K -  + Scattering from D Decays

1. K-+ Scattering from D Decays Brian Meadows University of Cincinnati

2. {12} {23} {13} 1 1 1 2 2 2 3 3 3 1 3 “Traditional” Dalitz Plot Analysis • The “isobar model”, with relativistic Breit-Wigner (RBW) resonant terms, is widely used in studying 3-body decays of heavy quark mesons. • Amplitude for channel {ij}: • Each resonance “R” (mass MR, width R) assumed to have form NR 2 NRConstant R form factor D form factor spin factor

3. Traditional E791 DD+!KK-p+p+ ~138 % c2/d.o.f. = 2.7 Flat “NR” term does not give good description of data. Phys.Rev.Lett.89:121801,2002

4. “Traditional”  Model for S-wave - E791 ~89 % c2/d.o.f. = 0.73 (95 %) Probability Mk = 797 § 19 § 42 MeV/c2 Gk = 410 § 43 § 85 MeV/c2 E. Aitala, et al, PRL 89 121801 (2002)

5. E791 (WMD) “Model-Independent”Partial-Wave Analysis (MIPWA) • Make partial-wave expansion of decay amplitude in angular momentum of K-+ system produced D form-factor • “Partial Wave:” • Describes invariant mass dependence of K-+ system • -> Related to K-+ • scattering ML(p,q) Watson Theorem holds that, up to elastic limit (K’ threshold for S-wave) K phases same as for elastic scattering.

6. Watson Theorem • The process P   + c can be thought of as Borrowed from M. Pennington (hep-ph/0608016) • The only channel open below elastic limit is elastic scattering, so  phase is same as for elastic scattering. • BUT the interaction between c and P introduces overall phase This might also depend on energy, in which case Watson theorem will not apply. FD FR means on mass shell.

7. MIPWA • Define S–wave amplitude at discrete points sK=sj. Interpolate elsewhere.  model-independent - two parameters (ccj, j) per point • P- and D-waves are defined by known K* resonances and act as analyzers for the S-wave.

8. MIPWA • Phases are relative to K*(890) resonance. • Un-binned maximum likelihood fit: • Use 40 (cj, j) points for S • Float complex coefficients of KK*(1680) and K2*(1430) resonances • 4 parameters (d1680, 1680) and (d1430, 1430) ! 40 x 2 + 4 = 84 free parameters.

9. MIPWA – E791 Mass Distributions E791 15,079 signal events 94% purity 2/NDF = 272/277 (48%) S Phys.Rev.D73:032004,2006

10. Phases for S-, P- and D-waves are compared with measurements from LASS. S-wave phase S for E791 is shifted by -750 wrt LASS fs energy dependence differs below 1100 MeV/c2. P-wave phase does not match very well above K*(892) Probably artifact of model used Lower arrow is at  threshold Upper arrow is at effective limit of elastic scattering observed by LASS. Watson Theorem - a direct test ? Elastic limit Kh’ threshold S P K1*(1410) D

11. A good fit was also made by constraining the shape of the S-wave phase to agree with that from K-+ scattering However: S-wave phase S for E791 still shifted by -750 wrt LASS fP match is even worse above K*(892) fD phase also shifts. Watson Theorem Enforced for S-wave S Elastic limit Kh’ threshold (1454 MeV/c2) P D

12. The BaBar Sample of D+K-++ • Skim carried out byRolf Andreassen. • A likelihood is based on PDFS (signal - MC) and PDFB (background - data sidebands) for each of the following quantities: • SignedD+ decay length l/sl • c2 probability for vertex • PLAB for D+ • Likelihood is product: Skim all with L>2

13. Rolf Andreassen’s Skim • K-p+p+ invariant mass vs. likelihood (L) (NOTE log scale). • Some purities:

14. D+ K-++ Dalitz Plot • Plot includes 500K events • of which 13K are background. • Obviously large S-wave content Interferes with K*(890) (and anything else in P-wave). • Some D-wave also present

