The Standard Model

The present precision of experimental particle physics faces theorists with the challenge of reproducing the experimental data.

The accepted theoretical model for the electromagnetic, weak and strong particle interactions is the Standard Model - an SU(3)xSU(2)xU(1) algebraic formalism explaining the multiplet structure of the hadrons, and allowing a perturbation expansion in the coupling constants, of the Green's function and scattering amplitudes. The electromagnetic part of this theory is called QED, the strong part is called QCD.

In the high energy regime, the phenomenon of asymptotic freedom allows treating the strong interaction as a perturbative expansion in the coupling constant.

Some virtues of the Standard Model are: 1) It unites 3 of the 4 known fundamental interactions in one theoretical framework. 2) It is renormalizable (higher order corrections can be consistently calculated). 3) There has been no disagreement with experiment so far.

Some drawbacks of the Standard Model are: 1) General relativity is disregarded. 2) Hadronic interactions at low energies cannot be treated by perturbative expansion in the strong coupling constant. It is consequently not possible to put QCD to a direct test with low energy phenomena.

Historically, in the low energy region, phenomenological models have been used, and this has not changed with the advent of the Standard Model. One thing has changed though: Instead of being more or less ignorant of how to model the strong interaction, according to the Standard Model, the strong interaction has to 'approximately' obey chiral symmetry. This piece of knowledge has systematically been put to use in the formalism known as Chiral Perturbation Theory.

Apart from more very high energy calculations and experiments, a practicable way of testing the Standard Model is more low energy experiments and careful phenomenological and computational investigations.

Computers and Feynman diagrams

Either in the Standard Model or in Chiral Perturbation Theory, the evaluation of Feynman loop diagrams is a tedious task. The large amount of calculational work necessarily introduces the possibility of errors. This is one argument for implementing these theories and the necessary mathematical tools in a computer program. Another argument is that a successful implementation of the many necessary symbolic manipulations leaves more time for physicists to deal with the physics instead of tedious work.

A large number of programs (see below) of varying generality exist, and some standardization would be nice. If one wishes an open project, the first thing to agree about is which programming language to use. Traditionally, Fortran is the language used in the physics community, and there exist large program libraries with Fortran programs, but there are alternatives. Especially for a project such as implementing QFT and the Feynman/Dyson/Wick formalism, it would be desirable to have: 1) A language that already knows most of the necessary mathematical formalism, and 2) a language that can handle symbolic manipulations and pattern matching. Mathematica fulfills these two demands. The drawback of Mathematica is that it is quite slow. The existing programs written in Fortran are mostly designed for drawing and evaluating tree order graphs and/or focusing on a particular model, and thus, though large, not too general. The Mathematica package FeynArts evaluates tree and loop diagrams, that is, it draws the graphs (and calculates the corresponding symmetry factors) and writes up the amplitudes as products of the propagators and Feynman rules (which are the input) integrated over internal momenta in the case of loops. No attempt is made to simplify the resulting expressions or evaluate the loop integrals. To reduce the resulting expressions one then needs analytical tools that handle integrals in D dimensions, Dirac traces etc. FeynCalc is a Mathematica package that provides such tools and moreover is an implementation of large parts of the formulas and rules constituting QFT and the Standard Model.