Measurement of the branching fraction and $CP$ asymmetry for $B\to\bar{D}^{0} \pi$ decays

We measure the branching fractions and $CP$ asymmetries for the decays $B^{0}\to \bar{D}^{0}\pi^{0}$ and $B^{+}\to \bar{D}^{0}\pi^{+}$, using a data sample of $772\times 10^{6}$ $B\bar{B}$ pairs collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB $e^{+}e^{-}$ collider. The branching fractions obtained and direct $CP$ asymmetries are $\mathcal{B}(B^{0}\to \bar{D}^{0}\pi^{0}) = [2.70 \pm 0.06~ \text{(stat.)} \pm 0.10~ \text{(syst.)}] \times 10^{-4}$, $\mathcal{B}(B^{+}\to \bar{D}^{0}\pi^{+}) = [4.53 \pm 0.02~ \text{(stat.)} \pm 0.15~ \text{(syst.)}] \times 10^{-3}$, $ {\cal A}_{CP}(B^{0}\to \bar{D}^{0}\pi^{0}) = [+0.42 \pm 2.05~ \text{(stat.)} \pm 1.22~ \text{(syst.)}]\%$, and $ {\cal A}_{CP}(B^{+}\to \bar{D}^{0}\pi^{+}) = [+0.19 \pm 0.36~ \text{(stat.)} \pm 0.57~ \text{(syst.)}]\%$. The measurements of $\mathcal{B}$ are the most precise to date and are in good agreement with previous results, as is the measurement of ${\cal A}_{CP}(B^{+}\to \bar{D}^{0}\pi^{+})$. The measurement of ${\cal A}_{CP}$ for $B^{0}\to \bar{D}^{0}\pi^{0}$ is the first for this mode, and the value is consistent with Standard Model expectations.


I. INTRODUCTION
The branching fraction (B) of the color-suppressed decay B 0 →D 0 π 0 [1] is measured [2][3][4] to be about a factor of four higher than theory predictions made using the "naive" factorization model, where final-state interactions (FSIs) are neglected [5,6]. This has led to a number of new theoretical descriptions of the process [7][8][9][10][11][12][13][14] that include FSI's and also treat isospin-related amplitudes of color-suppressed and color-allowed decays. The B 0 →D 0 π 0 process has been shown to have large nonfactorizable components [14], so precise measurements of its properties are valuable in comparing different theoretical models used to describe it. Many of these models predict a substantial strong phase in the final state. A non-vanishing strong phase difference between two amplitudes is necessary to give rise to direct CP violation [15]. The direct CP -violation parameter, A CP , for the B →D 0 π decay is defined as: where Γ is the partial decay width for the corresponding decay.
In the Standard Model (SM), B 0 →D 0 π 0 transitions proceed mainly via the tree-level diagram of Fig.1a. An exchange diagram (Fig.1b) with the same CKM factors is also present, but, due to OZI suppression, is expected to have a much smaller amplitude. As such, direct CP violation in this mode is expected to be small, even in the presence of a strong phase difference from FSI. Measurements of notable CP violation in this decay would be of significant interest and could hint at contributions from Beyond-the-Standard-Model (BSM) physics diagrams. Recently the BaBar and Belle collaborations performed time-dependent CP -violation analyses of the related modesB 0 → D ( * ) CP h 0 , where h ∈ π 0 , η, ω and D ( * ) CP refers to D or D * in a CP eigenstate [16]. They measure the CP -violation parameters C(= −A CP ), S CP and φ 1 [17], and obtain C(B 0 →D * 0 h 0 ) = (−2 ± 8) × 10 −2 . This value is consistent with the expectation of small A CP for B 0 →D 0 π 0 . However, this result does not exclude larger values up to 0.1, which would be much larger than SM predictions.
A high precision measurement of B and A CP for B 0 → D 0 π 0 has further utility in addition to comparison to theoretical predictions, as it is a common control mode for use in rare charmless B decays with a π 0 . The precise measurement of properties of a control mode is important to provide validation and refinement of analysis techniques.
