  The Lee-Nauenberg theorem is a fundamental quantum mechanical result which
provides the standard theoretical response to the problem of collinear and
infrared divergences. Its argument, that the divergences due to massless
charged particles can be removed by summing over degenerate states, has been
successfully applied to systems with final state degeneracies such as LEP
processes. If there are massless particles in both the initial and final
states, as will be the case at the LHC, the theorem requires the incorporation
of disconnected diagrams which produce connected interference effects at the
level of the cross-section. However, this aspect of the theory has never been
fully tested in the calculation of a cross-section. We show through explicit
examples that in such cases the theorem introduces a divergent series of
diagrams and hence fails to cancel the infrared divergences. It is also
demonstrated that the widespread practice of treating soft infrared divergences
by the Bloch-Nordsieck method and handling collinear divergences by the
Lee-Nauenberg method is not consistent in such cases.
