Basics

In this chapter, some basic quantities and some of the tools that are frequently used are introduced.

Since particles are tiny, their behaviour must be described through quantum mechanics. Their properties are usually described in scattering experiments, where beams of particles are brought to collision and eventually react. Such reactions are quantum mechanically described by the S-matrix, hence some of their properties will be recalled first. For a discussion of how to actually calculate S-matrices for specific reactions in particle physics, the reader is referred to later chapters. There, the Feynman diagram technique which forms the basic of perturbative calculations in quantum field theory will be presented in more detail. Having at hand the S-matrix for a process is not yet the end of the story, since it just represents the probability amplitude for a scattering of a well-defined initial state into a well-defined final state. In most cases, however, the states are dense - as an example for that consider the momentum eigenstates. Therefore, the initial and final states have to be averaged and/or summed over. This results in a quantity called cross section in case of scattering processes or width in the case of particle decays. The latter can be interpreted as being related to the lifetime of the particle. The meaning of the former becomes apparent when considering the collision of beams of particles, characterised by a quantity called luminosity. The respective definitons and connections will thus be discussed in due order.

Obviously, since in most cases particle physics deals with fast, i.e. relativistically, moving particles, a formulation through relativistic kinematics is unavoidable. Hence, after recalling some basic properties, different quantities are introduced that characterise kinematics, such as transverse momentum and mass, rapidity and pseudorapidity. Mandelstam variables will be introduced next. They are a common covariant way of describing 2->2 scattering and production processes. The meaning of these quantities will be illustrated by some simple examples.

Clearly, it is important to discuss very basic properties of detectors such as their acceptance. After all, in collider experiments, there must be "holes" in the detector to allow the beams to pass through them. Therefore, in most cases, detectors look like barrels, and the geometry of these barrels defines to some extent in which kinematical regions particles can be detected. Since detection is also a statistical process at that point some very basic methods of statistical data analysis (fitting etc.) are presented.

Then, the discussion will turn on the question of how to integrate over the four-momenta of outgoing particles, which is needed for the evaluation of cross sections or widths. The combination of all four-momenta of all outgoing particles is often called the phase space of a reaction, and ways of how to perform phase space integrations analytically are discussed first. Then, some simple (numerical) Monte Carlo methods to generate appropriate four-momenta in the framework of computer programmes will be discussed. These methods are in the heart of simulation codes. Starting from a simple parametrisation of a two-body decay phase space more involved algorithms will be developed and tested. They will then be used to discuss two important Monte Carlo techniques: Integration (sampling) and event generation (unweighting).