Dalitz Plot Analysis of B+- -->pi+-pi+-pi-+ Decays

We present a Dalitz plot analysis of charmless B+- decays to the final state pi+-pi+-pi-+ using a sample of (465+-5)x10^-6 BBbar pairs collected by the BABAR experiment at \sqrt{s} = 10.58 GeV. We measure the branching fractions BF(B+- -->pi+-pi+-pi-+) = (15.2+-0.6+-1.2+-0.4)x10^-6, BF(B+- -->rho0(770)pi+-) = (8.1+-0.7+-1.2^{+0.4}_{-1.1})x10^-6, BF(B+- -->f2(1270)pi+-) = (1.57+-0.42+-0.16^{+0.53}_{-0.19})x10^-6, and BF(B+- -->pi+-pi+-pi-+ nonresonant) = (5.3+-0.7+-0.6^{+1.1}_{-0.5})x10^-6, where the uncertainties are statistical, systematic, and model-dependent, respectively. Measurements of branching fractions for the modes B+- -->rho0(1450)pi+- and B+- -->f0(1370)pi+- are also presented. We observe no significant direct CP asymmetries for the above modes, and there is no evidence for the decays B+- -->chiC0 pi+-, or B+- -->chiC2 pi+-.


INTRODUCTION
Decays of B mesons to three-body charmless final states probe the properties of the weak interaction through their dependence on the complex quark couplings described in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2]. Furthermore, these decays test dynamical models for hadronic B decays.
One can measure direct CP asymmetries and constrain magnitudes and phases of the CKM matrix elements using individual channels that appear as intermediate resonances in the B ± → π ± π ± π ∓ decay. For example, the CKM angle γ could be extracted from the interference between the decay B ± → χ c0 π ± , which has no CPviolating phase (in the standard parametrization), and other modes such as B ± → ρ 0 (770)π ± [3,4,5,6,7,8].
Studies of B ± → π ± π ± π ∓ can also be useful for a precise measurement of the CKM angle α. A theoretically clean determination of this angle can be obtained from the decay-time dependence of the interference between B 0 → ρ + π − , B 0 → ρ − π + , and B 0 → ρ 0 π 0 via the analysis of the Dalitz plot for B 0 → π + π − π 0 decays [9] (recently implemented by BABAR [10] and Belle [11,12]). Charged B decays offer a large statistics sample with which to determine additional resonant or nonresonant contributions to the three-pion Dalitz plot that can affect the measurement of α. For example, the Dalitz plot analysis of B ± → π ± π ± π ∓ allows one to check for effects from B ± → ω(782)π ± , that could cause large direct CP violation due to ρ-ω mixing [13]. It is particularly important to limit the possible effects of broad scalar structures [including the so-called f 0 (600) or σ] and nonresonant contributions [14,15,16,17,18,19].
In this paper we present an amplitude analysis of B ± → π ± π ± π ∓ decays based on a 424 fb −1 data sample containing (465 ± 5) × 10 6 BB pairs (N BB ). The data were collected with the BABAR detector [30] at the PEP-II asymmetric-energy e + e − storage rings [31] operating at the Υ (4S) resonance with center-of-mass (CM) energy of √ s = 10.58 GeV. An additional total integrated luminosity of 44 fb −1 was recorded 40 MeV below the Υ (4S) resonance ("off-peak" data) and was used to study backgrounds. Compared to our previous publication [32], in addition to doubling the data sample we have included several improvements in reconstruction algorithms that enhance the signal efficiency, made numerous modifications to the analysis to increase the sensitivity to direct CP violation effects (for example, by including more discriminating variables in the maximum likelihood fit), and improved our model of the Dalitz plot structure. The remainder of the paper is organized as follows: Sec. II describes the amplitude analysis formalism, Secs. III and IV give details about the selection of signal B decays and how backgrounds are considered, Sec. V presents the results from the likelihood fit, Sec. VI gives an account of the various sources of systematic uncertainties, while Sec. VII summarizes the results.

