Measurement of branching fractions of B decays to K1(1270)pi and K1(1400)pi and determination of the CKM angle alpha from B0 -->a1(1260)+/- pi-/+

We report measurements of the branching fractions of neutral and charged B meson decays to final states containing a K_1(1270) or K_1(1400) meson and a charged pion. The data, collected with the BaBar detector at the SLAC National Accelerator Laboratory, correspond to 454 million B Bbar pairs produced in e^+ e^- annihilation. We measure the branching fractions BF(B^0 -->K_1(1270)^+ \pi^- + K_1(1400)^+ \pi^-) = 3.1^{+0.8}_{-0.7} x10^{-5} and BF(B^+ -->K_1(1270)^0 \pi^+ + K_1(1400)^0 \pi^+) = 2.9^{+2.9}_{-1.7} x10^{-5} (<8.2 x10^{-5} at 90% confidence level), where the errors are statistical and systematic combined. The B^0 decay mode is observed with a significance of 7.5\sigma, while a significance of 3.2\sigma is obtained for the B^+ decay mode. Based on these results, we estimate the weak phase \alpha = (79 +/- 7 +/- 11)^{\circ} from the time dependent CP asymmetries in B^0 -->a_1(1260)^{+/-} \pi^{-/+} decays.


I. INTRODUCTION
B meson decays to final states containing an axialvector meson (A) and a pseudoscalar meson (P ) have been studied both theoretically and experimentally. Theoretical predictions for the branching fractions (BFs) of these decays have been calculated assuming a naïve factorization hypothesis [1,2] and QCD factorization [3]. These decay modes are expected to occur with BFs of order 10 −6 . Branching fractions of B meson decays with an a 1 (1260) or b 1 (1235) meson plus a pion or a kaon in the final state have recently been measured [4,5].
The BABAR Collaboration has measured CP -violating asymmetries in B 0 → a 1 (1260) ± π ∓ decays and determined an effective value α eff [6] for the phase angle α of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [7]. In the absence of penguin (loop) contributions in these decay modes, α eff coincides with α.
The production of K 1 mesons in B decays has been previously observed in the B → J/ψK 1 , B → K 1 γ, and B → K 1 φ decay channels [15]. Here we present measurements of the B 0 → K + 1 π − and B + → K 0 1 π + branching fractions and estimate the weak phase α from the measurement of the time dependent CP asymmetries in B 0 → a 1 (1260) ± π ∓ decays and the branching fractions of SU(3) related modes. This paper is organized as follows. In Sec. II we describe the dataset and the detector. In Sec. III we introduce the K−matrix formalism used for the parameterization of the K 1 resonances. Section IV is devoted to a discussion of the reconstruction and selection of the B candidates. In Sec. V we describe the maximum likelihood fit for the signal branching fractions and the likelihood scan over the parameters that characterize the production of the K 1 system. In Sec. VI we discuss the systematic uncertainties. In Sec. VII we present the experimental results. Finally, in Sec. VIII, we use the experimental results to extract bounds on |∆α|.

II. THE BABAR DETECTOR AND DATASET
The results presented in this paper are based on data collected with the BABAR detector at the PEP-II asymmetric-energy e + e − storage ring, operating at the SLAC National Accelerator Laboratory. At PEP-II, 9.0 GeV electrons collide with 3.1 GeV positrons to yield a center-of-mass (CM) energy of √ s = 10.58 GeV, which corresponds to the mass of the Υ (4S) resonance. The asymmetric energies result in a boost from the laboratory to the CM frame of βγ ≈ 0.56. We analyze the final BABAR dataset collected at the Υ (4S) resonance, corresponding to an integrated luminosity of 413 fb −1 and N BB = (454.3 ± 5.0) × 10 6 produced BB pairs.
