Transport of covariance matrices in the inhomogeneous magnetic field of the ATLAS experiment by the application of a semi-analytical method

In this paper we study the transport of track parameter covariance matrices - the so-called error propagation - in the inhomogeneous magnetic field of the ATLAS experiment. The Jacobian elements are transported in parallel with the track parameters, avoiding the inherent need of any purely numerical scheme of propagating a set of auxiliary tracks. We evaluate the quality of the transported Jacobians by a very thorough, purely numerical approach of obtaining the same derivatives, providing a quantitative understanding of the effects of including gradients of energy loss and the magnetic field on the accuracy of the error propagation. Irrespective of the accuracy of the underlying track parameter propagation, the method of parallel integration of the derivatives is demonstrated to be significantly faster than even the simplest numerical scheme. The error propagation presented in this paper is part of the simultaneous track and error propagation (STEP) algorithm of the common ATLAS tracking software.