1. A. Švarc Rudjer Bošković Institute, Zagreb, Croatia
Partial wave Analysis
and
Baryon Resonances
A. Švarc
Rudjer Bošković Institute, Zagreb, Croatia
I will mostly talk about single energy (energy independent) partial wave analysis!
Boppard 2016

2. Some introductory remarks:
Start with the full set of formula for η - photoproduction
Discuss essential parts
Focus on knowledge which is not to be found in text-books
Problem of precise definitions of terms in use
Process or elimination errors: What is true and what is wrong when text-books do not give you the answer
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3. Partial wave analysis in η photoproduction
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4. Boppard 2016

5. How does one obtain partial waves?
One has to measure the COMPLETE set of observables
Extracting partial waves
Using theoretical models (STANDARD)
Performing single energy (energy independent) PWA
NO MODEL
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6. Extracting partial waves using theoretical models (STANDARD)
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7. In SE PWA with NO THEORETICAL MODEL these assumptions simply pop-up !
Problem:
In such a procedure many assumptions are hidden!
In SE PWA with NO THEORETICAL MODEL these assumptions simply pop-up !
I will focus my talk on SE PWA and stress what exactly which are additional assumptions which ARE AUTHOMATICALLY DEFINED if one uses theoretical models.
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8. Title contains two notions: Partial waves Baryon resonances
We defined partial waves.
Let us define the notion baryon resonances.
What are we looking for in partial waves?
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9. What is a resonance
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10. Various definitions !
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11. For me, the most transparent discussion is given in:
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12. "Adventures in Mathematical Physics" (Proceedings, Cergy-Pontoise 2006), Contemporary Mathematics, 447 (2007) 73-81
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13. Interrelation of scattering and resolvent resonances:
Concept of resolvent resonances is at length discussed in
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14. This is a reason for numerous disputes and controversies.
Both definitions of resonances are being used in the literature sometimes 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?
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15. 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...
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16. Light-cone approach How is it done? resonances
Hamiltonian proper values
Remember Dalitz-Moorhouse
Hamiltonian proper values
poles
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17. QCD is analyzing resolvent resonances !
So, in principle:
QCD is analyzing resolvent resonances !
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18. So we recently agreed that poles are the true quantities which quantify resonances!
Illustration: PDG
PDG 2015
PDG 2015 – encoder tools
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19. I. Continuum ambiguity
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20. PHASE AMBIGUITY - CONTINUUM AMBIGUITY
We start looking for poles.
Here comes the first ambiguity non-existing in theoretical models:
PHASE AMBIGUITY - CONTINUUM AMBIGUITY
What is continuum ambiguity?
In theoretical models : automatically defined
In SE PWA: FREE ENERGY AND ANGLE DEPENDENT PHASE
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21. Discrete and continuum ambiguities
Trento 2014

