1. Test of PWA Reliability at BES
(for BES PWA working group)
Yanping HUANG
Institute of High Energy Physics,BeiJing
Physics and methods in meson spectroscopy

2. Outline Motivation PWA fit methods tested Summary Bin-by-bin fit
Event-by-event fit
Bin-by-bin fit
Binned phase space input fit
Summary

3. Motivation PWA is a key technique in hadron spectroscopy studies
Spin-parity determination
Interference among different processes
PWA is complicated  Are PWA results reliable, especially with many parameters in the fit?
We use MC input/output checks to test the reliability of PWA methods used at BES.
We check: Mass, Width, Nevent and significance

4. Standard PWA procedure at BES
The probability density function:
ξ: four momentum of final particles;
σ: calculated by Covariant Tensor formalism (Eur.Phys.J.A 16, ,2003)
Likelihood function:
Event-by-event fit

5. Example 1 : Event-by-event fit
Input MC sample: One 0++ resonance & two 2++ resonances
Generated KK mass plot in J/K+K- (Nev=50000)
output
input

6. Inputs/outputs of Mass, Width and event number for each resonance
0++
2++ M
(MeV)
output
2071±1.7
2154±1.8
2237±1.6
input
2070
2156
2237 Γ
337 ±2.1
167±6.9
21 ±3.4
345
167 25
Nev
±180.6
±230.8
446.7
±54.8
494.4
Event-by-event fit can well reproduce the input information

7. We perform extensive checks on the event-by-event fit method, including MC input samples containing 7 resonances as in the real data of J/K+K-process.
All tests show that the output can reproduce the input.

8. Speed problem in event-by-event fit
We will have 200 times larger data sample at BESIII:
Typical size of a data sample at BESII: events. Usually it takes 1- 3 years to publish one PWA result (with more than 20 CPU fully used).
Naively, we would have 2M events for one data sample at BESIII  The speed will be about 100 times slower  How many years do we need?

9. 3 ways to speed up PWA Bin-by-bin fit Binned phase space input fit
GPU  N. Berger’s talk

10. PWA procedure of Bin-by-bin fit
Divide the mass spectrum into many (~100) bins.
In each bin, we only fit various JPC components without BW structure.
We can perform PWA fits for all bins on many parallel CPUs.
Get the mass, width and event number of each resonances by fitting mass spectrum of each component

11. Example 2: Bin-by-bin fit
Input MC sample :
One 0++ resonance & one 2++ resonance
0++
2++

12. It seems bin-by-bin fit can reproduce
0++
2++
Output
Fit
Nev
63874±814
39322±814
M(MeV)
1753.3±1.6
1690.1±1.1
Γ(MeV)
186.5±3.9
83.4±2.8
Input
65567
1750
200
38099
1690 80
It seems bin-by-bin fit can reproduce
the input information.

13. Example 3 : Bin-by-bin fit
Input MC sample: two 2++ resonances
2++: M=1970 MeV Г=180 MeV
2++: M=2040 MeV Г =22 MeV
0++
2++
output
input ??
2++ : cannot be reproduced : significantly inconsistent with zero.

14. What causes the problem?
7686
7684
7690
7688
-lnL
minimum solution
Input solution
Nev(0++)
Multi-local-minimum

15. Check in the event-by-event fit
2++
Event-by-event fit :
well reproduce the inputs.
Event-by-event fit results are more reliable than bin-by-bin fit.

16. Advantages and disadvantages of bin-by-bin fit
Model independent for each JPC component in each mass bin.
Phase shift measured automatically
Fast
Disadvantages
Detail mass information lost
The constraint on the phase in nearby mass bin lost.

17. PWA procedure of Binned PS input fit
φ : azimuth angle of K- in J/ (fix as 0)
cosθ0 : cosine of polar angle of K- in J/
cosθ1 : cosine of polar angle of K+ in  K+
φ : azimuth angle of K+ in  K+
M  K+ : invariant mass of  K+
4 independent Phase Space variables 
We divide above 4 variables into many bins
so that in each bin, all events have similar
4 momentum.
New Inputs of PWA:
4 momentum in each bin and number of events in each bin
The PWA speed depends on number of bins.

18. Example 4 : Binned PS input fit
Input MC sample: Two 2++ resonances
output
input
M(MeV)
1969.2±0.2
1970
2039.7±0.1
Γ(MeV)
178.0±0.2
180
22.0±0.4
Nev
±573.2
±584.9

19. ——Statistical Significance
Example 5 : Binned PS input fit
——Statistical Significance
Input MC sample: One 2++ resonance and Phase Space
The significance can be defined by Prob(c2 , d.o.f )
S : the change of 2lnL while adding 0++ component which is not produced in the input sample.
Its distribution should obey c2 distribution with d.o.f = 4.
Binned ps input fit
Event-by-event fit ΔS ΔS
Binned-ps-input fit over-estimates the significance

20. Compare the three PWA fit methods
Event-by-event fit: provide reliable solution, but the speed becomes very slow for huge sample.
Bin-by-bin fit: the speed is fast, but there exist multi-local-minimum.
Binned-PS-input fit: The speed is also fast, but the significance is over-estimated.
 We still need a PWA procedure which can provide both reliability and high speed.

21. Possible solutions and tests needed in the future
Other binned input methods, including Dalitz plot analysis (it only works for 3 body decays)
Combined solution with bin-by-bin and event-by-event fits
GPU and others?

22. Summary
The reliability of 3 different PWA fit procedures have been tested. There are advantages and disadvantages in each procedure.
New methods or fit procedures are needed at BES in order to obtain robust PWA results with high speed. More tests will be performed.
—— Suggestions of the solutions are welcome!

23. Thank you

24. PWA procedure at BES Fumili L1 C1 C2 L2 C3 L3 Different combination:Cn L2 L3
Different combination:
Component
Resonance
amplitude Ln

25. PWA procedure of Bin-by-bin fit
Divide the mass spectrum into many (~100) bins.
In each bin, we only fit various JPC components without BW structure.
We can perform PWA fits for all bins on 100 CPU.
Bin size
0++
2++
Fit mass spectrum of each component

26. PWA procedure of Binned PS input fit
Effective bin
φ : azimuth angle of K- in J/
cosθ0 : cosine of polar angle of K- in J/
cosθ1 : cosine of polar angle of K+ in  K+
φ : azimuth angle of K+ in  K+
M  K+ : invariant mass of  K+ 1 3 4 2 6
Rotation
φ0=0
four degrees of freedom
New Inputs of PWA:
4 momentum in each bin and number of events in each bin
The PWA speed depends on number of bins.

27. Fit result for at BESII
Event-by-event fit
Bin-by-bin fit