Quarkonium Polarization Fit: Toy Monte Carlo Studies

1. Quarkonium Polarization Fit:Toy Monte Carlo Studies Now in 3d! (CDF Note 10385) Matthew Jones Purdue University B Production and Decay Meeting

2. The ϒ Polarization Puzzle: ? or B Production and Decay Meeting

3. What we observe… B Production and Decay Meeting

4. ϒ Polarization (Transverse) (Longitudinal) (Unpolarized) B Production and Decay Meeting

5. Transverse: Longitudinal: Transverse/Longitudinal is not sufficient But an arbitrary rotation will preserve the transverse/longitudinal shape... B Production and Decay Meeting

6. Need for full polarization analysis • The templates for dN/dΩ are more complicated than simply 1 ± cos2θ. • Need to measure λθ, λφ and λθφ simultaneously. • Invariant under rotations: B Production and Decay Meeting

7. Toy Monte Carlo • Generate kinematics of • Gaussian mass distribution for signal, uniform for background • Uniform in pT from pTmin to pTmax • Uniform in rapidity, -2 < y() < 2 • Isotropic 2-body decay kinematics • Calculate (θ,ϕ) in S-channel helicity or Collins-Soper frame • Reject events to give desired angular distribution: B Production and Decay Meeting

8. Decay Angles • S-channel helicity frame is familiar to most people: • Z-axis is along ϒ momentum vector • X-axis is in the production plane (contains beam momentum vector and ϒ momentum) • Y-axis is perpendicular to Z and X. B Production and Decay Meeting

9. Decay Angles • Collins-Soper frame is more natural phenomenologically… • Boost beam vectors into ϒ rest frame • Z-axis bisects the angle between them • X-axis is in the production plane, to Z-axis • Y-axis is to Z- and Y-axes • Got it? B Production and Decay Meeting

10. Decay Angles • Check geometry using ROOT + X3D: B Production and Decay Meeting

11. Acceptance Cuts • With no acceptance cuts the distribution of (θ,ϕ) remains unchanged. • Cuts on and η restrict the accepted range of angles. • Example: • pT > 4 GeV/c • η < 1 Isotropic without acceptance cuts B Production and Decay Meeting

12. Acceptance Cuts • With no acceptance cuts the distribution of (θ,ϕ) remains unchanged. • Cuts on and η restrict the accepted range of angles. • Example: • pT > 4 GeV/c • η < 1 Very non-isotropic B Production and Decay Meeting

13. Fit Model • Acceptance depends on kinematics, not polarization. • Generate un-polarized Monte Carlo to calculate acceptance. • Re-weight un-polarized distributions after acceptance cuts • calculated using angular distribution • Vary weights to provide optimal description of the observed angular distribution. B Production and Decay Meeting

14. Fit Model • Complications: • Acceptance changes rapidly with mass • no obvious place to cut out edges of acceptance in all mass bins • This would also throw away valuable statistics B Production and Decay Meeting

15. Fit Model • Complications: • Acceptance changes rapidly with mass • no obvious place to cut out edges of acceptance in all mass bins • This would also throw away valuable statistics • Generate large Monte Carlo samples to reduce statistical fluctuations in bins with small acceptance? • Some bins have zero acceptance; no sample can be “large enough” • Ignore bins with small statistics (n<5? n<10?) • Ignoring information leads to biases. B Production and Decay Meeting

16. Acceptance is a nuisance parameter • Observables in each (θ,ϕ) bin are: • M: number of MC events generated • m: number that pass acceptance cuts • Their ratio is only an estimator of the acceptance • x: number of events observed • The acceptance corrected yield is usually computed using but even this is not correct when A≈0,1. B Production and Decay Meeting

17. Fitting with small MC statistics • Assume binomial distribution for acceptance: • The signal is a Poisson distribution, but this is almost a binomial distribution for large signal yields. • Apply Bayes’ theorem to calculate probability of observing given a yield and Monte Carlo statistics and : B Production and Decay Meeting

18. Fitting with Small Statistics • The integral has an analytic expression: • In the presence of background we have two independent processes with possibly different acceptances contributing to the observed number of events. = … B Production and Decay Meeting

19. (approaches a Gaussian as B Production and Decay Meeting

20. Likelihood function • Bins with only background: • : observed number of events • : Monte Carlo events generated • : Monte Carlo events accepted • : Background yield (parameter in fit) • : weights ( are parameters in fit) • Likelihood function: B Production and Decay Meeting

