Intheoretical physics,quantum chromodynamics (QCD) is the study of thestrong interaction betweenquarks mediated bygluons. Quarks are fundamental particles that make up compositehadrons such as theproton,neutron andpion. QCD is a type ofquantum field theory called anon-abelian gauge theory, with symmetry groupSU(3). The QCD analog of electric charge is a property calledcolor. Gluons are theforce carriers of the theory, just as photons are for the electromagnetic force inquantum electrodynamics. The theory is an important part of theStandard Model ofparticle physics. A large body ofexperimental evidence for QCD has been gathered over the years.
QCD exhibits three salient properties:
PhysicistMurray Gell-Mann coined the wordquark in its present sense. It originally comes from the phrase "Three quarks for Muster Mark" inFinnegans Wake byJames Joyce. On June 27, 1978, Gell-Mann wrote a private letter to the editor of theOxford English Dictionary, in which he related that he had been influenced by Joyce's words: "The allusion to three quarks seemed perfect." (Originally, only three quarks had been discovered.)[5]
The three kinds ofcharge in QCD (as opposed to one inquantum electrodynamics or QED) are usually referred to as "color charge" by loose analogy to the three kinds ofcolor (red, green and blue)perceived by humans. Other than this nomenclature, the quantum parameter "color" is completely unrelated to the everyday, familiar phenomenon of color.
The force between quarks is known as thecolour force[6] (orcolor force[7]) orstrong interaction, and is responsible for thenuclear force.
Since the theory of electric charge is dubbed "electrodynamics", theGreek wordχρῶμα (chrōma, "color") is applied to the theory of color charge, "chromodynamics".
With the invention ofbubble chambers andspark chambers in the 1950s, experimentalparticle physics discovered a large and ever-growing number of particles calledhadrons. It seemed that such a large number of particles could not all befundamental. First, the particles were classified bycharge andisospin byEugene Wigner andWerner Heisenberg; then, in 1953–56,[8][9][10] according tostrangeness byMurray Gell-Mann andKazuhiko Nishijima (seeGell-Mann–Nishijima formula). To gain greater insight, the hadrons were sorted into groups having similar properties and masses using theeightfold way, invented in 1961 by Gell-Mann[11] andYuval Ne'eman. Gell-Mann andGeorge Zweig, correcting an earlier approach ofShoichi Sakata, went on to propose in 1963 that the structure of the groups could be explained by the existence of threeflavors of smaller particles inside the hadrons: thequarks. Gell-Mann also briefly discussed a field theory model in which quarks interact with gluons.[12][13]
Perhaps the first remark that quarks should possess an additionalquantum number was made[14] as a short footnote in the preprint ofBoris Struminsky[15] in connection with the Ω−hyperon being composed of threestrange quarks with parallel spins (this situation was peculiar, because since quarks arefermions, such a combination is forbidden by thePauli exclusion principle):
Three identical quarks cannot form an antisymmetric S-state. In order to realize an antisymmetric orbital S-state, it is necessary for the quark to have an additional quantum number.
— B. V. Struminsky, Magnetic moments of barions in the quark model,JINR-Preprint P-1939, Dubna, Submitted on January 7, 1965
Boris Struminsky was a PhD student ofNikolay Bogolyubov. The problem considered in this preprint was suggested by Nikolay Bogolyubov, who advised Boris Struminsky in this research.[15] In the beginning of 1965,Nikolay Bogolyubov,Boris Struminsky andAlbert Tavkhelidze wrote a preprint with a more detailed discussion of the additional quark quantum degree of freedom.[16] This work was also presented by Albert Tavkhelidze without obtaining consent of his collaborators for doing so at an international conference inTrieste (Italy), in May 1965.[17][18]
A similar mysterious situation was with theΔ++ baryon; in the quark model, it is composed of threeup quarks with parallel spins. In 1964–65,Greenberg[19] andHan–Nambu[20] independently resolved the problem by proposing that quarks possess an additionalSU(3)gaugedegree of freedom, later called color charge. Han and Nambu noted that quarks might interact via an octet of vectorgauge bosons: thegluons.
Since free quark searches consistently failed to turn up any evidence for the new particles, and because an elementary particle back then wasdefined as a particle that could be separated and isolated, Gell-Mann often said that quarks were merely convenient mathematical constructs, not real particles. The meaning of this statement was usually clear in context: He meant quarks are confined, but he also was implying that the strong interactions could probably not be fully described by quantum field theory.
Richard Feynman argued that high energy experiments showed quarks are real particles: he called thempartons (since they were parts of hadrons). By particles, Feynman meant objects that travel along paths, elementary particles in a field theory.
