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Springer International Publishing AG 2018

Lorenzo Bianchini Selected Exercises in Particle and Nuclear Physics UNITEXT for Physics

1. Kinematics

Lorenzo Bianchini 1

(1)

Italian Institute for Nuclear Physics, Pisa, Italy

Lorenzo Bianchini

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Abstract

The first chapter is dedicated to the kinematics of relativistic particles. The starting point is the introduction of the Lorentz group through its representations. Large emphasis then is given to the transformation properties of velocities and angles. The centre-of-mass dynamics is studied in detail for two-to-two scattering and for two- and three-body decays. The last part of the chapter is devoted to the concept of cross section, which plays a central role in particle physics.

1.1 Lorentz Transformations

The set of space-time transformations under which the laws of physics are postulated to be invariant form the so-called Poincar group : they comprise four space-time translations, three spatial rotations, and three velocity transformations. Space rotations and velocity transformations (the latter are often referred to as boosts ) form the sub-group of Lorentz transformations. Rotations are determined by the three usual Euler angles, while boosts are determined by the three components of the velocity of the new reference frame as measured by an observed at rest in the initial reference frame , or, equivalently, by the dimensionless boost vector . Rotations and boosts change both the four-momentum and the spin vector of a particle. Since these transformations form a group, it is possible to find a representation of each element in terms of square matrices acting on vector spaces: the four-dimensional space of four-vectors and the -dimensional space of spin vectors for a particle of spin S .

The space of four-vectors in endowed with the Minkowski norm defined by:

(1.1)

with . By construction, the Lorentz transformations preserve the Minkowski norm of Eq. () provides an example.

The three-momentum of a particle can be embedded into a four-vector , called four-momentum . For a particle of mass m , the four-momentum is subject to the mass-shell constraint:

(1.2)

The velocity and gamma- factor of a particle are then defined as

(1.3)

In the four-momentum space, each Lorentz transformation is represented by a matrix which transforms p into a new four-momentum . The boost vector defines a privileged direction in space, and indeed the transformation distinguishes between the component parallel ( ) and orthogonal ( ) to . In terms of these two components, the generic boost transformation is given by:

(1.4)

with and . Notice that is a function of , although the explicit dependence is often omitted in the calculations. The variables at the left-hand side of Eq. ().

Given that and that , the first of Eq. () can be equivalently written as:

(1.5)

with . Were not for the imaginary angle and the same-sign off-diagonal elements, Eq. () would be the transformation of a normal vector under a spatial rotation. The differences accounts for the fact that the transformation has to preserve the Minkowski norm , and not the Euclidean norm , as done by ordinary rotations.

A final word of caution: and in Eq. () are the parameters of a transformation, and should not be confused with the velocity and gamma-factor of a particle as measured in a given reference frame: the suffix in the latter thus reminds that these quantities are different from the boost parameters. However, when no such ambiguity can arise, the suffix can be safely dropped to simplify the notation.

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