  Many models of electroweak symmetry breaking predict new particles with
masses at or just beyond LHC energies. Even if these particles are too massive
to be produced on-shell at the LHC, it may be possible to see evidence of their
existence through the use of integral dispersion relations (IDRs). Making use
of Cauchy's integral formula and the analyticity of the scattering amplitude,
IDRs are sensitive in principle to changes in the cross section at arbitrarily
large energies. We investigate some models of new physics. We find that a
sudden, order one increase in the cross section above new particle mass
thresholds can be inferred well below the threshold energy. On the other hand,
for two more physical models of particle production, we show that the reach in
energy and the signal strength of the IDR technique is greatly reduced. The
peak sensitivity for the IDR technique is shown to occur when the new particle
masses are near the machine energy, an energy where direct production of new
particles is kinematically disallowed, phase-space suppressed, or if
applicable, suppressed by the soft parton distribution functions. Thus, IDRs do
extend the reach of the LHC, but only to a window around $M_\chi\sim
\sqrt{s_{_{\rm LHC}}}$.
