Example of calculations

In few problems can the integrals that appear in the evaluation of cross sections be done analytically. Most of the time we have to revert to numerical methods to evaluate the integrals. There are many ways to do evaluate the integrals numerically. A good library is the CUBA package. It can be linked either with C/C++ or with Fortran programs. It has several methods with the same calling structure, so that one can substitute one method for another and check for accuracy. The manual comes with the package. I have done a gaussian integration subroutine gauss.f that uses the same calling conventions, and therefore can also be tested in the same way. To illustrate its use I have done an example in Fortran with a simple problem, the $e^- \mu^- \rightarrow e^- \mu^-$ in QED (that is, only the photon graph) that we studied before as an example of the use of the mathematica packages. In this case the integrations can be done analytically, but the integral diverges if we integrate in $\theta \in [0,\pi]$. So we have to apply a cut, say $\theta > \theta_{\rm min}$. This is the well-known collinear divergence that appears because the photon is massless. In terms of center of mass variables, we have (we do not neglect the fermion masses)

$$(p-p')^2 = -2 p_{\rm CM}^2\, (1 -\cos\theta_{\rm CM})$$

and we see that it diverges for $\theta \rightarrow 0$.

A Fortran program that calculates this is muemue.f. To use it you also need an header file with the physical parameters of the Standard Model smpc.h and the Cuba package compiled as library (we assume it in /usr/local/lib/Cuba/libcuba.a). It is useful to use a Makefile to compile the program. You can get a tar gzipped file Integrals.tar.gz with all the needed files. You should experiment with the integration methods and compare with the exact integration that is also included in the program.
Note added 22.04.2014: The current version is compatible with Cuba-3.3. We remind the reader that the calling of the routines was modified from previous versions.

The results are summarized in the following figure:

where we have applied a cut $\theta > 10^{\circ}$ and $p_{2 \rm lab}=50$ GeV. We have calculated in four different ways: a) Using the exact analytical results (black line) b) Using gaussian integration (red dots) c) Using Vegas integration (blue crosses) d) Using CalcHEP (green diamonds). We see that the results are in excellent agreement.