Angular analysis of $B^0 \rightarrow K^{*0} e^+e^-$ decays with the LHCb detector and upgrade of the electronics of the calorimeters

Flavor changing neutral current processes of the type $b \rightarrow s \gamma$ are forbidden at tree level in the Standard Model (SM). They are thus sensitive to potential New Physics (NP) effects occurring via loop processes which may manifest themselves through the modification of angular observables. In the SM, the photon is predominantly left handed. However several NP theories allow large right-handed currents. The angular analysis of $B^0 \rightarrow K^{*0} e^+e^-$ decays at very low $q^2$ (the dielectron pair invariant mass squared) allows to study the helicity structure of $b \rightarrow s \gamma$ transitions thanks to the dominant contribution from the virtual photon coupling to the dielectron pair at very low $q^2$. This thesis presents the angular analysis of $B^0 \rightarrow K^{*0} e^+e^-$ decays using the full Run 1 and Run 2 proton-proton collision datasets collected between 2011 and 2018 at the LHCb experiment, corresponding to an integrated luminosity of $9.1$ fb$^{-1}$ at centre-of-mass energies ranging from 7 to 13 TeV. The transverse asymmetries in the $q^2$ range $[0.0001, 0.25]$ GeV$^2$/c$^4$ are measured as $A_{T}^{(2)} = 0.106 \pm 0.103 {}^{+0.016}_{-0.017}$, $A_{T}^{Im} = 0.015 \pm 0.102 \pm 0.012$, $A_{T}^{Re} = -0.064 \pm 0.077 \pm 0.015$ and $F_L = 0.044 \pm 0.026 \pm 0.014$. These measurements provide the current world's best constraint on right handed contributions to the photon polarization in $b \rightarrow s \gamma$ transitions. The LHCb detector is undertaking a major upgrade in 2019/2020 to be able to run the experiment at an instantaneous luminosity of $2 \times 10^{33}$ cm$^{-2}$s$^{-1}$. In particular, the electronics of the calorimeters has to be upgraded with new Front-End boards (FEB) to cope with the $40$ MHz readout. This thesis also presents the upgrade of the Low Level Trigger module of the FEB, which is in particular responsible for identifying the maximum transverse energy cluster of each event.