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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 Recent results from BaBar D s +   -  +  + D s +  K - K +  + D s +  K s  +  - D 0 -D 0 Mixing measurements CP Violation measurements from D decays

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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 • Recent results from BaBar • Ds+-++ • Ds+K- K++ • Ds+Ks+- • D0-D0 Mixing measurements • CP Violation measurements from D decays • A way forward and D+K-+- ? • 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 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 and difficult to make work. Other hybrid schemes are also used. Brian Meadows, U. Cincinnati

  4. Isobar (Quasi-two Body) Model • Decays (e.g. D+K-¼+¼+) have amplitudes F(s), s=sK-¼+,are 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. NR - constant (L=0) D form factor spin factor R form factor Mass-Dependent width Resonance Mass Breit-Wigner Model “BWM” • The BWM ignores re-scattering, and problems of double-counting: • Amplitude is sum over channels {ij}and the isobarsR in each: • In the BWM each resonance “Rij” (mass mR, width R) has mass dependence: 2 {12} {23} {13} “NR” 1 1 1 2 2 2 3 3 3 1 3 2 Ak Lots of problems with this theoretically – especially in S- wave Brian Meadows, U. Cincinnati

  6. Make Tkf unitary: K is real  is phase space factor. Define production vector for complex couplings gk D + p+ Tkf k Pk f p - p+ K-Matrix Model • Designed to overcome problem in BWM of broad, overlapping resonances in a partial wave • Usual recipe: • Usually only applied to one partial wave in one channel I.J.R. Aitchison, NP A189, 417 (1972) T (s) is not unitary in BWM For each K-matrix partial wave Same as BWM but exclude isobars in KM waves Brian Meadows, U. Cincinnati

  7. PWA (Moments) Analysis B0K+¼-¼0 s(K+0) (GeV/c2)2 s(K+-) (GeV/c2)2 • Works only where just ONE channel has significant isobar contributions Would NOT work here due to 0, K *-,K2*-in other channels Would work here K2*  • Slice selected regions of Dalitz plot into invariant mass strips. • In each invariant mass strip: • Compute moments <YL0> =  YL0(cosµ) for all events in bin • Solve for partial waves S, P, ... Using the relationship: • Isobars from other two channels contribute many higher “moments”. Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  8. PWA (Moments) Analysis Moments with L > 2 are consistent with zero (for m < 1.2 GeV/c2)  Lmax = 2 • First used on 3-body D0 decays by Babar PRD 72, 052008 (2005) • Plot m(K+K-), weighting events by factors YL0(cos )/ to obtain “moments <YL0(m)>” K+K-CMS Helicity angle  in K+K- channel Brian Meadows, U. Cincinnati

  9. Moments Analysis 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 Brian Meadows, U. Cincinnati

  10. The E791 collaboration introduced the QMIPWA, to measure the K-+S-wave from D+K-++ decays in a way that required no knowledge of its isobar structure. The BWM for the S-wave was replaced by a spline defined by complex numbers (fit parameters) defined at discrete invariant masses. While the moments method can decompose multiple partial waves in small regions, the E791 method takes on other channels over the whole Dalitz plot as a kind of background, describing them by BWM isobars. E791 Quasi-Model-IndependentPartial Wave Analysis (QMIPWA) E791 Phys.Rev. D 73, 032004 (2006) The D+K-+p+ channel is characterized by presence of a large S-wave evidenced by clear asymmetry in the K*(890) helicity distribution D+K-++ Amplitude Analysis Workshop, Trento, Jan 26, 2011 Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati Brian Meadows, U. Cincinnati

  11. Partial Wave expansion in angular momentum L of K -+channel(s) from D+ K-a+b+ decays E791 Quasi-Model-IndependentPartial Wave Analysis (QMIPWA) E791 Phys.Rev. D 73, 032004 (2006) Bose Symmetrization Spin Factor 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). BUT S-wave values do depend onmodels forP- and D- waves. Amplitude Analysis Workshop, Trento, Jan 26, 2011 Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati Brian Meadows, U. Cincinnati

  12. Fitting Procedures BWM, K matrix and QMIPWA fits maximize log-likelihood with respect to the parameters (coefficients d, , m0’s, 0’s, etc) Ps,Pb are PDF’s for signal ( ) and background, respectively The PDF’s are normalized to unity over the Dalitz Plot area. fs,fb are fractions of signal and background, respectively Fraction of events attributed to any isobar or wave taken as To evaluate a fit, we compute the 2 (sum of squares of normalized residuals over 2D bins in the Dalitz plot) divided by , the number of d.o.f

