Search for b-->u transitions in B--->DK- and D*K- decays

We report results from an updated study of the suppressed decays B^-→DK^- and B^-→D^*K^- followed by D→K^+π^-, where D^(*) indicates a D^((*)0) or a D^((*)0) meson, and D^*→Dπ^0 or D^*→Dγ. These decays are sensitive to the Cabibbo-Kobayashi-Maskawa unitarity triangle angle γ due to interference between the b→c transition B^-→D^((*))0K^- followed by the doubly Cabibbo-suppressed decay D^0→K^+π^-, and the b→u transition B^-→D^((*)0)K^- followed by the Cabibbo-favored decay D^0→K^+π^-. We also report an analysis of the decay B^-→D^(*)π^- with the D decaying into the doubly Cabibbo-suppressed mode D→K^+π^-. Our results are based on 467× 10^6 Υ(4S)→BB decays collected with the BABAR detector at SLAC. We measure the ratios R^(*) of the suppressed ([K^+π^-]_DK^-/π^-) to favored ([K^-π^+]_DK^-/π^-) branching fractions as well as the CP asymmetries A^(*) of those modes. We see indications of signals for the B^-→DK^- and B^-→D_(Dπ0)^*K^- suppressed modes, with statistical significances of 2.1 and 2.2σ, respectively, and we measure: RDK=(1.1±0.6±0.2)×10^(-2), ADK=-0.86±0.47_(-0.16)^(+0.12), R_((Dπ0))K^*=(1.8±0.9± 0.4)×10^-2, A_((Dπ0))K^*=+0.77±0.35±0.12, R_((Dγ))K^*=(1.3±1.4±0.8)×10^-2,A_((Dγ))K^*=+0.36 ±0.94-_(0.41)^(+0.25), where the first uncertainty is statistical and the second is systematic. We use a frequentist approach to obtain the magnitude of the ratio r_B≡|A(B^-→D ^0K^-)/A(B^-→D^0K^-)|=(9.5_(-4.1)^(+5.1))%, with r_B<16.7% at 90% confidence level. In the case of B^-→D^*K^- we find r_B^*≡|A(B^-→D ^*0K^-)/A(B^-→D^*0K^-)|=(9.6_(-5.1)^(+3.5))%, with rB*<15.0% at 90% confidence level.

We report results from an updated study of the suppressed decays B − → DK − and B − → D * K − followed by D → K + π − , where D ( * ) indicates a D ( * )0 or aD ( * )0 meson, and D * → Dπ 0 or D * → Dγ. These decays are sensitive to the CKM unitarity triangle angle γ due to interference between the b → c transition B − → D ( * )0 K − followed by the doubly Cabibbo-suppressed decay D 0 → K + π − , and the b → u transition B − →D ( * )0 K − followed by the Cabibbo-favored decay D 0 → K + π − . We also report an analysis of the decay B − → D ( * ) π − with the D decaying into the doubly Cabibbo-suppressed mode D → K + π − . Our results are based on 467 million Υ (4S) → BB decays collected with the BABAR detector at SLAC. We measure the ratios R ( * ) of the suppressed ([K + π − ]DK − /π − ) to favored ([K − π + ]DK − /π − ) branching fractions as well as the CP asymmetries A ( * ) of those modes. We see indications of signals for the B − → DK − and B − → D * Dπ 0 K − suppressed modes, with statistical significances of 2.1 and 2.2σ, respectively, and we measure:

I. INTRODUCTION
The Standard Model accommodates CP violation through a single phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix V [1].