A detailed description of the BABAR detector can be found elsewhere [16]. Surrounding the interaction point is a five-layer double-sided silicon vertex tracker (SVT) that provides precision measurements near the collision point of charged particle tracks in the planes transverse to and along the beam direction. A 40-layer drift chamber surrounds the SVT. Both of these tracking devices operate in the 1.5 T magnetic field of a superconducting solenoid to provide measurements of the momenta of charged particles. Charged hadron identification is achieved through measurements of particle energy loss in the tracking system and the Cherenkov angle obtained from a detector of internally reflected Cherenkov light. A CsI(Tl) electromagnetic calorimeter provides photon detection and electron identification. Finally, the instrumented flux return (IFR) of the magnet allows discrimination of muons from pions and detection of K 0 L mesons. For the first 214 fb −1 of data, the IFR was composed of a resistive plate chamber system. For the most recent 199 fb −1 of data, a portion of the resistive plate chamber system has been replaced by limited streamer tubes [17].
We use a GEANT4-based Monte Carlo (MC) simulation to model the response of the detector [18], taking into account the varying accelerator and detector conditions. We generate large samples of signal and background for the modes considered in the analysis.

III. SIGNAL MODEL
In this analysis the signal is characterized by two nearby resonances, K 1 (1270) and K 1 (1400), which have the same quantum numbers, I(J P ) = 1/2(1 + ), and decay predominantly to the same Kππ final state. The world's largest sample of K 1 (1270) and K 1 (1400) events was collected by the ACCMOR Collaboration with the WA3 experiment [19]. The WA3 fixed target experiment accumulated data from the reaction K − p → K − π + π − p with an incident kaon energy of 63 GeV. These data were analyzed using a two-resonance, six-channel K-matrix model [20] to describe the resonant Kππ system. We base our parameterization of the K 1 resonances produced in B decays on a model derived from the K-matrix description of the scattering amplitudes in Ref. [19]. In Sec. III A we briefly outline the K-matrix formalism, which is then applied in Sec. III B to fit the ACCMOR data in order to determine the parameters describing the diffractive production of K 1 mesons and their decay. In Sec. III C we explain how we use the extracted values of the decay parameters and describe our model for K 1 production in B → (Kππ)π decays.
We parameterize the production amplitude for each channel in the reaction K − p → (K − π + π − )p as where the index i (and similarly j) represents the i th channel. The elements of the diagonal phase space matrix ρ(M ) for the decay chain are approximated with the form where M is the Kππ invariant mass, m 4 is the mass of the bachelor particle 4, and m * (∆) is the pole mass (half width) of the intermediate resonance state 3 [21]. In Eq.
(1), the δ i parameters are offset phases with respect to the (K * (892)π) S channel (δ 1 ≡ 0). The 6 × 6 K-matrix has the following form: where the labels a and b refer to K 1 (1400) and K 1 (1270), respectively. The decay constants f ai , f bi and the Kmatrix poles M a and M b are real. The production vector P consists of a background term D [23] and a direct production term R where τ is a constant. The background amplitudes are parameterized by for all channels but (K * (892)π) D and ωK. For the (K * (892)π) D channel we set D 5 = 0 as in the ACC-MOR analysis [19]. The parameters for the ωK channel are not fitted, as described later in this Section, and we set D 6 = 0. The results are not sensitive to this choice for the value of D 6 . R is given by where f pa and f pb represent the amplitude for producing the states K 1 (1400) and K 1 (1270), respectively, and are complex numbers. We assume f pa to be real. P-wave (ℓ = 1) and D-wave (ℓ = 2) centrifugal barrier factors are included in the K 1 decay couplings f ai and f bi and background amplitudes D i0 , and are given by: where q i is the breakup momentum in channel i. Typical values for the interaction radius squared R 2 are in the range 5 < R 2 < 100 GeV −2 [22] and the value R 2 = 25 GeV −2 is used. The physical resonances K 1 (1270) and K 1 (1400) are mixtures of the two SU(3) octet states K 1A and K 1B : Assuming that SU(3) violation manifests itself only in the mixing, we impose the following relations [19]: where γ + and γ − are the couplings of the SU(3) octet states to the (K * (892)π) S and ρK channels: The couplings for the ωK channel are fixed to 1/ √ 3 of the ρK couplings, as follows from the quark model [19].

