Pixel size studies for the Inner Pixel phase II upgrade

The high luminosity phase of the Large Hadron Collider (HL-LHC) will require the installation of a new Inner Pixel detector capable to withstand an unprecedented high instantaneous luminosity of $7 \times 10^{34}$~cm$^{-2}$s$^{-1}$ and integrated luminosity of 3000~fb$^{-1}$. This poses critical conditions in particular for the layer located closer to the interaction point, at about 3~cm, where hit rates up to 3~GHz/cm$^2$ are foreseen. To maintain high spatial resolution and two-tracks separation pixel unit cells of 2500~$\mu$m$^2$, smaller than those in use in the current detector, will be required. To improve the radiation tolerance the usage of 100-150~$\mu$m thick sensors is envisaged. As a consequence of the reduced signals, low-noise performance analogue electronics, allowing threshold as low as 1-2 ke minimum detectable charge, will be needed. In this document we investigate, using an ad-hoc version of the CMSSW software, the spatial resolution obtained in the first layer of the Inner Pixel detector for a set of plausible options for the parameters described above. The results presented should be considered as qualitative as relevant effects, as the inclusion of the radiation damage, are not taken into account.