Dalitz Plot Analyses of B D B D at BABAR...of interference effects. The uncertainty on each fk is...

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Proceedings of the DPF-2009 Conference, Detroit, MI, July 27-31, 2009 Dalitz Plot Analyses of B - D + π - π - , B + π + π - π + and D + s π + π - π + at B A B AR Liaoyuan Dong ∗† (On behalf of the B A B AR Collaboration) Iowa State University, Ames, Iowa 50011-3160, 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 B A B AR results are discussed. 1. 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 [1] and experi- mentally 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 B A B AR detector at the PEP-II asymmetric-energy e + e storage rings at SLAC. A detailed description of the B A B AR detector is given in Ref. [2]. Monte Carlo (MC) simulation is used to study the detector response, its acceptance, background, and to validate the analysis. 2. Dalitz Plot Analysis In the decay of a spin-0 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 2 Γ dxdy = |M| 2 256π 3 m 3 A , (1) where m A is the A particle mass, x and y are the invariant masses-squared of the ab and ac pairs, re- spectively. The Dalitz plot provides a graphical rep- resentation of the variation of the square of the matrix element, |M| 2 , 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 func- tion (PDF). The PDF is the sum of signal and back- * Email: [email protected] Current Address: Institute of High Energy Physics, Beijing 100049, China ground components and has the form: PDF(x,y)= f bg B(x,y) DP B(x,y)dxdy + (1 f bg ) [S(x,y) ⊗R] ǫ(x,y) DP [S(x,y) ⊗R] ǫ(x,y)dxdy , (2) where The integral is performed over the whole Dalitz plot (DP). S (= |M| 2 ) and B describe signal and back- ground 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 res- onances. 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 = Nevent i=1 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 (4) 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 ma- trix element M is given by M = k ρ k e k A k , (5) SLAC-PUB-14948 Work supported in part by US Department of Energy contract DE-AC02-76SF00515. SLAC National Accelerator Laboratory, Menlo Park, CA 94025

Transcript of Dalitz Plot Analyses of B D B D at BABAR...of interference effects. The uncertainty on each fk is...

  • Proceedings of the DPF-2009 Conference, Detroit, MI, July 27-31, 2009

    Dalitz Plot Analyses of B− → D+π−π−, B+ → π+π−π+ andD

    +s

    → π+π

    −π

    + at BABAR

    Liaoyuan Dong∗†

    (On behalf of the BABAR Collaboration)Iowa State University, Ames, Iowa 50011-3160, 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.

    1. Introduction

    Dalitz plot analysis is an excellent way to studythe dynamics of three-body decays. The decays understudy are expected to proceed predominantly throughintermediate quasi-two-body modes [1] and experi-mentally this is the observed pattern. Dalitz plotanalysis can explore resonances properties, measurecorresponding magnitudes and phases, and investigatecorresponding 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 atthe PEP-II asymmetric-energy e+e− storage rings atSLAC. A detailed description of the BABAR detectoris given in Ref. [2]. Monte Carlo (MC) simulation isused to study the detector response, its acceptance,background, and to validate the analysis.

    2. Dalitz Plot Analysis

    In the decay of a spin-0 particle A to a final statecomposed of three pseudo-scalar particles (abc), thedifferential decay rate is generally given in terms ofthe Lorentz-invariant matrix element M by

    d2Γ

    dxdy=

    |M|2

    256π3m3A, (1)

    where mA is the A particle mass, x and y are theinvariant masses-squared of the ab and ac pairs, re-spectively. The Dalitz plot provides a graphical rep-resentation of the variation of the square of the matrixelement, |M|2, over the kinematically accessible phasespace (x,y) of the process.

    The distribution of candidate events in the Dalitzplot is described in terms of a probability density func-tion (PDF). The PDF is the sum of signal and back-

    ∗Email: [email protected]†Current Address: Institute of High Energy Physics, Beijing

    100049, China

    ground components and has the form:

    PDF(x, y) = fbgB(x, y)

    DPB(x, y)dxdy

    + (1 − fbg)[S(x, y) ⊗R] ǫ(x, y)

    DP [S(x, y) ⊗R] ǫ(x, y)dxdy,

    (2)

    where

    • The integral is performed over the whole Dalitzplot (DP).

    • S (= |M|2) and B describe signal and back-ground amplitude contributions, respectively.

