Measurement of ${CP}\xspace$ Violation Parameters with a Dalitz Plot Analysis of $B^\pm \to D_{\pi^+\pi^-\pi^0} K^\pm$

We report the results of a CP violation analysis of the decay B+/- -->D(pipipi0)K+/-, where D(pipipi0) indicates a neutral D meson detected in the final state pi+pi-pi0, excluding K_S pi0. The analysis makes use of 324 million e+e- -->BB events recorded by the BaBar experiment at the PEP-II e+e- storage ring. By analyzing the pi+pi-pi0 Dalitz plot distribution and the B+/- -->D(pipipi0)K+/- branching fraction and decay rate asymmetry, we calculate parameters related to the phase gamma of the CKM unitarity triangle. We also measure the magnitudes and phases of the components of the D0 -->pi+pi-pi0 decay amplitude.

We report the results of a CP violation analysis of the decay B ± → D π + π − π 0 K ± , where D π + π − π 0 indicates a neutral D meson detected in the final state π + π − π 0 , excluding K 0 S π 0 . The analysis makes use of 324 million e + e − → BB events recorded by the BABAR experiment at the PEP-II e + e − storage ring. By analyzing the π + π − π 0 Dalitz plot distribution and the B ± → D π + π − π 0 K ± branching fraction and decay rate asymmetry, we calculate parameters related to the phase γ of the CKM unitarity triangle. We also measure the magnitudes and phases of the components of the D 0 → π + π − π 0 decay amplitude.
PACS numbers: 13.25.Hw,12.15.Hh,11.30.Er An important component of the program to study CP violation is the measurement of the angle γ = arg (−V ud V * ub /V cd V * cb ) of the unitarity triangle related to the Cabibbo-Kobayashi-Maskawa quark mixing matrix [1]. The decays B → D ( * )0 K ( * ) can be used to measure γ with essentially no hadronic uncertainties, exploiting interference between b → ucs and b → cus decay amplitudes [2]. In one of the measurement methods [3], γ is extracted by analyzing the D-decay Dalitz plot distribution in B ± → DK ± with multi-body D decays [4]. This method has only been used with the Cabibbo-favored decay D → K 0 S π + π − [5,6], and Cabibbo-suppressed decays are expected to be similarly sensitive to γ [7]. We present here the first CP -violation study of B ± → DK ± with a multibody, Cabibbo-suppressed D decay, D → π + π − π 0 .
The data used in this analysis were collected with the BABAR detector at the PEP-II e + e − storage ring, running on the Υ (4S) resonance. Samples of simulated Monte Carlo (MC) events were analyzed with the same reconstruction and analysis procedures. These samples include an e + e − → BB sample about five times larger than the data; a continuum e + e − → qq sample, where q is a u, d, s, or c quark, with luminosity equivalent to the data; and a signal sample about 300 times larger than the data, with both phase space D decays and decays generated according to the amplitudes measured by CLEO [8]. The BABAR detector and the methods used for particle reconstruction and identification are described in Ref. [9].
The reader is referred to Ref. [10] for details of the event selection criteria. Briefly, we use event-shape variables to suppress the continuum background, and identify kaon and pion candidates using specific ionization and Cherenkov radiation. The invariant mass of D candidates must satisfy 1830 < M D < 1895 MeV/c 2 . We require 5272 < m ES < 5300 MeV/c 2 , where m ES ≡ E 2 CM /4 − |p B | 2 , E CM is the total e + e − center-of-mass (CM) energy, and p B is the B candidate CM momentum. Events must satisfy −70 < ∆E < 60 MeV, where ∆E = E B − E CM /2 and E B is the B candidate CM energy. We exclude the decay mode D → K 0 S π 0 , which is a previously studied CP eigenstate not related to the method of Ref. [3], by rejecting candidates with 489 < M (π + π − ) < 508 MeV/c 2 or for which the distance between the π + π − vertex and the B − candidate decay vertex is more than 1.5 cm. We reject B ± → D π + π − π 0 K ± candidates in which the K ± π ∓ invariant mass satisfies 1840 < M (K ± π ∓ ) < 1890 MeV/c 2 , to suppress B − → D 0 K − π + ρ − decays. We require d > 0.25, where d [10] is a neural net variable that separates signal candidates (which peak toward d = 1) from those with a misreconstructed D (peaking toward d = 0). In events with multiple candidates, we keep the candidate whose m ES value is closest to the nominal B ± mass [11].