  13. Ds-->K+K-+ - Recent Result from Babar Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  14. Previous Analyses of Ds+K+K-π+ PLB 351:591-600 (1995) PRL 100:161804 (2008) arXiv:1011.4190v1 [hep-ex] • This channel represents the “standard” for measurements of Ds BF’s. • Prevous DP analyses by: • E687 with 300 events • CLEO-c (14,400 evs) • This analysis (96000 evs) Brian Meadows, U. Cincinnati 10 14

  15. Ds+K+K-π+ Event Selection ~101K Events 96% purity Background regions From ±6s ±10s Event Selection • BaBar tracking and particle ID • Select D*(2112)+Ds*to reduce background Dm=|m(K+K-p+g)–m(K+K-p+)-mPDG|< 2 • Cut on likelihood ratio based on • Momentum of Ds* in event CMS • c2 difference due to Ds flight length • Signed decay distance Backgrounds: • Remove D*+p+D0 (K-p+π0) • Remaining backrounds mostly combinatorial with some D+K-p+π+ • Model DP for these from sidebands. Brian Meadows, U. Cincinnati 10 15

  16. Dalitz Plot Overview f(1020) K*0(892) • f(1020) and K*(892) are clearly visible • Before making a fit to the full Dalitz Plot, PWA analyses were attempted in the two box regions • Parameterization of low mass K+K-S-wave was possible • No evidence for D-wave in K-p+ channel • No structure in K-p+ threshold region could be parameterized • Probably due to presence of f0(1370) in the box. Brian Meadows, U. Cincinnati 10 16

  17. K+K- Partial Wave Analysis θKK π+ Ds+ <Y00> <Y10> <Y20> Subtract background, correct for efficiency and phase-space factor at each m (K+K-) Weight each event by spherical harmonics to get coefficients <Yk0> of cosKK distribution. Brian Meadows, U. Cincinnati 10 17

  18. PWA for low mass K+K-S-Wave Ambiguity in ÁSP Resolved by direction of Á(1020) phase motion Extract |S| and |P| magnitudes and relative phase fSP as a function of m(K+K-) Fit distributions to a Flatte form for f0(980) for S-wave and an RBW for  (1020) for the P-wave (solid curves) Most of the phase motion comes from P-wave S-wave phase obtained by subtracting fitted P-wave phase. Brian Meadows, U. Cincinnati 10 18

  19. P-/S-wave Ratio in f(1020) Region Ds+fπ+ often used as normalization mode Presence of S-wave must be taken into account As a service, calculate S- and P-wave contributions as fractions of overall 3-body rate for different cuts on m(K+K-) These can be used to make normalization to the well-measured DsK+K- π + branching fraction instead. Brian Meadows, U. Cincinnati 10 19

  20. K+K-S-wave Comparison _ _ L. Maiani, A. D. Polosa and V. Riquer, Phys. Lett. B651, 129 (2007) K+K- S-wave shape compared with charm analyses of D0K+K-K0 and D0K+K-0 The K0K+ mode would be dominated by a0(980) For Ds, expect large f0(980) Similarity in form is evident for all  Evidence that they are both 4-quark states ? Brian Meadows, U. Cincinnati 10 20

  21. PWA Analysis in K-p+ System • Proceeding as for the K-K+ system, evidence exists for strong K*(890) and for S-wave interference. • |S|2 indicates that this wave is small • This makes it difficult to measure S-wave phase • ALSO, |S|2 goes negative • This indicates that other (f0?) amplitudes play a role in this region. Brian Meadows, U. Cincinnati

  22. Fit to the whole Dalitz Plot The Model: • S-wave K+K- channel – Use empirical parameterization off0(980) from fit to K+K- PWA. • S-wave K-p + channel – Use BWM forK0*(1430) with no non-resonant component. • P- and D-waves for K+K- and K-p + channels Modeled with BWM’s The Fit: • K*(892) mass and width are floated parameters • Unbinned maximum likelihood fit with complex coefficients floating • c2 computed by adaptive binning algorithm Brian Meadows, U. Cincinnati 22

  23. Dalitz Plot Projections Brian Meadows, U. Cincinnati 10 23

  24. Dalitz Plot m(K+K-)Moments Brian Meadows, U. Cincinnati 10 24

  25. Dalitz Plot m(K-p+)Moments Brian Meadows, U. Cincinnati 10 25

  26. Dalitz Plot Results Decay dominated by vector resonances K*(892) width is 5 MeV lower than PDG ‘08 (consistent with CLEO-c) Contributions from K*1(1410), K2(1430), κ(800), f0(1500), f2(1270), f2’(1525), NR - all consistent with zero Adding LASS non-resonant part of S-wave K-p+ system makes fit worse Brian Meadows, U. Cincinnati 10 26

