Measurement of the Cabibbo-Kobayashi-Maskawa angle gamma in B-+ ->D(*)K-+ decays with a Dalitz analysis of D ->K0s pi- pi+

We report on a measurement of the Cabibbo-Kobayashi-Maskawa CP-violating phase gamma through a Dalitz analysis of neutral D decays to K0s pi- pi+ in the processes B-+ ->D(*)K-+, D* ->D pi0, D gamma. Using a sample of 227 million BBbar pairs collected by the BaBar detector, we measure the amplitude ratios rB = 0.12+/-0.08+/-0.03+/-0.04 and rB* = 0.17+/-0.10+/-0.03+/-0.03, the relative strong phases deltaB = 104+/-45+17-21+16-24, and deltaB*= -64+/-41+14-12+/-15 between the amplitudes A(B- ->Dbar(*)0 K-) and A(B- ->D(*)0 K-), and gamma = 70+/-31+12-10+14-11. The first error is statistical, the second is the experimental systematic uncertainty and the third reflects the Dalitz model uncertainty. The results for the strong and weak phases have a two-fold ambiguity.

We report on a measurement of the Cabibbo-Kobayashi-Maskawa CP -violating phase γ through a Dalitz analysis of neutral D decays to K 0 S π − π + in the processes B ∓ → D ( * ) K ∓ , D * → Dπ 0 , Dγ. Using a sample of 227 million BB pairs collected by the BABAR detector, we measure the amplitude ratios rB = 0.12±0.08±0.03±0.04 and r * B = 0.17±0.10±0.03±0.03, the relative strong phases δB = 104 ± 45 +17 • . The first error is statistical, the second is the experimental systematic uncertainty and the third reflects the Dalitz model uncertainty. The results for the strong and weak phases have a two-fold ambiguity.
PACS numbers: 13.25.Hw,11.30.Er,12.15.Hh,13.25.Ft CP violation in the B meson system has been clearly established in recent years [1,2]. Although these results are in good agreement with Standard Model expectations, other and more precise measurements of CP violation in B decays are needed to over-constrain the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix [3] and search for new physics effects. The angle γ of the unitarity triangle [4] of the CKM matrix constitutes one of these crucial measurements.
Various methods using B − →D ( * )0 K − [5] decays have been proposed to measure γ [6][7][8]. Here,D 0 indicates either a D 0 or a D 0 meson and the symbol "( * )" refers to either a D or D * meson. All methods exploit the fact that a B − can decay into a D ( * )0 K − (D ( * )0 K − ) final state via b → cus (b → ucs) transitions. These decay amplitudes interfere when the D 0 and D 0 decay into the same final state, which can lead to different B + and B − decay rates (direct CP violation). In this Letter we report on a measurement of γ based on the analysis of the Dalitz distribution of the three-body decay D 0 → K 0 S π − π + [7,8]. The primary advantage of this method is that it involves the entire resonant structure of the three-body decay, with interference of doubly Cabibbo-suppressed (DCS), Cabibbo-allowed (CA), and CP eigenstate amplitudes, providing the sensitivity to γ. The analysis is based on an integrated luminosity of 205 fb −1 recorded at the Υ (4S) resonance (corresponding to 227 million BB decays) and 9.6 fb −1 collected at a center-of-mass (CM) energy 40 MeV below with the BABAR detector [9] at the SLAC PEP-II e + e − asymmetric-energy B Factory.
The small CP asymmetry in D decays allowed by the present experimental limits [10] has a negligible effect on this analysis. Thus, the B ∓ →D ( * )0 K ∓ , where m 2 − and m 2 + are the squared invariant masses of the K 0 S π − and K 0 S π + combinations, respectively, and A D (m 2 − , m 2 + ) is the D 0 → K 0 S π − π + decay amplitude. Here, r are the amplitude ra-tios and relative strong phases between the amplitudes A(B − → D ( * )0 K − ) and A(B − → D ( * )0 K − ). As a consequence of parity and angular momentum conservation in theD * 0 decay, the factor κ takes the value +1 for B − →D 0 K − and B − →D * 0 (D 0 π 0 )K − , and −1 for B − →D * 0 (D 0 γ)K − [11]. We first determine A D (m 2 − , m 2 + ) through a Dalitz analysis of a highstatistics sample of tagged D 0 mesons from inclusive D * + → D 0 π + decays reconstructed in data. We then perform a simultaneous fit to the |A B , and γ. We emphasize that in this analysis the Dalitz amplitude is only a means to extract the CP parameters.
