Alfred Švarc Ruđer Bošković Institute Croatia

1. The importance of inelastic channels in eliminating continuum ambiguities in pion-nucleon partial wave analyses Alfred Švarc Ruđer Bošković Institute Croatia

2. constraints from fixed t-analyticity … resolve the ambiguities Pg. 5 Pg. 6

4. The effects of discreet and continuumambiguities were separated…. Dispersion relations were used to solve ambiguities and to derive constraints…

7. 2005 1973 1975 1975 1973

8. 1985. 2005 1979 2005 1976 2005

9. What does it mean “continuum ambiguity”?

10. Differential cross section is not sufficient to determine the scattering amplitude: if The new function givesEXACTLY THE SAME CROSS SECTION then

11. S – matrix unitarity …………….. conservation of fluxRESTRICTS THE PHASE 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 behave exactly like that

12. there exists a whole family of functions F ,of limited magnitude but of infinite variety of functional form, which will behave exactly like that These family of functions, though containing a continuum infinity of points, are limited in extend. TheISLANDS OF AMBIGUITY are created.

13. I M P O R T A N T Once the three body channels open up, this way of eliminating continuum ambiguities (elastic channel arguments) become in principle impossible

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

18. The treatment of continuum ambiguity problems • Constraining the functional form F -mathematicalproblem • Implementing the partial wave T – matrix continuity (energy smoothing and search for uniqueness)

19. a. Finding the true boundaries of the phase function F(z) is a very difficult problem on which the very little progress has been made.

20. b.

21. Let us formulate what the continuum ambiguityproblem is in the language ofcoupled channel formalism

22. Continuum ambiguity/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

23. Basic idea: we wantto demonstrate the role of inelastic channels infully constraining the poles of the partial wave T-matrix,or, alternatively said, for eliminating continuum ambiguity which arises if only elastic channels a considered.

24. We want as well show that: supplying only scarce information for EACH channel is MUCH MORE CONSTRAINING then supplying the perfect information in ONE channel.

26. unitary fully analytic Coupled channel T matrix formalism

29. T-matrix poles are connected to the bare propagator poles, but shifted with the self energy term ! Important:real and imaginary parts of the self energy term are linked because of analyticity

30. Constraining data: Elastic channel: Pion elastic VPI SES solution FA02 Karlsruhe - Helsinki KH 80

32. Inelastic channel: recent pN ¨hN CC PWA Pittsburgh/ANL 2000 however NO P11 is offered! older p N ¨ hNCC PWA Zagreb/ANL 95/98 p N ¨ p2 NCC PWA Zagreb/ANL 95/98 gives P11

33. Both: three channel version of CMB model • p N elastic T matrices + p N¨h N data + dummy channel • Pittsburgh: VPI + data + dummy channel • Zagreb: KH80 + data + dummy channel • fitted all partal waves up to L = 4 • Pittsburgh: offers S11 only • Zagreb: offers P11 as well(nucl-th/9703023)

34. S11 Let us compare Pittsburgh¸/ Zagreb S11

35. Pittsburgh S11 is taken as an “experimentally constrained” partial wave

36. So we offer Zagreb P11 as the “experimentally constrained” partial wave as well. P11 from nucl-th/9703023

37. Two-channel-model

38. STEP 1 : Number of channels: 2 (pion elastic + effective) Number of GF propagator poles: 3 (2 background poles + 1 physical pole) ONLY ELASTIC CHANNEL IS FITTED

40. elastic channels is reproduced perfectly • inelastic channel is reproduced poorly • we identify one pole in the physical region

41. STEP 2 : Number of channels: 2 (pion elastic + effective) Number of GF propagator poles: 3 (2 background poles + 1 physical pole) ONLY INELASTIC CHANNEL IS FITTED

43. inelastic channel is reproduced perfectly • elastic channel is reproduced poorly • we identify two poles in the physical region, “Roper” and1700 MeV

44. STEP 3: Number of channels: 2 (pion elastic + effective) Number of GF propagator poles: 3 (2 background poles + 1 physical pole) ELASTIC + INELASTIC CHANNEL ARE FITTED

46. elastic channel is reproduced OK • inelastic channel is reproduced tolerably • we identify two poles in the physical region, but both are in the Roper –resonance region

48. We can not find a “single pole” solution which would simultaneously reproduce ELASTIC ANDINELASTIC CHANNELS

49. STEP 4: Number of channels: 2 (pion elastic + effective) Number of GF propagator poles: 4 (2 background poles + 2physical poles) ONLY ELASTIC CHANNEL IS FITTED