Dalitz plot analysis of B 0 → D ¯ ¯ ¯ ¯ 0 π + π − decays
LHCb collaboration, R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z. Ajaltouni, S. Akar, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. An, L. Anderlini, J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, P. d'Argent, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C. Baesso, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V. Batozskaya, V. Battista, A. Bay, L. Beaucourt, J. Beddow, F. Bedeschi, I. Bediaga, L.J. Bel, S. Belogurov, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, A. Bertolin, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Birnkraut, A. Bizzeti, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, M. Borsato, T.J.V. Bowcock, E. Bowen, C. Bozzi, D. Brett, M. Britsch, T. Britton, J. Brodzicka, N.H. Brook, A. Bursche, J. Buytaert, S. Cadeddu, R. Calabrese, M. Calvi, M. Calvo Gomez, P. Campana, D. Campora Perez, L. Capriotti, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, P. Carniti, L. Carson, K. Carvalho Akiba, et al. (614 additional authors not shown)
EEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2015-110LHCb-PAPER-2014-0707 May, 2015
Dalitz plot analysis of B → D π + π − decays The LHCb collaboration † Abstract
The resonant substructures of B → D π + π − decays are studied with the Dalitz plottechnique. In this study a data sample corresponding to an integrated luminosityof 3.0 fb − of pp collisions collected by the LHCb detector is used. The branchingfraction of the B → D π + π − decay in the region m ( D π ± ) > . /c is measuredto be (8 . ± . ± . ± . × − , where the first uncertainty is statistical,the second is systematic and the last arises from the normalisation channel B → D ∗ (2010) − π + . The π + π − S-wave components are modelled with the Isobar andK-matrix formalisms. Results of the Dalitz plot analyses using both models arepresented. A resonant structure at m ( D π − ) ≈ . /c is confirmed and itsspin-parity is determined for the first time as J P = 3 − . The branching fraction, massand width of this structure are determined together with those of the D ∗ (2400) − and D ∗ (2460) − resonances. The branching fractions of other B → D h decaycomponents with h → π + π − are also reported. Many of these branching fractionmeasurements are the most precise to date. The first observation of the decays B → D f (500), B → D f (980), B → D ρ (1450), B → D ∗ (2760) − π + and thefirst evidence of B → D f (2020) are presented. Published in Phys. Rev. D c (cid:13) CERN on behalf of the LHCb collaboration, license CC-BY-4.0. † Authors are listed at the end of this paper. a r X i v : . [ h e p - e x ] A ug i Introduction
The study of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism [1, 2] is a centraltopic in flavour physics. Accurate measurements of the various CKM matrix parametersthrough different processes provide sensitivity to new physics effects, by testing the globalconsistency of the Standard Model. Among them, the CKM angle β is expressed in termsof the CKM matrix elements as arg( − V cd V ∗ cb /V td V ∗ tb ). The most precise measurements havebeen obtained with the B → ( c ¯ c ) K ( ∗ )0 decays by BaBar [3], Belle [4] and more recentlyby LHCb [5]. The decay B → D π + π − through the b → c ¯ ud transition has sensitivityto the CKM angle β [6–10] and to new physics effects [11–14].The Dalitz plot analysis [15] of B → D π + π − decays, with the D → K + π − mode,is presented as the first step towards an alternative method to measure the CKM angle β . Two sets of results are given, where the π + π − S-wave components are modelledwith the Isobar [16–18] and K-matrix [19] formalisms. Dalitz plot analyses of the decay B → D π + π − have already been performed by Belle [20, 21] and BaBar [22]. Similarstudies for the charged B decays B − → D ( ∗ )+ π − π − have been published by the B -factories [23,24]. The LHCb dataset offers a larger and almost pure signal sample. Feynmandiagrams of the dominant tree level amplitudes contributing to the decay B → D π + π − are shown in Fig. 1. (a) (b)(c) Figure 1: Examples of tree diagrams via ¯ b → ¯ cu ¯ d transition to produce (a) π + π − resonances, (b)nonresonant three-body decay and (c) D π − resonances. In addition to the interest for the CKM parameter measurements, the analysis of theDalitz plot of the B → D π + π − decay is motivated by its rich resonant structure. Thedecay B → D π + π − contains information about excited D mesons decaying to Dπ , withnatural spin and parity J P = 0 + , 1 − , 2 + , ... A complementary Dalitz plot analysis ofthe decay B s → D K − π + was recently published by LHCb [25, 26], and constrains the The inclusion of charge conjugate states is implied throughout the paper. D K − ( D − sJ ) and K − π + states. The spectrum of excited D mesonsis predicted by theory [27,28] and contains the known states D ∗ (2010) , D ∗ (2400) , D ∗ (2460),as well as other unknown states not yet fully explored. An extensive discussion on theorypredictions for the c ¯ u , c ¯ d and c ¯ s mass spectra is provided in Refs. [26, 29]. More recentmeasurements performed in inclusive decays by BaBar [30] and LHCb [29], have led tothe observation of several new states: D ∗ (2650) , D ∗ (2760), and D ∗ (3000). However, theirspin and parity are difficult to determine from inclusive studies. Orbitally excited D mesons have also been studied in semi-leptonic B decays (see a review in Ref. [31]) withlimited precision. These are of prime interest both in the extraction of the CKM parameter | V cb | , where longstanding differences remain between exclusive and inclusive methods (seereview in Ref. [32]), and in recent studies of B → D ( ∗ ) τ ¯ ν τ [33] which have generated muchtheoretical discussion (see, e.g., Refs. [34, 35]).A measurement of the branching fraction of the decay B → D ρ is also presented. Thisstudy helps in understanding the effects of colour-suppression in B decays, which is due tothe requirement that the colour quantum numbers of the quarks produced from the virtual W boson must match those of the spectator quark to form a ρ meson [36–40]. Moreover,using isospin symmetry to relate the decay amplitudes of B → D ρ , B → D − ρ + and B + → D ρ + , effects of final state interactions (FSI) can be studied in those decays(see a review in Refs. [37, 41]). The previous measurement for the branching fractionof B → D ρ has limited precision, (3 . ± . × − [21], and is in agreement withtheoretical predictions that range from 1 . . × − [38, 42].Finally, a study of the π + π − system is performed on a broad phase-space range in B → D π + π − from 280 MeV /c ( ≈ m π ) to 3 . /c ( ≈ m B d − m D ), which is muchlarger than that accessible in charmed meson decays such as D → K π + π − [43–45] or in B decays such as B s ) → J/ψπ + π − [46–49]. The nature of the light scalar π + π − statesbelow 1 GeV /c ( J P C = 0 ++ ), and in particular the f (500) and f (980) states, has been alongstanding debate (see, e.g., Refs. [50–52]). Popular interpretations include tetraquarks,meson-meson bound states (molecules), or some other mixtures, where the iso-singlets f (500) and f (980) can mix, therefore leading to a non-trivial nature (e.g. pure s ¯ s state) ofthe f (980) and complicating the determination of the CKM phase φ s from B s → J/ψπ + π − decays [48, 53, 54]. In the tetraquark picture, the mixing angle, ω mix , between the f (980)and f (500) states is predicted to be | ω mix | ≈ ◦ [55, 56] (recomputed with the latestaverage of the mass of the κ meson 682 ±
