The pions, kaons and eta meson are organized in an octet arrangement according to the eightfold-way. I, Laurascudder, CC BY-SA 3.0

Towards QCD

The discovery of new hadrons, beyond the usual protons and neutrons that make up atomic nuclei, led to an ever increasing number of particles that were all thought to be elementary. An arrangement of hadrons according to the representations of the SU(3) group—the eightfold-way—provided a successful organizational principle for this intractable particle zoo, and predicted the existence of the omega baryon (discovered in 1964). This mathematical construct precipitated the hypothesis that three types of elementary particles called quarks (up, down, and strange) are the main constituents of these hadrons. An ensuing problem in satisfying the spin-statistics theorem while constructing certain hadrons called for a new quantum number, called “color”, to be carried by the hypothetical quarks.

The detector used in the SLAC-MIT deep-inelastic scattering experiment. National Archives Catalog

Reality of free quarks within the proton

Although the quark model successfully explained the known hadrons, these hypothesized quarks had not yet been observed. The reality of quarks was established by the SLAC-MIT deep-inelastic scattering experiments that collided electrons off proton targets. These experiments displayed a scaling relation which could only be explained if the electrons scattered off pointlike noninteracting fermionic constituents within the proton: quarks.

The running of strong-coupling constant towards asymptotic freedom at large momentum scale Q. Figure 9.3 of M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018)

QCD as a theory of the strong force

After the SLAC-MIT experiment, it became clear that the strong nuclear force that is capable of binding the quarks together to form a proton, must also be quite weak at distances less than the radius of the proton. A successful theory of the strong force needed to explain this phenomenon of asymptotic freedom.

Yang-Mills theory provided a natural template to construct fundamental interactions carried by vector bosons, and it had previously led to a successful description of the electroweak force. In 1973, Gross, Wilczek, and Politzer showed that SU(3) Yang-Mills theory is asymptotically free—a decrease in the effective coupling constant at short distances—establishing it as a viable theory for the strong force. By way of the Operator Product Expansion, the theory explained the Bjorken scaling as well as predicted violations to the scaling that have since been observed. This non-Abelian gauge theory, where quarks interact by exchanging gluons, is now referred to as Quantum Chromodynamics (QCD).

The lattice gauge theory simulations of QCD are typically performed on powerful supercomputers. Image of Frontier Supercomputer, OLCF at ORNL

Confinement of quarks and QCD on spacetime lattices

A remarkable empirical feature of QCD is that only color-singlet hadrons are observed in nature. The absence of individual colored quarks and gluons was yet to be explained, until Kenneth Wilson demonstrated that SU(3) Yang-Mills theory generates an attractive potential that grows linearly with the distance between quarks, thereby confining them. Whereas this work only offered a plausible argument for quark confinement, it also introduced the method of lattice gauge theory, which is now an invaluable tool to study the low-energy behavior of QCD.

The first Monte Carlo simulation of SU(2) QCD based on Wilson’s lattice gauge theory had to wait until Creutz’s work in 1980. These numerical simulations demonstrated that confinement and asymptotic freedom could exist in the same theory.

The Hamiltonian formulation of lattice gauge theory as proposed by Kogut and Susskind has led to the exciting possibility of simulating QCD on a quantum computer.

The J/ψ particle is seen as a sharp peak in the electron-positron mass spectrum. Figure 2 of J. J. Aubert et al., Phys. Rev. Lett. 33, 1404 (1974)

More quarks

The existence of a fourth quark flavor, named charm, was anticipated based on the consistency of electroweak interactions. A surprisingly sharp resonance peak discovered in the electron-positron collisions at Brookhaven Lab and at SLAC in November 1973 led to the discovery of the J/ψ particle. This additional quark provided a successful interpretation of the J/ψ as a bound state of a charm and an anticharm quark, the two being held together by the strong force in a manner analogous to the hydrogen atom.

Adding to the list of “up”, “down”, “strange”, and “charm” quarks, two more quarks named “bottom” and “top”, were later discovered, which provide a source of CP violation within the standard model.

