Mathematical Methods for $B^{0}\overline{B}^0$ Oscillation Analyses

The measurement of the $B^0_s \bar B^0_s$ mixing frequency $\Delta m_s$ requires the search for a periodic pattern in the time distribution of the data. Using Fourier analysis the consequences of vertex and boost resolution, mistag and statistical fluctuations are treated analytically and a general expression to estimate the significance of a $B^0 \bar B^0$ mixing analysis is derived. With the help of Fourier analysis the behavior of a classical maximum likelihood analysis in time space is studied, too. It can be shown that a naive maximum likelihood fit fails in general to give correct confidence levels. This is especially important if limits are calculated. Alternative methods, based on the likelihood, which give correct limits are discussed. A new method, the amplitude fit, is introduced which combines the advantages of a Fourier analysis with the power and simplicity of a maximum likelihood fit.