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3-Body Decays of D and B Mesons

3-Body Decays of D and B Mesons. Brian Meadows University of Cincinnati. Dalitz plot amplitude analyses The tools we use Scalar states What is known about S- wave  -  + and K -  + scattering and how this should apply to D (and B ) decays Recent analyses from Babar

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3-Body Decays of D and B Mesons

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  1. 3-Body Decays of D and B Mesons Brian Meadows University of Cincinnati • Dalitz plot amplitude analyses • The tools we use • Scalar states • What is known about S- wave -+ and K -+ scattering and how this should apply to D (and B ) decays • Recent analyses from Babar • CP Violation measurements from D decays • D0-D0 Mixing measurements • Summary Brian Meadows, U. Cincinnati

  2. B and DDalitz Plots • These decays are dominated by resonance structures • Analysis of their phase and amplitude structure can provide information on: • Presence of CP Violation in D decays • D0 Mixing parameters • Scalar (and other) light quark resonances • CP Violation parameters (e.g. CKM °) • Decay couplings • S-waves are both ubiquitous and “useful” • A suitable parametrization is to be found • Interference with P –wave resolves ambiguity in the sign of cos2¯ in BdÃK* analysis • Likewise, in K+K- system. Resolves sign of cos2¯s in BsÃÁ analysis • As larger data samples are examined, and rarer phenomena sought, more care will need to be taken with the analysis details BK+¼-¼0 DK-¼+¼0 Same Model as B Decay Brian Meadows, U. Cincinnati

  3. Analyses of 3-body Decays Analyses in use fall into two categories: • Model Dependent - assume isobar (quasi-two body) decay • Using Breit-Wigner forms for resonances (BWM) • K-matrix form for resonance and background • Model Independent (or quasi-independent) • Moments analysis (in restricted parts of the phase space) • Quasi Model independent partial wave analysis (QMIPWA) for a single wave. • Attempts to extend this to more than one wave are, so far, questionable. Other hybrid schemes are also used.

  4. Isobar (Quasi-two Body) Model • Decays (e.g. D+K-¼+¼+) have amplitudes F(s) related to final state scattering amplitude T(s)by: Ff (s)=Tfk (s)Pk (s) Intermediate states • Weak decay/fragmentation: • I-spinnot conserved • kscattering on +during • fragmentation can impart • an overall phase D + p+ T P k Scattering: kf f K- p+  Watson theorem: Up to elastic limit (for each L and I ) K -+phase has same dependence on s as elastic scattering but there can be an overall phase shift. Behaviour of P(s) is unknown. Brian Meadows, U. Cincinnati

  5. Breit-Wigner Model “BWM” • The BWM ignores re-scattering, and problems of double-counting: • Amplitude for channel {ij}with angular momentumL: • In the BWM each resonance “R” (mass mR, width R) described as: • Lots of problems with this theoretically – especially in S- wave 2 {12} {23} {13} “NR” 1 1 1 2 2 2 3 3 3 1 3 2 NR - constant (L=0) D form factor spin factor R form factor Mass-Dependent width Resonance Mass Brian Meadows, U. Cincinnati

  6. K-Matrix Model This was designed to overcome the major problem in the BWM of broad, overlapping resonances • T(s) is not unitary in BWM, but is in the K-matrix model For selected partial wave (usually only one) Introduce scattering, making T unitary: D + p+ I.J.R. Aitchison, Nucl. Phys. A189, 417 (1972) T P k Scattering: kf f p - p+ Ff=kPkTkf ; Tkf = [I – i½ Kkf]-1 K=RgRkf/ [mR2 –s]+ background ½ = phase space; g = coupling V.V. Anisovitch, A.V Sarantev Eur. Phys. Jour. A16, 229 (2003) Brian Meadows, U. Cincinnati

