1. A. Švarc Rudjer Bošković Institute, Zagreb, Croatia
Poles as a link between QCD and scattering theory
(old and contemporary knowledge)
A. Švarc
Rudjer Bošković Institute, Zagreb, Croatia

2. Several times here in GWU a statement has been made:
It is clear that the poles are the only resonance signal
However, it is still not clear whether:
Is it really completely true?
Is it a general knowledge?
How are the poles to be extracted from experiment?
Is that statement adequately represented in the secondary literature? (meaning PDG)

3. The main intention of this talk is to analyze the crucial ideas: definition of resonances, poles and Breit-Wigner parameters.
The talk is going to be referral and not educational.
The emphasis is going to be on existing, but sometimes forgotten knowledge about the field.
I will give information about issues as I see them, and at least one important reference which can serve as a starting point for further reading.

4. I What is a resonance

5. ? bound states resonances bound states resonances
How do I see what is our main task?
Experiment
Matching point
Theory
QCD
bound states resonances ?
bound states resonances
What is a resonance in QCD?
What is a resonance in experiment?

6. What is a resonance in experiment?
From the intuitive (heuristic) definition to the mathematical formulation
Heuristic definition

7. particle “gets trapped” (the “black hole” phenomenon)
a direct scattering event
the lifetime of the particle-target system in the region of interaction is larger than the collision time in a direct collision process

8. This is a very heuristic definition.
Question: What is the mathematical formulation of resonance signal?
Where do the Breit-Wigner parameters come in?

9. What is a resonance signal?
There is an uncertainty in literature.
A careful reader will notice that it is never explicitly stated:
the resonance is
Instead, introducing and defining the resonance is always much more delicate....

10. Examples
Introducing (and NOT defining) resonances varies from using rather undefined terms like:
follow from
are associated with
it is well known 1. 2.

11. 3.
Over strong statements that resonances are just a matter of convention “ ... simply an ad hoc hypothesis...” :

13. To full mathematical rigor

14. However, for me, the most transparent discussion is given in:

15. "Adventures in Mathematical Physics" (Proceedings, Cergy-Pontoise 2006), Contemporary Mathematics, 447 (2007) 73-81

16. Both definitions of resonances are being used in the literature without full awareness that they are different, and that both are in principle allowed.
This is a reason for numerous disputes and controversies.
Knowing that we are dealing with the two equivalent quantifications of the same phenomenon solves the issue.
Let us remember:
our task is not only describe resonances, but describe them in a way which is identical to QCD.
The final answer to which of the two resonance definitions should be used comes from analyzing what is precisely calculated in QCD!
Is it a time delay or a resolvent pole?

17. Scattering resonances:

18. They are aware that it is nowadays usual to identify resonances as resolvent resonances:
However they still advocate the idea of using scattering resonances instead:

19. Resolvent resonances:
Concept of resolvent resonances is at length discussed in

20. Highlights:
Equivalence of Hamiltonian eigenstates and resolvent poles

21. The conservation of probability requires that the resolvent poles lie in the complex energy plane

22. Importance of background

25. So, his final recommendation goes with Hamiltonian eigenstates – poles (resolvent resonance definition).

26. What is a resonance in QCD?
Talk presented at the Workshop "Light-cone Physics: Particles and Strings" at ECT* in Trento, Sep 3-11, 2001
...solving the bound-state problem in gauge filed theory, particularly QCD...

27. Light-cone approach How is it done? resonances
Hamiltonian proper values

28. discrete light-cone quantization

29. In all of these cases:
Resonance are identified with the Hamiltonian proper values
QCD is analyzing resolvent resonances only!
Remember Exner and Lipovsky definition of resolvent resonance:

30. So the answer to our question is:
 POLES

31. II How to extract T-matrix poles

32. Yesterday, Kanzu Nakayama discussed sets of poles coming from various analyses.
Let me show more precisely how I see the problem.

33. Is it enough that all models fit identical data base equally well?
Poles 1
Model 2
Poles 2
Poles 3
Model 3
Experimental data base
Under which conditions we have:
Poles 1 = Poles 2 = Poles 3 ?
Is it enough that all models fit identical data base equally well?
NO !

34. Illustration:
Red line: Juelich model
Black line: Zagreb CMB fit to Juelich model
Red line:
Complex N branch point  no P11(1710)
Black line:
Real 2N branch point  P11(1710) exists

35. In order to offer some insight into the problem we have used Zagreb CMB model to analyze a collection of well known PW amplitudes UNDERSTANDING THEM AS PARTIAL WAVE DATA
In that way we have effectively merged into one joint error
error of fitting different input data bases and
error due to different analytic structure of various models.
We have to keep in mind that for all analyzed PWA input data base was almost identical, and the reduced 2 comparable, the major part of the obtained difference in pole positions is due to the different analytic structure of the particular model.
We did not expect to reproduce the pole positions of each individual analysis because our analytic structure is different from their, we just wanted to extract poles using on and the same tool in each case.

36. And have produced a sequel of preprints illustrating how the method works for the S11 partial wave :1. 2.
Both publication are submitted to PRC.

39. III Is the problem well presented in PDG

40. I personally believe that PDG is the most important secondary publication in physics because it is read by millions!
It is read by:
professionals on a daily basis (just to see what is new because they know it by heart)
experts in physics for getting reliable info about facts in new fields they might be entering
professors in education to get the most precise, reliable, and worldly acceptable interpretation of some possibly controversial statements found in literature
students to get the condensed, precise new knowledge of unknown phenomena
layman to see what is the most current value of certain constant, and how the experts in physics think
And I believe that the responsibility of PDG is to sytisfy them all.

41. Therefore it is its duty to be
PRECISE (define all phenomena, all procedures, all assumptions…)
CONDENSED
CRITICAL
EXACT
Is it really so today in baryon spectroscopy sector?
I am afraid IT IS NOT.
The reasons for such a statement of mine is that in spite of strong recent progress in the field this section has barely changed in last 20 years:
the definition of resonance is NEVER GIVEN ! PDG is presenting chapters and chapters of the book relying on heuristic understanding of a reader what a resonance really is, and this creates ambiguities.
the difference between poles and Breit-Wigner parameters is never explicitly given
the definition, importance and role of poles is never given

42. Examples:
I am for quite some time objecting to the way how the nucleon resonances chapter is being written with respect to defining the resonance parameters (mass and width)
The situation is quite clear in the meson part:

43. However, it is NOT for the nucleon sector.
And what is interesting to me is that one of the leading PDG experts for nucleon resonances is since 1999 aware of the problem

44. Ron has announced some changes….

45. The situation remains the same even today!
Breit-Wigner parameters
PDG2010
Pole positions

46. III: How to proceed?

47. In Trento 2011
I have also discussed this issue, and I have shown the following two slides:

49. And I am happy that we have a chance to do it on Thursday!