Proposal to study pi- p --> pi0n and pi- p --> eta n at high energy Page: 27 of 37
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The main difference between the two dispersion calculations I and II
sets in around 100 BeV. At that point, the logarithmic part of Re A
in case II begins to dominate. Instead of going to zero, a+(V) becomes
positive and increases, while a_(v) turns over and becomes more negative.
The CEX forward cross section begins to rise again. On an absolute scale,
both effects are small. We should be able to see the CEX forward cross
section flattening, but for the real part to dominate the amplitude we
will need fantastically high energies. By that time, a new physics may
very well set in. It was pointed out in Ref. 2, as well as in Ref. 5,
that if Re A/Im A grows logarithmically, then one has to have the for-
ward elastic peak shrink like log s to avoid a conflict with unitarity.
Strictly speaking, such a conflict would arise only at such large values
of v that the whole problem looks rather academic. Nevertheless, the same
conclusion about the shrinkage arises of course from the assumption that
el does not rise with energy, which might very well be the case.
Finally, a word about errors and low-energy behavior. The cross
sections are accurate to about 1%. This leads to errors of approximately
0.003 in a+(v). A change in a above cutoff causes a bigger
correction. Varying the high-energy cross sections above 30 BeV does not
change the low-energy (v < 4 BeV) dispersion calculations. There, our
results agree with those of Ref. 3.
IV. REAL PARTS OF K-p AMPLITUDES
We calculated the real parts of K-p forward scattering amplitudes in
the same way as for ic p. The data between threshold and v = 3.3 BeV were
slightly smoothed. Above that point, the following fit was made:
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Tollestrup, A.V.; Walker, R.L. & /Caltech. Proposal to study pi- p --> pi0n and pi- p --> eta n at high energy, report, June 1, 1970; Batavia, Illinois. (digital.library.unt.edu/ark:/67531/metadc887653/m1/27/: accessed November 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.