AMPLITUDE ANALYSIS FORMALISM
A number of intermediate states contribute to the decay B ± → π ± π ± π ∓ . We determine their contributions with a maximum likelihood fit to the distribution of events in the Dalitz plot. This procedure has been described in detail in our previous publications [20,21,32].
The B ± → π ± π ± π ∓ decay contains two same-sign pions in the final state. We distinguish these particles according to the invariant mass they make when combined with the oppositely charged pion, and draw the Dalitz plot in terms of heavy and light invariant masses-squared of the π ± π ∓ systems (denoted m 2 max and m 2 min , respectively), so that each candidate has a uniquely defined position. Moreover, we explicitly enforce the symmetrization of the total amplitude under exchange of identical bosons.
The total signal amplitudes for B + and B − decays are given by The complex coefficients c j and c j for a given decay mode j contain all the weak phase dependence. Since the F j terms contain only strong dynamics, F j ≡ F j . We use the following parametrization [21] for the amplitude coefficients: In this approach, x j and y j (∆x j and ∆y j ) are the CPconserving (-violating) components of the decay amplitude.
The F j distributions describe the dynamics of the decay amplitudes and are written as the product of an invariant mass term R j , two Blatt-Weisskopf barrier form factors X J , and an angular function T j where m (J) is the mass (spin) of the resonance, p ⋆ is the momentum the bachelor pion that is not part of the resonance in the B meson rest frame, and q is the momentum of either daughter in the rest frame of the resonance (we use the c = 1 convention for all equations in this paper). The F j are normalized over the entire Dalitz plot: The Blatt-Weisskopf barrier form factors [33] are given by: where the meson radius parameter r BW is taken to be 4.0 ± 1.0 (GeV/c) −1 [34]. For most resonances in this analysis the R j are taken to be relativistic Breit-Wigner lineshapes where m 0 is the nominal mass of the resonance and Γ(m) is the mass-dependent width. In the general case of a spin-J resonance, the latter can be expressed as The symbol Γ 0 denotes the nominal width of the resonance. The values of m 0 and Γ 0 are obtained from standard tables [34] when they are well known. The symbol q 0 denotes the value of q when m = m 0 . The angular distribution terms T j in Eq. (3) follow the Zemach tensor formalism [35,36]. For the decay of a spin zero B-meson into a spin J resonance and a spin zero bachelor particle this gives [37] T J=0 where p is the momentum of the bachelor particle and q is the momentum of the resonance daughter with charge opposite from that of the bachelor particle, both measured in the rest frame of the resonance. The Gounaris-Sakurai parametrization [38] of the Pwave scattering amplitude for a broad resonance decaying to two pions is used for the ρ 0 (770) and ρ 0 (1450) lineshapes with The normalization condition at R j (0) fixes the parameter d = f (0)/(Γ 0 m 0 ). It is found to be We model the nonresonant component using an empirical function that has been found to accurately describe nonresonant contributions in other charmless three-body B decays [23,24,25,28,29]: We include this term in the coherent sum given by Eq. (1) when calculating the total signal amplitude over the Dalitz plot.
To allow comparison among experiments we present results also in terms of fit fractions (FF j ), defined as the integral of a single decay amplitude squared divided by the total matrix element squared for the complete Dalitz plot Note that the sum of all the fit fractions is not necessarily unity due to the possible presence of constructive or destructive interference. The CP asymmetry for each contributing resonance is determined from the fitted parameters The signal Dalitz plot probability density function (PDF) is formed from the total amplitude as follows: where q B is the charge of the B-meson candidate, and ε ≡ ε(m 2 max , m 2 min ) and ε ≡ ε(m 2 max , m 2 min ) are the signal reconstruction efficiencies for B + and B − events, respectively, defined for all points in the Dalitz plot.