In this paper, we present new measurements of B 0 → D 0 π 0 using the full data sample of (772 ± 10.6) × 10 6B B pairs (711 fb −1 ) collected with the Belle detector at the KEKB asymmetric-energy e + e − (3.5 GeV on 8.0 GeV) collider [18] operating near the Υ(4S) resonance. We also present corresponding measurements of B + →D 0 π + decays, which proceed via a simple spectator diagram with no color suppression.

II. BELLE DETECTOR
The Belle detector [19] is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrellike arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter (ECL) consisting of CsI(Tl) crystals. All these detector components are located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented with resistive plate chambers to detect K 0 L mesons and to identify muons. Two inner detector configurations were used: a 2.0 cm beam-pipe and a 3-layer SVD (SVD1) were used for the first sample of 152 × 10 6 BB pairs, while a 1.5 cm beam-pipe, a 4-layer SVD (SVD2), and small cells in the inner layers of the CDC were used to record the remaining 620 × 10 6 BB pairs [20].
The flavor of the neutral B-meson (B 0 orB 0 ) is determined by the charge of the reconstructed kaon (K + or K − ). This method of flavor tagging is not perfect, as the wrong-sign doubly Cabibbo-suppressed decays (DCS)B 0 →D 0 π 0 (Fig.1c) will result in wrongly tagged flavor. The same effect occurs with charm DCS decays D 0 → K + π − and D 0 → K + π − π 0 , and charm mixing, although the charm mixing effect is negligibly smaller. These effects can be calculated using the ratio of wrong-sign (Cabibbo-suppressed) to right-sign (Cabibbofavored) decay rates: where R(B 0 →D 0 π 0 ) has not been measured and thus is approximated with R(B 0 → D + π − ) = (2.92 ± 0.38 ± 0.31)×10 −4 [21], while R(D 0 → K + π − ) = (3.79±0.18)× 10 −3 and R(D 0 → K + π − π 0 ) = (2.12 ± 0.07) × 10 −3 are taken from the PDG [22]. These effects lead to the true value of B being (0.314 ± 0.008)% lower than the measured value, and the true A CP being (3.05 ± 0.08) × 10 −5 higher than the measured value. In the case of B this is corrected for; however, for A CP the correction is significantly smaller than our uncertainty and it is neglected. Photon candidates are mainly taken from clusters in the ECL but additionally are reconstructed from e + e − pairs resulting from photon conversion in the inner detector. Photons from π 0 decay must have an energy greater than 50 (100) MeV in the barrel (endcap) region of the ECL. The invariant mass of the two-photon combination must lie in the range 104 MeV/c 2 < M γγ < 165 MeV/c 2 , corresponding to ±3σ around the nominal π 0 mass [22]. We subsequently perform a mass-constrained fit with the requirement χ 2 < 50.
Charged tracks originating from a B decay are required to have a distance-of-closest-approach with respect to the interaction point of less than 4.0 cm along the zaxis (the direction opposite the positron beam), and of less than 0.3 cm in the plane transverse to the z-axis. Charged kaons and pions are identified using information from the CDC, ACC, and TOF detectors. This information is combined to form a K − π likelihood ratio R K/π = L K /(L K + L π ), where L K (L π ) is the likelihood of the track being a kaon (pion). Track candidates with R K/π > 0.6 (< 0.4) are classified as kaons (pions). The typical kaon (pion) identification efficiency is 83% (88%), with a pion (kaon) misidentification probability of 7% (11%).
Two kinematic variables are used to distinguish signal from background: the beam-energy-constrained mass, Here, p B and E B are the momentum and energy, respectively, of the B-meson candidate evaluated in the center-of-mass (CM) frame, and E beam is the beam energy in the CM frame.
