Dalitz Plot Analyses of B D + π π, B + π + π π + and D + s π+ π π + at BABAR

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1 Proceedings of the DPF-9 Conference, Detroit, MI, July 7-3, 9 SLAC-PUB-98 Dalitz Plot Analyses of B D + π π, B + π + π π + and D + s π+ π π + at BABAR Liaoyuan Dong (On behalf of the BABAR Collaboration Iowa State University, Ames, Iowa -36, USA We report on the Dalitz plot analyses of B D + π π, B + π + π π + and D + s π + π π +. The Dalitz plot method and the most recent BABAR results are discussed.. Introduction Dalitz plot analysis is an excellent way to study the dynamics of three-body decays. The decays under study are expected to proceed predominantly through intermediate quasi-two-body modes [] and experimentally this is the observed pattern. Dalitz plot analysis can explore resonances properties, measure corresponding magnitudes and phases, and investigate corresponding CP asymmetries. We report on the Dalitz plot analyses of B D + π π, B + π + π π + and D s + π + π π +. The data were collected with the BABAR detector at the PEP-II asymmetric-energy e + e storage rings at SLAC. A detailed description of the BABAR detector is given in Ref. []. Monte Carlo (MC simulation is used to study the detector response, its acceptance, background, and to validate the analysis.. Dalitz Plot Analysis In the decay of a spin- particle A to a final state composed of three pseudo-scalar particles (abc, the differential decay rate is generally given in terms of the Lorentz-invariant matrix element M by d Γ dxdy = M 6π 3 m 3, ( A where m A is the A particle mass, x and y are the invariant masses-squared of the ab and ac pairs, respectively. The Dalitz plot provides a graphical representation of the variation of the square of the matrix element, M, over the kinematically accessible phase space (x,y of the process. The distribution of candidate events in the Dalitz plot is described in terms of a probability density function (PDF. The PDF is the sum of signal and back- dongly@ihep.ac.cn Current Address: Institute of High Energy Physics, Beijing 9, China ground components and has the form: B(x, y PDF(x, y = f bg DP B(x, ydxdy [S(x, y R] ǫ(x, y + ( f bg DP [S(x, y R] ǫ(x, ydxdy, ( where The integral is performed over the whole Dalitz plot (DP. S (= M and B describe signal and background amplitude contributions, respectively. The S(x, y R is the signal term convolved with the signal resolution function, the signal resolution can be safely neglected for broad resonances. f bg is the fraction of background events. ǫ is the reconstruction efficiency. An unbinned maximum likelihood fit to the Dalitz plot is performed in order to maximize the value of L = N event i= PDF(x i, y i (3 with respect to the parameters used to describe S, where x i and y i are the values of x and y for event i respectively, and N event is the number of events in the Dalitz plot. In practice, the negative-log-likelihood (NLL value NLL = ln L ( is minimized in the fit. The isobar model formulation is used, in which the signal decays are described by a coherent sum of a number of two-body amplitudes. The total decay matrix element M is given by M = k ρ k e iφ k A k, ( Work supported in part by US Department of Energy contract DE-AC-76SF. SLAC National Accelerator Laboratory, Menlo Park, CA 9