In the Wolfenstein parameterization [2], the angle γ = arg (−V ud V * ub /V cd V * cb ) of the unitarity triangle is related to the complex phase of the CKM matrix element V ub through V ub = |V ub |e −iγ . A theoretically clean source of information on the angle γ is provided by B − → D ( * ) K − decays, where D ( * ) represents an admixture of D ( * )0 and D ( * )0 states. These decays exploit the interference between B − → D ( * )0 K − and B − → D ( * )0 K − (Fig. 1) that occurs when the D ( * )0 and the D ( * )0 decay to common final states. In the Atwood-Dunietz-Soni (ADS) method [3], the D 0 from the favored → c amplitude is reconstructed in the doubly Cabibbo-suppressed decay K + π − , while the D 0 from the → u suppressed amplitude is reconstructed in the favored decay K + π − . The product branching fractions for these final states, which we denote as [K + π − ] D K − (B − → DK − ) and [K + π − ] D * K − (B − → D * K − ), are small (∼ 10 −7 ), but the two interfering amplitudes are of the same order of magnitude, and large CP asymmetries are therefore possible. The favored decay mode B − → [K − π + ] D ( * ) K − is used to normalize the measurement and cancel many systematic uncertainties. Thus, ignoring possible small effects due to D mixing and assuming no CP violation in the normalization modes, we define the charge-specific ratios for B + and B − decay rates to the ADS final states as  [4][5][6][7] and r D = |A(D 0 → K + π − )/A(D 0 → K − π + )| = (5.78 ± 0.08)% [8] are the suppressed to favored B and D amplitude ratios. The rates in Eq. (1) depend on the relative weak phase γ and the relative strong phase δ ≡ δ B + δ D between the interfering amplitudes, where δ B and δ D are the strong phase differences between the two B and D decay amplitudes, respectively. The value of δ D has been measured to be δ D = (201.9 +11.3 −12.4 ) • [8], where we have accounted for a phase shift of 180 • in the definition of δ D between Ref. [8] and this analysis.
The main experimental observables are the chargeaveraged decay rate and the direct CP asymmetry, which can be written as The treatment for the D * K mode is identical to the DK one, but the parameters r * B and δ * B are not expected to be numerically the same as those of the DK mode. Taking into account the effective strong phase difference of π between the D * decays to Dγ and Dπ 0 [9], we define the charge-specific ratios for D * as: = r * 2 B + r 2 D + 2 r * B r D cos(±γ + δ * ), (4) is the strong phase difference between the two B decay amplitudes. The charge averaged ratios for D * → Dπ 0 and D * → Dγ are then: Definitions of the direct CP asymmetries A * (Dπ 0 )K and A * (Dγ)K follow Eq. (3). This paper is an update of our previous ADS analysis in Ref. [4], which used 232 × 10 6 BB pairs and set 90% C.L. upper limits R DK < 0.029, R * (Dπ 0 )K < 0.023 and R * (Dγ)K < 0.045. In addition to an increased data sample, new features in the analysis include a multidimensional fit involving the neural network output used to discriminate the signal from the continuum background, rather than a simple cut on this variable as was done in the previous analysis. We also include measurements of the ratios of the doubly Cabibbo-suppressed to Cabibbo-favored D ( * ) π decay rates, and of the corresponding asymmetries. These measurements are used as a check for the B − → [K + π − ] D ( * ) K − ADS analysis. In the D ( * ) π case, we expect that the ratio r of the V ub to V cb amplitudes is suppressed by a factor |V cd V us /V ud V cs | compared to the D ( * ) K case, if we assume the same color suppression factor for both decays. One expects therefore r where θ c is the Cabibbo angle and where we have assumed r ( * ) B = 10%. Neglecting higher order terms, R ( * ) B tan 2 θ c sin γ sin δ ( * ) /r D . Hence, the maximum asymmetry possible for D ( * ) π ADS decays is 2r

II. THE BABAR DETECTOR AND DATASET
The results presented in this paper are based on 467 × 10 6 Υ (4S) → BB decays, corresponding to an integrated luminosity of 426 fb −1 (on-peak data). The data were collected between 1999 and 2007 with the BABAR detector [10] at the PEP-II e + e − collider at SLAC. In addition, a 44 fb −1 data sample, with center-of-mass (CM) energy 40 MeV below the Υ (4S) resonance (offpeak data), is used to study backgrounds from continuum events, e + e − → qq (q = u, d, s, or c).
The BABAR detector response to various physics processes as well as to varying beam and environmental conditions is modeled with simulation software based on the Geant4 [11] tool kit. We use EVTGEN [12] to model the kinematics of B meson decays and JETSET [13] to model continuum processes e + e − → qq.

