1. Partial wave analysis of η photoproduction – experimental data
J. Stahov
University of Tuzla
for Mainz, Tuzla, Zagreb Collaboration*
*V. Kashevarov, K. Nikonov, M. Ostrick,
L. Tiator , M. Hadžimehmedović, R. Omerović,
H. Osmanović, J. Stahov, A. Švarc
Bled Mini Workshop 2017
Bled , 3-9 July, 2017

2. Outline of the talk
Analytic approximation theory- Pietarinen expansion method
Fixed-t amplitude analysis
Preparing input data for Fixed- t amplitude analysis and SE PWA
Spline smoothing and approximation
Results and discussion
Summary and conclusions

3. Application of the fixed-t analyticity of Invariant amplitudes to
partial wave analysis (PWA)
One of the main problems of single energy PWA (SE PWA) are ambiguities of partial waves
First attempt: Requiring smoothnes of invariant amplitudes or partial waves as a function of energy. It was shown that it was not enough.
One must impose more stringent constraints taking into account analyticity
of scattering amplitudes.
In this talk: Iterative procedure to impose fixed-t analyticity as an constraint in SE PWA
The method was fully developed and applied by Hoehler in KH80 SE
πN elastic PWA.
J. E. Bowcock and H. Burkhardt
Rep. Prog. Phys. 38 (1975)1099

4. How to built in analyticity of
scattering amplitudes?
Fixed-t AA

5. Analytic approximation theory
Consider scattering amplitude having following analytic structure at a fixed- t value in the complex plane.
i) is real analytic function having a cut from to
ii) is bounded on the v plane.
In most problems only informations on or some oher bilinear form are available .
For simplicity, to introduce our method let us suppose that a following information are available:
Real parts at M points with errors
b) Imaginary parts at N points with errors .
Task: Find an approximant of the function having the same analytic structure.
Standard procedure: find a minimum of quadratic form:
There are many approximants giving very small value of
Most of them are non-smooth inside or outside the region where data on are available.
Standard approach: Introduce a penalty function ( aka Convergence test function) and find a minimum of quadratic form:
This part of the talk is based on:
J. Hamilton, J. L. Petersen in
New developments in dispersion
theory Vol. 1,
Nordita, Copenhagen 1975
E. Pietarinen, Nuovo. Cim. 12 (1972) 522
Nucl. Phys. B49 (1972) 315
Nucl. Phys B107(1976) 21,
end references there

6. Digression - Conformal mapping method
w - plane
-b a
x x x x
z-plane x x
w(-b) w(a) x x
W. R. Fraser, Phys. Rev 123 (1961) 2180
See also example on page 567 in:
C. B. Lang, N. Pucker, Mathematische Methoden in der Physik,
2nd Ed., Springer Spektrum, Berlin 2005.

7. Pietarinen’s expansion method
5. For a large number of coefficients ( N>20) weight factor
λ can be calculated using simplified formula:
in an iterative procedure starting from some initial value.
To find Pietarinen proceeded as follows:
Conformal mapping
transforms a complex plane with physical cut along
to a unit circle
The physical cut is mapped on
Any function having properties i) and ii) may be represented
in the complex z- plane by a convergent series:
Using arguments from theory of functions and probability theory,
Pietarinen found a penalty function in the form:
Pietarinen has shown that higher coefficients are suppressed
and behave as:
Expansion in 2. may be truncated at some finite order
x x x z x x x x -1 +1

8. 1. Due to conformal maping z, a polynomial expansion in 2) of any order
reconstructs the analytic structure of the amplitude F at a fixed-t value.
It is a smoothest approximation of amplitude F in the region where the data
on amplitude F are available.
2. The conformal mapping z can be defined in such a way to assure a correct
crossing property of the amplitude.
3. The asympthotical behaviour of the amplitudes can be imposed explicitly.
4. Technically, the expansion can be evaluated fast and reliably using nested
multiplication

9. Bilinear structure of relations between observables and invariant amplitudes.
Minimisation of X2 becomes nonlinear, therefore more demanding.
Observables are expressed in terms of more amplitudes, each having its
own representation. All coefficients have to be determined simultaneously.
According to Pietarinen, one has to minimize quadratic form:
consists of sum of penalties of all amplitudes
In analysis of experimental data situation is a bit different :

11. Representation of invariant amplitudes in η photoproduction
Fixed-t amplitude analysis
To represent an analytic function having two branch points and
two corresponding cuts we use two conformal mappings with
conformal variables:
where
z1 maps the v 2 plane into the unit circle Cut is
maped onto the unit circle
z2 maps the v 2 plane into the unit circle Cut is
η photoproduction can be described by four independent,
crossing symmetric amplitudes B1, B2, B3 and B8/ν.
Their Pietarinen expansions are presented in Hedim’s talk.