CANDIDATE SELECTION
We reconstruct B candidates from events that have four or more charged tracks. Each track is required to be well measured and to originate from the beam spot. They must have a minimum transverse momentum of 50 MeV/c, and a distance of closest approach to the beam spot of less than 1.5 cm in the transverse plane and less than 2.5 cm along the detector axis. B candidates are formed from combinations of three charged tracks, and particle identification (PID) criteria are applied to reject electrons and to separate pions from kaons. In our final state, the average selection efficiency for pions that have passed the tracking and PID requirements is about 93% including geometrical acceptance, while the average misidentification probability of kaons as pions is close to 8%.
Two kinematic variables are used to identify signal B decays. The first variable is the difference between the reconstructed CM energy of the B-meson candidate (E * B ) and √ s/2, where √ s is the total CM energy. The second is the beam-energysubstituted mass where p * B is the B momentum measured in the CM frame. The m ES distribution for signal events peaks near the B mass with a resolution of around 2.5 MeV/c 2 , while the ∆E distribution peaks at zero with a resolution of approximately 20 MeV. We initially require events to lie in the region formed by the following selection criteria: 5.200 < m ES < 5.286 GeV/c 2 and −0.075 < ∆E < 0.300 GeV. The region of ∆E below −0.075 GeV is heavily contaminated by four-body B decay backgrounds and is not useful for studying the continuum background. The selected region is then subdivided into three areas: the "left sideband" (5.20 < m ES < 5.26 GeV/c 2 and |∆E| < 0.075 GeV) used to study the background ∆E and Dalitz plot distributions; the "upper sideband" (5.230 < m ES < 5.286 GeV/c 2 and 0.1 < ∆E < 0.3 GeV) used to study the background m ES distributions; and the "signal region" (5.272 < m ES < 5.286 GeV/c 2 and |∆E| < 0.075 GeV) with which the final fit to data is performed. Following the calculation of these kinematic variables, each of the B candidates is refitted with its mass constrained to the world-average value of the B meson mass [34] in order to improve the Dalitz plot position resolution and to make sure all events lie within the kinematic boundary of the Dalitz plot.
The dominant source of background comes from lightquark and charm continuum production (e + e − → qq, where q = u, d, s, c). This background is suppressed by requirements on event-shape variables calculated in the CM frame. We compute a neural network (NN) from the following five variables: the ratio of the secondand zeroth-order angular moments (L 2 /L 0 ), with L j = i p i | cos θ i | j , where θ i is the angle of the track or neutral cluster i with respect to the signal B thrust axis, p i is its momentum, and the sum excludes the daughters of the B candidate; the absolute value of the cosine of the angle between the direction of the B and the detector axis; the magnitude of the cosine of the angle between the signal B thrust axis and the detector axis; the output of a multivariate B-flavor tagging algorithm [39] multiplied by the charge of the B candidate; and the ratio of the measured proper time difference of the two B decay vertices and its statistical uncertainty. We train the NN using samples of off-peak data and signal Monte Carlo (MC) events generated with the phase-space distribution. A selection requirement is imposed on the NN output that accepts about 48% of signal events while rejecting 97% of continuum background events.
Dalitz plot distributions of the reconstruction efficiency for B + and B − events are modeled with twodimensional histograms formed from a sample of around 7 × 10 6 B ± → π ± π ± π ∓ phase-space MC events. All selection criteria are applied except for the exclusion of certain invariant-mass regions described below. We take the ratio of two histograms, the denominator containing the true Dalitz plot distribution of all generated MC events and the numerator containing the reconstructed MC events. The reconstructed events are weighted in order to correct for differences between data and MC simulations in the tracking and PID efficiencies. In order to give better resolution near the edges of the Dalitz plot, where most reconstructed events lie, the histograms are formed in the "square Dalitz plot" [10,32] coordinates. We use 50×50 bins and smooth these histograms by applying linear interpolation between neighboring bins. The efficiency shows very little variation across most of the Dalitz plot but decreases towards the corners where one of the particles has low momentum. The effect of experimental resolution on the signal model is neglected since the resonances under consideration are sufficiently broad. The average reconstruction efficiency for events in the signal region for the phase-space MC sample is about 15%. The fraction of misreconstructed signal events is only 5%, and MC studies indicate that there is no need for any explicit treatment of these events.

