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: $P$, $P^{\prime }$, $A_2$, $\rho$, and $\omega$. 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 $\surd s=2\mathrm{}$ GeV and $\surd s=2.15\mathrm{}$ 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 $t=0\mathrm{}$) and $B$, we find that assuming the $\rho$ 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 ($\frac{\nu B}{A}$) for the $A_2$, $P$, $P^{\prime }$, and $\omega$ contributions, where $\nu$ is $\frac{(s-u)}{4M}$, 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 $\omega$ 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 ${\alpha }_{\omega }$ for $-t\lesssim 0.8 {\left(\frac{\mathrm{GeV}}{c}\right)}^2$; we find ${\left(\frac{\nu B}{A}\right)}_{\omega }=+(1-3)\mathrm{}$ for $-t\lesssim 0.6 {\left(\frac{\mathrm{GeV}}{c}\right)}^2$. There is some evidence for an exchange degeneracy between the $\omega$ and the $P^{\prime }$ for this ratio, because we also find evidence for ${\left(\frac{\nu B}{A}\right)}_{P,P^{\prime }}\approx +1\mathrm{}$. There is some evidence for the no-compensation mechanism for the $P^{\prime }$, with ${\alpha }_{P^{\prime }}=0\mathrm{}$ at $-t\sim 0.5 {\left(\frac{\mathrm{GeV}}{c}\right)}^2$, which, however, would make the $\omega$ and $P^{\prime }$ trajectories quite nondegenerate. For the $A_2$, we find $\frac{\nu B}{A}\sim +10\mathrm{}$, which would be expected if the $A_2$ were degenerate with the $\rho$. Our determination of the signs of the spin-flip amplitudes $B$ allows us to predict the $K^{\pm }p$ polarizations semiquantitatively; our results agree with the available $K^{-}p$ 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 ${\rho }^{\prime }$ 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 $g_{\Lambda }^{}{}_{}{}^2$ and $g_{\Sigma }^{}{}_{}{}^2$ for the Born terms, a negligible $Y_1^{}{}_{}{}^{*}\mathrm{}\left(1385\right)\mathrm{}$ 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 $t=0\mathrm{}$.