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Low Mass S -wave K  and  Systems

Low Mass S -wave K  and  Systems. Brian Meadows University of Cincinnati. Outline. S- waves in Heavy flavour physics ? What is known about S-wave  -  + and K -  + scattering Information from Charm Decays Some new results Summary and Discussion.

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Low Mass S -wave K  and  Systems

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  1. Low MassS-wave K and  Systems Brian Meadows University of Cincinnati Brian Meadows, U. Cincinnati.

  2. Outline • S- waves in Heavy flavour physics ? • What is known about S-wave -+ and K-+ scattering • Information from Charm Decays • Some new results • Summary and Discussion 1See ArXiv:hep-ex/0507099 – E791 coll. & W.M. Dunwoodie - submitted to PRD. Brian Meadows, U. Cincinnati

  3. S-waves in Heavy flavour physics ? • Precise knowledge of low mass K and S- wave systems is important for understanding the spectroscopy of scalar mesons – existence of low mass  or  states in particular but this is not covered in this talk. • Interference in hadronic final states through Dalitz plot analyses plays a major role in studying much that is new in flavour physics, for example: • CKM  • D0-D0 mixing • Sign of cos2, etc…. General belief is that P-andD-waves are well described by resonance contributions, but that better ways to parameterize the S-wave systems are required. • Anyway, theS- wave nearly always seems to be wherever you want to look Brian Meadows, U. Cincinnati

  4. L = 0 L = 0 Phase 0 |T | Phase degrees |T | M (K-p+) (GeV/c2) M (K-p+) (GeV/c2) What is Known about K p Scattering ? SLAC/LASS experiment E135:K -p K -p+n (11 GeV/c) NPB 296, 493 (1988) +++ Total S-wave +++I = 1/2 +++I = 3/2 I =3/2 Phase 0 K +p K +p+n K -p K –p-D++ I- spins are separated using I=3/2 phases from K +p K +p+n andK -p K –p-D++(13 GeV/c) M (K§p§) (GeV/c2) No evidence for k(800) – yet ~no data below 825 MeV/c2 either. Estabrooks, et al, NP B133, 490 (1978) Brian Meadows, U. Cincinnati

  5. Effective Range Parametrization (LASS) NPB 296, 493 (1988) • Scattering amplitude is unitary (elastic) up to K’ threshold (for even L): • Used, with complex coefficient, in several published Dalitz plot analyses – yet only valid below ~1460 MeV/c2. where: • S-wave (I = 1/2): • S-wave (I = 3/2): No resonances: One resonance: M0 ~1435 ; 0 ~275 MeV/c2 a “scattering lengths” b “effective ranges” Some use  PT to extrapolate from 825 MeV/c2 to threshold. Brian Meadows, U. Cincinnati

  6. Im T B. Hyams, et al, NP B64, 134 (1973) KK Threshold Re T p S-wave Scattering (I = 0) Excellent Data from - p  - + n G. Greyer, et al, NP B75, 189-245 (1975) (several analyses - including other reactions) I = 0 Au, Morgan, Pennington, PR D35, 1633-1664 (1987) d00 (degrees) c PT KK Threshold M(pp) (MeV/c2 No evidence for s(500) – essentially no data below 500 MeV/c2 either. Brian Meadows, U. Cincinnati

  7. d02 02 (degrees) p S-wave Scattering (I = 2)from N. Achasov and G. Shestakov, PRD 67, 243 (2005) Data included in fit: + p  + + n (12.5 GeV/c) + d  - - ppspec(9 GeV/c) NOTE - d02is negative. Fit assumes amplitude to be unitary: W. Hoogland, et al, NP B69, 266-278 (1974) N. Durusoy, et al, PL B45, 517-520 (1973) Reasonable assumption up to r§r§ threshold Brian Meadows, U. Cincinnati

  8. {12} {23} {13} 1 1 1 2 2 2 3 3 3 1 3 S- waves in D Decays • The “isobar model”, with Breit-Wigner resonant terms (BWM), has been widely used in studying 3-body decays of heavy quark mesons. • Amplitude for channel {ij}: • Each resonance “R” described by Breit-Wigner form (mass mR, width R) NR 2 D form factor spin factor R form factor NR Constant Brian Meadows, U. Cincinnati