As in Ref. [10], we identify in the MC samples ten event types, one signal and nine different backgrounds. We list them here with the labels used to refer to them throughout the paper. DK sig : B ± → D π + π − π 0 K ± events that are correctly reconstructed; these are the only events considered to be signal. DK bgd : B ± → D π + π − π 0 K ± events that are misreconstructed; namely, some of the particles used to form the final state do not originate from the B ± → D π + π − π 0 K ± decay. Dπ D (Dπ D ): The measurement of the CP parameters proceeds in three steps, each involving an unbinned maximum likelihood fit. In step 1, we measure the complex Dalitz plot amplitude α(s + , s − ) for the decay D 0 → π + π − π 0 , where s ± = m 2 (π ± π 0 ) are the squared invariant masses of the π ± π 0 pairs. In step 2, we extract the numbers of B + and B − signal events and background yields. We obtain the CP parameters in step 3.
We parameterize α(s + , s − ) using the isobar model, α(s + , s − ) = [a NR e iφNR + r a r e iφr A r (s + , s − )]/N α , where the first term represents a nonresonant contri-bution, the sum is over all intermediate two-body resonances r, and N α is such that ds + ds − |α(s + , s − )| 2 = 1. The amplitude for the decay chain where m r is the peak mass of the resonance [11], M 2 AB is the squared invariant mass of the AB pair, F r is a spin-dependent form factor [12], and Γ r (M AB ) is the mass-dependent width for the resonance r [12]. The spin , and m i is the mass of particle i [11].
In step 1, we determine the parameters a NR , a r , φ NR , and φ r by fitting a large sample of D 0 and D 0 mesons, flavor-tagged through their production in the decay D * + → D 0 π + [13]. To select this sample, we require the CM momentum of the D * candidate to be greater than 2770 MeV/c, and The signal and background yields are obtained from a fit to the M D distribution, modeling the signal as a Gaussian and the background as an exponential. The signal Gaussian peaks at 1863.7 ± 0.4 MeV/c 2 and has a width of 17.4 ± 0.8 MeV/c 2 .
Of the D 0 candidates in the signal region 1848 < M D < 1880 MeV/c 2 , we obtain from the fit N S = 44780±250 signal and N B = 830±70 background events. To obtain the parameters of α(s ± , s ∓ ), we fit these candidates with the probability distribution function (PDF) where the background PDF f B (s + , s − ) is a binned distribution obtained from events in the sideband 1930 < M D < 1990 MeV/c 2 , and ǫ(s + , s − ) is an efficiency function, parameterized as a two-dimensional third-order polynomial determined from MC. To within the MC-signal statistical uncertainty, ǫ(s + , s − ) = ǫ(s − , s + ). The region M D < 1848 MeV/c 2 , which contains D 0 → K − π + π 0 events that are absent from the signal region, is not used. Table I summarizes the results of this fit, with systematic errors obtained by varying the masses and widths of the ρ(1700) and σ resonances, setting F r = 1, and varying ǫ(s + , s − ) to account for uncertainties in reconstruction and particle identification. The Dalitz plot distribution of the data is shown in Fig. 1(a-c). The distribution is marked by three destructively interfering ρπ amplitudes, suggesting an I = 0-dominated final state [14].
The fit for step i ∈ {2, 3} uses the PDF where ξ i is the set of n i event variables ξ 1 = {∆E, q ′ , d ′ }, ξ 2 = {∆E, q ′ , s − , s + }, t corresponds to one of the ten event types listed above, N t = N + t + N − t is the number of events of type t, A t = (N − t − N + t )/N t is their charge asymmetry, C = ±1 is the electric charge of the B can- I: Result of the fit to the D * + → D 0 π + sample, showing the amplitudes ratios Rr ≡ ar/a ρ + (770) , phase differences ∆φr ≡ φr − φ ρ + (770) , and fit fractions fr ≡ |arAr(s+, s−)| 2 ds−ds+. The first (second) errors are statistical (systematic). We take the mass (width) of the σ meson to be 400 (600) MeV/c 2 .

