Dalitz-plot Analysis of B0 -->D0bar pi+ pi-

We report preliminary results from a study of the decay B0 -->D0bar pi+ pi- using a data sample of 470.9 +/- 2.8 million BBbar events collected with the BaBar detector at the Y(4S) resonance. Using the Dalitz-plot analysis technique, we find contributions from the intermediate resonances D*_2(2460)-, D*_0(2400)-, rho(770)0 and f_2(1270) as well as a pi+ pi- S-wave term, a D0bar pi- nonresonant S-wave term and a virtual D*(2010)- amplitude. We measure the branching fractions of the contributing decays.


INTRODUCTION
The study of the Dalitz plot [1] of B 0 → D 0 π + π − decays is motivated by several factors. The branching fractions of B → D * * transitions 1 are of interest to help address a conflict between theoretical predictions [2] and experimental results [3,4] in semileptonic B → D * * lν decays. The D 0 π + π − final state allows relatively clean studies of the J P = 0 + and 2 + D * * states, since the 1 + mesons cannot decay to Dπ. Measurements of these decays test theoretical models including quark models [5], QCD sum rules [6][7][8] and lattice QCD [9]. Similarly, measurement of the branching fraction of the B 0 → D 0 ρ 0 decay will help to test the dynamics of "color-suppression" in B decays (related to the fact that the color quantum numbers of the quarks produced from the virtual W boson must match that of the spectator quark in order for a ρ 0 meson to be formed) [10][11][12][13]. Moreover, using isospin symmetry to relate the decay amplitudes of B 0 → D 0 ρ 0 , B 0 → D − ρ + and B + → D 0 ρ + , it is possible to study effects of final state interactions in these decays [11,14].
Another motivation is that the B 0 → D 0 ρ 0 decay can be used to measure sin2β, where β is the CKM unitarity triangle angle [15,16], if the D 0 meson is reconstructed in a CP eigenstate. The measurement of this angle in thē b →cud quark-level transition is theoretically cleaner than the commonly usedb →ccs decays (such as B 0 → J/ψ K 0 S ) [17,18] and comparisons of the values measured in different quark-level transitions can be used to search for the influence of physics beyond the Standard Model [19]. The time-dependent analysis of the B 0 → D 0 π + π − Dalitz plot not only allows a proper handling of effects due to interference between broad resonances, but also enables an improved measurement of β since terms proportional to cos2β as well as sin2β can be measured [20,21]. For such an analysis, it is necessary to have a good understanding of the population of the B 0 → D 0 π + π − Dalitz plot. This can be best studied in the D 0 → K + π − decay, which is the subject of this study.
The B 0 → D 0 π + π − decay has been previously studied by Belle [22] and the related B + → D − π + π + decay has been studied by both BABAR [23] and Belle [24]. In this paper we present preliminary results from the first study of the B 0 → D 0 π + π − decay by BABAR. The data used in the analysis, collected with the BABAR detector [25] at the PEP-II asymmetric energy e + e − collider at SLAC, consist of an integrated luminosity of 429 fb −1 recorded at the Υ (4S) resonance ("on-peak") and 45 fb −1 collected 40 MeV below the resonance ("off-peak"). The on-peak data sample contains the whole BABAR dataset of 470.9 ± 2.8 million BB events. SELECTION We reconstruct B 0 → D 0 π + π − candidates (the inclusion of charge conjugate reactions is implied throughout this paper) by combining a D 0 candidate with two oppositely charged pion candidates. The charged pion candidates are required to satisfy particle identification requirements that have efficiency above 97 % and kaon misidentification probability below 20 %. We reconstruct D 0 mesons in the decay channel K + π − . For the D 0 daughters, the charged kaon candidates are required to satisfy particle identification requirements that have efficiency above 97 % and pion misidentification probability below 15 %, while the charged pion candidates are required to pass slightly looser criteria than those for the bachelor pions. The D candidates are required to have an invariant mass within 15 MeV/c 2 of the nominal D 0 mass [26]; this requirement is 85 % efficient for signal Monte Carlo (MC) events.
