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J. Stahov University of Tuzla

4 th Pion- Nucleon Pwa Workshop, Helsinki, 2007. J. Stahov University of Tuzla. Partial wave dispersion relations and partial wave relations as a test of PWA. Motivations Analytic Structure of the partial waves PWDR HPWR Examples Conclusions. Motivations.

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J. Stahov University of Tuzla

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  1. 4th Pion- Nucleon Pwa Workshop, Helsinki, 2007. J. Stahov University of Tuzla Partial wave dispersion relations and partial wave relations as a test of PWA

  2. Motivations • Analytic Structure of the partial waves • PWDR • HPWR • Examples • Conclusions

  3. Motivations • Since 1980. there was no pion-nucleon PWA which incorporates • analytic properties of the pion-nucleon invariant amplitudes as a strong constraint • in all kinematic regions where experimental data exist. • VPI/GW- up to 2.1 GeV/c, constrained up to 0.8 GeV/c • Low energy PWA • Petersburg • Helsinki group • ACU-UnTz • Higher partial waves ( d waves and higher) at low energies can not be determined • from experimental data only. • These pw are needed to perform reliable analytic continuation into unphysical parts • of the Mandelstam plane below s- channel threshold. • - Without analytic constraints it is hard to obtain unique solution.

  4. Analytic structure of the piN partial waves

  5. PWDR Circle cut+Left hand cut+ short cut+ s-channel cut+ Discrepancy Discrepancy function is slowly changing function in the physical region, without structure, because it describes the contributions of the distant parts of the cuts.

  6. Input • Results frompiN PWA:KH80, VPI/GW ( 0.02 GeV/c-10 GeV/c) • s-channel cut • The partial waves ( ) • part of the t-channelcircle cut • Leading contributions • t-channel • short cut

  7. How to test results from PWA? Discrepancy function describes contributions from the distant parts of the cuts- must be slowly variable function

  8. Partial wave relations -PWR from the FTDR: • Valid up to k=450 MeV/c • Slow convergence • Kernels reproduce s and u channl cuts

  9. PWR from dispersion relations along hyperbolas in the Mandelstam s-u plane • Hite and Steiner, 1973. • Dispersion realtions along hyperbolas in • the Mandelstam plane. • (s-a)(u-a)=b • - PWR from DR along hyperbolas =N-exch+s-chan+ u-chan+t-chan

  10. Properties • Valid up to 450 MeV/c • Kernels reproduce analytic • structures of PW • s and u-channels • t-channel • Fast convergence • Leading contributions: t- • channel, short cut • The t-channel kernels behave • as . Partial waves up to • enough to describe • t- channel contributions

  11. Conclusion (s) • PWDR and PWR are good test of consistency of • results of PWA with Mandelstam analyticity. • Higher partial waves obtained from HPWR and PWR • may be used as a part of the input in PWA.

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