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

Motivations Analytic Structure of the partial waves PWDR HPWR Examples Conclusions

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.

Analytic structure of the piN partial waves

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.

part of the t-channel circle cut Input Results from piN PWA: KH80, VPI/GW ( 0.02 GeV/c-10 GeV/c) s-channel cut The partial waves ( ) part of the t-channel circle cut Leading contributions t-channel short cut

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

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

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

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

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.