B. Fit to WA3 data
Only some of the K-matrix parameters extracted in the ACCMOR analysis have been reported in the literature [19]. In particular, the results for most of the decay couplings f ai and f bi are not available. The ACCMOR Collaboration performed a partial-wave analysis of the WA3 data. The original WA3 paper [19] provides the results of the partial-wave analysis of the Kππ system in the form of plots for the intensity in the (K * (892)π) S , ρK, K * 0 (1430)π, f 0 (1370)K, and (K * (892)π) D channels, together with the phases of the corresponding amplitudes, measured relative to the (K * (892)π) S amplitude. The ωK data were not analyzed. In order to obtain an estimate of the parameters that enter the Kmatrix model, we perform a χ 2 fit of this model to the 0 ≤ |t ′ | ≤ 0.05 GeV 2 WA3 data for the intensity of the m = 0 Kππ channels and the relative phases. Here |t ′ | is the four momentum transfer squared with respect to the recoiling proton in the reaction K − p → K − π + π − p, and m denotes the magnetic substate of the Kππ system. Since the results of the analysis performed by the ACCMOR Collaboration are not sensitive to the choice of the value for the constant τ in Eq. (5), we set τ = 0. We seek solutions corresponding to positive values of the γ ± parameters, as found in the ACCMOR analysis [19]. The data sample consists of 215 bins. The results of this fit are displayed in Fig. 1 and show a good qualitative agreement with the results obtained by the ACCMOR Collaboration [19]. We obtain χ 2 = 855, with 26 free parameters, while the ACCMOR Collaboration obtained χ 2 = 529. Although neither fit is formally a good one, the model succeeds in reproducing the relevant features of the data.

C. Model for K1 production in B decays
We apply the above formalism to the parameterization of the signal component for the production of K 1 resonances in B decays. The propagation of the uncertainties in the K−matrix description of the ACCMOR data to the model for K 1 production in B decays is a source of systematic uncertainty and is taken into account as described in Sec. VI.
In order to parameterize the signal component for the analysis of B decays, we set the background amplitudes D, whose contribution should be small in the non-diffractive case, to 0. The backgrounds arising from resonant and non-resonant B decays to the (Kππ)π final state are taken into account by separate components in the fit, as described in Sec. V. The parameters of K and the offset phases δ i are assumed to be independent from the production process and are fixed to the values extracted from the fit to WA3 data (Table I). Finally, we express the production couplings f pa and f pb in terms of two real production parameters ζ = (ϑ, φ): f pa ≡ cos ϑ, f pb ≡ sin ϑe iφ , where ϑ ∈ [0, π/2], φ ∈ [0, 2π]. In this parameterization, tan ϑ represents the magnitude of the production constant for the K 1 (1270) resonance relative to that for the K 1 (1400) resonance, while φ is the relative phase.
The dependence of the selection efficiencies and of the distribution of the discriminating variables (described in Sec. V) on the production parameters ζ are derived from Monte Carlo studies. For given values of ζ, signal MC samples for B decays to the (Kππ)π final states are generated by weighting the (Kππ)π population according to the amplitude i =ωK Kππ|i F i , where the term Kππ|i consists of a factor describing the angular distribution of the Kππ system resulting from K 1 decay, an amplitude for the resonant ππ and Kπ systems, and isospin factors, and is calculated using the formalism described in Refs. [19,24]. The ωK channel is excluded from the sum, since the ω → π + π − branching fraction is only (1.53 +0.11 −0.13 ) %, compared to the branching fraction (89.2 ± 0.7) % of the dominant decay ω → π + π − π 0 [12]. Most of the K 1 → ωK decays therefore result in a different final state than the simulated one. We account for the K 1 → ωK transitions with a correction to the overall efficiency. In Fig. 2 we show the reference frame chosen to evaluate the distributions of the products of FIG. 2: Definition of (a) the coordinate axes in the K1 rest frame, (b) the angles Θ and Φ in the K1 rest frame, and (c) the angle β in the rest frame of the X s,d intermediate resonance.
