A new fast track-fit algorithm based on broken lines

The determination of the particle momentum in HEP tracking chambers requires a fit of a parametrization to the measured points. Various effects can result in deviations to the ideal helix curve in the magnetic field of a solenoid, and the fit with a pure helix parametrization is not optimal. One effect is multiple scattering, which causes a “random walk” of the particle and affects especially low-momentum tracks and very accurate measurements. Optimal values of track parameters close to the vertex can be determined by standard least squares methods either by including multiple scattering in the covariance matrix, which becomes non-diagonal, or by the introduction of additional parameters; both global methods work with large matrices and are generally not acceptable due to the large execution time O(n3) for n measurements along the track. A popular method of track reconstruction is the Kalman filter, an algorithm known from time-series analysis and signal processing, and with results mathematically equivalent to global least squares fits. It is recursive, includes measurements one after the other and has an execution time O(n), because large matrices are avoided. Drawbacks are the facts that parameter values are needed already to start the filter fit and that the track parameters are known with optimal precision only after the last step of the fit. The proposed method based on broken lines is non-recursive and allows to reconstruct the particle trajectory taking into account details of the multiple scattering. It provides optimal parameters and their covariance matrices at track start and end, and optimal values at each measured point along the trajectory including the variances. The method is constructed to allow the use of sparse-matrix techniques with a total execution time O(n), and is not slower than the Kalman filter.