Studies of the rare decays ${B^{0}(B^{0}_{s}) \rightarrow \pi^{+} \pi^{-} \mu^{+} \mu^{-}}$ with the LHCb experiment

Flavour changing neutral currents only occur at loop level within the Standard Model of particle physics. This makes these processes extremely rare and potentially sensitive to interference with currently undiscovered physics that lies beyond the scope of the Standard Model. To date, tensions with the Standard Model have been found in the studies of $b \rightarrow s l^{+} l^{-}$ processes at large hadronic recoil. This thesis presents a branching fraction analysis of the $b \rightarrow d(s) \mu^{+}\mu^{-}$ processes $B^{0}(B^{0}_{s}) \rightarrow \pi^{+} \pi^{-} \mu^{+} \mu^{-}$ using the data recorded by the LHCb collaboration during Run 1 and Run 2 of the Large Hadron Collider. The presence of the $B^{0}(B^{0}_{s}) \rightarrow \pi^{+} \pi^{-} \mu^{+} \mu^{-}$ decays are confirmed with a statistical significance in excess of 12 standard deviations. The branching fraction of the $B^{0} \rightarrow \pi^{+} \pi^{-} \mu^{+} \mu^{-}$ process is measured to be $(2.35 \pm0.26\pm0.18) \times 10^{-8}$ while the branching fraction of the $B^{0}_{s} \rightarrow \pi^{+} \pi^{-} \mu^{+} \mu^{-}$ process is measured to be $(5.35 \pm0.64\pm0.24) \times 10^{-8}$ where the first uncertainties are statistical and the second systematic. In addition, this thesis presents for the first time the branching fraction measurements for the decay modes in two separate regions of the invariant mass of the dimuon pair, $q^{2}$. The $B^{0}$ decay is observed for the first time at both large and small hadronic recoil while the $B^{0}_{s}$ decay is observed for the first time at large hadronic recoil.