|Yang–Mills theory in the non-perturbative regime:|
|Quantum field theory|
Yang–Mills theory is a gauge theory based on the SU(N) group, or more generally any compact, semi-simple Lie group. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-Abelian Lie groups and is at the core of the unification of the Weak and Electromagnetic force (i.e. U(1) × SU(2)) as well as Quantum Chromodynamics, the theory of the Strong force (based on SU(3)). Thus it forms the basis of our current understanding of particle physics, the Standard Model.
History and theoretical description 
In a private correspondence, Wolfgang Pauli formulated in 1953 a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher dimensional internal space. However, there is no evidence that Pauli developed the 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. Although Pauli did not publish his six-dimensional theory, he gave two talks about it in Zürich. Recent research shows that an extended Kaluza–Klein theory is in general not equivalent to Yang–Mills theory, as the former contains additional terms.
In early 1954, Chen Ning Yang and Robert Mills  extended the concept of gauge theory for abelian groups, e.g. quantum electrodynamics, to nonabelian groups to provide an explanation for strong interactions. The idea by Yang–Mills was criticized by Pauli, as the quanta of the Yang–Mills field must be massless in order to maintain gauge invariance. 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 unification and quantum chromodynamics (QCD). The electroweak interaction is described by SU(2) × U(1) group while QCD is a SU(3) Yang–Mills theory. The electroweak theory is obtained by combining SU(2) with U(1), where quantum electrodynamics (QED) is described by a U(1) group, and is replaced in the unified electroweak theory by a U(1) group representing a weak hypercharge rather than electric charge. The massless bosons from the SU(2) × U(1) theory mix after spontaneous symmetry breaking to produce the 3 massive weak bosons, and the photon field. The Standard Model combines the strong interaction, with the unified electroweak interaction (unifying the weak and electromagnetic interaction) through the symmetry group SU(2) × U(1) × SU(3). In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observed running of the coupling constants it is believed they all converge to a single value at very high energies.
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 is the reason confinement has not been theoretically proven, though it is a consistent experimental observation. Proof that QCD confines at low energy is a mathematical problem of great relevance, and an award has been proposed by the Clay Mathematics Institute for whoever is also able to show that the Yang–Mills theory has a mass gap and its existence.
Mathematical overview 
and the covariant derivative defined as
can be derived by the commutator
The field has the property of being self-interacting and 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.
Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for a indices (e.g. ), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature, .
From the given Lagrangian one can derive the equations of motion given by
Putting , these can be rewritten as
A Bianchi identity holds
which is equivalent to the Jacobi identity
since . Define the dual strength tensor , then the Bianchi identity can be rewritten as
A source enters into the equations of motion as
Note that the currents must properly change under gauge group transformations.
We give here some comments about the physical dimensions of the coupling. We note that, in D dimensions, the field scales as and so the coupling must scale as . This implies that Yang–Mills theory is not renormalizable for dimensions greater than four. Further, we note that, for 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 of Yang–Mills theory 
A method of quantizing the Yang–Mills theory is by functional methods, i.e. path integrals. One introduces a generating functional for n-point functions as
but this integral has no meaning as 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
for the field,
for the gauge fixing and
for the ghost. This is the expression commonly used to derive Feynman's rules (see Feynman diagram). Here we have ca for the ghost field while α fixes the gauge's choice for the quantization. Feynman's rules obtained from this functional are the following
These rules for Feynman diagrams can be obtained when the generating functional given above is rewritten as
being the generating functional of the free theory. Expanding in g and computing the functional derivatives, we are able to obtain all the n-point functions with perturbation theory. Using LSZ reduction formula we get from the n-point functions the corresponding process amplitudes, cross sections and decay rates. 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 . For the abelian case, all the structure constants 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 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 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 current 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 hybrids spectra are yet a questioned matter in view of the experimental observation of such exotic states. Indeed, the σ resonance is not seen in any of such lattice computations and contrasting interpretations have been put forward. This is currently a hotly debated issue.
Beta function 
One of the key properties of a quantum field theory is the behavior over all the energy range of the running coupling. Such a behavior can be obtained from a theory once its beta function is known. Our ability to extract results from a quantum field theory relies on perturbation theory. Once the beta function is known, the behavior at all energy scales of the running coupling is obtained through the equation
being . Yang–Mills theory has the property of being asymptotically free in the large energy limit (ultraviolet limit). This means that, in this limit, the beta function has a minus sign driving the behavior of the running coupling toward even smaller values as the energy increases. Perturbation theory permits to evaluate beta function in this limit producing the following result for SU(N)
In the opposite limit of low energies (infrared limit), the beta function is not known. It is note the exact one for a supersymmetric Yang–Mills theory. This has been obtained by Novikov, Shifman, Vainshtein and Zakharov and can be written as
As can be seen from the beta function of the supersymmetric theory, the limit of a large coupling (infrared limit) implies
and so the running coupling in the deep infrared limit goes to zero making this theory trivial. This implies that the coupling reaches a maximum at some value of the energy turning again to zero as the energy is lowered. Then, if Ryttov and Sannino hypothesis is correct, the same should be true for ordinary Yang–Mills theory. This would be in agreement with recent lattice computations.