    • The S(x, y) ⊗ R is the signal term convolvedwith the signal resolution function, the signalresolution can be safely neglected for broad res-onances.

    • fbg is the fraction of background events.

    • ǫ is the reconstruction efficiency.

    An unbinned maximum likelihood fit to the Dalitzplot is performed in order to maximize the value of

    L =

    Nevent∏

    i=1

    PDF(xi, yi) (3)

    with respect to the parameters used to describe S,where xi and yi are the values of x and y for event irespectively, and Nevent is the number of events in theDalitz plot. In practice, the negative-log-likelihood(NLL) value

    NLL = − lnL (4)

    is minimized in the fit.The isobar model formulation is used, in which the

    signal decays are described by a coherent sum of anumber of two-body amplitudes. The total decay ma-trix element M is given by

    M =∑

    k

    ρkeiφkAk, (5)

    SLAC-PUB-14948

    Work supported in part by US Department of Energy contract DE-AC02-76SF00515.

    SLAC National Accelerator Laboratory, Menlo Park, CA 94025

  • 2 Proceedings of the DPF-2009 Conference, Detroit, MI, July 27-31, 2009

    where ρk and φk are the magnitudes and phases of thekth decay mode respectively, the Ak distributions forresonant contribution describe the dynamics of the de-cay amplitudes and are written as the product of a rel-ativistic Breit-Wigner function, two Blatt-Weisskopfbarrier form factors [3] and an angular function [4].

    The efficiency-corrected fraction due to the resonantor non-resonant contribution k is defined as follows:

    fk =|ρk|

    2∫

    DP |Ak|2dxdy

    DP | M |2 dxdy

    . (6)

    The fk values do not necessarily add to 1 becauseof interference effects. The uncertainty on each fk isevaluated by propagating the full covariance matrixobtained from the fit.

    3. Results from BABAR

    3.1. B− → D+π−π− [5]

    We reconstruct the decays B− → D+π−π− [6] withD+ → K−π+π+ based on a sample of about 383 ×106 Υ(4S) → BB decays. The resulting Dalitz plotdistribution for data is shown in Fig. 1. The numberof signal events in the Dalitz plot is 3414 ± 85. Thebackground fraction is about 30.4%. The backgroundis parametrized using the MC simulation.

    )4

    /c2

    ) (GeV 1 π(D 2

    m5 10 15 20 25

    )4

    /c2

    ) (G

    eV 2

    π

    (D

    2 m

    5

    10

    15

    20

    25

    Figure 1: Dalitz plot for B− → D+π−π−.

    The nominal fit model contains D∗02 , D∗00 ,

    D∗(2007)0, B∗ and P-wave non-resonant contribu-tions. The S-wave and D-wave non-resonant contri-butions are found to be small and can be ignored.Replacing the D∗(2007)0 by a Dπ S-wave [7] state isfound to give a much worse fit.

    Fig. 2a, 2b and 2c show the m2min(Dπ), m2max(Dπ)

    and m2(ππ) projections respectively. Here m2max(Dπ)

    and m2min(Dπ) are the heavy and light invariantmasses-squared of the Dπ systems, respectively. Thedistributions in Fig. 2 show good agreement betweenthe data and the fit.

    We have performed some tests with the broad reso-nance D∗00 excluded or with the J

    P of the broad res-onance replaced by other quantum numbers. In allcases, the NLL values are significantly worse than thatof the nominal fit. We conclude that a broad spin-0state D∗00 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−) = (1.08 ± 0.03 ± 0.05)× 10−3,

    where the first error is statistical and the second is sys-tematic. The systematic error on the measurement ofthe total B− → D+π−π− branching fraction is dueto the uncertainties on the number of B+B− events,the charged track reconstruction and identification ef-ficiencies, theD+ → K−π+π+ branching fraction, thebackground shape and the fit models.

    The mass and width of D∗02 are determined to be:

    mD∗02

    = (2460.4 ± 1.2 ± 1.2 ± 1.9)MeV/c2 and

    ΓD∗02

    = (41.8 ± 2.5 ± 2.1 ± 2.0)MeV,

    respectively, while for the D∗00 they are:

    mD∗00

    = (2297± 8 ± 5 ± 19)MeV/c2 and

    ΓD∗00

    = (273 ± 12 ± 17 ± 45)MeV,

    where the first and second errors reflect the statisticaland systematic uncertainties, respectively, the thirdone is the uncertainty related to the fit models andthe Blatt-Weisskopf barrier factors.