15. Max. Likelihood Fit • Likelihood function covers 3-dimensions: • sK1, sK2 and also the reconstructed 3-body mass MK • Factorize MK dependence: • All events used in signal as well as sidebands have a D+ mass constraint. • Makes it possible to overlay Dalitz plot for sideband data directly on signal • Greatly simplifies computation of efficiency. • is efficiency Subscript s is signal Subscript b is background

16. Background Model • K-p+p+ invariant mass distribution from sample with L > 3 • Dalitz plot distributions in lower side-band, signal region and upper side-band (log. Scale) • Used directly as input to background function. PDF1b - bin-by-bin interpolation

17. Second Background • Probable origin • PDF2b = g(MK) x Gauss (M2K) Lost

18. Efficiency • Efficiency (%) over the Dalitz plot for various laboratory momentum ranges.

19. Efficiency vs. pLAB • Efficiency (%) vs laboratory momentum. • Lab. momentum for Data (black). • Lab. momentum for reconstructed, signal MC (red).  No need to use efficiency as function of pLAB

20. “Traditional”  Model for S-wave - BaBar 2/NDF = 20.1x103 / 15.6x103 - a very poor fit

21. Partial Waves from  Model Fit Phase Magnitude NOTE – no K*(1410) Width of lines represents 1

22. E791 S-Wave Fit (on BaBar data) • S-wave is spline with 30 equally spaced points • P-wave is as in  model fit, with complex coefficients floated. • D-wave also as in  model fit – complex coefficient floated.

23. E791 S-Wave Fit (on BaBar data) 2/NDF = 1007/574 – better, but still a poor fit

24. The P-wave Problem • Problem is – the S-wave solution depends on assumptions made about the reference P-wave. • Models: • Add K1*(1410) RBW - this crowds the wave - SHOWN HERE • Could use K-matrix to avoid this – TO BE TRIED • Use LASS phases up to elastic limit (~110 MeV/c2) – TO BE TRIED BUT these all just transfer the ignorance. • Parametrize as table of complex values (spline) as for S-wave: • Sn(s) = splinen(s) - spline defined by n points. • Pm(s) = RBW[K*(890)] x splinem(s) • Not much progress on this yet.  Uniqueness problem ? – can it be done at all?

25. Add K*(1410) to P-wave - BaBar 2/NDF = 18.8x103 / 15.5x103 – Better, but still a very poor fit

26. Spline for P-wave - Fit Procedures • Make E791 fit • Sn(s) from a table of n points. • Fix S-wave and fit P-wave same way • Pm(s) from a table of n points. • Fix P-wave and re-fit S-wave • Repeat cycle several times • SIMPLEX • Errors from likelihood scan • Alternatives • Cycle Magnitudes (both waves) and phases (both waves). • Use Binned likelihood – WORKS WELL • 2 fit. – WORKS LESS WELL Antimo’s procedure

27. |S| S phase S P phase |P| P Antimo’s Result

28. Some MC Tests • Generate toy MC corresponding to  model fit to data • No background • Look for self-consistency between fit and generated quantities

29. Kappa Model Test Output Input

30. MC Test – S-wave Only • S-wave is fitted tospline with 40 equally spaced points • P-wave is fixed as in  model fit (but defined as a spline). • D-wave complex coefficient floated.

31. MC Test – P-wave Only • S-wave is fixed as in  model fit. • P-wave is fitted tospline with 40 equally spaced points • D-wave complex coefficient floated.

32. MC Test – Migrad S-P Cycles Cycling does work, but convergence seems far away even after 16 cycles! S -2lnL P S etc # Function Calls

33. Mag -2lnL Phase Mag etc # Function Calls MC Test –Magn./Phase Cycles Cycling does work, but convergence seems far away even after 16 cycles!

34. BUT – Float S- and P-waves Together !! Maybe it is not possible tofindboth S- and P-wave amplitudes without a definite form for one of them.

35. MIPWA for Both S- and P-waves? • Regions exist where the P-wave is much smaller than the S-wave This makes phase measurements more difficult to make

36. Summary • So far, we have gone as far as E791 did, but the next steps need some more work. • We conclude that the S-wave can be well determined if the P-wave is known • Understanding the P-wave is a challenging problem. • New ideas how to parametrize the P-wave need be considered.