2 Proceedings of the DPF-9 Conference, Detroit, MI, July 7-3, 9 where ρ k and φ k are the magnitudes and phases of the k th decay mode respectively, the A k distributions for resonant contribution describe the dynamics of the decay amplitudes and are written as the product of a relativistic Breit-Wigner function, two Blatt-Weisskopf barrier form factors [3] and an angular function []. The efficiency-corrected fraction due to the resonant or non-resonant contribution k is defined as follows: f k = ρ k DP A k dxdy DP M dxdy. (6 The f k values do not necessarily add to because of interference effects. The uncertainty on each f k is evaluated by propagating the full covariance matrix obtained from the fit. 3. Results from BABAR 3.. B D + π π [] We reconstruct the decays B D + π π [6] with D + K π + π + based on a sample of about Υ(S BB decays. The resulting Dalitz plot distribution for data is shown in Fig.. The number of signal events in the Dalitz plot is 3 ± 8. The background fraction is about 3.%. The background is parametrized using the MC simulation. and m min (Dπ are the heavy and light invariant masses-squared of the Dπ systems, respectively. The distributions in Fig. show good agreement between the data and the fit. We have performed some tests with the broad resonance D excluded or with the J P of the broad resonance replaced by other quantum numbers. In all cases, the NLL values are significantly worse than that of the nominal fit. We conclude that a broad spin- state D is required in the fit to the data. The total branching fraction of the B D + π π decay is measured to be B(B D + π π = (.8 ±.3 ±. 3, where the first error is statistical and the second is systematic. The systematic error on the measurement of the total B D + π π branching fraction is due to the uncertainties on the number of B + B events, the charged track reconstruction and identification efficiencies, the D + K π + π + branching fraction, the background shape and the fit models. The mass and width of D are determined to be: m D Γ D = (6. ±. ±. ±.9MeV and = (.8 ±. ±. ±.MeV, respectively, while for the D they are: (GeV π (D m m D Γ D = (97 ± 8 ± ± 9MeV and = (73 ± ± 7 ± MeV, where the first and second errors reflect the statistical and systematic uncertainties, respectively, the third one is the uncertainty related to the fit models and the Blatt-Weisskopf barrier factors. We have also obtained exclusive branching fractions for D and D production: m (D (GeV Figure : Dalitz plot for B D + π π. The nominal fit model contains D, D, D (7, B and P-wave non-resonant contributions. The S-wave and D-wave non-resonant contributions are found to be small and can be ignored. Replacing the D (7 by a Dπ S-wave [7] state is found to give a much worse fit. Fig. a, b and c show the m min (Dπ, m max(dπ and m (ππ projections respectively. Here m max (Dπ π B(B B(B D π B(D D+ π = (3. ±. ±. ±. and D π B(D D + π = (6.8 ±.3 ±. ±.. The relative phase of the scalar and tensor amplitude is measured to be φ D =.7 ±.6 ±.9 ±.8 rad. The systematic uncertainties that affect the results of Dalitz plot analysis are due to the background parametrization, the selection criteria, and the possible fit bias. Our results for the masses, widths and branching fractions are consistent with but more precise than previous measurements performed by Belle [8].

3 Proceedings of the DPF-9 Conference, Detroit, MI, July 7-3, 9 3 Events/(. GeV Events/(. GeV Events/(. GeV 3 (a mmin (Dπ (GeV m (Dπ (GeV max (b (c 6 8 m (ππ (GeV Figure : Result of the nominal fit to the data: projections on (a m min(dπ, (b m max(dπ and (c m (ππ. The points with error bars are data, the solid curves represent the nominal fit. The shaded areas show the D contribution, the dashed curves show the D signal, the dash-dotted curves show the D (7 and B signals, and the dotted curves show the background. 3.. B + π + π π + [9] We fit the B and B + samples independently in order to determine the CP asymmetry. The total signal amplitudes for B + and B decays are given by M = j c j A j (m max, m min, M = j c j Ā j (m max, m min, (7 where m max and m min are the heavy and light invariant masses-squared of the π ± π systems, respectively. The complex coefficients c j and c j for a given decay mode j contain all the weak phase dependence. Since the A j terms contain only strong dynamics, A j Āj. The CP asymmetry for each contributing resonance is determined from the fitted parameters A CP, j = c j c j c j + c j. (8 We reconstruct the decays B + π + π π + based on a sample of about 6 6 Υ(S BB decays. We reject background from two-body decays of D meson and charmonium states (J/ψ and ψ(s by excluding invariant masses (in units of GeV in the ranges:.66 < m π + π <.9, 3. < m π + π < 3., and 3.66 < m π + π < 3.8. The backgroundsubtracted Dalitz plot distribution for data is shown in Fig. 3. We fit 33 B candidates to calculate the fit fractions and CP asymmetries. The background is parametrized using the MC simulation. (GeV max m 6 8 (GeV mmin Figure 3: Background-subtracted Dalitz plot for B + π + π π +. The plot shows bins with greater than zero entries. The depleted bands are the charm and charmonia exclusion regions. The nominal signal Dalitz plot model comprises a momentum-dependent non-resonant component and four intermediate resonance states: ρ (77π ±, ρ (π ±, f (7π ±, and f (37π ±. We do not find any significant contributions from f (98π ±, χ c π ±, or χ c π ±. The absence of the charmonium contributions precludes the extraction of the unitarity triangle angle γ. We find no evidence for direct CP violation. The inclusive CP asymmetry is determined to be ( +3. ±. ± %, where the uncertainties are statistical, systematic, and model-dependent, respectively. Projections of the data, with the fit result overlaid, as π ± π invariant-mass distributions can be seen in