The D 0 candidate and the two bachelor pion candidates are required to originate from a common vertex. Signal events are distinguished from background using two almost uncorrelated kinematic variables: the difference ∆E between the CM energy of the B candidate and √ s/2, and the beam-energy-substituted mass m ES = s/4 − p 2 B , where √ s is the total CM energy and p B is the momentum of the candidate B meson in the CM frame. We apply preselection criteria of −0.075 GeV < ∆E < 0.075 GeV and 5.272 GeV/c 2 < m ES < 5.286 GeV/c 2 ; these requirements are 86 % efficient for signal MC events. We make further use of these kinematic variables to discriminate signal from background in the fit described below. We exclude candidates consistent with the abundant B 0 → D * (2010) − π + decay by rejecting events which contain a candidate with D 0 π − invariant mass below 2.02 GeV/c 2 (to maintain the symmetry of the Dalitz plot, we also remove the region with D 0 π + invariant mass below the same value). These events are used as a control sample to monitor differences between data and MC.
To suppress the background contribution from continuum e + e − → qq (q = u, d, s, c) events, we construct a neural network (NN) discriminant that combines four variables commonly used to separate jet-like qq events from the more spherical BB events. These are: the 0 th order momentum-weighted monomial moment, 2 L 0 ; the ratio of the 2 nd order momentum-weighted monomial (L 2 ) to that of 0 th order (L 0 ), L 2 /L 0 ; the absolute value of the cosine of the angle between the B direction and the beam (z) axis, |cos θ B mom |; and the absolute value of the cosine of the angle between the B thrust axis and the beam (z) axis, |cos θ B thr |. All these variables are evaluated in the e + e − center-of-mass frame. We apply a requirement on the NN output that retains approximately 88 % of the signal and rejects ∼ 52 % of the continuum background. Most of the remaining background originates from B decays, and is discussed below.
After applying all selection criteria, we retain 26334 events with candidate B 0 → D 0 π + π − decays. Around 20% of these events have multiple candidates. When an event has multiple candidates we retain the candidate with the best geometrical B-vertex probability.
The efficiency for signal events to pass all the selection criteria is determined as a function of position in the Dalitz plot (DP). Using a Monte Carlo (MC) simulation in which events uniformly populate the phase-space, we obtain an average efficiency of approximately 35 %. The efficiency is shown as a function of phase-space in Figure 1, both in terms of the conventional DP (for which we choose axes m 2 + = m 2 D 0 π + and m 2 − = m 2 D 0 π − ), and in terms of the "square Dalitz plot" (SDP). The latter is described by the variables M and Θ, where m π + π − is the invariant mass of the two pions, m max π + π − = m B 0 − m D 0 and m min π + π − = 2m π are the kinematical limits of m π + π − , and θ π + π − is the angle between the D 0 and the π + in the π + π − rest frame. While the conventional DP representation provides a useful visual representation of the physics of the signal decay, the SDP allows closer scrutiny of the most densely populated regions of the phase-space, and hence is appropriate for studies of background distributions, for example.

BACKGROUNDS
In addition to the background from continuum processes, we expect backgrounds from other BB decays. These are studied using large MC samples in which the B mesons decay generically according to our current knowledge of their branching fractions. We classify backgrounds from BB decays in six categories based on their ∆E and m ES distributions as determined from MC samples. The different BB background categories also have different DP distributions. Table I lists the expected number of events and the dominant contributing mode for each category. Background categories 1 and 2 have four track final states, and peak in both ∆E and m ES -category 1 has signal-like peaks in both, while category 2 has a ∆E peak shifted to positive values due to pion→kaon misidentification. The decay modes that contribute to these categories do not contain real D mesons, with the exception of D 0 K 0 S , which contributes to category 1. Background categories 3-6, which are dominant, do contain real D mesons. Category 3 peaks in m ES and has a ∆E distribution that is shifted to negative values due to kaon→pion misidentification. Categories 4-6 do not peak strongly in either ∆E or m ES . Category 4 has a broad m ES distribution and a slight peak in ∆E, and includes background from D * (2010) − π + events that escape the veto due to misreconstruction. Category 5 has a broad m ES distribution (similar to category 4) and an approximately linear ∆E shape. Category 6 has combinatorial distributions for both m ES and ∆E. The continuum background shape is combinatorial and does not peak strongly in either ∆E or m ES . A summary of the backgrounds in given in Table I.

