Jets and Fragmentation

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JETS AND FRAGMENTATION

The quantum field theory quantum chromodynamics (QCD) is the theory of quarks and gluons, which are collectively referred to as partons. The most direct evidence for the existence and properties of partons is found in jets.

What are jets? When particles collide in very-high-energy accelerators, they usually produce many secondary particles, some of which travel at wide angles from the initial directions. Particle detectors are designed to identify and measure the directions and energies of these particles. When particles emerge from a collision at high energies, they often appear grouped into a few, highly collimated sprays. This happens with a frequency much greater than can be accounted for by chance. Such groups of nearly parallel-moving particles are called jets. In QCD jets are understood as the visible manifestations of partons. These partons may have collided and been scattered, created in a collision, or emitted in the decay of short-lived particles, such as electroweak bosons. Each such parton evolves into a collection of hadrons, a process known as fragmentation. This term is a little misleading because in QCD the partons are elementary, without substructure, and the hadrons are composite, made up of partons. A special case of interest is the very heavy top quark, which itself decays into lighter quarks, which then produce jets.

The situation in electron-positron (e⁺e^-) collisions is particularly well suited to illustrate these processes. An electron and a positron collide head on and combine (annihilate) to form a photon of considerable energy but with little momentum. This is impossible for a photon in classical physics, whose momentum and energy must be related by p = E/c , with c being the speed of light. In quantum field theory, however, the uncertainty principle allows such a photon to exist for a short period of time. This virtual photon quickly transforms itself (decays) into a pair of electrically charged particles: particle plus antiparticle. This process is illustrated in Figure 1. In quantum field theory, it is not possible to say beforehand which species of charged particles will appear. In fact, every kind is possible, as long as the total energy E of the colliding electron and positron is large enough to create the pair. Sometimes, the charged pair will be a pair of quarks.

FIGURE 1

FIGURE 2

If one takes a closer look at energies that are much higher than the masses of the quarks, the majority of e⁺e^- annihilation events appear like the one in Figure 2, which shows particles produced at the Opal detector at the European Laboratory for Particle Physics (CERN) in Geneva, Switzerland. This event has two, highly collimated, and nearly back-to-back jets of particles. The tracks in the middle of the picture are made by charged particles, primarily hadrons, and the histograms show how much energy was deposited in the outer shells of the detector. The probabilities and angles at which these jets appear are given to good approximation by the process of Figure 1. It is as if each quark simply becomes a jet.

The fragmentation process, through which a quark turns into a jet, is illustrated in Figure 3. A newly produced quark is not yet surrounded by a color field, and it begins to emit the quanta of that field, the gluons, in much the same way that an electron emits photons when it is accelerated, from radio waves at low energies to gamma rays under extreme circumstances. Unlike the photons of quantum electrodynamics

FIGURE 3

(QED), however, gluons themselves carry color charge, and when they are created, they also radiate. This process of rapid particle creation is known as a partonic cascade. In a short time, the original quark is surrounded by a cloud of partons: quarks, anti-quarks, and gluons.

QCD predicts how the cascade develops over time. With fairly good accuracy, it reduces to individual steps of the sort shown in the Figure 3. For example, let P_q→qg(E , θ) be the probability per unit energy and unit angle for a quark to emit a gluon of energy E at angle θ to its own momentum. Then P_q→qg(E , θ) is given approximately by where α_s is the QCD coupling, the analog of the fine-structure constant of quantum electrodynamics. Equation 1 shows that the probability of emitting gluons increases as the angle between the gluon's and quark's momenta decreases. One can understand this effect in the following manner.

If one could move alongside the quark just as it came into being, it would appear to establish its color field by sending out waves of the strong force more or less equally in all directions. On the other hand, as seen "in the laboratory," the quark is moving rapidly, and the waves it emits, made up of gluons, are extremely Doppler-shifted.

They appear to have much higher energy and are primarily moving in the same direction as the quark. This collection of Doppler-shifted radiation is the jet.

As the high-energy partons produced in the cascade travel further and further outward, they separate, penetrating into the surrounding vacuum, which actually repels their color fields, by raising the energies of gluons with wavelengths larger than about 1 fermi (10^-13 cm). The wavelength associated with a gluon of energy E is λ(E ) = hc /E , with h being Planck's constant and c the speed of light. For the first few gluon emissions at an event such as the one shown in Figure 2, E can easily be of order 10¹⁰electron volts (10 GeV), corresponding to wavelengths much less than a fermi. Thus, in the first few steps of these cascades, the unfriendly vacuum is not too important, and Equation 1 can be used. Eventually, however, as the energies of the emitted gluons begin to fall, and their wavelengths grow, an extra energy is required for them to penetrate the vacuum, compared to E = hc /λ. For such long wavelengths and low energies, the approximations implicit in Equation 1 fail. Nevertheless, as the color charges of the remaining energetic partons grow farther and farther apart, they must be connected by lines of the color field, just as electric charges are connected by lines of the electric field.

Although it is not yet possible to describe this process quantitatively, it is certain that the energy of the color field between a quark and antiquark grows without bound as they separate, unless their color charges are neutralized. For this reason, it is always energetically favorable to create pairs of the lightest quarks and antiquarks, until all partons are grouped into color-neutral hadrons. This stage of jet formation is known as hadronization. The process of hadronization, although inexorable, hardly ever uses up more than a small fraction of the total energy. After hadronization the particles can separate freely, and it is the hadrons that create the tracks that are seen in detectors. The numbers and energies of hadrons within a given jet depend only on whether the jet started out as a quark or a gluon. This property of jets is known as factorization. Thus, quark jets in electron-proton collisions are indistinguishable from quark jets of the same energy in proton-antiproton collisions.

Despite the complexity of the cascade and of hadronization, it is possible to compute the probability of finding a jet with a specified total energy and direction because when gluons are emitted at small energies or angles, where Equation 1 is not useful, a jet's total energy and overall direction are unchanged. Properties like the total energy, which are insensitive to low-angle and low-energy gluons, are sometimes said to be infrared safe, a term chosen to emphasize their independence of long wavelength gluon emission. Infrared safe quantities can be calculated with analogs of Equation 1.

Equation 1 implies that the likelihood for extra jets to appear from gluon radiation is proportional to α_s. Because QCD is asymptotically free, the coupling α_s in Equation 1 decreases as E sin θ increases. At the highest-energy accelerators, jet energies can exceed 100 GeV, and at this scale, α_s is approximately

FIGURE 4

equal to 0.1. This means that about one out of every ten events with a quark whose energy is more than 100 GeV includes a gluon with a comparable energy, separated from the quark by a substantial angle. Compelling evidence for the gluon was first provided in "three-jet" events in e⁺e^- annihilation, originating from a quark, an antiquark, and an energetic gluon emitted at wide angles. An example of such an event is shown in Figure 4.

Figure 5 shows the relative numbers of events for jets observed at the Tevatron accelerator at Fermilab in Batavia, Illinois, as a function of their energy. Over many orders of magnitude, calculations based on more elaborate versions of Equation 1 track the data. When, as in this case, the collisions are of protons and antiprotons, the observed jets come mainly from the scattering of partons already present in these particles. The data then reveal how energy is shared among the partons inside a proton.

Jets are at the center of QCD studies at high energy, as well as in the search for new particles created at high energy. In the formation of jets also lies an essential challenge for the theory of QCD: to

FIGURE 5

create a quantitative description of how quarks and gluons evolve into hadrons.

See also:Quantum Chromodynamics; Quarks