Improved Measurement of $B^+\to\rho^+\rho^0$ and Determination of the Quark-Mixing Phase Angle $\alpha$

We present improved measurements of the branching fraction ${\cal B}$, the longitudinal polarization fraction $f_L$, and the direct {\ensuremath{CP}\xspace} asymmetry {\ensuremath{{\cal A}_{CP}}\xspace} in the $B$ meson decay channel $B^+\to\rho^+\rho^0$. The data sample was collected with the {{\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R}}} detector at SLAC. The results are ${\cal B} (\Bp\ra\rprz)=(23.7\pm1.4\pm1.4)\times10^{-6}$, $f_L=0.950\pm0.015\pm0.006$, and $\Acp=-0.054\pm0.055\pm0.010$, where the uncertainties are statistical and systematic, respectively. Based on these results, we perform an isospin analysis and determine the CKM weak phase angle $\alpha$ to be $(92.4^{+6.0}_{-6.5})^{\circ}$.

PACS numbers: 13.25.Hw, 12.15.Hh,11.30.Er In the Standard Model (SM), the weak interaction cou-plings of quarks are described by elements V ij of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1], where i = u, c, t and j = d, s, b are quark indices. The CKM elements are complex, introducing violation of chargeparity (CP ) symmetry. Unitarity of the CKM matrix yields a relationship between the V ij that can be represented as a triangle in the complex plane. The SM mechanism for CP violations can be tested through measurement of the sides and angles of this unitarity triangle (UT) [2]. An approximate result α eff for the UT angle α = arg (−V td V * tb /V ud V * ub ) can be obtained from B meson decays to CP eigenstates dominated by treelevel b → uūd amplitudes, such as B → ρρ decays (see, e.g., Refs. [2,3]). The correction ∆α = α − α eff , which accounts for loop amplitudes, can be extracted from an analysis of the branching fractions and CP asymmetries of the full set of isospin-related b → uūd channels [4]. One of the most favorable methods to determine α is through an isospin analysis of the B → ρρ system [2,3].
Here, we present updated results for the B + → ρ + ρ 0 channel, with ρ + → π + π 0 and ρ 0 → π + π − , leading to an improved determination of α. Previous studies are presented in Refs. [5,6]. We measure the branching fraction B, the longitudinal polarization fraction f L , and the di- with Γ B ± the B ± decay width. Significant deviation of A CP from the SM prediction of zero could indicate new physics. We also search for the as-yet-unobserved decay B + → ρ + f 0 (980), with f 0 → π + π − . The use of charge conjugate reactions is implied throughout.
The analysis is based on (465 ± 5) × 10 6 BB events (424 fb −1 ) collected on the Υ (4S) resonance [center-ofmass (CM) energy √ s = 10.58 GeV] with the BABAR detector [7] at the PEP-II asymmetric energy e + e − collider at SLAC. Compared to our previous study [5], the analysis incorporates higher signal efficiency and background rejection, twice as much data, and improved procedures to reconstruct charged particles and to account for correlations in the backgrounds. Simulated event samples based on Monte Carlo (MC) event generation are used to determine signal and background characteristics, optimize selection criteria, and evaluate efficiencies. B + → ρ + ρ 0 decays are described by a superposition of two transversely (helicity ±1) and one longitudinally (helicity 0) polarized amplitudes. Our acceptance is independent of the angle between the two ρ decay planes in the B rest frame. We integrate over this angle to obtain an expression for (1/Γ) d 2 Γ/ d cos θ ρ 0 d cos θ ρ + : with f L ≡ Γ L /Γ, where Γ is the total decay width, Γ L is the partial width to the longitudinally-polarized mode, and the ρ 0 (ρ + ) helicity angle θ ρ 0 (θ ρ + ) is the angle between the daughter π + in the ρ 0 (ρ + ) rest frame and the direction of the boost from the B + rest frame.
A B meson candidate is kinematically characterized by the beam-energy-substituted mass m ES ≡ s/4 − (p * B c) 2 /c 2 and energy difference ∆E ≡ E * B − √ s/2, where E * B and p * B are the CM energy and momentum of the B candidate, respectively. Signal events peak at the nominal B mass for m ES and at zero for ∆E, with resolutions of 3 MeV/c 2 and 30 MeV, respectively.
The π 0 mesons are reconstructed through π 0 → γγ. The γ is required to be consistent with a single electromagnetic shower. The γ and π 0 laboratory energies must be larger than 30 MeV and 0.2 GeV, respectively. The mass of a π 0 candidate (resolution 6 MeV/c 2 ) is required to lie within [0.115, 0.150] GeV/c 2 and is subsequently constrained to its nominal value [2].