BACKGROUNDS
In addition to the continuum (qq) background we also have backgrounds from BB events. There are four main sources: (i) combinatorial background from three unrelated tracks; (ii) three-and four-body B decays involving an intermediate D meson; (iii) charmless twoand four-body decays with an extra or missing parti-cle; and (iv) three-body decays with one or more particles misidentified. We reject background from two-body decays of D mesons and charmonium states by excluding invariant masses (in units of GeV/c 2 ) in the ranges: 1.660 < m π + π − < 1.920, 3.051 < m π + π − < 3.222, and 3.660 < m π + π − < 3.820. These ranges reject decays from D 0 → K + π − (or π + π − ), J/ψ → ℓ + ℓ − , and ψ(2S) → ℓ + ℓ − respectively, where ℓ is a lepton that has been misidentified as a pion. We also employ a special requirement to reject the decay process B ± → K 0 S π ± ; K 0 S → π + π − , by excluding candidates where the vertexed mass of two oppositely charged pions lies in the range of [478, 516] MeV/c 2 .
We use a large sample of MC-simulated BB decays, equivalent to approximately 3 times the integrated luminosity of the data sample, to identify the important B backgrounds that survive the invariant-mass exclusion requirements described above. In total, 53 B-meson decay modes are identified for which larger samples of exclusive MC events are used for further study. We combine modes that have similar behavior in the discriminating variables m ES and ∆E into a B-background category. There are four such categories: the first contains the two-body decays B 0 → π + π − and B 0 → K − π + , the second is dominated by B ± → K ± π ± π ∓ and contains other decays with similar topologies, the third contains only B 0 → π + π − π 0 , and the fourth contains the remaining backgrounds from B decays that are combinatorial in nature. For each B-background category the combined m ES , ∆E, and Dalitz plot distributions are created where the relative contributions of various decay modes in a specific category are calculated from the reconstruction efficiencies from MC simulations and the branching fractions listed by the Particle Data Group [34] and the Heavy Flavor Averaging Group [40]. These distributions are used in the likelihood fit described below.
Background Dalitz plot distributions are included in the likelihood fit through the use of two-dimensional histograms. For backgrounds from B decays these histograms are formed from the various MC samples. For the continuum background the left sideband data sample is used. Since this data sideband also contains events from B decays, MC samples are used to subtract these events. To these B-subtracted sideband events, we add off-peak data events from across the whole range of m ES and ∆E in order to enhance statistics. We have verified that the shapes of various discriminating variables are compatible between the sideband and off-peak events. As for the reconstruction efficiency histograms, the background Dalitz plot distributions are formed in the square Dalitz plot coordinates and are smoothed by linear interpolation applied between neighboring bins. Separate histograms are constructed for B + and B − events. The qqand B-background PDFs are identical in their con-struction, and the qq PDF is shown here as an example: where A qq is the charge asymmetry in the background, and Q(m 2 max , m 2 min ) and Q(m 2 max , m 2 min ) are the Dalitz plot distributions of qq events in selected B + and B − samples, respectively.

MAXIMUM LIKELIHOOD FIT
To provide further discrimination between the signal and background hypotheses in the likelihood fit, we include PDFs for the kinematic variables m ES and ∆E, which multiply that of the Dalitz plot. The signal m ES shape is modeled with the sum of a Gaussian function and a Crystal-Ball lineshape [41], and the ∆E shape is modeled with a double Gaussian function. The parameters of these functions are obtained from a sample of B ± → π ± π ± π ∓ MC events, modeled according to the Dalitz plot distribution from Ref. [32], and are appropriately adjusted to account for possible differences between data and MC simulations determined with a control sample of B + → D 0 π + ; D 0 → K + π − decays. These parameters are fixed in the fit to data.