The difference between Feynman's and Gell-Mann's approaches reflected a deep split in the theoretical physics community. Feynman thought the quarks have a distribution of position or momentum, like any other particle, and he (correctly) believed that the diffusion of parton momentum explaineddiffractive scattering. Although Gell-Mann believed that certain quark charges could be localized, he was open to the possibility that the quarks themselves could not be localized because space and time break down. This was the more radical approach ofS-matrix theory.
James Bjorken proposed that pointlike partons would imply certain relations indeep inelastic scattering ofelectrons and protons, which were verified in experiments atSLAC in 1969. This led physicists to abandon the S-matrix approach for the strong interactions.
In 1973 the concept ofcolor as the source of a "strong field" was developed into the theory of QCD by physicistsHarald Fritzsch andHeinrich Leutwyler, together with physicistMurray Gell-Mann.[21] In particular, they employed the general field theory developed in 1954 byChen Ning Yang andRobert Mills[22] (seeYang–Mills theory), in which the carrier particles of a force can themselves radiate further carrier particles. (This is different from QED, where the photons that carry the electromagnetic force do not radiate further photons.)
The discovery ofasymptotic freedom in the strong interactions byDavid Gross,David Politzer andFrank Wilczek allowed physicists to make precise predictions of the results of many high energy experiments using the quantum field theory technique ofperturbation theory. Evidence of gluons was discovered inthree-jet events atPETRA in 1979. These experiments became more and more precise, culminating in the verification ofperturbative QCD at the level of a few percent atLEP, atCERN.
The other side of asymptotic freedom isconfinement. Since the force between color charges does not decrease with distance, it is believed that quarks and gluons can never be liberated from hadrons. This aspect of the theory is verified withinlattice QCD computations, but is not mathematically proven. One of theMillennium Prize Problems announced by theClay Mathematics Institute requires a claimant to produce such a proof. Other aspects ofnon-perturbative QCD are the exploration of phases ofquark matter, including thequark–gluon plasma.
Every field theory ofparticle physics is based on certain symmetries of nature whose existence is deduced from observations. These can be
QCD is a non-abelian gauge theory (orYang–Mills theory) of theSU(3) gauge group obtained by taking thecolor charge to define a local symmetry.
Since the strong interaction does not discriminate between different flavors of quark, QCD has approximateflavor symmetry, which is broken by the differing masses of the quarks.
There are additional global symmetries whose definitions require the notion ofchirality, discrimination between left and right-handed. If thespin of a particle has a positiveprojection on its direction of motion then it is called right-handed; otherwise, it is left-handed. Chirality and handedness are not the same, but become approximately equivalent at high energies.
As mentioned,asymptotic freedom means that at large energy – this corresponds also toshort distances – there is practically no interaction between the particles. This is in contrast – more precisely one would saydual– to what one is used to, since usually one connects the absence of interactions withlarge distances. However, as already mentioned in the original paper of Franz Wegner,[23] a solid state theorist who introduced 1971 simple gauge invariant lattice models, the high-temperature behaviour of theoriginal model, e.g. the strong decay of correlations at large distances, corresponds to the low-temperature behaviour of the (usually ordered!)dual model, namely the asymptotic decay of non-trivial correlations, e.g. short-range deviations from almost perfect arrangements, for short distances. Here, in contrast to Wegner, we have only the dual model, which is that one described in this article.[24]
The color group SU(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group U(1), which is gauged to giveQED: this is anabelian group. If one considers a version of QCD withNf flavors of massless quarks, then there is a global (chiral) flavor symmetry group SUL(Nf) × SUR(Nf) × UB(1) × UA(1). The chiral symmetry isspontaneously broken by theQCD vacuum to the vector (L+R) SUV(Nf) with the formation of achiral condensate. The vector symmetry, UB(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry UA(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called ananomaly. Gluon field configurations calledinstantons are closely related to this anomaly.
There are two different types of SU(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry that rotates different flavors of quarks to each other, orflavor SU(3). Flavor SU(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks.
In theQCD vacuum there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under SU(2)isospin rotations of up and down, and to a lesser extent under rotations of up, down, and strange, or full flavor group SU(3), and the observed particles make isospin and SU(3) multiplets.
The approximate flavor symmetries do have associated gauge bosons, observed particles like the rho and the omega, but these particles are nothing like the gluons and they are not massless. They are emergent gauge bosons in an approximatestring description of QCD.
The dynamics of the quarks and gluons are defined by the quantum chromodynamicsLagrangian. Thegauge invariant QCD Lagrangian is
where is the quark field, a dynamical function of spacetime, in thefundamental representation of theSU(3) gaugegroup, indexed by and running from to; is theDirac adjoint of; is thegauge covariant derivative; the γμ areGamma matrices connecting the spinor representation to the vector representation of theLorentz group.