21. Likelihood function • Bins with signal + background: • : observed number of events • : Monte Carlo signal events generated • : Monte Carlo signal events accepted • : Monte Carlo background events generated • : Monte Carlo background events accepted • : Background yield (parameter in fit) • : Signal yield (parameter in fit) • : signal weights ( are parameters in fit) • : signal weights ( are parameters in fit) • Likelihood function constructed using “q” pdf’s. B Production and Decay Meeting

22. Example: • Generate toy Monte Carlo with • 9.5 < pT< 10.5 GeV/c • Mass resolution of 50 MeV on signals • 200,000 background events • 100,000 ϒ(1S) events • 50,000 ϒ(2S) events • 25,000 ϒ(3S) events • All events un-polarized • Acceptance averages about 10% so the statistics is similar to what we have to work with at CDF. B Production and Decay Meeting

23. Toy Experiment Mass bins used for fit. B Production and Decay Meeting

24. Fit Parameters • Assume linear function for background yield • Assume background polarization varies linearly with mass • Number of parameters: • 4 for each signal (yield, ) • 8 for background (linear in yield and each polarization parameter) • Straight forward to use a more complex background description • But we expect Drell-Yan expected to be like spin-1 B Production and Decay Meeting

25. Fitted yields from 664 toy experiments No significant bias, but uncertainties underestimated by about 50%. B Production and Decay Meeting

26. Fitted parameter B Production and Decay Meeting

27. Fitted parameter: B Production and Decay Meeting

28. Fitted parameter B Production and Decay Meeting

29. Other polarizations Just for illustration – only one toy experiment for each ϒ(1S) polarization point. The same un-polarized ϒ(2S) and ϒ(3S) samples used in all cases. ϒ(3S) ϒ(2S) B Production and Decay Meeting

30. High pT bin: 15 < pT < 20 GeV/c More precise because of larger acceptance and greater angular coverage of accepted events. B Production and Decay Meeting

31. Analysis in S-channel Helicity Frame • The fitting procedure does not depend on which reference frame we use • Although it may not converge when the frame is not well defined (as in the case when pT≈0) • Acceptance in SH frame: B Production and Decay Meeting

32. Frame Independent Parameters is the same in all frames Pure L and T states generated in S-channel helicity frame. Then fit in the S-channel helicity frame and in the Collins-Soper frame. Pure transverse SH CS CS SH Pure longitudinal B Production and Decay Meeting

33. Realistic Projections for CDF Data • UPSILON_CMUP_CMU and UPSILON_CMUP_CMX: Example: N(1S) = 22780 ± 169 N(2S) = 6998 ± 105 N(3S) = 3980 ± 86 B Production and Decay Meeting

34. Toy Trigger Model Simulated CMUP muons Reconstructed CMUP muons • Approximates the geometric coverage of CMUP. • Similar geometric treatment for CMU and CMX. B Production and Decay Meeting

35. “Realistic” pT spectra • Measured pT spectra fit gamma functions quite well. • Use fitted parameters to generate pTfor 1S, 2S and 3S. B Production and Decay Meeting

36. Tune signal and background yields • Signal yields after acceptance • Background level • Slope of background B Production and Decay Meeting

37. Hypothetical Scenario • Always pure transverse polarization • Angle of quantization axis rotates as pT increases GeV/c UPSILON_CMUP_CMU only GeV/c Both UPSILON_CMUP_CMU and UPSILON_CMUP_CMX B Production and Decay Meeting

38. Enough Statistics? • Yields scaled to match those in CDF 10154: Green contours are with CDF 10154 yields. Fits become less precise and less stable. Increased fraction of fits don’t converge when calculating contours. B Production and Decay Meeting

39. Application to real data • Monte Carlo samples • Desirable to have x10 statistics for unpolarized templates. This 100 times the observed signals. • Trigger efficiency can be applied to weight the acceptance templates • Need to re-weight pT spectrum of templates • Straight forward in this framework • Could also tune pT spectrum by iterating • Background description: • Could get more elaborate than linear • Consider comparison with spin-2 distribution? B Production and Decay Meeting

40. Conclusions • At this time nobody knows what polarization state vector mesons have when they are produced. • We have enough data on disk to make the first 3D polarization measurement. • The LHC experiments understand that meaningful polarization measurements must be done in 3D. • It would be nice to do it first. B Production and Decay Meeting