Probability Distributions for Track Fitting and the Advanced Lucky Model

Probability distributions for the center of gravity are fundamental tools for track fitting. The center of gravity is a widespread algorithm for position reconstruction in tracker detectors for particle physics. Its standard use is always accompanied by an easy guess (Gaussian) for the probability distribution of the positioning errors. This incorrect assumption degrades the results of the fit. The explicit error forms evident Cauchy–Agnesi tails that render the use of variance minimizations problematic. Therefore, it is important to report probability distributions for some combinations of random variables essential for track fitting: x=ξ/(ξ+μ), y=(ξ−μ)/[2(ξ+μ)], w=ξ/μ, x=θ(x3−x1)(−x3)/(x3+x2)+θ(x1−x3)x1/(x1+x2) and x=(x1−x3)/(x1+x2+x3). The first two are partial forms of the two strip center of gravity. The fourth is the complete two strip center of gravity, and the fifth is a partial form of the three strip center of gravity. For the complexity of the forth equation, only approximate expressions of the probability are allowed. Analytical expressions are calculated assuming ξ, μ, x1, x2 and x3 independent Gaussian random variables. The analytical form of the probability for the two strip center of gravity allows one to construct an approximate proof for the lucky model of our previous paper. This proof also suggests how to complete the lucky model by its absence of a scaling constant, relevant to combine different detector types. This advanced lucky model (the super-lucky model) can be directly used in trackers composed of non-identical detectors. The construction of the super-lucky model is very simple. Simulations with this upgraded tool also show resolution improvements for a combination of two types of very different detectors, near to the resolutions of the schematic model.