Due to energy leakage in the ECL, the reconstructed π 0 energy is typically lower than its true value. To compensate for this, we rescale the reconstructed π 0 momentum to give E π 0 ≡ E beam − E D , specifically: Using this we calculate a new B-meson momentum, p B,corr , then calculate a corrected M bc (from now on simply referred to as M bc ), Rescaling M bc in this way improves the mass resolution and removes some correlations between M bc and ∆E. This procedure is only applied to the π 0 that is the direct daughter of the B 0 . All candidates satisfying M bc > 5.25 GeV/c 2 and −0.2 GeV < ∆E < 0.2 GeV are retained for further analysis. We find that 16% (47%) of events have more than one B 0 candidate in the B 2b (B 3b ) reconstruction modes. In these cases, we select one of the reconstructed B 0 mesons based on the mass difference ∆m(X) = m PDG (X) − M (X), where m PDG (X) is the mass reported by the Particle Data Group (PDG) [22] for particle X, and M (X) is the reconstructed mass. The best candidate is selected as the B 0 or B + that minimizes ∆m(D 0 ). If there are multiple candidates with the same minimal ∆m(D 0 ), the one that minimizes ∆m(π 0 ) is selected. Monte Carlo simulation (MC) studies show that this procedure selects the correct B 0 in 96% (86%) of cases for the B 2b (B 3b ) samples.

III. BELLE DETECTOR AND SIGNAL SELECTION
Backgrounds to our signal are studied using MC simulation. These simulations use EvtGen [23] and PYTHIA [24] to generate the physics interactions at the quark level, and employ GEANT3 [25] to simulate the detector response.
The largest background arises from e + e − → qq (q ∈ {u, d, s, c}) continuum events. A neural network [26] is used to distinguish the spherical BB signal from the jetlike continuum background. It combines the following five observables based on the event topology: a Fisher discriminant formed from 17 modified Fox-Wolfram moments [27]; the cosine of the angle between the B-meson candidate direction and the beam axis; the cosine of the angle between the thrust axis [28] of the B-meson candidate and that of the rest of the event (all of these quantities being calculated in the CM frame); the separation along the z-axis between the vertex of the B-meson candidate and that of the remaining tracks in the event; and the tagging quality variable from a B-meson flavortagging algorithm [29]. The training and optimization of the neural network are performed with signal and continuum MC samples. These are divided into five training samples and one verification sample. The output of the neural net (C NN ) has a range of (−1, 1), with 1 being the most signal-like and −1 being the most background-like.
In order to maximally use C NN information, we impose only a loose requirement on C NN and use C NN as a variable in the fit. We require C NN > −0.05 for both the B 2b and B 3b modes. This results in 86% background reduction and 87% signal efficiency. To facilitate modelling C NN analytically with Gaussian functions, we transform it to an alternative variable C NN via the formula where C min NN is the minimum value of −0.05, and C max NN is the maximum value of C NN obtained from the signal MC sample used to verify the training.
There is a significant background arising from b → c transitions, which we refer to as "generic B" decays. The main components of the generic B background are incorrectly assigned tracks, combinatorial backgrounds, B 0 →D 0 ρ 0 , and B 0 →D 0 * π 0 with eitherD 0 * →D 0 γ orD 0 * →D 0 π 0 . These are investigated with MC simulations of BB decays. To reduce this background, signal candidates are selected within ±3 standard deviations of the mean values for M bc , ∆E, and the reconstructed π 0 and D 0 mass distributions. Low final-state momentum events are excluded with selection criteria on the labframe momentum of the final-state particles: P (K ± ), P (π ± ), P (π 0 B 0 ), and P (π 0 D 0 ). These requirements are listed in Table I.   TABLE I. Requirements on kinematic variables employed in the reconstruction of B 0 →D 0 π 0 decays, to minimize generic B-decay background. The subscript identifies the origin of the particle, and theD 0 decay mode is shown in square brackets.