29 MeV /c [32]). Other theory models based onQCD factorisation and its extensions [57, 58] predict that the f (500) and f (980) mixingangle ϕ mix for the q ¯ q model is 20 ◦ (cid:46) ϕ mix (cid:46) ◦ . The LHCb experiment, in the studyof B s ) → J/ψπ + π − decays [47–49], has already set stringent upper bounds on ϕ mix in B ( B s ) decay: ϕ mix < ◦ ( < . ◦ ) at 90% CL. For the first time, the f (500) − f (980)mixing in the B → D π + π − decay, both in q ¯ q and tetraquark pictures, is studied.The analysis of the decay B → D π + π − presented in this paper is based on a datasample corresponding to an integrated luminosity of 3 . − of pp collision data collectedwith the LHCb detector. Approximately one third of the data was obtained during 2011when the collision centre-of-mass energy was √ s = 7 TeV and the rest during 2012 with √ s = 8 TeV. 2he paper is organised as follows. A brief description of the LHCb detector as wellas the reconstruction and simulation software is given in Sec. 2. The selection of signalcandidates and the fit to the B candidate invariant mass distribution used to separateand to measure signal and background yields are described in Sec. 3. An overview of theDalitz plot analysis formalism is given in Sec. 4. Details and results of the amplitudeanalysis fits are presented in Sec. 5. In Sec. 6 the measurement of the B → D π + π − branching fraction is documented. The evaluation of systematic uncertainties is describedin Sec. 7. The results are given in Sec. 8, and a summary concludes the paper in Sec. 9. The LHCb detector [59] is a single-arm forward spectrometer covering the pseudorapidityrange 2 < η <
5, designed for the study of particles containing b or c quarks. Thedetector includes a high-precision tracking system consisting of a silicon-strip vertexdetector surrounding the pp interaction region [60], a large-area silicon-strip detectorlocated upstream of a dipole magnet with a bending power of about 4 Tm, and threestations of silicon-strip detectors and straw drift tubes [61] placed downstream of themagnet. The tracking system provides a measurement of momentum, p , with a relativeuncertainty that varies from 0.4% at low momentum to 0.6% at 100 GeV /c . The minimumdistance of a track to a primary vertex, the impact parameter (IP), is measured with aresolution of (15 + 29 /p T ) µ m, where p T is the component of p transverse to the beam,in GeV /c . Different types of charged hadrons are distinguished using information fromtwo ring-imaging Cherenkov detectors [62]. Photon, electron and hadron candidates areidentified by a calorimeter system consisting of scintillating-pad and preshower detectors,an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by asystem composed of alternating layers of iron and multiwire proportional chambers [63].The online event selection is performed by a trigger which consists of a hardware stage,based on information from the calorimeter and muon systems, followed by a softwarestage, which applies a full event reconstruction. At the hardware trigger stage, events arerequired to have a muon with high p T or a hadron, photon or electron with high transverseenergy in the calorimeters. For hadrons, the transverse energy threshold is 3.5 GeV. Thesoftware trigger requires a two-, three- or four-track secondary vertex with a significantdisplacement from the primary pp interaction vertices (PVs). At least one charged particlemust have a transverse momentum p T > . /c and be inconsistent with originatingfrom a PV. A multivariate algorithm [64] is used for the identification of secondary verticesconsistent with the decay of a b hadron. The p T of the photon from D ∗− s decay is too lowto contribute to the trigger decision.Simulated events are used to characterise the detector response to signal and certaintypes of background events. In the simulation, pp collisions are generated using Pythia [65]with a specific LHCb configuration [66]. Decays of hadronic particles are describedby
EvtGen [67], in which final-state radiation is generated using
Photos [68]. Theinteraction of the generated particles with the detector, and its response, are implemented3sing the
Geant4 toolkit [69] as described in Ref. [70].
Signal B candidates are formed by combining D candidates, reconstructed in the decaychannel K + π − , with two additional pion candidates of opposite charge. Reconstructedtracks are required to be of good quality and to be inconsistent with originating from a PV.They are also required to have sufficiently high p and p T and to be within kinematic regionswhere reasonable particle identification (PID) performance is achieved, as determinedby calibration samples of D ∗ + → D ( K − π + ) π + decays. The four final state tracks arerequired to be positively identified by the PID system. The D daughters are required toform a good quality vertex and to have an invariant mass within 100 MeV /c of the known D mass [32]. The D candidates and the two charged pion candidates are required toform a good vertex. The reconstructed D and B vertices are required to be significantlydisplaced from the PV. To improve the B candidate invariant mass resolution, a kinematicfit [71] is used, constraining the D candidate to its known mass [32].By requiring the reconstructed D vertex to be displaced downstream from the re-constructed B vertex, backgrounds from both charmless B decays and direct promptcharm production coming from the PV are reduced to a negligible level. Background from D ∗ (2010) − decays is removed by requiring m ( D π ± ) > . /c . Backgrounds fromdoubly mis-identified D → K + π − or doubly Cabibbo-suppressed D → K − π + decays arealso removed by this requirement.To further distinguish signal from combinatorial background, a multivariate analysisbased on a Fisher discriminant [72] is applied. The sPlot technique [73] is used tostatistically separate signal and background events with the B candidate mass used as thediscriminating variable. Weights obtained from this procedure are applied to the candidatesto obtain signal and background distributions that are used to train the discriminant. TheFisher discriminant uses information about the event kinematic properties, vertex quality,IP and p T of the tracks and flight distance from the PV. It is optimised by maximisingthe purity of the signal events.Signal candidates are retained for the Dalitz plot analysis if the invariant mass of the B meson lies in the range [5250, 5310] MeV /c and that of the D meson in the range[1840, 1890] MeV /c (called the signal region). Once all selection requirements are applied,less than 1 % of the events contain multiple candidates, and in those cases one candidateis chosen randomly.Background contributions from decays with the same topology, but having one ortwo mis-identified particles, are estimated to be less than 1 % and are not considered inthe Dalitz analysis. These background contributions include decays like B → D K + π − , B s → D K − π + [74], Λ b → D pπ − [75] and B → D π + π − with D → π + π − or D → K + K − .Partially reconstructed decays of the type B → D π + π − X , where one or more particlesare not reconstructed, have similar efficiencies to the signal channel decays. They are4 ) [MeV/c - p + p Dm( ) E v e n t s / ( M e V / c LHCb
Figure 2: Invariant mass distribution of B → D π + π − candidates. Data points are shown inblack. The fit is shown as a solid (red) line with the background component displayed as dashed(green) line. distributed in the region below the B mass. By requiring the invariant mass of B candidates to be larger than 5250 MeV /c , these backgrounds are reduced to a negligiblelevel, as determined by simulated samples of B → D ∗ π + π − and B → D ∗ ρ (770) with D ∗ decaying into D γ or D π under different hypotheses for the D ∗ helicity.The signal and combinatorial background yields are determined using an unbinnedextended maximum likelihood fit to the invariant mass distribution of B candidates. Theinvariant mass distribution is shown in Fig. 2, with the fit result superimposed. The fituses a Crystal Ball (CB) function [76] convoluted with a Gaussian function for the signaldistribution and a linear function for the combinatorial background distribution in themass range of [5250, 5500] MeV /c . Simulated studies validate this choice of signal shapeand the tail parameters of the CB function are fixed to those determined from simulation.Table 1 summarises the fit results on the free parameters, where µ B is the mean peakposition and σ G is the width of the Gaussian function. The parameter σ CB is the width ofthe Gaussian core of the CB function. The parameters f CB and p give the fit fractionof the CB function and the slope of the linear function that describes the backgrounddistribution. The yields of signal ( ν s ) and background ( ν b ) events given in Table 1 arecalculated within the signal region. The purity is (97 . ± .