MAXIMUM LIKELIHOOD FIT
We perform an extended unbinned maximum likelihood fit using the variables ∆E, m ES and the DP co-ordinates in order to determine the signal yield and the properties of the Dalitz plot. The complete likelihood function is given Combinatoric 7336 ± 99 qq 5352 ± 226 by: where N k is the event yield for species k, the index i runs over the N e events in the data sample and P k is the probability density function (PDF) for species k, which consists of a product of the DP, m ES and ∆E PDFs. The different species k are signal, qq background and six BB background categories. The function − ln L is minimized to obtain the preferred values of the free parameters of the fit. For each of the BB background categories, the ∆E, m ES and DP PDFs are described with histograms obtained using MC. For qq background, the ∆E and m ES PDFs are a 1 st -order polynomial and an ARGUS function [27], respectively. The parameters of the ARGUS function are fixed to values determined using off-peak data, while the slope of the qq ∆E PDF is a free parameter of the fit. The continuum background DP PDF is modelled with a histogram obtained from data in a sideband region of m ES , after subtraction of the (MC-based) expected contribution from BB decays in this region. We have verified the consistency of our background PDFs in off-peak data, in background MC samples, and in on-peak data sidebands. All histograms used in the fit are in the square Dalitz plot format.
The signal component is composed of two parts which are distinguished by whether or not the kinematics of the daughter particles are well reconstructed. We refer to the well reconstructed events as "correctly reconstructed" (CR) and the misreconstructed events as "self-cross-feed" (SCF). The fraction of SCF events as a function of DP position f SCF (m 2 + , m 2 − ) is determined from MC, and is shown in Figure 2. Its value is typically below 10 % but is larger in the corners of the Dalitz plot where one of the pions has low momentum. Both CR and SCF events have the same underlying physics PDF, but due to misreconstruction SCF events have reconstructed DP positions that differ from their true values. This smearing is implemented by convoluting the PDF with a resolution function R SCF (m 2 + , m 2 − ;m 2 + ,m 2 − ) that gives the probability that an event with true DP position (m 2 + ,m 2 − ) is reconstructed at (m 2 + , m 2 − ), and is described by a histogram in the square Dalitz plot co-ordinates that is itself a function of position in the phase-space. For correctly reconstructed events, DP resolution effects are negligible.
The signal Dalitz plot PDF is thus written as where P phys (m 2 + , m 2 − ) is the underlying physics PDF (discussed below), ǫ(m 2 + , m 2 − ) is the efficiency (Figure 1), and f SCF (m 2 + , m 2 − ) is the SCF fraction ( Figure 2). The integral is over the Dalitz plot. The normalization factor N ensures that P sig (m 2 + , m 2 − ) gives unity when integrated over the phase-space. The CR and SCF signal events have different distributions in ∆E and m ES . For m ES , both CR and SCF PDFs are described by double Gaussian functions where the widths of the two Gaussians are constrained to be the same. For ∆E, the CR PDF is again a double Gaussian function (in which the two Gaussians have different widths) while the SCF PDF is represented by an asymmetric Gaussian with power-law tails. The two Gaussian widths of the ∆E PDF for the CR component are given by linear functions of m min We determine a nominal signal DP model using information from previous studies of B 0 → D 0 π + π − [22] and B + → D − π + π + [23,24], and the change in the fit likelihood value observed when omitting or adding resonances. We use the isobar model [28][29][30], which models the total amplitude as resulting from a sum of amplitudes from the individual decay channels: where F j (m 2 + , m 2 − ) are the dynamical amplitudes and c j are complex coefficients describing the relative magnitude and phase of the different decay channels. All the weak phase dependence is contained in the c j coefficients, which we express in terms of their real and imaginary parts: c j = x j + iy j , so F j (m 2 + , m 2 − ) contains kinematics and strong dynamics only. We treat the D 0 → K + π − decay as flavour-specific and neglect contributions from the doubly-Cabibbo-suppressed D 0 → K + π − decay. We assume direct CP violation is negligible and hence use the same model for B 0 → D 0 π + π − and its conjugate decay. We also neglect possible contributions from b → u mediated, and hence highly suppressed, transitions (e.g. B 0 → D * 2 (2460) + π − ). In the Dπ spectrum previous studies [22][23][24] have observed contributions from D * 2 (2460) and D * 0 (2400), as well as the effect of a virtual D * (D * v (2010)) amplitude. The latter amplitude is described as virtual since although the region around the narrow D * (2010) pole is vetoed, off-shell production can contribute to the amplitude -the effect is similar to a nonresonant P-wave term. We find that an additional nonresonant (S-wave) Dπ contribution is necessary to fit the data; we describe the nonresonant (NR) term using an empirical shape, first introduced in Ref. [31], proportional to e −iαm 2 − , where the shape parameter is determined from the data to be α = 0.60 ± 0.15 (statistical uncertainty only). In the π + π − spectrum previous studies [22] have observed contributions from ρ(770) 0 and f 2 (1270). We find it is necessary to include S-wave terms and hence include a contribution using the K-matrix formalism [32][33][34], described in more detail in the Appendix. To our knowledge, this is the first use of the K-matrix formalism in B meson decays. All other resonances are described using relativistic Breit-Wigner (RBW) shapes, with Blatt-Weisskopf barrier form factors [35] and angular distributions given in the Zemach tensor formalism [36,37]. The Dalitz plot formalism used in this analysis is the same as that described in more detail in several previous publications [38][39][40][41]. The masses and widths of all resonances are constrained to world-average values [26], while K-matrix parameters are fixed to the values tabulated in the Appendix.
