Regge Poles and Finite-Energy Sum Rules for Kaon-Nucleon Scattering

Abstract

Generalized finite-energy sum rules (FESR) for kaon-nucleon scattering are evaluated to determine the $t$ dependence and other properties of the relevant Regge-exchange amplitudes: $P$, ${P}^{\ensuremath{'}}$, ${A}_{2}$, $\ensuremath{\rho}$, and $\ensuremath{\omega}$. The FESR's have been evaluated (a) with the available phase-shift analyses for the low-energy $\mathrm{KN}$ system as input and (b) in the resonance-saturation approximation with all the appropriate resonances of which ${J}^{P}$ is known. For the ${K}^{+}p$ system, the phase-shift analysis of Lea et al. has been used; for the $\overline{K}N$ system, the multichannel effective-range analysis of Kim, and the resonance-plus-background analysis of Armenteros et al. Matching energies of $\ensuremath{\surd}s=2$ GeV and $\ensuremath{\surd}s=2.15$ GeV have been used for the cases (a) and (b), respectively. In terms of the definite helicity flip amplitudes, $A$ (which is the full forward amplitude at $t=0$) and $B$, we find that assuming the $\ensuremath{\rho}$ contribution to be known, the non-spin-flip contributions for the various Regge poles are similar to those deduced from high-energy fits; however, the spin-flip contributions of the high-energy fits are inconsistent with our FESR results. For example, the factorization ratio ($\frac{\ensuremath{\nu}B}{A}$) for the ${A}_{2}$, $P$, ${P}^{\ensuremath{'}}$, and $\ensuremath{\omega}$ contributions, where $\ensuremath{\nu}$ is $\frac{(s\ensuremath{-}u)}{4M}$, is found to have the opposite sign to that used in previous high-energy fits. As far as our results go, the FESR's are consistent with the usual explanation of the crossover phenomenon in terms of a single genuine $\ensuremath{\omega}$ Regge pole, though we cannot regard this conclusion as very strong, because of the poor available input data. We find no evidence of a wrong-signature nonsense zero in ${\ensuremath{\alpha}}_{\ensuremath{\omega}}$ for $\ensuremath{-}t\ensuremath{\lesssim}0.8 {(\frac{\mathrm{GeV}}{c})}^{2}$; we find ${(\frac{\ensuremath{\nu}B}{A})}_{\ensuremath{\omega}}=+(1\ensuremath{-}3)$ for $\ensuremath{-}t\ensuremath{\lesssim}0.6 {(\frac{\mathrm{GeV}}{c})}^{2}$. There is some evidence for an exchange degeneracy between the $\ensuremath{\omega}$ and the ${P}^{\ensuremath{'}}$ for this ratio, because we also find evidence for ${(\frac{\ensuremath{\nu}B}{A})}_{P,{P}^{\ensuremath{'}}}\ensuremath{\approx}+1$. There is some evidence for the no-compensation mechanism for the ${P}^{\ensuremath{'}}$, with ${\ensuremath{\alpha}}_{{P}^{\ensuremath{'}}}=0$ at $\ensuremath{-}t\ensuremath{\sim}0.5 {(\frac{\mathrm{GeV}}{c})}^{2}$, which, however, would make the $\ensuremath{\omega}$ and ${P}^{\ensuremath{'}}$ trajectories quite nondegenerate. For the ${A}_{2}$, we find $\frac{\ensuremath{\nu}B}{A}\ensuremath{\sim}+10$, which would be expected if the ${A}_{2}$ were degenerate with the $\ensuremath{\rho}$. Our determination of the signs of the spin-flip amplitudes $B$ allows us to predict the ${K}^{\ifmmode\pm\else\textpm\fi{}}p$ polarizations semiquantitatively; our results agree with the available ${K}^{\ensuremath{-}}p$ polarization data, while the previous Regge models gave the wrong sign of this polarization. Our new signs for the $B$ amplitudes also improve the agreement of the conventional Regge model with the available ${K}^{+}n$ charge-exchange cross section without invoking a ${\ensuremath{\rho}}^{\ensuremath{'}}$ contribution. On the basis of getting good agreement between the FESR results and the Regge expectations, we are able to choose a particular set of low-energy input data as our favored one: We prefer Kim's coupling constants $g_{\ensuremath{\Lambda}}^{}{}_{}{}^{2}$ and $g_{\ensuremath{\Sigma}}^{}{}_{}{}^{2}$ for the Born terms, a negligible $Y_{1}^{}{}_{}{}^{*}(1385)$ coupling (as also found by Kim), and the nonresonant-type solution IV for the ${K}^{+}p$ phase-shift analysis. We have also considered FESR's for amplitudes with the wrong crossing properties, generalizing the Schwarz superconvergence relations. A simple model to remove the infinities expected in the case of Schwarz FESR's is seen to be in good agreement with the low-energy data, at least at $t=0$.