2) %.
The analysis of the distribution of decays across the Dalitz plot [15] allows a determinationof the amplitudes contributing to the three-body B → D π + π − decay. Two of the three5 able 1: Results of the fit to the invariant mass distribution of B → D π + π − candidates.Uncertainties are statistical only. Parameter Value µ B . ± . /c σ G . ± . /c σ CB . ± . /c f CB . ± . p − . ± .
035 ( GeV /c ) − ν s ± ν b ± m ( D π + ) + m ( D π − ) + m ( π + π − ) = m B + m D + 2 m π , (1)are sufficient to describe the kinematics of the system. The two observables m ( D π − )and m ( π + π − ), where resonances are expected to appear, are chosen in this paper. Theseobservables are calculated with the masses of the B and D mesons constrained to theirknown values [32]. The invariant mass resolution has negligible effect and therefore it isnot modeled in the Dalitz plot analysis.The total decay amplitude is described by a coherent sum of amplitudes from resonantor nonresonant intermediate processes as M ( (cid:126)x ) = (cid:88) i c i A i ( (cid:126)x ) . (2)The complex coefficient c i and amplitude A i ( (cid:126)x ) describe the relative contribution anddynamics of the i -th intermediate state, where (cid:126)x represents the ( m ( D π − ) , m ( π + π − ))coordinates in the Dalitz plot. The Dalitz plot analysis determines the coefficients c i . Inaddition, fit fractions and interference fit fractions are also calculated to give a convention-independent representation of the population of the Dalitz plot. The fit fractions aredefined as F i = (cid:82) | c i A i ( (cid:126)x ) | d(cid:126)x (cid:82) | (cid:80) i c i A i ( (cid:126)x ) | d(cid:126)x , (3)and the interference fit fractions between the resonances i and j ( i < j ) are defined as F ij = (cid:82) c i c ∗ j A i ( (cid:126)x ) A ∗ j ( (cid:126)x )] d(cid:126)x (cid:82) | (cid:80) i c i A i ( (cid:126)x ) | d(cid:126)x , (4)where the integration is performed over the full Dalitz plot with m ( D π ± ) > . /c .Due to these interferences between different contributions, the sum of the fit fractions isnot necessarily equal to unity. 6he amplitude A i ( (cid:126)x ) for a specific resonance r with spin L is written as A i ( (cid:126)x ) = F ( L ) B ( q, q ) × F ( L ) r ( p, p ) × T L ( (cid:126)x ) × R ( (cid:126)x ) . (5)The functions F ( L ) B ( q, q ) and F ( L ) r ( p, p ) are the Blatt-Weisskopf barrier factors [77] forthe production, B → rh , and the decay, r → h h , of the resonance, respectively. Theparameters p and q are the momenta of one of the resonance daughters ( h or h ) and ofthe bachelor particle ( h ), respectively, both evaluated in the rest frame of the resonance.The value p ( q ) represents the value of p ( q ) when the invariant mass of the resonance isequal to its pole mass. The spin-dependent F B and F r functions are defined as L = 0 : F (0) ( z, z ) = 1 ,L = 1 : F (1) ( z, z ) = (cid:114) z z ,L = 2 : F (2) ( z, z ) = (cid:115) ( z − + 9 z ( z − + 9 z , (6) L = 3 : F (3) ( z, z ) = (cid:115) z ( z − + 9(2 z − z ( z − + 9(2 z − ,L = 4 : F (4) ( z, z ) = (cid:115) ( z − z + 105) + 25 z (2 z − ( z − z + 105) + 25 z (2 z − , where z (0) is equal to ( r BW × q (0) ) or ( r BW × p (0) ) . The value for the radius of theresonance, r BW , is taken to be 1.6 GeV − × (cid:126) c (= 0 . T L ( (cid:126)x ) represents the angular distribution for the decay of a spin L resonance. It is defined as L = 0 : T = 1 ,L = 1 : T = (cid:112) y cos θ × qp,L = 2 : T = ( y + 3 / θ − / × q p , (7) L = 3 : T = (cid:112) y (1 + 2 y / θ − θ ) / × q p ,L = 4 : T = (8 y /
35 + 40 y /
35 + 1)(cos θ −
30 cos ( θ ) /
35 + 3 / × q p . The helicity angle, θ , of the resonance is defined as the angle between the direction of themomenta p and q . The y dependence accounts for relativistic transformations between the B and the resonance rest frames [79, 80], where1 + y = m B + m ( h h ) − m h m ( h h ) m B . (8)Finally, R ( (cid:126)x ) is the resonant lineshape and is described by the relativistic Breit-Wigner(RBW) function unless specified otherwise,RBW( s ) = 1 m r − s − im r Γ ( L ) ( s ) , (9)7here s = m ( h h ) and m r is the pole mass of the resonance; Γ ( L ) ( s ), the mass-dependentwidth, is defined as Γ ( L ) ( s ) = Γ (cid:18) pp (cid:19) L +1 (cid:18) m r √ s (cid:19) F ( L ) r ( p, p ) , (10)where Γ is the partial width of the resonance, i.e., the width at the peak mass s = m r .The lineshapes of ρ (770), ρ (1450) and ρ (1700) are described by the Gounaris-Sakurai(GS) function [81], GS( s ) = m r (1 + Γ g/m r ) m r − s + f ( s ) − im r Γ ρ ( s ) , (11)where f ( s ) = Γ m r p (cid:34) ( h ( s ) − h ( m r )) p + ( m r − s ) p dhds (cid:12)(cid:12)(cid:12)(cid:12) s = m r (cid:35) ,h ( s ) = 2 π p √ s log (cid:18) √ s + 2 p m π (cid:19) ,g = 3 π m π p log (cid:18) m r + 2 p m π (cid:19) + m r πp − m π m r πp , and Γ ρ ( s ) = Γ (cid:20) pp (cid:21) (cid:20) m r s (cid:21) / . (12)The ρ − ω interference is taken into account by R ρ − ω ( s ) = GS ρ (770) ( s ) × (1 + ae iθ RBW ω (782) ( s )) , (13)where Γ is used, instead of the mass-dependent width Γ ( L ) ( s ), for ω (782) [82].The D ∗ (2010) − contribution is vetoed as described in Sec. 3. Possible remainingcontributions from the D ∗ (2010) − RBW tail or general D π − P-waves are modelled as R D ∗ (2010) ( m ( D π + )) = e − ( β + iβ ) m ( D π + ) , (14)where β and β are free parameters.The π + π − S-wave contribution is modelled using two alternative approaches, theIsobar model [16–18] or the K-matrix model [19]. Contributions from the f (500), f (980), f (2020) resonances and a nonresonant component are parametrised separately in theIsobar model and globally by one amplitude in the K-matrix model.In the Isobar model, the f (2020) resonance is modelled by a RBW function and themodelling of the f (500), f (980) resonances and the nonresonant contribution are describedas follows. The Bugg resonant lineshape [83] is employed for the f (500) contribution, R f (500) ( s ) = 1 / (cid:20) m r − s − g s − s A m r − s A z ( s ) − im r Γ tot ( s ) (cid:21) , (15)8here m r Γ ( s ) = g s − s A m r − s A ρ ( s ) ,g ( s ) = m r ( b + b s ) exp( − ( s − m r ) /A ) ,z ( s ) = j ( s ) − j ( m r ) ,j ( s ) = 1 π (cid:20) ρ log (cid:18) − ρ ρ (cid:19)(cid:21) ,m r Γ ( s ) = 0 . g ( s )( s/m r ) exp( − α | s − m K | ) ρ ( s ) ,m r Γ ( s ) = 0 . g ( s )( s/m r ) exp( − α | s − m η | ) ρ ( s ) ,m r Γ ( s ) = m r g π ρ π ( s ) /ρ π ( m r ) ,ρ π ( s ) = 1 / [1 + exp(7 . − . s )] , and Γ tot ( s ) = (cid:88) i =1 Γ i ( s ) . (16)The parameters are fixed to m r = 0 .