There is also a very small background component from b → u and b → s transitions that consists mainly of combinatorial background, with some non-resonant B decays to the same final states (K + π − π 0 , K + π − π 0 π 0 , K + π − π + and K + π − π + π 0 ). These are studied using a large MC sample corresponding to 50 times the number of BB events recorded by Belle. We refer to these background events as "rare." The yield of these rare events is fixed when fitting for the signal yield based on the most recent branching fractions from the PDG [22].
After all selections have been made for B 0 →D 0 π 0 , the reconstruction efficiencies are (

IV. FITTING STRATEGY
The signal yield and A CP are extracted via an unbinned extended maximum-likelihood fit to the variables M bc , ∆E, and C NN . There are four categories of events fitted: B 0 →D 0 π 0 or B + →D 0 π + signal events (s), continuum events (c), generic BB events (b), and rare B-decay backgrounds (r). These events are described by probability density functions (PDFs) denoted as P s , P c , P b , and P r , respectively. Separate PDFs are constructed for the B 2b and B 3b reconstruction modes, which are fitted as two separate data sets. The data are further divided into events tagged as B 0 andB 0 , defined as having flavor q = +1 and q = −1, respectively, based on the charge of the kaon.
The physics parameters are determined via a simultaneous fit to the four data sets. The total likelihood is given by where N q,d is the number of events with flavor tag q for the data set d (d ∈ B 2b , B 3b ), and N j is the number of events in the j th category (j ∈ s, c, b, r) contributing to the total yield. The fraction of events in the data set d corresponds to the j th category in the d data set for flavor q, measured at M i bc , ∆E i , and C i NN for the i th event. The PDF for each component is given by: The model accounts for a possible direct CP asymmetry, A j CP , and the fractions of signal and backgrounds expected in each reconstruction mode B 2b (B 3b ). In Eq. (6), the fraction f s d is determined via MC studies of the B 2b and B 3b modes and is fixed in the fit to data, and A j CP (j ∈ c, b, r) is fixed based on studies of detector bias using sideband data (see Section V).
The 20 free parameters in the fit are: the number of signal events N s , signal asymmetry A s CP , the number of continuum events (N c ) and generic B-decay events (N b ), fractions of backgrounds expected in each reconstruction mode f j d (j ∈ c, b, r), shape parameters of the continuum M bc and ∆E PDFs, and the mean and width of the B 2b signal M bc and ∆E PDFs. The number of rare background events (N r ) is fixed to that expected from MC studies. The effects of these assumptions are included in the systematic uncertainties.
The PDFs used for the M bc and ∆E distributions for the various event types are as follows.
• Signal: for the B 2b mode, the M bc PDF is a Crystal Ball function [30], while the ∆E PDF is the sum of a Crystal Ball function and a Gaussian with the same mean. The Gaussian component is small and included to handle the tails of the distribution. For the B 3b mode, there is a strong correlation between M bc and ∆E, and no analytic 2D PDF could be found to fit the data satisfactorily. Instead a 2D kernel density estimation (KEST) PDF [31] is used.
• Generic B-decay background: similarly to B 3b signal, there exist complex correlations between M bc and ∆E, so a 2D KEST PDF obtained from MC simulations in both modes.
• Continuum background: M bc is fitted as an AR-GUS function [32], and ∆E as a 3rd-order Chebyshev polynomial, in both modes.
• Rare B-decay background: as with generic B and B 3b signal, the M bc and ∆E distributions for rare B-decay background are modelled with a 2D KEST PDF in both modes. This PDF is determined using MC simulations corresponding to 50 times the luminosity of the Belle data set.
To fit C NN , three summed Gaussians are used for all components except continuum background, which employed two summed Gaussians. The PDFs for all event types are summarized in Table II. The fitting procedure and accuracy of the various PDF models are extensively investigated using MC 'pseudoexperiments'. In these studies, the signal and rare B background events are selected from large samples of simulated events. Events for e + e − → qq and generic B-decay are generated from their respective PDF shapes. For B 0 →D 0 π 0 , we observe a small bias of (+0.6 ± 0.3)% in signal yield, and (+0.04 ± 0.05)% in A CP . We correct for this bias in our final measurements and include a corresponding systematic uncertainty for it. No significant bias is observed for B + →D 0 π + , i.e., only (+0.06 ± 0.16)% in signal yield and (−0.02 ± 0.02)% in A CP .
