Optimizing Momentum Resolution with a New Fitting Method for Silicon-Strip Detectors

Abstract

A new fitting method is explored for momentum reconstruction. The tracker model reproduces a set of silicon micro-strip detectors in a constant magnetic field. The new fitting method gives substantial increases of momentum resolution respect to standard fit. The key point is the use of a realistic probability distribution for each hit (heteroscedasticity). Two different methods are used for the fits, the first method introduces an effective variance as weight for each hit, the second method uses the search of the maximum likelihood. The tracker model is similar to the PAMELA tracker with its two sided detectors. The two detector sides have very different properties and quality. Each side is simulated as momentum reconstruction device. One of the two is similar to silicon micro-strip detectors of large use in running experiments. Two different position reconstructions are used for the standard fits, the η-algorithm (the best one) and the two-strip center of gravity. The gain obtained in momentum resolution is measured as the virtual magnetic field and the virtual signal-to-noise ratio required by the two standard fits to reach an overlap with the best of two new methods. For the low noise side, the virtual magnetic field must be increased 1.5 times respect to the real field to reach the overlap and 1.8 for the other. For the high noise side, the increases must be 1.8 and 2.0. The virtual signal-to-noise ratio has to be increased by 1.6 for the low noise side and 2.2 for the high noise side (η-algorithms). Changes of the signal-to-noise ratio has no effect on the fits with the center of gravity as position-algorithm. The momentum resolution is simulated even in function of the number N of the detection layers. A very rapid linear increase with N is observed for our two methods, the two standard fits has the usual grow as √ N. Other interesting effects are obtained selecting tracks with good or excellent hits. KEYWORDS: Performance of High Energy Physics Detectors, Pattern recognition, cluster finding, calibration and fitting methods, Si microstrip and pad detectors, Analysis and statistical methods. * Corresponding author. the most recent ones of a very large set) are addressed to the first three effects. The handling of the full non linearity of the particle path is clearly described in ref. [3].This work is addressed to an accurate study of the statistical effects in the positioning algorithms of minimum ionizing particle (MIP) and how to reduce at minimum their influence in the momentum reconstruction. For this task, realistic probability density functions (PDFs; probability distributions in the physicist jargon) for the errors of the positioning algorithms must be used. This strategy allows to go beyond the results of the standard least squares method or other fitting method that neglects the hit-statistical properties, or the heteroscedasticity (as it called in literature). For our task, we will simulate momentum values where the multiple scattering is unimportant, similarly for the energy loss (principally fo...

The Cramer—Rao Inequality to Improve the Resolution of the Least-Squares Method in Track Fitting

“…As we said above, a binomial PDF suffices in the disordered tracker detector for the models of refs. [2,3], just a mean disorder of half strip (≈30 µm)…”

“…This search on products of PDFs for the hits of each track gives very large improvements in the resolution of all the track parameters and shows insensitivity to outliers. Among the new fitting results, one looks particularly attractive [2], it shows a linear growth of the momentum distributions with the number N of detecting layers. This linear growth is much faster than the growth of the standard least-squares method that cannot increase faster than √ N. This effect could not be discussed with the due details in ref.…”

“…Track reconstruction is a fundamental tool in experimental particle physics, which justifies the large efforts in hardware and software dedicated to its improvements. As part of these efforts, we started to develop new fitting tools based on studies of the signal patterns released by minimum ionizing particles (MIPs) in silicon detectors [1,2]. It is evident that any information acquired by a detector is precious to improve the track reconstruction.…”

“…This linear growth is much faster than the growth of the standard least-squares method that cannot increase faster than √ N. This effect could not be discussed with the due details in ref. [2], it contained too many different arguments to add other. Therefore, ref.…”

“…Even if the essential equations are very similar, and our tracks are stochastic processes, it is better to develop our approach in direct connection to our problem with a strict reference to the equations used in refs. [1][2][3]. If other authors did similar developments we apologize for not citing them.…”