The book often cites MAPLE routines and encourages the reader to use MAPLE for own calculations. In my home university students had access to MAPLE and could also buy it at a low rate. This is not true anymore. Fortunately there is a noncommercial alternative: Python/Sympy. The very popular language Python is part of the curriculum at many universities. Sympy is a computer algebra programme based on Python. Used with the jupyter notebook it provides an elegant frame for doing computer algebra. Please consult the Python and Jupyter pages for installing these programmes on your system.
A direct link to jupyter notebook files is at the moment not allowed by the web site provider. Here is a link to a Google Drive folder, where notebooks and other python routines will be stored. For those, who don’t want to run the notebooks pdf files of the results are also stored. The file heppackv0.py provides necessary subroutines. In case of problems contact me via email (berger@rwth-aachen.de)
Cross sections like electron positron annihilation into muon pairs are calculated by first determining the (16) helicity amplitudes and subsequently squaring and adding these amplitudes. Evaluating the spin average of squared amplitudes the textbooks favour the traditional method of using traces of gamma matrices. However, having the amplitudes at hand also polarization effects can be calculated in an transparent and intuitive way.
It is strongly recommended to change the parameters of the cells in order to start some own investigations. The following notebooks are available:
- Dirac-tutorial-vb.ipynb treats gamma matrices, Dirac Spinors, Spin and helicity operators, charge conjugation, parity and time reversal operators and ends with a look at the Weyl representation.
- eemumu.ipynb The amplitudes for electron positron annihilation into muon pairs are calculated step by step including mass terms. Then the cross section in invariant form and expressed by CM variables is calculated. Furthermore muon pair production by polarised beams is investigated and finally the transition to electron muon scattering including the limiting case of Mott-scattering is discussed. The figure Kap2fig8.jpg should also be downloaded.
- eepipi-tutorial-vb.ipynb Like the eemumu notebok but this time treating pion pair production and electron pion scattering.
- Rosenbluthandmore-vb.ipynb The notebook starts calculating the amplitudes for electron scattering off a Dirac-proton and then continues to include the anomalous magnetic moment (Pauli-terms) and the form factors F1 and F2, respectively GE and GM. Subsequently the famous Rosenbluth formula is derived in invariant form and in the traditional lab system form. We then calculate the ratio of transverse and longitudinal polarization of the recoil proton. This observable is important because it is proportional to the ratio of the electric and magnetic form factor.
- Compton(HE)-vb.ipynb The amplitudes for Compton scattering are calculated in the high energy limit, i.e. for massless electrons. The cross section is given in invariant form.
- Compton.ipynb A rather long notebook starting with the kinematics of Compton scattering. In section 2 the eight independent amplitudes are derived and expressed by invariants. In section 3 follows the formula for the differential cross section expressed by the invariants s,t,u and alternatively by two dimensionless invariants x and y. The cross section seems to be determined by so called large and small amplitudes, but the discussion in section 4 shows, that this is more complicated. In section 5 polarization effects are discussed. In particular the use of Compton scattering to measure the polarization of the electron beam in the proposed linear collider is discussed. Finally in section 6 the cross sections for electron positron annihilation into photons and two photon annihilation into electro positron pairs are derived.
- eeee.ipynb The notebook treats Bhabha and Moller scattering , i.e. elastic electron positron and electron electron scattering. Because final state and initial state are identical two Feynman diagrams have to be evaluated for each helicity amplitude. Don’t forget to also download the figures eetoee.jpg and Moller.jpg
- LeptonWeak-vb.ipynb The notebook starts in section 1 with discussing muon-neutrino electron scattering including all mass terms. The unitarity of Fermi’s theory is thoroughly investigated followed by standard model amplitudes and cross section containing the W boson propagator. Section 2 derives the muon lifetime including the electron mass terms from crossing symmetry arguments. Section 3 finally derives the muon lifetime and the cross section for antineutrino electron scattering by explicit calculation of the amplitudes. The figures mudecay.jpg and nubarelectron.jpg should also be downloaded.
- NeutronDecay-vb.ipynb In the first chapter neutrons are treated as elementary fermions. The resulting decay-width depends strongly on including the electron mass. In chapter 2 the hadronic nature of neutrons is included in the ansatz for the weak currents. Chapter 3 then contains a plot of the momentum spectrum and the formula for the polarization of the decay electrons.
- Compvirt2020.ipynb treats the scattering of virtual photons off electrons where the outgoing photon is on the mass shell. This notebook proves formula 3.238 of the book.
- ggtogg.ipynb The rather long notebook calculates the QCD cross section for elastic gluon gluon scattering gg→gg from the four underlying Feynman-Diagrams in file ggtogg.jpg, For running the notebook this file should also be downloaded. The notebook is necessary for understanding chapter 4.2 of the planned fourth edition of the book.
Professor Christoph Berger 