  Generalized dispersion relations are discussed for unphysical particles, e.g.
confined degrees of freedom that are not present in the physical spectra but
can give rise to observable bound states. While in general the propagator of
the unphysical particles can have complex poles and cannot be reconstructed
from the knowledge of the imaginary part, under reasonable assumptions the
missing piece of information is shown to be in the rational function that
contains the poles and must be added to the integral representation. For pure
Yang-Mills theory, the rational part and the spectral term are identified in
the explicit analytical expressions provided by the massive expansion of the
gluon propagator. The multi particle spectral term turns out to be very small
and the simple rational part provides, from first principles, an approximate
propagator that is equivalent to the tree-level result of simple
phenomenological models like the refined Gribov-Zwanziger model.