    We have also obtained exclusive branching fractionsfor D∗02 and D

    ∗00 production:

    B(B− → D∗02 π−) × B(D∗02 → D

    +π−)

    = (3.5 ± 0.2 ± 0.2 ± 0.4) × 10−4 and

    B(B− → D∗00 π−) × B(D∗00 → D

    +π−)

    = (6.8 ± 0.3 ± 0.4 ± 2.0) × 10−4.

    The relative phase of the scalar and tensor ampli-tude is measured to be

    φD∗00

    = −2.07± 0.06 ± 0.09 ± 0.18 rad.

    The systematic uncertainties that affect the resultsof Dalitz plot analysis are due to the backgroundparametrization, the selection criteria, and the pos-sible fit bias.

    Our results for the masses, widths and branchingfractions are consistent with but more precise thanprevious measurements performed by Belle [8].

  • Proceedings of the DPF-2009 Conference, Detroit, MI, July 27-31, 2009 3

    )4 /c2) (GeVπ(Dmin2m

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    )4 /c2

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    Figure 2: Result of the nominal fit to the data: projec-tions on (a) m2min(Dπ), (b) m

    2max(Dπ) and (c) m

    2(ππ).The points with error bars are data, the solid curves rep-resent the nominal fit. The shaded areas show the D∗02contribution, the dashed curves show the D∗00 signal, thedash-dotted curves show the D∗(2007)0 and B∗ signals,and the dotted curves show the background.

    3.2. B+ → π+π−π+ [9]

    We fit the B− and B+ samples independently in or-der to determine the CP asymmetry. The total signalamplitudes for B+ and B− decays are given by

    M =∑

    j

    cjAj(m2max,m

    2min) ,

    M̄ =∑

    j

    cjĀj(m2max,m

    2min) , (7)

    where m2max and m2min are the heavy and light invari-

    ant masses-squared of the π±π∓ systems, respectively.The complex coefficients cj and cj for a given decaymode j contain all the weak phase dependence. Sincethe Aj terms contain only strong dynamics, Aj ≡ Āj .The CP asymmetry for each contributing resonance isdetermined from the fitted parameters

    ACP, j =|cj |

    2− |cj |

    2

    |cj |2 + |cj |

    2 . (8)

    We reconstruct the decays B+ → π+π−π+ basedon a sample of about 465 × 106 Υ(4S) → BB decays.We reject background from two-body decays of D0

    meson and charmonium states (J/ψ and ψ(2S)) byexcluding invariant masses (in units of GeV/c2) in theranges: 1.660 < mπ+π− < 1.920, 3.051 < mπ+π− <3.222, and 3.660 < mπ+π− < 3.820. The background-subtracted Dalitz plot distribution for data is shownin Fig. 3. We fit 4335 B candidates to calculate thefit fractions and CP asymmetries. The background isparametrized using the MC simulation.

    )4/c2 (GeVmin2m0 2 4 6 8 10 12 14

    )4/c2

    (G

    eVm

    ax2

    m

    0

    5

    10

    15

    20

    25

    Figure 3: Background-subtracted Dalitz plot for B+ →π+π−π+. The plot shows bins with greater than zero en-tries. The depleted bands are the charm and charmoniaexclusion regions.

    The nominal signal Dalitz plot model comprisesa momentum-dependent non-resonant componentand four intermediate resonance states: ρ0(770)π±,ρ0(1450)π±, f2(1270)π

    ±, and f0(1370)π±. We do

    not find any significant contributions from f0(980)π±,

    χc0π±, or χc2π

    ±. The absence of the charmoniumcontributions precludes the extraction of the unitar-ity triangle angle γ. We find no evidence for direct CPviolation. The inclusive CP asymmetry is determinedto be

    (

    +3.2 ± 4.4 ± 3.1 +2.5−2.0)

    %, where the uncertain-ties 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 Proceedings of the DPF-2009 Conference, Detroit, MI, July 27-31, 2009