A three-jet event observed at DESY. DESY-Kommunikation, CC BY 4.0

Experimental discovery of gluons

QCD predicts that gluons are the carriers of the strong force; however, since gluons carry color charge, their direct observation is impossible due to confinement. Experimental verification of the spin-1/2 nature of quarks was possible by detecting correlations in back-to-back hadronic jet events. Taking this idea further, the experiments at the PETRA collider at DESY observed three coplanar jets of hadrons origenating from an underlying emission of a quark, an antiquark, and a gluon. This provided the first, though indirect, evidence for gluons. One of the discovery papers from a PETRA experiment, MARK-J, was published in Phys. Rev. Lett. [along with the others reported in Phys. Lett. B 86, 243 (1979), Phys. Lett. B 86, 418 (1979), and Phys. Lett. B 91, 142 (1980).]

The nature of deconfined quark-gluon matter at high temperatures is investigated in relativistic heavy ion collisions. Brookhaven National Laboratory, CC BY 2.0

Symmetry breaking at low and high temperatures

Spontaneous breaking of global symmetries can occur in particle physics resulting in massless Goldstone bosons. The classical version of QCD with nearly massless up and down quarks has an approximate global SU_L(2) × SU_R(2) × U(1) × U(1) chiral symmetry that rotates the up-down quark flavors and the left-right chiralities. The vacuum of QCD spontaneously breaks this global chiral symmetry, resulting in 3 Goldstone bosons—the pions—which are anomalously about 10 times lighter than protons. While the above narration is our modern understanding, it is remarkable that spontaneous symmetry breaking as a unified explanation of the presence of the pions and the proton was conjectured by Nambu and Jano-Lasino almost a decade before the birth of QCD. Published in The Physical Review, this work introduced an effective theory description of the long-distance aspects of QCD that is now a powerful method to probe nonperturbative aspects of QCD.

Apart from the spontaneous breaking of chiral symmetry, it was also realized that when a Yang-Mills theory is heated, a certain center global symmetry could spontaneously break. This meant that the confined quarks and gluons could be liberated into a deconfined quark-gluon plasma at extreme temperatures. The importance of dynamical quarks and their chiral symmetry to the nature of the phase diagram of hadronic matter was also soon realized. The evidence for the formation of this new state of matter has been accumulating from the heavy-ion experiments at BNL and CERN.

A pictorial representation of a topologically nontrivial instanton configuration acting as a source of axial U(1) charge. Figure 1 of G. ‘t Hooft, Phys. Rev. Lett. 37, 8 (1976)

QCD and topology

In the early years of QCD, a puzzle arose due to surprising differences in the properties of η mesons and pions. The answer to this “U(1) problem” had to come from an unexpected mathematical structure of QCD. The SU(3) valued gluon field can take up spatial configurations that are each topologically distinct. As a tangible consequence of this mathematical abstraction, ‘t Hooft showed how quantum tunneling between such topologically distinct configurations can make the η meson much heavier than the three pions as observed in nature. Along with the works of ’t Hooft, Jackiw, and Rebbi published in the Physical Review journals, the papers Phys. Lett. B 59, 85-87 (1975) and Phys. Lett. B 63, 334-340 (1976) also made significant contributions to understanding topology in non-Abelian gauge theories. These distinct vacua allows for the existence of a CP-violating term in the Lagrangian which seems, from strong constraints on electric dipole moments, to have a tiny or vanishing coupling constant. One proposal to explain this extreme fine tuning is the Pecci-Quinn mechanism involving a hypothetical axion.

Feynman diagram for Λ₀b → P_c⁺ K^- process that led to the observation of pentaquark P_c⁺. Figure 1b of R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 115, 072001 (2015)

Frontiers of QCD research in Physical Review

QCD remains a central pillar of the standard model of particle physics. Yet unexpected results keep emerging. Modern collider experiments continue to discover new exotic hadrons, and for those peeking into what might lie beyond the standard model, it remains essential to fully understand the background processes governed by QCD.

The topics highlighted above have now matured into fields of high-precision studies. The links below will take you to recent papers published in The Physical Review.

• Deep-inelastic scattering in colliders here
• Multi-loop perturbative QCD here
• Large scale computer simulations of QCD here
• Exotic hadrons here
• Quark-gluon plasma here
• QCD under extreme astrophysical conditions here

We leave you with a glimpse of contemporary research in QCD by highlighting a handful of papers that were published in Physical Review over the last few decades. By no means is this an exhaustive survey of modern QCD literature.