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. (Their work was recognized by the 1999 Nobel prize in physics.) Renormalizability obtains 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.
Concerning the mathematics, it should be noted that presently, i.e. in 2011, the Yang–Mills theory is a very active field of research, yielding e.g. a classification of differentiable structures of four-dimensional manifolds by Simon Donaldson. Furthermore, the field of Yang–Mills theories was included in the Clay Mathematics Institute's list of "Millennium Prize Problems". Here 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 confinement property in the presence of additional Fermion particles.
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 theories.
See also 
- Yang–Mills existence and mass gap
- Aharonov–Bohm effect
- Coulomb gauge
- Electroweak theory
- Field theoretical formulation of the standard model
- Gauge covariant derivative
- Kaluza–Klein theory
- Lorenz gauge
- Quantum chromodynamics
- Quantum gauge theory
- Symmetry in physics
- Weyl gauge
- Yang–Mills–Higgs equations
- Straumann, N (2000). "On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953". arXiv:gr-qc/0012054 [gr-qc].
- See Abraham Pais' account of this period as well as L. Susskind's "Superstrings, Physics World on the first non-abelian gauge theory" where Susskind wrote that Yang–Mills was "rediscovered" only because Pauli had chosen not to publish.
- Reifler, N (2007). "Conditions for exact equivalence of Kaluza-Klein and Yang–Mills theories". arXiv:gr-qc/0707.3790 [gr-qc].
- Yang, C. N.; Mills, R. (1954). "Conservation of Isotopic Spin and Isotopic Gauge Invariance". Physical Review 96 (1): 191–195. Bibcode:1954PhRv...96..191Y. doi:10.1103/PhysRev.96.191.
- Caprini, I.; Colangelo, G.; Leutwyler, H. (2006). "Mass and width of the lowest resonance in QCD". Physical Review Letters 96 (13): 132001. arXiv:hep-ph/0512364. Bibcode:2006PhRvL..96m2001C. doi:10.1103/PhysRevLett.96.132001.
- Yndurain, F. J.; Garcia-Martin, R.; Pelaez, J. R. (2007). "Experimental status of the ππ isoscalar S wave at low energy: f0(600) pole and scattering length". Physical Review D 76 (7): 074034. arXiv:hep-ph/0701025. Bibcode:2007PhRvD..76g4034G. doi:10.1103/PhysRevD.76.074034.
- Novikov, V. A.; Shifman, M. A.; A. I. Vainshtein, A. I.; Zakharov, V. I. (1983). "Exact Gell-Mann-Low Function Of Supersymmetric Yang-Mills Theories From Instanton Calculus". Nuclear Physics B 229 (2): 381–393. Bibcode:1983NuPhB.229..381N. doi:10.1016/0550-3213(83)90338-3.
- Ryttov, T.; Sannino, F. (2008). "Supersymmetry Inspired QCD Beta Function". Physical Review D 78 (6): 065001. arXiv:0711.3745. Bibcode:2008PhRvD..78f5001R. doi:10.1103/PhysRevD.78.065001.
- Bogolubsky, I. L.; Ilgenfritz, E.-M.; A. I. Müller-Preussker, M.; Sternbeck, A. (2009). "Lattice gluodynamics computation of Landau-gauge Green's functions in the deep infrared". Physics Letters B 676 (1-3): 69–73. arXiv:0901.0736. Bibcode:2009PhLB..676...69B. doi:10.1016/j.physletb.2009.04.076.
- 't Hooft, G.; Veltman, M. (1972). "Regularization and renormalization of gauge fields". Nuclear Physics B 44: 189. doi:10.1016/0550-3213(72)90279-9.
Further reading 
- Frampton, P. (2008). Gauge Field Theories (3rd ed.). Wiley-VCH. ISBN 978-3-527-40835-1.
- Cheng, T.-P.; Li, L.-F. (1983). Gauge Theory of Elementary Particle Physics. Oxford University Press. ISBN 0-19-851961-3.
- 't Hooft, Gerardus (2005). 50 Years of Yang-Mills theory. World Scientific. ISBN 981-238-934-2.
- Svetlichny, George (1999). "Preparation for Gauge Theory". arXiv:math-ph/9902027 [math-ph].
- Gross, D. (1992). "Gauge theory - Past, Present and Future". Retrieved 2009-04-23.