Final state particles are labeled with a subscript {k, l, m, n}, according to the following scheme: n for charged B meson decays. The angular distribution for the K 1 system produced in B decays can be expressed in terms of three independent angles (Θ, β, Φ). In the K 1 rest frame, we define the Y axis as the normal to the decay plane of the K 1 , and orient the Z axis along the momentum of l (Fig. 2a). Θ and Φ are then the polar and azimuthal angles of the momentum of k, re-spectively, in the K 1 rest frame (Fig. 2b). We define β as the polar angle of the flight direction of m relative to the direction of the momentum of l (Fig. 2c). The resulting angular parts of the transition amplitudes for S-, P -, and D-wave decays of the K 1 axial vector (J P = 1 + ) mesons with scalar (J P = 0 + ) and vector (J P = 1 − ) intermediate resonances X s,d are given by: (−2 cos Θ cos β + sin Θ sin β cos Φ) . (17) For the ππ and Kπ resonances, the following ℓ-wave Breit-Wigner parameterization is used [24]: where m 0 is the nominal mass of the resonance, Γ(m) is the mass-dependent width, Γ(m 0 ) is the nominal width of the resonance, q is the breakup momentum of the resonance into the two-particle final state, and R 2 = 25 GeV −2 . The K * 0 (1430) and f 0 (1370) amplitudes are also parameterized as Breit-Wigner functions. For the K * 0 (1430) we assume a mass of 1.250 GeV and a width of 0.600 GeV [19], while for the f 0 (1370) we use a mass of 1.256 GeV and a width of 0.400 GeV [25]. This parameterization is varied in Sec. VI and a systematic uncertainty evaluated.

IV. EVENT RECONSTRUCTION AND SELECTION
The B 0 → K + 1 π − candidates are reconstructed in the K + 1 → K + π + π − decay mode by means of a vertex fit of all combinations of four charged tracks having a zero net charge. Similarly we reconstruct B + → K 0 1 π + candidates, with K 0 1 → K 0 S π + π − , by combining K 0 S candidates with three charged tracks. We require the reconstructed mass m Kππ to lie in the range [1.1, 1.8] GeV. Charged particles are identified as either pions or kaons, and must not be consistent with the electron, muon or proton hypotheses. The K 0 S candidates are reconstructed from pairs of oppositely-charged pions with an invariant mass in the range [486, 510] MeV, whose decay vertex is required to be displaced from the K 1 vertex by at least 3 standard deviations.
The reconstructed B candidates are characterized by two almost uncorrelated variables, the energysubstituted mass 20) and the energy difference where (E 0 , p 0 ) and (E B , p B ) are the laboratory fourmomenta of the Υ (4S) and the B candidate, respectively, and the asterisk denotes the CM frame. We require 5.25 < m ES < 5.29 GeV and |∆E| < 0.15 GeV. For correctly reconstructed B candidates, the distribution of m ES peaks at the B-meson mass and ∆E at zero. Background events arise primarily from random combinations of particles in continuum e + e − → qq events (q = u, d, s, c). We also consider cross feed from other B meson decay modes than those in the signal.
To separate continuum from BB events we use variables that characterize the event shape. We define the angle θ T between the thrust axis [26] of the B candidate in the Υ (4S) frame and that of the charged tracks and neutral calorimeter clusters in the rest of the event. The distribution of | cos θ T | is sharply peaked near 1 for qq jet pairs and nearly uniform for B-meson decays. We require | cos θ T | < 0.8. We construct a Fisher discriminant F from a linear combination of four topological variables: 27]. Here, p * i and θ * i are the CM momentum and the angle of the remaining tracks and clusters in the event with respect to the B candidate thrust axis. θ * C and θ * B are the CM polar angles of the Bcandidate thrust axis and B-momentum vector, respectively, relative to the beam axis. In order to improve the accuracy in the determination of the event shape variables, we require a minimum of 5 tracks in each event.