In total there are 43 free parameters of the fit. These are the yields of signal, qq and the 6 BB background categories; the real and imaginary parts of 5 intermediate contributions to the signal DP model (not counting those of D * 2 (2460) − π + which are fixed as a reference); the real and imaginary parts of 10 complex coefficients in the production vector of the K-matrix parametrization of the π + π − S-wave; 2 parameters each of the CR signal ∆E and m ES PDFs and the slope of the continuum ∆E distribution.

RESULTS
The fit returns 5098 ± 102 signal events. For this and all other quantities the statistical uncertainties are calculated from an MC study where the events are generated from the PDFs and the PDF parameters are the central values from the fit to data. Yields of the various background categories are broadly in line with expectation, although there appears to be some cross-feed between BB categories. Projections of the fit result onto m ES and ∆E are shown in Figure 3, while projections onto each of the two-particle invariant masses are shown in Figure 4 and projections onto the cosines of the helicity angles, defined as the direction of one of the two daughters of the resonance relative to the direction of the third particle in the rest frame of the resonance, are shown in Figure 5. The signal distribution across the phase-space, in both conventional and square Dalitz plot co-ordinates, calculated using the s Plot technique [42], is shown in Figure 6. Structures due to the D * 2 (2460) − , ρ(770) 0 and f 2 (1270) resonances are clearly visible. Figures 4 and 5 show that our DP model gives an excellent representation of the data in most regions of the Dalitz plot. The only region where discrepancies between the data and the fit result are apparent is at low values of m − , where a sharp rise near threshold is observed. This structure also appears as a reflection in the m + and cos θ − distributions. We discuss this further when we consider model uncertainties, below. We calculate the fit fractions and interference fit fractions, shown as a matrix in Table II. The fit fractions are the elements along the diagonal, and are given by while the interference fraction are the off-diagonal elements and are given by for i < j only. Note that, with this definition, FF jj = 2FF j . These give a convention independent representation of the population of the DP. Although the sum of fit fractions can be greater than unity -in this case it is (148 ± 5) % (statistical uncertainty only) -the sum including interference fit fractions must be identically equal to one. The largest interference effect is between D * 0 (2400) − π + and the Dπ nonresonant amplitude. In Table III we give results for the branching fractions. The inclusive B 0 → D 0 π + π − branching fraction is calculated by dividing the signal yield by the average efficiency determined from the nominal model, by the number of BB pairs in the data sample, and by the branching fraction for the D decay (B(D 0 → K + π − ) = (3.91 ± 0.05) × 10 −2 [26]). The average efficiency is found to be 30.6 % and is further corrected for the measured data/MC differences (discussed under systematic uncertainties below). Our result compares well to that of Belle: B(B 0 → D 0 π + π − ) = (8.4 ± 0.4 ± 0.8) × 10 −4 [22]. The product branching fractions for the contributing decay modes are obtained by multiplying the 2047 · · · · · · · · · · · · · · · · · · D * 0 (2400) − π + 0.0000 0.2481 · · · · · · · · · · · · · · · ρ(770) 0 D 0 −0.0133 0.0264 0.3343 · · · · · · · · · · · · f2(1270)D 0 −0.0130 0.0223 0.0000 0.0983 · · · · · · · · · D * v (2010) − π + 0.0000 −0.0001 −0.0565 −0.0347 0.1579 · · · · · · Dπ NR 0.0000 inclusive branching fraction by the relevant fit fraction. Where possible, these have also been corrected for subdecay branching fractions (B(ρ(770) 0 → π + π − ) = (98.9 ± 0.16)%, B(f 2 (1270) → π + π − ) = (84.8 +2.4 −1.2 )% [26]). We are not able to perform such a calculation for D * 2 (2460) − π + since, although decay modes other than Dπ have been seen, the relative branching fractions are not known. The D * 0 (2400) has only been observed to decay into Dπ, but it may be presumptuous to conclude that its branching fraction is 100 %. Our results for D * 2 (2460) − π + , ρ(770) 0 D 0 and f 2 (1270)D 0 are consistent with those of Belle, while we see a somewhat larger branching fraction for D * v (2010) − π + and a much larger branching fraction for D * 0 (2400

SYSTEMATIC UNCERTAINTIES
We consider the following systematic effects on the values of the fit fractions.
