Software Alignment of the LHCb Inner Tracker Sensors

This work uses the Millepede linear alignment method, which is essentially a $\chi^{2}$ minimization algorithm, to determine simultaneously between 76 and 476 alignment parameters and several million track parameters. For the case of non-linear alignment models, Millepede is embedded in a Newton-Raphson iterative procedure. If needed, a more robust approach is provided by adding quasi-Newton steps that minimize the approximated $\chi^{2}$ model function. The alignment apparatus is applied to locally align the LHCb's Inner Tracker sensors in an a priori fixed system of coordinate. An analytic measurement model was derived - as function of the track parameters and alignment parameters - for the two cases: null and non-null magnetic field. The alignment problem is equivalent to solving a linear system of equations, and usually a matrix inversion is required. In general, as consequence of global degrees of freedom or poorly constrained modes, the alignment matrix is singular or near-singular. The global degrees of freedom are obtained: directly from the $\chi^{2}$ function invariant transformations, and, in parallel, by a diagonalization of the alignment matrix followed by an extraction of the least constrained modes. The procedure allows to properly define the local alignment of the Inner Tracker (IT). Using Monte Carlo data, the outlined procedure reconstructs the position of the IT sensors within micrometer precision or better. For rotations, an equivalent precision was obtained.