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Amplitude Analysis of the D0 Dalitz Plot
G. Mancinelli, B.T. Meadows, K. Mishra, M.D. SokoloffUniversity of Cincinnati
BaBar Coll. Meeting, 9/12/2006
Motivation• Theorist community has expressed interest [ see J.L. Rosner,
hep-ph/0608102 ] in an amplitude analysis of D0K-K+π0 decay which will be useful in understanding the behavior of Kπ S-wave below K’ threshold.
• The K±π0 system from this decay can also provide information relevant to the existence of (800). Evidence for such a state has been reported only for the neutral state. If is an I = 1/2 particle, then it should also be observed in the charged state.
• These decays are also interesting because one needs to analyze several D0 decay modes in B±DK± decays in order to be able to constrain (3). At present the only CS mode exploited so far is D0π-π+π0 [ under internal BaBar review ].
• 3-body CS decays of D0 are especially interesting because of their sensitivity to direct CP violation. Such a analysis is already underway.
Event Selection
Events used to obtain Bkg shape
Use events in 1 mass window for DP analysis: ≈ 7000 events with purity ≈ 97 %
• We use decays D*+D0 [K-K+0]πs
+
• Integrated Lumi 232 fb-1
• | mD* - mD0 - 145.5 | < 0.6 MeV/c2
• PCM > 2.77 GeV/c2
m2(K-π0)
m2(K
+π
0) ~ 3 % bkg
Isobar Model
2
NRConstant
D formfactor
R formfactor
spinfactor
1 1
12
2
3 3 3
{12} {13} {23}1
23
NR
Schematically:
Amplitude for the [ij] channel:
Each resonance “R” (mass MR, width R) typically has a form
p, q are momenta in ij rest frame. rD, rR meson radii
S-, P-, D- wave Amplitudes
The Decay Processes are of type :
Parent [P] bachelor [b] + Resonant System [R]
Write amplitude schematically as : < (R)L | P b >
L = angular momentum
Introduce a complete set of intermediate states for each L :
for L = 0, S-wave
for L = 1, P-wave
for L = 2, D-wave, …..
The interference between these waves can be viewed as the addition of angular momenta and can be described by spherical harmonics Yl
0 (cos H).
Dalitz plot and Fit ModelDalitz plot and Fit Model
o K+π0 and K-π0 S-wave: LASS parameters
o K+K- S-wave: f0(980) : Flatte (with BES parameters)
o P- and D- waves: relativistic Breit Wigner
PW: K*(892), K*(1410), (1020) DW: f2’(1525)
Kπ s-wave parameterization- Apart from the K*0(1430), resonant structure in the S-
wave K system in the mass range 0.6 – 1.4 GeV/c2 is not well-understood.
- A possible state ~ 800 MeV/c2 has been conjectured, but this has only been reported in the neutral state. Its existence is not established and is controversial.
- The best results on Kπ S-wave parameters come from the LASS experiment. Recently, the E791 collaboration has come up with a model independent parameterization of Kπ S-wave.
- We try three different models: LASS Kπ scattering results, E791 shape and model.
Generalized LASS Parameterization(W. M. Dunwoodie notation)
• Kπ S-wave amplitude is described by:
S = B sin(B+ B) ei(B + B) Non-resonant Term
+ R eiR e2i (B + B) sinR eiR Resonance Term
B, B, R, R are constants, phases B and R depend on Kπ mass.
B = cot-1 [ 1/aq + rq/2 ], R= cot -1 [ (m2R-s)/(mR R ) ]
a = scat. length, r = eff. range, mR = mass of K*0(1430), R= widthFor Kπ scattering, S-wave is elastic up to K' threshold (1.45 GeV).• Original LASS parameterization: B = R =1; B = R =0
S = sin(R+B) ei (R + B)
We use : B = R = 1; B = 90, R = 0
S = sin(R+B+ π/2 ) . ei (R + B + π/2)
s–wave from D+ K-++ Dalitz Plot
• 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) points:
• Model P and D.
[ E791 Collaboration, slide from Brian Meadow’s Moriond 2005 talk ]
S (“partial wave”)
Comparison of Kπ S-wave Models
∆ E791 MIPWAO LASS Original
This analysis
LASS phase is shifted by -900 and phase in our parameterization is shifted by -1800.
S-wave Modeled on D0K decay
• The E791 collaboration needed a broad scalar resonance to get a good fit in their first D+K-π+π+ DP analysis (2002).
• We formulate as a I = 1/2 particle with parameters taken from E791, mass = 797 ± 47 MeV and = 410 ± 97 MeV.