  7. Moments Analysis Plot m(KK)0, weighting events by factors YL0(cos )/ to obtain “moments <YL0(m)>” ( is “efficiency”) • First used on D0K0K+K- decays by Babar PRD 72, 052008 (2005) • Key assumption is that, in low mass K+K- region, resonances only from that channel play role • Resonances from other two channels would contribute to many higher “moments”. K+K- CMS Helicity angle  in K+K- channel

  8. Moments Analysis Higher moments are consistent with zero Moments<YL0> for L=0, 1 and 2 are related to S- and P-wave amplitudes S and P, with relative phase SP by S |P|2 S*P P-wave phase assumed to be dominated by  meson Breit-Wigner form Same technique also applied to K+K0 channel

  9. S-waves in Heavy Flavor Decays • The S- wave is both ubiquitous and “useful” • Interference in hadronic final states through Dalitz plot analyses plays a major role in studying much that is new in flavor physics: • CKM  • D0-D0 mixing • Sign of cos2, etc…. • Low mass K and S- wave systems are of intrinsic interest and important for understanding the spectroscopy of scalar mesons – existence of low mass  or  states in particular • This is not covered in this talk, though a review of recent theoretical and experimental efforts focussing on pole parameters for (476–628)− i (226–346) and of  (694-841)-i(300-400) MeV/c2 cites many of the relevant references: D. V. Bugg, J. Phys. G 34, 151 (2007). • 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 as our targets become more precise. • This talk focusses on recent attempts to improve on this situation. Brian Meadows, U. Cincinnati

  10. S-waves in Heavy Flavor Decays • The S- wave is both ubiquitous and “useful” • Interference in hadronic final states through Dalitz plot analyses plays a major role in studying much that is new in flavour physics: • CKM  • D0-D0 mixing • Sign of cos2, etc…. • Low mass K and S- wave systems are of intrinsic interest and important for understanding the spectroscopy of scalar mesons – existence of low mass  or  states in particular • This is not covered in this talk, though a review of recent theoretical and experimental efforts focussing on pole parameters for (476–628)− i (226–346) and of  (694-841)-i(300-400) MeV/c2 cites many of the relevant references: D. V. Bugg, J. Phys. G 34, 151 (2007). • 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 as our targets become more precise. • This talk focusses on recent attempts to improve on this situation. Brian Meadows, U. Cincinnati

  11. 3-Body Decays of D and B Mesons • These decays provide information on • D0-D0 mixing • CKM  • Sign of cos2, etc…. • Scalar mesons • 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 as our targets become more precise. • This talk focusses on recent attempts to improve on this situation. Brian Meadows, U. Cincinnati

  12. 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

  13. Effective Range Parametrization (LASS) NPB 296, 493 (1988) • Scattering amplitude is unitary (elastic) up to K’ threshold (for even L): • Strictly, 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” Brian Meadows, U. Cincinnati

  14. 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

  15. 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

  16. How This Should Apply to 3-body D Decays • Decays have amplitudes F(s) related to scattering amplitude T(s)by: Ff (s)=Tfk (s)Qk (s) Intermediate states • Weak decay/fragmentation: • I-spinnot conserved • kscattering on +during • fragmentation can impart • an overall phase D + p+ Q T k Scattering: kf f K- p+  Watson theorem: Up to elastic limit (for each L and I ) K -+phase has same dependence on s as elastic scattering but there can be an overall phase shift. Behaviour of Q(s) is unknown. Brian Meadows, U. Cincinnati

  17. Conventional Approach – Breit-Wigner Model “BWM” • The “isobar model” ignores all this, and problems of double-counting: • Amplitude for channel {ij}with angular momentumL: • In the BWM each resonance “R” (mass mR, width R) described as: • Lots of problems with this theoretically – especially in S- wave 2 {12} {23} {13} “NR” 1 1 1 2 2 2 3 3 3 1 3 2 NR - constant (L=0) D form factor spin factor R form factor Brian Meadows, U. Cincinnati

  18. Study D Decay Channels withLarge 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 E791 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

