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.
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Notes
- 1.
It can be proved that the sign of the 0th component is also a Lorentz invariant .
- 2.
The fact that j is a Lorentz-vector can be proved by noticing that the continuity equation \(\partial _t \rho + \text{ div }_{\mathbf {x}} \rho \mathbf {v}=0\) has to be invariant since it states the conservation of mass, which as to hold for any frame. The latter can be written in covariant notation as \(\partial _\mu j^\mu =0\), hence \(j_\mu \) has to transform as a covariant vector since \(\partial _\mu \) is contravariant.
- 3.
\(\mathbf {K}\) is a vector under rotations, see e.g. the second of Eq. (1.29), and it also transforms as a vector under parity transformations, since a parity operation must change the direction of the boost.
- 4.
For example, using the input values \(\beta =0.8\), \(\beta ^*_1=0.3\) and \(\beta ^*_2=0.5\), one gets \(\cos \theta ^*_\mathrm {max}=0.392\), which is in agreement with the numerical evaluation in Fig. 1.3.
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Appendices
Appendix 1
We report here a simple computer program in Python which implements Newton’s method for finding the roots of a real-valued function f, specialised here to the case where f is the first derivative of \(\tan (\phi )\) in Eq. (1.67) with respect to \(x\equiv \cos \theta ^*\).
After initialising the program with the values of \(\beta \), \(\beta ^*_1\), and \(\beta ^*_2\), the roots of f are searched for by iteratively incrementing the variable x as:
starting from an initial value \(x_0\). The loop stops when the desired accuracy is attained, i.e. \(\varDelta x_i/x_{i-1}<\varepsilon \), or the maximum number of iterations is exceeded.
A critical point of such a method applied to the case of interest arises from the fact that \(x\in [-1,1]\), while the intermediate values \(x_i\) may occasionally fall outside of this range. When this happens, one can try tuning the starting value \(x_0\) until the convergence is attained.
Appendix 2
The computer program below illustrates the generation of toy MC events where an unpolarised resonance of mass \(M\) decays into a pair of massless particles. This routine profits from a number of built-in functions available in ROOT that implement a good deal of four-vectors algebra.
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Bianchini, L. (2018). Kinematics. In: Selected Exercises in Particle and Nuclear Physics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-70494-4_1
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