1. Dalitz plot analysis in D meson decays
Hai-Bo Li
IHEP
Workshop on Partial Wave Analysis and Dalitz Plot Analysis
Beijing, Jan
2018/11/19

2. Rich structures in D three body decays
D0 ! K
B0 ! K
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.
2018/11/19

3. 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
2018/11/19

4. 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
2018/11/19

5. 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KSK+K-
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6. BESIII Opportunity in Dalitz plot analysis
e+e-  (3770)  DD, 18 M D0D0 , 14M D+D- /year
1. D mesons are produced almost at rest; clean events;
2. Flavor tag
3. CP tag
modes D  P1P2P3 ( P= p, K) good for DP analysis
e+e-  s=4170  DSD*S, 2 M DSD*S /year
DS Dalitz plot can also be done at BESIII.
Huge C0, C from (2S) and J/ Decays.
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7. 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 (chi^2, likelihood, adaptive binning).
How to tabulate Results (fractions, amplitudes and phases).
Common systematic errors evaluation.
Validations (use of toyMC, of simulated data).
2018/11/19

8. 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
2018/11/19

9. 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. – Let’s start there
Recommend reading D. Asner’s review
hep-ex/ v1, 5 Oct 2004
Isobar Model 2 a b c H
d2G FH Fk H a b c k
+  ak e ifk /
dm2dm2 k bc ab NR
Isobars
2018/11/19
From D. Asner

10. Isobar Model Mk = (-2pq)L YL0(cosq bc) TL(m2) x FH(q,rH) x Fk(m2)
Each isobar term can be written as
Mk = (-2pq)L YL0(cosq bc) TL(m2) x FH(q,rH) x Fk(m2)
(for resonance J=L decay to 0+0)
TL(m2) = BWL(m2) = sin dk eidk
tan dk = M0G(m)/(M02-m2)
G(m) = G0 (p/p0)2L+1 (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 .
2018/11/19

11. K Matrix Fits  Klaus Peters talk in this meeting
Best (only?) method for coupled channel analyses
Single pole in single channel -> BW
Distant, narrow poles -> sum of BW’s
One pole in >1 channel -> Flatte BW
Production vector allocates poles produced in three body decays
Recommend:
D. Asner, hep-ex/ v1, 5 Oct 2004 (again)
S U. Chung et al., Annalen Phys.4: ,1995
k
j 
Klaus Peters talk in this meeting
2018/11/19

12. K matrix parameterization
For review see: Partial wave analysis in K matrix formalism, S.U. Chung et al.,
Annalen Phys.4: ,1995
2018/11/19

13. 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 !?
2018/11/19

14. Interest in K-+ s-wave—E791
hep-ex/ B. Meadows for E791
~138 %
c2/d.o.f. = 2.7
Flat “NR” term does not give
good description of data.
2018/11/19
Traditional iso-bar model

15. E791 D+ ! K-p+p+ 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 = § 43 § 85 MeV/c2
The precise line-shape of K* is very important for this analysis.
J/ KKp0 can definitely do that at BESIII.
2018/11/19

16. E791 Model Independent PWA
Compare MIPWA & BW Isobar
Compare MIPWA & LASS Data
Phase
Amplitude
Phase
S-wave
P-wave
D-wave
2018/11/19

17. E791 D+ ! p-p+p+
PRL, 86, 770 (2001) E791 collaboration
2018/11/19

18. D+  p+p-p+ analysis from CLEOc
hep-ex/ CLEO collaboration
281 pb-1/1.8M D pairs
KS removed
Leading fractions shown on the projection plots.
2018/11/19

19. K-matrix Analysis D0KSp+p- 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 %
BW fit with s1, s2 – 30.4%
A/S K-matrix fit %
BaBar Data: K-matrix Model
2018/11/19

20. 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 <> versus bin area.
Efficiency variations from PID/tracking.
2018/11/19

21. Efficiency on the DP
2018/11/19

22. The momentum resolution effect
Events does not respect dalitz kinematics boundary!
Due to the momentum resolution, some signal events fall outside the Kinematics boundary.
Since some signals are being cut away, the amplitude/Branching Fraction is being altered!
2018/11/19

23. Alternative Representation of the Dalitz Plot
2018/11/19

24. Rectangular Dalitz Plot
Homogenization
B0 p+p-p0 BaBar
Easy to do efficiency correction, and bin by bin fit.
2018/11/19

25. Efficiency and Resolution on B0p+p-p0 Dalitz Plot
Selection efficiency on Dalitz plane
Large variations
in the Dalitz
plot corners
Fraction of mis-reconstructed events in 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: 7÷8 MeV/c2
Truth-matched events
Misreconstructed events
SCF events:
large variations in
resolution over the DP
 4D-convolution
functions introduced G(m’rec, q’rec, m’true, q’true)
2018/11/19

26. D0-D0 mixing from D0K+p-p0 Decay (1)
Charm2006
Event-level tagged
Prominent K* peak in DCS Mode
2018/11/19

27. D0-D0 mixing from D0K-p+p0 Decay (2)
Charm2006
Event-level tagged
Prominent r peak in CF Mode
2018/11/19

28. 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-p+p0 the main resonance is K- r+
In D0  K+p-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.
2018/11/19

29. D0K0+- decay & CKM angle γ
γ can be measured from the interference between decays with bcus and bucs transitions
bu transition
Vub
bc transition
Vcb e
i δB
-i γ
Interference occurs when some final state is accessible by both D0 and D Giri-Grossman-Soffer-Zupan: PRD68, (2003): Final state = Ks0π+π-  Dalitz Plot Analysis
2018/11/19

30. BESIII Impact on g
Simultaneous CP tagged & flavor tagged analysis of D0Ks+-
[correlated D’s,(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.
2018/11/19

31. CP-tagged D KSp+p- decay at BESIII
Toy study from A. Bondar
BESIII sensitivity
A model independent way is proposed
by A. Bondar and A. Poluektov hep-ph/
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
10, 000, KS p+p- ,7,500 p+ p- p0, 1,900 KSK+K-
The d(cos(dD)) 2%  10 –20
2018/11/19

32. Dalitz Fitter--Tools RooFit BaBar ……..many RooFit based fitters
Charmfitter Miniut based, BaBar ….
Charmfitter
2018/11/19

33. Summary Learn more experiences from BaBar, Belle,
CLEO, FOCUS, E791 …..
Towards Uniformity
Discussions with CLEO and BELLE
Methodology
2018/11/19