A. Basic Requirements
We reconstruct B − → D ( * ) K − and B − → D ( * ) π − with the D decaying to K − π + (right-sign (RS) decays) and K + π − (wrong-sign (WS) decays). For de-cays involving a D * , both D * → Dπ 0 and D * → Dγ modes are reconstructed. Charged kaon and pion candidates must satisfy identification criteria that are typically 85% efficient, depending on momentum and polar angle. The misidentification rates are at the few percent level. We select D candidates with an invariant mass within 20 MeV/c 2 (about 3 standard deviations) of the known D 0 mass [14]. All D candidates are mass and vertex constrained. For modes with D * → Dπ 0 or D * → Dγ, the mass difference ∆m between the D * and the D must be within 4 MeV/c 2 (≃ 4σ) or 15 MeV/c 2 (≃ 2σ), respectively, of the nominal mass difference [14].
For the WS decays B ± → [K ∓ π ± ] D K ± , two important sources of background arise: the first from B ± → [π ∓ K ± ] D K ± (in which the K and π in the D decay are misidentified as π and K) and the second from B ± → [K ∓ K ± ] D π ± (when the K ∓ π ± pair has an invariant mass within 20 MeV/c 2 of the nominal D 0 mass). To eliminate the first background, we recompute the invariant mass (M switch ) of the h + h ′− pair in D 0 → h + h ′− switching the mass assumptions on the h + and the h ′− . We veto candidates with M switch within 20 MeV/c 2 of the D 0 mass [14]. To eliminate the second background, we also veto any candidate where the KK invariant mass is within 20 MeV/c 2 of the D 0 mass. To ensure the same selection efficiencies, these criteria are applied both to candidates. These veto cuts are 88% efficient on signal decays.
We identify B candidates using two nearly independent kinematic variables that are customarily used when reconstructing B-meson decays at the Υ (4S). These variables are the energy-substituted mass, m ES ≡ where E and p are energy and momentum, the asterisk denotes the CM frame, the subscripts 0 and B refer to the Υ (4S) and B candidate, respectively, and s is the square of the CM energy. For signal events m ES = m B + [14] and ∆E = 0 within the resolutions of about 2.6 MeV/c 2 and 17 MeV, respectively. We require that all candidates have |∆E| < 40 MeV and we use m ES in the fit to extract the number of signal events.
The average number of B → D ( * ) K candidates reconstructed per selected event is about 1.4 in B → DK signal Monte Carlo (MC) events and about 2 for B → D * K signal MC events. This is mostly due to the crossfeed between the DK and the D * K final states. For all events with multiple B → D ( * ) K candidates, we retain only one candidate per event, based on the smallest value of |∆E|. This method does not bias the sample since ∆E is not used to extract the number of signal events. After this arbitration, less than 0.4% (0.  The B → D ( * ) π analysis is performed independently of the B → D ( * ) K analysis, but uses the same multiple candidate selection algorithm. A summary of the selection efficiencies for the WS modes [K ± π ∓ ] D ( * ) h ∓ (h=K, π) and the RS modes [π ± K ∓ ] D ( * ) h ∓ is given in Table I.

B. Neural Network
After these initial requirements, backgrounds dominantly arise from continuum events, especially e + e − → cc, withc → D 0 X, D 0 → K + π − and c → D 0 X, D 0 → K − +anything. The continuum background is reduced by using neural network techniques. To select the discriminating variables used in the neural network, we rely on a study performed for the previous version of this analysis [4], and we consider the seven quantities listed below: 1. Two event shape moments L 0 = i p i , and L 2 = i p i cos 2 θ i , calculated in the CM frame. Here, p i is the momentum and θ i is the angle with respect to the thrust axis of the B candidate; the index i runs over all tracks and clusters not used to reconstruct the B meson (rest of the event). These variables are sensitive to the shape of the event, separating jet-like continuum events from more spherical BB events.
2. The absolute value of the cosine of the angle in the CM frame between the thrust axes of the B candidate and the detected remainder of the event, | cos θ T |. The distribution of | cos θ T | is approximately uniform for signal and strongly peaked at one for continuum background.
3. The absolute value of the cosine of the CM angle between the B candidate momentum and the beam axis, | cos θ B |. In this variable, the signal follows a 1 − cos 2 θ B distribution, while the background is approximately uniform.
