Measurement of the $B^{+}\rightarrow \rho^{0} \mu^{+} \nu_{\mu}$ differential branching fraction

The tension between different determinations of the CKM matrix element $|V_{ub}|$ using measurements of either exclusive or inclusive semileptonic decays represents a long-standing puzzle in flavour physics. To resolve this, measurements of new exclusive channels as well as more precise measurements are needed. This thesis presents the first measurement of the exclusive semileptonic decay $B^{+}\to \rho^{0} \mu^{+} \nu_{\mu}$ at the LHCb experiment, using data collected during 2018. Profiting from the large number of $B$ mesons produced in $pp$ collisions at the LHC, a precise measurement of the $B^{+}\to \rho^{0} \mu^{+} \nu_{\mu}$ differential branching fraction as a function of the dilepton invariant mass squared is performed relative to the decay $B^{+}\to \bar{D^{0}} \mu^{+} \nu_{\mu}$ with $\bar{D^{0}} \to \pi^+ \pi^-$. This measurement will be used to obtain a determination of the $|V_{ub}|$ matrix element from the $B^{+}\to \rho^{0} \mu^{+} \nu_{\mu}$ channel. All steps leading to the final measurement are documented and the current precision of the result is given, with the numerical values of the result still kept blind to avoid accidental biases before the final publication. The measurement presented in this thesis does not only pave the way for obtaining $|V_{ub}|$, but also for measuring the full differential branching fraction that expresses the rich helicity structure of the $B^{+}\to \rho^{0} \mu^{+} \nu_{\mu}$ decay and provides more variables to test the Standard Model and to probe new physics.