V. B + →D 0 π + We first select and fit a sample of B + →D 0 π + decays. This sample is not color-suppressed and thus has much larger statistics and lower background than the sample of B 0 →D 0 π 0 decays. As well as ensuring the fitted B and A CP are consistent with existing measurements, this mode is used to obtain calibration factors for the fixed shape parameters of the PDFs used to fit B 0 → D 0 π 0 decays, to account for any differences between MC and data. In addition, this mode provides a data-driven estimation of the systematic uncertainty associated with the A CP correction for a detection asymmetry (discussed below).
To account for potential differences in the distribution of fitting variables between MC and data, additional parameters (calibration factors) are included in the fit to enable small adjustments in the fitted PDF shapes. These calibration factors are applied as mean shifts and width factors to the C NN Gaussians.
To account for small differences between MC and data in the width of the ∆E distribution, the 2D KEST PDF for M bc and ∆E in B 3b is modified slightly. This is done by modifying each ∆E data point in the MC data set with a random shift based on a Gaussian distribution with a mean of 0 GeV and a width of 7 MeV (chosen after testing with a range of widths) and generating a new KEST PDF from this modified data sample.
Fits to the B + →D 0 π + sample are performed to determine the signal yield, A CP,raw , the continuum background yield, the generic B-decay background yield, and the calibration factors. The rare B-decay background yield is fixed to the value expected from MC simulations. From the fit we obtain N sig = 84537 ± 306 and A CP,raw = (1.97 ± 0.36)%. The uncertainties listed are statistical. A CP,raw is the output of the fit without a correction to account for sources of bias. Figure 2 shows the fits to data in M bc , ∆E and C NN .
To account for possible bias in A CP , we perform an analysis over a "sideband" region of data, defined as 0.1 GeV < ∆E < 0.4 GeV and 5.255 GeV/c 2 < M bc < 5.27 GeV/c 2 . This region consists almost entirely of continuum events and has an expected A CP of zero. Counting the number of events in this region we find A CP,sideband = (1.78 ± 0.38)%. We subtract this value from A CP,raw to correct for the detection asymmetry bias.
The branching fraction is calculated as: where N B ± is the number of charged B-mesons in the data set, based on the PDG average value of B(Υ(4S) → B + B − ) = (51.4 ± 0.6%) [22]; f s d is the fraction of signal events in data set d = 2b or 3b (f s 2b = 0.51, f s 3b = 0.49); and d is the product of the reconstruction efficiency, thē D 0 branching fraction B d , and small corrections for particle identification (PID), and charged track and π 0 reconstruction efficiencies (see Section VII), for mode d. The π 0 → γγ branching fraction is accounted for in the MC simulation. The resulting values for 2b and 3b are (1.19 ± 0.03) × 10 −2 and (1.16 ± 0.05) × 10 −2 , respectively. The mean is calculated as a generalized weighted mean [33] [34], taking into account correlated and uncorrelated uncertainties in a covariance matrix. This approach is used because the difference in systematic uncertainties between the twoD 0 decay modes leads to the need to weight them in order to calculate the final branching fraction and uncertainty correctly. Finally, the correction due to DCS decays discussed in Section II is made.
The results for B and A CP for B + →D 0 π + are: The uncertainties quoted are statistical and systematic, respectively. The systematic uncertainties associated with the measurement of B and A CP are explained in detail in Section VII, and the contributions of each of these are listed in Table V and VI, respectively. These results are in agreement with the PDG values [22] of B = (4.68 ± 0.13) × 10 −3 and A CP = (−0.7 ± 0.7)%. As a cross-check, we determined B and A CP for each of the B 2b and B 3d modes separately, and for just the SVD1 data set. All are in agreement within statistical uncertainties. The fitted yield for each respective category is listed in Table III.
