  The problem of the origin of the mu parameter in the Minimal Supersymmetric
Standard Model can be solved by introducing singlet supermultiplets with
non-renormalizable couplings to the ordinary Higgs supermultiplets. The
Peccei-Quinn symmetry is broken at a scale which is the geometric mean between
the weak scale and the Planck scale, yielding a mu term of the right order of
magnitude and an invisible axion. These models also predict one or more singlet
fermions which have electroweak-scale masses and suppressed couplings to MSSM
states. I consider the case that such a singlet fermion, containing the axino
as an admixture, is the lightest supersymmetric particle. I work out the
relevant couplings in several of the simplest models of this type, and compute
the partial decay widths of the next-to-lightest supersymmetric particle
involving leptons or jets. Although these decays will have an average proper
decay length which is most likely much larger than a typical collider detector,
they can occasionally occur within the detector, providing a striking signal.
With a large sample of supersymmetric events, there will be an opportunity to
observe these decays, and so gain direct information about physics at very high
energy scales.
