Scattering (Wave Optical Model)
(click on the above title for a GIF version of the notes)
The cross section for any distinguishable scattering process have a cross section less than
Examples:
(1) e+e- annihilation into point fermion-antifermion pair
No problem for muons (Qm=1) or quarks (Qu= Qc= Qt=2/3, Qd= Qs= Qb=1/3), but what if a point fermion existed with charge Qf=475?
Answer: Perturbation theory would break down, and the QED cross section would be reduced below the unitarity bound.
(2) billiard balls (or neutron-proton scattering at high energies) have a essentially constant total cross section
This violates unitarity if pcm>2/R!
Answer:: In billiard ball scattering, the number of orbital angular momentum states is proportional to s, so the constant cross section is maintained because the number of scattering states increases at a rate which cancels out the reduction in the maximum cross section per scattering state.
(3) neutrino-proton interactions
This cross section formula cannot be true for c.o.m. energies above 150 GeV!
A Yukawa force mediated by a massive particle
gives a cross section
in the low energy limit. In the high energy limit (corresponding to x very small) this potential is just a colomb-like potential and the cross section turns over and falls as 1/s just like the e+e- annihilation cross-section. The boson must have a mass less than 150 GeV (Prediction!), so that the turn-over occurs before the unitarity violation would occur.
(4) WW scattering violates unitarity at about 1.5 TeV
Theoretical solution: Higgs or supersymmetry or W substructure or …
Experimental solution: Build the biggest accelerator you can convince your government(s) to afford (SSC - nope, LHC yes), and see what happens.