Search for the rare decays $B^0_{(s)} \rightarrow \mu^+\mu^-$ and $K^0_{\rm S} \rightarrow \mu^+\mu^-$ with 1 fb$^{-1}$ at LHCb

This thesis is devoted to the explanation of the searches for the very rare decays $B^0_{(s)} \rightarrow \mu^+\mu^-$ and $K^0_{\rm S} \rightarrow \mu^+\mu^-$, performed using the 1 fb$^{-1}$ of data taken by LHCb during 2011 from the LHC proton-proton collisions at an energy of $\sqrt{s}$=7 TeV. The most relevant theory aspects concerning these searches will be presented. Furthermore, the LHCb experiment will be described, and some of its most interesting results in the rare decays area reported. The $B^0_{(s)} \rightarrow \mu^+\mu^-$ search has not resulted in any signal excess above background, but has produced world best upper limits in the branching ratios of both channels: ${\cal B}(B^0 \rightarrow \mu^+\mu^-)<1.0 \times 10^{-9}$ and ${\cal B}(B^0_s \rightarrow \mu^+\mu^-)<4.5 \times 10^{-9}$, both at 95% confidence level. Finally, no significant signal excess has been found in the search for $K^0_{\rm S} \rightarrow \mu^+\mu^-$ neither, and again an upper limit on its branching ratio has been set: ${\cal B}(K^0_{\rm S} \rightarrow \mu^+\mu^-)<11.2 \times 10^{-9}$ at 95% confidence level. This upper limit is a factor of 35 better than the previous world best.