4 parametrization, the charged track reconstruction and identification efficiencies, and the possible fit bias. Our results will be useful to reduce model uncertainties in the extraction of the CKM angle α from time-dependent Dalitz plot analysis of B π + π π. 6 Events/( MeV 3.3. Ds+ π + π π + [] χc...6 mπ+π- (GeV χc We reconstruct Ds+ π + π π + based on 38 fb data sample. We use the tag of Ds (+ Ds+ γ to reduce the combinatorial background. The signal PDFs are build using the Ds+ K + K π +. The reconstructed Ds+ sample contains 379 events with a purity of 8%. The background shape is obtained by fitting the Ds+ sidebands. The Dalitz plot distribution for data is shown in Fig mπ+π- (GeV Figure : Dipion invariant mass projections: (upper in the ρ (77 region; and (lower in the regions of χc and χc. The data are the points with statistical error bars, the dark-shaded (red histogram is the q q component, the light-shaded (green histogram is the BB background contribution, while the upper (blue histogram shows the total fit result. The dip near. GeV in the upper plot is due to the rejection of events containing KS candidates. m(π+π- (GeV Events/( MeV Proceedings of the DPF-9 Conference, Detroit, MI, July 7-3, m (π π (GeV + - Figure : Dalitz plot for Ds+ π + π π +. Fig.. The agreement between the fit results and the data is generally good. The Dalitz plot is dominated by the ρ (77 resonance and a non-resonant contribution. The mass and width of the f (37 are determined to be mf (37 = ± MeV and Γf (37 = 3 ± 8 MeV (statistical uncertainties only. We measure the branching fractions B(B ± π ± π ± π = (. ±.6 ±. ±. 6, B(B ± ρ (77π ± = 6 (8. ±.7 ±.+., B(B ± f (7π ± =. +.3 (.7 ±. ±.6.9 6, and B(B ± π ± π ± π non resonant = (.3 ±.7 ± , where the uncertainties are statistical, systematic, and model-dependent, respectively. We have made the first observation of the decay B ± f (7π ± with a statistical significance of 6.σ. The systematic uncertainties that affect the results of Dalitz plot analysis are due to the the number of B +B events, signal and background PDF For the π + π S-wave amplitude, we use a modelindependent partial wave analysis: instead of including the S-wave amplitude as a superposition of relativistic Breit-Wigner functions, we divide the π + π mass spectrum into 9 slices and we parametrize the S-wave by an interpolation between the 3 endpoints in the complex plane: AS wave (mππ = Interp(ck (mππ eiφk (mππ k=,..,3. (9 The amplitude and phase of each endpoint are free parameters. The width of each slice is tuned to get approximately the same number of π + π combinations ( 379 /9. Interpolation is implemented by a Relaxed Cubic Spline []. The phase is not constrained in a specific range in order to allow the spline to be a continuous function. The nominal signal Dalitz plot model includes f (7, ρ(77, ρ( and ππ S-wave. The Dalitz