    )2 (GeV/c-π+πm0.4 0.6 0.8 1 1.2 1.4 1.6

    )2E

    vent

    s/(2

    5 M

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    0

    40

    80

    120

    160

    200

    )2 (GeV/c-π+πm3.25 3.35 3.45 3.55 3.65

    )2E

    vent

    s/(1

    0 M

    eV/c

    0

    5

    10

    15

    20

    25 c0χ

    c2χ

    Figure 4: Dipion invariant mass projections: (upper) inthe ρ0(770) region; and (lower) in the regions of χc0 andχc2. The data are the points with statistical error bars,the dark-shaded (red) histogram is the qq̄ component, thelight-shaded (green) histogram is the BB background con-tribution, while the upper (blue) histogram shows the totalfit result. The dip near 0.5 GeV/c2 in the upper plot is dueto the rejection of events containing K0S candidates.

    Fig.4. The agreement between the fit results and thedata is generally good.

    The Dalitz plot is dominated by the ρ0(770) res-onance and a non-resonant contribution. The massand width of the f0(1370) are determined to bemf0(1370) = 1400 ± 40 MeV/c

    2 and Γf0(1370) = 300 ±80 MeV (statistical uncertainties only). We mea-sure the branching fractions B(B± → π±π±π∓) =(15.2±0.6±1.2±0.4)×10−6, B(B± → ρ0(770)π±) =(8.1 ± 0.7 ± 1.2+0.4−1.1) × 10

    −6, B(B± → f2(1270)π±) =

    (1.57 ± 0.42 ± 0.16 +0.53−0.19) × 10−6, and B(B± →

    π±π±π∓ non − resonant) = (5.3 ± 0.7 ± 0.6+1.1−0.5) ×

    10−6, where the uncertainties are statistical, sys-tematic, and model-dependent, respectively. Wehave made the first observation of the decay B± →f2(1270)π

    ± with a statistical significance of 6.1σ.

    The systematic uncertainties that affect the re-sults of Dalitz plot analysis are due to the the num-ber of B+B− events, signal and background PDF

    parametrization, the charged track reconstruction andidentification efficiencies, and the possible fit bias.

    Our results will be useful to reduce model uncer-tainties in the extraction of the CKM angle α fromtime-dependent Dalitz plot analysis of B0 → π+π−π0.

    3.3. D+s→ π+π−π+ [10]

    We reconstruct D+s → π+π−π+ based on 384 fb−1

    data sample. We use the tag of D∗s(2112)+ → D+s γ

    to reduce the combinatorial background. The signalPDFs are build using the D+s → K

    +K−π+. The re-constructed D+s sample contains 13179 events with apurity of 80%. The background shape is obtained byfitting the D+s sidebands. The Dalitz plot distributionfor data is shown in Fig. 5.

    0

    1

    2

    3

    0 0.5 1 1.5 2 2.5 3 3.5

    m2(π+ π-) (GeV2/c4)

    m2 (

    π+π-

    ) (G

    eV2 /

    c4)

    Figure 5: Dalitz plot for D+s→ π+π−π+.

    For the π+π− S-wave amplitude, we use a model-independent partial wave analysis: instead of includ-ing the S-wave amplitude as a superposition of rela-tivistic Breit-Wigner functions, we divide the π+π−

    mass spectrum into 29 slices and we parametrize theS-wave by an interpolation between the 30 endpointsin the complex plane:

    AS−wave(mππ) = Interp(ck(mππ)eiφk(mππ))k=1,..,30 .

    (9)

    The amplitude and phase of each endpoint are freeparameters. The width of each slice is tuned to getapproximately the same number of π+π− combina-tions (≃ 13179× 2/29). Interpolation is implementedby a Relaxed Cubic Spline [11]. The phase is not con-strained in a specific range in order to allow the splineto be a continuous function.

    The nominal signal Dalitz plot model includesf2(1270), ρ(770), ρ(1450) and ππ S-wave. The Dalitz

  • Proceedings of the DPF-2009 Conference, Detroit, MI, July 27-31, 2009 5

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    Figure 6: (a) S-wave amplitude extracted from the best fit, (b) corresponding S-wave phase.

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    Figure 7: Dalitz plot projections (points with error bars) and fit results (solid histogram). (a) m2(π+π−)low, (b)m2(π+π−)high, (c) total m

    2(π+π−), (d) m2(π+π+). The hatched histograms show the background distribution.

    plot is dominated by the Ds → (π+π−)S−waveπ

    +

    contribution. The S-wave shows, in both amplitudeand phase, the expected behavior for the f0(980) res-onance, and further activity in the regions of thef0(1370) and f0(1500) resonances. The S-wave issmall in the f0(600) region, indicating that this reso-nance has a small coupling to ss̄.