  9. Channel with Large S-wave Component D + K -++ (shown to right) Prominent feature: • Strong asymmetry inK*(892) bands • F-B asymmetry vs. K*(892)Breit-Wigner phase (inset) is zero at 560. • (Differs from LASS where this is zero at 135.50  Interference with large S–wave component.  Shift in S–Prelative phase wrt elastic scattering by -79.50 Asymmetry M 2(K -+) BW M 2(K -+) Another channel with similar features w.r.t. the 0(770) is D+ -++ Brian Meadows, U. Cincinnati

  10. E791 Require s(500) in D+ p-p+p+ E. Aitala, et al, PRL 86:770-774 (2001) Withouts(500): • NR ~ 40% dominates • r (1400) > r (770) !! • Very Poor fit (10-5 %) Observations: • NR and s phases differ by ~ 1800 • Inclusion of k makes K0*(1430) parameters differ greatly from PDG or LASS values. Phase 0 Fraction % With  S~116 % No  c2/d.o.f. = 0.90 (76 %) This caught the attention of our theorist friends ! Brian Meadows, U. Cincinnati

  11. … and k(800) in D+ K-p+p+ E. Aitala, et al, PRL 89 121801 (2002) Withoutk(800): • NR ~ 90% • Sum of fractions 130% • Very Poor fit (10-5 %) Observations: • NR and k phases differ by ~ 1800 • Inclusion of k makes K0*(1430) parameters differ greatly from PDG or LASS values. Phase 0 Fraction % S~89 % M1430 = 1459§7§12 MeV/c2 G1430 = 175 § 12 § 12 MeV/c2 Mk = 797§19§42 MeV/c2 Gk = 410 § 43 § 85 MeV/c2 c2/d.o.f. = 0.73 (95 %) This also caught the attention of our theorist friends ! Brian Meadows, U. Cincinnati

  12. Recent BWM Fits to D0 K-p+p+ • NEW RESULTS from both FOCUS and CLEO c support similar conclusions: •  required (destructively interferes with NR) to obtain acceptable fit. • K0*(1430) parameters significantly different from LASS. These BW parameters are not physically meaningful ways to describe true poles in the T- matrix. Can no longer describe S-wave by a single BW resonance and flat NR term for either K -+ or for -+ systems.  Search for more sophisticated ways to describe S- waves. Brian Meadows, U. Cincinnati

  13. Charged (800) ? Babar: D0 K-K+0 Tried three recipes for K§0S-wave: • LASS parametrization • E791 fit • NR and BW’s for  and K0*(1430) • Best fit from #1. • No need for + nor -, though not excluded: Fitted with: M = (870§ 30) MeV/c2,  = (150§ 20) MeV/c2 11,278 § 110 events (98% purity) 385 fb-1: PRC-RC 76, 011102 (2007) Brian Meadows, U. Cincinnati

  14. Partial Wave Analysis in D0 K-K+0 • Region under  meson is ~free from cross channel signals: allows Legendre polynomial moments analysis inK-K+channel: (Cannot do this is K channels) p- s | S | | P | (in K –K + CMS) where • |S| consistent with either • a0(980) or f0(980) lineshapes. Babar: 385 fb-1: PRC-RC 76, 011102 (2007) Brian Meadows, U. Cincinnati

  15. E791 Quasi-Model-IndependentPartial Wave Analysis (QMIPWA) • Re-arrange sum in BWM amplitude into terms of same angular momentum L of K -+channel D form factor spin factor Decay amplitude: S-wave (L = 0): ReplaceBWMby discrete points cnein P-orD-wave:Define as inBWM Parameters (cn, n) provide quasi-model independent estimate of total S- wave (both I- spins). These values do depend onP-orD-wave models. Brian Meadows, U. Cincinnati

  16. Best fit floats: S-wave: All (c, g)values P- & D-waves: All (d, d) coeffs. Red curves delineate BWM fit. Completely flexible S-wave changes P- & D-waves to accommodate. arg{F(s)} S P D E791 Phys.Rev. D 73, 032004 (2006) (Quasi-) Model-Independent PWA Similar fit by FOCUS confirms these results: Brian Meadows, U. Cincinnati

