Continuum ambiguities as a limitation factor in single-channel PW analysis

1. Continuum ambiguities as a limitation factor insingle-channel PW analysis A. Švarc Rudjer Bošković Institute, Zagreb, Croatia INT-09-3 The Jefferson Laboratory Upgrade to 12 GeV (Friday, November 13, 2009)

2. Continuum ambiguity is an old problem

3. Tallahassee 2005

4. Today 1984 1985 1978 1973 1981 Zgb 1978 Zgb Zgb Nothing much changed

5. However, people are encountering problems when performing single channel PWA. Illustration:

7. Possible explanation of the problems: continuum ambiguities because they have single channel fit

8. However, people are (in principle) aware of the existence of continuum ambiguities!

9. - Hoehler Pg. 5 Pg. 6

11. What does it mean “continuum ambiguity”?

12. Simplified definition: In a single-channel case, phase shifts (partial wave poles) are not always uniquely defined! Unfortunately it turns out that this is the case as soon as inelastic channels open.

13. Differential cross section (or any bilinear of scattering functions) is not sufficient to determine the scattering amplitude: if The new function givesEXACTLY THE SAME CROSS SECTION then

14. S – matrix unitarity …………….. conservation of fluxRESTRICTS THE PHASE HOW? elastic region ……. unitarity relates real and imaginary part of each partial wave – equality constraint each partial wave must lie upon its unitary circle inelastic region ……. unitarity provides only an inequality constraint between real and imaginary part each partial wave must lie uponor insideits unitary circle there exists a whole family of functions F ,of limited magnitude but of infinite variety of functional form, which will indeed lie uponor inside its unitary circle

15. there exists a whole family of functions F ,of limited magnitude but of infinite variety of functional form, which will indeed lie uponor inside its unitary circle These family of functions, though containing a continuum infinity of points, are limited in extend. TheISLANDS OF AMBIGUITY are created.

16. I M P O R T A N T DISTINCTION theoretical islands of ambiguity / experimental uncertainties

17. Let us illustrate this on a simple example!

19. The treatment of continuum ambiguity problems The issues are: How to obtain continuity in energy? How to achieve uniqueness? In original publications several methods are suggested.

21. However, there is another way to restore uniqueness: by restoring unitarity in a coupled channel formalism

23. Let us formulate what the continuum ambiguityproblem means in the language of coupled channel formalism

24. Continuumambiguity/T-matrix poles T matrix is an analytic function in s,t. Each analytic function is uniquely defined with its poles and cuts. If an analytic function contains a continuum ambiguity it is not uniquely defined. If an analytic function is not uniquely defined, we do not have a complete knowledge about its poles and cuts. Consequentlyfully constraining poles and cuts means eliminating continuum ambiguity

25. Basic idea:we wantto demonstrate the role and importance of inelastic channels in fully constraining the poles of the partial wave T-matrix,or, alternatively said, for eliminating continuum ambiguity which arises if only elastic channels a considered. Statement: We need ALL channels, elastic AND as much inelastic ones as possible in order to uniquely define ALL scattering matrix poles.

26. What is the procedure? Having a coupled-channel formalism and fitting data only in one channel we may“mimic” single channel case. By fitting one channel only we shall reveal those poles (resonant states) which dominantly couple to this channel. Poles (resonant states) which do not couple to this channel will remain undetected. Consequently, we have not been able to discover ALL analytic function poles, consequently the partial wave analytic function is ambiguous. If we add data for the second inelastic channel, we constrain other set of poles which dominantly couple to this channel. This set of poles is overlapping with the first one, but not necessarily identical. We have established a new, enlarged set of poles which is somewhat more constraining the unknown analytic function We add new inelastic channels until we have found all scattering matrix poles, and uniquely identified the type of analytic PW function

28. Example 1: The role of inelastic channels in N (1710) P11 Published:

30. CMB coupled-channel model • All coupled channel models are based on solving Dyson-Schwinger integral type equations, and they all have the same general structure: • full = bare + bare * interaction* full

31. Carnagie-Melon-Berkely (CMB) model Instead of solving Lipmann-Schwinger equation of the type: with microscopic description of interaction term we solve the equivalent Dyson-Schwinger equation for the Green function with representing the whole interaction term effectively.

32. We represent the full T-matrix in the form where the channel-resonance interaction is not calculated but effectively parameterized: bare particle propagator channel-resonance mixing matrix channel propagator

33. Assumption: The imaginary part of the channel propagator is defined as: And we require its analyticity through the dispersion relation: where qa(s) is the meson-nucleon cms momentum:

34. we obtain the full propagator G by solving Dyson-Schwinger equation where we obtain the final expression 34 34

35. We use: • CMB model for 3 channels: • p N, h N, and dummy channel p2N • p N elastic T matrices , PDG: SES Ar06 • p N¨h N T matrices, PDG:Batinic 95 We fit: πNelastic only p N¨h N only both channels

36. Results for extracted pole positions:

40. Conclusions • Continuum ambiguities appear in single channel PWA, and have to be eliminated. • A new way, based on reinstalling unitarity is possible within the framework of couple-channel models. • T matrix poles, invisible when only elastic channel is analyzed, may spontaneously appear when inelastic channels are added. • It is demonstrated that: • the N(1710) P11 state exists • thepole is hidden in the continuum ambiguity of VPI/GWU FA02 • it spontaneously appears when inelastic channels are introduced in addition to the elastic ones.

41. A few transparencies from NSTAR2005 talk: