1. From Experimental Data to Pole Parameters in a Model Independent Way
(Angle Dependent Continuum Ambiguity and Laurent + Pietarinen Expansion)
A. Švarc
Rudjer Bošković Institute, Zagreb, Croatia
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3. Poles as resonance signal
Main task
Poles as resonance signal
Experiment
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4. Poles as resonance signal
Main task
Poles as resonance signal
PWA
Experiment
SE PWA
MODEL
MODEL
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5. Poles as resonance signal
Main task
Poles as resonance signal
PWA
Experiment
SE PWA
MODEL
MODEL
Phase
ambiguity
Laurent
+ Pietarinen
From Experimental Data to Pole Parameters in a Model Independent Way
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6. Continuum ambiguity 𝑶 ≡ 𝒇( 𝑨 𝒊 ∙𝑩 𝒋 ∗ )
All observables in single-channel processes are described as bilinears of invariant amplitudes!
𝑶 ≡ 𝒇( 𝑨 𝒊 ∙𝑩 𝒋 ∗ )
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7. Example: η photoproduction
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8. Therefore, the simultaneous transformation
1-fold rotation does not change anything!
A consequence:
ALL SC OBSERVABLES ARE INVARIANT WITH RESPECT TO
ENERGY AND ANGLE DEPENDENT PHASE ROTATION
where Φ(W,θ) is an arbitrary energy and ANGLE dependent real function!
This is called CONTINUUM AMBIGUITY!
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9. It is usually done on the level of partial waves:
Energy dependent phase rotation has been discussed very often when two different models are compared!
It is usually done on the level of partial waves:
We choose one multipole
Match the phase of that multipole
Multiply all other multipoles with that phase
Angle dependent phase rotation has hardly been ever discussed!
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10. For η photoproduction partial wave decomposition is defined as
Let us see what is the influence of angle dependent continuum ambiguity upon a partial wave decomposition!
Example
For η photoproduction partial wave decomposition is defined as
Let us simplify things somewhat!
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11. Legendre polynomials satisfy recursive relations
All derivatives of Legendre polynomials can be reduced to lower index Legendre polynomials.
So, in practice we deal with the general problem
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12. So, our problem is:
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14. Angle dependent phase rotations mix multipoles
is a long time ago given in
However, Yannick Wunderlich has in his thesis
Derived a very nice closed formula:
Angle dependent phase rotations mix multipoles
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15. Explicitly:
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16. Importance of angle dependent phase
rotations in PWA
Influence upon unconstrained SE PWA
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17. Fitted invariant amplitudes
General testing scheme of a procedure
by using numeric or pseudo data
Theoretical
multipoles
Invariant amplitudes
Observables
(complete set)
SE PWA
I D E N T I C A L
Fitted invariant amplitudes
Fitted multipoles
Fitted observables
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18. truncated PWA of η photoproduction pseudo-data
We perform
SINGLE ENERGY
UNCONSTRAINED
LMAX = 8
truncated PWA of η photoproduction
pseudo-data
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19. G
dσ/dΩ
Data F P
Oz’ T Σ
Cx’
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20. Two different initial values two different solutions with the same χ2
Sol 1: MAID16a
Sol 2: Bon-Gatchina
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22. PROBLEM ! Complex analysis
Sol 1 and Sol 2 must be identical to the generating model as we fitted a complete set ….
Continuum ambiguity
Sol 1 and Sol 2 must be identical up to a phase
Conclusion
There must exist a phase which connects Sol 1 and Sol 2.
And the phase MUST be defined at the level of AMPLITUDES
and NOT partial waves
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23. Let me illustrate the problem
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24. Let me illustrate the problem
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25. Let us plot THE PHASE of helicity amplitude H2
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26. generating
model
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28. Let me solve the problem
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31. PWA9/ATHOS Bad Honnef 2017 31
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32. This means that all three SE solutions: Generating solution MAID15a
can be transformed into each other by a phase rotation
at arbitrary energy and arbitrary ANGLE!
SO THEY ARE EQUIVALENT!
MAID15a
Sol 1
Sol 2
Question: Can we use it to stabilize unconstrained SE PWA?
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33. 𝑯 𝒌 𝑺𝑬𝟏𝟔𝒂 (W, 𝒙)= 𝑯 𝒌 𝑺𝑬𝟏𝟔𝒂 𝒆 𝒊 𝚽 𝑯 𝟐 𝟏𝟓𝒂 𝑾,𝒙 −𝒊 𝚽 𝑯 𝟐 𝑺𝑬𝟏𝟔𝒂 𝑾,𝒙
Let us introduce the following angle dependent phase rotation
Sol 1:
𝑯 𝒌 𝑺𝑬𝟏𝟔𝒂 (W, 𝒙)= 𝑯 𝒌 𝑺𝑬𝟏𝟔𝒂 𝒆 𝒊 𝚽 𝑯 𝟐 𝟏𝟓𝒂 𝑾,𝒙 −𝒊 𝚽 𝑯 𝟐 𝑺𝑬𝟏𝟔𝒂 𝑾,𝒙
Sol 2:
𝑯 𝒌 𝑺𝑬𝑩𝒐𝑮𝒂 (W, 𝒙)= 𝑯 𝒌 𝑺𝑬𝑩𝒐𝑮𝒂 𝒆 𝒊 𝚽 𝑯 𝟐 𝟏𝟓𝒂 𝑾,𝒙 −𝒊 𝚽 𝑯 𝟐 𝑺𝑬𝑩𝒐𝑮𝒂 𝑾,𝒙
Both rotated SE solutions now have the same H2 phase;
phase of generating 15a model
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34. Problem solved
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35. we obtain the „anchor” point
VERY IMPORTANT!
we obtain the „anchor” point
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38. References:
CLAS
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39. II How to extract T-matrix poles from partial waves
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40. 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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41. 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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42. Local approach
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43. 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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44. Padé expansion
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45. Regularization method
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46. PROBLEMS for local solutions !
In both cases we have n-TH DERIVATIVE of the function
PROBLEMS for local solutions !
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47. 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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48. 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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49. 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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50. Expansion of the T-matrix in terms of constant coefficients
cannot be valid everywhere in C 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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51. 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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52. 1. Analyticity
Analyticity is introduced via generalized Laurent’s decomposition
(Mittag-Leffler theorem)
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53. 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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54. 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 EXPLICT analytic form!
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55. 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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56. By L. Tiator 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...”
By L. Tiator
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57. In practice this means:
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58. 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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59. Illustration:
Powes series for Z(ω) =
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60. Z(ω)
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61. Z(ω)2
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62. Z(ω)3
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63. A resonance CANNOT be well described by Pietarinen series.
Important!
A resonance CANNOT be well described by Pietarinen series.
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64. Finally, the area of convergence for Laurent expansion of P11 partial wave
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65. The model
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66. 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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67. 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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68. 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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69. 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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70. 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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71. Conclusion for L+P
The L+P method defined as:
WORKS
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74. L+P method is tested and well documented
Conformal-mapping generated fast converging expansion of regular part of Laurent decomposition around physical branchpoints in the limited part of 2-nd Riemann sheet of complex energy plane near real axes
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75. SUM
MARY
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76. Future prospects
We need hard work…
We have the tool
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