A Study of B0 to rho+rho- Decays and Constraints on the CKM Angle alpha

We present results from an analysis of B0 to rho+rho- decays using 383.6 million BB-bar pairs collected by the BaBar detector at the PEP-II asymmetric-energy B Factory at SLAC. The measurements of the B0 to rho+rho- branching fraction, longitudinal polarization fraction f_L, and the CP-violating parameters S_{long} and C_{long} are: B(B0->rho+rho-) = (25.5 +/- 2.1 (stat) +3.6/-3.9 (syst))*10^-6 f_L = 0.992 +/- 0.024 (stat) +0.026/-0.013 (syst) S_{long} = -0.17 +/- 0.20 (stat) +0.05/-0.06 (syst) C_{long} = 0.01 +/- 0.15 (stat) +/- 0.06 (syst) We determine the unitarity triangle angle alpha, using an isospin analysis of B to rhorho decays. One of the two solutions, alpha = [73.1, 117.0] degrees at 68% CL is compatible with standard model-based fits of existing data. Constraints on the unitarity triangle are also evaluated using an SU(3) symmetry based approach.


I. INTRODUCTION
Charge conjugation-parity (CP ) violation was first seen in the decay of neutral kaons [1]. It was shown some forty years ago that CP violation is a necessary but insufficient condition required to generate a net baryon anti-baryon asymmetry in the universe [2]. The standard model (SM) of electroweak interactions describes CP violation as a consequence of a complex phase in the threegeneration Cabibbo-Kobayashi-Maskawa (CKM) quarkmixing matrix [3,4]: (1) V CKM describes the couplings of the u, c and t quarks to d, s and b quarks, which are mediated by the exchange of a W boson. In B-meson decays the CP violating parameters of the SM are most directly related to the angles and sides of the so-called Unitarity Triangle (UT), shown in Fig. 1. The angles α, β, and γ are defined as Any non-trivial phase in V ij is CP violating. CP violating phases originating from the CKM matrix are related to weak interactions, and therefore referred to as weak phases. In the Wolfenstein convention [5], the angle γ is the phase of V ub , β is the phase of V td , and α is the phase difference between V ub and V td constrained to satisfy α = π − β − γ through the unitarity of V CKM . In Fig. 1 the UT is shown in the complex (ρ, η) plane, where the apex is given by the phase-convention independent definition [6]: (5) * Deceased † Also with Università di Perugia, Dipartimento di Fisica, Perugia, Italy ‡ Also with Università della Basilicata, Potenza, Italy § Also with IPPP, Physics Department, Durham University, Durham DH1 3LE, United Kingdom The quest to understand CP violation remains, despite its observation in the B meson system by both the BABAR [7] and Belle experiments [8], since the SM does not, through the CKM phase, incorporate enough CP violation to explain the current matter-antimatter asymmetry [9,10]. The CKM angle β is measured to a 1 • precision with b → ccs decays [11,12] and is consistent with current predictions [13][14][15]. A significant deviation from results of SM-based fits of existing data for any of the UT angles would be a clear indication of new physics [16,17].
In principle, direct experimental measurements of α can be made from decays that proceed mainly through a b → uud tree diagram such as B 0 → ρ + ρ − , ρ ± π ∓ , π + π − and a 1 π [18,19]. Interference between the direct decay and decay after B 0 B 0 mixing in each of these decays results in a time-dependent decay-rate asymmetry that is sensitive to the angle α. Figure 2 shows the leading order tree and gluonic penguin loop contributions to the decay B 0 → ρ + ρ − . The presence of penguin contributions complicates the extraction of α from these decays. Using isospin relations [20], measurements of the B + → ρ + ρ 0 [21,22] and B 0 → ρ 0 ρ 0 [23] branching fractions show that the penguin contribution in B → ρρ is smaller than the leading tree diagram. The use of SU (3) flavor symmetry to increase the precision on the weak phase constrained using B 0 → ρ + ρ − and B + → K * 0 ρ + decays has also been proposed [24]. Both of these approaches are discussed in Section X. Section X and Ref. [25] describe a number of possible sources of theoretical uncertainty. In B 0 → ρ + ρ − decays, a spin 0 particle (the B 0 meson) decays into two spin 1 particles (ρ ± mesons), as shown in Fig. 3. Subsequently each ρ ± meson decays into a π ± π 0 pair. As a result, the CP analysis of B 0 → ρ + ρ − is complicated by the presence of one amplitude with longitudinal polarization and two amplitudes with transverse polarization. The longitudinal amplitude is CP -even, while the transverse amplitudes contain CP -even and CP -odd states. The decay is observed to be dominated by the longitudinal polarization [26,27], with a fraction f L defined as the fraction of the helicity zero state in the decay. Integrating over the angle between the ρ decay planes φ, the angular decay rate is d 2 Γ Γd cos θ 1 d cos θ 2 = 9 4 f L cos 2 θ 1 cos 2 θ 2 where the helicity angles θ i=1,2 are the angles between the π 0 momentum and the direction opposite to that of the B 0 in the ρ rest frame. We identify (tag) the initial flavor of the reconstructed B candidate, B rec , using information from the other B meson, B tag , in the event. The decay rate f + (f − ) for a neutral B meson decaying into a CP eigenstate, where the B tag is a B 0 (B 0 ), is given by: Here ∆t is the proper time difference between the decay of the B rec and B tag mesons in an event, τ B 0 is the mean B 0 lifetime, ∆m d is the B 0 -B 0 mixing frequency [28], and η f is the CP eigenvalue of the decay. This assumes that there is no difference between the B 0 and B 0 lifetimes, ∆Γ = 0. The parameters S and C describe B-mixing induced and direct CP violation, respectively. For the longitudinal polarization η f = +1, S = S long and C = C long , are defined as where λ CP = q p A A [29], q and p are complex constants that relate the B meson flavor eigenstates to the mass eigenstates, and A/A is the ratio of amplitudes of the decay of a B 0 or B 0 to the final state under study. CP violation is probed by studying the time-dependent decay-rate asymmetry where Γ(Γ) is the decay-rate for B 0 (B 0 ) tagged events. This asymmetry has the form The transverse polarization is an admixture of CP -even (η f = +1) and CP -odd (η f = −1) parts, where each CP eigenstate has a decay-rate distribution of the form of Eq. (7). As the longitudinal polarization dominates this decay, we describe the CP admixture of the transverse polarization with common parameters S tran and C tran . We set S tran = C tran = 0, and vary these parameters when evaluating systematic uncertainties.
In the absence of penguin contributions in B 0 → ρ + ρ − , S long = sin 2α and C long = 0. The presence of penguin contributions with different weak phases to the tree level amplitude shifts the experimentally measurable parameter α eff away from the value of α. In the presence of penguin contributions α eff = α + δα, where and C long can be non-zero. For B + → ρ + ρ 0 one measures a time-integrated charge asymmetry, which is analogous to Eq. (10) and is defined as where N (N ) is the number of B (B) decays to this final state. Table I summarizes the most recent measurements of the complete set of experimental inputs needed to extract α from the B → ρρ decays.
In this article, we present an update of the timedependent analysis of B 0 → ρ + ρ − and measurement of the CKM angle α reported in Ref. [26] and branching fraction reported in Ref. [30]. Improvements to the analysis reported here include an increase in data sample analyzed, a tighter constraint on the proper time difference between the two B meson decays in selected events, an improved algorithm to determine the flavor of B mesons, a modified multivariate analyzer for background suppression, and an improved description of the signal and background model.  (B) are in units of 10 −6 . The column marked C long (ACP ) denotes the time dependent charge asymmetry C long in the case of the B 0 → ρ + ρ − decay, and the time integrated charge asymmetry ACP in the case of B ± → ρ ± ρ 0 .

