Improved measurement of CP observables in B- -->D0_{CP} K- decays

We present a study of the decay B- ->D0_CP K- and its charge conjugate, where D0_CP is reconstructed in both a non-CP flavor eigenstate and in CP (CP-even and CP-odd) eigenstates, based on a sample of 382 million Y(4S) ->BBbar decays collected with the BABAR detector at the PEP-II e+e- storage ring. We measure the direct CP asymmetries A_CP\pm and the ratios of the branching fractions R_CP\pm: A_CP+ = 0.27\pm 0.09(stat)\pm 0.04(syst), A_CP- =-0.09\pm 0.09(stat)\pm 0.02(syst), R_CP+ = 1.06\pm 0.10(stat)\pm 0.05(syst), R_CP- = 1.03\pm 0.10(stat)\pm 0.05(syst). We also express the results in terms of the so called Cartesian coordinates x+, x-, and r^2: x+ = -0.09\pm 0.05(stat)\pm 0.02(syst)}, x- = 0.10\pm 0.05(stat)\pm 0.03(syst)}, r^2 = 0.05\pm 0.07(stat)\pm 0.03(syst)}. These results will help to better constrain the phase parameter $\gamma=\arg(-V_{ud}V_{ub}^*/V_{cd}V_{cb}^*)$ of the Cabibbo-Kobayashi-Maskawa quark mixing matrix.

PACS numbers: 11.30.Er, 13.25.Hw,14.40.Nd The angle γ = arg(−V ud V * ub /V cd V * cb ) is one of the least precisely known parameters of the corresponding unitarity triangle of the Cabibbo-Kobayashi-Maskawa matrix [1]. There are many proposals on how to measure γ involving charged B decays. The B − →D ( * )0 K ( * )− decay mode [2], which exploits the interference between b → cūs and b → ucs decay amplitudes, is one of the most important of these [3,4]. In this paper we use a theoretically clean measurement technique suggested by Gronau, London, and Wyler (GLW). It exploits the interference between B − →D 0 K − and B − →D 0 K − decay amplitudes, where the D 0 and D 0 mesons decay to the same CP eigenstate [3]. We express the results in terms of the commonly used ratios R CP ± of charge-averaged partial rates and of the partial-rate charge asymmetries A CP ± , Here, D 0 CP ± = (D 0 ± D 0 )/ √ 2 are the CP eigenstates of the neutral D meson system, following the notation in Ref. [5]. Neglecting D 0 −D 0 mixing [6], the observables R CP ± and A CP ± are related to the angle γ, the magnitude ratio r of the amplitudes for the processes B − →D 0 K − and B − →D 0 K − , and the relative strong phase δ of these amplitudes through the relations R CP ± = 1 + r 2 ± 2r cos δ cos γ and A CP ± = ±2r sin δ sin γ/R CP ± [3]. Theoretical predictions for r are on the order of 0.1 [3], in agreement with recent results by BABAR (r = 0.091 ± 0.059 [7]) and Belle (r = 0.159 ± 0.074 [8]), obtained through the study of This analysis, based on 348 fb −1 of data collected at the Υ (4S) resonance, updates a previous BABAR study based on 211 fb −1 of data [9]. Belle recently presented a similar measurement of R CP ± and A CP ± based on 251 fb −1 of data [10].
The ratios R CP ± are computed under the assumption R CP ± = R ± /R, which holds neglecting a factor of r π < ∼ 0.012 as discussed later. The quantities R + , R − , and R are defined as: .
(3) Several systematic uncertainties affect the D 0 K and D 0 π final states in the same way and therefore cancel in the double ratios R CP+ and R CP− , for instance the uncertainties on charged particle reconstruction efficiencies, and the uncertainties on the secondary branching ratios of the D 0 decays. We express the CP -sensitive observables in terms of three independent quantities x + , x − , and r: where x ± = r cos(δ ± γ) and y ± = r sin(δ ± γ) are the so called Cartesian coordinates related to the CP parameters that are measured using a Dalitz analysis of [8,11]. This choice allows the results of the two measurements to be expressed in a consistent manner.
