Dalitz plot analysis in D meson decays

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Dalitz plot analysis in D meson decays

• Hai-Bo Li
• IHEP
• Workshop on Partial Wave Analysis and Dalitz Plot
  Analysis
• Beijing, Jan. 25-26

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Rich structures in D three body decays
Interference effects More important in D than in
B decay (usually!), since there are many
resonance below D mass region (scalar)! Or rich
final state interaction.
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History of D Dalitz decays
PDG2006 by D. Asner

• Charm three body analyses from several
  experiments summarized in PDG2006 review by D.
  Asner.
• Variety of Physics probed
• Doubly-Cabibbo suppressed decays
• CP Violation
• charm mixing
• Properties of light mesons
• Properties pp Kp S-wave
• Usual analysis technique for P ? 3P decays is
  Dalitz-plot analysis technique

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Dalitz-plot Analysis Technique(1)

• Want to describe the internal dynamics of
  
• D0 ? ABC (P ? 3P) decay
• Daughter 4-momenta 12 parameters
• Conservation of 4-momentum 4 constraints
• Masses of Decay products 3 constraints
• D is spin-0 3 orientations uninteresting
• Decay described by 2 degrees of freedom
• 3 Lorentz invariants (MAB)2, (MAC)2, (MBC)2
• Related by
• (MD)2 (MA)2 (MB)2 (MC)2 (MAB)2 (MAC)2
  (MBC)2
• Dalitz plot is (MAB)2 vs (MBC)2
• Phase space is flat in these variables

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Dalitz-plot Analysis Technique(2)

• Structure on Dalitz plot is due to
• Non-uniform reconstruction efficiency
• Background
• Internal dynamics of the decay D ? ABC
• Analyze structure on the Dalitz plot to
  elucidate a broad range of physics topics

Phase space triangular for massless daughters
becomes rounder as Q decreases
B?KSKK-
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BESIII Opportunity in Dalitz plot analysis

• ee- ? ?(3770) ? DD, 18 M D0D0 , 14M DD-
  /year
• 1. D mesons are produced almost at rest
  clean events
• 2. Flavor tag
• 3. CP tag
• 4. 20 modes D ? P1P2P3 ( P p, K) good for
  DP analysis
• ee- ? s4170 ? DSDS, 2 M DSDS /year
• DS Dalitz plot can also be done at BESIII.
• Huge ?C0, ?C from ?(2S) and J/? Decays.

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Technical Issues on Dalitz Plot analyses

• Parameterization of the angular distributions
  for 3-body Dalitz plots.
• Parameterization of the angular distributions
  for 4-body Dalitz plots.
• Parameterization of the resonant line-shapes.
• Parameterization of Non-Resonant contributions.
• Normalization of the amplitudes (analytic? toyMC
  technique? )
• Efficiency.
• Backgrounds.
• Mass resolution.
• Visualization of fit results (mass projections
  and angular distributions).
• Estimators of the fit quality (chi2, likelihood,
  adaptive binning).
• How to tabulate Results (fractions, amplitudes
  and phases).
• Common systematic errors evaluation.
• Validations (use of toyMC, of simulated data).

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Dalitz Model ---Parameterization of Dynamics

• But very often not just an isolated resonance
• Mass dependence
• Behaviour near decay-channel thresholds
• Strongly overlapping resonances
• and non-resonant dynamics
• (e.g. crossed channel)

By S. Spanier
K matrix Isobar Model starting from 2-Body
Scattering LASS parameterizations for S
wave Relativistic Breit-Wigner form
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Methodology?

• We made no recommendation on the type of fit to
  make, but we do expect that the isobar model will
  still play a large role in many analyses. Lets
  start there
• Recommend reading D. Asners review
• hep-ex/0410014 v1, 5 Oct 2004
• Isobar Model

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d2G
? ak e ifk
/
dm2dm2
k
bc
ab
Isobars
NR
From D. Asner
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Isobar Model

• Each isobar term can be written as
• Mk (-2pq)L YL0(cosq bc) TL(m2) x FH(q,rH) x
  Fk(m2)
• (for resonance JL decay to 00)
• TL(m2) BWL(m2) sin dk eidk
• tan dk M0G(m)/(M02-m2)
• G(m) G0 (p/p0)2L1 (M0/m) Fk(p)2
• Form factor Fk usually taken as Blatt-Weisskopf
  damping factor.
• A suggestion define it to be unity at M0 for
  each resonance.
• ! Easier comparison of complex coefficients ak .

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K Matrix Fits

• Best (only?) method for coupled channel analyses
• Single pole in single channel -gt BW
• Distant, narrow poles -gt sum of BWs
• One pole in gt1 channel -gt Flatte BW
• Production vector allocates poles produced in
  three body decays
• Recommend
• D. Asner, hep-ex/0410014 v1, 5 Oct 2004 (again)
• S U. Chung et al., Annalen
  Phys.4404-430,1995

Klaus Peters talk in this meeting
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K matrix parameterization
For review see Partial wave analysis in K matrix
formalism, S.U. Chung et al., Annalen
Phys.4404-430,1995
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Resonances in K-Matrix Formalism

• The Breit-Wigner form is an approximation for
  isolated resonances far from
• the opening of thresholds.
• The K-matrix parameterization of multi-body
  interaction in terms of well
• constrained 2-body interactions (isobar model)
  is a way to build on existing
• amplitude measurements.
• The K-matrix can be parameterized as an
  extension of the relativistic Breit
• -Wigner form for s-channel resonance
  dominance.
• The K-matrix is a good starting point to
  construct Lorentz invariant, 2-body
• unitary, analytic (crossing) amplitudes.
• Rem
• In some models even K-matrix poles are
  interpreted
• as particles !?

