First differential measurements of t$Z$q production and luminosity determination using $Z$ boson rates at the LHC

In this thesis, measurements are presented using proton-proton collision data at $\sqrt{s}$ = 13 TeV collected by the CMS experiment in the years 2016, 2017, and 2018, corresponding to an integrated luminosity of 138 fb$^{-1}$. New results on single top quark production in association with a $Z$ boson (t$Z$q) are presented. The inclusive cross section is measured with unprecedented accuracy. Differential cross sections are presented for the first time, measuring 11 kinematic observables that are sensitive to modeling aspects and possible effects of new physics. Absolute and normalized distributions are obtained for experiment independent parton- and particle-level definitions. The spin asymmetry of the top quark, a measure of its polarization, is measured for the first time in the t$Z$q process. The results are obtained utilizing binned maximum likelihood fits which directly account for backgrounds and systematic uncertainties. The results are compared to theory predictions with different modeling assumptions on the parton content of the proton. No significant deviations from standard model predictions are observed. Using the same data, the measurement of $Z$ boson production is presented as a means to determine the integrated luminosity. It is demonstrated that the measurement of $Z$ boson rates constitutes a precise and complementary method to monitor and transfer the luminosity between data sets and that the method leads to an improved measurement of the integrated luminosity.