1. Dalitz Plot Analysis of D0K–K+0
Kalanand Mishra
University of Cincinnati
BaBar Collaboration

2. 3-Particle Phase Space + 2 Observables Usual choice 
Dalitz plot provides information
about two-body amplitude structure.
2 Observables
From four vectors
Conservation laws
Final state particle masses
Free rotation in decay plane -3 Σ
Usual choice
Invariant mass squared m212
Invariant mass squared m213
m2+0 + m2-0 + m2+- =
m2+ + m2- + m20 + m2D0
D0-+0 Dalitz plot
m2(+–) +
=0
= π1  π3 D0
m (+0)  cos π2
{ π1 , π2, π3 }  { +, 0, - }
Hadron07, Frascati, October 8-13
Kalanand Mishra

3. Isobar Model Formalism
three-body decay DABC decaying through an [ rAB ] resonance 1 2 3
{12}
{13}
{23} NR 2
D decay three-body amplitude
NR term(direct 3 body decay)
a0, δ0, ar, δr : Free parameters of fit
Relativistic Breit-Wigner
f0(980)
a0(980)
Angular distribution
D and r form factors
Hadron07, Frascati, October 8-13
Kalanand Mishra

4. Introducing Angular Distributions
Schrödinger‘s Equation
Angular Amplitude
Dynamic Amplitude (BW, Flatte, S-wave)
In case only l = 0 (S-wave) and 1 (P-wave) amplitudes are present :
For S- and P- waves only, in the absence of cross-feeds from other channels, the amplitudes and the relative phase are given by these relations.
Hadron07, Frascati, October 8-13
Kalanand Mishra

5. D0K–K+0 Dalitz Plot Analysis
Interference among three types of singly Cabibbo-suppressed amplitudes
Motivation
Nature of Kπ S-wave below 1.4 GeV.
Is there a charged  state?
- Nature of f0/a0 (980): KK S-wave.
(I)
(II) —
D0 decay reconstruction K- 
(III)  K+ 0
Color-suppressed D0
+soft
Hadron07, Frascati, October 8-13
Kalanand Mishra

6. Dalitz Plot for D0K-K+0
 (1020)
Approx. isolated
±1 region:
≈ 11 k events,
purity ≈ 98 % 
K-K+ channel;
can attmpt
model ind.
analysis.
m2(K+0) (GeV2/c4)
Events used
for bg shape
K*+
385 fb-1
K*-
m2(K-0) (GeV2/c4)
Define amplitude for the D0K-K+0 decay as:
Event Selection
PCM ( D0) > 2.77 GeV/c
|mD* - m D |
< 0.6 MeV/c2
Dalitz plot intensity  | f |2
Phys. Rev. D74, (2006)
Hadron07, Frascati, October 8-13
Kalanand Mishra

7. K and K+K- S-wave Amplitudes
LASS
Nucl. Phys. B296, 493 (1988);
For K S-wave
E791
Phys. Rev. D73, (2006); 
- The LASS amplitude gives the best fit.
E-791 fit worse at low mass.
 model yields
mass ± 30 MeV/c2
width ± 20 MeV/c2
significantly different from the values reported for 0.
-  with E-791 parameters does not give a satisfactory fit.
K invariant mass range available in D0K–K+0 decay.
Use LASS amplitude for nominal fit and E-791 amplitude for syst. uncertainty.
For K-K+ S-wave
- f0(980) and a0(980) virtually indistinguishable from each other.
Both f0(980) and a0(980) give satisfactory fits.
Since they are so similar, we try each as a description of the KK S-wave amplitude. —
Hadron07, Frascati, October 8-13
Kalanand Mishra

8. Fit Results for D0K-K+0 Dalitz Plot
: 19 %
f0/a0: 7-10%
K*(892)-
φ(1020)
K*(892)+   
Ambiguity between a large K+0 S-wave & K*(1410),f2’(1525).
2 Prob = 62 %
2 Prob = 48 %
Hadron07, Frascati, October 8-13
Kalanand Mishra

9.  Partial Wave Analysis in K-K+ channel    Impose  
Look into the distributions of the spherical harmonic
functions (l=0,1,2,…..).
[ in the region where the K cross-channels have little effect ] _
S-wave as in D0K-K+K0.
f0(980)
a0(980) 
 (1020) 
Solve these equations to extract |S|, |P|, and cos SP.
Data
Data 
Because of the interference from the crossing K channels, the model independent partial-wave analysis performed here seems valid only up to about GeV/c2.
Impose 
Breit-Wigner phase for P-wave, (1020)  
Data
 Model-I
 Model-II
Choose the solution with increasing S-wave phase
(Wigner causality)
Hadron07, Frascati, October 8-13
Kalanand Mishra

10. Back up slides

11. BaBar: B and charm Factory
Electromagnetic Calorimeter
6580 CsI crystals
e+ ID, π0 and γ reco
Instrumented Flux Return
12-18 layers of RPC/LST
μ ID
e+ [3.1 GeV]
Cherenkov Detector
144 quartz bars
K, π, p separation
Drift Chamber
40 layers
tracking + dE/dx
e- [9 GeV]
Silicon Vertex Tracker
5 layers (double-sided Si strips)
vertexing + tracking (+ dE/dx)
1.5T Magnet
Hadron07, Frascati, October 8-13
Kalanand Mishra

