Searches for lepton number violation, and flavour violation beyond the Yukawa couplings at LHCb

The Standard Model does not describe several phenomena, such as gravity and dark matter, and therefore is an incomplete description of nature. This demands the existence of new physics beyond the Standard Model. Two searches for new physics are presented in this thesis, along with a sensitivity study for a third analysis sensitive to new physics. The $\nu MSM$ model motivates a search for lepton number violation using $B^{+} \rightarrow h^{-} \mu^{+} \mu^{+}$ decays, where $h = (\pi, K$). No $B^{+} \rightarrow h^{-} \mu^{+} \mu^{+}$ candidates are seen in $\sim36$ pb$^-1$ of LHCb data and limits are set of $ B (B^{+} \rightarrow K^{-} \mu^{+} \mu^{+})$ < 4.1 x $10^{-8}$ and $ B (B^{+} \rightarrow \pi^{-} \mu^{+} \mu^{+})$ < 4.4 x 10$^{-8}$ at 90% C.L. These improve the previous best limits by a factor 40 and 30, respectively. Using $\sim1$ fb$^{-1}$ of LHCb data, the $B^{+} \rightarrow \pi^{+} \mu^{+} \mu^{-}$ decay is observed for the first time with $5.2\sigma$ significance. This is the first $b \rightarrow d \mu^+ \mu^-$ transition to be observed. The $B^{+} \rightarrow \pi^{+} \mu^{+} \mu^{-}$ branching fraction is measured to be $(2.3~\pm~0.6~ (stat)~\pm~0.1~(syst))\times 10^{-8}$. The ratio of branching fractions between $B^{+} \rightarrow \pi^{+} \mu^{+} \mu^{-}$ and $B^{+} \rightarrow K^{+} \mu^{+} \mu^{-}$ is measured to be $0.053~\pm~0.014~(stat)~\pm~0.001~(syst)$, and this is used to determine a value of the ratio of quark mixing matrix elements $|V_{td}|/|V_{ts}|$ = $0.266 ~\pm~0.035~(stat)~\pm~0.003~(syst)$. All of these results are compatible with the Standard Model expectations. Previous measurements of the ratio of $B \rightarrow D^{(*)} \tau^+ \nu$ and $B \rightarrow D^{(*)} \mu^+ \nu$ branching fractions exceed the Standard Model expectations by more than $3\sigma$, combining $D$ and $D^{*}$. These decays are challenging to measure at a hadron collider, due to the presence of neutrinos in the final state. A sensitivity study is presented for a measurement of the ratio of $B^0 \rightarrow D^{*} \tau^+ \nu$ and $B^0 \rightarrow D^{*} \mu^+ \nu$ branching fractions at LHCb. This study includes a novel fit method, and two new algorithms which enable the backgrounds to be controlled, and control samples to be isolated. The estimated uncertainty on $R_{D^{*}}$, including the largest systematic uncertainties, is $\sim8\%$, competitive with the $9\%$ uncertainty on the present best measurement of $R_{D^{*}}$.