The measurements use a sample of 382 million Υ (4S) decays into BB pairs collected with the BABAR detector [12] at the PEP-II asymmetric-energy B factory. Charged-particle tracking is provided by a fivelayer double-sided silicon vertex tracker and a 40-layer drift chamber (DCH). A ring-imaging Cherenkov detector (DIRC) provides additional particle identification (PID). Photons are identified by the electromagnetic calorimeter (EMC), which is comprised of 6580 thalliumdoped CsI crystals. These systems are mounted inside a 1.5 T solenoidal superconducting magnet. We use the GEANT [13] software to simulate interactions of particles traversing the detector, taking into account the varying accelerator and detector conditions. We reconstruct B − →D 0 h − decays, where the prompt track h − is either a kaon or a pion. The D 0 candidates are reconstructed in the CP -even eigenstates π − π + and K − K + (D 0 CP+ ), in the CP -odd eigenstates K 0 S π 0 and K 0 S ω (D 0 CP− ), and in the (non-CP ) flavor eigenstate K − π + . The ω candidates are reconstructed in the π − π + π 0 channel, and K 0 S candidates in the π + π − channel. Compared to the previous analysis [9], the current study does not include the decay mode D 0 → K 0 S φ, since it is going to be explored by a BABAR Dalitz analysis of Excluding the K 0 S φ channel from the present analysis will allow the results of both studies to be more easily combined in the future.
We optimize our event selection to minimize the statistical error on the B − →D 0 K − signal yield, determined for each D 0 decay channel using simulated signal and background events. We reject a candidate track if its Cherenkov angle does not agree within four standard deviations (σ) with either the pion or kaon hypothesis [14], or if it is identified as an electron by the DCH and the EMC. Neutral pions are reconstructed by combining pairs of photon candidates with energy deposits larger than 30 MeV that are not matched to charged tracks. The photon pair invariant mass is required to be in the range 115-150 MeV/c 2 and the total π 0 energy must be greater than 200 MeV in the laboratory frame. To improve momentum resolution, the invariant mass of the two photons from candidate π 0 's is constrained to the nominal π 0 mass [14]. Neutral kaons are reconstructed from pairs of oppositely charged tracks with invariant mass within 7.8 MeV/c 2 (∼ 3 σ) of the nominal K 0 S mass. The ratio between the candidate K 0 S flight length and its uncertainty must be greater than 2. The ω mesons are reconstructed from π + π − π 0 combinations with invariant mass in the range 0.763 < M (π + π − π 0 ) < 0.799 GeV/c 2 . We define θ N as the angle between the normal to the ω decay plane and the D 0 momentum in the ω rest frame, and θ ππ as the angle between the flight direction of one of the three pions in the ω rest frame and the flight direction of one of the other two pions in the two-pion rest frame. The quantities cos θ N and cos θ ππ follow cos 2 θ N and sin 2 θ ππ distributions for the signal and are almost flat for wrongly reconstructed or false ω candidates. We require the product cos 2 θ N sin 2 θ ππ > 0.08. The invariant mass of a D 0 candidate M (D 0 ) must be within 2.5 σ of the mean fitted mass, with σ ranging from 4 to 20 MeV/c 2 depending on the D 0 decay mode. To improve the D 0 momentum resolution, the candidate invariant mass is then constrained to the nominal D 0 mass [14] for all D 0 decay channels. For D 0 →π − π + , the invariant mass of the (h − π + ) system, where π + is the pion from the D 0 and h − is the prompt track from B − taken with the kaon mass hypothesis [14], must be greater than 1.9 GeV/c 2 to reject background from B − →D 0 π − , D 0 →K − π + and B − →K * 0 π − , K * 0 →K − π + decays. We reconstruct B meson candidates by combining a D 0 candidate with a track h. For the D 0 →K − π + mode, the charge of the track h must match that of the kaon from the D 0 meson decay, selecting b → c mediated B decays.
We select B meson candidates using the energy difference ∆E = E * B −E * ee /2 and the beam-energy-substituted where the subscripts ee and B refer to the initial e + e − system and the B candidate, respectively, and the asterisk denotes the e + e − center-of-mass (CM) frame. The m ES distributions for B − →D 0 h − signals are Gaussian functions centered at the B mass with a resolution of 2.6 MeV/c 2 , and do not depend on the D 0 decay mode or on the nature of the prompt track. In contrast, the ∆E distributions depend on the mass assigned to the prompt track. We evaluate ∆E with the kaon mass hypothesis so that the peaks of the distributions are centered near zero for B − →D 0 K − events and shifted by approximately 50 MeV for B − →D 0 π − events. The ∆E resolution depends on the momentum resolutions of the D 0 meson and the prompt track h − , and is typically 16 MeV for all D 0 decay modes under study. All B candidates are selected with m ES within 2.5 σ of the mean value and with ∆E in the range −0.15 < ∆E < 0.20 GeV.