  17. Scattering and 3-body Decays • Decay amplitude for channel k is related to scattering amplitude T(s): Ff (s)=Tfk (s)Qk (s) • Weak decay/fragmentation: • I-spinnot conserved • kscattering on +during • fragmentation can impart • an overall phase D + p+ Q T Scattering: kf f k K- p+  Watson theorem: Up to elastic limit (for each L and I ): K -+phase has same s- dependence as elastic scattering apart from overall phase. Brian Meadows, U. Cincinnati

  18. S-wave phase for E791 is shifted by –750 wrt LASS. Energy dependence compatible above ~1100 MeV/c2. Parameters for K*0(1430) are very similar – unlike the BWM Complex form-factor for the D+ 1.0 at ~1100 MeV/c2? Compare D+ K-p+p+ with LASS |F0 (s) | arg{F0 (s)} E791 LASS Not obvious if Watson theorem is broken in these decays ? Brian Meadows, U. Cincinnati

  19. … Is Watson Theorem Broken ? • E791 concludes: “If the data are mostlyI= 1/2, this observation indicates that the Watson theorem, which requires these phases to have the same dependence on invariant mass, may not apply to these decays without allowing for some interaction with the other pion.” • Point out that their measurement can include an I =3/2 contribution that may influence any conclusion. • Note: • They also make a perfectly satisfactory fit (c2 / n = 0.99) in which the S-wave phase variation is constrained to follow the LASS shape up to Kh’ threshold. Brian Meadows, U. Cincinnati

  20. FOCUS / Pennington: D K-++ arXiv:0705.2248v1 [hep-ex] May 15, 2007 • Use K-matrix formalism to separate I-spins in S-wave. • The K-matrix comes from their fit to scattering data T(s) from LASS and Estabrooks, et al: Extend T(s) toKthreshold usingPT I= 1/2: 2-channels (K and K’ ) one pole (K *1430) I= 3/2: 1-channel (K only) no poles • This defines the D+ decay amplitudes for each I-spin: where T- pole is at: 1.408 – i 0.011 GeV/c2 Brian Meadows, U. Cincinnati

  21. FOCUS / Pennington: D K-++ arXiv:0705.2248v1 [hep-ex] May 15, 2007 • Amplitude used in fit: • P- vectors are of form: that can have s-dependent phase except far from pole. Usual BWM model for P- and D- waves I- spin 1/2 and 3/2 K-+ S-wave k=1K ; k=2K’ Same as pole in K-matrix Brian Meadows, U. Cincinnati

  22. FOCUS / Pennington: D K-++ • Fit without I= 3/2 much worse - P(2) 1.2%  10-5. • P-and D-wave fractions & phases almost same as BWMfit. TotalK-+ S-wave I=1/2K-+ S-wave Large Data sample: 52,460§245 events (96.4% purity) S- wave phase(deg.) arXiv:0705.2248v1 [hep-ex] 2007 LASS I=1/2 phase s 1/2 (GeV/c2) S-wave fractions (%):I=1/2:207.25 § 24.45 § 1.81 § 12.23 I=3/2: 40.50 § 9.63 § 0.55 § 3.15 stat. syst. Model Brian Meadows, U. Cincinnati

  23. FOCUS / Pennington: D K-++ arXiv:0705.2248v1 [hep-ex] May 15, 2007 Observations: • I=½ phase does agree well with LASS as required by Watson theorem Except near pole (1.408 GeV/c2) • BUT is this just be built in to the fit model ? • Huge fractions of each I- spin interfere destructively. Brian Meadows, U. Cincinnati

  24. CLEO c: D K-++ arXiv:0707.3060v1 [hep-ex] Jul 20, 2007 • Very clean sample from (3770) data: 67,086 events with 98.9 % purity. • BWM fit similar to E791 • K* (1410) in P- wave not required • However, (800) in S- wave (as a Breit-Wigner) is. Brian Meadows, U. Cincinnati

  25. CLEO c: D K-++ arXiv:0707.3060v1 [hep-ex] Jul 20, 2007 • BWM fit is also significantly improved by adding I=2 ++ amplitude – repairs poor fit to ++ inv. mass spectrum. • Best fit uses a modification of E791 QMIPWA method … QIMPWAfit BWMfit Brian Meadows, U. Cincinnati