    The resulting S-wave π+π− amplitude and phase isshown in Fig. 6a and 6b. Our results on the amplitudeand phase of the π+π− S-wave agree better (withinuncertainties) with the results from FOCUS [12] thanthose from E791 [13]. The Dalitz plot projections to-gether with the fit results are shown in Fig. 7.

    Since D+s → π+π−π+ and D+s → K

    +K−π+ havesimilar topologies, the ratio of branching fractions isexpected to have a reduced systematic uncertainty.We therefore select events from the two Ds decaymodes using similar selection criteria for the D∗+s se-lection. The ratio of branching fractions is evaluatedas:

    B(D+s → π+π−π+)

    B(D+s → K+K−π+)= 0.199 ± 0.004± 0.009,

    where the first error is statistical and the second issystematic. The systematic uncertainty comes fromthe MC statistics and from the selection criteria used.

  • 6 Proceedings of the DPF-2009 Conference, Detroit, MI, July 27-31, 2009

    4. Conclusion

    We reviewed the recent results from the Dalitz plotanalyses of B− → D+π−π−, B+ → π+π−π+ andD+s → π

    +π−π+ performed by the BABAR Collabora-tion. The results can be summarized as follows:

    • B− → D+π−π−

    We find the total branching fraction of the three-body decay: B(B− → D+π−π−) = (1.08 ±0.03± 0.05)× 10−3. We observe the establishedD∗02 and confirm the existence of D

    ∗00 in their

    decays to D+π−, we measure the masses andwidths of D∗02 and D

    ∗00 to be: mD∗02 = (2460.4±

    1.2±1.2±1.9)MeV/c2, ΓD∗02

    = (41.8±2.5±2.1±

    2.0)MeV, mD∗00

    = (2297±8±5±19)MeV/c2 and

    ΓD∗00

    = (273 ± 12 ± 17 ± 45)MeV.

    • B+ → π+π−π+

    We measure the branching fractions B(B± →π±π±π∓) = (15.2 ± 0.6 ± 1.2 ± 0.4) ×10−6, B(B± → ρ0(770)π±) = (8.1 ± 0.7 ±1.2+0.4−1.1) × 10

    −6, B(B± → f2(1270)π±) =

    (1.57 ± 0.42 ± 0.16 +0.53−0.19) × 10−6, and B(B± →

    π±π±π∓ non − resonant) = (5.3 ± 0.7 ±0.6+1.1−0.5)×10

    −6. We observe no significant directCP asymmetries for the above modes, and thereis no evidence for the decays B± → f0(980)π

    ±,B± → χc0π

    ±, or B± → χc2π±.

    • D+s → π+π−π+

    We perform a high precision measurement ofthe ratio of branching fractions: B(Ds →π+π−π+)/B(Ds → K

    +K−π+) = 0.199 ±0.004 ± 0.009. The amplitude and phase of

    the π+π− S-wave is extracted in a model-independent way for the first time.

    References

    [1] M. Bauer, B. Stech, and M. Wirbel, Z. Phys. C34, 103 (1987).

    [2] B. Aubert et al., [BABAR Collaboration], Nucl. In-strum. Methods A 479, 1 (2002).

    [3] J. Blatt and V. Weisskopf, Theoretical NuclearPhysics, p.361, New York, John Wiley & Sons(1952).

    [4] C. Zemach, Phys. Rev. 140, B97 (1965).[5] B. Aubert et al. [BABAR Collaboration], Phys.

    Rev. D 79, 112004 (2009).[6] Charged conjugate states are implied throughout

    the paper.[7] D. V. Bugg, J. Phys. G 36, 075003 (2009).[8] K. Abe et al., [Belle Collaboration], Phys. Rev.

    D 69, 112002 (2004).[9] B. Aubert et al. [BABAR Collaboration], Phys.

    Rev. D 79, 072006 (2009).[10] B. Aubert et al. [BABAR Collaboration], Phys.

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    Lett. B 585, 200 (2004).[13] E. M. Aitala et al. [E791 Collaboration], Phys.

    Rev. Lett. 86, 765 (2001).

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