  The vacuum energies corresponding to massive Dirac fields with the boundary
conditions of the MIT bag model are obtained. The calculations are done with
the fields occupying the regions inside and outside the bag, separately. The
renormalization procedure for each of the situations is studied in detail, in
particular the differences occurring with respect to the case when the field
extends over the whole space. The final result contains several constants
undergoing renormalization, which can be determined only experimentally. The
non-trivial finite parts which appear in the massive case are found exactly,
providing a precise determination of the complete, renormalized zero-point
energy for the first time, in the fermionic case. The vacuum energy behaves
like inverse powers of the mass for large masses.