4. The charge difference ∆Q between the sum of the charges of tracks in the D ( * ) hemisphere and the sum of the charges of the tracks in the opposite hemisphere, excluding the tracks used in the reconstructed B, and where the partitioning of the event into two hemispheres is done in the CM frame. This variable exploits the correlation occurring in cc events between the charge of the c (orc) in a given hemisphere and the sum of the charges of all particles in that hemisphere. For signal events, the average charge difference is ∆Q = 0, whereas for the cc background ∆Q ≈ 7 3 × Q B , where Q B is the charge of the B candidate.
5. The product Q B · Q K , where Q K is the sum of the charges of all kaons in the rest of the event. In many signal events, there is a charged kaon among the decay products of the other B in the event.
The charge of this kaon tends to be highly correlated with the charge of the B. Thus, signal events tend to have Q B · Q K ≤ −1. On the other hand, most continuum events have no kaons outside of the reconstructed B, and therefore Q K = 0.
6. A quantity M Kℓ , defined to be zero if there are no leptons (e or µ) in the event, and, if a lepton is found, taken to be equal to the invariant mass of this lepton and the kaon from B (bachelor K). This quantity differentiates between continuum background and signal because continuum events have fewer leptons than BB events. Furthermore, a large fraction of leptons in cc background events are from D → Kℓν, where the kaon becomes the bachelor kaon candidate, so that the average M Kℓ in cc events is lower than in B signal events.
7. The absolute value of the measured proper time interval between the two B decays, |∆t|. This is calculated from the measured separation, ∆z, between the decay points of the reconstructed B and the other B along the beam direction, and the known Lorentz boost of the initial e + e − state. For continuum background, |∆t| is peaked at 0, with most events having |∆t| < 2 ps, while it is less peaked and can extend beyond 5 ps for B ± → D ( * ) h ± signal events.
The neural network is trained with simulated continuum and signal [K ± π ∓ ] D ( * ) K ∓ events. Only wrong-sign D ( * ) K candidates are used in the training, but the neural network is used in the analysis of all the D ( * ) h ∓ channels. The distributions of the neural network output (N N ) for signal-enriched right-sign control samples are compared with expectations from the MC simulation in Fig. 2(a) (DK) and Fig. 2(d) (Dπ). The agreement is satisfactory. In the same figure, the N N spectra of background control samples (off-peak data) are compared with expectations from continuum qq MC. Since we do not expect these distributions to be exactly the same for the right-sign and wrong-sign background samples, they are shown separately for the [K ± π ∓ ] D ( * ) K ∓ (Fig. 2 Fig. 2(e)) and [K ∓ π ± ] D ( * ) π ∓ and ∆E requirements on the off-resonance and continuum MC events have been relaxed, and the Dh ± and D * h ± contributions have been summed, after checking that they are in agreement with each other. Good agreement between data and the simulation is observed in all channels. Good agreement between the D ( * ) K and the D ( * ) π background N N distributions is also visible in Fig. 2, while on the contrary the background N N distribution of wrong-sign decays is clearly different from the background N N distribution of right-sign decays. We have examined the distributions of all variables used in the neural network, and found good agreement between the simulation and the data control samples. Finally, we examined the N N distributions in the signal MC for the different B signal channels, right-sign and wrong-sign separately (Dπ, D * π, DK, D * K) and did not observe any significant difference between these channels.

C. Fitting for event yields and R ( * )
The ratios R ( * ) are extracted by performing extended unbinned maximum likelihood fits to the set of variables m ES , N N , and I sign , where I sign is a discrete variable equal to 0 for WS events and to 1 for RS events. We write the extended likelihood L as where the vector x indicates the variables (m ES , N N , and I sign ) and θ indicates the set of parameters which are fitted from the data. N is the total number of signal and background events, and N ′ = i N i is the expectation value for the total number of events. The sum runs over the different signal and background categories i which will be detailed below. The probability density function (PDF) f (x j |θ) is written as the sum over the different signal and background categories and a two-bin histogram H(I sign ) set to (1,0) for the WS category and (0,1) for the RS category. The N N distributions are all modeled by histograms with 102 bins between −1.02 and 1.02. The fits are performed separately to each of the Dπ, D * Dπ 0 π, D * Dγ π, DK, D * Dπ 0 K and D * Dγ K samples. They are configured in such a way that R ( * ) is an explicit fit parameter: for the B signal, we fit for the number of rightsign decays N RS and the ratio where N W S is the number of wrong-sign signal events and c is the ratio of the wrong-sign to right-sign selection efficiencies. For B → D ( * ) K, the factor c is consistent with unity within the statistical precision of the simulation (Table I) and is set to this value in the fits. For B → D ( * ) π, c differs slightly from unity due to different particle identification cuts applied at an early stage of the event selection and we use therefore the values of Table I in the fits.