  27. Ds +-+ - Recent Result from Babar 384 fb-1 Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  28. Dalitz Plot Preliminaries 13K events f0(980) f0(980) Sample Selection: • Again, use Ds(2112)Ds • Cut on likelihood ratio based on • CMS momentum of Ds • c2 probability difference due to Ds flight length • Signed decay distance • Dalitz plot is symmetrized (each event plotted twice) • Efficiency and backgrounds modeled from MC simulated uniformly over the Dalitz plot Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati 10 28

  29. Decay Amplitude Modeling Magnitude f0(980) Phase (radians) The analysis closely follows the QMIPWA of D+K-++ decays by the E791 collaboration The p+p-S-wave is described by a spline defined at 29 invariant mass values by complex numbers (58 parameters in the fit) P- and D-waves are modeled as linear combinations of BWM contributions with complex coefficients that are also determined in the fit Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati 10 29

  30. Fit Results - Dalitz Plot Projections Brian Meadows, U. Cincinnati 10 30

  31. Fit Results - Dalitz Plot m(p-p+)Moments p- p+ µ p+ Agreement between data and model is generally good Unknown reflection? Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati 10 31

  32. Dalitz Plot Results BaBar 3 2 1 4 2/DOF = 437/(422-64)  P(2)=1.2% Derived from sum of S-wave isobar fractions 1Phys.Rev.D 79 032003,2009 2 Phys.Lett.B407:79-91,1997 3 Phys.Rev.Lett.86:765-769,2001 4 Phys.Lett.B585:200-212,2004 Interesting to compare different modeling of S-wave from E791 (BWM model) and FOCUS (K Matrix formalism) S-wave component is Large Significant D-wave f2(1270) Brian Meadows, U. Cincinnati 10 32

  33. S-wave Comparison The f0(980) behavior is evident and as expected Activity at f0(1370) and f0(1520) consistent with E791 fit This wave is small in f0(600) region Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati 10 33

  34. Other Observations _ *L. Maiani, A. D. Polosa, and V. Riquer, Phys. Lett.B651, 129 (2007) Line shapes for K+K-, K+K0and+-S-wave (both f0(980) and a0(980) modes) are in striking agreement. Does this suggest these are 4-quark states*? Presence of  signals in Ds decay suggests either W-annihilation process or significant re-scattering Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati 34

  35. Ds Ks-+ - Recent Result from Babar 468 fb-1 Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  36. D0 Ks+- A large, clean sample (540K events in the signal at 98.5% purity) was identified as D0 or D0 from charge of slow pion from D*D0s+ decays For this, we choose events where m=m(Ks+-s)-m(Ks+-)¼145.5 Mev/c2. Backgrounds mostly from mis-reconstructed D’s or combinatorial sources. Other, readily identifiable backgrounds were from D0K0K0 or from D0+-+- decays. Self-conjugate channel is used in measurements of CKM phase AND D0-D0 mixing parameters _ _ _

  37. D0 Ks+- Time-integrated population s(Ks+) (GeV/c2)2 s(Ks-) (GeV/c2)2 • The Dalitz plot has rich structure: • Cabibbo-favoured (CF) “right sign” (RS)K*-(890) – vertical band • “Wrong sign” (WS)K*(890), horizontal band,is upside-down” wrt S-wave background as expected for Doubly Cabibbo Suppressed (DCS) decay • The P-wave+-system includes (770) and states at higher mass, • S-wave +- structures including f0(980) and possible higher mass isobars and overall S-wave background. • Also in the +- system is the K0 from D0K0K0. Brian Meadows, U. Cincinnati

  38. A Decay model for D0 Ks+- Decays include isobars in all three channels TheS-wave +- system is parameterized using the K-matrix form derived from much +- data* For theS-wave K0§ systems, the form introduced by the LASS collaboration is used. For P- and D-waves the BWM is used. Time-integrated Fit Results * Anisovitch, Sarantev, EPJ:A16, 229 (2003) • Sum of fractions 103% • WS • Decays • << RS Mass K *(890)- 893.7 § 0.1  K *(890)- 46.7 § 0.1 Mass K0(1430)- 1422 § 2  K0(1430)- 247 § 3 * D. Aston, et.al., NP:B296, 493 (1988) BUT 2/ = 10,429/8,585 = 1.21 Very poor 2 probability !

  39. D0 Mixing and D0 Ks+- Mixing D0 _ • This (self-conjugate) channel provides information from the time-dependence of on D0-D0 mixing parameters xD, yD and q/p. • Each point on the DP can be reached either by direct decay OR by mixing followed by decay • The plot shows the mean decay time as function of position in the plot. Note, for theWS K*band, where mixing is a significant contributor and mean decay time is long, this time is BELOW the average. This results in a mean WS decay time that depends on the time-dependent quantity Time-dependence of  determines xD, yD and q/p Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  40. With so much data, the “best” fit has very small probability ! Variations in the model lead to no improvement. E.g.: Several other models also yield comparable 2/ values Which model (if any) is right ? Which can be ignored ? Obviously, this impacts seriously on how much we can learn about LQ spectroscopy from HQ decays like these. A Significant Problem All fitsareBAD – some are just more so than others !