953 GeV /c , s A = 0 . m π , b = 1 .
302 GeV /c , b = 0 . A = 2 .
426 GeV /c and g π = 0 .
011 GeV /c [83]. The phase-space factors of thedecay channels ππ , KK and ηη correspond to ρ , , ( s ), respectively and are defined as ρ , , ( s ) = (cid:115) − m , , s , , , and 3 = π, K and η. (17)The Flatt´e formula [84] is used to describe the f (980) lineshape, R f (980) ( s ) = 1 m r − s − im r ( ρ ππ ( s ) g + ρ KK ( s ) g ) , (18)where ρ ππ ( s ) = 23 (cid:114) − m π ± s + 13 (cid:114) − m π s , and ρ KK ( s ) = 12 (cid:114) − m K ± s + 12 (cid:114) − m K s . (19)The parameters g and m r [46] are m r = 939 . ± . /c , g = 199 ±
30 MeV and g /g = 3 . ± . R NR ( m ( π + π − ) , m ( D π + )) = e iαm ( π + π − ) . (20)Its modulus equals unity, and a slowly varying phase over m ( π + π − ) accounts for rescat-tering effects of the π + π − final state and α is a free parameter of the model.9he K-matrix formalism [19] describes the production, rescattering and decay of the π + π − S-wave in a coherent way. The scattering matrix S , from an initial state to a finalstate, is S = I + 2 i (cid:0) ρ † (cid:1) / T ρ / , (21)where I is the identity matrix, ρ is a diagonal phase-space matrix and T is the transitionmatrix. The unitarity requirement SS † = I gives( T − + iρ ) † = T − + iρ. (22)The K-matrix is a Lorentz-invariant Hermitian matrix, defined as K − = T − + iρ . Theamplitude for a decay process, A i = ( I − i K ρ ) − ij P j , (23)is computed by combining the K-matrix obtained from scattering experiment with aproduction vector to describe process-dependent contributions. The K-matrix is modelledas a five-pole structure,K ij ( s ) = (cid:32)(cid:88) α g αi g αj m α − s + f scatt ij − s scatt0 s − s scatt0 (cid:33) − s A s − s A (cid:18) s − s A m π (cid:19) , (24)where the indexes i, j = 1 , , , , ππ , KK , ηη , ηη (cid:48) and multi-meson (mainly 4 π states) respectively. The coupling constant of the bare state α to the decay channel i , g αi , is obtained from a global fit of scattering data and is listed inTable 2. The mass m α is the bare pole mass and is in general different from the resonantmass of the RBW function. The parameters f scatt ij and s scatt0 are used to describe smoothscattering processes. The last factor of the K-matrix, − s A s − s A (cid:16) s − s A m π (cid:17) , regulates thesingularities near the π + π − threshold, the so-called “Adler zero” [85, 86]. The Hermitianproperty of the K-matrix imposes the relation f scatt ij = f scatt ji , and since only π + π − decaysare considered, if i (cid:54) = 1 and j (cid:54) = 1, f scatt ij is set to 0. The production vector is modelledwith P j = (cid:34) f prod1 j − s prod0 s − s prod0 + (cid:88) α β α g αj m α − s (cid:35) , (25)where f prod1 j and β α are free parameters. The singularities in the K-matrix and theproduction vector cancel when calculating the amplitude matrix element. An unbinned extended maximum likelihood fit is performed to the Dalitz plot distribution.The likelihood function is defined by L = L (cid:48) × √ πσ s exp (cid:18) − ( ν s − ν s ) σ s (cid:19) × √ πσ b exp (cid:18) − ( ν b − ν b ) σ b (cid:19) , (26)10 able 2: The K-matrix parameters used in this paper are taken from a global analysis of π + π − scattering data [22]. Masses and coupling constants are in units of GeV /c . m α g απ + π − g αKK g α π g αηη g αηη (cid:48) − . − . − . − . − . − . − . f scatt11 f scatt12 f scatt13 f scatt14 f scatt15 − . s scatt0 s prod0 s A s A − . − . − .