B − candidates are formed by combining a massconstrained D ( * )0 candidate with a track identified as a kaon [9]. We accept K 0 S → π + π − candidates that have a two-pion invariant mass within 9 MeV/c 2 of the K 0 S mass [4] and a cosine of the angle between the line connecting the D 0 and K 0 S decay vertices and the K 0 S momentum (in the plane transverse to the beam) greater than 0.99. D 0 candidates are selected by requiring the K 0 S π − π + invariant mass to be within 12 MeV/c 2 of the D 0 mass [4]. The π 0 candidates from D * 0 → D 0 π 0 are formed from pairs of photons with invariant mass in the range [115,150] MeV/c 2 , and with photon energy greater than 30 MeV. Photon candidates from D * 0 → D 0 γ are selected if their energy is greater than 100 MeV. D * 0 → D 0 π 0 (D 0 γ) candidates are required to have a D * 0 -D 0 mass difference within 2.5 (10) MeV/c 2 of its nominal value [4].
The beam-energy substituted B mass m ES [12] (Fig. 1) and the difference ∆E between the reconstructed energy of the B − candidate and the beam energy in the e + e − CM frame are used to identify signal B − decays. We require m ES > 5.2 GeV/c 2 and |∆E| < 30 MeV. Since the background is dominated by random combinations of tracks arising from e + e − → qq, q = {u, d, s, c} (continuum) events, we require | cos θ * T | < 0.8, where θ * T is the CM angle between the thrust axis of the B − candidate and that of the remaining particles in the event.
The curves represent the fit projections for signal plus background (solid lines) and background (dotted lines). The peaking structure of the background is due to remaining B − →D ( * )0 π − events.
The reconstruction efficiencies (purities in the signal re- The cross-feed among the different samples is negligible. The D 0 decay amplitude is determined from an unbinned maximum-likelihood Dalitz fit to a high-purity (97%) sample of 81496 D * + → D 0 π + decays reconstructed in 91.5 fb −1 of data (Fig. 2). We use the isobar formalism described in Ref. [13] to express A D as a sum of two-body decay-matrix elements (subscript r) and a non-resonant (subscript NR) contribution, where each term is parameterized with an amplitude a r and a phase φ r . The function A r (m 2 − , m 2 + ) is the Lorentz-invariant expression for the matrix element of a D 0 meson decaying into K 0 S π − π + through an intermediate resonance r, parameterized as a function of the position in the Dalitz plane. Table I summarizes the values of a r and φ r obtained using a model consisting of 16 two-body elements comprising 13 distinct resonances and accounting for efficiency variations across the Dalitz plane and the small background contribution. For r = ρ(770), ρ(1450) we use the functional form suggested in Ref. [14], while the remaining resonances are parameterized by a spin-dependent relativistic Breit-Wigner distribution. For intermediate states with a K * , the regions of interference between DCS and CA decays are particularly sensitive to γ, and we include the DCS component when a significant contribution is expected. In addition, we find that the inclusion of the scalar ππ resonances σ and σ ′ significantly improves the quality of the fit [15]. Since the two σ resonances are not well established and are only introduced to improve the description of our data, the uncertainty on their existence is considered in the systematic errors. We estimate the goodness of fit through a two-dimensional χ 2 test and  I: Amplitudes ar, phases φr and fit fractions obtained from the fit of the D 0 → K 0 S π − π + Dalitz distribution from D * + → D 0 π + events. Errors are statistical only. Masses and widths of all resonances except σ and σ ′ are taken from [4]. The fit fraction is defined as the integral of a 2 r |Ar(m 2 − , m 2 + )| 2 over the Dalitz plane divided by the integral of |AD(m 2 − , m 2 + )| 2 . The sum of fit fractions is 1.24. obtain χ 2 = 3824 for 3054 − 32 degrees of freedom. We simultaneously fit the B − →D ( * )0 K − samples using an unbinned extended maximum-likelihood fit to extract the CP -violating parameters along with the signal and background yields. Three different background components are considered: continuum events, B − →D ( * )0 π − and Υ (4S) → BB (other than B − → D ( * )0 π − ) decays. In addition to m ES , the fit uses ∆E and a Fisher discriminant [12] to distinguish signal from B − →D ( * )0 π − and continuum background, respectively. The log-likelihood is

