Feynman Diagrams and the Sinc Function Representation

The Sinc function representation is a novel approach to the numerical evalutation of Feynman integrals. It was introduced in our paper Fast Evaluation of Feynman Diagrams, and in The Sinc Function Representation and Three-Loop Master Diagrams we used it to evalaute the master diagrams, whose analytic properties are described by Broadhurst.

To illustrate how the method works in practice (and to help anyone planning to use the Sinc function representation for their own calculations) we are providing the codes we used to perform the calculations described in our papers.

However, please take note of the following points:

• Anyone who wants to use, copy or redistribute these codes is free to do so, but please mention us when (and if) you do. The code is copyrighted, and if you want to distribute a modified version, please check with us first.

• These is prototypes, not finished products, and we can see a number of possible improvements. Furthermore, the parameter values are included directly in the code itself, rather than being input by the user. Thus to change the "input" you need to edit the source and recompile (or add a couple of lines of code that lets you input parameter values when the program runs).

• The code is written in Fortran 77, and while it is documented we do not claim that it is pretty! In particular, the nested summations could be tackled much more elegantly using recursive subroutines (in Fortran 90), or in C/C++ which does not skip a beat when a procedure calls itself.

• Declarations for common blocks (and their associated variables) are inserted via an include 'header' command. This makes it easy to keep the common blocks properly aligned, but it also means that each subroutine ``sees'' more variables than it actually uses. If you were careless, this could cause problems.

• The codes definitely works on a Sun Ultrasparc II (using versions 4.2 and 5.0 of the Sun Fortran 77 compiler, to be truly precise). Your mileage may vary. We compiled with "f77 -fast", which aggressively optimizes the resulting binary.

• Some warnings (underflow errors, mainly) are generated as the code runs. Ideally we would trap these explicitly, but they are not doing any harm.

• If you have suggestions, comments or questions, send me email.

These programs were used to evaluate the diagrams discussed in the paper Fast Evaluation of Feynman Diagrams.

• The renormalized version of the sunset diagram.
• The renormalized sunset diagram (as above), but using interpolation to compute the innermost sum.
• The three loop diagram with cut-off, via the Sinc function representation.
• The four loop diagram with cut-off, via the Sinc function representation.

You will also need the following header files (which must be edited slightly, see the comments in the source). You have to save the files with the appropriate names (header and headerint).

• This is the header for the Sinc function calculations. Comment out the appropriate lines before compiling.
• This is headerint, used for the version of the sunset code which interpolates the inner sum.

This program evaluates the Sinc function representation of the master diagrams, described in The Sinc Function Representation and Three-Loop Master Diagrams.

• This is the master diagram code, which requires the following header file.