  The hidden sector of heterotic $M$-theory vacua whose gauge bundle contains
an anomalous $U(1)$ factor is discussed in detail. The mathematical formalism
for computing the inhomogeneous transformation of the dilaton and K\"ahler
moduli axions under an anomalous $U(1)$ transformation is presented. Along with
matter multiplets, which transform homogeneously under $U(1)$, the relevant
part of the $U(1)$ invariant low energy hidden sector Lagrangian is presented
and analyzed. A detailed mathematical formalism is given for rotating these
field to a new basis of chiral superfields with normalized kinetic energy and a
diagonal mass matrix. It is shown that the scalar and fermionic components of
one such chiral superfield give rise to a massive $U(1)$ vector superfield,
whose mass is composed of both anomalous and Higgs contributions associated
with the inhomogeneous and homogeneous transformations respectively. Two
explicit examples are presented, with vanishing and non-zero Fayet-Iliopoulos
term respectively. The canonically normalized hidden sector Lagrangian given
here is restricted to gauge interactions only. A study of higher order
interactions of the moduli and matter multiplets, as well as the coupling to
supergravity, will be presented elsewhere.