The following signal and background categories are used to describe each sample in the fits: its m ES spectrum is modeled by a Gaussian function G sig (m ES ) whose mean and width are determined from the fit to data.
its m ES and N N spectra have the same parametrizations G sig (m ES ) and N N sig as the right-sign signal.
3. The right-sign combinatorial background from qq [15] A qq (m ES ) whose shape and endpoint parameters, ζ qq and m 0 , are allowed to vary in the fit. The N N PDF N N Dπ) candidates in the qq continuum MC (Figs. 2c and 2f), where the ∆E requirement has been extended to |∆E| < 200 MeV and the DK and D * K (or Dπ and D * π) samples have been summed to increase the statistics. 4. The wrong-sign combinatorial background from qq (Figs. 2b and 2e). fixed to its value determined from BB MC, after removal of the B → D ( * ) K/π signal events. The N N PDF used to describe this background is the PDF N N sig describing the N N spectrum of the B − → D ( * ) h − signal MC. The number of BB right-sign combinatorial background events is allowed to vary in the Dh − fits but is fixed to the MC prediction in the D * h − fits (see below). 6. The wrong-sign combinatorial background from BB events into [K + π − ]K − (DK) or [K + π − ]π − (Dπ), excluding the peaking background which is considered in category 8: its m ES component is described by an ARGUS function [15] A

The right-sign combinatorial background from
fixed to its value determined from the BB MC, after removal of the B → D ( * ) K/π signal events. The N N PDF used to describe this background is the PDF N N sig describing the N N spectrum of the B − → D ( * ) h − signal MC. The number of BB wrong-sign combinatorial background events is allowed to vary in the Dh − fits but is fixed in the D * h − fits (see below).
7. The background from BB events in the right-sign component peaking in m ES inside the signal region (peaking background): this background is discussed in more detail in Section IV. For the DK ± , Dπ ± and D * Dπ 0 K ± categories, the peaking part of the BB background m ES spectrum is described by the same Gaussian function G sig (m ES ) as the signal. This component is therefore indistinguishable from the signal and its rate has to be fixed to the MC predictions. For the D * Dπ 0 π ± , D * Dγ π ± and the D * Dγ K ± categories, the m ES component is described by an asymmetric Gaussian whose shape parameters and amplitude for each category are determined from a fit to the m ES spectrum of BB MC events, after vetoing the B ± → D ( * ) h ± signal component. For all categories, the N N PDF used to describe this background is the PDF N N sig describing the N N spectra of the B → D ( * ) h ± signal MC.
8. The peaking background from BB events in the wrong-sign component: the treatment is similar to the previous component but G sig (m ES ) is used to describe the m ES spectrum of the DK ± , Dπ ± , D * Dπ 0 K ± and D * Dγ K ± categories, while an asymmetric Gaussian is used to describe the m ES spectrum of the D * Dπ 0 π ± and D * Dγ π ± categories.