To reduce background from e + e − → qq events (with q = u, d, s, c), denoted qq in the following, we construct a linear Fisher discriminant [15] based on the four eventshape quantities L ROE 2 , | cos θ * T |, | cos θ * B | and R ROE 2 . The ratio L ROE 2 between L 2 = i p i cos 2 θ i and L 0 = i p i is evaluated in the CM frame, where the p i are the momenta of charged tracks and neutral clusters not used to reconstruct the B (i.e., the rest of the event, ROE), and the θ i are their angles with respect to the thrust axis of the B candidate's decay products. The angle θ * T is measured between the thrust axis of the B candidate's decay products and the beam axis, and is evaluated in the CM frame. The angle θ * B is measured between the B candidate momentum and the beam axis, again evaluated in the CM frame. The ratio R ROE 2 of the Fox-Wolfram moments H 2 and H 0 , is computed using tracks and photons in the ROE [16]. The efficiency of the requirement on the value of the Fisher discriminant ranges from 74% to 78% for B − →D 0 K − signal events and from 17% to 23% for qq background events. For the Kπ channel, the values are 87% for signal and 42% for background events.
For events with multiple B − →D 0 h − candidates (0.4%-7.7% of the selected events, depending on the D 0 decay mode), we choose the B candidate with the smallest Mc + Γ 2 c ) formed from the measured and true masses of the composite candidates c, M c and M c , scaled by the resolution σ Mc and width Γ c of the reconstructed mass distributions. Composite candidates considered are the B candidate itself (m ES ), D 0 , π 0 , and ω candidates. Also Γ ω is the only non-negligible width.
The total reconstruction efficiencies, based on simulated B → D 0 K events, are 36% (K − π + ), 29% (K − K + ), 29% (π − π + ), 15% (K 0 S π 0 ), and 6% (K 0 S ω). The main contributions to the background from BB events come from the processes B − →D * h − , B − →D 0 ρ − , misreconstructed B − →D 0 h − , and from charmless B decays to the same final state as the signal: for instance, the process B − →K − K + K − is a background for B − →D 0 K − , D 0 →K − K + . These charmless backgrounds have similar ∆E and m ES distributions as the D 0 K − signal and are referred to in the following as peaking BB backgrounds (B − → X 1 X 2 K − ).
We determine the signal and background yields for each D 0 decay mode independently from a twodimensional extended unbinned maximum-likelihood fit to the selected data events. The fit is performed simultaneously on the B + and B − subsamples. The input variables to the fit are ∆E and the Cherenkov angle θ C of the prompt track as measured by the DIRC. The extended likelihood L for N candidates is given by the product of the probabilities for each individual candidate i and a Poisson factor: The probability P i is the sum of the signal and back-ground terms, where J denotes the seven signal and background hypotheses D 0 h, qq(h), BB(h), and X 1 X 2 K. N ′ is the total event yield estimated by the fit, and N J is the event yield in each category. We fit directly for the ratios R ′ ≡ R (±) and asymmetries A CP ± , as appropriate to the decay mode; they enter Eq. (7) through where N D 0 π = N D 0 π + + N D 0 π − and A D 0 π CP ± is defined analogously to Eq. 2.
The ∆E distribution for B ± →D 0 K ± signal is parameterized with a double Gaussian function. The fraction of the wide component of the signal shape, its offset from the narrow component and the ratio between the widths of the two components are fixed to values obtained from simulation. The ∆E probability density function (PDF) for B ± →D 0 π ± is the same as the B ± →D 0 K ± one, but with an additional shift, ∆E shift , which arises from the wrong mass assignment to the prompt track. The shift is computed event by event as a function of the prompt track momentum p and a Lorentz factor γ PEP-II = E ee /E * ee characterizing the boost to the e + e − CM frame: The ∆E distributions for the continuum background are parameterized with a straight line. The ∆E distribution for the BB background is empirically parametrized with a Gaussian peak with an exponential tail [17]. The parameters of the background shapes are determined from simulated events (BB) and off-resonance data (qq) and are fixed in the fit. The number of peaking background events N X1X2K is fixed to values obtained from a study of the D 0 mass sidebands. The particle identification PDF is a double Gaussian as a function of θ pull C , which is the difference between the measured Cherenkov angle θ C and its expected value for a given mass hypothesis, divided by the estimated error. The PID shape parameters are obtained from simulation. To summarize, the floating parameters in each of the five the fits are the D 0 K and D 0 π signal yield asymmetries, the total number of signal events in D 0 π, the appropriate ratios R and R ± , eight background yields (one for each charge), and two parameters of the ∆E signal shape (common for positive and negative samples).
