Reconstruction of electrons with the Gaussian-sum filter in the CMS tracker at LHC

TL;DR

The paper addresses the non-Gaussian bremsstrahlung energy loss in electron tracking by replacing the Kalman filter's single Gaussian with a Gaussian-sum filter (GSF) that models energy loss as a Gaussian mixture. It develops methods to approximate the Bethe-Heitler distribution via CDF- and KL-distance minimization, and propagates multiple mixture components through the track, with practical component-reduction schemes to control complexity. The study shows that GSF improves momentum resolution and yields more accurate error estimates compared to the standard Kalman filter, both in simplified simulations with known material and in full CMSIM simulations, where the gains persist even with imperfect knowledge of energy loss. These results support the GSF as a valuable enhancement for electron reconstruction in the CMS tracker and potentially for other detectors with strong non-Gaussian energy-loss effects.

Abstract

The bremsstrahlung energy loss distribution of electrons propagating in matter is highly non Gaussian. Because the Kalman filter relies solely on Gaussian probability density functions, it might not be an optimal reconstruction algorithm for electron tracks. A Gaussian-sum filter (GSF) algorithm for electron track reconstruction in the CMS tracker has therefore been developed. The basic idea is to model the bremsstrahlung energy loss distribution by a Gaussian mixture rather than a single Gaussian. It is shown that the GSF is able to improve the momentum resolution of electrons compared to the standard Kalman filter. The momentum resolution and the quality of the estimated error are studied with various types of mixture models of the energy loss distribution.

Figures (14)

• Figure 1: Probability density function $f(z)$ for different thickness values.
• Figure 2: The distances $D_{\hbox{\scriptsize KL}}$ and $D_{\hbox{\scriptsize CDF}}$ as a function of the thickness $t$, for CDF-mixtures, with different numbers of components.
• Figure 3: The distances $D_{\hbox{\scriptsize KL}}$ and $D_{\hbox{\scriptsize CDF}}$ as a function of the thickness $t$, for KL-mixtures, with different numbers of components.
• Figure 4: Estimated $q/p$ of one single track for the GSF (solid), the KF (dashed) and the combined GSF state (dotted). The combined GSF state refers to the first and the second moments of the GSF estimate, here visualized as a single Gaussian. The arrow denotes the true value of $q/p$. It can be seen that the estimated PDF of the GSF is a non-Gaussian function.
• Figure 5: Probability distribution for the estimated $q/p$ for the KF (solid) and the GSF with a maximum of six (dashed-dotted), twelve (dashed), $18$ (solid) and $36$ (dotted) components kept during reconstruction. In this case the same six-component CDF-mixture has been used both in the simulation of the disturbance of the momentum in a detector unit and in reconstruction. Keeping 36 components yields estimates quite close to the correct distribution of the parameter.