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Interest in K-? s-waveE791
hep-ex/0506040 B. Meadows for E791
138
c2/d.o.f. 2.7
Flat NR term does not give good description of
data.
Traditional iso-bar model
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E791 D ! K-pp
PRL 89, 121801
hep-ex/0506040 B. Meadows for E791
89
c2/d.o.f. 0.73 (95 )
Probability
Mk 797 19 42 MeV/c2 Gk 410 43 85
MeV/c2
The precise line-shape of K is very important
for this analysis. J/?? KKp0 can definitely do
that at BESIII.
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E791 Model Independent PWA
Compare MIPWA LASS Data
Compare MIPWA BW Isobar
Amplitude
Phase
Phase
S-wave
P-wave
D-wave
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E791 D ! p-pp
PRL, 86, 770 (2001) E791 collaboration
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D ? pp-p analysis from CLEOc
hep-ex/0607069 CLEO collaboration
281 pb-1/1.8M D pairs KS removed Leading
fractions shown on the projection plots.
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K-matrix Analysis D0?KSpp- BaBar
AS param

• 81496 events with
• 91.5 fb-1(EPS05 paper)
• Much better fit quality than BW fit with only
  known resonances
• Similar quality to BW fit that included extra pp
  S-wave resonances
• Compare pp S-wave content
• CLEO BW Model - 15.1
• BW fit with s1, s2 30.4
• A/S K-matrix fit - 16.2

BaBar Data K-matrix Model
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Efficiency over the DP
Branching fractions B N_signal/N_DD? Is the
efficiency is constant across the DP? In general,
no, there can be significant efficiency
variation across the DP, from PID/tracking
efficiency.
Solution Signal events and efficiency found
bin by bin. However, we need to find the number
of bins such that 1. Not to few efficiency
variation are covered 2. Not too many bins
Fewer statistics. Look at the variation of lt?gt
versus bin area. Efficiency variations from
PID/tracking.
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Efficiency on the DP
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The momentum resolution effect
Events does not respect dalitz kinematics
boundary!
Due to the momentumresolution, some signal
events fall outside the Kinematics boundary.
Since some signals are being cut away, the
amplitude/Branching Fraction is being altered!
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Alternative Representation of the Dalitz Plot
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Rectangular Dalitz Plot
Homogenization
B0? pp-p0 BaBar
Easy to do efficiency correction, and bin by bin
fit.
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Efficiency and Resolution on B0?pp-p0 Dalitz Plot
Fraction of mis-reconstructed events in Dalitz
plane
Large variations in the Dalitz plot corners
Selection efficiency on Dalitz plane
?0?0
Efficiency roughly flat Fraction of SCF
events highly dependent on the Dalitz plot
position
??
?0?0
??
??
??
Truth-matched mass res. double Gaussian core
78 MeV/c2
Truth-matched events
Misreconstructed events
SCF events large variations in resolution over
the DP ? 4D-convolution functions
introduced G(mrec, qrec, mtrue, qtrue)
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D0-D0 mixing from D0?Kp-p0 Decay (1)
Charm2006

• Event-level tagged
• Prominent K peak in DCS Mode

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D0-D0 mixing from D0?K-pp0 Decay (2)
Charm2006

• Event-level tagged
• Prominent r peak in CF Mode

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Resonant structures on the DP sensitive to D
mixing

• The resonance amplitudes are different for DCS
  and CF- there is more sensitivity to mixing
• In D0 ? K-pp0 the main resonance is K- r
• In D0 ? Kp-p0 the main resonance is K p-

The ratio will depend on the point On the DP,
full DP analysis will be very sensitive to
mixing parameters, especially RM.
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D0?K0??- decay CKM angle ?
? can be measured from the interference between
decays with b?cus and b?ucs transitions
Interference occurs when some final state is
accessible by both D0 and D0
Giri-Grossman-Soffer-Zupan
PRD68, 054018 (2003) Final state Ks0pp- ?
Dalitz Plot Analysis
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BESIII Impact on g

• Simultaneous CP tagged flavor tagged analysis
  of D0?Ks??-
• correlated Ds,?(3770)?DD, ?(4140)?DD(n)g(m)p0
  
• One can write
• We will extract as well as in
  a model independent way.
• This is exactly what the g analyses need.
• CLEO-c has enough data in
• hand to reduce systematic
• error to ?2o with 20 fb-1 at BESIII.

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CP-tagged D? KSpp- decay at BESIII
CP-1
CP1
Toy study from A. Bondar
BESIII sensitivity
A model independent way is proposed by A.
Bondar and A. Poluektov hep-ph/0510246 50 ab-1
at super-B will allow a model independent g/f3
measurement at accuracy 2o. However, 10 fb-1 at
y(3770/4170) data needed to accompany the
analysis _at_BES-III 10, 000, KS pp- ,7,500 p
p- p0, 1,900 KSKK- The d(cos(dD)) 2 ? 10 20
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Dalitz Fitter--Tools
RooFit BaBar ..many RooFit based
fitters Charmfitter Miniut based, BaBar .
Charmfitter
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Summary

• Learn more experiences from BaBar, Belle,
• CLEO, FOCUS, E791 ..
• Towards Uniformity
• Discussions with CLEO and BELLE
• Methodology