A finite-transform sum rule and duality

Abstract

Explicit relationships are established between the finite-transform sum rules derived previously and the finite-energy sum rule. The sine sum rule is applied to the pion-nucleon charge-exchange scattering in the forward direction. The assumption that the ρ trajectory alone dominates the charge-exchange amplitude leads to conflicts with analyticity even at 6 GeV. In the limit of the zero-mass pion the Born term in the sum rule is shown to be cancelled for all values of the continuous moment by a secondary trajectory with the intercept α′=−1 and the reduced residue γ′=−πf2, wheref2 is the pion-nucleon coupling constant. This secondary trajectory removes the conflict with analyticity at 6 GeV. Such a cancellation is a common feature of at least five sum rules. The local and global dualities are found to be violated whether or not the systematic errors are used and whether or not the Born term is subtracted out by means of the secondary trajectory. However, the usual low-mass resonances with well-determined quantum numbers satisfy the dualities better than the higher-mass resonances with mass >2.62 GeV. The CERN-Serpukhov data are amenable to the Reggepole description and are consistent with the Pomeranchuk theorem due to their large errors.