Towards a simultaneous measurement of $W$ boson mass and production properties with the CMS detector

The Compact Muon Solenoid (CMS) is one of the four main detectors which operate at the Large Hadron Collider (LHC) at CERN, Geneva. The large amount of events delivered by LHC and the excellent detector calibration achieved turned CMS from discovery to a precision experiment. The $W$ boson mass ($m_W$) is a precision electroweak observable of major interest. The combination of $m_W$ measurement from ATLAS and Tevatron has a precision of 12 MeV and it is limited by systematic uncertainty. The $m_W$ theoretical prediction from the electroweak fit has an uncertainty of 7 MeV and there is a tension of $1.5\sigma$ with respect to the experimental result. One of the goals of CMS for Run 2 is to perform a measurement of $m_W$ with precision at the level of the theoretical prediction. The $m_W$ is measured using the ${W^\pm\to\mu^\pm\nu}$ channel, because of the excellent muon reconstruction performance of CMS, in terms of efficiency, momentum calibration and background discrimination power. When $m_W$ is extracted with a template fit from muon transverse momentum ($p_T^\mu$) distribution, the limited knowledge of the $W$ production model (transverse momentum $q_T^W$, rapidity $Y_W$ and polarization) induces a systematic uncertainty the value of $m_W$. In particular, the finite $\eta$ acceptance of the CMS detector introduces a dependence of $p_T^\mu$ distribution on the $Y_W$. A precise measurement of $Y_W$ spectrum and the $W$ polarization can constrain these uncertainties, as demonstrated by a recent CMS result. The content of this thesis goes beyond the previously described perspective. The entire statistics collected by CMS during Run 2 of LHC allows to extract the value of $m_W$ \textit{simultaneously} to $q_T^W$, $Y_W$ and polarization spectra, to finally obtain a measurement independent from production model assumptions. The $m_W$, $Y_W$, $q_T^W$ and 5 angular coefficients can be extracted from the muon kinematics with a $\eta^\mu \times p_T^\mu$ template fit in bins of $Y_W$, $q_T^W$ for each angular coefficient, unfolding the underlying boson distribution from the charged lepton kinematics alone. The muon is the only reconstructed particle in the final state and so the control of the momentum scale calibration is one of the central ingredients to reach the required precision. To deliver a precision on the muon scale at $10^{-4}$ level, the $Z$ boson mass must be used as a standard candle. The momentum distributions of the colliding partons (Parton Distribution Functions, PDFs) produce a distortion on the $Z$ boson lineshape, shifting in particular the mode of the invariant mass distribution. This effect can in principle produce a bias in the momentum scale evaluation and has been studied in detail and reported in the thesis. Despite the $W^\pm\to \mu^\pm\nu$ decay channel having very high purity, two possible sources of background must be taken into account. The first is composed of electroweak processes which fall in the signal kinematic acceptance because they mimic the single muon signature. They are subtracted using the Monte Carlo (MC) because of the accurate description of these processes in the simulation. The second and most relevant source of background, are the energetic muons from multi-jet production which falls in signal region (QCD background). A data-driven background estimation approach, called \textit{fake rate} method, has been chosen to assess the QCD yield in the signal region because the MC description of multi-jet production spectra has limited precision. The analysis framework is challenging due to the high level of complexity of the template fit (which has order of $10^4$ templates) and the large dataset involved for signal and background estimation. A custom framework, based on RDataFrame ROOT package has been developed, which runs on NanoAOD ntuples, the state-of-the-art compressed CMS data format. Unfortunately, at the time of this work, the required calibrations are not available for the entire Run 2 dataset and only the data and MC samples corresponding to 2016 data taking can be used. With this reduced sample, the results are therefore presented as a proof of feasibility for a future measurement. The final result on $Y_W$, $q_T^W$ and angular coefficients has not been extracted from data, but the analysis has been performed on the CMS simulation, keeping the real parameters of interest blinded. The results of this work are currently limited by the statistical uncertainty only. This approach demonstrates the capability to convert a systematically limited measurement into a statistically limited measurement. In particular with this strategy it is possible to measure the $W$ boson transverse momentum spectra with a granularity of 2 GeV, with 4%-8% precision. In this thesis it has been also shown how the same fitting framework has the capability to simultaneous fit $m_W$ and the $W$ production properties, reducing the systematic uncertainty related to production properties well below 10 MeV.