Sleptonium at the linear collider and the slepton co-next-to-lightest supersymmetric particle scenario in gauge mediated symmetry breaking models

We discuss the possibility of formation and subsequent detection of a supersymmetric bound state composed of a slepton--antislepton pair at the next linear collider. The Green function method is used within a non-relativistic approximation to estimate the threshold production cross-section of the $2P$ bound state. The parameter space of Gauge Mediated Symmetry Breaking (GMSB) models allow a particular scenario in which a charged slepton ($\widetilde{e}_R, \widetilde{\mu}_R$ or $\widetilde{\tau}_1$) is the NLSP. Within this scenario the produced $2P$ bound-state decays, through a dipole transition, into the $1S$ ground-state with branching ratio $\approx 100% $ emitting a very soft ($\approx 1 $ MeV) photon which goes undetected. The spectroscopy of the $1S$-state shows that it decays into two photons with $Br \approx 0.5$ up to $m_{NLSP} \approx 1$ TeV. Thus NLSP sleptonium threshold production gives rise to the signal $e^+e^- \to 2P \to 1S + "soft \gamma " \to \gamma \gamma$ which when compared with the standard model two-photon process ($ e^+e^- \to \gamma \gamma$) has a statistical significance ($SS$=signal/noise) which, at an energy offset from threshold of E=20 GeV, goes from $SS=11$ to $SS=2$ when the mass of the NLSP ranges in the interval $[100,200]$ GeV.