The qq m ES distribution is modeled with the experimentally motivated ARGUS function [42]. The end point for the ARGUS function is fixed to 5.289 GeV/c 2 , and the parameter describing the shape is fixed to the value determined from the combined sample of upper sideband and off-peak data. We model the continuum ∆E shape using a linear function, the slope of which is fixed to the value determined from the left sideband and off-peak data. The BB background distributions are modeled with histograms obtained from the mixture of BB MC samples. The yields of signal and qq events are allowed to vary in the final fit to the data while the yields of BB backgrounds are fixed to 11 (two-body decays), 195 (B ± → K ± π ± π ∓ type), 117 (B 0 → π + π − π 0 ), and 495 (combinatorial) events.
The complete likelihood function is given by: Ne j k where N is equal to k N k , N k is the yield for the event category k, N e is the total number of events in the data sample, and P j k is the PDF for the category k for event j, which consists of a product of the Dalitz plot, m ES , and ∆E PDFs. The function −2 ln L is minimized in an unbinned fit to the data.
Our nominal signal Dalitz plot model comprises a momentum-dependent nonresonant component and four intermediate resonance states: ρ 0 (770)π ± , ρ 0 (1450)π ± , f 2 (1270)π ± , and f 0 (1370)π ± . The parameters used to describe these states are summarized in Table I. We fit 4335 B candidates in the signal region selected from the data to obtain the central values of the x j , ∆x j , y j , and ∆y j parameters for each component, and use Eqs. (15) and (16) to calculate the fit fractions and CP asymmetries. We use ρ 0 (770)π ± as the reference amplitude, fixing its x, y, and ∆y parameters to unity, zero, and zero, respectively. The signal yield, qq background yield and asymmetry are also free parameters of the fit, giving a total of 20 free parameters. The Dalitz plot model was determined using the results of our previous analysis [32] and the changes in the fit likelihood and χ 2 values when omitting or adding resonances. The latter is calculated from the projection of the fit results onto the Dalitz plot using the formula where y i is the number of data events in bin i and f (x i ) is the number of events in that bin as predicted by the fit result. The number of degrees of freedom is calculated as n b −h−1, where n b is the total number of bins used and h is the number of free parameters in the fit. A minimum of 20 entries in each bin is required; if this requirement is not met then the neighboring bins are combined. Typically, n b takes values around 100. In our previous study we found significant contributions from ρ 0 (770)π ± and f 2 (1270)π ± ; with f 0 (980)π ± , ρ 0 (1450)π ± , and a uniform nonresonant term also included in the model. Because of the larger data sample and many improvements to the analysis, we find it necessary to include an additional contribution from f 0 (1370)π ± , and to use a momentum-dependent nonresonant amplitude [see Eq. (14)] in order to achieve a reasonable agreement of the fit with the data. We do not find any significant signal from f 0 (980)π ± , so we exclude this channel from our nominal model and calculate an upper limit for its fit fraction. The statistical significance of the presence of a component is estimated by evaluating the difference ∆ ln L between the negative log-likelihood of the nominal fit and that of a fit where all of the x, y, ∆x, and ∆y parameters for the given component are fixed to zero. This is then used to evaluate a p value where f (z; n d ) is the PDF of the χ 2 distribution and n d is the number of degrees of freedom, four in this case. We then determine the equivalent one-dimensional significance from this p value. We find that the f 2 (1270) contribution has a statistical significance of 6.1σ, the ρ 0 (1450) 4.6σ and the f 0 (1370) 3.9σ.
Since the mass and width of the f 0 (1370) state are not well known [34], we determine the preferred values from data by scanning the likelihood values obtained with different parameters. The mass and width are determined to be m f0(1370) = 1400 ± 40 MeV/c 2 and Γ f0(1370) = 300 ± 80 MeV, with a correlation of (−39 ± 4)%, where the errors are statistical only, and are obtained from a fit to the two-dimensional likelihood profile. Similarly, we determine the parameter of the nonresonant lineshape to be α nr = 0.28 ± 0.06 GeV −2 c 4 (statistical uncertainties only).