II. THE DATASET AND BABAR DETECTOR
The results presented in this paper are based on data collected with the BABAR detector at the PEP-II asymmetric e + e − storage ring [32] operating at the Stanford Linear Accelerator Center. At PEP-II, 9.0 GeV electrons and 3.1 GeV positrons are collided at a centerof-mass energy of 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 center-ofmass (CM) frame of βγ ≈ 0.56. The dataset analyzed has an integrated luminosity of 349 fb −1 corresponding to (383.6 ± 4.2)×10 6 BB pairs recorded at the Υ (4S) resonance (on-peak). An additional 27.2 fb −1 of data were recorded about 40 MeV below this energy (off-peak) for the study of continuum background, where light quarks are produced in the process e + e − → qq (q = u, d, s, c).
The BABAR detector is described in detail elsewhere [33]. Surrounding the interaction point is a five double-sided layer silicon vertex tracker (SVT) which measures the impact parameters of charged particle tracks in both the plane transverse to, and along the beam direction. A 40-layer drift chamber (DCH) surrounds the SVT and provides measurements of the momenta for charged particles. Both the SVT and DCH are surrounded by a solenoid magnet, that provides a 1.5 T magnetic field. 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 (EMC) provides photon detection, electron identification, and π 0 reconstruction. Finally, the instrumented flux return of the magnet allows discrimination of muons from pions. For the most recent 134.0 fb −1 of data, a portion of the resistive plate chambers constituting the muon system has been replaced by limited streamer tubes [34][35][36].
We use a right-handed coordinate system with the z axis along the electron beam direction and the y axis upward, with the origin at the nominal beam interaction point. Unless otherwise stated, kinematic quantities are calculated in the laboratory rest frame. The other reference frame which we commonly use is the CM frame of the colliding electrons and positrons.
We use Monte Carlo (MC) simulated events generated using the GEANT4 [37] based BABAR simulation.

III. RECONSTRUCTION OF B CANDIDATES
A. Photon and π 0 reconstruction Photons are reconstructed from localized energy deposits in the EMC that are not associated with a charged track. We require photon candidates to have an energy greater than 50 MeV, and a lateral shower profile [38] to be consistent with the photon hypothesis. We reconstruct neutral pions from combinations of two distinct photon candidates where the invariant γγ mass m γγ satisfies 0.10 < m γγ < 0.16 GeV/c 2 .

B. ρ ± reconstruction
We combine reconstructed π 0 mesons with charged tracks that are consistent with the π ± hypothesis to form ρ ± candidates. The invariant mass m π ± π 0 of the reconstructed ρ ± is required to lie between 0.5 and 1.0 GeV/c 2 , to select events in the vicinity of the ρ resonance. We require that the helicity angle of each ρ meson satisfies −0.90 < cos θ i < 0.98. This selection criteria suppresses continuum and B backgrounds.

C. B 0 reconstruction and event selection
We combine two oppositely charged ρ candidates to form the B candidate B rec . All particles in the rest of the event (ROE) are combined to form the other B meson in the event B tag . In addition to the two charged tracks in the B rec , we require that there is at least one track in the B tag .
In order to suppress potential backgrounds from e + e − → e + e − , µ + µ − events, we require the secondto-zeroth Fox-Wolfram moment R 2 [39] of the event to be less than 0.98. Continuum events are the dominant background which is reduced by requiring the absolute value of the cosine of the angle between the B rec thrust axis (T B) and that of the ROE (T R) to satisfy | cos(T B, T R)| < 0.8. We retain 17.1% and 20.1% of longitudinal and transverse signal, respectively, by requiring the aforementioned selection criteria.
We calculate ∆t = ∆z/βγc from the measured separation ∆z between the B rec and B tag vertices [40]. We determine the B rec vertex from the two charged-pion tracks in its decay. The B tag decay vertex is obtained by fitting the other tracks in the event, with constraints from the B rec momentum and the beam-spot location. The RMS resolution on ∆t is 1.1 ps. We only use events that satisfy |∆t| < 15 ps and for which the error on ∆t (σ ∆t ) is less than 2.5 ps.
Signal events are identified using two kinematic variables, the difference ∆E between the CM energy of the B rec , E * B , and √ s/2, and the beam-energy-substituted mass, where √ s is the total CM energy. The B rec momentum p B and four-momentum of the initial state (E i , p i ) are defined in the laboratory frame. We accept candidates that satisfy 5.25 < m ES < 5.29 GeV/c 2 and −0.12 < ∆E < 0. 15 GeV. An asymmetric ∆E selection is used in order to reduce backgrounds from higher-multiplicity B decays. The resolution on m ES is dominated by the knowledge of the energy of the e + and e − beams, and the resolution on ∆E is dominated by the reconstruction performance of the EMC.
After the selection criteria mentioned above have been applied, the average number of candidates per event is approximately 1.5. In events with more than one candidate, we select the candidate that minimizes the sum of (m γγ − m π 0 ) 2 where m π 0 is the true π 0 mass [28]. In 0.3% of events, the same π 0 mesons are used by multiple B candidates. In such an event we randomly select the candidate to keep.

