LHC Experiments

The chapter gives an overview of the track and vertex reconstruction methods of the LHC experiments that were used in production during Run 2 of the LHC, which ended in autumn of 2018.

Vertex Finding

Vertex finding is the search for clusters of tracks that originate at the same point in space. The chapter discusses a variety of methods for finding primary vertices, first in one and then in three dimensions. Details are given on model-based clustering, the EM algorithm and clustering by deterministic annealing in 1D, and greedy clustering, iterated estimators, topological vertex finding, and a vertex finder based on medical imaging in 3D.

Track Fitting

Track fitting is an application of established statistical estimation procedures with well-known properties. For a long time, estimators based on the least-squares principle were—with some notable exceptions—the principal methods for track fitting. More recently, robust and adaptive methods have found their way into the reconstruction programs. The first section of the chapter presents least-squares regression, the extended Kalman filter, regression with breakpoints, general broken lines and the triplet fit. The following section discusses robust regression by the M-estimator, the deterministic annealing filter, and the Gaussian-sum filter for electron reconstruction. The next section deals with linearized fits of space points to circles and helices. The chapter concludes with a section on track quality and shows how to test the track hypothesis, how to detect outliers, and how to find kinks in a track.

Belle II and CBM

The chapter gives an overview of the tracking and vertexing algorithms of two experiments not at the LHC, Belle II at SuperKEKB in Japan and CBM at FAIR in Germany.

Tracking Detectors

The chapter gives an overview of particle detectors, with the emphasis on tracking detectors. The working principles and the calibration of gaseous, semiconductor, and fiber detectors are explained, followed by a brief review of detector alignment. As an illustration, the tracking systems of the four experiments at the LHC and two non-LHC experiments, Belle II and CBM, are presented.

Track Finding

There is no systematic theory of track finding yet. Therefore, the first section of this chapter presents a list of basic techniques which have been successfully used, stand-alone or in combination, in past and present experiments. Among them are the conformal transformation, the Hough and the Legendre transform, cellular automata and neural networks, pattern matching, and track following by the combinatorial Kalman filter. The following section gives a brief excursion into online or real-time track finding in the collider experiments CDF, ATLAS, and CMS. As track finding in most cases delivers some candidates that do not correspond to actual particle tracks, the concluding section discusses methods for an efficient selection of valid candidates.

Vertex Fitting

The methods used for vertex fitting are closely related to the ones used in track fitting. The chapter describes least-squares estimators as well as robust and adaptive estimators. Furthermore, it is shown how the vertex fit can be extended to a kinematic fit by imposing additional constraints on the tracks participating in the fit.

Secondary Vertex Reconstruction

The chapter reviews methods for the search for secondary vertices. Four types of secondary vertices are discussed in detail: decays of short-lived particles, decays of long-lived particles, photon conversions, and hadronic interactions in the detector material.

Statistics and Numerical Methods

The chapter gives an outline of some statistical and numerical methods that will be applied in later chapters. The first section deals with the minimization of functions. Several gradient-based methods and a popular non-gradient method are discussed. The following section discusses statistical models and the estimation of model parameters. The basics of linear and nonlinear regression models and state space models are presented, including least-squares estimation and the (extended) Kalman filter. The final section gives a brief overview of clustering and different types of clustering algorithms.

Event Reconstruction

The chapter gives an outline of the event reconstruction chain of a typical large experiment, from the trigger to the physics object reconstruction. The concept of the trigger is illustrated by two examples, CMS and LHCb, followed by a discussion of track reconstruction and its stages: hit generation, local reconstruction, and global reconstruction. The section on vertex reconstruction introduces a classification of vertices and sets the scene for the dedicated chapters on vertex finding and vertex fitting. The chapter concludes with some remarks on particle identification and reconstruction of physics objects such as electrons, muons, photons, jets,τleptons, and missing energy.