1 Quarks, Gluons and QCD

1.1 Brief History

The quark model was independently proposed by theoretical/mathematical physicists (Nobel Laureate) Murray Gell–Mann and George Zweig in 1964. However, there was little experimental evidence for the physical reality of quarks until 1968, when electron–proton scattering experiments indicated that the electrons were scattering off three point–like constituents ‘inside’ the proton.

Gell–Mann ‘borrowed’ the word quark from James Joyce’s book “Finnegans Wake”:

“Three quarks for Muster Mark!

Sure he has not got much of a bark

And sure any he has it’s all beside the mark.”

By 1995, when the top quark was detected in high–energy experiments at the Fermilab in Illinois, all six quark flavors have been finally observed. Gell–Mann and Zweig proposed in 1964, without any substantial experimental evidence, that hadrons were not elementary particles, but they were instead composed of specific (triplet) combinations of quarks and antiquarks. They also postulated independently that there are only three flavors of quarks: up, down and strange, to which there also ascribed known properties such as, mass, spin and Electric Charge.

However, within a year, two extensions of the Gell–Mann–Zweig model were proposed when two other physicists, Sheldon Lee Glashow and James Bjorken, predicted the existence of a fourth flavor of quark, which they referred to as charm. This addition was needed because it expanded the power and self–consistency of the theory: it allowed a much improved and consistent description of the weak interaction when it was relaized that it provided the mechanism that causes the quarks to decay; interestingly, this new theoretical prediction also equalized the number of quarks with the number of known leptons, and led to a formula for predicting correctly the mass of known (π) mesons (that are hadrons with integer spin, or bosons, previously predicted theoretically by Yukawa in 1934 as the carriers of the nuclear interaction via their exchange).

In 1968, deep inelastic electron scattering experiments at the Stanford Linear Accelerator Center (SLAC) showed that the proton was not an elementary particle, but instead contained much smaller, ‘point–like’ objects, that were not so hastily identified with quarks. While this showed that hadrons indeed had a substructure, as predicted by the quark model, physicists remained reluctant to identify these smaller objects with quarks. Instead, they became known as ‘partons’ (a term proposed by Richard Feynman, and supported by some experimental project reports). Such partons were later identified as the up and down quarks when other flavors were also detected. Their discovery is claimed to have ‘validated’ the existence of the strange quark, because it was necessary in the predictions made by the Gell–Mann/Zweig model.

In a 1970 paper, Glashow, John Iliopoulos and Luciano Maiani gave much more compelling theoretical arguments for the prediction of the as–yet undiscovered quark that had charm. The number of the predicted quark flavors thus grew from two to the current six in 1973, following the more complete predictions by Makoto Kobayashi and Toshihide Maskawa who noted that the experimental observation of CP violation could only be explained if there were another pair of quarks with different flavor from the ones already observed. These two new quarks became known as ‘beauty’ and ‘truth’, but later were re–named as ‘bottom’ , b, and ‘top’, t, respectively.

Following a decade without experimental evidence supporting the actual existence of charm quarks, they were finally produced and observed almost simultaneously by two teams in November 1974 : one team working at the Stanford Linear Accelerator Center (SLAC) supervised by Burton Richter and the other at the Brookhaven National Laboratory supervised by Samuel Ting. The two teams had assigned the discovered particle two different names, the J and the ψ. The particle hence became formally known as the J∕ψ ‘meson’ and it was considered a quark–antiquark pair with the charm flavor that Glashow and Bjorken had predicted, called the ‘charmonium’ particle by the latter theoreticians.

In 1977, the bottom quark was observed by Leon Lederman’s team at Fermilab in Illinois. This indicated that a ‘top’ quark should also exist, because the bottom quark would have been most strangely without a partner if it had not existed. However, it was not until 1995, that the top quark was finally detected after much effort and lengthy high–energy experimentation. The top quark’s discovery was crucial; furthermore, it showed that the top quark was significantly more massive than predicted, ‘almost as heavy as a gold atom’, and thus its presence was found at higher energies than those expected. The actual theoretical reasons for the top quark’s larger mass remain to be determined.

1.2 Quarks, Anti-quarks, Nucleons and Hadrons

The building blocks of the atomic nucleus, called also ‘nucleons’–the proton and the neutron–are baryons. Stable quarks are then considered at present as the ‘elementary particles’ found in nucleons, that is, ‘inside’ protons and neutrons of atomic nuclei, as well as in mesons where they appear as quark–pairs. Unstable, high-energy quarks are present in the ‘physical’ vacuum in virtual states, and also in other subatomic particles generated in particle accelerators. They are major constituents of matter, along with leptons (such as electrons and neutrinos). In theoretical physicsl terms, quarks are elementary fermions (of spin 1∕2) because they are subject to Fermi statistics and the Pauli exclusion principle.