IV. CONTINUUM BACKGROUND SUPPRESSION
In addition to the cuts on cos θ i , R 2 , and | cos(T B, T R)| that directly remove background events, we use an artificial neural network in order to discriminate between continuum background and signal events.
For this purpose we combine the following eight variables into a single output, N .
• The coefficients, L 0 , L 2 , split into sums over the ROE for neutral and charged particles; L 0,n , L 2,n and L 0,c , L 2,c . The coefficients are defined as L k = p j | cos(ψ j )| k , where k = 0, 2, p j is the particle momentum and ψ j is the angle of the particle direction relative to the thrust axis of the B candidate. Both p j and ψ j are defined in the CM frame.
• | cos(B, Z)|, the absolute value of the cosine of the angle between the direction of the B and z axis in the CM frame. This variable follows a sine squared distribution for BB events, whereas it is almost uniform for qq.
• | cos(T B, T R)|. This variable, previously defined in Section III C, is strongly peaked at unity for qq events. BB events are more isotropic as the B mesons are produced close to the kinematic threshold.
• | cos(T B, Z)|, the absolute value of the cosine of the angle between the B thrust and the z axis.
• The scalar sum of the transverse momenta p t in the ROE. This sum includes neutral and charged particles.
The distributions of these input variables are shown in Fig. 4. Figure 5 shows N for signal MC simulated events and continuum background samples (off-peak data) and the efficiency for signal and qq background as a function of cut on the minimum value of N . We require N to be greater than −0.4. Note that later, we use this variable in the maximum likelihood fit described in Section VII. The samples used to train the neural net were correctly reconstructed MC simulated events and off-peak data. To avoid over-training, we used an independent sample of these data (i.e. distinct from the sample used for the training) to check the performance of the network. The training is stopped when the separation between the signal and continuum background is stable.

V. MIS-RECONSTRUCTED SIGNAL AND SELECTION EFFICIENCY
Mis-reconstructed signal candidates, referred to as selfcross-feed (SCF) signal, may pass the selection requirements even if one or more of the pions assigned to the ρ + ρ − state belongs to the other B in the event. These SCF candidates constitute 50.7% (27.9%) of the accepted longitudinally (transversely) polarized signal. The majority of SCF events have both charged pions from the ρ + ρ − final state, and unbiased CP information. These correct (right) track SCF events are denoted by RT SCF. There is a SCF component (13.8% of the signal) where at least one track in B rec is from the ROE. These wrong track (WT) events are used to determine the signal yield N signal and f L but have biased CP information, and are treated separately for the CP result. A systematic error is assigned to the CP results from this type of signal event. The total selection efficiency for longitudinally (transversely) polarized signal is 7.7% (10.5%).

VI. B MESON FLAVOR TAGGING
A key ingredient in the measurement of timedependent CP asymmetries is to determine whether at the time of decay of the B tag , corresponding to ∆t = 0, the B rec was a B 0 or a B 0 . This 'flavor tagging' is achieved with the analysis of the decay products of the recoiling B meson B tag . The overwhelming majority of B mesons decay to a final state that is flavor-specific, i.e. only accessible from either a B 0 or a B 0 , but not from both. The purpose of the flavor tagging algorithm is to determine the flavor of B tag with the highest possible efficiency ǫ tag and lowest possible probability w of assigning a wrong flavor to B tag . It is not necessary to fully reconstruct B tag in order to determine its flavor.
In the presence of a finite mistag probability w, the CP asymmetry is reduced by a dilution factor 1 − 2w, so Eq.
The figure of merit for the performance of the tagging algorithm is the effective tagging efficiency which is related to the statistical uncertainty σ in the coefficients S and C through We use a neural network based technique [11,40] that isolates primary leptons, kaons and pions from B decays to final states containing D * mesons, and high momentum charged particles from B decays, to determine the flavor of the B tag . The output of this algorithm is divided into seven mutually-exclusive categories. These are (in order of decreasing signal purity) Lepton, Kaon I, Kaon II, Kaon-Pion, Pion, Other and Untagged. The performance of this algorithm is determined using fully reconstructed neutral B decays to D ( * )− (π + , ρ + , a + 1 ) and is summarized in Table II. The categories assigned correspond to events with leptons, kaons and pions in the decay products of B tag . The Untagged category of events contain no flavor information and therefore carry no weight in the time-dependent analysis. In addition, these events are not considered useful for the branching fraction measurement since they are dominated by continuum background. The total Q of this algorithm is 32.7 ± 0.7%.

VII. LIKELIHOOD FIT METHOD
On applying the selection criteria described above, we obtain a sample of 37424 events that enter the fit. These events are dominated by backgrounds from qq (81.4%) and BB (16.6%) events. The remaining 2% of events are considered as signal. We distinguish between the following components in the fit: (i) correctly reconstructed signal, (ii) SCF signal, split into RT and WT parts, The dominant B backgrounds come from components (iii) and (iv). The signal, continuum and B background models are described in Sections VII A, VII B, and VII C, respectively. We use an unbinned, extended maximum likelihood (ML) fit to extract N signal , f L , S long , and C long . The likelihood function incorporates the following previously defined eight discriminating variables to distinguish signal from the backgrounds: m ES , ∆E, ∆t, N , and the m k and cos θ k values of the two ρ mesons, where k = 1, 2. For each of the aforementioned components j we construct a probability density function (PDF) that is the product of one-dimensional PDFs for each of the variables. The PDFs do not account for all possible correlations among the discriminating variables and we account for possible biases as a result of neglecting these correlations as discussed in Section VIII. For each event i, the PDFs can be written as We determine the PDFs for signal and each of the B background components from MC distributions for each discriminating variable. The PDFs for the continuum background are determined from on-peak and off-peak data. The likelihood function is where n j are the event yields of hypotheses j (determined from the ML fit) and N is the observed number of events in the sample. The normalized exponent takes into account Poisson fluctuations in the total number of events. We simultaneously fit events in the mutually-exclusive flavor tag categories listed in Table II, excluding events in the Untagged category. There are six continuum background yields, one for each flavor tag category, and we use a single yield for the signal and each of the B backgrounds, while accounting for the relative fractions of events expected in each of the flavor tag categories. The fit results are obtained by minimizing the value of − ln L.