To summarize, we fit for the number of right-sign signal events N RS , the ratio R = N W S /(c × N RS ) of wrongsign to right-sign events, the number of wrong-sign and right-sign qq combinatorial background events, N , as well as the number of BB combinatorial background events for D * h ± . The other parameters fitted are the reconstructed m ES peak and resolution, m B and σ mB , and the qq continuum background shape parameter and endpoint, ζ qq and m 0 .  [14] 67.1±9.7 K + π − π − Dπ WS < 0.9 [16] <1.1 K − π + K − DK RS < 0.2 [16] <0.2 K + π − K − DK WS 5.0 ± 0.7 [17] 6.0±0.8

IV. STUDY OF BB BACKGROUNDS
We study the BB background for each signal category (Dπ, D * π DK, D * K) and charge combination (right-sign and wrong-sign) using a sample of e + e − → Υ (4S) → BB MC events corresponding to about 3 times the data luminosity. In addition, dedicated Monte Carlo signal samples are used to estimate the background from B − → Dh − events and the background from the charmless decay B − → K + π − K − . We identify three main classes of background events which can peak in m ES inside the signal region and mimic the D ( * ) π and D ( * ) K signal: we list in Table II the 3-body charmless decays affecting our analysis, their branching fractions [14] and the numbers of reconstructed events expected in the affected modes after the selection. Due to the particle identification criteria used in the analysis only decays with the same final state particles as our signal modes contribute significantly to the background. These events are indistinguishable from the Dh ± signal if the K − π + invariant mass is consistent with the D mass. The two decays affected by a significant charmless background are right-sign B − → [K − π + ] D π − and wrong-sign events selected in the BB Monte Carlo sample, we estimate the efficiency of B − → K − π + π − events to be reconstructed as a [K − π + ] D π − candidate as (0.26 ± 0.02)%. The corresponding background is estimated to be 67.1±9.7 events, where the error is dominated by the statistical uncertainty on the B − → K − π + π − branching fraction. The efficiency of B − → K + π − K − events to be reconstructed as [K + π − ] D K − WS candidates is determined from a high statistics dedicated B − → K + π − K − signal Monte Carlo sample, and is found to be (0.27 ± 0.01)%. The corresponding peaking background from B − → K + π − K − events mimicking B − → [K + π−] D K − WS decays is estimated to be 6.0±0.8 events, where the error is dominated by the statistical uncertainty on the B − → K + π − K − branching fraction. From a fit to data selected in the D mass sidebands, we cross-check this prediction and find 6.5±4.0 peaking events, in good agreement with the MC prediction. We also check that, because of the tight ∆m cut applied to the D * decay products, the B − → D * h − channels are not affected by charmless peaking backgrounds.  Table III.

Events
where the e + is missed, faking the WS signal A summary of the BB background studies is given in Table III, for B → D ( * ) π and B → D ( * ) K. For each channel, the m ES spectra of events selected in the BB MC simulation (after removing the corresponding signal) were fitted by the sum of a combinatorial background component and a peaking component, using the same parametrization described in Sec. III C. The average number of BB combinatorial and peaking background events predicted by the simulation are given in Table III, together with the main sources of peaking events and the   Table III in the maximum likelihood fit. Statistical uncertainties in the expected yields are incorporated in the corresponding systematic uncertainties.

A. Results for B → D ( * ) π
The results for B → D ( * ) π are displayed in Fig. 3 (right-sign modes) and Fig. 4 (wrong-sign modes). They are summarized in Table IV. Clear signals are observed in the B → Dπ and in the B → D * Dπ 0 π WS modes, with statistical significances of 7σ and 4.8σ, respectively. The significance is defined as −2 ln(L 0 /L max ), where L max and L 0 are the likelihood values with the nominal and with zero WS signal yield, respectively. For B → D * Dγ π WS decays, the significance is only 2σ, due to the large peaking background. Below we discuss the sources of systematic uncertainties that contribute to our R ( * ) Dπ measurements: 1. Signal N N shape: in the nominal fit, we use the N N PDF from the B signal MC. To estimate the related systematics, we refit the data using a signal N N PDF extracted from the high purity and high statistics B → Dπ RS data, after subtracting the residual continuum background contamination predicted by the simulation. We set the systematic uncertainty to the difference with the nominal fit result.
2. B background N N shape: from a study of generic BB MC, it appears that the N N spectra of B background events in the m ES -∆E signal box are similar to the signal (but suffer from very low statistics), while the N N spectra of background events in an enlarged m ES -∆E region differ significantly from the signal and show less peaking close to 1. In the nominal fit we assumed that both the peaking and the non-peaking BB background components could be described by the B → Dπ signal N N PDF. To estimate the related systematic error, we used BB generic background events selected in a ∆E-m ES enlarged window |∆E| < 200 MeV and m ES > 5.20 GeV/c 2 to build the N N PDF of the non-peaking part of the BB background (keeping the signal N N PDF to describe the peaking part of this background) and repeated the fits, taking the difference of the results as the associated systematic uncertainty.