A critical limitation to the experimental and theoretical studies of quarks is the fact that quarks are never found as isolated, single particles; rather, they are bound either as quark–pairs or bound together in composite particles named hadrons, (with the most common hadrons being protons and neutrons, which are the basic building blocks of all atomic nuclei). For this reason, much of what is known about quarks has been inferred from observations on the hadrons themselves and observations of quark jets or pairs that are generated in particle accelerators at very high energies. Quarks (and antiquarks) are the only known particles whose electric charge comes as exactly one third of the elementary charge of the electron or proton. However this can never be directly observed as hadrons because they latter have always an integer charge. There are two known types of hadrons: baryons, formed of three quarks, and mesons, formed of a quark and an antiquark pair. The quarks (and antiquarks) which determine the quantum numbers of hadrons are called valence quarks. Apart from these, any hadron may contain an indefinite number of virtual quarks, antiquarks and gluons which do not influence their quantum numbers. Such virtual quarks are called sea quarks (vide infra).

Remarkably, quarks are the only particles in the current Standard Model of physics (SUSY) to experience all four fundamental forces: strong, electromagnetic, electroweak and gravitational.

There are currently six known different types of quarks, that are defined by their flavor: up (symbols: u), down (d), charm (c), strange (s), top (t) and bottom (b). Furthermore, the QCD theory holds the view that these are the only possible types of quarks found in nature or in the high-energy laboratory.

The quarks with the lowest masses, the up and the down quark, are stable within nucleons of atomic nuclei where they are coupled with each other and interact strongly also via gluons- the nuclear field carrier particles. The heavier charm, strange, top and bottom quarks are unstable and decay extremely rapidly; these can only be produced in high energy collisions, such as in particle accelerators and in cosmic rays. Quarks have defining property in addition to mass, electric charge, and spin which is unique to nuclear interactions–the color ‘charge’–which behaves somewhat like a very strong ‘magnetic’ interaction, but with three ‘poles’ instead of the ‘North and South’ characteristic magnetic poles of the classical magnets derived from electron magnetic moment/spin interactions.

For every quark flavor there is a corresponding antiparticle, called its antiquark flavor, which differs from the quark only in that its electrical charge has the opposite sign. Such antiparticles of quarks–called antiquarks– are denoted by a bar over the designating letter for the quark, such as u for a quark and u for an up antiquark. As with all antimatter, general, antiquarks have the same mass, lifetime and spin as their respective quarks, but the electric charge and other charges have the opposite sign.

Having electric charge, mass, spin, flavor and color charge, the quarks are the only known elementary particles that engage in all four Fundamental interactions of contemporary physics: Electromagnetism, weak interaction, strong interaction and gravitation. Gravitation, however, is not included in the theoretical Standard Model, because quantum gravity developments are yet to be completed, and also because gravitational interactions are extremely weak in comparison with all of the other three fundamental interactions.

1.3 Quark’s Electric Charge

A quark has a precise fractional electric charge value from that of the electron or proton (considered as unity), that is, either −1∕3 or +2∕3 times the elementary charge of the latter. More specifically, the up, charm and top quarks –that are collectively referred to as up or u-quarks– have a charge of +2∕3each, whereas the down, strange and bottom quarks (down–type , or d-quarks) have a charge value of −1∕3. The antiquarks, as explained above, have the opposite charge of their corresponding quark (the up–type antiquarks have charges of −2∕3, and the down–type antiquarks have a charge value of +1∕3). Since the electric charge of a hadron is the sum of the charges of the constituent quarks, the combinations of three quarks, or three anti–quarks, or a quark with an anti–quark, always result in an integer charge.

The electric charge of quarks is important in the formation of atomic nuclei. The two stable hadron constituents of the atom, the neutron and the proton, have respectively charge values of 0 and +1; thus, the quark model expects that the neutron contains two down quarks and one up quark, whereas the proton contains two up quarks and a down quark. The total electric charge of a nucleus is given by the number of protons present inside the atomic nucleus, is known as the atomic number that spans the periodic table of elements.

1.4 Strong (Nuclear) Interactions and Color Charges

As indicated above, quarks do possess a remarkable property called ‘color charge’; individual quarks obtain their color charge and interact in this manner via nuclear field carrier particles known as gluons.