A. Signal model
The signal has contributions from true and SCF longitudinally (long) and transversely (tran) polarized events. In addition to this, the longitudinally polarized SCF signal PDF is further sub-divided into the two categories of mis-reconstructed signal: RT and WT SCF signal, and all signal PDFs are sub-divided according to the B tag flavor tag category. The total signal PDF is given by where f long RT(WT) SCF is the fraction of longitudinally polarized RT(WT) SCF signal and f tran SCF is the fraction of transversely polarized SCF signal. The PDFs P long RT(WT) SCF and P tran SCF are defined accordingly. In order to extract the observable f L from the fit, we account for the different reconstruction efficiencies for longitudinally and transversely polarized signal using where f obs L is the fraction of signal from longitudinally polarized events in the data, and R is the ratio of longitudinal to transverse polarized signal efficiencies.
All of the SCF PDFs are further sub-divided into parts that contain correctly reconstructed and misreconstructed ρ mesons. Four sub-divisions are used, corresponding to both mesons being correctly reconstructed (T T ), one being correctly reconstructed and the other being mis-reconstructed (T F or F T ), and both mesons being mis-reconstructed (F F ). Here the first (second) ρ meson is positively (negatively) charged. The SCF PDFs have the form where T and F are correctly and mis-reconstructed ρ mesons, respectively, f T T , f T F , f F T , f F F are the fractions of each type of mis-reconstructed event, and the PDFs for m ES , ∆E, and the helicity and mass for each reconstructed ρ meson are the products of one dimensional PDFs. The signal PDFs for N and ∆t have different shapes according to the B tag flavor tag category assigned to an event.
The longitudinally polarized true and RT SCF signal m ES distribution are described by a Gaussian with a power law tail [41][42][43] which takes the following form where m 0 is the mean, σ is the width of the Gaussian part, and the parameters ξ and n describe the exponential tail. The remaining PDFs for the signal m ES distributions are the sum of a Gaussian and a Gaussian with an exponential tail. The signal ∆E distributions for the longitudinally polarized SCF events are described by first and second order polynomials, with the exception of T T events, which are modeled with the sum of a second order polynomial and a Gaussian. The other ∆E PDFs are described by the sum of Gaussian and a Gaussian with an exponential tail. The N distributions are modeled using smoothed histograms of MC simulated events. The signal ∆t distribution is described by Eq. (16) convolved with a triple Gaussian resolution function given by where G is a Gaussian with means µ l and width σ l for l = core, tail, and outlier. The parameters σ tail , σ outlier and µ outlier are set to 3.0 ps, 8.0 ps and 0.0 ps, respectively. The remaining parameters of the resolution function are determined from signal MC simulated events scaled by the differences between data and MC simulated events for large samples of exclusively reconstructed B decays to D ( * )− (π + , ρ + , a + 1 ) final states described in Section VI. The values of the mean and width of the core Gaussian contribution to the resolution function depend on the flavor tagging category of an event. There are three signal ∆t PDFs, one for the true and RT SCF longitudinally polarized signal, one for the WT SCF longitudinally polarized signal and one for the transversely polarized signal. Each of these PDFs has distinct values of S and C as described in Sections I and V. The WT SCF longitudinally polarized signal has a different resolution function with respect to the rest of the signal. The cos θ i distribution for true signal events is described by the expected cosine squared or sine squared distribution multiplied by a polynomial acceptance function. The cos θ i PDFs for T ρ mesons in longitudinally (transversely) polarized SCF signal are described by sixth order polynomials (smoothed histograms of MC simulated events). The cos θ i PDFs for mis-reconstructed ρ mesons are described by smoothed histograms of MC simulated events. The longitudinally polarized signal ρ mass PDFs are described using relativistic Breit-Wigners and third order polynomials for T and F ρ contributions, respectively. The transversely polarized signal SCF ρ mass PDF distributions are described using smoothed histograms of MC simulated events.
In addition to N signal , f L , S long , and C long , we determine the mean of the core Gaussian part of the m ES and ∆E distributions from the fit. We obtain parameters consistent with expectations from MC simulated events.

B. Continuum background model
The continuum background PDFs are also sub-divided according to the mis-reconstruction of ρ mesons in the final state and have the form P = P(m ES )P(∆E)P(N )P(∆t) where helicity and mass PDFs for each reconstructed ρ meson are the products of one dimensional PDFs. The continuum distribution for m ES is described by a phase-space-motivated distribution [44] with the following form where θ(m ES ) = 1 for m ES ≤ √ s/2 and θ(m ES ) = 0 for m ES > √ s/2 and x = 2m ES / √ s. The ∆E and N shapes are modeled with third and fourth order polynomials, respectively. The parameters of the m ES , ∆E and N shapes are allowed to vary in the fit to the on-peak data. The continuum ρ mass distribution is described using a relativistic Breit-Wigner and a third order polynomial PDF for T and F ρ contributions, respectively. The cos θ i distribution is described by a third order polynomial. The continuum ∆t distribution has a prompt lifetime component convolved with a triple Gaussian resolution function. The parameters of the ρ mass and helicity distributions are obtained from a fit to the off-peak data, and the remaining parameters are determined in the fit.
C. B background model

Charm B backgrounds
Combinatorial events from b → c transitions are the dominant B backgrounds. These components have shapes similar to continuum and do not peak in the signal region for the discriminating variables. The functional form used for the PDFs of these background components is given by Eq. (26). We parameterize the m ES and ∆E distributions of these backgrounds using the phasespace-motivated distribution of Eq. (27), and a third order polynomial, respectively. The ρ mass distribution is described using a relativistic Breit-Wigner and a third order polynomial PDF for T and F ρ contributions, respectively. The remaining PDFs are described using smoothed histograms of MC simulated events. Each of the PDF parameters for these backgrounds are determined from samples of MC simulated events, and the yields of these components are determined in the fit. When studying systematic uncertainties we modify the PDF used for ∆t so that it has a form similar to the signal which uses the resolution function of Eq. (25). An effective lifetime that is smaller than τ B 0 is used to account for mis-reconstruction of these events and the finite charm meson lifetime. The value of this parameter is obtained by fitting MC simulated events for this category of events.

