Search for $B^0_s \to \mu^+ \mu^-$ and $B^0_d \to \mu^+ \mu^-$

Within the Standard Model (SM), the two flavour changing neutral current transitions, $B^0_s \to \mu^+ \mu^-$ and $B^0_d \to \mu^+ \mu^-$, occur only via loop diagrams and are helicity suppressed. These processes are therefore very rare with branching fractions ($\mathcal{B}$) predictions [1] of $\mathcal{B}(B^0_s \to \mu^+ \mu^-)$=(0.32$\pm$ 0.2)x 10$^{-8}$ and $\mathcal{B}(B^0_d \to \mu^+ \mu^-)$=(0.010$\pm$0.001)x 10$^{-8}$. These precise predictions allow the SM to be tested against New Physics scenarios, for example the Minimal Supersymmetric SM [2], in which $\mathcal{B}$ are significantly enhanced. The most restrictive experimental preliminary limits obtained by CDF [4]} are: $\mathcal{B}(B^0_s \to \mu^+ \mu^-)$ <4.3 x 10$^{-8}$ and $\mathcal{B}(B^0_d \to \mu^+ \mu^-)$ <0.76 x 10$^{-8}$ at 95% confidence level (C.L.). The 37 pb$^{-1}$ of data collected by the LHCb detector [3] already allow for upper limits to be set close to the world best ones.