Herein, thegauge covariant derivativecouples the quark field with a coupling strengthto the gluon fields via the infinitesimal SU(3) generatorsin the fundamental representation. An explicit representation of these generators is given by, wherein theare theGell-Mann matrices.
The symbol represents the gauge invariantgluon field strength tensor, analogous to theelectromagnetic field strength tensor,Fμν, inquantum electrodynamics. It is given by:[25]
where are thegluon fields, dynamical functions of spacetime, in theadjoint representation of the SU(3) gauge group, indexed bya,b andc running from to; andfabc are thestructure constants of SU(3) (the generators of the adjoint representation). Note that the rules to move-up or pull-down thea,b, orc indices aretrivial, (+, ..., +), so thatfabc =fabc =fabc whereas for theμ orν indices one has the non-trivialrelativistic rules corresponding to themetric signature (+ − − −).
The variablesm andg correspond to the quark mass and coupling of the theory, respectively, which are subject to renormalization.
An important theoretical concept is theWilson loop (named afterKenneth G. Wilson). In lattice QCD, the final term of the above Lagrangian is discretized via Wilson loops, and more generally the behavior of Wilson loops can distinguishconfined and deconfined phases.
Quarks are massive spin-1⁄2fermions that carry acolor charge whose gauging is the content of QCD. Quarks are represented byDirac fields in thefundamental representation3 of thegauge groupSU(3). They also carry electric charge (either −1⁄3 or +2⁄3) and participate inweak interactions as part ofweak isospin doublets. They carry global quantum numbers including thebaryon number, which is1⁄3 for each quark,hypercharge and one of theflavor quantum numbers.
Gluons are spin-1bosons that also carrycolor charges, since they lie in theadjoint representation8 of SU(3). They have no electric charge, do not participate in the weak interactions, and have no flavor. They lie in thesinglet representation1 of all these symmetry groups.
Each type of quark has a corresponding antiquark, of which the charge is exactly opposite. They transform in theconjugate representation to quarks, denoted.
According to the rules ofquantum field theory, and the associatedFeynman diagrams, the above theory gives rise to three basic interactions: a quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. This contrasts withQED, in which only the first kind of interaction occurs, sincephotons have no charge. Diagrams involvingFaddeev–Popov ghosts must be considered too (except in theunitarity gauge).
Detailed computations with the above-mentioned Lagrangian[26] show that the effective potential between a quark and its anti-quark in ameson contains a term that increases in proportion to the distance between the quark and anti-quark (), which represents some kind of "stiffness" of the interaction between the particle and its anti-particle at large distances, similar to theentropic elasticity of arubber band (see below). This leads toconfinement [27] of the quarks to the interior of hadrons, i.e.mesons andnucleons, with typical radiiRc, corresponding to former "Bag models" of the hadrons[28] The order of magnitude of the "bag radius" is 1 fm (= 10−15 m). Moreover, the above-mentioned stiffness is quantitatively related to the so-called "area law" behavior of the expectation value of the Wilson loop productPW of the ordered coupling constants around a closed loopW; i.e. is proportional to thearea enclosed by the loop. For this behavior the non-abelian behavior of the gauge group is essential.
Further analysis of the content of the theory is complicated. Various techniques have been developed to work with QCD. Some of them are discussed briefly below.
This approach is based on asymptotic freedom, which allowsperturbation theory to be used accurately in experiments performed at very high energies. Although limited in scope, this approach has resulted in the most precise tests of QCD to date.
Among non-perturbative approaches to QCD, the most well established islattice QCD. This approach uses a discrete set of spacetime points (called the lattice) to reduce the analytically intractable path integrals of the continuum theory to a very difficult numerical computation that is then carried out onsupercomputers like theQCDOC, which was constructed for precisely this purpose. While it is a slow and resource-intensive approach, it has wide applicability, giving insight into parts of the theory inaccessible by other means, in particular into the explicit forces acting between quarks and antiquarks in a meson. However, thenumerical sign problem makes it difficult to use lattice methods to study QCD at high density and low temperature (e.g. nuclear matter or the interior of neutron stars).
A well-known approximation scheme, the1⁄N expansion, starts from the idea that the number of colors is infinite, and makes a series of corrections to account for the fact that it is not. Until now, it has been the source of qualitative insight rather than a method for quantitative predictions. Modern variants include theAdS/CFT approach.
For specific problems, effective theories may be written down that give qualitatively correct results in certain limits. In the best of cases, these may then be obtained as systematic expansions in some parameters of the QCD Lagrangian. One sucheffective field theory ischiral perturbation theory or ChiPT, which is the QCD effective theory at low energies. More precisely, it is a low energy expansion based on the spontaneous chiral symmetry breaking of QCD, which is an exact symmetry when quark masses are equal to zero, but for the u, d and s quark, which have small mass, it is still a good approximate symmetry. Depending on the number of quarks that are treated as light, one uses either SU(2) ChiPT or SU(3) ChiPT. Other effective theories areheavy quark effective theory (which expands around heavy quark mass near infinity), andsoft-collinear effective theory (which expands around large ratios of energy scales). In addition to effective theories, models like theNambu–Jona-Lasinio model and thechiral model are often used when discussing general features.