Charmless B backgrounds
Some of the charmless B backgrounds have PDFs similar to the signal for one or more of the discriminating variables, so it is important to correctly account for such events in the fit. We consider the 20 exclusive and 2 inclusive components of this type of B background listed in Table III. If a charmless B background decay contributes an event yield more than 1% of the expected signal yield, we model that mode exclusively. The remaining 140 charmless background components considered were combined with the appropriate weightings to form the neutral and charged inclusive charmless B background components. Where possible we use branching fractions from existing measurements. Where measurements are not available, we have either tried to use SU (2) and SU (3) flavor symmetries to relate an unmeasured decay to a measured one, or where this is not possible, we have assigned a branching fraction of 10 ×10 −6 to a decay mode. An uncertainty of 100% is assumed on all  [45,46] 55 ± 3 11 ± 1 B + → π + π 0 π 0 10 ± 10 15 ± 15 B + → π 0 ρ + [47][48][49] 11 ± 2 73 ± 10 B + → ρ + ρ 0 [21,22] † 17 ± 3 71 ± 12 B + → ρ + π + π − 10 ± 10 9 ± 8 B + → K * 2 (1770)ρ 10 ± 10 9 ± 9 B + → K * 0 π[45, 46, 49] 11 ± 1 9 ± 1 B + → a + 1 π 0 20 ± 20 60 ± 60 B + → ρ + π 0 π 0 10 ± 10 12 ± 12 extrapolated branching fractions. We assume that the a 1 meson decays into a three pion final state via ρπ. The decay B 0 → a 0 1 π 0 is penguin dominated and is expected to have a small branching fraction, as is the case for the penguin dominated decays B 0 → ρ 0 π 0 and B 0 → ρ 0 ρ 0 . As a result, we have neglected possible contributions from this potential source of background. Although we don't see evidence for B 0 → ρ ∓ π ± π 0 , we do consider this channel as a potential source of background. Contribution from B 0 → 4π is expected to be even smaller and is neglected. When considering the systematic uncertainty arising from interference between signal and other π + π − π 0 π 0 final states, we assume that the non-resonant π + π − π 0 π 0 final state has the same branching fraction as that of B 0 → ρ ∓ π ± π 0 , where B(B 0 → ρ ∓ π ± π 0 ) is calculated from the yield determined in the B 0 → ρ + ρ − nominal fit. This is discussed further in the Appendix.
The functional form used for the PDFs of these charmless B background components is given by Eq. (26). The ∆E distributions are described by third order polynomials, except for non-resonant B 0 → ρ ∓ π ± π 0 which uses smoothed histograms of MC simulated events. The m ρ distributions for true ρ mesons are parameterized using a relativistic Breit-Wigner, and the fake m ρ (combinatorial π ± π 0 ) distribution is described using third order polyno-mials. The remaining background shapes are described using smoothed histograms of MC simulated events. The yield for B 0 → ρ ∓ π ± π 0 decays is allowed to vary in the fit. All other charmless background yields are fixed to expectations. This constraint is relaxed when studying possible sources of systematic uncertainties. When studying systematic uncertainties from possible CP violation in the B background, we modify the PDF used for ∆t so that it has a form similar to the signal which uses the signal resolution function in Eq. (25).

VIII. LIKELIHOOD FIT VALIDATION
Before applying the fitting procedure to the data, we subject it to various tests. The aim of these tests are to verify that one can extract the signal observables N signal , f L , S long , and C long in a controlled way. Consistency of the likelihood fit is checked with ensembles of experiments simulated from the PDFs. The event yields are generated according to a Poisson distribution with mean n j for each category in these ensembles. In each of these tests we verify that the values generated for the signal observables are reproduced with the expected resolution. The distribution of − ln L for the ensemble of experiments in comparison to that obtained when fitting the data provides an additional, but limited, cross-check of the consistency.
The PDFs used in the likelihood do not account for all possible correlations among the discriminating variables. We account for possible biases as a result of neglecting these correlations by fitting ensembles of experiments obtained from samples of signal and the charmless B background MC simulated events combined with charm backgrounds and qq background events simulated from the PDFs. The MC simulated events used in these ensembles have these correlations modeled. We find a positive bias of 38.5 ± 5.3 events on N signal , and a negative bias of 0.016 ± 0.002 on f L , and we do not observe a significant bias on S long and C long .
As continuum events are the dominant background, we apply the fitting procedure to the off-peak data (after correcting for the difference in √ s) to confirm that we do not find a fake signal in this control sample of events. We fit 790 off-peak events and extract signal and continuum yields of 8 ± 7 and 782 ± 28 events, respectively. A blind analysis technique has been used for the extraction of S long and C long , where the actual values of these observables have been hidden by an offset. With the values of S long and C long hidden we perform the following fit cross-checks. We first verify that the uncertainties on the signal observables, and the value of − ln L obtained from the fit to data, are compatible with the ensembles of simulated experiments described above. We then validate the stability of our results by introducing a variety of modifications to the fitting procedure. In addition to the nominal set of variables determined from data, we allow the physics parameters τ B 0 and ∆m d to vary in turn. The signal observables do not change significantly when doing this, and the results obtained for τ B 0 and ∆m d are 1.72 ± 0.16 ps and 0.36 ± 0.22h/ ps, respectively, consistent with the reported world average [28].
The Lepton and Kaon I tagged events have the highest signal purity, and lowest w probability, and dominate our CP results. We check that the results obtained from fitting only these categories of events are compatible with our nominal blind results. When doing this, we observe shifts of −0.08 (−0.05) on S long (C long ) relative to the nominal result.
Given that there are a number of B backgrounds that contribute to the data that have yields fixed to expectations, we validate this assumption by allowing each fixed B background yield to vary in the fit. We obtain background yields consistent with our expectations, and observe the shifts on signal parameters listed in Table IV. A systematic uncertainty corresponding to the largest observed deviation is assigned on our results.
Once these checks have been completed, the fitting procedure is frozen and we extract the values of S long and C long by removing the offset.

