Measurement of the Relative Branching Fractions of Charmless Three-Body $ B^{+}$ Decays at LHCb

This thesis documents a measurement of the relative branching fractions of the charmless three-body decays $B^+ \to K^+ K^+ K^-$, $B^+ \to K^+ \pi^+ K^-$, $B^+ \to K^+ \pi^+ \pi^-$, and $B^+ \to \pi^+ \pi^+ \pi^-$, performed using the dataset collected by the LHCb experiment in the years 2011 and 2012 and corresponding to a total integrated luminosity of $3\,\mathrm{fb}^{-1}$. The branching fractions relative to that of the $B^+ \to K^+ K^+ K^-$ decay are determined to be \begin{align*} \mathcal{B}(B^+ \to K^+ \pi^+ K^-)\,/\,\mathcal{B}(B^+ \to K^+ K^+ K^-) &= 0.1521 \pm 0.0040 ~\text{(stat)} \pm 0.0087~\text{(syst)} \,,\\ \mathcal{B}(B^+ \to K^+ \pi^+ \pi^-)\,/\,\mathcal{B}(B^+ \to K^+ K^+ K^-) &= 1.714\phantom{0} \pm 0.011~\text{(stat)} \phantom{0} \pm 0.061~\text{(syst)} \,,\\ \mathcal{B}(B^+ \to \pi^+ \pi^+ \pi^-)\,/\,\mathcal{B}(B^+ \to K^+ K^+ K^-) &= 0.501\phantom{0} \pm 0.005 ~\text{(stat)} \phantom{0} \pm 0.015 ~\text{(syst)} \,. \end{align*} The branching fractions relative to the other decay channels are also determined. These measurements show a significant improvement in precision over the current world average values.