Baryon Exchanges fromu-Channel Finite-Energy Sum Rules

Abstract

Finite-energy sum rules (FESR) are used to study the $u$-channel baryon Regge exchanges in $\pi N$ and $\mathrm{KN}$ elastic scattering. The low-energy integrals over the $\overline{N}N\to \pi \pi$ and $\overline{N}N\to \overline{K}K$ cuts are evaluated by resonance saturation, including $\sigma \left(0^{+}\right)$, $\rho \left(1^{-}\right)$, $\omega \left(1^{-}\right)$, $A_2\left(2^{+}\right)$, $f^0\left(2^{+}\right)$, $\rho \left(3^{-}\right)$, and $\omega \left(3^{-}\right)$ meson poles. The integral over the $\pi N\to \pi N$ cut is evaluated in both the phase-shift and the resonance-dominance approximation, whereas resonance saturation only is used for the $\overline{K}N\to \overline{K}N$ and $\mathrm{KN}\to \mathrm{KN}$ discontinuities. The FESR integrals thus obtained are in reasonable agreement with existing Regge-exchange fits to high-energy scattering data. In particular, the following interesting results are obtained from the FESR: (i) The $N_{\alpha }$ and ${\Lambda }_{\alpha }$ exchange amplitudes change sign at $u$ values appropriate to $\alpha =-\frac{1}{2}$, as required by conventional Regge theory. (ii) The spin dependence of the $N_{\alpha }$ and ${\Lambda }_{\alpha }$ exchange amplitudes is predicted in the scattering region. (iii) The $\Lambda$ exchange contributions dominate over $\Sigma$ exchange contributions in $\mathrm{KN}$ backward elastic scattering.