Background from B decays to final states containing charm or charmonium mesons is suppressed by means of vetos. A signal candidate is rejected if it shares at least one track with a B candidate reconstructed in the B 0 → D − π + , B 0 → D * − π + , B + →D 0 π + , or B + →D * 0 π + decay modes, where the D meson in the final states decays hadronically. A signal candidate is also discarded if any π + π − combination consisting of the primary pion from the B decay together with an oppositely charged pion from the K 1 decay has an invariant mass consistent with the cc mesons χ c0 (1P ) or χ c1 (1P ) decaying to a pair of oppositely charged pions, or J/ψ and ψ(2S) decaying to muons where the muons are misidentified as pions.
We define H as the cosine of the angle between the direction of the primary pion from the B decay and the normal to the plane defined by the K 1 daughter momenta in the K 1 rest frame. We require |H| < 0.95 to reduce background from B → V π decay modes, where V is a vector meson decaying to Kππ, such as K * (1410) or K * (1680).
The average number of candidates in events containing at least one candidate is 1.2. In events with multiple candidates, we select the candidate with the highest χ 2 probability of the B vertex fit.
We classify the events according to the invariant masses of the π + π − and K + π − (K 0 S π + ) systems in the K + 1 (K 0 1 ) decay for B 0 (B + ) candidates: events that satisfy the requirement 0.846 < m Kπ < 0.946 GeV belong to class 1 ("K * band"); events not included in class 1 for which 0.500 < m ππ < 0.800 GeV belong to class 2 ("ρ band"); all other events are rejected. The fractions of selected signal events in class 1 and class 2 range from 33% to 73% and from 16% to 49%, respectively, depending on the production parameters ζ. About 11% to 19% of the signal events are rejected at this stage. For combinatorial background, the fractions of selected events in class 1 and class 2 are 22% and 39%, respectively, while 39% of the events are rejected.
The signal reconstruction and selection efficiencies depend on the production parameters ζ. For B 0 modes these efficiencies range from 5 to 12% and from 3 to 8% for events in class 1 and class 2, respectively. For B + modes the corresponding values are 4-9% and 2-7%.

V. MAXIMUM LIKELIHOOD FIT
We use an unbinned, extended maximum-likelihood (ML) fit to extract the event yields n s,r and the parameters of the probability density function (PDF) P s,r . The subscript r = {1, 2} corresponds to one of the resonance band classes defined in Sec. IV. The index s represents the event categories used in our fit model. For the analysis of B 0 modes, these are 1. signal, 2. combinatorial background, The likelihood L e,r for a candidate e to belong to class r is defined as L e,r = s n s,r P s,r (x e ; ζ, ξ), where the PDFs are formed using the set of observables x e = {∆E, m ES , F , m Kππ , |H|} and the dependence on the production parameters ζ is relevant only for the signal PDF. ξ represents all other PDF parameters.
In the definition of L e,r the yields of the signal category for the two classes are expressed as a function of the signal branching fraction B as n 1,1 = B × N BB × ǫ 1 (ζ) and n 1,2 = B × N BB × ǫ 2 (ζ), where the total selection efficiency ǫ r (ζ) includes the daughter branching fractions and the reconstruction efficiency obtained from MC samples as a function of the production parameters.
For the B 0 modes we perform a negative log-likelihood scan with respect to ϑ and φ. Although the events in class r = 2 are characterized by a smaller signal-to-background ratio with respect to the events in class r = 1, MC studies show that including these events in the fit for the B 0 modes helps to resolve ambiguities in the determination of φ in cases where a signal is observed. At each point of the scan, a simultaneous fit to the event classes r = 1, 2 is performed.
For the B + modes, simulations show that, due to a less favorable signal-to-background ratio and increased background from B decays, we are not sensitive to φ over a wide range of possible values of the signal BF. We therefore assume φ = π and restrict the scan to ϑ. At each point of the scan, we perform a fit to the events in class r = 1 only. The choice φ = π minimizes the variations in the fit results associated with differences between the m Kππ PDFs for different values of φ. This source of systematic uncertainty is accounted for as described in Sec. VI. The variations in the efficiency ǫ 1 as a function of φ for a given ϑ can be as large as 30 %, and are taken into account in deriving the branching fraction results as discussed in Sec. VII.