12. Coefficients in Pietarinen’s expansions of invariant amplitudes are obtained by minimizing quadratic form:
NO - number of observables, with corresponding
errors ,
NE - number of data for observable Oa at a given t- value.
Result of Fixed-t AA: a set of coefficients ,
corresponding invariant amplitudes and ,consequently,
helicity amplitudes H1FT , H2FT , H3FT , H4FT
Invariant and halicity amplitudes at a given t-value may be calculated at any energy W and any scattering angle inside the physical region :

14. ... Until Single energy partial wave analysis
Iterative minimisation scheme
In a single energy partial wave analysis ( SEPWA) we minimize the quadratic form:
containes all data at a given energy W:
is number of data on observable Oa .
makes a soft cut-off of higher multipoles. Effective at low energies.
Result: Multipoles up to Lmax at a given energy and helicity amplitudes H1SE , H2SE , H3SE , H4SE
... Until

15. Experimental data base
Incomplete data set
Statistical and systematic uncertainties much larger
than in pseudo data set
Limitation in the kinematical coverage
The best coverage in energy and angles in unpolarized
cross sections
Very good angular coverage for polarisation data
Expected soon:
Unpolarized differential cross section
Single beam polarisation
T Single target polarisation asymmetry
F Double beam- target polarisation with circular
polarized photons

16. Preparation of the fixed- s input data set
Cubic spline smoothing and approximation
Much more energy points for dxs then for
polarisation observables (120 vs 15,12,12)
In order to avoid the dxs dominance in our fitting procedure,
we use the same kinematical points also for polarisation observable.
Interpolate to energies where is available
We use spline smoothing method, which was simmilarly applied in
the KH analysis. Errors of interpolated data are taken to be equal to
errors of neares data points.
Detailes about splines, spline interpolation, spline smoothing and
spline approximation can be found in the book:
C. de Boor, A practical Guide to Splines, Springer-Verlag, Heidelberg,
1978, revised 2001.
Original de Boor’s well documented Fortran programs might be downloaded from the web page:
Given experimental values measured
at ponts with errors ,
find a cubic spline fp that minimizes a form
Parameter p can be found in such a way that :
Minimisation of establishes a compromise
between goodnes of the fit and a smoothness
of the cubic spline fp .
Recommended values of are in the range:
Subroutines csspline and splifit, which we routinely
use, are well tested using examples from de
Boor’s book.

17. Preparation of the fixed-t input data set
Create the fixed-t input data set from the fixed energy data set
For a given ti at energy Wi interpolate data at angle:
It was done using cubic spline smoothing method.
Our fixed-t amplitude analysis is performed at 20 equidistant values in the range
Example of interpolated fixed-t data base at t=-0.2 GeV2 and t=-0.5 GeV2
are shown in the figure below. Physical region starts at Wph =1486 MeV for t=-0.2 GeV2 and Wph =1486 MeV for t=-0.5 GeV2 .
t= GeV2
t= GeV2

18. In order to explore the model dependence
of our solution, we start the analysis with MAID
Solutions – Solution II and Solution III
References in Hedim’s talk

19. For small amplitudes H1 and H2 predictions from Solutions II and III
are significantly different.
Predictions for largest amplitudes H2 and H4
S- wave dominates

20. 1 st iteration

21. Pietarinen’s fit of the interpolated data.
Continuum Ambiguity in Ft AA:
Even identical fits are obtained with very different amplitudes
Pietarinen’s fit of the interpolated data.
Results obtained starting with Solution II- Dashed black line
Results obtained starting with Solution III- solid blue line

22. 1st iteration
The fit parameters in this step are
multipoles up to Lmax=5

23. Constraint from Ft AA in SE PWA
Starting solution: II dashed lines
Starting solution III solid lines
W=1554 MeV
W=1602 MeV
W=1775 MeV
W=1840 MeV

24. Solution II Solution III
The blue and red points are results of FtAA and are independent from point to point.
The blue and red lines are obtained from
SE PWA and are continuous over angular range.
W=1602 MeV
Solution II
W=1840 MeV
Solution III

25. Iterative procedure converges quickly...

26. After three iterations
W=1602 MeV
At higher energies the Ft AA
loses its constraining power
1st iteration
W=1840 MeV
3rd iteration

27. S, P and D waves:
Initial Solution II solid blue line
Final solution blue dots
S vave- almost does
not change
In all other pw an
evolution from
initial to final solution
can be observed

28. S, P and D waves
Initial Solution III solid red
line
Final solution red dots
S wave almost does not
change
A litle change in Re M1+ ,
E2- ,M2-

29. Real and imaginary parts of S, P, and D waves
Analytical constraints:
Solution II: Blue dots
Solution III: Red dots
S wawe- almost no
difference
For most other PW two
solutions are consistent
within their statistical
uncertainties
Fore some PW
considerable differences
in certain kinematical
regions show clearly
non unique solution
Im E 1+, Im E 2-, Re M 2-

30. Predictions for observables not fitted Start Solution II dashed line
Start Solution III solid line
W=1554 MeV
W=1765 MeV

31. Summary and Conclusions
Folowing the ideas in the Karlsruhe Helsinki PWA of πN scattering data, we have developed
a new method for PWA of meson photoproduction data and apply it to PWA of η
photoproduction data from the threshold WηTh =1487 MeV up to W=1850 MeV.
In an iterative procedure we perform fixed-t amplitude analysis using Pietarinen expansion
method and single energy PWA with mutual support and constraint of each others.
Iterative procedure leads to final solution after three iterations.
Presently available, and analysed, data are still limited to the incomplete set of 4 observables
(σ0 , Σ, F, T).
We find solutions for multipoles up to Lmax =5 which are continuous in energy.
Remaining ambiguities are found, mainly at higher energies
New data from ELSA, JLab and MAMI, expected in the near future will ( hopefully) help to
resolve these ambiguities.