# Bachelor thesis with Priv.-Doz. Worek in 2016

Numerical Methods of Phase Space Integration

An important problem in particle physics is the computation of cross sections. Those are usually a very complicated integrals over the square matrix element and the phase-space volume of momenta of the final state particles. Such cross sections exhibit strong peaks in many different regions of the phase space. Additionally, the presence of complicated kinematical cuts render an analytic treatment impossible. The Monte Carlo methods of integration are often used instead. In the Monte Carlo approach, a huge effort must be made to reduce the variance of the integrand. One of the popular approaches of variance reduction is the so-called stratified sampling technique, another approach is that of importance sampling. The goal of this bachelor thesis will be to calculate the phase-space integral using Monte Carlo techniques together with different methods to minimalise the integration error. Those approaches will be applied to the real-life calculation of a cross section, namely that of electron-positron annihilation, at LEP energies, into an electron-positron pair. A comparative study of various methods for variance reduction will be made for this process. Finally, a comparison to existing Monte Carlo programs will be performed to cross-check the efficiency of the implemented methods.

**The student will learn:**

- The basic of Monte Carlo methods
- How to reduce the variance of the integrand (stratified sampling, importance sampling, other methods)
- How to efficiently calculate phase-space volume of momenta of the final state particles

**Requirements:**

- Basic understanding of particle physics is required and can be obtained during the work on the thesis
- Programming skills e.g. in Fortran that can be easily learned during the time of the thesis