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 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.

5. Analytic structure of the piN partial waves

6. 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.

7. 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

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

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

22. 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

23. 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

28. 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.