• The parameterization of as a BW is an ad hoc formulation.
D0+K- D0-K+
KK S-wave: f0(980)
• Coupled-channel BW to the K+K- and KS0KS
0 states (Flatte) :
BW(s) = 1/ [ mr2 - s - i mr (π + K) ]
π = gπ . [ s/4 - mπ2 ]1/2
K = (gK /2). [ (s/4 - mK2 )1/2 + (s/4 - mK0
2 )1/2 ]
• BES parameter values for gπ and gK:
mr = 0.975 ± 0.010 GeV/c2
gπ = 0.165 ± 0.018
gK / gπ = 4.21 ± 0.33
BES is the only experiment which has good amount of
data on f0(980) decays to both π+π- (from J/π+π-) and
K+K- (from J/K+K-) . The BES measurements of these parameters have made E791
and WA76 measurements obsolete.
Nominal Fit
Data Fit
(Data-Fit)/Poisson (Data-Fit)/Poisson2/= 1.03 for = 705
Normalized Residual Normalized Residual
Nominal FitGen. LASS parameterization for Kπ S-wave
Fit Components: 1) K*+(892) (fixed amp & phase) 4) K*- (892) 7) K-π0 S-wave
2) K*+(1410) 5) K*-(1410) 8) f0(980) 3) (1020) 6) K+π0 S-wave 9) f2’(1525)
m2(K+π0) m2(K-π0) m2(K+K-)
Fit Results
2 / = 1.05
Fit with Kπ S-wave from E791
FIT FRACTIONS: 1) K*+ : 0.41 6) K+pi0 SW : 0.08 2) K*1410+ : 0.006 7) K-pi0 SW : 0.07 3) Phi : 0.19 8) f0(980) : 0.03 4) K*- : 0.17 9) f2’1525 : 0.006 5) K*1410- : 0.05
S-wave Amplitude using S-P interference in D+K- + +
m2(K+π0) m2(K-π0) m2(K+K-)
Fit with S-wave Modeled on D0K decay
K*-_amp 0.57 ± 0.02
K*- phase -28.5 ± 3.1
K*1410+ amp 1.41 ± 0.12
K*1410+ phase -136.2 ± 11.0
K*1410- amp 1.80 ± 0.22
K*1410- phase 186.6 ± 7.3
Fit Fractions K*+ : 0.43 + : 0.16 K*(1410)+ : 0.01
Phi : 0.2 K*- : 0.14
- : 0.13
K*(1410)- : 0.02 2 / = 1.35428
+ amp 1.60 ± 0.08
+ phase 104.0 ± 3.2
- amp 1.46 ± 0.08
- phase 174.0 ± 3.4
amp 0.68 ± 0.01
phase -0.4 ± 4.7
m2(K+π0)m2(K-π0)
m2(K+K-)
Moments Analysis
pq
cos = p. q
K-
K+
0
Helicity angle in K-+ system.Similar definitions applies to
the two Kπ channels.
• Several different fit models provide good description of data in terms of 2/ and NLL values.
• We plot the moments of the helicity angles, defined as the invariant mass distributions of events when weighted by spherical harmonic functions Y0
l (cosH).
• These angular moments provide further information on the structure of the decays, nature of the solution and agreement between data and fit.
Angular Moments & Partial Waves
• We notice a strong S-P interference in both Kπ and KK channels, evidenced by the rapid motion of Y0
1 at the K*(892) and masses.
• The Y02 moment is
proportional to P2 which can be seen in the background-free (1020) signal region.
√4π <Y00> = S2 + P2
√4π <Y01> = 2 |S| |P| cosSP
√4π <Y02> = 0.894 P2
Higher moments = 0
In case of S- and P- waves only and in absence of cross-feeds from other channels:
With cross-feeds or presence of D-waves, higher moments ≠ 0 .
Wrong fit models tend to give rise to higher moments, as seen in the
moments plots earlier, thus creating disagreement with data.
Angular Moments (K-K+)Nominal Fit : Excellent agreement with data
Y01Y0
0
Y02
Y04
Y06
Y03
Y05
Y07
Angular Moments (K-K+) -wrong
Fit with K2*(1430)included!
Y01Y0
0
Y02
Y04
Y06
Y03
Y05
Y07
Angular Moments (K-K+) - wrong
NoKK SW !