3. Continuum background N N shape: to account for possible differences between the simulation and the data, we used the N N spectrum from off-peak data instead of qq MC (q = u, d, s, c) to model this component. We set the associated systematic uncertainty to the difference of the two results, but the error is dominated by the large statistical uncertainty on the off-peak data sample.   determined from BB simulated events. To account for possible disagreement between data and simulation, we repeated the fits varying these parameters in a conservative range.  6. Uncertainty on the number of BB combinatorial background events: in the D * π (and D * K) fits where this component has been fixed, we vary it by ±25% (the level of agreement between data and simulation observed in the Dπ and DK fits) and we take the difference with the nominal fit result as a systematic uncertainty.

The shape parameters ζ
The resulting systematic uncertainties are listed in Table V. We add them in quadrature and quote the results: where the first uncertainty is statistical and the second is systematic. The values of R ( * ) Dπ are in good agreement with the world average R D = r 2 D = B(D 0 → K + π − )/B(D 0 → K − π + ), R D = (3.36 ± 0.08) × 10 −3 [8].
A separate fit to B + and B − candidates provides a measurement of the corresponding asymmetries. We obtain the following results: A Dπ = 0.03 ± 0.17 ± 0.04, A * (Dπ 0 )π = −0.09 ± 0.27 ± 0.05, A * (Dγ)π = −0.65 ± 0.55 ± 0.22, where the uncertainties are dominated by the statistical error. No significant asymmetry is observed for the D ( * ) π WS decays. The largest source of systematic uncertainty on the D ( * ) π asymmetries is from the uncertainty on the B background peaking component.  Table VI. Indications of signals are observed in the B → DK and in the B → D * Dπ 0 K WS modes, with statistical significances of 2.2σ and 2.4σ, respectively (Fig. 7). Accounting for the systematic uncertainties, the significances become 2.1σ and 2.2σ, respectively. For B → D * Dγ K WS, no significant signal is observed. The systematic uncertainties have been estimated by testing different fit models and recomputing R ( * ) DK , as explained in Section V A. A summary of the different systematic uncertainties is given in Table VII. The uncertainties on the N N describing the BB combinatorial background and the uncertainties on the BB peaking background are the two main contributions. For B ± → DK ± , we find for the ratio of the WS to RS decay rates Expressed in terms of event yields, the fit result is 19.4 ± 9.6 ± 3.5 WS events. The results of fits to separate B + → DK + and B − → DK − data samples are given in Table VI. Projections of the fits to B + and B − data are shown in Figs. 8 and 9, respectively. We fit R + DK = (2.2 ± 0.9 ± 0.3) × 10 −2 for the B + sample, corresponding to 19.2 ± 7.9 ± 2.6 events. On the contrary, no significant WS signal is observed for the B − sample, and we fit R − DK = (0.2 ± 0.6 ± 0.2) × 10 −2 . The statistical correlation between R + DK and R − DK (or R DK and A DK ) is insignificant.
The systematic errors on the asymmetries are estimated using the method discussed previously. The main systematic error on A DK is from the uncertainty on the number of peaking B background events for the WS channel. This source contributes +0.11 −0.14 to A DK , and ±0.08 × 10 −2 to R DK , where the changes in the two  quantities are 100% negatively correlated (increasing the peaking background increases A DK but decreases R DK ). The other sources of systematic uncertainty considered in Table VII are 100% correlated between R + and R − , and mostly cancel in the asymmetry calculation. By com-paring the number of B + and B − events reconstructed in the [K ± π ∓ ] D π ± analysis, where no significant asymmetry is expected, the uncertainty due to the detector charge asymmetry is estimated to be below the 1% level. Finally, we also account for a possible asymmetry of the 21.9 ± 9.0 4.9 ± 7.9 9 ± 16 B − Ratio of rates, R DK for B ± → DK ± (left), B ± → D * Dπ 0 K ± (center) and B ± → D * Dγ K ± (right). Systematic uncertainties are not included. charmless B ± → K ± K ∓ π ± peaking background. The asymmetry of this background has been measured to be 0 ± 10% [17] and we estimate the corresponding systematic uncertainty by assuming a ±10% asymmetry of this background. The final result for the asymmetry is: For B ± → D * Dπ 0 K ± , we find for the ratio of the WS to RS decay rates R * (Dπ 0 )K = (1.8 ± 0.9 ± 0.4) × 10 −2 .