  41. The Problem with Significance Each model also produces a different set of values for D0 mixing parameters xD, yD , |q/p| and Arg{q/p}. Creates an Irreducible model uncertainty (IMU) ~10-4 in xD and yD . This channel is also used to measure CKM (see Tim Gershon’s talk). In this case  IMU ~ §40 in . Neither effect is comparable to statistical uncertainty yet, but LHCb and SuperB/Belle 2 will alter that. The more data we have, the less we realize we know !! • Either we need a better way to model • CLEO-c, BES3, SuperB • Or we must measure amplitudes in a more model-independent way.

  42. CPV Tests – D0-+0 384 fb-1 Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  43. CPV in D0-+0 and K-K+0 Potential for extended search within these 3-body modes: CPV is unlikely to be in all channels – perhaps in one Search each channel - e.g. D0 0 + 0 Each channel can be normalized to whole Dalitz plot. Systematic uncertainties from s+ tagging or from production asymmetries become 2nd order effects CPV signalled by different D0 and D0 phase behavior Dalitz plot for these 3-body final states yields information on phase behaviour between channels. BaBar used three search strategies Two model-independent searches for CPV in exclusive parts of DP. A model-dependent search based on fit to the DP distributions

  44. Two Model-Independent Searches for CPV in D0-+0 and K-K+0 _ Dalitz Plots for D 0 and for D 0 are normalized and differences plotted Unbiassed frequentist test 16.6% conf. level there is no difference. _ Legendre polynomial moments up to order 8 for D 0 and for D 0 are normalized and compared, in each channel. Unbiassed frequentist test indicates 23-66% conf. level there are no differences in the various channels. [+-]+ 0 channel [+0]+ - channel

  45. Model-dependent Search for CPV in D0-+0 and K-K+0 384 fb-1 _ Dalitz plots for D0 and for D0 were fitted to isobar model expansions of interfering amplitudes in each channel. Differences (+ for D0 and – for D0) in magnitudes a and phasesfor each of theNR=15 isobars were insignificantly small. _

  46. Towards Model-Independence - D+ K-++ Much interest in this S-wave dominant channel In particular, E791 observed a need for “(800)-like” scalar to explain low mass K-+ data Three QMIPWA fits have now been made Agreement is quite good – especially the phase in the elastic region. Notably, 2/ are all of acceptable quality CLEO-c claim that I=2 (++) amplitude is also required. A physics issue arising from these fits is why the S-wave phase does not match the LASS I=1/2 data in the elastic region (Watson theorem) 1Phys. Rev. D73, 032004 (2006) 2 Phys.Rev.D78:052001 (2008) 3Phys.Lett.B681:14-21 (2009)

  47. Isobar Model and D+K-++ +or e+e s c K- _ _ + d d • A physics issue arising from the QMIPWA fits is why the S-wave phase does not match LASS I=1/2 data in the elastic region (Watson theorem). Possible reasons are • The QMIPWA measures the total K-+ wave that can include an I=3/2 component • FOCUS has checked for evidence of this, but is inconclusive1. • The model for the P-wave does not match the LASS I=1/2 data either. The S-wave phase is measured relative to this. • Perhaps the isobar model, does not work well in this hadronic final state, and three-body scattering is involved. • We note that recent evidence from D+K-+e+e decays show good agreement with the LASS I=1/2 data • Suggests that I=3/2 production is small if external spectator diagram dominates 1Phys.Lett.B653:1-11,2007 Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  48. K-+ S-wave Phase from D+K -+e+ºe New data on from Babar extract S-wave phase information in 5-variable fit Use P-wave model: K*(890) + K*(1410) Preliminary arXiv:1012.1810 (347.5 fb-1) Amplitude Analysis Workshop, Trento, Jan 26, 2011 Brian Meadows, U. Cincinnati

  49. A Way Forward ? It seems important to improve on parameterization of the reference waves. For D+K-++, the P-wave should be modelled on LASS data rather than using the RBW model, for example Is there a way to make a QMIPWA for more than one wave at a time? Probably, but only if another wave provides a good reference phase This could work in cases with Bose symmetrization In their D+K-++, CLEOc claim to have done this, BUT it is difficult to understand what they used as a reference phase.

  50. Summary Dalitz plots for D decays hold much information on hadron and flavor physics, but we are getting close to the limit of what we can extract from them It is clear that, with the large data samples expected from LHCb, BES3 and the new B factories, that better models, or new model-independent techniques will be required Analyses of decays to 4-body states could add much, but they have yet to be tried This is an area where theorists and experimentalists need to work together.

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