15 1where L (cid:48) = e − ( ν s + ν b ) ( ν s + ν b ) N N ! N (cid:89) i =1 (cid:20) ν s ν s + ν b f s ( (cid:126)x i ; θ s ) + ν b ν s + ν b f b ( (cid:126)x i ; θ b ) (cid:21) . (27)The background probability density function (PDF) is given by f b ( (cid:126)x ; θ b ) and is describedin Sec. 5.1. The signal PDF, f s ( (cid:126)x i ; θ s ), is described by M ( (cid:126)x i ; θ s ) ε ( (cid:126)x i ) (cid:82) M ( (cid:126)x ; θ s ) ε ( (cid:126)x ) d(cid:126)x , (28)where the decay amplitude, M ( (cid:126)x ; θ s ), is described in Sec. 4 and the efficiency variationover the Dalitz plot, ε ( (cid:126)x ), is described in Sec. 5.2. The fit parameters, θ s and θ b , includecomplex coefficients and resonant parameters like masses and widths. The value N isthe total number of reconstructed candidates in the signal region. The number of signaland background events, ν s and ν b , are floated and constrained by the yields, ν s and ν b ,determined by the D π + π − mass fit and shown with their statistical uncertainties inTable 1. The only significant source of candidates in the signal region, other than B → D π + π − decays, is from combinatorial background. It is modelled using candidates in the upper m ( D π + π − ) sideband ([5350, 5450] MeV /c ) with a looser requirement on the Fisherdiscriminant, and is shown in Fig. 3. The looser requirement gives a similar distribution inthe Dalitz plane but with lower statistical fluctuations. The Dalitz plot distribution of thecombinatorial background events lying in the upper-mass sideband is considered to providea reliable description of that in the signal region, as no dependence on m ( D π + π − ) isfound by studying the Dalitz distribution in a different upper-mass sideband region. Thecombinatorial background is modelled with an interpolated non-parametric PDF [87, 88]using an adaptive kernel-estimation algorithm [89].11 /c ) [GeV - p D( m ] / c ) [ G e V -p + p ( m E n t r i e s LHCb
Figure 3: Density profile of the combinatorial background events in the Dalitz plane obtainedfrom the upper m ( D π + π − ) sideband with a looser selection applied on the Fisher discriminant. The efficiency function ε ( (cid:126)x ) accounts for effects of reconstruction, triggering and selectionof the B → D π + π − signal events, and varies across the Dalitz plane. Two simulatedsamples are generated to describe its variation with several data-driven corrections. Oneis uniformly distributed over the phase space of the Dalitz plot and the other is uniformlydistributed over the square Dalitz plot, which models efficiencies more precisely at thekinematic boundaries. The square Dalitz plot is parametrised by two variables m (cid:48) and θ (cid:48) that each varies between 0 and 1 and are defined as m (cid:48) = 1 π arccos (cid:18) m ( π + π − ) − m ( π + π − ) min m ( π + π − ) max − m ( π + π − ) min − (cid:19) and θ (cid:48) = 1 π θ ( π + π − ) , (29)where m ( π + π − ) max = m B − m D , m ( π + π − ) min = 2 m π and θ ( π + π − ) is the helicity angleof the π + π − system.The two samples are fitted simultaneously with common fit parameters. A 4th-orderpolynomial function is used to describe the efficiency variation over the Dalitz plot. As theefficiency in the simulation is approximately symmetric over m ( D π + ) and m ( D π − ),the polynomial function is defined as ε ( x, y ) ∝ . a ( x + y ) + a ( x + y ) + a ( xy ) + a ( x + y ) + a ( x + y ) xy + a ( x + y ) + a ( x + y ) xy + a x y , (30)where x = m ( D π + ) − m ( m B − m π ) − m and y = m ( D π − ) − m ( m B − m π ) − m , (31)with m defined as [( m D + m π ) + ( m B − m π ) ] /
2. The fitted efficiency distribution overthe Dalitz plane is shown in Fig. 4. 12 igure 4: Efficiency function for the Dalitz variables obtained in a fit to the LHCb simulatedsamples.
The efficiency is corrected using dedicated control samples with data-driven methods.The corrections applied to the simulated samples include known differences betweensimulation and data that originate from the trigger, PID and tracking.
The Dalitz plot distribution from data in the signal region is shown in Fig. 5. The analysisis performed using the Isobar model and the K-matrix model. The nominal fit model ineach case is defined by considering many possible resonances and removing those that donot significantly contribute to the Dalitz plot analysis. The resulting resonant contributionsare given in Table 3 while the projections of the fit results are shown in Fig. 6 (Fig. 7) forthe Isobar (K-matrix) model.The comparisons of the S-wave results for the Isobar model and the K-matrix model areshown in Fig. 8. The results from the two models agree reasonably well for the amplitudesand phases. In the π + π − mass-squared region of [1 . , .
0] GeV /c , small structures areseen in the K-matrix model, indicating possible contributions from f (1370) and f (1500)states. These contributions are not significant in the Isobar model and are thus notincluded in the nominal fit: adding them results in marginal changes and shows similarqualitative behaviour to the K-matrix model as displayed on Fig. 8. The measured S-wavesfrom both models qualitatively agree with predictions given in Ref. [91].To see more clearly the resonant contributions in the region of the ρ (770) resonance,the data are plotted in the π + π − invariant mass-squared region [0 . , .
1] GeV /c in Fig. 9.In the region around 0.6 GeV /c , interference between the ρ (770) and ω (782) resonancesis evident. In the π + π − S-wave distributions of both the Isobar model and the K-matrixmodel, a peaking structure is seen in the region [0 . , .
0] GeV /c , which corresponds tothe f (980) resonance. The structure in the region [1 . , .
8] GeV /c corresponds to thespin-2 f (1270) resonance. 13 /c ) [GeV - p D( m ] / c ) [ G e V -p + p ( m E n t r i e s Figure 5: Dalitz plot distribution of candidates in the signal region, including backgroundcontributions. The red line shows the Dalitz plot kinematic boundary.Table 3: Resonant contributions to the nominal fit models and their properties. Parameters anduncertainties of ρ (770), ω (782), ρ (1450) and ρ (1700) come from Ref. [90], and those of f (1270)and f (2020) come from Ref. [32]. Parameters of f (500), f (980) and K-matrix formalism aredescribed in Sec. 4. Resonance Spin Model m r ( MeV /c ) Γ ( MeV) D π − P-wave 1 Eq. 14 Floated D ∗ (2400) − D ∗ (2460) − D ∗ J (2760) − ρ (770) 1 GS 775 . ± .
35 149 . ± . ω (782) 1 Eq. 13 781 . ± .
24 8 . ± . ρ (1450) 1 GS 1493 ±
15 427 ± ρ (1700) 1 GS 1861 ±
17 316 ± f (1270) 2 RBW 1275 . ± . . + 2 . − . ππ S-wave 0 K-matrix See Sec. 4 f (500) 0 Eq. 15 See Sec. 4 f (980) 0 Eq. 18 See Sec. 4 f (2020) 0 RBW 1992 ±
16 442 ± . , .