Angular moments (K+π0)Nominal Fit : Excellent agreement with data
Y01Y0
0
Y02
Y04
Y06
Y03
Y05
Y07
Angular Moments (K-π0)
m2(K-π0) [GeV/c2 ] m2(K-π0) [GeV/c2 ]
Nominal Fit : Excellent agreement with data
Y00 Y0
1
Y02
Y04
Y06
Y03
Y05
Y07
Strong Phase Difference, D and rD
• The strong phase difference D and relative amplitude rD between the decays D0K*-K+ and D0K*+ K- are defined, neglecting direct CP violation in D0 decays, by the equation :
rD eiD = [aK*-K+/ aK*+K-] exp[ i(K*-K+ - K*-K+) ]
• We find
D = -37.0o ± 2.2o (stat) ± 0.7o (exp syst) ± 4.2o (model syst)
rD = 0.64 ± 0.01 (stat) ± 0.01 (exp syst) ± 0.00 (model syst). • These can be compared to CLEO’s recent results:
D = -28o ± 8o (stat) ± 2.9o (exp syst) ± 10.6o (model syst)
rD = 0.52 ± 0.05 (stat) ± 0.02 (exp syst) ± 0.04 (model syst).
Summary
• The resonance structure is largely dominated by various P-wave resonances, with small but significant contributions from S-wave components.
• The Kπ S-wave modeled by a ±(800) resonance does not fit the data well, 2/ being 1.35 for = 706.
• The E791 model-independent amplitude for a Kπ system describes the data well except near the threshold.
• The generalized LASS parameterization shifted by +900 gives the best agreement with data and we use it in our nominal fits.
• A small but statistically significant contribution comes from KK D-wave component f2’(1525).
• The D0K*+(892)K- decay dominates over D0K*-(892)K+. This may be related to the dominance of the external spectator diagram.
• But the order is reversed for the next p-wave state K*(1410).
Summary continued ….
• The f0(980) with Flatte shape and the BES parameters is enough to parameterize the KK S-wave.
• A good 2 value does not guarantee a robust fit. One needs to also look at angular moments to understand localized effects produced by interference from cross-channels.
• We have measured rD and D.
Backup Slides
Resonance Shapes
K*(892)+ K*(892)- (1020) NR
K*(1410)+ K(1410)*- Kappa+ Kappa-
P-wave NR(+) P-wave NR(-)
P-wave NR(0) K*0(1430)+ K*0(1430)-
Fit with CLEO PDF
1 Nonres_amp 4.80848e+00 8.76759e-02 (5.6 in CLEO results) 2 Nonres_phase 2.45715e+02 1.41802e+00 (220 in CLEO results) 3 K*- amp 5.21620e-01 1.26111e-02 4 K*-_phase -2.51342e+01 2.09421e+00 5 amp 6.03842e-01 1.11649e-02 6 phase -3.30354e+01 2.89297e+00
2 / = 1.83342
Fit with p-wave NR
1 K*-_amp 6.13060e-01 1.98369e-02
2 K*-_phase -4.28001e+01 3.65266e+00
3 K*1410+_amp 3.46743e+00 4.76307e-01
4 K*1410+_phase 3.99550e+01 8.05654e+00
5 K*1410-_amp 2.67283e+00 4.14485e-01
6 K*1410-_phase 1.65986e+02 1.19152e+01
7 Kappa+_amp 7.30570e-01 2.10914e-01
8 Kappa+_phase 8.81885e+01 1.80236e+01
9 Kappa-_amp 6.05465e-01 1.68914e-01
10 Kappa-_phase 1.08270e+02 2.16174e+01
11 NRPW_P_amp 4.88345e+00 1.64838e+00
12 NRPW_P_phase 8.97154e+01 2.37566e+01
13 NRPW_M_amp -4.66088e+00 1.66335e+00
14 NRPW_M_phase -1.02777e+02 2.27370e+01
15 NRPW_0_amp 1.23893e+01 2.76792e+00
16 NRPW_0_phase 7.53007e+01 1.38116e+01
17 Nonres_amp 2.60086e+00 2.58137e-01
18 Nonres_phase 2.80830e+02 7.04073e+00
19 Phi_amp 6.49647e-01 1.52032e-02
20 Phi_phase 7.74845e+01 7.16402e+00
Fit Fractions
K*+ : 0.45507 K*1410+ : 0.090682 Kappa+ : 0.035070
P-wave NR+ : 0.15697 Phi : 0.19792
P-wave NR0 : 0.63210 K*- : 0.17685 K*1410- : 0.053947
Kappa- : 0.023975 P-wave NR- : 0.14484
Nonres : 0.090031
2 /nDOF = 1.00708