1. PWA with huge statistics at BESIII
H.X. Yang
(For PWA working group)
BES-Belle-CLEO-Babar 2007 Joint
workshop on Charm Physics
26-27th,Nov. 2007

2. PWA working group Bai Yu, Ge Junyan, Huang Yanping,
Ji Xiaobin, Jin Shan, Li Xiaoling,
Liao Hongbo, Liu Fang, Liu Hongbang,
Yang Hongxun, Zhu Xingwang

3. Outline Problems of PWA at BESIII PWA with bin-by-bin method
PWA with Bined PS method
Summary

4. PWA is crucial to most analyses at BESIII
Not only to the spin-parity determination of new hadrons
But also to the measurement of decay BR:
e.g.: BR(J/K*K)
BR(J/K*(1410) K)
From interference

5. We will discuss how to speed up PWA in this talk!
With huge data samples at BESIII, PWA is possible.
However , there are many difficult problems to be solved.
How to speed up the PWA fit?
How to treat background?
How to parameterize the intermediate states?
How to treat small components?
How can we include data/MC differences in the PWA fit?
How to judge whether we have reached a real minimum in the PWA fit rather than local ones?
How to get reasonable errors?
How to treat resolution?
……………………………
We will discuss how to speed up PWA in this talk!

6. How to speed up the PWA 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 more than 100 times slower  How many years do we need?

7. PWA procedure at BESII: Global fit
The PWA input contains 4-momentum of all events (the whole mass range).
Various fits are tried with different combinations of the possible components/resonances/amplitudes, to find minimum –lnL of all these combinations.
One possible solution at BESIII: 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.

8. Bin-by-Bin Fit Advantages Disadvantages
Model independent for each JPC component in each mass bin.
Phase shift measured automatically
Fast (by parallel calculation on many CPU)
Disadvantages
Detail mass information lost
The constraint on the phase in nearby mass bin lost.

9. MC Input/Output Checks of Bin-by-Bin Fit
We checked whether the bin-by-bin fit can reproduce the input values based on extensive MC studies.
Some examples 

10. Example 1: One 0++ resonance & one 2++ resonance
Generated KK mass plot in J/KK (160K evts)
int
All
0++
2++

11. 0++ 2++ Error bar: bin-by-bin fit result
Histogram: generated mass distribution
0++
2++
MKK

12. Fit mass plot based on bin-by-bin fit
0++
Gen
Fit
Dif
Nevent
65567
63874±814
2.2σ
Mass
1750
1753.3±1.6
2.1σ
Width
200
186.5±3.9
3.5σ
2++
Gen
Fit
Dif
Nevent
38099
39322±814
1.5σ
Mass
1690
1690.1±1.1
0.1σ
Width 80
83.4±2.8
1.2σ

13. Input : 0++: m=2.5GeV, =0.2GeV, =60 2++: Phase Space =0
Example 2: Phase Shift Measurement
0++
Input : : m=2.5GeV, =0.2GeV, = : Phase Space =0
Output (fitting the phase shift curve):
=61.49±6.90
M(0++)=2.5012±0.005GeV
(0++)=0.206±0.012GeV
M(2++)=1.227±0.403GeV
(2++)=32.273±17.906GeV
2++

14. Bin-by-bin approach looks promising…
However 

15. Example3 : Two 2++ resonances (300K events)
Error bar: bin-by-bin fit result
Histogram: generated mass distribution
Resonance 1:
M=1970 MeV
Г=180 MeV
2++
Resonance 2:
M=2040 MeV
Г=22 MeV
2++ component cannot be reproduced.

16. Binned Phase Space Method
If bin-by-bin approach cannot obtain robust PWA results, what’s other solutions to speed up PWA fit?
Global fit
Binned Phase Space Method

17. Binned Phase Space Method
Divide 4-dimension phase space into a number of cells.
We still use 4-momentum as input of PWA. However:
Parameters for PWA
Original
Binned PS
4-momentum of each event
4-momentum of each cell
number of events (Nevent)
number of effective (weighted) cells (Ncell)
the weight of each cell =
the event number in the cell
Speed can be significantly enhanced
Fast as a Ncell events sample.

