Test of PWA Reliability at BES

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 2008-10-25

2. Outline • Motivation • PWA fit methods tested • 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 16,537-547,2003) Likelihood function: Event-by-event fit

5. output input 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)

6. Inputs/outputs of Mass, Width and event number for each resonance 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: 10000 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. Input 65567 1750 200 Input 38099 1690 80 It seems bin-by-bin fit can reproduce the input information.

13. output input ?? 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++ 2++ : cannot be reproduced. 0++ : significantly inconsistent with zero.

14. 7690 7688 minimum solution 7686 7684 0 1000 2000 3000 4000 What causes the problem? -lnL Input solution Nev(0++) Multi-local-minimum

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

16. Advantages and disadvantagesof bin-by-bin fit • Advantages • 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 • φ0 : 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+ • φ1 : 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

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

25. Bin size 2++ 0++ 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. Fit mass spectrum of each component

26. 4 3 1 3 4 3 4 2 3 PWA procedure of Binned PS input fit Effective bin • φ0 : azimuth angle of K- in J/ • cosθ0 : cosine of polar angle of K- in J/ • cosθ1 : cosine of polar angle of K+ in  K+ • φ1 : azimuth angle of K+ in  K+ • M  K+: invariant mass of  K+ 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