22. Continuum ambiguities
Trento 2014

23. Question: What does it do to our problem?
Direct consequence:
Unconstrained SE PWA is discontinuous in energy
We need stabilization procedures !
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24. One way: fixed-t analiticitity
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25. t – analyticity in fixed-t minimization
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26. s –channel penalty function
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27. II. Cut direction
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28. parameterize partial amplitudes directly in term of poles
How to extract poles?
First idea:
parameterize partial amplitudes directly in term of poles
B(W) …………. background
Unfortunately, this is IMPOSSIBLE!
In scattering theory resonances are poles on the II - Riemann sheet, and the only poles on the I-st Riemann are bound states.
Unfortunately, constant poles appear on all Riemann sheets, hence on the I-st one as well. FORBIDDEN!
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29. For more details see:
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30. Therefore, we have to create poles with more complicated functions!
The simplest choice is to have functions with energy dependent parameters and cuts in the complex energy plane.
Lothar Tiator constructed a very simple, educative model with only square root cuts!
See at:
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32. Where do you rotate cuts?
Question:
Where do you rotate cuts?
Mathematics says that you can rotate cuts ANY WAY YOU LIKE!
Does physics agree with that?
Let us show what is happening with poles if square root cut is rotated in different directions!
Simple rules how to define the direction of the cut!
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33. Example:
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34. We see that we have 4 possible branch-points and a lot of combinations of directions!
(D, D, D, D)
(L, D, D, D)
(D, D, D, R)
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35. And 3D plot for
(D, D, D, D)
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36. Model is simple, and produces unphysical poles below threshold!
Let us focus on:
Physical range
Only on the physical cut m + mπ
Three possible directions of rotations (L,D,R)
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37. (D, D, D, L)
(D, D, D, D)
(D, D, D, R)
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38. HAS TO BE ROTATED TO THE RIGHT!
Important!
If we want to agree with scattering theory demand that only bounds states can have poles on the I-st Riemann sheet, and that resonances are poles on the II-nd Riemann sheet, physical cut
HAS TO BE ROTATED TO THE RIGHT!
Discussion! When can we rotate cuts?
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39. New example: elastic cut + inelastic cut on the real axes
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40. Elastic cut to the right
Elastic cut down
Elastic cut to the right
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41. ? 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?
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42. So the answer to our question is:
 POLES
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43. II How to extract T-matrix poles from partial waves
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44. Speed plot, expansions in power series, etc
The usual answer was:
Do it globally
One first has to make a model which fits the data, SOLVE IT, and obtain an explicit analytic function in the full complex energy plane. Second, one has to look for the complex poles of the obtained analytic functions.
Do it locally
Speed plot, expansions in power series, etc
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45. Direct problems for global solutions: Many models
Complicated and different analytic structure
Elaborated method for solving the problem
SINGLE USER RESULTS
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46. Local approach
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47. Eliminate or reduce the dependence upon background contribution
Speed plot
Idea behind it?
Eliminate or reduce the dependence upon background contribution
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48. Padé expansion
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49. Regularization method
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50. PROBLEMS for local solutions !
In both cases we have n-TH DERIVATIVE of the function
PROBLEMS for local solutions !
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51. I have tryed to do it starting from very general principles:
Is it possible to create universal approach, usable for everyone, and above all REPRODUCIBLE?
I have tryed to do it starting from very general principles:
Analyticity
Unitarity
Idea: TRADING ADVANTAGES
GLOBALITY FOR SIMPLICITY
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52. THEORETICAL MODELS
If you create a model, the advantage is that your solution is absolutely global, valid in the full complex energy plane (all Riemann sheets). The drawback is that the solution is complicated, pole positions are usually energy dependent otherwise you cannot ensure simple physical requirements like absence of the poles on the first, physical Riemann sheet, Schwartz reflection principle, etc. It is complicated and demanding to solve it.
WE PROPOSE
Construct an analytic function NOT in the full complex energy plane, but CLOSE to the real axes in the area of dominant nucleon resonances, which is fitting the data by using
LAURENT EXPANSION.
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53. Why Laurent’s decomposition?
It is a unique representation of the complex analytic function on a dense set in terms of pole parts and regular background
It explicitly seperates pole terms from regular part
It has constant pole parameters
It is not a representation in the full complex energy plane, but has its well defined area of convergence
IMPORTANT TO UNDERSTAND:
It is not an expansion in pole positions with constant coefficients (as some referees reproached), because it is defined only in a part of the complex energy plane.
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54. Expansion of the T-matrix in terms of constant coefficients
cannot be valid in principle.
Namely, poles with constant coefficients have poles on ALL physical sheets, and that violates common sense because only bound states are allowed to be located on the physical sheet.
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55. However, even this function has its Laurent decomposition