  19. k(800) in BWM Fit to D+ K-p+p+ E791: E. Aitala, et al, PRL 89 121801 (2002) Withoutk(800): • NR ~ 90% • Sum of fractions 130% • Very Poor fit (10-5 %) BUT • 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 Similarly, (500) is required in D+ -++ E791: E. Aitala, et al, PRL 86:770-774 (2001) c2/d.o.f. = 0.73 (95 %) Can no longer describe S-wave by a single BW resonance and constant NR term for either K -+ or for -+ systems. Search for more sophisticated ways to describe S- waves Brian Meadows, U. Cincinnati

  20. New BWM Fits Agree • 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. FOCUS - arXiv:0705.2248v1 [hep-ex] 2007 CLEO c - arXiv:0707.3060v1 [hep-ex] 2007 Brian Meadows, U. Cincinnati

  21. E791 Quasi-Model-IndependentPartial Wave Analysis (QMIPWA) E791 Phys.Rev. D 73, 032004 (2006) • Partial Wave expansion in angular momentum L of K -+channels from D+ K-1+2+ decays Bose Symmetrization Decay amplitude : S-wave (L = 0): ReplaceBWMby discrete pointscne in P-orD-wave: Define as inBWM Parameters (cn, n) provide quasi-model independent estimate of total S- wave (sum of both I- spins). (S-wave values do depend onP- and D- wave models). Brian Meadows, U. Cincinnati

  22. 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 QMIPWA with LASS for S-wave |F0 (s) | arg{F0(s)} E791 LASS Not obvious if Watson theorem is broken in these decays ? Brian Meadows, U. Cincinnati

  23. Watson Theorem Applicability or I = 3/2 ? FOCUS / Pennington: D K-++arXiv:0705.2248v1 [hep-ex] 2007 K-matrix fit using LASS Data ForI=1/2production vector: Includes separateI=3/2wave  Bigimprovement in 2. LASS I=1/2 phase S- wave phase(deg.) TotalK-+ S-wave I =1/2K-+ S-wave Large Data sample: 52,460 ± 245 events (96.4% purity) s 1/2 (GeV/c2) Observations: I=½ phase does agree well with LASS as required by Watson theorem except near pole (1.408 GeV/c2) This possibility is built in to the fit model Huge fractions of each I- spin interfere destructively. What about P- wave ? 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 P-and D-wave fractions & phases ~same as BWMfit. 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 • (800) in S- wave is required (as a Breit-Wigner) with NR. • K* (1410) in P- wave not required 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. 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

  27. 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) = C(s) + ae i 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) • Is final fit converged. (Errors?) • Is solution unique? • Is I=2 wave over-constraint? BUT – only float C(s) for one wave at a time. Brian Meadows, U. Cincinnati

  28. Data from CLEO c: D -++ arXiv:0704.3965v2 [hep-ex] Jul 20, 2007 BWM fits • 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

  29. 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

  30. 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

  31. 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

  32. BW Isobar Model Analysis of D0 KsK-K+ BaBar: Phys.Rev.Lett. 105 081803 (2010) 79,900 Events, 468.5 fb-1. Isobar Model Parameters: a0(980)+ and a0(980) Parameters: a0/f0(980)0 • Primary goal was to determine CKM  and D0 mixing parameters • No distinction between a0/f0(980) was attempted. Brian Meadows, U. Cincinnati

  33. Earlier Moments Analysis of D0 KsK-K+ S |P|2 S*P BaBar: Phys.Rev.D72:052008,2005 12,500 Events, 91.5 fb-1. _ Low KKinvariantmass parts of Dalitz plot allow PWA of S- and P-waves: |S|2 _ (in KK CMS) where Shapes of K+K- and K+K0 are much the same ! Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  34. This suggests that EITHER Both S-wave charges + and 0 are from a0(980)  there is no f0(980) OR The the a0 and f0(980) have the same line shape However, if there is no f0(980), then it is difficult to understand why a 3f0(1370) signal is required. Perhaps, indeed  The a0(980) and f0(980) have the same line shape (but different couplings to KK, , ) ?