All types of hadrons always have a net (or total) color charge value of zero, that is they are ‘white’. There are three types of color charge, called ‘blue’, ‘green’ and ‘red’. Each of these color charges is complemented by an ‘anti–color‘, such as: ‘antiblue’, ‘antigreen’ and ‘antired’, respectively. Whereas each quarks carries a single color, each antiquark carries a single anticolor.

The balance of attractive and repulsive interactions between quarks color-charged with any of the three colors is the cause of what is called ‘strong interaction’ in nuclear and high-energy physics. The area of physics that studies such strong interactions is called Quantum Chromodynamics or QCD. Thus, a quark charged with one color value is bound with an antiquark carrying the corresponding anticolor; also three quarks,(or anti–quarks) all charged with the three different colors will similarly be bound stably together. In any other case, composite particles that are not ‘white’ but charge–colored may not be able to form. The three color types play a role in the process of hadronization, which is the process of hadron formation out of quarks and gluons. The result of two attracting quarks that form a stable quark–antiquark pair will be color neutrality: a quark with ξ color charge plus an antiquark of -ξ color charge will result in a net color charge of 0, or ‘white’ color, and in the formation of a meson. Analogous to the additive color charge model, the combination of all three color charges will similarly result in a ‘white’ color charge. This is what also happens when three quarks combine to form a baryon.

The properties of the color charge are explained by a gauge symmetry (a type of symmetry group) known as the “special unitary group” SU(3). This gauge symmetry is at the core of quantum chromodynamics and the details of its mathematical structure explain why only ‘white’–colored particles can be observed. Each quark is in the basic triplet of the SU(3) group, whose three components corresponds to each color charge (red, green, and blue). The gluons are then described by the adjoint representation of this group, which explains why gluons carry at the same time both the color and anticolor charges. The strong interaction is thus the only one whose field carrier quantum particles, the gluons, also carry the charge for the (strong) force they mediate. This feature of gluons makes the strong interaction very difficult to study and prevents the succesful use of the perturbative techniques previously employed with such great accuracy in Feynman’s QED approach, for example, in both electromagnetism ( with U(1) symmetry) and subsequently for the electroweak interaction (with SU(2) symmetry). This close connection between quantum group symmetries and interactions is present in the three forces described by the current Standard Model of physics (SUSY). The general mathematical approach involving these connections is called the unified quantum Yang-Mills theory.

1.5 Color Confinement and Gluons

A key phenomenon called ‘color confinement’ is thought to trap the quarks within the stable hadrons. This refers to any individual quark’s inability to escape as a single particle from its parent hadron, thereby rendering impossible the actual observation of any isolated, single quark. As already explainede, the color confinement is primarily the result of the strong interactions with the gluon color field, and also the gluon exchange between quarks. Color confinement applies to all quarks, except for the case of the top quark where the actual escape mechanism at extremely high energies is uncertain. One method used was to compare two hadrons that have all but one quark in common. The properties of the differing quarks are then inferred from the difference in values between the two hadrons.

Quarks have inherent relationships with the gluons, that can be theoretically defined as ‘massless vector gauge bosons’. Gluons are therefore responsible for the color field, or the strong interaction, which ensures that quarks remain bound in hadrons through color confinement.

Gluons are constantly exchanged between quarks through a virtual emission and re-absorption process. When a gluon is transferred between one quark and another, a color change occurs in the receiving and emitting quark. For example, if a red quark emits a red–antigreen gluon, it then becomes green, and if a green quark absorbs it, it then becomes red. Therefore, although the color of each quark may always be changing, a bound hadron will ‘confine’ single quarks, thus preventing single quarks to leave the hadron, or exist in isolation. The color field carried by the gluon contributes most significantly to such a hadron’s indivisibility into single quarks, or the color confinement. This is demonstrated by the varying strength of the chromodynamic binding force between the constituent quarks of a hadron; as quarks get closer to each other, the chromodynamic binding force actually weakens through a process called ‘asymptotic freedom’, suggesting the presence also of repulsive forces at extremely close range, somewhat similar to van der Waals repulsive interactions, but of quite different, and yet unknown nature. However, when quarks move further apart, the binding strength dramatically increases via gluon as well as quark-quark color interactions. The color field ‘becomes stressed’ by the drifting away of the quarks, ‘much as an elastic band is stressed when pulled apart’, because a multitude of gluons of appropriate color charges are being generated in order to strengthen the ‘stretched’ field lines that are holding the quarks inside the stable hadron. In this way, a very large amount of energy would be required to ‘wrench a quark from its ‘hadronized state’ through the formation of a quark pair, rather than of a single isolated quark. Thus, in a high–energy experiment as soon as enough energy has been spent to overcome some of the restraining quark–gluon interactions, either a quark–antiquark pair, or quark–pair ‘jets’ would be produced from the original hadron, but never a single quark.