State
Rr ( didate, and η ≡ t N t . Using MC, we verify that the ξ i and ξ j (i = j) distributions are uncorrelated for each event type. Therefore, the PDFs P (C) i,t are the products The parameters of the Dalitz plot PDF D ′ C DKsig (s + , s − ) are obtained from the data as described below. Those of all other functions in Eq. (2) are obtained from the MC samples. The functions E t (∆E) are parameterized as the sum of a Gaussian and a second-order polynomial. The PDFs Q t (q ′ ) and C t (d ′ ) are the sum of a Gaussian and an asymmetric Gaussian. The PDF parameters are different for each event type. Assuming no CP violation in the background, we take D ′ + t (s + , s − ) = D ′ − t (s − , s + ) and A t = 0 for t = DK sig . The functions D ′ C DπX (s + , s − ) and D ′ C DK bgd (s + , s − ) are binned histograms obtained from the MC. For other event types, where the efficiency function ǫ(s + , s − ) has different parameters for wellreconstructed and misreconstructed D candidates.
The signal Dalitz PDF accounts for interference between the b → ucs and b → cus amplitudes A u and A c : where z ± = |A u /A c |e i(δ±γ) and δ is a CP -even phase.
In the step-2 fit, we extract the B ± → D π + π − π 0 K ± signal yield and asymmetry, as well as some background yields, as described in Ref. [10]. From this fit we find N DKsig = 170 ± 29 signal events and a decay rate asymmetry A DKsig = −0.02 ± 0.15. Errors are statistical only.
Only the complex parameters z ± are free in the step-3 fit. This fit minimizes the function where N ev is the number of events in the data sample. The term χ 2 = 2 u,v=1 X u V −1 uv X v increases the sensitivity of the fit by using the results of the step-2 fit via where are the expected numbers of B ± signal events. In Eq. (6), N 0 is the product of the number N B + B − of charged B + B − pairs in the dataset, the branching fractions B(B − → D 0 K − ) [11] and B(D 0 → π + π − π 0 ) [13], and the total reconstruction efficiency ǫ = 11.4%. The error matrix V uv is the sum of two components: the step-2 fit error matrix V stat uv , which is almost diagonal (the correlation coefficient is −2.8%), and the N 0 systematic error matrix V syst uv . Here V syst where σ rel c are the relative errors on the four components N B + B − (1.1%), ǫ (3.3%), B(D → π + π − π 0 ) (3.8%) [13], and B(B − → D 0 K − ) (5.9%) [11].
shown in Fig. 1(d). Projections of the data and the PDF onto s + and s − are shown in Fig. 1(e-f). Additional systematic errors due to the analysis procedure are evaluated for the signal branching fraction, charge asymmetry, ρ ± , and θ ± . The uncertainty in the model used for α(s + , s − ) is the largest source of error on the CP parameters: σ model This error is evaluated by removing all but the ρ(770), ρ(1450), f 0 (980), and nonresonant terms in α(s + , s − ); adding an f ′ 2 (1525), an ω, and a nonresonant P-wave contribution; varying the meson "radius" parameter in F r [12]; and propagating the errors from Table I. Uncertainties due to the masses and widths of the ρ(1700) and σ resonances are small by comparison. Other errors are due to uncertainties on background yields that are fixed in the fits [10], finite MC sample size, a possible reconstruction efficiency charge asymmetry, and uncertainties in the background PDF shapes, evaluated by comparing MC and data in signal-free sidebands of the variables M D , ∆E, and m ES . We also evaluate errors due to possible charge asymmetries in DKX and DK bgd , uncertainties in particle identification and the efficiency functions, the finite s ± measurement resolution, the background PDF f B in the D * sample, D-flavor mistagging in the D * sample, and correlations between the D flavor and the kaon charge in qq D events. These errors add in quadrature to σ syst ρ± = 0.05, σ syst θ− = 19 • , σ syst θ+ = 13 • , and are combined with the systematic errors of Eqs. (9).