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$R$-parity Violating Decays of Bino Neutralino LSPs at the LHC

The $R$-parity violating decays of Bino neutralino LSPs are analyzed within the context of the $B-L$ MSSM "heterotic standard model". These LSPs correspond to statistically determined initial soft supersymmetry breaking parameters which, when evolved using the renormalization group equations, lead to an effective theory satisfying all phenomenological requirements; including the observed electroweak vector boson masses and the Higgs mass. The explicit RPV decay channels of these LSPs into standard model particles, the analytic and numerical decay rates and the associated branching ratios are presented. The analysis of these quantities breaks into two separate calculations; first, for Bino neutralino LSPs with mass larger than $M_{W^{\pm}}$ and, second, when the Bino neutralino mass is smaller than the electroweak scale. The RPV decay processes in both of these regions is analyzed in detail. The decay lengths of these RPV interactions are discussed. It is shown that for heavy Bino neutralino LSPs the vast majority of these decays are "prompt", although a small, but calculable, number correspond to "displaced" decays of various lengths. The situation is reversed for light Bino LSPs, only a small number of which can RPV decay promptly. The relation of these results to the neutrino hierarchy--either normal or inverted--is discussed in detail.