After applying the calibration factors determined from studies of the B + →D 0 π + mode to the PDFs, we fit the signal B 0 →D 0 π 0 PDFs to data and find 4448±97 signal events and A CP,raw = (1.48 ± 2.05)%. The uncertainties quoted are statistical. As was the case for B + →D 0 π + , A CP,raw is the value returned from the fit without a correction for sources of bias. Figure 3 shows the signal-  B 3b (bottom). The blue short-dashed curve shows the signal PDF, red dotted curve shows the BB background PDF, green dash-dotted curve shows the continuum background PDF, pink long-dashed curve shows the (almost negligible) rare background PDF, black line is the fit result, points are data. Also shown underneath each graph is the residual pulls between the data points and fitted PDF. enhanced projections of the fits. Figure 4 shows signalenhanced projections of M bc separated into B 0 andB 0 decays.
The A CP correction for the B 0 →D 0 π 0 decay is measured in the same way as for the B + →D 0 π + mode. A sideband region of data is defined as 0.1 GeV < ∆E < 0.4 GeV and 5.255 GeV/c 2 < M bc < 5.27 GeV/c 2 . Events in this region consist almost entirely of continuum with an expected A CP of zero. In this region we find A CP,sideband = (1.02 ± 0.64)%, and we subtract this value and the fit bias (0.04%) from A CP,raw to correct for detector bias.
The direct CP -violation parameter is thus measured to be: The uncertainties quoted are statistical and systematic, respectively.

VII. SYSTEMATIC UNCERTAINTIES
The systematic uncertainties associated with the measurement of B and A CP are as follows. The blue short-dashed curve shows the signal PDF, red dotted curve shows the BB background PDF, green dash-dotted curve shows the continuum background PDF, pink long-dashed curve shows the (almost negligible) rare background PDF, black line is the fit result. Also shown underneath each graph is the residual pulls between the data points and fitted PDF.
• Number of BB pairs: the uncertainty associated with the measured number of BB pairs in the full data set collected at Belle is 1.37% [35].
• DCS mode correction: the uncertainty due to the correction for doubly Cabibbo-suppressed decays is 0.01% for both B 0 →D 0 π 0 and B + →D 0 π + (see Section II).
• Charged track efficiency: the uncertainty associated with a possible difference in efficiency between MC and data for charged-track reconstruction is found to be 0.35% per track using partially reconstructed D * + → D 0 (→ π + π − π 0 )π + events [35].
•D 0 subdecay branching fraction and A CP : from the PDG average [22].
• PID efficiency: systematic error associated with a small difference in PID efficiency between MC and data. This is based on an inclusive D * + → D 0 (K − π + )π + study [35].
• Signal decay mode yield ratio f s d : the ratio between the D 0 decay modes in signal, f s d , is fixed based on the expected yields from MC. To account for the uncertainty, we perform two fits, varying The blue short-dashed curve shows the signal PDF, red dotted curve shows the BB background PDF, green dash-dotted curve shows the continuum background PDF, pink long-dashed curve shows the (almost negligible) rare background PDF, black line is the fit result. Also shown underneath each graph is the residual pulls between the data points and fitted PDF. the fixed value by ±1σ (based on MC statistics of the simulation). This variation gives changes of [−0.38, +0.31]% and [−0.08, +0.19]% in B for B 0 → D 0 π 0 and B + →D 0 π + respectively. The uncertainty in A CP is [−0.02, +0.03] for B 0 →D 0 π 0 and < 0.01 for B + →D 0 π + .
• C NN calibration factors: we fit with and without the calibration factors applied to the PDFs. The difference between the yields and A CP of these fits is quoted as the uncertainty. The uncertainty in B is 0.34% and 0.06% for B 0 →D 0 π 0 and B + → D 0 π + , respectively. For A CP it is 0.06 and < 0.01, respectively.