The dominant background, from random combinations of particles in continuum events (e + e − → qq, with q = u, d, s, c), is suppressed by requiring | cos θ T | < 0.8 [9], with θ T the angle between the thrust axis of the B candidate's decay products and the thrust axis of the remaining particles in the event (ROE), evaluated in the CM frame, and by employing a neural network algorithm based on 11 variables calculated in the CM: | cos θ T |; the cosines of the angles with respect to the beam axis of the B momentum and B thrust axis (we use the absolute value for the latter variable); the momentum-weighted sums L 0 and L 2 [9], determined with charged and neutral particles separately; the sum of transverse momenta of the ROE particles with respect to the beam axis; the ratio of the second to zeroth Fox-Wolfram moments [10]; the proper time difference between the B and B candidates divided by its uncertainty; and B-tagging information from ROE particles [8]. The neural network output N N peaks near 0 and 1 for continuum and signal events, respectively. We require N N > 0.2, which rejects about 5% of the signal and 60% of the continuum events.
The likelihood function is with N the number of events, n j the yield of component j, P j (x i ) the probability density function (PDF) for event i to be associated with component j, and x i the seven experimental observables specified in Eq. (2) below. The signal ρ + ρ 0 , ρ + f 0 , continuum and non-peaking BB background yields are allowed to vary in the fit. The ρ + ρ 0 SxF yield is fixed to its expected value based on the MC prediction for the SxF rate and the B + → ρ + ρ 0 branching fraction determined here (we iterate the fit to find this result). The relative contributions of the ρ + ρ 0 longitudinal and transverse polarization components are determined by allowing f L to vary, with f L common to the signal and SxF events. The three ρππ yields are varied under the requirement that they have the same branching fraction. The π 0 a + 1 , π + a 0 1 , ωρ + , and η ′ ρ + yields are fixed according to their known branching fractions [2]. The π 0 π − π + π + and f 0 π 0 π + yields are fixed assuming their branching fractions to be 10 −5 , consistent with or larger than the limits [11,12] for B 0 → π + π − π + π + and f 0 π + π − decays.
About 85% of continuum events, and 90% of nonpeaking BB background events, contain at least one misreconstructed ρ. For these events, we find correlations of order 10% between the N N , m ππ , and cos θ ρ variables, and -to account for these correlations -construct threedimensional (3D) PDF's of the five variables based on conditional PDF's P(x|y) of variable x given the value of variable y: P 3D = [P(m π + π − | cos θ ρ 0 )×P(cos θ ρ 0 |N N )]× [P(m π + π 0 | cos θ ρ + ) × P(cos θ ρ + |N N )] × P(N N ). For example, P(m π + π 0 | cos θ ρ + ) is constructed by examining the m π + π 0 distribution in nine bins of cos θ ρ + , fitting a second order polynomial to each bin, and parameterizing how the coefficients of the polynomial vary between bins. The fraction of events with a correctly reconstructed ρ + and ρ 0 is fixed to the MC prediction for the non-peaking BB background and allowed to vary for the continuum background. For all other components, the overall PDF's are defined as the product of seven 1D PDF's, one for each observable. The PDF's of the ρ + ρ 0 signal and SxF helicity angles take the form of Eq. (1), with detector resolution and acceptance incorporated, by summing the longitudinal (L) and transverse (T ) components with a relative fraction f L ǫ L /[f L ǫ L + (1 − f L )ǫ T ], with ǫ L and ǫ T the respective reconstruction efficiencies, leading to an effective 2D PDF in cos θ ρ + and | cos θ ρ 0 |: The continuum background m ES and ∆E PDF's are derived from a 44 fb −1 data sample collected 40 MeV below the Υ (4S) mass. All other PDF's are derived from simulation. For m ES , the PDF's of signal and continuum are parameterized by a Crystal Ball [13] and an ARGUS function [14], respectively. A relativistic Breit-Wigner function with a p-wave Blatt-Weisskopf form factor is used for the m ππ distributions in ρ + ρ 0 signal events. For the background, m ππ is modeled by a combination of a polynomial and the signal function. Slowly varying distributions (∆E for non-peaking backgrounds, and cos θ ρ ) are modeled by polynomials. High statistics histograms are used for the N N distributions. The remaining variables are parameterized with sums of Gaussians, e.g., the m ππ distribution in f 0 decays is modeled with a sum of three Gaussians. A large data control sample of B + → D 0 π + (D 0 → K 0 S π 0 , K 0 S → π + π − ) events is used to verify that the resolution and peak position of the signal m ES and ∆E PDF's are accurately simulated.