4] GeV /c of m ( D π − )are shown in Fig. 10. There is a significant contribution from the D ∗ J (2760) − resonanceobserved in Ref. [29] and a spin-3 assignment gives the best description. A detaileddiscussion on the determination of the spin of D ∗ J (2760) is provided in Sec. 8.2.The fit quality is evaluated by determining a χ value by comparing the data and thefit model in N bins = 256 bins that are defined adaptively to ensure approximately equal14 /c [GeV ) - p + p m( ) / c E v e n t s / ( . G e V DataIsobar fit - p + p - p D S-wave - p + p Background LHCb(a) ] /c [GeV ) - p + p m( ) / c E v e n t s / ( . G e V LHCb(b) ] /c [GeV ) - p Dm( ) / c E v e n t s / ( . G e V DataIsobar fit - p + p - p D S-wave - p + p Background LHCb(c) ] /c [GeV ) - p Dm( ) / c E v e n t s / ( . G e V LHCb(d)
Figure 6: Projections of the data and Isobar fit onto (a) m ( π + π − ) and (c) m ( D π − ) with alinear scale. Same projections shown in (b) and (d) with a logarithmic scale. Components aredescribed in the legend. The lines denoted D π − and π + π − include the coherent sums of all D π − resonances, π + π − resonances, and π + π − S-wave resonances. The various contributions donot add linearly due to interference effects. population with a minimum bin content of 37 entries. A value of 287 (296) is found for theIsobar (K-Matrix) model based on statistical uncertainties only. The effective number ofdegrees of freedom (nDoF) of the χ is bounded by N bins − N bins − N pars −
1, where N pars is the number of parameters determined by the data. Pseudo experiments give aneffective number of 234 (235) nDoF.Further checks of the consistency between the fitted models and the data are performedwith the unnormalised Legendre polynomial weighted moments as a function of m ( D π − )and m ( π + π − ). The corresponding distributions are shown in Appendix A.15 /c [GeV ) - p + p m( ) / c E v e n t s / ( . G e V DataK-matrix fit - p + p - p D S-wave - p + p Background LHCb(a) ] /c [GeV ) - p + p m( ) / c E v e n t s / ( . G e V LHCb(b) ] /c [GeV ) - p Dm( ) / c E v e n t s / ( . G e V DataK-matrix fit - p + p - p D S-wave - p + p Background LHCb(c) ] /c [GeV ) - p Dm( ) / c E v e n t s / ( . G e V LHCb(d)
Figure 7: Projections of the data and K-matrix fit onto (a) m ( π + π − ) and (c) m ( D π − ) with alinear scale. Same projections shown in (b) and (d) with a logarithmic scale. Components aredescribed in the legend. The lines denoted D π − and π + π − include the coherent sums of all D π − resonances, π + π − resonances, and π + π − S-wave resonances. The various contributions donot add linearly due to interference effects. B → D π + π − branching frac-tion Measuring the branching fractions of the different resonant contributions requires knowledgeof the B → D π + π − branching fraction. This branching fraction is normalised relative tothe B → D ∗ (2010) − π + decay that has the same final state, so systematic uncertainties arereduced. Identical selections are applied to select B → D ∗ (2010) − π + and B → D π + π − candidates, the only difference being that m ( D π − ) < . /c is used to select D ∗ (2010) − candidates. The kinematic constraints remove backgrounds from doublymis-identified D → K + π − or doubly Cabibbo-suppressed D → K − π + decays and norequirement is applied on m ( D π + ).The invariant mass distributions of m ( D π − ) and m ( D π + π − ) for the B → D ∗ (2010) − π + candidates are shown in Fig. 11 and are fitted simultaneously to deter-mine the signal and background contributions. The D ∗ (2010) − signal distribution is16 /c ) [GeV - p + p ( m , A r b it a r y s ca l e A m p lit ud e - - - - - K-matrixIsobar, nominal (1500) (1370), f Isobar with f LHCb(a) ] /c ) [GeV - p + p ( m ] (cid:176) P h a s e [ - K-matrixIsobar, nominal (1500) (1370), f Isobar with f LHCb(b)
Figure 8: Comparison of the π + π − S-wave obtained from the Isobar model and the K-matrixmodel, for (a) amplitudes and (b) phases. The K-matrix model is shown by the red solid line,two scenarios for the Isobar model with (black long dashed line) and without (blue dashed line) f (1370) and f (1500) are shown. ] /c [GeV ) - p + p m( ) / c E v e n t s / ( . G e V DataIsobar fit(770) r (782) w (980) f (1270) fS-waveBackground LHCb(a) ] /c [GeV ) − π + π m( ) / c E v e n t s / ( . G e V DataK matrix fit(770) ρ (782) ω S wave(1270) fBackground LHCb(b) Figure 9: Distributions of m ( π + π − ) in the ρ (770) mass region. The different fit componentsare described in the legend. Results from (a) the Isobar model and (b) the K-matrix model areshown. modelled by three Gaussian functions to account for resolution effects while its backgroundis modelled by a phase-space factor. The modelling of the signal and background shapesin the m ( D π + π − ) distribution are described in Sec. 3. The B → D ∗ (2010) − π + yield inthe signal region is 7327 ± B → D ∗ (2010) − π + and B → D π + π − decays areobtained from simulated samples. To take into account the resonant distributions in theDalitz plot, the B → D π + π − simulated sample is weighted using the model described inthe previous sections. The average efficiencies are (1 . ± . × − and (4 . ± . × − for the B → D ∗ (2010) − π + and B → D π + π − decays.Using the branching fractions of B ( B → D ∗ (2010) − π + ) = (2 . ± . × − /c [GeV ) - p Dm( ) / c E v e n t s / ( . G e V DataIsobar fit(2760) *J D * Other D - p + p Background LHCb(a) ] /c [GeV ) - p Dm( ) / c E v e n t s / ( . G e V DataK-matrix fit(2760) *J D * Other D - p + p Background LHCb(b)
Figure 10: Distributions of m ( D π − ) in the D ∗ J (2760) − mass region. The different fit componentsare described in the legend. Both results from (a) the Isobar model and (b) the K-matrix modelare shown. ] ) [MeV/c - p + p Dm( ) E v e n t s / ( M e V / c - DataFit + p - (2010) * D fi Signal: B + p - (2010) * Background: D - p + p D fi Background: BCombinatorial background
LHCb(a) ] ) [MeV/c - p Dm( ) E v e n t s / ( . M e V / c - LHCb(b)
Figure 11: Invariant mass distributions of (a) m ( D π + π − ) and (b) m ( D π − ) for B → D ∗ (2010) − π + candidates. The data is shown as black points with the fit superimposed asred solid lines. and B ( D ∗ (2010) − → D π − ) = (67 . ± . B → D π + π − in the kinematic region m ( D π ± ) > . /c is (8 . ± . ± . × − ,where the first uncertainty is statistical and the second uncertainty comes from thebranching fraction of the normalisation channel. Two categories of systematic uncertainties are considered, each of which is quoted separately.They originate from the imperfect knowledge of the experimental conditions and from the18ssumptions made in the Dalitz plot fit model. The Dalitz model-dependent uncertaintiesalso account for the precision on the external parameters. The various sources are assumedto be independent and summed in quadrature to give the total.Experimental systematic uncertainties arise from the efficiency and background mod-elling and from the veto on the D ∗ (2010) − resonance. Those corresponding to the signalefficiency are due to imperfect estimations of PID, trigger, tracking reconstruction effects,and to the finite size of the simulated samples. Each of these effects is evaluated by thedifferences between the results using efficiencies computed from the simulation and fromthe data-driven methods. The systematic uncertainties corresponding to the modelling ofthe small residual background are estimated by using different sub-samples of backgrounds.The systematic uncertainty due to the veto on the D ∗ (2010) − resonance is assigned bychanging the selection requirement from m ( D π ± ) > .
