1. Data Interpretation of Baryon Resonances
A. Švarc
Rudjer Bošković Institute, Zagreb, Croatia
CLAS Collaboration Meeting June 2017

2. CLAS Collaboration Meeting 13-16 June 2017

3. Poles as resonance signal CLAS Collaboration Meeting 13-16 June 2017
Main task
Poles as resonance signal
Experiment
CLAS Collaboration Meeting June 2017

4. Poles as resonance signal CLAS Collaboration Meeting 13-16 June 2017
Main task
Poles as resonance signal
PWA
Experiment
SE PWA
MODEL
MODEL
CLAS Collaboration Meeting June 2017

5. Poles as resonance signal CLAS Collaboration Meeting 13-16 June 2017
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
CLAS Collaboration Meeting June 2017

6. CLAS Collaboration Meeting 13-16 June 2017
Continuum ambiguity
All observables in single-channel processes are described as bilinears of invariant amplitudes!
𝑶 ≡ 𝒇( 𝑨 𝒊 ∙𝑩 𝒋 ∗ )
CLAS Collaboration Meeting June 2017

7. CLAS Collaboration Meeting 13-16 June 2017
Example: η photoproduction
CLAS Collaboration Meeting June 2017

8. CLAS Collaboration Meeting 13-16 June 2017
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!
CLAS Collaboration Meeting June 2017

9. CLAS Collaboration Meeting 13-16 June 2017
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!
CLAS Collaboration Meeting June 2017

10. CLAS Collaboration Meeting 13-16 June 2017
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!
CLAS Collaboration Meeting June 2017

11. CLAS Collaboration Meeting 13-16 June 2017
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
CLAS Collaboration Meeting June 2017

12. CLAS Collaboration Meeting 13-16 June 2017
So, our problem is:
CLAS Collaboration Meeting June 2017

13. CLAS Collaboration Meeting 13-16 June 2017

14. CLAS Collaboration Meeting 13-16 June 2017
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
CLAS Collaboration Meeting June 2017

15. CLAS Collaboration Meeting 13-16 June 2017
Explicitly:
CLAS Collaboration Meeting June 2017

16. Importance of angle dependent phase rotations in PWA
Influence upon unconstrained SE PWA
CLAS Collaboration Meeting June 2017

17. Fitted invariant amplitudes CLAS Collaboration Meeting 13-16 June 2017
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
CLAS Collaboration Meeting June 2017

18. CLAS Collaboration Meeting 13-16 June 2017
We perform
SINGLE ENERGY
UNCONSTRAINED
LMAX = 8
truncated PWA of η photoproduction
pseudo-data
CLAS Collaboration Meeting June 2017

19. CLAS Collaboration Meeting 13-16 June 2017
dσ/dΩ
Data F P
Oz’ T Σ
Cx’
CLAS Collaboration Meeting June 2017

20. CLAS Collaboration Meeting 13-16 June 2017
Two
different
initial
values
two
different solutions
with the same χ2
Sol 1: MAID16a
Sol 2: Bon-Gatchina
CLAS Collaboration Meeting June 2017

21. CLAS Collaboration Meeting 13-16 June 2017

22. CLAS Collaboration Meeting 13-16 June 2017
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
CLAS Collaboration Meeting June 2017

23. CLAS Collaboration Meeting 13-16 June 2017
Let me illustrate the problem
CLAS Collaboration Meeting June 2017

24. CLAS Collaboration Meeting 13-16 June 2017
Let me illustrate the problem
CLAS Collaboration Meeting June 2017

25. CLAS Collaboration Meeting 13-16 June 2017
generating
model
CLAS Collaboration Meeting June 2017

26. CLAS Collaboration Meeting 13-16 June 2017

27. CLAS Collaboration Meeting 13-16 June 2017
Let me solve the problem
CLAS Collaboration Meeting June 2017

28. CLAS Collaboration Meeting 13-16 June 2017

29. CLAS Collaboration Meeting 13-16 June 2017

30. CLAS Collaboration Meeting 13-16 June 2017
PWA9/ATHOS Bad Honnef 2017 30
CLAS Collaboration Meeting June 2017

31. CLAS Collaboration Meeting 13-16 June 2017
This means that all three SE solutions:
Generating solution MAID15a
Sol 1
Sol 2
can be transformed into each other by an angle dependent phase rotation!
SO THEY ARE EQUIVALENT!
MAID15a
Sol 1
Sol 2
Question: Can we use it to stabilize unconstrained SE PWA?
CLAS Collaboration Meeting June 2017

32. CLAS Collaboration Meeting 13-16 June 2017
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
CLAS Collaboration Meeting June 2017

33. CLAS Collaboration Meeting 13-16 June 2017
Problem solved
CLAS Collaboration Meeting June 2017

34. CLAS Collaboration Meeting 13-16 June 2017
VERY IMPORTANT!
we obtain the „anchor” point
CLAS Collaboration Meeting June 2017

35. CLAS Collaboration Meeting 13-16 June 2017

36. CLAS Collaboration Meeting 13-16 June 2017

37. CLAS Collaboration Meeting 13-16 June 2017
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
CLAS Collaboration Meeting June 2017

38. CLAS Collaboration Meeting 13-16 June 2017
References:
CLAS
CLAS Collaboration Meeting June 2017

39. CLAS Collaboration Meeting 13-16 June 2017
CLAS seminar at JLab
Friday, August 18th, at 11.00
CLAS Collaboration Meeting June 2017

40. CLAS Collaboration Meeting 13-16 June 2017
SUMMARY
CLAS Collaboration Meeting June 2017

41. CLAS Collaboration Meeting 13-16 June 2017
Future prospects
We need hard work…
We have the tool
CLAS Collaboration Meeting June 2017

42. CLAS Collaboration Meeting 13-16 June 2017