5 Proceedings of the DPF-9 Conference, Detroit, MI, July 7-3, 9 3 (a 7 (b Amplitude Phase (rad m(π + π - (GeV -.. m(π + π - (GeV Figure 6: (a S-wave amplitude extracted from the best fit, (b corresponding S-wave phase. events/. GeV 8 6 (a events/.87 GeV 8 6 (b.. m (π + π - low (GeV 3 m (π + π - high (GeV events/.87 GeV (c events/.87 GeV 8 6 (d 3 m (π + π - (GeV 3 m (π + π + (GeV Figure 7: Dalitz plot projections (points with error bars and fit results (solid histogram. (a m (π + π low, (b m (π + π high, (c total m (π + π, (d m (π + π +. The hatched histograms show the background distribution. plot is dominated by the D s (π + π S wave π + contribution. The S-wave shows, in both amplitude and phase, the expected behavior for the f (98 resonance, and further activity in the regions of the f (37 and f ( resonances. The S-wave is small in the f (6 region, indicating that this resonance has a small coupling to s s. The resulting S-wave π + π amplitude and phase is shown in Fig. 6a and 6b. Our results on the amplitude and phase of the π + π S-wave agree better (within uncertainties with the results from FOCUS [] than those from E79 [3]. The Dalitz plot projections together with the fit results are shown in Fig. 7. Since D s + π+ π π + and D s + K+ K π + have similar topologies, the ratio of branching fractions is expected to have a reduced systematic uncertainty. We therefore select events from the two D s decay modes using similar selection criteria for the Ds + selection. The ratio of branching fractions is evaluated as: B(D s + π+ π π + B(D s + =.99 ±. ±.9, K + K π + where the first error is statistical and the second is systematic. The systematic uncertainty comes from the MC statistics and from the selection criteria used.

6 6 Proceedings of the DPF-9 Conference, Detroit, MI, July 7-3, 9. Conclusion We reviewed the recent results from the Dalitz plot analyses of B D + π π, B + π + π π + and D s + π + π π + performed by the BABAR Collaboration. The results can be summarized as follows: B D + π π We find the total branching fraction of the threebody decay: B(B D + π π = (.8 ±.3 ±. 3. We observe the established D and confirm the existence of D in their decays to D + π, we measure the masses and widths of D and D to be: m D = (6. ±.±.±.9MeV, Γ D = (.8±.±.±.MeV, m D = (97±8±±9MeV and = (73 ± ± 7 ± MeV. Γ D B + π + π π + We measure the branching fractions B(B ± π ± π ± π = (. ±.6 ±. ±. 6, B(B ± ρ (77π ± = (8. ±.7 ± , B(B ± f (7π ± = (.7 ±. ± , and B(B ± π ± π ± π non resonant = (.3 ±.7 ± We observe no significant direct CP asymmetries for the above modes, and there is no evidence for the decays B ± f (98π ±, B ± χ c π ±, or B ± χ c π ±. D + s π + π π + We perform a high precision measurement of the ratio of branching fractions: B(D s π + π π + /B(D s K + K π + =.99 ±. ±.9. The amplitude and phase of the π + π S-wave is extracted in a modelindependent way for the first time. References [] M. Bauer, B. Stech, and M. Wirbel, Z. Phys. C 3, 3 (987. [] B. Aubert et al., [BABAR Collaboration], Nucl. Instrum. Methods A 79, (. [3] J. Blatt and V. Weisskopf, Theoretical Nuclear Physics, p.36, New York, John Wiley & Sons (9. [] C. Zemach, Phys. Rev., B97 (96. [] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 79, (9. [6] Charged conjugate states are implied throughout the paper. [7] D. V. Bugg, J. Phys. G 36, 73 (9. [8] K. Abe et al., [Belle Collaboration], Phys. Rev. D 69, (. [9] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 79, 76 (9. [] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 79, 33 (9. [] K. S. Kölbig and H. Lipps, Cubic Splines and Their Integrals, CERN Program Library, E. [] J. M. Link et al. [FOCUS Collaboration], Phys. Lett. B 8, (. [3] E. M. Aitala et al. [E79 Collaboration], Phys. Rev. Lett. 86, 76 (.

7 events/. GeV 8 6 (a events/.87 GeV m (π + π - low (GeV m (π + π - high events/.87 GeV (c events/.87 GeV m (π + π - (GeV m (π + π + (G