12. Kaon/Pion Discrimination: DIRC
LAYOUT
Cherenkov angle vs. momentum
for pions and kaons
>4 s separation
at 3 GeV/c
Hadron07, Frascati, October 8-13
Kalanand Mishra

13. D0-+0, K-K+0 Background Sources D* Reconstruction
D0h-h+0 Reconstruction
Background Sources
Charged track combinatoric
Mis-reconstructed 0
Real D0, fake s
K0 reflection in 0
and KK0 modes
h- and h+ tracks are fit to a vertex
Mass of 0 candidate is constrained
to m0 at h-h+ vertex
PCM ( D0 ) > GeV/c
D* Reconstruction
D*+ candidate is made by fitting the D0 and the s+ to a vertex constrained in x and y to the measured beam-spot for the run.
|mD* - m D | < 0.6 MeV/c2
Vertex 2 probability > 0.01
Choose a single best candidate with smallest 2 for the whole decay chain ( multiplicity = 1.03 ).
Hadron07, Frascati, October 8-13
Kalanand Mishra

14. _ BaBar CLEO KK0: Strong-phase Diff. & Amp. Ratio
The strong phase difference D and relative amplitude rD between the decays of D0 and D0 to K*(892)+ K- state are defined, neglecting direct CP violation in D decays, by the equation:
We find
                 
BaBar
CLEO
rD =  (stat)  (syst)
δD = -35.5° (stat) ± 1.9° ± 2.2° (syst)
rD = 0.52  0.05 (stat)  0.04 (syst)
δD = -28° ± 8° (stat) ± 11°(syst)
Phys. Rev. D76, (2007)
Phys. Rev. D74, (2006)
These measurements are consistent with each other.
Hadron07, Frascati, October 8-13
Kalanand Mishra

15. KK0: K S-wave Parametrization
KpS-wave in mass range 0.6–1.4 GeV/c2 is not well-understood.
A possible  state ~ 800 MeV/c2 has been conjectured, but has only been reported in the neutral state.
For the K+0 and K-0 S-wave amplitudes, we try three models:
Amplitude from LASS K-+ K-+ scattering.
K-+ amplitude from a model-independent analysis of D+K-++ data by the E791 collaboration.
[ coherent sum of (800) + uniform NR + K*0(1430) ].
Nucl. Phys. B296, 493 (1988);
Phys. Rev. D73, (2006);
{No evidence in K elastic scattering.}  
E-791
LASS
- 800
Normalized to arbitrary scale for m > 1.15 Gev/c2 for easy comparison.
Hadron07, Frascati, October 8-13
Kalanand Mishra

16. LASS K S-wave Parameterization
Kπ S-wave amplitude is described by the coherent sum of an effective range term and the K*0(1430) resonance:
S(s) = (√s/ p). sin . ei
 = cot-1 [ 1/ap + rp/2 ] cot -1 [ (m2R-s)/(mR R ) ]
Effective Range (NR) term K*0(1430) resonance term
a = scat. length, r = eff. range, mR = mass of K*0(1430), R= width
p = momentum of either daughter in the Kπ rest frame.
Phase
space
factor
For Kπ scattering, S-wave is elastic up to K' threshold (1.45 GeV).
Hadron07, Frascati, October 8-13
Kalanand Mishra

17. [ E791 Collaboration, slide from Brian Meadow’s Moriond 2005 talk ]
K S-wave from D0K-++ DP
[ E791 Collaboration, slide from Brian Meadow’s Moriond 2005 talk ]
Divide m2(K-+) into slices
Find s–wave amplitude in each slice (two parameters)
Use remainder of Dalitz plot as an interferometer
For s-wave:
- Interpolate between (ck,k).
Model P and D waves.
S (“partial wave”)
Hadron07, Frascati, October 8-13
Kalanand Mishra

18. Excellent agreement between data & models.
Analysis of Angular Moments
Excellent agreement between data & models.
Each event is weighted by the spherical harmonic
functions (l=0,1,2,…..).
Large interference between S and P waves. 
m2(K-K+) (GeV2/c4)
m2(K-K+) (GeV2/c4)
m2(K+0) (GeV2/c4)
m2(K+0) (GeV2/c4)
Higher moments above 1.1 GeV are coming from cross channels.
For S- and P- waves only, in the absence of cross-feeds from other channels, the amplitudes and the relative phase are given by:
We solve these equations for the K-K+ system in a limited mass range (where the above conditions are satisfied) to extract |S|, |P|, and cos SP.
Hadron07, Frascati, October 8-13
Kalanand Mishra

19. Partial Wave Analysis in K-K+ channel
f0(980)
Solve the equations on the previous slide to extract |S|, |P|, and cos SP.
a0(980) 
 (1020) 
Data
Data
KK S-wave amplitude, extracted in a model independent analysis of the decay D0K-K+K0. –
Phys. Rev. D 72, (2005)
Two solutions for SP  upper one is the physical  
Because of the interference from the crossing K channels, the model independent partial-wave analysis performed here is valid only up to about GeV/c2
Breit-Wigner phase for P-wave, (1020) 
Data
 Model-I
 Model-II 
Hadron07, Frascati, October 8-13
Kalanand Mishra