Expressed in terms of event yields, the fit result is 10.3 ± 5.5 ± 2.4 WS events. The results of fits to separate B + → D * K + and B − → D * K − data samples are given in Table VI. Projections of the fits to B + and B − data are shown in Figs. 8 and 9, respectively. We find R * − (Dπ 0 )K = (3.7 ± 1.8 ± 0.9) × 10 −2 for the B − sample, corresponding to 10.2 ± 4.8 ± 2.4 events. On the contrary, no significant WS signal is observed for the B + sample, and we find R * + (Dπ 0 )K = (0.5 ± 0.8 ± 0.3) × 10 −2 . The systematic errors are estimated using the same method as for B ± → DK ± , separately for B + and B − events. The main systematic error on the asymmetry A * (Dπ 0 )K is from the uncertainty on the number of peaking B background events for the WS channel. This source contributes ±0.09 to A * (Dπ 0 )K , and ∓0.3 × 10 −2 to R * (Dπ 0 )K , where the two quantities are anti-correlated. The other sources of systematic uncertainties mostly cancel in the asymmetry calculation, because they induce relative changes on R * + and R * − which are 100% correlated. The final result for the asymmetry is: A * (Dπ 0 )K = +0.77 ± 0.35 ± 0.12.
The asymmetry for D * Dπ 0 K has the opposite sign to the asymmetry for DK, in agreement with the shift of approximately 180 • between δ B and δ * B suggested by the measurements of Refs. [5,7]. For B → D * Dγ K, we have no significant signal and fit Expressed in terms of event yields, this result corresponds to 5.9 ± 6.4 ± 3.2 events D * Dγ K WS. We fit 211 ± 19 RS B − events and 244 ± 20 RS B + events, and find for the WS to RS ratios R * − (Dγ)K = (1.9 ± 2.3 ± 1.2) × 10 −2 and R * + (Dγ)K = (0.9 ± 1.6 ± 0.7) × 10 −2 . The corresponding asymmetry is

VI. DISCUSSION
We use the B − → D ( * ) K − analysis results and a frequentist statistical approach [18] to extract information on r B and r ( * ) B . In this technique a χ 2 is calculated using the differences between the measured and theoretical values (including systematic errors) of the various ADS quantities from Eqs. (1), (4) and (5). We assume Gaussian measurement uncertainties. This assumption was checked to be valid and conservative at low r B values with a full frequentist approach [5]. For B − → DK − , we have for instance where R ±(th) DK (r B , γ, δ B , r D , δ D ) is given by Eq. (1), and where the two last terms constrain r D and δ D to the values r (m) D and δ (m) D of Ref. [8] within their errors σ r and σ δ . The choice of (R + DK , R − DK ) rather than (R DK , A DK ) is motivated by the fact that the set of variables (R DK , A DK ) is not well-behaved (the uncertainty on A DK depends on the central value of R DK ), while (R + DK , R − DK ) are two statistically independent observables. In the same way, the two pairs of ADS observables (R * + (Dπ 0 )K , R * − (Dπ 0 )K ) and (R * + (Dγ)K , R * − (Dγ)K ) are used to extract r * B , while accounting for the relative phase difference in the two D * decays [9]. We allow 0 ≤ r summarized in Tab. VIII. For B − → [Kπ] D K − , we find the minimum χ 2 at r B = (9.5 +5.1 −4.1 )%. This leads to the upper limit: r B < 16.7% at 90% C.L., to be compared to r B < 23% at 90% C.L. for the previous ADS analysis as performed by BABAR [4] with 232 × 10 6 BB pairs, and to r B < 19% at 90% C.L. for the corresponding ADS analysis as performed by Belle [6] with 657 × 10 6 BB pairs. We exclude r B = 0 with a C.L. of 95.3%. Similarly, for B − → [Kπ] D * K − we find r * B = (9.6 +3.5 −5.1 )%. This leads to the upper limit: r * B < 15.0% at 90% C.L., to be compared to r * B < 16% at 90% C.L. for the previous BABAR ADS analysis [4]. We exclude r * B = 0 with a C.L. of 83.9%.
Using the above procedure we also determine the 2D confidence intervals for γ vs δ  [5,7]. Finally, Fig.13 shows the C.L. curve as a function of γ when combining the DK and D * K results.

VIII. ACKNOWLEDGMENTS
We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminos- ity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them.