A. Fit results
From the ML fit described above, we obtain the following results N (signal) = 729 ± 60(stat), f L = 0.992 ± 0.024(stat), S long = −0.17 ± 0.20(stat), C long = 0.01 ± 0.15(stat), after correction for a +38.5 ± 5.3 event fit bias (see Section VIII), a −76 event bias from SCF on the signal yield (see Section IX B) and a correction for a −0.016 ± 0.002 fit bias on f L . The correlation between S long and C long is −0.035. We discuss the origin of these fit biases in Section IX B. We calculate the branching fraction of this decay using B = N signal /(ǫN bb ), where ǫ is the efficiency for signal corresponding to the observed f L , and N B pairs is the number of BB pairs analyzed. We obtain The B 0 → ρ ∓ π ± π 0 background yield obtained from the fit is 9.2 ± 53.6 events. Figure 6 shows distributions of m ES , ∆E, cos θ i and m π ± π 0 for the Lepton and Kaon I tagged events with a loose requirement on N . Relative to the total number of events in the fit, the plot of m ES contains 15.6% of the signal and 1.1% of the total background. For the other plots there is an added constraint that m ES > 5.27 GeV/c 2 ; these requirements retain 13.9% of the signal and 0.4% of the total background. Figure 7 shows the ∆t distribution for B 0 and B 0 tagged events, as well as the time-dependent decayrate asymmetry of Eq. (10). Here we apply the same selection criteria as in Fig. 6(b)-(d). Table V lists the possible sources of systematic uncertainties on the values of the N signal , f L , S long , and C long that have been studied, and are described in the following.

B. Systematic uncertainty studies
• The uncertainty from PDF parameterisation is obtained by varying PDF shape parameters by ±1σ, in turn. The deviations obtained are added in quadrature to give the quoted uncertainty.
• The systematic uncertainty from the fraction of SCF events is obtained from the difference between the nominal fit result and that obtained when fitting the data and removing the SCF from the fit model. In the case of the signal yield we correct for half of the difference observed, and use 100% of the correction as an uncertainty. The uncertainty on the other signal observables comes from the difference observed between the two sets of fit results.  • The uncertainty from the m ES and ∆E widths is obtained from the observed shifts relative to our nominal result, when allowing these parameters to vary independently in the fit to data.
• We vary the B background normalization within expectations for each background in turn. The deviations obtained are added in quadrature to give the quoted uncertainty from this source.
• As the branching fractions of some of the B backgrounds are not well known, we assign an additional uncertainty coming from the maximum shifts obtained when allowing each of the fixed backgrounds to vary in turn in the fit to data.
• Additional uncertainties on the CP results come from possible CP violation in the B background. We replace the ∆t PDFs used by each of the B backgrounds in turn by one resembling the signal. Charged B backgrounds can have non-zero values of A CP , and neutral B backgrounds can have nonzero values of S and C. We use existing experimental constraints where possible, otherwise we allow for a CP asymmetry up to 10% in B decays to final states with charm, and up to 50% in B decays to charmless final states.
• The tagging and mistag fractions for signal and the B backgrounds are corrected for data/MC differences observed in samples of fully reconstructed hadronic B decays. Each of the tagging and mistag parameters is varied in turn by the uncertainty from the correction. The deviations obtained are added in quadrature to give the quoted uncertainty from this source.
• Allowing for possible CP violation in the transverse polarization, and in the WT longitudinally polarized signal SCF events results in additional uncertainties on signal observables. We vary S and C by ±0.5 (±1.0) for the transverse polarization (WT SCF).
• Possible CP violation from interference in doubly Cabibbo-suppressed decays (DCSD) on the tag side of the event [53] contribute to systematic uncertainties on S long and C long .
• We estimate the systematic error on our results coming from neglecting the interference between B 0 → ρ + ρ − and other 4π final states: B → a 1 π, ρππ 0 and πππ 0 π 0 . Details of this calculation are given in the Appendix.
• As the PDFs used in the ML fit do not account for all of the correlations between discriminating variables used in the fit, the results have a small bias. We calculate the fit bias on the signal observables as described in Section VIII. The observed bias on N signal , and f L is corrected, and 100% of the correction is assigned as a systematic uncertainty. We do not observe a significant bias on S long and C long , and conservatively assign a systematic uncertainty from fit bias based on the statistical precision of this test.
• Small imperfections in the knowledge of the geometry of the SVT over time can affect the measurement of S long and C long . We vary the alignment according to the results obtained from the study of e + e − → e + e − , µ + µ − events in order to estimate the magnitude of this systematic error on our CP results.
The branching fraction has multiplicative systematic uncertainties from the reconstruction of π 0 mesons in the detector (6%), uncertainties in the reconstruction of charged particles (0.8%), and the discrimination of π ± from other types of charged particles (1%). In addition to these uncertainties, there is a 1.1% uncertainty on the number of BB pairs in the data sample. The statistical uncertainties arising from the MC samples used in this analysis are negligible.