The fit is applied to the sample of 82,224 selected events. We allow 11 parameters to vary in the fit: five parameters of continuum background PDF's, f L , and five yields as mentioned above. We find 1122 ± 63 (stat.) ρ + ρ 0 signal events, 50 ± 30 (stat.) ρ + f 0 events, and f L = 0.945 ± 0.015 (stat.). The fit provides a simultaneous determination of the number of B + → ρ + ρ 0 and B − → ρ − ρ 0 signal events. These fitted yields are used to determine A CP = −0.054 ± 0.055 (stat.). Fig. 1 shows projections of the m ES and m π + π − distributions. To enhance the visibility of the signal, events are required to satisfy is the sum of the likelihood functions for ρ + ρ 0 and ρ + f 0 signal events excluding the PDF of the plotted variable i, and L i (B) is the corresponding sum of all other components.
A possible bias, from unmodeled correlations, is evaluated by applying the ML fit to an ensemble of simulated experiments, where the numbers of signal and background events in each component correspond to those observed or fixed in the fit to data. The continuum events are drawn from the PDF's, while events for all other components are drawn from MC samples. The biases are determined to be 71 ± 3 and −31 ± 1 events for the signal ρ + ρ 0 and ρ + f 0 yields, and −0.005 ± 0.001 for f L , where the uncertainties are statistical. The signal yields and f L are then corrected by subtracting these biases.
The branching fractions are given by the bias-corrected m π + π − variables. A requirement on the likelihood ratio that retains 38% of the signal, 0.1% of the continuum background, and 1.3% of the BB background has been applied. The peak in the BB background at m π + π − ≈ 0.78 GeV/c 2 is from B + → ρ + ω events with ω → π + π − . yields divided by the reconstruction efficiencies and initial number of BB pairs N BB . From the simulations, the ρ + ρ 0 signal efficiencies including the π 0 daughter branching fraction [2] are ǫ L =[9.12±0.02 (stat.)]% and ǫ T =[17.45±0.03 (stat.)]%. The corresponding result for ρ + f 0 is [14.20±0.08 (stat.)]%. We assume that the Υ (4S) decays to each of B + B − and B 0 B 0 50% of the time.
The principal systematic uncertainties associated with the ML fit are listed in Table I. Uncertainties from the fit biases are defined by the quadratic sum of half the biases themselves (for f L , the full bias) and the statistical uncertainties of the biases. The uncertainties related to the signal and non-peaking BB background PDF's are assessed by varying the PDF parameters within their uncertainties. For the signal, the uncertainties of the PDF parameters are determined from the B + → D 0 π + data control sample. Variations of the π 0 a + 1 , π + a 0 1 , ωρ + , and η ′ ρ + branching fractions within their measured uncertainties, and of the assumed π + π − π + π 0 and f 0 π + π 0 branching fractions by ±100%, define the systematic uncertainty associated with the peaking BB background. The uncertainty associated with the SxF fraction is assessed by varying the fixed SxF yield by ±10%. The other principal sources of systematic uncertainty are the π 0 reconstruction efficiency (3.0%), the track reconstruction efficiency (1.1%), the π ± identification efficiency (1.5%), the uncertainty of N BB (1.1%), and the selection requirements on | cos θ T | (1.0%). The individual terms are added in quadrature to define the total systematic uncertainties.
We perform an isospin analysis of B → ρρ decays by minimizing a χ 2 that includes the measured quantities expressed as the lengths of the sides of the B and B isospin triangles [4]. We use the B + → ρ + ρ 0 branching fraction and f L results presented here, with the branching fractions, polarizations, and CP -violating parameters in B 0 → ρ + ρ − [15] and B 0 → ρ 0 ρ 0 [11] decays. We assume the uncertainties to be Gaussian-distributed and neglect potential isospin I = 1 and electroweak-loop amplitudes, which are expected to be small [3].
In summary, we have improved the precision of the measurements of the B + → ρ + ρ 0 decay branching and longitudinal polarization fractions, leading to a significant improvement in the determination of the CKM phase angle α based on the favored B → ρρ isospin method. We set a 90% CL upper limit of 2.0×10 −6 on the branching fraction of B + → ρ + f 0 (980) with f 0 → π + π − .
We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and kind hospitality.