1. Partial Wave Analysis with Complete Experiments
L. Tiator, Mainz
SFB/S3/MTZ Collaboration Meeting, Mainz, February 15-19, 2016 1

2. what is a complete experiment?
a complete experiment is
a set of polarization observables
that is sufficient to exactly determine
all other possible experiments
and all underlying (complex) amplitudes up to 1 phase

3. complete experiment in photoproduction
set
observables
single S
ds/dW T P
beam-target BT G H E F
beam-recoil BR
Ox´
Oz´
Cx´
Cz´
target-recoil TR
Tx´
Tz´
Lx´
Lz´
Barker, Donnachie, Storrow (1975): 9 obs. needed ?
Keaton, Workman (1996) and Chiang,Tabakin (1997):
a carefully chosen set of 8 observables is sufficient.

4. choose 8 out of 16 observables
this set does not work!
set
observables
single S
ds/dW T P
beam-target BT G H E F
beam-recoil BR
Ox´
Oz´
Cx´
Cz´
target-recoil TR
Tx´
Tz´
Lx´
Lz´

5. choose 8 out of 16 observables
this set works!
set
observables
single S
ds/dW T P
beam-target BT G H E F
beam-recoil BR
Ox´
Oz´
Cx´
Cz´
target-recoil TR
Tx´
Tz´
Lx´
Lz´
theoretical activities with pseudodata: Gent for p(g,K)L Mainz for p(g,p)p
present activities with experimental data: JLab/Gent for p(g,K)L

6. truncated partial wave analysis TPWA
mathematical studies:
Omelaenko (1981) for a truncated partial wave analysis (TPWA) with lmax waves only 5 observables are necessary, e.g. the 4 from group S and 1 additional from any other group
Wunderlich (2014) revisited Omelaenko formalism and tested modern PWA the need for only 5 observables is confirmed, but with rising lmax, large amount of accidental ambiguities will appear, making numerical analysis difficult
experimental applications:
Grushin (1989) applied it for a PWA in the D(1232) region with only S+P waves (lmax = 1)
present activities in Mainz, Bonn, JLab and PWA collaborations with Zagreb, Tuzla, Gent, Nikosia

7. is this a paradox ? determination of amplitudes Ai(W,q)
requires 8 observables including recoil polarization
determination of partial waves Mj(W)
requires only 5 observables without recoil
The answer is found in the higher partial waves, which are neglected in the second approach. If all higher partial waves are exactly zero, the complete experiment can also be done with 5 observables at a certain set of angles, depending on L. In total 8L - 1 measurements are required. (Omelaenko)

8. How can we get partial waves from complete experiments?
still for the purely mathematical problem (infinite precision, zero errors)
suppose the complete experiment is done with 8 observables and ~ 20 angles are measured (good up to L ~ 10)
this gives us a discrete set of energies and angles for helicity or CGLN amplitudes, which are exact, but reduced by an unknown phase, depending on energy and angle
Can these reduced amplitudes be decomposed in partial waves?
If we do this just straightforwardly, can we use them for N*,D resonance analysis?
from my point of view: No!
Omelaenko (1981) suggested the following method, but I am skeptic about it:

9. Omelaenko‘s statement about Complete Experiments

10. is this correct?

11. my point of view for the PWA from complete experiments
the complete experiment gives us the most condensed form of experimental informations on meson photoproduction in terms of 7 real amplitudes, e.g.:
instead of doing the TPWA analysis with 5, 6 7, 8,...12,...16 observables,
we can now do it with these 7 real amplitudes, which is equivalent to 16 observables
and this analysis is again non-linear !
I don‘t believe, that one can get the same information by a simple linear partial wave decomposition.
The only way I see is the constrained PWA that we are already doing, the most model-independent constraints are the analytical constraints, like FT, but we may need to refine them!

12. reduced (CGLN) amplitudes
normal CGLN or helicity amplitudes can be expanded in partial waves by a simple linear expansion or linear fit, e.g. for F1(E,q):
reduced amplitudes are amplitudes with one arbitrary phase:
they can be written in the following way (here F1 is chosen to be real) and it becomes obvious, that the expansion/fit becomes non-linear here the arbitrary phase is chosen as the phase of F1(E,q): f(E,q) = - f1(E,q)

13. weak constraints
a) un-constrained (discussed in detail by Yannick Wunderlich)
try to find an exact solution (often impossible)
use random start parameters in a certain range, e.g. given by total c.s.
repeat such local minimizations a few 1000 times to find the global minimum in addition second, third, etc. minima can be investigated
this only works, if the data have no higher partial waves beyond L (L ≤ 3) or if we know the higher partial waves precisely
b) weakly constrained:
use higher partial waves from a model, e.g. Born terms
use start parameters from a „good“ model either exactly from the model or within a range of e.g. ± 30%
try to find a better model by first improving the ED fit the average c2/d.o.f may not exceed values ≈ 5
this is the method, that Grushin et al. used in their complete analysis

14. hard constraints c) hard constraints with penalty terms:
if a „good“ model exists, penalty terms can be used to bind the SE solutions to the model (ED solution)
this is the standard method, that people are usually using
d) analytical constraints with penalty terms:
instead of constraining parameters of a model, or multipoles directly, also Invariant amplitudes, Helicity amplitudes or CGLN amplitudes can be constrained to a model or to fixed-t dispersion relations, successfully applied in the Karlsruhe-Helsinki pN PWA
this is probably the best method, in the past we have already applied this method at pion threshold and in the D region with unitarity constraints, beyond the D region we can use fixed-t constraints

15. fixed-t kinematics t = (qh - kg)² = mh2 - 2kg (wh-qhcos qh)
in our energy range of interest: thresh < W < 1.85 GeV, t runs from 0 to -1.5 GeV²
hN threshold