl-access = free
}}
</ref>
Renormalizability is obtained even if the gauge bosons described by this theory are massive, as in the electroweak theory, provided the mass is only an "acquired" one, generated by the [[Higgs mechanism]].

The mathematics of the Yang–Mills theory is a very active field of research, yielding e.g. invariants of differentiable structures on four-dimensional manifolds via work of [[Simon Donaldson]]. Furthermore, the field of Yang–Mills theories was included in the [[Clay Mathematics Institute]]'s list of "[[Millennium Prize Problems]]". Here [[Yang–Mills existence and mass gap|the prize-problem]] consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang–Mills theory (i.e. without matter fields) have a finite mass-gap with regard to the vacuum state. Another open problem, connected with this conjecture, is a proof of the [[Color confinement|confinement]] property in the presence of additional fermions.

In physics the survey of Yang–Mills theories does not usually start from perturbation analysis or analytical methods, but more recently from systematic application of numerical methods to [[lattice gauge theory|lattice gauge theories]].

== See also ==
{{div col|colwidth=12em}}

* [[Aharonov–Bohm effect]]
* [[Coulomb gauge]]
* [[Deformed Hermitian Yang–Mills equations]]
* [[Gauge covariant derivative]]
* [[Gauge theory (mathematics)]]
* [[Hermitian Yang–Mills equations]]
* [[Kaluza–Klein theory]]
* [[Lattice gauge theory]]
* [[Lorenz gauge]]
* [[N = 4 supersymmetric Yang–Mills theory|{{mvar|n}} = 4 supersymmetric Yang–Mills theory]]
* [[Propagator]]
* [[Quantum gauge theory]]
* [[Standard model (basic details)|Field theoretical formulation of the standard model]]
* [[Symmetry in physics]]
* [[Two-dimensional Yang–Mills theory]]
* [[Weyl gauge]]
* [[Yang–Mills equations]]
* [[F-Yang–Mills equations]]
* [[Bi-Yang–Mills equations]]
* [[Yang–Mills existence and mass gap]]
* [[Yang–Mills–Higgs equations]]

{{div col end}}

== References ==
{{reflist}}

== Further reading ==
{{refbegin|small=y}}
; Books
* {{cite book
 |last = Frampton |first = P. |author-link = Paul Frampton
 |year=2008
 |title = Gauge Field Theories |edition=3rd
 |publisher=[[Wiley-VCH]]
 |isbn = 978-3-527-40835-1
}}
* {{cite book
 |last1 = Cheng |first1 = T.-P.
 |last2 = Li    |first2 = L.-F. <!-- |author2-link=Ling-Fong Li -->
 |year=1983
 |title = Gauge Theory of Elementary Particle Physics
 |publisher = [[Oxford University Press]]
 |isbn = 0-19-851961-3
}}
* {{cite book
 |editor-last = 't&nbsp;Hooft |editor-first = G.  |editor-link = Gerard 't Hooft
 |year = 2005
 |title = 50&nbsp;Years of Yang–Mills theory
 |location = Singapore
 |publisher = [[World Scientific]]
 |isbn = 981-238-934-2
}}

; Articles
* {{cite arXiv
 |last = Svetlichny  |first = George
 |year=1999
 |title = Preparation for Gauge Theory
 |eprint=math-ph/9902027
}}
* {{cite web
 |last=Gross |first=D. |author-link = David Gross
 |year=1992
 |title = Gauge theory – Past, Present and Future
 |url = https://scholar.google.se/scholar_url?url=http://www.colin-baxter.com/academic/bib/downloads/955.pdf&hl=sv&sa=X&scisig=AAGBfm0MBLXQuVbYC78zgUqhhplqMGAWfw&nossl=1&oi=scholarr&ei=-DRJVafYG8KUsgHOuIDYBQ&ved=0CB8QgAMoADAA
 |access-date = 2015-05-05
}}

{{refend}}

== External links ==
* {{springer|title=Yang-Mills field|id=p/y099030}}
* {{cite web
 |title=Yang–Mills theory
 |website=DispersiveWiki
 |url=https://dispersivewiki.org/DispersiveWiki/index.php?title=Yang-Mills_equations
 |url-status=dead
 |access-date=2018-08-30
 |archive-date=2021-06-03
 |archive-url=https://web.archive.org/web/20210603082507/https://dispersivewiki.org/DispersiveWiki/index.php?title=Yang-Mills_equations
 }}
* {{cite web
 |title=The Millennium Prize Problems
 |publisher=The [[Clay Mathematics Institute]]
 |url=http://www.claymath.org/prizeproblems
 |access-date=2008-11-24
 |archive-date=2009-01-16
 |archive-url=https://web.archive.org/web/20090116001545/http://www.claymath.org/prizeproblems/
 |url-status=dead
 }}

{{Quantum field theories}}

{{DEFAULTSORT:Yang-Mills Theory}}
[[Category:Gauge theories]]
[[Category:Symmetry]]