Strong interactions are highly non-linear, because gluons can emit gluons and also exchange gluons with other gluons. This property has led to speculations regarding the possible existence of a ‘glueball’, that is, a particle that is purely made of gluons, despite previous observations indicating that gluons cannot exist without ‘their attached’ quarks.

1.6 Sea Quarks and Gluon Fluxes

The quarks that contribute to the quantum numbers of the hadrons are called valence quarks (q_v). Hadrons also contain virtual quark–antiquark (qq) pairs, known as sea quarks (qs), originating from the gluons’ strong interaction field. Such sea quarks are much less stable, and they annihilate each other very quickly within the interior of the hadron. When a gluon is split, sea quarks are formed, and this process also works in reverse in that the annihilation of two sea quarks will emit a gluon. There would be then a fluctuating quantum flux of sea quarks born from the vacuum, and this would also allow for a steady cycle of gluon splits and/or re–births. Such a gluon flux is called “the sea”.

1.7 Quark Masses

There are presently two different terms in use when one describes the quark masses: the current quark mass refers to the mass of a quark ‘by itself’, while ‘constituent quark mass’ refers to the current quark mass plus the mass of the gluon particle field(s) surrounding the quark. The two values are typically quite different for several reasons the will be explained next. In a hadron, like a proton, most of the mass comes from the gluons that bind the constituent quarks together, rather than from the individual quarks. The mass of the quarks ‘in themselves’, or ‘by themselves’ is quite low (about one third) compared to the mass derived from the gluons’ energy. While gluons are inherently massless, they possess energy, and it is this energy that contributes so greatly to the overall mass of the hadron through relativistic effects. This is readily demonstrated for the proton- the most common hadron. Composed of one d and two u quarks, the proton has an overall mass of approximately 938MeV∕c², of which the mass of three valence quarks contributes around 11MeV∕c², with the remainder coming from the quantum chromodynamics binding energy (QCBE) provided by sea quarks and gluons. This makes ‘direct’ calculations of quark masses based on quantum chromodynamics quite difficult, and very often quite unreliable, as quantum perturbation methods that were very successful in quantum electrodynamics, fail most of the time in QCD. Mass value estimates can be however derived after obtaining from experimental data the difference in mass between two related hadrons that have opposing or complementary quark components. For example, by comparing the proton with the neutron, where the difference between the two particles is one down quark to one up quark, the relative masses and the mass differences can be measured by the difference in the overall mass of the two hadrons. The masses of most quarks were within such ‘predicted ranges’ at the time of their discovery, with the notable exception of the top quark, which was found to have a mass approximately equal to that of a gold nucleus, significantly heavier than what was expected from the QCD theoretical estimates. Several hypotheses have been suggested to explain this very large mass miscalculation. The extended Standard Model postulates that elementary particles derive their masses through the Higgs mechanism, thus related to a so far unobserved “Higgs boson”– a hypothetical particle.

1.8 Flavor Quantum Numbers

In order to explain the phenomenology of strong and weak interactions, particle physicists assigned quantum numbers to the known baryons and mesons. The first such quantum number is known as the isospin, related to the symmetry properties determined by the Lie group SU(2). This was introduced by Werner Heisenberg in 1932 to represent the remarkable similarity between the properties of the protons and neutrons other than their electric charge value, and the presence of three types of pions. The z-component, commonly denoted I_z, is related to the electric charge Q and the baryon number (+1 for baryons, 0 for mesons) of these particles.

An additional quantum number, strangeness (S, which is not to be confused with the spin), was introduced in 1954 to explain the unexpectedly long lifetimes of particles such as K mesons and ξ baryons. This new, (strangeness) quantum number was unchanged by strong interactions, but not by the weak ones, which would explains the anomalously long life-times of the particles in question that can be pair-produced by the strong force, but can only decay via the electro-weak interactions. The formula chosen for the new hypercharge Y was then :

where B is the baryon number and S is the strangeness value.

