Search For the Rare Decay $K_{L}\rightarrow\pi^{0}\pi^{0}\mu^{+}\mu^{-}$

Using data collected by the KTeV Experiment at Fermi National Accelerator Laboratory in Batavia, Illinois, this study will be the first experimental analysis of $K_{L}\rightarrow\pi^{0}\pi^{0}\mu^{+}\mu^{-}$. Although this decay mode is possible within the Standard Model, it is limited to a very narrow band of phase space. The HyperCP Experiment has recently observed three $\Sigma^{+}\rightarrow{\it p}\mu^{+}\mu^{-}$ events within a narrow dimuon mass range of 213.8 MeV/$c^{2}$ to 214.8 MeV/$c^{2}$. This suggests that the process occurs via a neutral intermediary particle, $\Sigma^{+}\rightarrow{\it p}X^{0}\rightarrow{\it p}\mu^{+}\mu^{-}$, with an $X^{0}$ mass of 214.3 MeV/$c^{2}$$\pm$0.5 MeV/$c^{2}$. Since the $X^{0}$ has a light mass and a low interaction probability, then it is not feasible within the Standard Model. However, the $X^{0}$ could be explained by a theory known as the ``Next-to-Minimal Supersymmetric Standard Model'' (NMSSM). In NMSSM, there are seven Higgs bosons and theorists believe that the $X^{0}$ may be the lightest of this group. Recent theoretical predictions suggest that the decay mode $K_{L}\rightarrow\pi^{0}\pi^{0}\mu^{+}\mu^{-}$ can also occur via the aforementioned neutral intermediary particle: $K_{L}\rightarrow\pi^{0}\pi^{0}X^{0}\rightarrow\pi^{0}\pi^{0}\mu^{+}\mu^{-}$. Therefore, in addition to a Standard Model measurement, the search for $K_{L}\rightarrow\pi^{0}\pi^{0}\mu^{+}\mu^{-}$ is also carried out in an effort to address the viability of $X^{0}$ in explaining the HyperCP phenomena.