Study of the decay B+ → K+π0 at LHCb and mechanical development for the design of the Upstream Tracker

The LHCb experiment at the Large Hadron Collider (LHC) is designed to measure the properties of particles containing charm ($c$) and bottom ($b$) quarks. This dissertation documents two major studies I have completed, one analyzing data collected by the LHCb detector, and another contributing to the design and development of an extensive upgrade to the detector. The pattern of CP asymmetry measurements of the \mbox{$B\rightarrow K\pi$} family of decays deviates from expectations derived from the standard model (SM), a contradiction known as the ``$\text{K}\pi$ puzzle.'' The present size of the experimental errors are such that more precise measurements in the $B^{+}\rightarrow K^{+}\pi^0$ decay channel are especially important. An analysis of the $B^{+}\rightarrow K^{+}\pi^0$ decay using data collected during Run 1 is performed. Despite low reconstruction and trigger efficiencies and enormous combinatorial backgrounds, a signal is found with a statistical significance of $3.7\sigma$. This achievement has led to the creation of a dedicated $B^{+}\rightarrow K^{+}\pi^0$ trigger, and has inspired the creation of a number of dedicated triggers for decay modes with similar topologies. A preliminary analysis of data collected during Run 2 demonstrates that the new trigger is a major success, with excellent prospects for making the world's best measurements in the $B^{+}\rightarrow K^{+}\pi^0$ decay channel using the entire Run 2 data set. Run 2 of the LHC will conclude at the end of 2018, and will be followed by Run 3, scheduled to begin in early 2021. In the interim, the LHCb detector will be upgraded to be read-out in real-time at 40 MHz, and to withstand the radiation damage associated with collecting 50 fb$^{-1}$ of integrated luminosity by the conclusion of Run 4. A key part of this upgrade is the design and construction of a new silicon-strip tracking detector---the upstream tracker (UT). Regions at the periphery of the UT suffer from severe electrical and mechanical constraints, making a high-fidelity CAD model a critical element of the design process. The result is a mechanical integration solution that is entirely non-trivial, and which has had significant influences on the UT design. This solution and the constraints that influence it are shown in detail.