Possible contributions from χ c0 π ± and χ c2 π ± are not significant so we set upper limits on their branching fractions. Furthermore, we do not find any evidence for a very broad enhancement at low π + π − invariant mass such as could be caused by the decay B ± → σπ ± .   candidate event [43]. The background-subtracted Dalitz plot of the data in the signal region can be seen in Fig. 2.
The χ 2 per number of degrees of freedom of the projection of the fit result onto the Dalitz plot is 82/84. Using the fitted signal distribution we calculate the average reconstruction efficiency for our signal sample to be 18%. We generate a large number of MC experiments with the fitted parameters, and from the spread of results of fits to those experiments we determine the statistical uncertainties on the parameters, FF j , and A CP, j . This procedure takes into account correlations between the x j , ∆x j , y j , and ∆y j parameters. The linear correlation coefficients between the FF j and A CP, j parameters are also obtained and are presented in Appendix . In order to calculate the branching fraction for an intermediate mode, we multiply the fit fraction of the latter by the total inclusive B ± → π ± π ± π ∓ branching fraction. They are needed for comparison with previous measurements and theoretical predictions. The π ± π ± π ∓ signal yield is found to be 1219 ± 50 ± 75 +29 −24 events and the inclusive CP asymmetry to be +3.2 ± 4.4 ± 3.1 +2.5 −2.0 %, where the uncertainties are statistical, systematic, and model-dependent, respectively. Additionally, the total yield and CP asymmetry of the continuum background are found to be 2337 ± 62 events and (+0.2 ± 2.7)%, respectively, where the uncertainties are statistical only. Further results are shown in Tables II and III. Projections of the data, with the fit result overlaid, as π ± π ∓ invariant-mass distributions can be seen in Fig. 3. A detailed examination of possible direct CP violation effects in the low π ± π ∓ invariant-mass region is shown in Fig. 4, where we have subdivided the data into positive and negative values of cos θ H = p· q /(| p || q |), where θ H is the helicity angle, p is the momentum of the bachelor par-ticle and q is the momentum of the resonance daughter with charge opposite from that of the bachelor particle, both measured in the rest frame of the resonance. The agreement between the fit result and the data is generally good; the χ 2 per number of non-zero bins for these plots varies between 35/46 and 34/24. We calculate 90% confidence-level (CL) upper limits for components not included in the nominal Dalitz plot model. These are obtained by generating many MC experiments from the results of fits to the data where the extra component is added to the nominal Dalitz plot model, with all major systematic sources varied within their 1σ uncertainties. We fit these MC samples and plot the fit fraction distributions. The 90% CL upper limit for each fit fraction is the value which includes 90% of the MC experiments. The branching fraction upper limit is then the product of the fit fraction upper limit and the total branching fraction for B ± → π ± π ± π ∓ . We have searched for the presence of multiple solutions in the fit to data with the nominal model. We find a second solution with a value of −2 ln L about ten units higher than our nominal fit, and with a χ 2 of the Dalitz plot projection increased by four units. A comparison of the results between the two solutions is given in Appendix . The most significant difference is seen in the f 0 (1370) fit fraction, which is much smaller in the second solution. In Dalitz plot analyses of Kππ and KKK final states [20,21,22,23,24,25,28,29], similar phenomena relating to multiple solutions have been observed, and interpreted as being due to differences in the possible interference pattern -constructive or destructive interference -between nonresonant and broad amplitudes. Interference between the nonresonant and f 0 (1370) amplitudes appears to be a plausible explanation for the effect in this analysis. As can be seen in Fig. 4, the data exhibit a non-trivial interference pattern, and the difficulty in modeling this effect leads to model uncertainties in our results.

SYSTEMATIC UNCERTAINTIES
Systematic uncertainties that affect the measurement of fit fractions, phases, event yields, and CP asymmetries are summarized in Table IV. The fixed BB-background yields and asymmetries are allowed to vary and the variations of the other fitted parameters are taken as the uncertainties. The effect of limited statistics of the data sideband and MC samples used to obtain the fixed shapes of all the histogram PDFs is accounted for by fluctuating independently the histogram bin contents in accordance with their errors and repeating the nominal fit. The uncertainties on how well the samples model these distributions are also taken into account through various crosschecks, including variation of the mass rejection ranges and comparison of continuum shapes between sideband and signal region in MC samples.