This equation is known as the (original) Gell–Mann–Nishijima formula. The connection with group theory become clear only in 1961 when Gell-Mann and Ne’emann showed that all the proposed quantum numbers could be explained by relating the fundamental SU(3) triplet to the three lightest quarks: the up, down and strange quarks. Further advances through both theory and High energy physics experiments has led to the introduction of a three flavor quantum numbers, charmness (C), bottomness (B’) and topness (T), corresponding the charm, bottom and top quark respectively. An enlarged flavor symmetry group, SU(6), and also unified SU(5), or SU(3) × SU(2) × U(1) groups are being considered to provide a unified ‘symmetry’ for electromagnetic, electroweak and strong interactions. The modified Gell–Man–Nishijima formula generalizes the equation for all of the flavor quantum numbers and the electrical charge, with the modified hypercharge formula being

that includes also the charm, the ‘truth and beauty’ numbers, with the following notations: J = spin, B = baryon number, Q = electric charge, I_z = isospin, S = strangeness, C =  charmness, B′ =  bottomness, and T =  topness.

1.9 Quark Spin and Spin Parton Distributions

The spin is a intrinsic symmetry property, or quantum observable of all quantum particles, and its orientation is an important degree of freedom. Roughly speaking, the spin of a particle is a contribution to its angular momentum that is not due to its motion but whose correct calculation requires relativistic quantum field theory. Unlike the classical momentum of rotation of a sphere, the spin of a particle takes only discrete values as a result of momentum quantization in quantum physics, and its observation requires the presence of an external field gradient such as magnetic or gravitational that ‘raises the degeneration of the spin levels’, thus splitting quantum particle beams according to their spin values. Spin group representations are known as Pauli matrices and are being extensively used for computations of spin hamiltonians and any other interactions or phenomena that involve the spin property of quantum particles; a widely known, and also very useful example, is that of the Nuclear Magnetic resonance (NMR) phenomenon with wide applications in Chemistry, Physics, agriculture and medical imaging (MRI and 2D-FT).

Spin is measured in units of h∕(2π), where h is the Planck constant. This unit is often denoted by h (“h-bar”), called the “reduced Planck constant”. The result of a measurement of the component of the spin of a quark along any axis- always in the presence of an external field gradient- is always either h∕2 or −h∕2; for this reason quarks are classified as spin-1∕2 particles, called fermions. The component of spin along any given axis– which is by convention the z-axis–is denoted by an upward-pointing arrow ↑ for the value +1∕2 and down arrow pointing arrow ↓ for −1∕2; the symbol also follows the up and down values for the flavor, although one must note that the flavor is ‘not determined’ by the spin, or related to the latter. For example, an up quark with a spin of +1∕2 along the z-axis is denoted by u ↑.

The quark’s spin value contributes to the overall spin of the parent hadron, much as quark’s electrical charge does to the overall charge of the hadron. Varying combinations of quark spins result in the total spin value that can be assigned to the hadron. However, because of the vacuum polarization in QCD and the presence of valence and sea quarks, the spin distribution, or the ‘spin fine structure’ of nuclei involves spin distributions that are not simple additions of spin values for the individual nucleonic spin constituent quarks.

1.10 QCD Matter and Asymptotically-Free Quarks

A novel concept that has recently been proposed is that of quark matter, or QCD matter, a number of theorized phases of matter that might contain only a mixture of strongly interacting ‘free quarks and gluons’. One of such model is called a quark-gluon plasma. This model assumes that, at sufficiently high temperatures and densities, quarks and gluons could potentially become deconfined and degenerate into a plasma which is ‘fluid-like’ consisting of an inhomogeneous mix of gluons and quarks. The precise extreme conditions needed to give rise to such a ‘quark plasma’ state are yet unspecified, and have been the subject of a great deal of speculation; CERN made many attempts to produce such conditions in the 1980s and 1990s. The characteristic signatures of such quark plasma states might include a marked increase in the number of heavier quark pairs compared to the volume of pairs of up and down quarks. It is also believed by some that, in the period from a picosecond to the first microsecond after the Big Bang ( called “the quark epoch”), the Universe was filled by a quark-gluon plasma, because the temperatures were much too high for hadrons to be formed, or condense, as stable particles. It is also hypothesized that strangematter, that is non-nuclear matter containing relatively equal numbers of up, down and strange quarks, might also be stable at ‘ordinary’ temperatures and pressures, in atomic nucleus-sized ‘strangelets’ or kilometer-sized “quark stars”.

Acknowledgement The use of partial content from the author’s contribution to a related entry on the internet under the GNUL license terms is hereby acknowledged.

References