The properties of elementary particles and their interactions are probed in relativistic scattering experiments. In accelerator-based experiments, either two beams of particles collide or a single beam of particles hits a fixed target that is made up of particles at rest. In both cases, the particle beams carry well-defined energy and momentum, and these quantities, as well as electric charge, are conserved in the scattering process. The more energy an incoming particle beam carries, the more massive the final state that can be produced. (Recall that in the theory of relativity, energy, momentum, and mass are related.) Energy and distance are inversely related; in optical language, a particle's wavelength λ is inversly proportional to its momentum k, λ = 2π/k . Smaller distances are thus probed in scattering experiments with higher-energy beams.
The probability of any particular final state being produced in a scattering experiment is expressed in terms of a quantity called a cross section, denoted by σ. A cross section has units of area. A pictorial description is given by the likelihood of hitting the side of a barn with a baseball; the bigger the barn, the more likely that the baseball will hit it. In fact, the unit of measure for a cross section is called a barn. One barn is 10-24 cm2. The event rate, the number of events produced per second, is given by the product of the cross section for a reaction with the incident luminosity of the particle beams. The beam luminosity is simply the number of particles per second per unit area traveling in the beam. The total number of events collected in the lifetime of an experiment is given by integrating the event rate over the time that the experiment has been operating. In this analogy, the total number of baseballs that hit the barn is given by the area of the barn times the frequency with which the baseballs were tossed, which is then summed, or integrated, over the length of time during which the baseballs were tossed.
In 2002 high-energy accelerator experiments routinely measured cross sections with a magnitude of pico-barns, or 10-36 cm2. Typical collider luminosities were roughly 1031 cm-2s-1. This yields 100 events produced for a pico-barn sized cross section when the experiment operates for 107 seconds, which is on the order of a year. Clearly, higher luminosities result in a larger number of events.
The quantum theory of scattering is described by a quantity known as the S -matrix. The possible set of incoming particles in a reaction, or initial states, are known as in states and are denoted as |in >. Likewise, the set of possible final, or outgoing, states are out states, <out |. Both the in and out states are complete sets, meaning that they include all possible initial and final states. The probability of a specific incoming state from the full set of in states producing a particular final state from the possible out states is encoded in a transition matrix. This matrix is called the S -matrix, and the transition is written as <out |S |in >. There are two contributions to the S -matrix: the cases when the particles do not interact and those when they do. Note that the possibility always exists that the particles do not interact, or simply miss each other. The S -matrix is then written as S = I + iT , where I represents the identity matrix (the diagonal elements are unity and the nondiagonal elements vanish), and all the particle interactions are contained in the quantity T. The invariant matrix element for a scattering amplitude is thus written as It contains all information about the interaction between the incoming and outgoing particles. It is Lorentz invariant, meaning that it has the same value in all reference frames.
The cross section for a reaction is obtained by integrating the square of this invariant matrix element over the available phase space, which is the final state momentum space that is kinematically available in a reaction, and dividing this by the incident
flux, which is proportional to the relative velocity between the two incoming particles.
High-energy physicists compute S -matrix elements from Feynman diagrams using a set of Feynman rules. The diagrams pictorially represent a reaction, and the rules encode all the information from quantum field theory that is needed for the computation. Typical diagrams for two body → two body scattering are displayed in Figure 1, where the arrows indicate the direction of momentum flow. These graphs represent the contributions to the process known as Bhabha scattering, e+e- → e+e-. In the Standard Model this process is mediated by the virtual exchange of V = γ, Z bosons. Each diagram represents a different virtual momentum being carried by the exchanged bosons. The graph on the left is known as an s -channel diagram, where the total initial momentum, or incoming center-of-mass energy, is transferred to the gauge boson. The graph on the right is known as a t -channel diagram, where the exchanged boson carries the difference between the initial and final state electron momenta. These quantities, s and t, as well as a third, u, are the so-called invariant Mandelstam variables. As above, they are invariant as they have the same value in all reference frames. They are specifically defined as where qi, pi, ki represent the particle's four-momentum as labeled in the figure. They satisfy the relation s + t + u = m1 + m2 + m3 + m4, where mi represents the mass of the particles in the reaction. The S -matrix elements are expressed in terms of these variables as they convey the kinematic information of the process, in particular, the angular dependence of the final states. A matrix element is computed for each channel that contributes to a reaction, and then they are added coherently to obtain the full invariant matrix element.
s -channel scattering possesses special properties. The denominator of an s -channel S -matrix element is proportional to s - M2, where M is the mass of the particle being exchanged. When the value of the center-of-mass energy √s approaches the mass of the exchanged particle, a resonance known as a Breit-Wigner resonance appears. This results in a substantial increase, usually a factor of 10 to 1,000, in the production cross section. As the cross section is mapped out as a function of √s , one sees that this resonant enhancement has a finite width, that is, it occurs for a finite range of √s , and the resonance looks like a bump. The width of this resonance is the total decay width of the exchanged particle. The appearance of resonances is a key tool in the discovery of new particles. Resonance structures do not occur in t - and u -channel processes.
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