Measurements of the branching fractions of $\Lambda_{c}^{+} \rightarrow p \pi^{-} \pi^{+}$, $\Lambda_{c}^{+} \rightarrow p K^{-} K^{+}$, and $\Lambda_{c}^{+} \rightarrow p \pi^{-} K^{+}$

The ratios of the branching fractions of the decays $\Lambda_{c}^{+} \rightarrow p \pi^{-} \pi^{+}$, $\Lambda_{c}^{+} \rightarrow p K^{-} K^{+}$, and $\Lambda_{c}^{+} \rightarrow p \pi^{-} K^{+}$ with respect to the Cabibbo-favoured $\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+}$ decay are measured using proton-proton collision data collected with the LHCb experiment at a 7 TeV centre-of-mass energy and corresponding to an integrated luminosity of 1.0 fb$^{-1}$: \begin{align*} \frac{\mathcal{B}(\Lambda_{c}^{+} \rightarrow p \pi^{-} \pi^{+})}{\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})} & = (7.44 \pm 0.08 \pm 0.18)\,\%, \frac{\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} K^{+})}{\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})} &= (1.70 \pm 0.03 \pm 0.03)\,\%, \frac{\mathcal{B}(\Lambda_{c}^{+} \rightarrow p \pi^{-} K^{+})}{\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})} & = (0.165 \pm 0.015 \pm 0.005 )\,\%, \end{align*} where the uncertainties are statistical and systematic, respectively. These results are the most precise measurements of these quantities to date. When multiplied by the world-average value for $\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$, the corresponding branching fractions are \begin{align*} \mathcal{B}(\Lambda_{c}^{+} \rightarrow p \pi^{-} \pi^{+}) &= (4.72 \pm 0.05 \pm 0.11 \pm 0.25) \times 10^{-3}, \mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} K^{+}) &= (1.08 \pm 0.02 \pm 0.02 \pm 0.06) \times 10^{-3}, \mathcal{B}(\Lambda_{c}^{+} \rightarrow p \pi^{-} K^{+}) &= (1.04 \pm 0.09 \pm 0.03 \pm 0.05) \times 10^{-4}, \end{align*} where the final uncertainty is due to $\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$.