**Scattering (Wave Optical Model)**

(click on the above title for a GIF version
of the notes)

**Unitarity**

The universe never fails, but
our theories do!
The cross section for any distinguishable
scattering process have a cross section less than

s_{max}
= 4p/p_{cm}^{2}

= 16p/s
= (19.6 mb GeV^{2})/s (in high energy limit)

Examples:

(1)
e^{+}e^{-} annihilation into point fermion-antifermion
pair

No problem for muons (Q_{m}=1)
or quarks (Q_{u}= Q_{c}= Q_{t}=2/3, Q_{d}=
Q_{s}= Q_{b}=1/3), but what
if a point fermion existed with charge Q_{f}=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

s_{billiards}
= 4pR^{2}
This violates unitarity if p_{cm}>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

s(n_{m}p)
= 0.4x10^{-38}cm^{2}/GeV^{2}
= s_{max}
s^{2}/(150 GeV)^{4}
This cross section formula cannot be
true for c.o.m. energies above 150 GeV!

A Yukawa force mediated by a massive
particle

V(x) = (-g^{2}/x) exp(-xM_{B})
gives a cross section

s = 4pg^{4}s/M_{B}^{4}
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.

#### Copyright 1998 David Bailey, University of Toronto