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\section{\usemenu{slacpub7205::context::slacpub72050011}{Scattering}}\label{section::slacpub72050011}
We ask the reader at this point to refer back to our section on the
Handy Dandy Formula when needed. There we saw (Eq. 17)
that the unitary scattering amplitude $T(s)$ for systems of
angular momentum zero can be computed at
a single energy if we know the phase shift $\delta (s)$ at that energy.
For the following analysis, it is more convenient to work with
the dimensionless amplitude $a(s)\equiv \sqrt{s(m_a+m_b)^2}T(s)$,which
is related to the tangent of the phase shift by
\begin{equation}
a(s)=e^{i \delta(s)}tan\ \delta (s) = {\tan \delta (s)\over 1 + i\ tan \
\delta (s)} \equiv {t(s)\over 1 + it(s)}
\end{equation}
In a conventional treatment, given a real
interaction potential $V(s,s')=V(s',s)$, $T(s)$ can be obtained by solving
the Lippmann Schwinger equation $T(s,s';z)=V(s,s')$\break
+ $\int ds'' T(s,s'';z)R(s'',s';z)T(s'',s';z)$ with a singular resolvent $R$ and taking
the limit $T(s) = {lim \atop z\rightarrow s+i0^+,s' \rightarrow s}
T(s,s';z)$.
Here we replace this integral equation by an {\it algebraic}
equation for $t(s)$:
\begin{equation}
t(s) = g(s) + g(s)t(s) ={g(s)\over 1  g(s)}
\end{equation}
One can think of this equation as a sequence of scatterings each
with probability $g(s)$ which is summed by solving the equation.
Here $g(s)$ will be our model of a {\it running coupling constant},
which we assume known as a function of energy. We see that if
$g(s_0)=0$ there is no scattering at the energy corresponding to $s_0$,
while if $g(s_0) = +1$, the phase shift is ${\pi \over 2}$ at the corresponding
energy and $a(s_0) = i$; otherwise the scattering is finite.
The above remarks apply in the physical region $s> (m_a +m_b)^2$, where
in the singular case a phase shift of ${\pi \over 2}$ causes the
cross section $4\pi sin^2 \delta/k^2$ to reach the unitarity limit
$4\pi \lambda_0^2$ where $\lambda_0=\hbar / p_0$ is the de Broglie
wavelength at that energy; this is called a resonance and the
cross section goes through a maximum value at that energy. If, as in
Smatrix theory, we analyticly continue our equation below
elastic scattering threshold, the scattering amplitude is real
and the singular case corresponds to a bound state pole in
which the two particles are replaced by a single coherent particle
of mass $\mu$, within which the particles keep on scattering
until some third interaction supplies the energy and momentum
needed to liberate them. There can also be a singularity corresponding
to a repulsion rather than attraction, which is called a ``CDD pole''
in Smatrix dispersion theory\cite{17}. The corresponding situation in the
physical region is a cross section which
never reaches the unitarity limit. To cut a long story short,
these four cases correspond to the four roots of the quartic
equation (Eq. 19) called the handydandy formula, which we repeat here,
replacing the running coupling constant by its value at the singularity
which we call $g_0=g(s_0)$
\begin{equation}
(g_0)^4\mu^2= (m_a+m_b)^2 \mu^2
\end{equation}
Again to cut a long story short, the model for a running coupling
constant which Ed Jones and I are exploring\cite{75}
is simply
\begin{equation}
g_{m_a,m_b;\mu}(s)=\sqrt{ { \pm [(m_a+m_b)^2 \mu^2](m_a+m_b)^2
\over[k^2(s)(m_a+m_b)^2]s} }g_{m_am_b;\mu}(0)
\end{equation}
The singularity at $s=0$ is included only when $m_a$ and $m_b$
have a bound state of zero mass, usually called a quantum.
We have seen that when $m_a=m_e$, $m_b=m_p$ and $\mu = m_H$
the handydandy formula gives the relativistic
Bohr formula for the hydrogen spectrum. Replacing $m_p$ by $m_e$
in the formula gives the corresponding formula for positronium (i.e
the bound state of an electronpositron pair). But for that system,
one can think of the photons produced in electronpositron
annihilation as bound states of the pair with zero rest mass.
This interaction is important in high energy electronpositron
scattering, where it is called ``Bhabha scattering''. Introducing
the $s^{{1\over 2}}$ in this way is supposed to insure that our
theory gives the correct Feynman diagram (and hence cross section)
for this effect, but until we have checked the detailed derivation and
predictions I warn the reader to treat this formula (Eq.42)
as a guess rather than as a result actually derived from the theory.
In my paper at ANPA WEST 13\cite{70}, I started to explore the
connections of this type of scattering theory
to bitstrings {\it and} to Etter's Link Theory by making
the hypothesis that \begin{equation}
tan \delta_{ab} = {\pm ({\bf a}\oplus {\bf b})\cdot
({\bf a}\oplus {\bf b})\over {\bf a}\cdot {\bf b}}
\end{equation}
Unfortunately the details are about a sketchy as presented here, but at
least should provide insight into where I am headed.
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