Based on anOperator product expansion one can derive sets of relations that connect different observables with each other.
The notion of quarkflavors was prompted by the necessity of explaining the properties of hadrons during the development of thequark model. The notion of color was necessitated by the puzzle of the
Δ++
. This has been dealt with in the section onthe history of QCD.
The first evidence for quarks as real constituent elements of hadrons was obtained indeep inelastic scattering experiments atSLAC. The first evidence for gluons came inthree-jet events atPETRA.[30]
Several good quantitative tests of perturbative QCD exist:
Quantitative tests of non-perturbative QCD are fewer, because the predictions are harder to make. The best is probably the running of the QCD coupling as probed throughlattice computations of heavy-quarkonium spectra. There is a recent claim about the mass of the heavy meson Bc . Other non-perturbative tests are currently at the level of 5% at best. Continuing work on masses andform factors of hadrons and their weak matrix elements are promising candidates for future quantitative tests. The whole subject ofquark matter and thequark–gluon plasma is a non-perturbative test bed for QCD that still remains to be properly exploited.[citation needed]
One qualitative prediction of QCD is that there exist composite particles made solely ofgluons calledglueballs that have not yet been definitively observed experimentally. A definitive observation of a glueball with the properties predicted by QCD would strongly confirm the theory. In principle, if glueballs could be definitively ruled out, this would be a serious experimental blow to QCD. But, as of 2013, scientists are unable to confirm or deny the existence of glueballs definitively, despite the fact that particle accelerators have sufficient energy to generate them.
There are unexpected cross-relations tocondensed matter physics. For example, the notion ofgauge invariance forms the basis of the well-known Mattisspin glasses,[31] which are systems with the usual spin degrees of freedom fori =1,...,N, with the special fixed "random" couplings Here the εi and εk quantities can independently and "randomly" take the values ±1, which corresponds to a most-simple gauge transformation This means that thermodynamic expectation values of measurable quantities, e.g. of the energy are invariant.
However, here thecoupling degrees of freedom, which in the QCD correspond to thegluons, are "frozen" to fixed values (quenching). In contrast, in the QCD they "fluctuate" (annealing), and through the large number of gauge degrees of freedom theentropy plays an important role (see below).
For positiveJ0 the thermodynamics of the Mattis spin glass corresponds in fact simply to a "ferromagnet in disguise", just because these systems have no "frustration" at all. This term is a basic measure in spin glass theory.[32] Quantitatively it is identical with the loop product along a closed loopW. However, for a Mattis spin glass – in contrast to "genuine" spin glasses – the quantityPW never becomes negative.
The basic notion "frustration" of the spin-glass is actually similar to the Wilson loop quantity of the QCD. The only difference is again that in the QCD one is dealing with SU(3) matrices, and that one is dealing with a "fluctuating" quantity. Energetically, perfect absence of frustration should be non-favorable and atypical for a spin glass, which means that one should add the loop product to the Hamiltonian, by some kind of term representing a "punishment". In the QCD the Wilson loop is essential for the Lagrangian rightaway.
The relation between the QCD and "disordered magnetic systems" (the spin glasses belong to them) were additionally stressed in a paper by Fradkin, Huberman and Shenker,[33] which also stresses the notion ofduality.
A further analogy consists in the already mentioned similarity topolymer physics, where, analogously to Wilson loops, so-called "entangled nets" appear, which are important for the formation of theentropy-elasticity (force proportional to the length) of a rubber band. The non-abelian character of the SU(3) corresponds thereby to the non-trivial "chemical links", which glue different loop segments together, and "asymptotic freedom" means in the polymer analogy simply the fact that in the short-wave limit, i.e. for (whereRc is a characteristic correlation length for the glued loops, corresponding to the above-mentioned "bag radius", while λw is the wavelength of an excitation) any non-trivial correlation vanishes totally, as if the system had crystallized.[34]
There is also a correspondence between confinement in QCD – the fact that the color field is only different from zero in the interior of hadrons – and the behaviour of the usual magnetic field in the theory oftype-II superconductors: there the magnetism is confined to the interior of theAbrikosov flux-line lattice,[35] i.e., the London penetration depthλ of that theory is analogous to the confinement radiusRc of quantum chromodynamics. Mathematically, this correspondendence is supported by the second term, on the r.h.s. of the Lagrangian.
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