• Modification of the B 3d M bc × ∆E KEST PDF: the uncertainty from the ∆E modification to thē D 0 → K + π − π 0 M bc ×∆E KEST PDF is evaluated by comparing the fit results obtained using the corrected and uncorrected PDF. The difference in the fitted yields of 0.63% for B 0 →D 0 π 0 and 0.24% for B + →D 0 π + is quoted as the uncertainty. For A CP this is 0.06 and < 0.01, respectively.
• 2D KEST PDFs: BB, rare, and B 3b signal M bc × ∆E PDFs all use a fixed 2D KEST PDF. To estimate the uncertainty from this, an ensemble test is performed running 1000 fits over the data, with each fit using a different Gaussian-fluctuated KEST PDF based on bin statistics. The uncertainty is quoted as the RMS of the resulting yield and A CP distributions. This contributes 0.35% and 0.05% to the uncertainty in B for B 0 →D 0 π 0 and B + →D 0 π + , respectively. For A CP the contribution is 0.15 and < 0.01, respectively.
• Fixed rare B-decay background yield: the uncertainty due to the rare B yield is the quadratic sum of the statistical uncertainty based on the size of the MC dataset and the uncertainty in the branching fractions used to generate the MC. For modes with three body final states (K + π − π 0 and K + π − π + ), this latter component is taken from the uncertainty in the PDG branching fractions B(B 0 → K + π − π 0 ) = (37.8 ± 3.2) × 10 −6 and B(B + → K + π − π + ) = (51.0 ± 2.9) × 10 −6 [22]. For D 0 → K + π − π 0 modes, this latter component is taken as the difference between the PDG values for the decays with experimentally measured branching fractions and the branching fractions used in the MC generator (or the uncertainty on the PDG value if that is larger). To estimate the effect on signal yield, the data is refitted, varying the rare yield by ±1σ. The uncertainty in B is 0.47% for B 0 →D 0 π 0 and 0.03% for B + →D 0 π + .
• Fit bias: the uncertainty in the fit bias obtained from the signal MC ensemble tests is quoted as an uncertainty. For B this is 0.30% and 0.16% for B 0 →D 0 π 0 and B + →D 0 π + , respectively, and for A CP it is 0.05 and 0.02, respectively.
• A CP detector bias correction: uncertainty on the correction made to A CP is the statistical uncertainty on the A CP,sideband measurement, summed in quadrature with the deviation of the A CP of the B + →D 0 π + mode from the expected value of A CP = 0. This is 0.66 for B 0 →D 0 π 0 and 0.42 for B + →D 0 π + .
• Fixed background A CP : uncertainties from background A CP being fixed in fits are estimated by varying them by ±1σ (based on sideband data) and comparing the A CP in the resultant fits. This is found to be 0.49 for B 0 →D 0 π 0 and 0.03 for B + →D 0 π + . This is correlated with the A CP detector bias correction uncertainty.
In order to accurately calculate the uncertainty in B, theD 0 decay-mode-dependent factors are combined in a generalized weighted mean as shown in Eq. 8. The absolute uncertainties for charged track efficiency, π 0 detection efficiency, reconstruction efficiency, PID efficiency, andD 0 branching fraction are combined into a covariance matrix, Σ, that accounts for their correlations between the two-body and three-body modes. For B 0 →D 0 π 0 , Σ = 1.47 2.40 2.40 6.76 , and for B + →D 0 π + Σ = 1.17 1.05 1.05 4.03 . The combined value, which we call "mean efficiency", is The relative uncertainty on this is found to be 2.43% for B 0 →D 0 π 0 , and 2.54% for B + →D 0 π + . The values of all contributions to the branching fractions are listed in Table V. The quadratic sum of these terms is quoted as the total systematic uncertainty for B. The values of all contributions to the A CP measurements are listed in Table VI. The quadratic sum of these terms is quoted as the total systematic uncertainty for A CP .