The fixed parameters of the signal m ES and ∆E PDFs are studied in the control sample B + → D 0 π + ; D 0 → K + π − . The parameters are determined from data and MC samples, from which shift and scale factors are calculated and used to adjust the parameters for the nominal fit. The parameters are then varied in accordance with the errors on these shift and scale factors and the fits are repeated. Uncertainties due to the m ES distribution for the qq background, which is fixed in the fit, are assessed to be negligible. The fit confirms the value of the single parameter of the ARGUS function that is taken as input.
To confirm the fitting procedure, we perform 500 MC experiments in which the events are generated from the PDFs used in the fit to data. We repeat the exercise with qq events alone drawn from the PDF into which we embed signal and BB background events randomly extracted from the MC samples. Small fit biases are observed for some of the fit parameters and are included in the systematic uncertainties.
Relative uncertainties in the efficiency, due to PID and tracking efficiency corrections are 4.2% and 2.4% respectively; while N BB has an associated error of 1.1%. The efficiency correction due to the selection requirement on the NN output has also been calculated from B + → D 0 π + ; D 0 → K + π − data and MC samples, and is found to be (96.2 ± 1.2)%. The error on this correction is incorporated into the branching fraction systematic uncertainties.
Measured CP asymmetries could be affected by detector charge bias. We include a systematic uncertainty of 0.005 to account for this effect [21]. Furthermore, some  of our selection requirements, for example that on the NN output, may induce an asymmetry. We estimate the possible size of such an effect as 0.020 based on the study of our control sample. In addition to the above systematic uncertainties we also estimate uncertainties from two sources related to the signal Dalitz plot model. The first of these elements consists of the parameters of the various components of the signal model: the masses and widths of all intermediate resonances, the value of the parameter that characterizes the nonresonant shape, and the value of the Blatt-Weisskopf barrier radius. The associated uncertainties are evaluated by adjusting the parameters within their experimental errors and refitting. The second element is the uncertainty due to the composition of the signal model. It reflects observed changes in the parameters of the components when the data are fitted with the less significant f 0 (1370) component removed from the model and with one of the states ω(782), f 0 (980), χ c0 , or χ c2 added to the model. The uncertainties from each of these elements are added in quadrature to obtain the final model-dependence.

DISCUSSION AND SUMMARY
Our results are shown in Tables II and III. The Dalitz plot is dominated by the ρ 0 (770) resonance and a nonresonant contribution which, as seen in other charmless three-body hadronic B decays, is well modeled with an exponential form-factor. The measured branching fraction for the decay B ± → ρ 0 (770)π ± agrees with the world-average value [34] and is consistent with theoretical predictions based on QCD factorization models [44] and SU(3) flavor symmetry [45]. The measured branching fraction of the nonresonant component is consistent with some theoretical predictions [14,16,17]. We find the parameter of the nonresonant lineshape to be α nr = 0.28±0.06 GeV −2 c 4 (statistical uncertainties only), which is comparable with values found in analyses of other charmless decay modes such as B → Kππ [23,24,25] and B → KKK [23,28].
We do not find any signal for the decay B ± → f 0 (980)π ± . The branching fraction upper limit we obtain is consistent with the prediction of a recent perturbative QCD calculation if the f 0 (980) meson is dominated by an ss component [47].
We do not find any statistically significant CP asymmetries for the components in the nominal Dalitz plot model. The CP asymmetry in B ± → ρ 0 (770)π ± has a dependence on the presence or absence of the f 0 (1370) term in the model. The CP asymmetry of the f 0 (1370) term itself appears highly sensitive to the Dalitz plot model, varying dramatically between the favored and the second solution. Since the presence of this component is not established, especially given its insignificant contribution in the second solution, we set a 90% CL upper limit on its product branching fraction at 4.0 × 10 −6 .