Evidence on Duality and Exchange Degeneracy from Finite-Energy Sum Rules:K−n→π−Λandπ+n→K+Λ

Using finite-energy sum rules for the reactions $K^{-}n\to {\pi }^{-}\Lambda$ and ${\pi }^{+}n\to K^{+}\Lambda$, we determine the effective "pole" parameters of the $K^{*}$ and $K^{**}$ Regge trajectories from a knowledge of the low-energy resonances and their couplings. The resonance parameters and the $\frac{D}{\mathrm{}(D+F)\mathrm{}}$ ratio for the ${\mathrm{½}}^{+}$ baryon octet are varied somewhat to test the sensitivity of the high-energy predictions; ${\mathrm{½}}^{+}$ octet couplings within the range of values found empirically in other reactions are preferred in our solution. We find that the $s$-channel resonances in $K^{-}n\to {\pi }^{-}\Lambda$ do add in such a way as to produce predominantly real amplitudes at high energies as predicted by duality diagrams. We find, however, that these predictions are not satisfied exactly. Although the phases of both $A^{\prime }$ and $B$ are small and independent of $t$ for $\mathrm{}\left|t\right|<0.5\mathrm{} {\left(\frac{\mathrm{GeV}}{c}\right)}^2$, the residues of the even- and odd-signature Regge poles are closely exchange-degenerate only for the $B$ amplitudes, and not for the $A^{\prime }$ amplitudes, thereby allowing an appreciable polarization for $K^{-}n\to {\pi }^{-}\Lambda$ as is observed experimentally. The Regge-pole parameters determined from the sum rules give a good fit to the reaction $K^{-}n\to {\pi }^{-}\Lambda$ over a wide range of energies, whereas they are unable to fit ${\pi }^{+}n\to K^{+}\Lambda$ at intermediate energies. Comparison of the resonance contributions to $K^{-}n\to {\pi }^{-}\Lambda$ and ${\pi }^{+}n\to K^{+}\Lambda$ shows that "peripheral" resonances dominate the sum rules in the first reaction, while "nonperipheral" states are important in the second. By supposing that "peripheral" resonances are dual to the leading Regge singularities in the $t$ channel, while "nonperipheral" resonances are dual to lower-lying singularities, we are led to a rationale of why the simple model of two effective Regge poles is adequate for $K^{-}n\to {\pi }^{-}\Lambda$ even at intermediate energies, but inadequate there for ${\pi }^{+}n\to K^{+}\Lambda$.