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Innovative Analysis of K-π+ Scattering via D+ Decays from BABAR Experiment

Explore novel approaches to study 3-body decays using isobar models and model-independent partial-wave analysis. The research involves detailed analysis of K-π+ scattering through D+ decays from the BABAR experiment, emphasizing the significance of traditional versus spline models. By incorporating various resonant terms and form factors, the study aims to enhance the understanding of heavy quark meson decays. Discover the complexities of amplitude descriptions, form factors, and spin factors in relation to the invariant mass dependence of the K-π+ system. Uncover the insights gained from MIPWA analysis and the challenges encountered in fitting S-wave, P-wave, and D-wave components. Marvel at the precision of the undertaken analysis and the implications for future research in the field of particle physics.

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Innovative Analysis of K-π+ Scattering via D+ Decays from BABAR Experiment

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  1. K-+ Scattering Using D+ Decays from BABAR 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)

  6. 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.

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

  8. BaBar Sample • K-p+p+ invariant mass distribution from Rolf’s sample. • A likelihood is based on PDFS (signal - MC) and PDFB (background - data sidebands) for each of the following quantities: • Signed D+ decay length SDZ= l¥ l/sl • c2 probability for vertex • PLAB for D+ • Likelihood is product: Skim all with L>2

  9. Rolf’s Skim • K-p+p+ invariant mass vs. likelihood (L) (NOTE log scale).

  10. 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

  11. 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

  12. Second Background • Probable origin Lost

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

  14. 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

  15. D+ K-++ Dalitz Plot • Obviously large S-wave content Interferes with K*(890) (and anything else in P-wave). • D-wave also present

  16. “Traditional”  Model for S-wave - BaBar 2/NDF = 1443/624 – poor fit

  17. Partial Waves from  Model Fit Phase Magnitude Width of lines represents 1

  18. E791 S-Wave Fit (on BaBar data) • S-wave is spline with 30 equally spaced points • P-wave is as in  model fit. • D-wave also as in  model fit.

  19. Spline Model for S-wave - BaBar 2/NDF = 1007/574 – still a poor fit

  20. What to do with P-wave? • S-wave solution depends on P-wave reference. • Could add K1*(1410) • BUT this crowds the wave. • Try a spline: • Sn(s) = splinen(s) - spline defined by n points. • Pm(s) = RBW[K*(890)] x splinem(s) • Not much progress yet •  Uniqueness problem ??

  21. Double Spline Fit - n x m = 40 x 20 2/NDF = 843/574 – better, but still a poor fit

  22. Double Spline Fit - n x m = 40 x 20 • S-wave is spline with 40 equally spaced points • P-wave is also a spline with 20 equally spaced points x RBW[K*(890)]. • D-wave just as in  model fit.

  23. Double Spline Fit - n x m = 50 x 15 2/NDF = 867/518 – just better, but still a poor fit

  24. Double Spline Fit - n x m = 50 x 15 • S-wave is spline with 40 equally spaced points • P-wave is also a spline with 20 equally spaced points x RBW[K*(890)]. • D-wave just as in  model fit.

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