Software Alignment of the LHC$b$ Outer Tracker Chambers

This work presents an alignment algorithm that was developed to precisely determine the positions of the LHC$b$ Outer Tracker detector elements. The algorithm is based on the reconstruction of tracks and exploits that misalignments of the detector change the residual between a measured hit and the reconstructed track. It considers different levels of granularities of the Outer Tracker geometry and fully accounts for correlations of all elements which are imposed by particle trajectories. In extensive tests, simulated shifts and rotations for different levels of the detector granularity have been used as input to the track reconstruction and alignment procedure. With about 260 000 tracks the misalignments are recovered with a statistical precision of $\cal{O}$ $(10-100\mu m)$ for the translational degrees of freedom and of $\cal{O}$ $(10^{-2}-10^{-1}$ mrad) for rotations. A study has been performed to determine the impact of Outer Tracker misalignments on the performance of the track reconstruction algorithms. It shows that the achieved statistical precision does not decrease the track reconstruction performance in a significant way. During the commissioning of the LHC$b$ detector, cosmic ray muon events have been collected. The events have been analysed and used for the first alignment of the 216 Outer Tracker modules. The module positions have been determined within $\sim{90}\mu m$. The developed track based alignment algorithm has demonstrated its reliability and is one of the core algorithms which are used for the precise determination of the positions of the LHC$b$ Outer Tracker elements.