To determine the efficiency and contamination of a given TPC cut as well as finding a useful cut in the first place, it is necessary to have a sufficiently acurate fit of the data. To achieve this, individual momentum bins were fitted separately. Apart from electrons, the main contributions from other particles after a TOF cut are pions, muons, kaons and protons. Within a small momentum bin, the distribution of the ionization of each particle type is approximately Gaussian.
The fitting of 5 Gaussians of different width, amplitude and center gives 15 free parameters for the fit. The certainty of convergence of the fit to the correct solution can be increased by constraining the space within which the parameters may vary. Two types of constraints are applied to achieve this. One constraint is gained through a prefit: The data with an applied TRD cut is fitted first. This data set has a cleaner peak for the electrons. From this fit constraints for the center and width of the electrons and pions can be gained.
A second set of constraints can be taken from the spline approximation of the dE/dx of the particles. By comparing the centersof electrons and pions with the expected dE/dx values, a guess for the centers of the other particles can be obtained. A constraint for the widths is gained from the approximate proportionality of center and width of the distributions in the dE/dx-plot.
In a momentum bin of finite width the center of a ionization distribution varies over the length of the bin. This leads to a widening of the distribution. To take this into account, a sum of several Gaussians is used for non-electrons. The Gaussians have the same amplitude and width and the centers are equidistant. The total variation of the centers is approximated via the data from the splines.
Apart from the data and the fits, the plots contain to additional graphs. These can be used to determine how well the fit works. The light blue line gives the ratio of the data and the fit.
The data within each bin should follow Poisson-statistics. To get a measure of the systematic error of the fit, the absolute of the difference between data and fit is divided by the square root of the amplitude (of the fit). The result is the deviation relative to the expected deviation. This is given by the pink line in the plots. If there is no systematic error and the fit works perfectly, the line should remain below 1 in most bins.