  35. Ds -++ f0(980)0 BaBar: Phys.Rev.D79:032003,2009 13,000 Events, 384 fb-1. Isobar Model Parameters: and a0(980) Parameters: • Primary goal was to determine CKM  and D0 mixing parameters • No distinction between a0/f0(980) was attempted. Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  36. Ds -++ BaBar: Phys.Rev.D79:032003,2009 13,000 Events, 384 fb-1. • S-wave amplitude extracted using E791-style MIPWA analysis • P-wave parametrized as linear combination of RBW’s Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  37. Ds K –K ++ f0(980)0 BaBar: Phys.Rev.D79:032003,2009 13,000 Events, 384 fb-1. Isobar Model Parameters: and a0(980) Parameters: • Primary goal was to determine CKM  and D0 mixing parameters • No distinction between a0/f0(980) was attempted. Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  38. Moments Analysis in D+ K-K++ K++channel has no resonances Remove  meson in K+K+channel Allows Legendre polynomial moments analysis inK-+channel free from cross-channel: where Focus: hep-ex/0612032v1 (2007) 6400 Events before  cut. • |S|similar to LASS • Phase was not computed, but appears to be shifted~900 wrt LASS. (in K –+ CMS) |S|2 |P|2 S*P Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  39. S- Wave in B J/K+- • Similar analysis (more complex due to vector nature of J/) on K-p+ system • Mass dependence of S- and P-wave relative phase in K-+ system was used to determine 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 PRD 71: 032005 (2005) 89 fb-1 Brian Meadows, U. Cincinnati

  40. S- Wave in D+ +K+- Phys.Lett.B621:72-80,2005 • The FOCUS experiment first pointed out the FB asymmetry in K-p+ system in these decays arising from interference with the S-wave. This was observed to follow the LASS behaviour. Clearly an interesting way to probe the K- +S- wave Brian Meadows, U. Cincinnati

  41. K-+ S-wave Phase from D+K -+e+ºe and D+K -++ Decays • New data on from Babar extract S-wave phase information in 5-variable fit • Use P-wave model: F_{11} Brian Meadows, U. Cincinnati

  42. K-K+ S-wave from Ds+K -K+e+e • Babar data see some evidence for a small S-wave background under the . • S-wave amplitude information could not be extracted with Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  43. Some Kp S-wave MeasurementsCompared to LASS Amplitude Use of LASS S- wave parametrization or determination of relative S-P phase in various Dalitz plot analyses leads to a confusing picture. More channels are needed to understand any pattern. Adapted from W.M. Dunwoodie, Workshop on 3-Body Charmless B Decays, LPHNE, Paris, Feb. 1-3, 2006 Brian Meadows, U. Cincinnati

  44. 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 • New techniques seem to yield information on the S- wave in various decay modes, BUT it is not yet obvious how to carry that over information from one decay to another. • Understanding this will require a systematic study of many more D and B decays • This should remain a goal before it becomes a limiting systematic uncertainty in other heavy flavour analyses. Brian Meadows, U. Cincinnati

  45. Back Up Slides Brian Meadows, U. Cincinnati

  46. 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 rotated by ~-900. • No need for + nor -, though not excluded: Fitted with: M = (870± 30) MeV/c2,  = (150± 20) MeV/c2 ? 11,278 ± 110 events (98% purity) ? Not consistent With “” 385 fb-1: PRC-RC 76, 011102 (2007) Brian Meadows, U. Cincinnati

  47. 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

  48. Red curves are ±1 bounds on BWM fit. Black curves are ±1 bounds on QMIPWA fit. Completely flexibleS-wave changesP-&D-waves. Compare QMIPWA with BWM Fit arg{F(s)} S P D E791 Phys.Rev. D 73, 032004 (2006) (S-wave values do depend on P- and D- wave models). Brian Meadows, U. Cincinnati

  49. 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

  50. 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

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