10 GeV /c to 2 .
05 GeV /c .The systematic uncertainties related to the Dalitz models considered (see Sec. 4) includeeffects from other possible resonant contributions that are not included in the nominal fit,from the modelling of resonant lineshapes and from imperfect knowledge of the parametersof the modelling, i.e. , the masses and widths of the π + π − resonances considered, and theresonant radius.The non-significant resonances added to the model for systematic studies are the f (1300), f (1500), f (cid:48) (1525), and D ∗ (2650) − ( f (cid:48) (1525) and D ∗ (2650) − ) mesons for theIsobar (K-matrix) model [29, 32, 48, 49]. The spin of the D ∗ (2650) − resonance is set to 1.The differences between each alternative model and the nominal model are conservativelyassigned as systematic uncertainties.The radius of the resonances ( r BW ) is set to a unique value of 1 . − × (cid:126) c in thenominal fit. In the systematic studies, it is floated as a free parameter and its best fitvalue is 1 . ± .
05 GeV − × (cid:126) c (1 . ± .
31 GeV − × (cid:126) c ) for the Isobar (K-matrix) model.The value 1.85 GeV − × (cid:126) c is chosen to estimate the systematic uncertainties due to theimperfect knowledge of this parameter.The masses and widths of the π + π − resonances considered are treated as free parameterswith Gaussian constraints according to the inputs listed in Table 3. The differencesbetween the results from those fits and those of the nominal fits are assigned as systematicuncertainties.For the Isobar model, additional systematic uncertainties due to the modelling ofthe f (500) and f (980) resonances are considered. The Bugg model [83] for the f (500)resonance and the Flatt´e model [84] for the f (980) resonance, used in the nominal fit,are replaced by more conventional RBW functions. The masses and widths, left asfree parameters, give 553 ±
15 MeV /c and 562 ±
39 MeV, for the f (500) meson and981 ±
13 MeV /c and 191 ±
39 MeV, for the f (980) meson. The resulting differences tothe nominal fit are assigned as systematic uncertainties.The kinematic variables are calculated with the masses of the D and B mesonsconstrained to their known values [32]. These kinematic constraints affect the extraction ofthe masses and widths of the D π − resonances. The current world average value for the B meson mass is 5279 . ± .
17 MeV /c and for the D meson is 1864 . ± .
07 MeV /c [32].A conservative and direct estimation of the systematic uncertainties on the masses and19 able 4: Systematic uncertainties on B ( B → D π + π − ). Source Uncertainty ( × − )PID 0 . . < . . B , D ∗ (2010) − mass model < . . . D π − resonances is provided by the sum in quadrature of the B and D mass uncertainties. The effects of mass constraints are found to be negligible for the fitfractions, moduli and phases of the complex coefficients.The systematic uncertainties are summarised for the Isobar (K-matrix) model Dalitzanalysis in Appendix B. Systematic uncertainties related to the measurements performedwith the Isobar formalism are listed in Tables 14 to 17, while those for the K-matrixformalism are given in Tables 18 to 21. In most of cases, the dominant systematicuncertainties are due to the D ∗ (2010) − veto and the model uncertainties related to otherresonances not considered in the nominal fit. In the Isobar model, the modelling of the f (500) and f (980) resonances also have non-negligible systematic effects.Several cross-checks have been performed to study the stability of the results. Theanalysis was repeated for different Fisher discriminant selection criteria, different triggerrequirements and different sub-samples, corresponding to the two data-taking periods andto the two half-parts of the D π + π − invariant mass signal region, above and below the B mass [32]. Results from those checks demonstrate good consistency with respect to thenominal fit results. No bias is seen, therefore no correction is applied, nor is any relateduncertainty assigned. B → D π + π − branchingfraction The systematic uncertainties related to the measurement of the B → D π + π − branchingfraction are listed in Table 4. The systematic uncertainties on the PID, trigger, reconstruc-tion and statistics of the simulated samples are calculated in a similar way to those of theDalitz plot analysis. Other systematic uncertainties are discussed below.The systematic uncertainty on the modelling of the D π − and D ∗ (2010) − π + invariantmass distributions is estimated by counting the number of signal events in the B signalregion assuming a flat background contribution. The D ∗ (2010) − mass region is restrictedto the range [2007, 2013] MeV /c for this estimate. The calculated branching fraction isnearly identical to that from the mass fit and thus has a negligible contribution to thesystematic uncertainty. The signal purity of B → D ∗ (2010) − π + is more than 99%.20 able 5: Statistical significance ( σ ) of π + π − resonances in the Dalitz plot analysis. For thestatistically significant resonances, the effect of adding dominant systematic uncertainties isshown (see text). Resonances ω (782) f (980) f (1370) ρ (1450) f (1500) f (cid:48) (1525) ρ (1700) f (2020)Isobar 8.0 10.7 1.1 8.7 1.1 3.6 4.5 10.2K-matrix 8.1 n/a n/a 8.6 n/a 2.6 2.2 n/aWith syst. 7.7 7.0 n/a 8.7 n/a n/a n/a 4.3 To account for the effect of resonant structures on the signal efficiency, the data sampleis divided using an adaptive binning scheme. The average efficiency is calculated in amodel independent way as ε ave = (cid:80) i N i (cid:80) i N i /ε i , (32)where N i is the number of events in bin i and ε i is the average efficiency in bin i calculatedfrom the efficiency model. The difference between this model-independent method andthe nominal is assigned as a systematic uncertainty. The Isobar and K-matrix models employed to describe the Dalitz plot of the B → D π + π − decay include all of the resonances listed in Table 3. The statistical significances of well-established π + π − resonances are calculated directly with their masses and widths fixedto the world averages. They are computed as the relative change of the minimum of thenegative logarithm of the likelihood (NLL) function with and without a given resonance.Besides the π + π − resonances listed in Table 3, the significances of the f (1370), f (1500)and f (cid:48) (1525) are also given. The results, expressed as multiples of Gaussian standarddeviations ( σ ), are summarised in Table 5. All of the other π + π − resonances not listed inthis Table have large statistical significances, well above fivestandard deviations.To test the significance of the D ∗ J (2760) − state, where J = 3 (see Sec. 8.2), an ensembleof pseudo experiments is generated with the same number of events as in the data sample,using parameters obtained from the fit with the D ∗ J (2760) − resonance excluded. Thedifference of the minima of the NLL when fitting with and without D ∗ J (2760) − is used asa test statistic. It corresponds to 11.4 σ (11.5 σ ) for the for the Isobar (K-matrix) modeland confirms the observation of D ∗ J (2760) − reported in Ref. [29]. The two other orbitallyexcited D resonances, D ∗ J (2650) − and D ∗ J (3000) − , whose observations are presented inthe same paper, are added into the nominal fit model with different spin hypotheses andtiny improvements are found. They also do not describe the data in the absence of the D ∗ J (2760) − . Those resonances are thus not confirmed by this analysis. Finally, an extra D π − resonance, with different spin hypotheses ( J = 0 , , , ,