X. CONSTRAINTS ON THE UNITARITY TRIANGLE
A. The SU (2) isospin analysis In SU (2) isospin symmetry, the amplitudes of B decays to ρρ final states [20] are related by: for the longitudinal polarization and each of the CP eigenstates of the transverse polarization, where A ij (A ij ) are the amplitudes of B(B) decays to the final state with charge ij, where i = +, −, 0 and j = −, 0. These two relations correspond to triangles in a complex plane as shown in Fig. 8. In the usual phase convention [20], the amplitudesĀ are rotated toÃ in order to align the base of the triangles. After aligning A +0 and A −0 , the phase difference between A +− and A +− is 2δα. There are two sources of ambiguities on the measurement of α.
There is a four-fold ambiguity coming from the orientation of the isospin triangles, and a two-fold ambiguity from the trigonometric conversion in Eq. (12). In order to measure α one must measure the branching fractions and charge asymmetries of B decays to ρ + ρ − , ρ ± ρ 0 , ρ 0 ρ 0 . There are several assumptions implicitly used in the isospin-based direct measurements of α: • This approach only considers tree and gluonic penguin contributions. Possible contributions from electroweak penguins (EWP) are neglected as they do not obey SU (2) isospin symmetry. These EWPs have the same topology as the gluonic penguin diagram in Fig. 2, with the gluon replaced by γ or Z 0 bosons. In the absence of EWP contributions |A +0 | = |A −0 |, and A CP is zero for B + → ρ + ρ 0 .
Several groups have estimated the correction due to the SU (2) breaking effect of EWP contributions to be 1.5 − 2.0 degrees [54,55]. These estimates consider contributions from the two EWP operators assumed to be dominant in the effective Hamiltonian.
• The possible effect on the isospin analysis from ρ 0 − ω mixing [55] is neglected.
• Other SU (2) symmetry breaking effects are neglected. Estimates of the magnitude of these effects are much less than the current experimental precision [56,57]. Possible isospin 5/2 amplitudes also break the SU (2) triangle construction [58].
• The isospin analysis outlined above neglects possible I=1 amplitudes [59]. The presence of I=1 amplitudes in B 0 → ρ + ρ − can be tested by measuring S long and C long for different ranges of the invariant π ± π 0 mass.
We constrain the CKM angle α and penguin contribution δα from an isospin analysis of B → ρρ decays. The inputs to the isospin analysis are the amplitudes of the CP -even longitudinal polarization of the ρρ final state, as well as the measured values of S long and C long for B 0 → ρ + ρ − . We use the following numerical inputs in the isospin analysis: • The measurements of B( B 0 → ρ + ρ − ), f L , S long , and C long presented here.
• The measurement of B(B 0 → ρ 0 ρ 0 ) from Ref. [23]. : Confidence level on α obtained from the isospin analysis with the statistical method described in [14]. The dashed lines correspond to the 68% (top) and 90% (bottom) CL intervals.
To interpret our results in terms of a constraint on α from the isospin relations, we construct a χ 2 that includes the measured quantities expressed as the lengths of the sides of the isospin triangles and we determine the minimum χ 2 0 . We have adopted a simulated-experiment technique to compute the confidence level (CL) on α; our method is similar to the approach proposed in Ref. [60]. For each value of α, scanned between 0 and 180 • , we determine the difference ∆χ 2 DATA (α) between the minimum of χ 2 (α) and χ 2 0 . We then generate MC experiments around the central values obtained from the fit to data with the given value of α and we apply the same procedure. The fraction of these experiments in which ∆χ 2 MC (α) is smaller than ∆χ 2 DATA (α) is interpreted as the CL on α. Figure 9 shows 1 − CL for α obtained from this method.
It is possible to obtain a prediction of α from indirect constraints by combining measurements of the CKM matrix elements |V us |, |V ud |, |V ub |, and |V cb |, CP violation in mixing from neutral kaons, B-B mixing in B d and B s mesons and the measurement of sin 2β from b → ccs decays. The indirect constraint on α from the UTfit [15] and CKMfitter [14] groups are (92.9 ± 5.7) • and (100 +4.5 −7.3 ) • respectively. Recent calculations using QCD Factorization predict α = (85.6 +7.4 −7.3 ) • [61]. Selecting the solution closest to the CKM combined fit average we find α = [73.1, 117.0] • at 68% CL, where the error is dominated by |δα| which is 18 • at 68% CL. The constraint obtained on α is worse than the previous result in Ref. [26] for the following reasons: (i) the central value of the branching fraction for B + → ρ + ρ 0 [21] is smaller than the value previously used [22,62], and (ii) the results of the latest search for B 0 → ρ 0 ρ 0 show evidence for a signal [23], while the previous constraint on α used the central value corresponding to the upper limit available at that time [63]. Both of these factors lead to an increase in the penguin contribution to the total uncertainty on α when using an isospin analysis.
A future extension to the SU (2) isospin method to measure α described here will be possible when there are sufficient statistics to perform a time-dependent analysis of B 0 → ρ 0 ρ 0 . The parameters S long and C long measured in B 0 → ρ 0 ρ 0 can be incorporated into the isospin analysis, allowing one to over-constrain the isospin triangles of Eq. (28) and Eq. (29).

B. Flavor SU (3) analysis
There has been progress on constraining the penguin contribution in B 0 → ρ + ρ − using a flavor SU (3) based approach and experimental constraints from B + → K * 0 ρ + [24]. The amplitude for B 0 → ρ + ρ − can be written as where T and P are the magnitudes of tree and penguin amplitudes, γ is the phase of V ub , and δ TP is the strong phase difference between T and P . The approach of Ref. [24] relates the penguin amplitude in B 0 → ρ + ρ − to the amplitude of the penguin decay B + → K * 0 ρ + giving three relations: S long = sin 2α + 2r cos δ TP sin(β − α) − r 2 sin 2β 1 − 2r cos δ TP cos(β + α) + r 2 , with three unknowns, α, r, and δ TP . The parameter r = |P/T |, and β is the phase of V td . We use the value of β obtained from b → ccs decays [7,8]. The CP averaged decay rates of the longitudinal components of B ± → K * 0 ρ + and B 0 → ρ + ρ − (Γ L ) are related by the squared ratio of CKM matrix elements and decay constants (f i , where ρ or K * ) as shown in Eq. (33). The factor F accounts for additional sources of SU (3) breaking not described by the decay constants. The assumptions used in this approach are: • The amplitude in the penguin dominated decay B + → K * 0 ρ + is related to the penguin amplitude • The dominant SU (3) breaking correction accounted for by F is the neglect of annihilation diagrams in the B + → K * 0 ρ + decay. We use the same value of F as Ref. [24]; F = 0.9 ± 0.6.