Resonance
where ξ j = {m ES , ∆E, F} j and η j = (m 2 − , m 2 + ) j characterize the event j. Here, P c ( ξ) and P Dalitz c ( η) are the probability density functions (PDF's), and N c the event yield for signal or background component c. For signal events, P Dalitz c ( η) is given by |A ( * ) ∓ ( η)| 2 corrected by the efficiency variations. All PDF shape parameters used to describe signal, continuum and B − →D ( * )0 π − components are determined directly from B − →D ( * )0 K − and B − →D ( * )0 π − signal, sideband regions, and off-peak data, and are fixed in the final fit for CP parameters and event yields. Only the m ES , ∆E and Dalitz PDF's for BB background events are determined from a detailed Monte Carlo simulation. B − →D ( * )0 π − candidates have been selected using criteria similar to those applied for B − →D ( * )0 K − but requiring the bachelor pion not to be consistent with the kaon hypothesis.
The CP fit yields 282 ± 20, 90 ± 11, and 44 ± 8 signal B ± e i(δ ( * ) B ±γ) , respectively, are summarized in Table II. Here, r ( * ) B ± is the amplitude ratio between the amplitudes b → u and b → c, separately for B + and B − . The only non-zero statistical correlations involving the CP parameters are for the pairs z − , z + , z * − , and z * + , which amount to 3%, 6%, −17%, and −27%, respectively. The z ( * ) ± variables are more suitable fit parameters than r ( * ) B and γ because they are better behaved near the origin, especially in low-statistics samples. Figures 3(a,b) show the one-and two-standard deviation confidence-level contours (statistical only) in the z ( * ) planes forD 0 K − andD * 0 K − , and separately for B − and B + . The separation between the B − and B + regions in these planes is an indication of direct CP violation.
The largest single contribution to the systematic uncertainties in the CP parameters comes from the choice of the Dalitz model used to describe the D 0 → K 0 S π − π + decay amplitudes. To evaluate this uncertainty we use the nominal Dalitz model (Table I)   ± obtained from the CP fit to the B − →D ( * )0 K − samples. The first error is statistical, the second is the experimental systematic uncertainty and the third reflects the Dalitz model uncertainty.
The experimental systematic uncertainties include the errors on the m ES , ∆E, and F PDF parameters for signal and background, the uncertainties in the knowledge of the Dalitz distribution of background events, the effi-ciency variations across the Dalitz plane, and the uncertainty in the fraction of events with a real D 0 produced in a back-to-back configuration with a negatively-charged kaon. Less significant systematic uncertainties originate from the imprecise knowledge of the fraction of real D 0 's, the invariant mass resolution, and the statistical errors in the Dalitz amplitudes and phases from the fit to the tagged D 0 sample. The possible effect of CP violation in B − →D ( * )0 π − decays and BB background was found to be negligible.
A frequentist (Neyman) construction of the confidence regions of p ≡ (r B , r * B , δ B , δ * B , γ) based on the constraints on z ( * ) ± has been adopted [4]. Using a large number of pseudo-experiments corresponding to the nominal CP fit model but with many different values of the CP fit parameters, we construct an analytical (Gaussian) parameterization of the PDF of z ( * ) ± as a function of p. For a given p, the five-dimensional confidence level C = 1 − α is calculated by integrating over all points in the fit parameter space closer (larger PDF) to p than the fitted data values. The one-(two-) standard deviation region of the CP parameters is defined as the set of p values for which α is smaller than 3.7% (45.1%). Figures 3(c,d) show the two-dimensional projections in the r The constraint on γ is consistent with that reported by the Belle Collaboration [8], which has a slightly better statistical precision since our r ( * ) B constraint favors smaller values.
We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues,