PION PION SCATTERING
S-wave scattering and the "Universal curve"
D.Morgan and G.Shaw, Nucl.Phys. B10(1969) 261 [40 citations]
D.Morgan and G.Shaw, Phys.Rev. D2(1970)520 [100 citations]
The first paper set up the dispersion theory methods and derived some of
the results on which the conclusions of the second paper were
based. Two main points were addressed:
- I = 0 S-wave phase shifts:
At the time there were four
different families of experimental phase shifts in the energy region of the
rho(770) meson(including a family with a narrow resonances) corresponding to
the so-called Up-Down ambiguities. These were
resolved once and for all with the help of the dispersion theory results in
favour of the family with a slowly increasing phase in the region of
90 degrees. This behaviour has been confirmed and more precisely delineated
by subsequent experiments.
- The Universal Curve:
The most important result emerged on
examining the low energy behaviour of dispersion theory solutions compatible
with the phase shifts in the rho region and reasonable Regge estimates for the
high energy behaviour. For all solutions found, the s-wave scattering lengths
a_0, a_2 with isospin 0 and 2 respectively were found to lie in a narrow
band in the (a_0, a_2) plane as shown below.Furthermore, all
other threshold parameters (p-wave scattering length, s-wave ranges etc.)
were found to be determined with small errors for a given position on this
so-called "universal curve."
This result was subsequently refined and improved by other authors using the
Roy relations and more precise input data.
( For a recent re-evaluation, see
Ollsen.)
It remains basic to the
determination of the low energy parameters of pi-pi scattering from
experiment for comparison with the predictions of chiral symmetry.