The only way how to accomodate both, requirements of absence of poles on the physical sheet, and Schwartz principle requires that pole positions become energy dependent:
However, even this function has its Laurent decomposition
But it is valid only in the part of the complex energy plane
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56. 1. Analyticity
Analyticity is introduced via generalized Laurent’s decomposition
(Mittag-Leffler theorem)
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57. Now, we have two parts¸of Laurent’s decomposition: Poles Regular part
Assumption:
We are working with first order poles so all negative powers in Laurent’s expansion lower than
n -1
are suppressed
Now, we have two parts¸of Laurent’s decomposition:
Poles
Regular part
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58. The problem is how to determine regular function B(w).
The problem is how to determine regular function B(w).
What do we know about it?
We know it’s analytic structure for each partial wave!
We do not know its EXPLICIT analytic form!
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59. So, instead of „guessing” its exact form by using model assumptions we
EXPAND IT IN FASTLY CONVERGENT POWER SERIES OF PIETARINEN („Z”) FUNCTIONS WITH WELL KNOWN BRANCH-POINTS!
S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961)
I. Ciulli, S. Ciulli, and J. Fisher, Nuovo Cimento 23, 1129 (1962).
Original idea:
S. Ciulli and J. Fischer in Nucl. Phys. 24, 465 (1961)
Detailed proof in I. Caprini and J. Fischer: "Expansion functions in perturbative QCD and the determination of αs", Phys.Rev. D84 (2011) ,
Convergence proven in:
E. Pietarinen, Nuovo Cimento Soc. Ital. Fis. 12A, 522 (1972).
Hoehler – Landolt Boernstein „BIBLE” (1983)
Applied in πN scattering
on the level of invariant amplitudes
PENALTY FUNCTION INTRODUCED
NAMING !
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60. What is Pitarinen’s expansion?
In principle, in mathematical language, it is ” ...a conformal mapping which maps the physical sheet of the ω-plane onto the interior of the unit circle in the Z-plane...”
In practice this means:
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61. Or in another words, Pietarinen functions Z(ω) are a complet set of functions for an arbitrary function F(ω) which HAS A BRANCH POINT AT xP !
Observe:
Pietarinen functions do not form a complete set of functions for any function, but only for the function having a well defined branch point.
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62. Illustration:
Powes series for Z(ω) =
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63. Z(ω)
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64. Z(ω)2
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65. Z(ω)3
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66. A resonance CANNOT be well described by Pietarinen series.
Important!
A resonance CANNOT be well described by Pietarinen series.
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67. Finally, the area of convergence for Laurent expansion of P11 partial wave
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68. The model
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69. Pietarinen power series
We use Mittag-Leffleur decomposition of „analyzed” function:
regular background
k - simple poles
We know analytic properties (number and position of cuts) of analyzed function
ONE
Pietarinen power series
per cut
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70. Method has problems, and the one of them definitely is:
There is a lot of cuts, so it is difficult to imagine that we shall be able to represent each cut with one Pietarinen series (too many possibly interfering terms).
Answer:
We shall use „effective” cuts to represent dominant effects.
We use three Pietarinen series:
One to represent subthreshold, unphysical contributions
Two in physical region to represent all inelastic channel openings
Strategy of choosing branchpoint positions is extremely important and will be discussed later
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71. The method is „self-checking” ! It might not work.
Advantage:
The method is „self-checking” !
It might not work.
But, if it works, and if we obtain a good χ2, then we have obtained
AN ANALYTIC FUNCTION WITH WELL KNOWN POLES AND CUTS WHICH DEFINITELY DESCRIBES THE INPUT!
So, if we have disagreements with other methods, then we are looking at two different analytic functions which are almost identical on a discrete set, so we may discuss the general stability of the problem.
However, our solution definitely IS A SOLUTION!
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72. What can we do with this model?
We may analyze various kinds of inputs
Theoretical curves coming from ANY model
but also
Information coming directly from experiment (partial wave data)
Observe:
Partial wave data are much more convenient to analyze!
To fit „theoretical input” we have to „guess” both:
pole position AND analyticity structure of the background imposed by the analyzed model
exact
To fit „experimental input” we have to „guess” only:
pole position AND analyticity structure of the background as no information about functional type is imposed
the simplest
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73. Testing is a very simple procedure. It comes to:
Does it work?
Testing is a very simple procedure. It comes to:
Doesn’t work
Works
TESTING
Testing on a toy model:
Testing and application on realistic amplitudes
πN elastic scattering
ED PW amplitudes (some solutions from GWU/SAID)
ED PW amplitudes (some solutions from Dubna-Mainz-Taipei)
Photo – and electroproduction on nucleon
ED multipoles (all solutions from MAID and SAID)
SES multipoles (all solutions from MAID and SAID)
arXiv nucl-th
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74. a. Toy model
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75. We have constructed a toy model using two poles and two cuts, used it to construct the input data set, attributed error bars of 5%, and tried to use L+P method to extract pole parameters under different conditions.
C1, C2, B1, B2 = -1, 0, 1
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78. b. Testing on realistic amplitude πN elastic GWU/SAID FA02
GWU/SAID SP06
GWU/SAID WI08
DMT
Photoproduction
GWU/SAID ZN11 ED
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79. Quality of the fit
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80. πN elastic scattering SAID FA02 ED
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81. Conclusion
The L+P method defined as:
WORKS
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82. World recognition
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83. Graz 2015

84. Boppard 2016