18. Why not do Dalitz fit? Dalitz fit Binned PS variables 2 5 information
More information reserved with Bined PS method(No lost)
For 3-body decay process. Phase Space can be determined by 5 independently kinematic variables.
Considering the symmetry around the beam (Z axis). there are four parameters remained.
Dalitz fit
Binned PS
variables 2 5
information
lost
All reserved

19. How to divide phase space?
Base on J/ K+ K - process
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+ .
Thanks Prof. Zhang Dahua.
The 5 variables are determined after we discussed with him.
At first, we rotate the event around Z axis to make φ0 = 0 and get a new system （symmetry around the beam）;
Then, we use cosθ0, cosθ1, φ1 and MγK+ of new system to divide phase space;
A data file is output with weight and 4-momentum of Each cell
Weighted PWA will be done with the output data file.

20. Example : One 0++ resonance and two 2++ resonances
variable
bins of variable
cosθ0 10
cosθ1 40 φ1 18
M  K+ 22
original events
effective cells
620444
54562

21. Error bar: original sample
Histogram: binned PS sample

23. Project of global fit. Error bars : sample; Histogram : project
Binned PS sample

24. The global fit of binned PS sample can well reproduce the inputs.
The scaned mass and width of each resonance in binned PS sample
gen
f0(2070)
f2(2156)
f2(2237)
M(MeV)
Γ(MeV)
2070
345
2156
167
2237 25
fit
2070.0±0.5
342.5 ±1.2
±0.6
167.3
±1.8
2238.5
±0.5
23.5
±0.9
The global fit of binned PS sample can well reproduce the inputs.

25. Error bar: Fast fit result Histogram: original mass plot
Two 2++ resonances sample fails in bin-by-bin fit but succeeds with this method
Error bar: Fast fit result
Histogram: original mass plot
2++
Resonance 1:
M=1970 MeV
Г=180 MeV
Bin by Bin
Bined PS
Resonance 2:
M=2040 MeV
Г=20 MeV

26. The fit results agree with the input distribution very well for each distribution

27. Summary —— Suggestions of the solutions are welcome!
PWA is crucial to most physics results at BESIII. There are many difficult problems to be solved in PWA;
We found that binned PS method can speed up the PWA fit and it also can reproduce the inputs.
Much more to be done on errors, checks …
—— Suggestions of the solutions are welcome!

28. THANK YOU！

29. The event number of each resonance in binned PS sample
Gen
6134.5
Fit ± ±
5173.5± 211.9
Diff
2.05 σ
3.32 σ
4.54 σ

31. Are the errors in the global fit of binned PS sample reasonable?
It should be compared with the global fit results of original sample. But, global fit is very difficult to deal with so large a sample directly (620444). We have to use a sub-sample (50000) of original sample to check the errors of binned PS sample. The ratios of relative errors between the sub-sample and binned PS sample should be approximately equal 3.52.
Project of global fit
Sub-sample (50000) of original sample
Error bars : sample Histogram : project

32. The event number of each resonance in the two samples
binned PS
sample
Gen
6134.5
Fit
±833.0
(1034.6)
±1088.9
(1107.7)
5173.5±211.8
(211.9)
Rela. errσ1
0.21%
0.69%
4.10%
Diff
2.05σ
3.32σ
4.54 σ
50000 events of original sample
494.4
±180.6
(250.9)
±230.8
(238.2)
446.7±54.8
(54.8)
Rela.
err σ2
0.64%
1.81%
12.3%
0.04 σ
0.13 σ
0.87 σ
σ2/ σ1(3.52)
3.05
2.62
3.00

33. 50000 events of original sample
The mass and width of each resonance in the two samples
f0(2070)
f2(2156)
f2(2237)
M(MeV)
Γ(MeV)
2070
345
2156
167
2237 25
events of original sample
2071.0±1.7
337.0 ±2.1
2154.0±1.8
167.0±6.9
2237.0±1.6
21.0 ±3.4
binned PS sample
2070.0±0.5
342.5±1.2
±0.6
167.3±1.8
2238.5±0.5
23.5±0.9
σ1/σ2
(3.52)
3.40
1.78
3.00
3.84
3.20
4.22
The errors of binned PS sample are reasonable.

34. Global fit results are more reliable than bin-by-bin fit.
How to understand the bin-by-bin fit results?
Multi-solution
minimum solution
Input solution