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