Optimized estimation of energy loss rate for charged particles from energy deposit measurements in tracking detectors
Introduction
The identification of charged particles is crucial in several fields of particle and nuclear physics: particle spectra, correlations, selection of daughters of resonance decays, and also for reducing the background of rare physics processes [1], [2], [3]. Tracker detectors, both semiconductor and gaseous, can be employed for particle identification, or yield extraction in the statistical sense, by proper use of energy deposit measurements along the trajectory of the particle. While for gaseous detectors a wide momentum range is available [4], [5], in semiconductors there is practically no logarithmic rise of the average energy loss rate at high momentum, thus only momenta below the minimum ionization region are accessible [6], [7]. In this work two representative materials, silicon and neon, are studied.
Energy loss of charged particles in matter is a complicated process. For a detailed description see Refs. [8], [9]. All calculations presented in this study are based on Monte Carlo methods. While the energy lost and deposited differ, they will be used interchangeably in the discussion. It is also clear that the energy determined from the measured data is affected by noise and digitization effects.
This article is organized as follows: Section 2 describes the microscopic energy loss simulation used in this study. Section 3 introduces the method of truncated means to estimate the energy loss from a number of energy deposit measurements along the particle trajectory, while Section 4 deals with the optimization of weights for the methods of arithmetically and of geometrically weighted means. Section 5 discusses how the effects of different segment lengths are treated. Results of the simulation and applications of the optimized weighted mean are shown in Section 6. The work ends with conclusions and it is supplemented by four appendices with interesting results, such as optimal weights in case of few (Appendix A) and many measurements (Appendix B); some theoretical insights (Appendix C); also on connection to maximum likelihood estimation (Appendix D).
Section snippets
Simulation
When a charged particle passes through material it loses energy in several collisions (Fig. 1). The conditional probability density , energy deposit along a given segment length t, can be built using a microscopic simulation. The mean free path between collisions is given by the mean number of collisions per unit path length . The distance to the next collision was determined by sampling an exponential distribution with the rate parameter . In case of a given segment length t, the
Truncated mean
There are several possibilities to determine the energy loss rate of a charged particle. An approach using an analytical model of energy loss would permit to use advanced methods such as maximum likelihood estimation. However, especially at startup, particle detectors are not expected to be understood to the degree that would enable the use of such estimator.
One of the robust and simple estimators is the so-called truncated mean that is traditionally used in gas filled detector chambers [16],
Weighted means
It is possible to generalize this estimator, and optimize the weights, by looking at some measures of its distribution. The generalization can be twofold. Instead of the 50% truncated mean, the more general weighted mean, linear combination, can be examined where the weights are allowed to take on different values, not just 0, 1/2 or 1. In addition, it is possible that the performance of the weighted mean is more beneficial when averaging a monotonic function of the measurements than
Weighted mean with different segment lengths
Up to this point, we assumed that the lengths of each segment in the sensitive detector are the same. In case of real particle trajectories, there are differences due to bending in the magnetic field, also due to the placement of detector units. The energy deposits can be corrected towards a reference path length. The distribution of energy deposit depends on the velocity of the particle and the segment length t. To a good approximation,2
Results
Particle identification and extraction of particle yields are particularly difficult at those momenta where the energy loss rates of different types of particles are close. For hadrons, the pion–kaon resolution gets problematic above about 0.8 GeV/c, while for the pion-proton case it happens above about 1.6 GeV/c. Hence the relevant region is 1–10.
In this study, charged particles with , 3.16 and 10.0 are studied, with the number of segments 2–50. Both semiconductor and gaseous detectors
Conclusions
The estimation of energy loss rate for charged particles in tracker detectors was studied. It was shown that the truncated mean method can be generalized to the linear combination of the energy deposit measurements. The optimized weights are rather independent of particle momentum and segment length, allowing for a robust estimation. Weighted arithmetic and geometric means result in better particle separation power for both semiconductor and gaseous detectors. Further inspections showed that
Acknowledgements
The authors wish to thank to Krisztián Krajczár for helpful discussions. This work was supported by the Hungarian Scientific Research Fund with the National Office for Research and Technology (K 48898, K 81614, H07-B 74296), and the CERN Summer Student Program.
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