The signal branching fractions are free parameters in the fit. The yields for event categories s = 5, 6 (B 0 modes) and s = 5 (B + modes) are fixed to the values estimated from MC simulated data and based on their previously measured branching fractions [12,28]. The yields for the other background components are determined from the fit. The PDF parameters for combinatorial background are left free to vary in the fit, while those for the other event categories are fixed to the values extracted from MC samples.
The signal and background PDFs are constructed as products of PDFs describing the distribution of each observable. The assumption of negligible correlations in the selected data samples among the discriminating variables has been tested with MC samples. The PDFs for ∆E and m ES of the categories 1, 3, 4, and 5 are each parameterized as a sum of a Gaussian function to describe the core of each distribution, plus an empirical function determined from MC simulated data to account for the tails of each distribution. For the combinatorial background we use a first degree Chebyshev polynomial for ∆E and an empirical phase-space function [29] for m ES : where x ≡ 2m ES / √ s and ξ 1 is a parameter that is determined from the fit. The combinatorial background PDF is found to describe well both the dominant quarkantiquark background and the background from random combinations of B tracks.
For all categories the F distribution is well described by a Gaussian function with different widths to the left and right of the mean. A second Gaussian function with a larger width accounts for a small tail in the distribution and prevents the background probability from becoming too small in the signal F region.
The m Kππ distribution for signal depends on ζ. To each point of the ζ scan, we therefore associate a different nonparametric template, modeled upon signal MC samples reweighted according to the corresponding values of the production parameters ϑ and φ. Production of K * (1410) and a 1 (1260) resonances occurs in B background and is taken into account in the m Kππ and |H| PDFs. For all components, the PDFs for |H| are parameterized with polynomials.
We use large control samples to verify the m ES , ∆E, and F PDF shapes, which are initially determined from MC samples. We use the B 0 → D − π + decay with D − → K + π − π − , and the B + →D 0 π − decay with D 0 → K 0 S π + π − , which have similar topology to the signal B 0 and B + modes, respectively. We select these samples by applying loose requirements on m ES and ∆E, and requiring for the D candidate mass 1848 < m D − < 1890 MeV and 1843 < m D 0 < 1885 MeV. The selection requirements on the B and D daughters are very similar to those of our signal modes. These selection criteria are applied both to the data and to the MC events. There is good agreement between data and MC samples: the deviations in the means of the distributions are about 0.5 MeV for m ES , 3 MeV for ∆E, and negligible for F .

VI. SYSTEMATIC UNCERTAINTIES
The main sources of systematic uncertainties are summarized in Table II. For the branching fractions, the errors that affect the result only through efficiencies are called "multiplicative" and given in percentage. All other errors are labeled "additive" and expressed in units of 10 −6 .
We repeat the fit by varying the PDF parameters ξ, within their uncertainties, that are not left floating in the fit. The signal PDF model excludes fake combinations originating from misreconstructed signal events. Potential biases due to the presence of fake combinations, or other imperfections in the signal PDF model, are estimated with MC simulated data. We also account for possible bias introduced by the finite resolution of the (ϑ, φ) likelihood scan. A systematic error is evaluated by varying the K 1 (1270) and K 1 (1400) mass poles and K−matrix parameters in the signal model, the parameterization of the intermediate resonances in K 1 decay, and the offset phases δ i . We test the stability of the fit results against variations in the selection of the "K * " and "ρ bands," and evaluate a corresponding systematic error. An additional systematic uncertainty originates from potential peaking BB background, including B → K * 2 (1430)π and B → K * (1680)π, and is evaluated by introducing the corresponding components in the definition of the likelihood and repeating the fit with their yields fixed to values estimated from the available experimental information [12]. We vary the yields of the B 0 → a 1 (1260) ± π ∓ and B 0 → D − K + π − π − π + (for the B 0 modes) and B + → K * + ρ 0 (for the B + modes) event categories by their uncertainties and take the resulting change in results as a systematic error. For B + modes, we introduce an additional systematic uncertainty to account for the variations of the φ parameter. The above systematic uncertainties do not scale with the event yield and are included in the calculation of the significance of the result.