  26. CLEO c: D K-++ arXiv:0707.3060v1 [hep-ex] Jul 20, 2007 • QMIPWA (E791 method applied to all waves and channels!) Define wave in each channel as: F(s) = aei C(s) + bei R(s) • Total of ~ 170 parameters: Breit-Wigner type of propagator: K-+ S- wave – K0*(1430) K-+P- wave – K*(890) D- wave – K2*(1420) ++S- wave – R = 0 Interpolation table (26 complex values) BUT – only float C(s) for one wave at a time. Brian Meadows, U. Cincinnati

  27. Total S- wave from D+ K-++ Decays • General agreement • is good • All differ from LASS • (blue curves, 2nd row) CLEO c (Solid line) arXiv:0707.3060v1, 2007 E791 (Error bars) Phys.Rev.D73:032004, 2006 FOCUS (Range) arXiv:0705.2248v1, 2007 M(K- +) (GeV/c2) Brian Meadows, U. Cincinnati

  28. CLEO c: D K-++ arXiv:0707.3060v1 [hep-ex] Jul 20, 2007 • This is an ambitious fit indeed ! • All waves, including I=2 ++ S-wave, are treated as interpolation tables. • Two questions arise: 1. Are we seeing true convergence? (If not, can we believe the uncertainties?) 2. Are there other solutions? Brian Meadows, U. Cincinnati

  29. New Data from CLEO c: D -++ arXiv:0704.3965v2 [hep-ex] Jul 20, 2007 • Use 281 pb-1 sample (3770): • ~4,086 events including background. • Had to remove large slice in m+- invariant mass corresponding to D+ Ks+ • General morpholgy similar to E791 and FOCUS • Standard BWM fit requires a  amplitude much the same • Introduced several variations in S- wave parametrization: ………………….. FOCUS: Phys.Lett.B585:200-212,2004 E. Aitala, et al, PRL 89 121801 (2002) CLEO c Brian Meadows, U. Cincinnati

  30. Complex Pole for : J. Oller: PRD 71, 054030 (2005) • Replace S- wave Breit-Wigner for by complex pole: • Best fit: arXiv:0704.3965v2 [hep-ex] Jul 20, 2007 Brian Meadows, U. Cincinnati

  31. Linear  Model inspired Production Model Black, et al. PRD 64, 014031 (2001), J. Schecter et al., Int.J.Mod.Phys. A20, 6149 (2005) Replace S- wave  and f0 (980)by weakly mixed complex poles: • Full recipe includes both weak and strong mixing between and f0(980) – 7 parameters in all arXiv:0704.3965v2, 2007 Weakly mixed Poles  and f0(980) Unitary . . .+ usual BW terms for f0 (1350) and f0 (1500) % % % • Excellent fit: Brian Meadows, U. Cincinnati

  32. CLEO c: D -++ arXiv:0704.3965v2 [hep-ex] Jul 20, 2007 • A fourth, “custom model” for S- wave (Achasov, et. Al., priv. comm.) also gave excellent fit • All models tried (including BWM): • Give essentially the same non S- wave parameters • Provide excellent descriptions of the data Brian Meadows, U. Cincinnati

  33. PRD 71: 032005 (2005) 89 fb-1 CKM Angle  and K-p+ S-wave Babar: • Mass dependence of S- and P-wave relative phase in K-+ system determines sign: cos 2b > 0 • A clear choice agrees with the LASS data with overall shift +p radians. Clearly an interesting way to probe the K- +S- wave Brian Meadows, U. Cincinnati

  34. Measured (or Observed) K p S-wave Compared to LASS Amplitude Use of LASS S-wave parametrization in various Dalitz plot analyses. More channels are needed to understand any pattern. W.M. Dunwoodie, Workshop on 3-Body Charmless B Decays LPHNE, Paris, Feb. 1-3, 2006 Brian Meadows, U. Cincinnati

  35. Conclusions • The most reliable data on S- wave scattering are still from LASS or CERN-Munich data. • More information on very low mass data may be accessible through study of • semi-leptonic D decays • larger samples of B  J/ K-(-)+ decays • Extracting model-free wave information should remain the goal in Dalitz plot analyses. • New techniques seem to yield information on the S- wave in a single decay, BUT it is not yet obvious how to carry that over to other decays. • Understanding this will require a systematic study of many more D and B decays. Brian Meadows, U. Cincinnati