• Fixed shapes of the efficiency, qq and BB Dalitz-plot histograms: The contents of all bins of square Dalitz plot histograms used to describe these shapes are fluctuated in accordance with the uncertainties. This procedure is repeated many times and the RMS of the distribution of the change in the fit results is taken as the associated systematic uncertainty.
• Fixed m ES and ∆E PDF parameters (or histograms): We vary any fixed parameters in the PDF descriptions by their uncertainties, taking correlations into account.  The variation in the fit results is taken as the systematic uncertainty. For most parameters, their values and uncertainties are determined from data control samples. An exception is the self-cross-feed fraction which is obtained from Monte Carlo. To conservatively allow for possible data/MC differences in the behaviour of the SCF component, we apply a Dalitz-plot independent scale factor that alternately increases and decreases the SCF fraction by a factor of two, and take the larger difference compared to the nominal result as the uncertainty. The contents of the histograms used to describe the BB background m ES and ∆E PDFs are varied using the same prescription as described above.
• Fit bias: We generate large ensembles of pseudo-experiments, containing fully simulated signal events, using the parameters returned by the fit to data. From the distribution of results of these ensembles, we evaluate biases on the fit parameters. All biases are found to be small compared to the statistical uncertainties. We assign systematic uncertainties of the sum in quadrature of half the bias and its uncertainty. Branching fraction results from the fit to data. The first uncertainty is statistical, the second is systematic, the third is due to the Dalitz-plot model, and the fourth (where present) is due to secondary branching fractions. The third column gives the product of the branching fraction of the B decay to the mode listed in the leftmost column with that of the intermediate resonance decay to the final state particles.  These sources of systematic uncertainty are summarized in Table IV. The total is obtained by combining all sources in quadrature. We consider additional systematic effects on the values of the branching fractions. These are uncertainties on the differences between the efficiencies of selection requirements on data and MC for tracking (1.0%), particle identification (4.0%), the neural network cut (3.2%), and the number of BB pairs (0.6%). Furthermore, where we have divided by a daughter branching fraction in order to isolate the B decay branching fraction, any uncertainty in the world average value used in the division also contributes systematic uncertainty.

Resonance Fit Fraction
An additional source of uncertainty in Dalitz-plot analyses arises due to the composition of the Dalitz plot. We consider the following sources of model uncertainty: • Fixed parameters of contributing amplitudes: We vary the masses and widths of all resonances described by RBW shapes according to the uncertainties of the world average values [26] (with the exception of the D * 0 (2400) mass, which we vary by ±100 MeV/c 2 to account for the discrepancy in the measured masses of charged and neutral isospin partners). We vary the α parameter of the Dπ nonresonant contribution within its uncertainty. We change the radius parameter of the Blatt-Weisskopf factors from its nominal value of 4 GeV −1 to both 3 GeV −1 and 5 GeV −1 .
• Alternative parameterisations: We use the Gounaris-Sakurai lineshape [43] as an alternative description for the ρ(770) resonance. We replace the π + π − S-wave K-matrix term with contributions used in the analysis of B 0 → D 0 π + π − by Belle [22], namely σ (described as in Ref. [44]), f 0 (980) (described by the Flatté distribution [45]) and f 0 (1370) (RBW). To address the possible discrepancy between the data and the fit result at low values of m − , we replace the Dπ nonresonant contribution with a functional form proposed for a putative "dabba" state [46]. We have also performed a fit in which the background from D * − (2010)π + events escaping the veto is treated as a separate (seventh) BB background category, and include the deviation in the results as a source of model uncertainty.