We estimate the systematic uncertainty due to the interference between the B → K 1 π and the B → K * ππ + ρKπ decays using simulated samples in which the decay amplitudes are generated according to the results of the likelihood scans. The overall phases and relative contribution for the K * ππ and ρKπ interfering states are assumed to be constant across phase space and varied between zero and a maximum value using uniform prior distributions. We calculate the systematic uncertainty from the RMS variation of the average signal branching fraction and parameters. This uncertainty is assumed to scale as the square root of the signal branching fraction and does not affect the significance. The systematic uncertainties in efficiencies include those associated with track finding, particle identification and, for the B + modes, K 0 S reconstruction. Other systematic effects arise from event selection criteria, such as track multiplicity and thrust angle, and the number of B mesons.

Figures 3 and 4 show the distributions of ∆E, m ES
and m Kππ for the signal and combinatorial background events, respectively, obtained by the event-weighting technique s Plot [30]. For each event, signal and background weights are derived according to the results of the fit to all variables and the probability distributions in the restricted set of variables in which the projection variable is omitted. Using these weights, the data are then plotted as a function of the projection variable.
The results of the likelihood scans are shown in Table III and Fig. 5. At each point of the ζ scan the −2 ln L(B; ζ) function is minimized with respect to the signal branching fraction B. Contours for the value B max (ζ) that maximizes L(B; ζ) are shown in Fig. 5c and   Fig. 5d as a function of the production parameters ζ, for B 0 and B + modes, respectively. The associated statistical error σ B (ζ) at each point ζ, given by the change in B when the quantity −2 ln L(B; ζ) increases by one unit, is displayed in Fig. 5e and Fig. 5f. Systematics are included by convolving the experimental two-dimensional likelihood for ϑ and φ, L ≡ L(B max (ζ); ζ), with a twodimensional Gaussian that accounts for the systematic uncertainties. In Fig. 6a and Fig. 6b we show the resulting distributions in ϑ and φ. The 68% and 90% prob- Results of the ML fit at the absolute minimum of the − ln L scan. The first two rows report the values of the production parameters (ϑ, φ) that maximize the likelihood. The third and fourth rows are the reconstruction efficiencies, including the daughter branching fractions, for class 1 and class 2 events. The fifth row is the correction for the fit bias to the signal branching fraction. The sixth row reports the results for the B → K1(1270)π + K1(1400)π branching fraction and its error (statistical only).   ability regions are shown in dark and light shading, respectively, and are defined as the regions consisting of all the points that satisfy the condition L(r) > x, where the value x is such that L(r)>x L(ϑ, φ)dϑdφ = 68% (90%). The significance is calculated from a likelihood ratio test ∆(−2 ln L) evaluated at the value of ϑ that maximizes the likelihood averaged over φ. Here ∆(−2 ln L) is the difference between the value of −2 ln L (convolved with systematic uncertainties) for zero signal and the value at its minimum for given values of ζ. We calculate the significance from a χ 2 distribution for ∆(−2 ln L) with 2 degrees of freedom. We observe nonzero B 0 → K + 1 π − and B + → K 0 1 π + branching fractions with 7.5σ and 3.2σ significance, respectively.
At each point in the ζ plane we calculate the distributions for the branching fractions, given by f (B; ζ) = cL(B; ζ), where c is a normalization constant. Systematics are included by convolving the experimental one-dimensional likelihood L(B; ζ) with a Gaussian that represents systematic uncertainties. Branching fraction results are obtained by means of a weighted average of the branching fraction distributions defined above, with weights calculated from the experimental two-dimensional likelihood for ϑ and φ.