  36. Back Up Slides Brian Meadows, U. Cincinnati

  37. Another Solution? Phase Magnitude • Qualitatively good agreement with data • BUT does not give acceptable c2. • This solution also violates the Wigner causality condition. S P S E. P. Wigner, Phys. Rev. 98, 145 (1955) D Brian Meadows, U. Cincinnati

  38. E791 D+!K-p+p+ ~138 % c2/d.o.f. = 2.7 Flat “NR” term does not give good description of data. Brian Meadows, U. Cincinnati

  39. Some Comments • Should the S-wave phase be constrained to that observed in K-+ scattering (Watson theorem)? • Are models of hadron scattering other than a sum of Breit-Wigner terms a better way to treat the S-wave1 • We decided to measure the S-wave phase (and magnitude) rather than use any model for it. 1S. Gardner, U. Meissner, Phys. Rev. D65, 094004 (2002), J. Oller, Phys. Rev. D71, 054030 (2005) Brian Meadows, U. Cincinnati

  40. Helicity angle q in K-p+ system Asymmetry: Asymmetry in K*(892) K- q + p q cosq = p¢q + = tan-1m00/(m02-sK ) LASS finds =0 when BW» 135± !P - s is -750 relative to elastic scattering Brian Meadows, U. Cincinnati

  41. Model-IndependentPartial-Wave Analyses • Make partial-wave expansion of decay amplitude in angular momentum L of produced K-+ system • CL(sK) describes scattering of produced K-+. • Related to amplitudes TL(sK) measured by LASS D form factor spin factor Brian Meadows, U. Cincinnati

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

  43. Does this Work? Phase Magnitude Fit the E791 data: • P and D fixed from isobar model fit with  • Find S(sj) • S and D fixed from isobar model fit with  • Find P(sj) • S and P fixed from isobar model fit with  • Find D(sj)  The method works. S P S D Brian Meadows, U. Cincinnati

  44. Fit E791 Data for S-wave Phase Magnitude • Find S. Allow P and D parameters to float • General appearance of all three waves very similar to isobar model fit. • Contribution of P-wave in region between K*(892) and K*(1680) differs slightly – balanced by shift in low mass S-wave. S P D Brian Meadows, U. Cincinnati

  45. Comparison with Data – Mass Distributions 2/NDF = 272/277 (48%) S Brian Meadows, U. Cincinnati

  46. K- q 2+ p q 1+ Comparison with Data - Moments • Mean values of YL0(cos ) • Exclude K*(890) in K-2+ S Brian Meadows, U. Cincinnati

  47. Main Systematic Uncertainty S-wave Amplitudes: • Even with 15K events, fluctuations in P- and D-wave contributions reflect into S-wave solution. Many 15K samples simulated using the isobar model fit from E. Aitala, et al, PRL 89, 12801 (2002) (first few shown here) • Solutions similar to those observed in data are common. Phase Magnitude S Brian Meadows, U. Cincinnati

  48. Comparison with K-+ Scattering (LASS) • S (sK) is related to K-p+ scattering amplitude T (sK) : • In elastic scattering K-p+!K-p+ the amplitude is unitary • Watson theorem requires that 0(sK) be real: • Phase of TL(sK) should match that of CL(sK). • Applies to each partial wave (L=0, 1, 2, …) Production factor for Ksystem 2-body phase space Measured by LASS K.M. Watson, Phys. Rev. 88, 1163 (1952) Brian Meadows, U. Cincinnati

  49. Production of K-+ Systems • Production factor 0 (sK) is • Value for 0 found by minimizing Summed over measured j’s • 0 = (-123.3 § 3.9 )0 ; Q = 0.74 § 0.01 (GeV/c2)-2. Brian Meadows, U. Cincinnati

  50. Production of K-+ Systems • Plot quantities (sj), evaluated at each sj value, using measured j there. • Roughly constant up to about 1.250 GeV/c2 • Constant = 0.74§ 0.01 (GeV/c2)2. Brian Meadows, U. Cincinnati

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