A summary is given in Table V. The total model uncertainty is obtained by combining all sources in quadrature.  [11,14,48,49]: A(D − ρ + ) = 1/3A 3/2 + 2/3A 1/2 , where A 3/2 and A 1/2 are the amplitudes for isospin 3/2 and 1/2 final states respectively. These equations give the triangle relation This relation can be used to determine cos δ Dρ , where δ Dρ is the phase between the A 3/2 and A 1/2 amplitudes, and R Dρ = A 1/2 / √ 2A 3/2 . In QCD factorization, both of these are expected to be unity up to corrections due to final state interactions of O(Λ QCD /m Q ), where Λ QCD is the QCD scale and m Q is either m c or m b [11]. We obtain constraints on these parameters using the same approach previously used in the D ( * ) π system [50,51]. Using our result for B(B 0 → D 0 ρ 0 ), together with world average values of B(B 0 → D − ρ + ), B(B + → D 0 ρ + ) and the ratio of lifetimes τ (B + )/τ (B 0 ) [26], we find cos δ Dρ = 0.998 +0.133 −0.062 , R Dρ = 0.68 +0. 15 −0.16 , where all sources of uncertainty are combined. These results suggest the presence of non-factorizable final state interaction effects that, in contrast to the D ( * ) π system, do not introduce a significant non-zero phase difference between the isospin amplitudes.

SUMMARY
We have performed a Dalitz-plot analysis of B 0 → D 0 π + π − decays using the whole BABAR dataset of 470.9 ± 2.8 million BB events. We measure the inclusive branching fraction B(B 0 → D 0 π + π − ) = (8.81 ± 0.18 ± 0.76 ± 0.78 ± 0.11) × 10 −4 where the first uncertainty is statistical, the second is systematic, the third is due to the Dalitz-plot model, and the fourth is due to secondary branching fractions. We find the Dalitz plot to be composed of contributions from D * 2 (2460) − , D * 0 (2400) − , ρ(770) 0 and f 2 (1270) as well as a π + π − S-wave, a Dπ nonresonant S-wave term and a virtual D * v (2010) − contribution. We determine their branching fractions: Our Dalitz plot model differs from that obtained in a previous study of B 0 → D 0 π + π − by Belle [22] in that (i) we use the K-matrix description of the π + π − S-wave, instead of including separate contributions from the f 0 (600) (σ), f 0 (980) and f 0 (1370) scalar resonances; (ii) we include an additional Dπ nonresonant S-wave term. Our results for the inclusive branching fraction and for the color-suppressed decays B 0 → ρ(770) 0 D 0 and B 0 → f 2 (1270)D 0 are consistent with those from Belle and (for ρ(770) 0 D 0 ) with theoretical predictions [12,52]. However, we find the product branching fractions for the broad and narrow D * * states (D * 0 (2400) and D * 2 (2460), respectively) to have similar values. This result disagrees with the analysis by Belle, which found a much smaller value for the D * 0 (2400) branching fraction.
We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariatà l'Energie Atomique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Research on Matter (The Netherlands), the Research Council of Norway, the Ministry of Education and Science of the Russian Federation, Ministerio de Ciencia e Innovación (Spain), and the Science and Technology Facilities Council (United Kingdom). Individuals have received support from the Marie-Curie IEF program (European Union), the A. P. Sloan Foundation (USA) and the Binational Science Foundation (USA-Israel).
The K matrix is expressed as where the factor g (α) i is the real coupling constant of the K matrix pole α (with mass m α ) to meson channel i, the parameters f scatt ij and s scatt 0 describe a smooth part for the K-matrix elements, and the last factor accounts for the so-called "Adler zero", and suppresses kinematically fake singularities near π + π − production threshold (s represents the square of the π + π − invariant mass). The K-matrix parameters are determined from global fits to scattering data experiments below 1900 MeV/c 2 [34]. Note that the phase-space for B 0 → D 0 π + π − extends beyond this limit, and that the K-matrix amplitude in this high-π + π − invariant mass region is therefore an extrapolation.
The parameters unique to the production vector, by contrast, must be determined from our data. The P vector is given by where as before the first term in the square brackets is nonresonant-like ("slowly varying"), and the second term is resonant-like. Hence the free parameters in the Dalitz plot fit are the complex coupling and production vector parameters β α and f prod 1j (we use a fixed value of s prod 0 ). The index j runs over the open channels for the ππ S-wave, which are: ππ, KK, ηη, ηη ′ and 4π (or multi-meson). At higher masses there are in principle more open channels, but this is not expected to affect the results significantly. Global fits to the scattering data determine the number of poles and their parameters. We use a 5 pole approximation, and give the values of all fixed parameters in the K-matrix model in Table VI. Note that all f prod ij = 0 for i = 1 since we are interested only in the ππ final state.