From the resulting distributions f (B) we calculate the corresponding two-sided intervals at 68% probability, which consist of all the points B > 0 that satisfy the condition f (B) > x, where x is such that f (B)>x, B>0 f (B)dB = 68%. The upper limits (UL) at 90% probability are calculated as 0<B<UL f (B)dB = 90%. The results are summarized in Table IV (statistical only) and Table V (including systematics).

VIII. BOUNDS ON |∆α|
We use the measurements presented in this work to derive bounds on the model uncertainty |∆α| on the weak phase α extracted in B 0 → a 1 (1260) ± π ∓ decays. We use the previously measured branching fractions of B 0 → a 1 (1260) ± π ∓ , B 0 → a 1 (1260) − K + and B + → a 1 (1260) + K 0 decays [4] and the CP −violation asymmetries [6] as input to the method of Ref. [8]. The values used are summarized in Tables VI and VII.
The CP asymmetries A ± CP in B 0 → a ± 1 π ∓ decays are related to the time-and flavor-integrated charge asymmetry A a1π CP [6] by C and ∆C parameterize the flavor-dependent direct CP violation and the asymmetry between the CP -averaged ratesB(a + 1 π − ) andB(a − 1 π + ), respectively [8]:  , and corresponding confidence levels (C.L., statistical uncertainties only). For each branching fraction we provide the mean of the probability distribution, the most probable value (MPV), the two-sided interval at 68% probability, and the upper limit at 90% probability.
The CP -averaged rates are calculated as where B(a ± 1 π ∓ ) is the flavor-averaged branching fraction of neutral B decays to a 1 (1260) ± π ∓ [4].
For the constantλ = |V us |/|V ud | = |V cd |/|V cs | we take the value 0.23 [12]. The decay constants f K = 155.5 ± 0.9 MeV and f π = 130.4 ± 0.2 MeV [12] are experimentally known with small uncertainties. For the decay constants of the a 1 and K 1A mesons the values f a1 = 203 ± 18 MeV [31] and f K1A = 207 MeV [3] are used. For f K1A we assume an uncertainty of 20 MeV. The value assumed for the f K1A decay constant is based on a mixing angle θ = 58 • [3], because f K1A is not available for the value θ = 72 • used here (see Table I); this discrepancy is likely accommodated within the accuracy of the present experimental constraints on the mixing angle. Using naïve arguments based on SU(3) relations and the mixing formulae, we have verified that the dependence of f K1A on the mixing angle is rather mild in the θ range [58, 72] • . It should be noted that due to a different choice of notation, a positive mixing angle in the formalism used by the ACCMOR Collaboration [19] and in this paper corresponds to a negative mixing angle with the notation of Ref. [3].
We use a Monte Carlo technique to estimate a probability region for the bound on |α eff − α|. All the CPaveraged rates and CP -violation parameters participating in the estimation of the bound are generated according to the experimental distributions, taking into account the statistical correlations among A a1π CP , C, and ∆C [28]. For each set of generated values we solve the system of inequalities in Eq. (24), which involve |α + eff − α| and |α − eff − α|, and calculate the bound on |α eff − α| from |α eff − α| ≤ (|α + eff − α| + |α − eff − α|)/2.
The probability regions are obtained by a counting method: we estimate the fraction of experiments with a value of the bound on |α eff − α| greater than a given value. We obtain |α eff − α| < 11 • (13 • ) at 68% (90%) probability.
Finally, we combine the results presented in this paper with existing experimental information to derive an independent estimate for the CKM angle α, based on the time-dependent analysis of CP -violating asymmetries in B 0 → a 1 (1260) ± π ∓ , and find α = (79 ± 7 ± 11) • .

X. ACKNOWLEDGEMENTS
We thank Ian Aitchison for helpful discussions and suggestions. We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity 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. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariatà l'Energie Atomique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Research on Matter (The Netherlands), the Research Council of Norway, the Ministry of Education and Science of the Russian Federation, Ministerio de Educación y Ciencia (Spain), and the Science and Technology Facilities Council (United Kingdom). Individuals have received support from the Marie-Curie IEF program (European Union) and the A. P. Sloan Foundation.