  This paper discusses the dilaton, K\"ahler moduli and hidden sector matter
chiral superfields of heterotic $M$-theory vacua in which the hidden sector
gauge bundle is chosen to be a line bundle with an anomalous U(1) structure
group. For simplicity of notation, the theory is compactified on a Calabi-Yau
threefold with $h^{1,1}=1$, although all methods and results apply to more
general heterotic compactifications. After introducing a non-perturbative
$F$-term potential and coupling to supergravity, the canonically normalized
scalar and fermion mass eigenstates, evaluated around a fixed supersymmetry
breaking vacuum, are computed and the explicit expressions for their masses
presented. In addition, the relevant couplings of these eigenstates to
themselves and to chiral matter in the observable sector are evaluated. The
decay rates of generic observable sector scalars into both moduli and hidden
sector matter scalars and fermions are then presented. This opens the door to
explicit calculations of the decay of an observable sector cosmological
inflaton into moduli and hidden sector dark matter candidates. Finally, an
explicit flux and gaugino condensate induced non-perturbative superpotential is
introduced which is shown to stabilize three of the four real components of the
moduli fields.