The results of the fits, expressed in terms of signal yields, are summarized in Table I. Figure 1 shows the distributions of ∆E for the K − π + , CP+ and CP− modes after enhancing the B − →D 0 K − purity by requiring that the prompt track be consistent with the kaon hypothesis. This requirement is 88% (1%) efficient for h − = K − (h − = π − ).
12745 ± 120 12338 ± 120 954 ± 36 918 ± 36 K − K + + 1109 ± 36 1051 ± 35 51 ± 10 113 ± 13 π − π + + 390 ± 24 378 ± 24 39 ± 9 36 ± 9 K 0 S π 0 − 1102 ± 37 1134 ± 38 100 ± 13 88 ± 12 The ratios R (±) , as measured by each fit, are corrected to take into account small differences in the selection efficiency between B→DK and B→Dπ. The efficiency ratios range from 1.013 ± 0.006 to 1.037 ± 0.010. Their uncertainties are due to the statistics of the simulated samples and are considered in the study of systematic uncertainties. In the case of CP − need to be corrected to take into account a possible dilution from a nonresonant CP -even background arising from There is little information on this background. We estimate the corrections using a fit to the ω helicity angle in the selected data events and find the correction factors to be 1.12 ± 0.14 for A K 0 S ω CP − and 1.00 ± 0.01 for R K 0 S ω CP − . The uncertainties in the correction factors are included in the systematic errors. After applying all corrections, the quantities R ± /R and A CP ± are computed by means of a weighted average over the CP+ and CP− modes. The results for the CPeven and CP -odd combinations are reported in Table II. Systematic uncertainties in R CP ± and A CP ± are listed in Table III. The uncertainties on the fitted signal yields are due to the imperfect knowledge of the ∆E and PID PDFs and of the peaking background yields, and are evaluated in test fits by varying the parameters of the PDFs and the peaking background yields by ±1σ and taking the difference in the fit results. A possible ±20% CP asymmetry in the peaking background is considered in the same way. In the K 0 S ω channel we also take into account the uncertainties in the correction factors due to the CP -even backgrounds from D 0 → K 0 S (π − π + π 0 ) non−ω decays. A possible bias in the measured A CP ± comes from an intrinsic detector charge asymmetry due to asymmetries in acceptance or tracking and particle identification efficiencies. An upper limit on this bias is obtained from the measured asymmetries in the processes B − →D 0 h − , D 0 →K − π + and B − →D 0 CP ± π − , where CP violation is expected to be negligible. From the average  asymmetry, −(1.6 ± 0.6)%, we obtain the limit ±2.2% for the bias. For the branching fraction ratios R CP ± , an additional source of uncertainty is associated with the assumption that R CP ± = R ± /R. This assumption holds only if the magnitude of the ratio r π between the amplitudes of the B − →D 0 π − and B − →D 0 π − processes is neglected [18]. r π is expected to be small: r π ∼ r λ 2 1−λ 2 < ∼ 0.012, where λ ≈ 0.22 [14] is the sine of the Cabibbo angle. This introduces a relative uncertainty ±2r π cos δ π cos γ on R CP ± , where δ π is the relative strong phase between the amplitudes A(B − →D 0 π − ) and A(B − →D 0 π − ). Since | cos δ π cos γ| ≤ 1 and r π < ∼ 0.012, we assign a relative uncertainty ±2.4% to R CP ± , which is completely anti-correlated between R CP+ and R CP− . We quote the measurements in terms of x ± and r 2 , The correlations between the different sources of systematic errors, when non-negligible, are considered when calculating x ± and r 2 . The measured values of x ± are consistent with those found from B − →D 0 K − , D 0 → K 0 S π − π + decays, and the precision is comparable [11].  In conclusion, we have reconstructed B − →D 0 K − decays with D 0 mesons decaying to non-CP , CP -even and CP -odd eigenstates. The combined uncertainties we find for A CP ± (R CP ± ) are smaller by a factor of 0.7 (0.9) and 0.6 (0.6) than the previous BABAR [9] and Belle [10] measurements, respectively. We find A CP+ to deviate by 2.8 standard deviations from zero. We express the results in terms of the Cartesian coordinates x ± and r 2 (Eqs. 4,5). These measurements, combined with the existing measurements from B − →D 0 K − decays, will improve our knowledge of the angle γ and the parameter r.
We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and kind hospitality.