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References
V. De Alfaro, S. Fubini, G. Furlan andC. Rossetti: preprint, Torino (Dec. 1966).
V. De Alfaro, S. Fubini, G. Furlan andC. Rossetti:Phys. Lett.,21, 576 (1966);R. F. Dashen andM. Gell-Mann:Proc. of the Third Coral Gables Conference on Symmetry Principles at High Energies, University of Miami (1966).
This result is proved in ref. (1)V. De Alfaro, S. Fubini, G. Furlan andC. Rossetti: preprint, Torino (Dec. 1966). in the case of spin-flip pion-nucleon scattering without charge exchange. In the case of charge exchange the proof fails; however the saturation in terms of a finite number of resonances, if even possible, appears to give a physically unacceptable solution. This point is commented further in Sect. 2 on a specific example.
R. F. Dashen andM. Gell-Mann:Phys. Rev. Lett.,17, 340 (1966) and lecture notes given at the 1966 International School of Physics “Ettore Majorana”, Erice (1966).
We acknowledge interesting discussions with Prof.B. Sakita on the difficulties encountered in the case of nondegenerate masses.
S. Fubini andG. Segrè:Nuovo Cimento,45, 641 (1967);J. B. Bronzan, I. S. Gerstein, B. W. Lee andF. E. Low:Phys. Rev. Lett.,18, 32 (1967);V. Singh:Phys. Rev. Lett.,18, 36 (1967);H. R. Rubinstein andG. Veneziano:Phys. Rev. (to be published).
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A. A. Logunov, L. D. Soloviev andA. N. Tavkhelidze:Phys. Lett.,24 B, 181 (1967);R. Gatto:Phys. Rev. Lett.,18, 803 (1967);K. Igi andS. Matsuda:Phys. Rev. Lett.,18, 625 (1967);D. Horn andC. Schmid: preprint CALT-68-127 (1967).
seee.g. V. Barger: Talk presented at the Symposium on Regge Poles, Argonne National Laboratory (1966). Nevertheless it is worth noticing that the tail of the resonant amplitude does not decrease fast enough to fit the high-energy data.
We negleet the ϕ contribution according to the maximal ω-ϕ mixing and to the experimental evidence for ϕ+>ρπ. We also neglect other resonances, like the possible partners of B(1210), whose data are not available. On the other hand these contributions would simply add to the 1+ terms without affecting our conclusions.
Mandelstam andWang (Phys. Rev., to be published) have shown that β does not vanish at the nonsense wrong signature point, if the third spectral function is nonzero. However we assume, as these authors do, that the presence of the dips in the cross-sections is enough to make one believe that the β is indeed small.
We are not willing to useSU 3 to relate pion to kaon coupling constants as we know that the symmetry is badly broken there. For example from the experimental rate of K*→Kπ we getg 2K*Kπ /4π≃0.84, whileSU 3 givesg 2K*Kπ /4π=1/4g 2ρππ /4π≃0.68.
Due to the numerical values of the masses, the above condition is almost equivalent tog 2 L −(m2/m 21 )g 2 T =0, predicting equal rates for longitudinal and transverse polarization of the ρ emitted in the A1 decay. This is in disagreement with the prediction ofgT=0 given byF. J. Gilman andH. Harari:Phys. Rev. Lett.,18, 1150 (1967), and also with that ofP. H. Framiton andJ. C. Taylor:Nuovo Cimento,49 A, 152 (1967) (in their notations we findC≃0 instead ofC≃10). All the above authors neglect the high-energy Regge contribution neart=0.
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Ademollo, M., Rubinstein, H.R., Veneziano, G. et al. Saturation of superconvergent sum rules at nonzero momentum transfer. Nuovo Cimento A (1965-1970) 51, 227–231 (1967). https://doi.org/10.1007/BF02740002
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DOI: https://doi.org/10.1007/BF02740002