XI. CONCLUSIONS
We report the measurement of the branching fraction, f L , and CP violation parameters, S long and C long , for the decay B 0 → ρ + ρ − using a data sample of (383.6 ± 4.2)×10 6 BB pairs. We obtain the following results: Using these results, and experimental knowledge of the other B → ρρ final states, we perform an isospin analysis to obtain a measurement of the CKM angle α. The confidence level distribution for α is shown in Fig. 9. The solution obtained that is compatible with the results of SMbased fits of existing data is α = [73.1, 117.0] • at 68% CL. The uncertainty on the measurement of α from the isospin analysis is dominated by penguin pollution. If one uses the flavor SU (3) approach described in the text to constrain α, one obtains the constraint α = [83.3, 105.8] • at 68% CL.

XII. ACKNOWLEDGMENTS
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 An extensive study of the B backgrounds, and the associated systematic error, has been presented earlier in this paper. A number of B backgrounds decay into the same final state as the signal. These are: • B 0 → π + π − π 0 π 0 , non resonant.
The systematic error associated with these modes was evaluated by propagating the uncertainty on the branching ratio and CP asymmetries to the final measurements, with the appropriate description of acceptance, SCF, resolutions and other reconstruction biases (see Section IX B). The likelihood described in Section VII does not account for possible interference between B 0 → ρ + ρ − and other B 0 → 4π final states.
Acceptance, SCF, and other reconstruction biases are not taken into account in this study, and instead, the ρ + ρ − measurements are averaged in a perfect region of the phase space referred to as the "ρ + ρ − band", on which the analysis technique is assumed to have a uniform sensitivity. This "ρ + ρ − band" corresponds to the kinematic selection of ρ ± (see Section III B): 0.5 < m π ± π 0 < 1.0 GeV/c 2 and −0.9 < cos θ i < 0.98.
The remainder of this Appendix describes the final state wave function, decay amplitudes, effective CP asymmetries and the estimate of the systematic uncertainty from neglecting interference between the signal and other B 0 → 4π final states.

The final state wave functions
This section summarizes the kinematic dependence of the decay amplitudes.

a. Lorentz invariant phase space
The four-particle final state can be described completely in terms of the following five variables: • m 1 and m 2 : the masses of the π + π 0 and π − π 0 pairs • cos θ 1 and cos θ 2 : the angle between the π ± π 0 pair line of flight (as seen in the B frame) and the π 0 in the pion pair frame.
• φ: the angle between the two planes defined by the π + π 0 and π − π 0 pairs in the B frame.
We write each phase space dependent quantity dΦ 4 as a function of these five variables. These are: where P π + π 0 B denotes the momentum of the pion pair in the B frame and M B is the mass of the B (and similarly for the other quantities in this expression).
b. Kinematic dependence of ρ + ρ − We assume that the longitudinal polarization of ρ + ρ − is dominant, so the kinematic dependence of the ρ + ρ − wave function is: where BW denotes a Breit-Wigner function (the simple non-relativistic form is assumed), and Y m l are the spherical harmonics.

c. Kinematic dependence of a1π
In order to write the wave function of a + 1 π − , we need to define the following three variables, which are functions of the variables found in part 1a of this Appendix: • θ ρ : the angle between the π 0 and the ρ line of flight (as seen from the a 1 ) in the ρ frame, • θ a1 : the angle between the ρ and a 1 line of flight (as seen from the B) in the a 1 frame, • φ a1 : the angle between the two decay planes of the a 1 and ρ mesons, in the a 1 frame.
For a 1 π we assume that all of these parameters are constant over the whole of phase space.

Effective CP asymmetries
Considering the two final states ρ + ρ − and a 1 π, the phase-space dependent four-pion decay amplitude is: with a similar expression for A(B 0 → 4π). Similar equations exist for ρππ and 4π final states. We can define the phase-space dependent quantity: where λ 4π (Φ 4 ) is a function of m 1 , m 2 , cos θ 1 , cos θ 2 , and φ. Thus define the phase-space dependent CP -violating asymmetries: that are analogous to Eq. (8) and Eq. (9). In principle it is possible to measure the value of C 4π (Φ 4 ) and S 4π (Φ 4 ) at each point of phase space, however this would be experimentally challenging. Instead, this measurement uses integrated values C eff and S eff , which can be seen as the averages of C and S across the previously defined "ρ + ρ − band", weighted over the number of events found at each location of the phase space which corresponds to: We can also define the values of C and S that would be observed in the absence of interference effects, denoted by C I 4π (Φ 4 ) and S I 4π (Φ 4 ), and similarly the integrated values across the "ρ + ρ − band" C I eff and S I eff . Finally, for the evaluation of the systematic error, we are really interested in the biases on C and S purely due to interference which are the two differences δC = C eff − C I eff and δS = S eff − S I eff . We adopt the same strategy for the branching fraction B(ρ + ρ − ) by defining δB = B eff − B I eff in a similar way.

The systematic error
As the 8 parameters describing the amplitude A(B 0 (B 0 ) → a 1 π) are not yet measured, to estimate the systematic error on ρ + ρ − parameters, we need to propagate the uncertainty on these 8 parameters. For this, we use a MC technique. For each experiment, we generate the CP parameters C(a 1 π), ∆C(a 1 π), S(a 1 π), ∆S(a 1 π), A CP (a 1 π), the phases δ(a + 1 π − ) and δ(a − 1 π + ), and the branching fraction B(a 1 π). The values used to generate the a 1 π contribution have Gaussian PDFs for measured quantities (with mean and width corresponding to the measurement), and uniform PDFs for unmeasured quantities (sampled over the allowed values). Then, we estimate the value of δC and δS for each generated value of the a 1 π parameters. The systematic error are taken as the RMS of the δC, δS and δB distributions.
We estimate the effect of interference between signal and either ρ ∓ π ± π 0 or π + π − π 0 π 0 in a similar way, where we assume the branching fraction for these two nonresonant backgrounds is (5.7 ± 13.0) × 10 −6 . This value is obtained by modeling the ρ ∓ π ± π 0 component in the likelihood fit and by fitting a free yield to the data, and we assume that ρ ∓ π ± π 0 and π + π − π 0 π 0 have the same branching fraction.
We summarize the systematic uncertainty obtained for the different 4π final states considered here in Table VI. The total uncertainty on B is used to calculate the quoted systematic uncertainty on f L from this source. We con-clude that interference between signal and other 4π final states has a negligible impact on the measurement.