4) and with its mass and21idth allowed to vary, is added to the nominal fit model and no significant contribution isfound.The significance of each of the significant ω (782), f (980), ρ (1450), f (2020) and D ∗ J (2760) − states is checked while including the dominant systematic uncertainties (seeSec. 7.1), namely, the modelling of the f (500) and f (980) resonances, the addition ofother resonant contributions and the modification of the D ∗ (2010) − veto criteria. In allconfigurations, the significances of the ω (782), f (980), ρ (1450) and D ∗ J (2760) − resonancesare greater than 7 . σ , 7 . σ , 8 . σ , and 10 . σ , respectively. The significance of the f (2020)drops to 4 . σ when using a RBW lineshape for the f (500) resonance. The abundant f (500) contribution is highly significant under all of the applied changes. As described in Sec. 5.3, a spin-3 D ∗ J (2760) − contribution gives the best description ofthe data. To obtain the significance of the spin-3 hypothesis with respect to other spinhypotheses ( J = 0 , , , D ∗ J (2760) − resonance are floatedin all the cases. Pseudo experiments are generated using the fit parameters obtainedusing the other spin hypotheses. Significances are calculated according to the distributionsobtained from the pseudo experiments of the test statistic and its values from data. Thesestudies indicate that data are inconsistent with other spin hypotheses by more than 10 σ .Following the discovery of the D ∗ sJ (2860) − meson, which is interpreted as the superpositionof two particles with spin 1 and spin 3 [25, 26], a similar configuration for the D ∗ J (2760) − has been tested and is found to give no significant improvement in the description of thedata. To illustrate the preference of the spin-3 hypothesis, the cosine of the helicity angledistributions in the mass-squared region of [7.4, 8.2] GeV /c for m ( D π − ) are shown inFig. 12 under the various scenarios. Based on our result, D ∗ J (2760) − is interpreted as the D ∗ (2760) − meson. Recently, LHCb observed a neutral spin-1 D ∗ (2760) state [92]. Thecurrent analysis does not preclude a charged spin-1 D ∗ state at around the same mass,but it is not sensitive to it with the current data sample size.Studies have also been performed to validate the spin-0 hypothesis of the D ∗ (2400) − resonance, as the spin of this state has never previously been confirmed in experiment [32].When moving to other spin hypotheses, the minimum of the NLL increases by more than250 units in all cases, which confirms the expectation of spin 0 unambiguously. The shape parameters of the π + π − resonances are fixed from previous measurements exceptfor the nonresonant contribution in the Isobar model. The fitted value of the parameter α defined in Eq. (20) is − . ± . σ statistical significancecompared to the case where there is no varying phase. An expansion of the model byincluding a varying phase in the D π − axis is also investigated but no significantly varying22 - p D( q cos - E v e n t s / . *J No DSpin 0Spin 1 Spin 2Spin 3Spin 4
LHCb(a)Isobar model ) - p D( q cos - E v e n t s / . *J No DSpin 0Spin 1 Spin 2Spin 3Spin 4
LHCb(b)K-matrix model
Figure 12: Cosine of the helicity angle distributions in the m ( D π − ) range [7.4, 8.2] GeV /c for (a) the Isobar model and (b) the K-matrix model. The data are shown as black points. Thehelicity angle distributions of the Dalitz plot fit results, without the D ∗ J (2760) − and with thedifferent spin hypotheses of D ∗ J (2760) − , are superimposed. phase in that system is seen. The results indicate a weak, but non-negligible, rescatteringeffect in the π + π − states, while the rescattering in the D π − states is not significant. Themasses, widths and other shape parameters of the D π − contributions are allowed tovary in the analysis. The values of the shape parameters of the D π − P-wave component,defined in Eq. (14), are β = 0 . ± .
05 (0 . ± .
04) and β = 0 . ± .
06 (0 . ± . D ∗ (2400) − , D ∗ (2460) − and D ∗ (2760) − are listed in Table 6. The present precision on the mass andwidth of the D ∗ (2400) − resonance is improved with respect to Refs. [29, 32]. The result forthe width of the D ∗ (2460) − meson is consistent with previous measurements, whereas theresult for the mass is above the world average which is dominated by the measurement usinginclusive production by LHCb [29]. In the previous LHCb inclusive analysis, the broad D ∗ (2400) − component was excluded from the fit model due to a high correlation with thebackground lineshape parameters, while here it is included. The present result supersedesthe former measurement. The Dalitz plot analysis used in this paper ensures that thebackground under the D ∗ (2460) − peak and the effect on the efficiency are under control,resulting in much lower systematic uncertainties compared to the inclusive approach.The moduli and the phases of the complex coefficients of the resonant contributions,defined in Eq. (2), are displayed in Tables 7 and 8. Compatible results are obtained usingboth the Isobar and K-matrix models. The results for the fit fractions are given in Table 9,while results for the interference fit fractions are given in Appendix C. Pseudo experimentsare used to validate the fitting procedure and no biases are found in the determination ofparameter values. 23 able 6: Measured masses ( m in MeV /c ) and widths (Γ in MeV) of the D ∗ (2400) − , D ∗ (2460) − and D ∗ (2760) − resonances, where the first uncertainty is statistical, the second and the third areexperimental and model-dependent systematic uncertainties, respectively.Isobar K-matrix D ∗ (2400) m ± ± ± ± ± ±
2Γ 217 ± ± ±
12 230 ± ± ± D ∗ (2460) m . ± . ± . ± . . ± . ± . ± .
3Γ 47 . ± . ± . ± . . ± . ± . ± . D ∗ (2760) m ± ± ± ± ± ±
3Γ 105 ± ± ±
23 154 ± ± ± | c i | ) K-matrix ( | c i | )Nonresonance 3 . ± . ± . ± .
51 n/a f (500) 18 . ± . ± . ± .
80 n/a f (980) 2 . ± . ± . ± .
46 n/a f (2020) 4 . ± . ± . ± .
78 n/a ρ (770) 1.0 (fixed) 1.0 (fixed) ω (782) 0 . ± . ± . ± .
01 0 . ± . ± . ± . ρ (1450) 0 . ± . ± . ± .
02 0 . ± . ± . ± . ρ (1700) 0 . ± . ± . ± .
008 0 . ± . ± . ± . f (1270) 0 . ± . ± . ± .
005 0 . ± . ± . ± . D π − P-wave 18 . ± . ± . ± . . ± . ± . ± . D ∗ (2400) − . ± . ± . ± . . ± . ± . ± . D ∗ (2460) − . ± . ± . ± .
02 1 . ± . ± . ± . D ∗ (2760) − . + 0 . − . ± . ± .
008 0 . + 0 . − . ± . ± . The measured branching fraction of the B → D π + π − decay in the phase-space region m ( D π ± ) > . /c is B ( B → D π + π − ) = (8 . ± . ± . ± . × − , (33)taking into account the systematic uncertainties reported in Table 4. The first uncertaintyis statistical, the second systematic, and the third the uncertainty from the branchingfraction of the B → D ∗ (2010) − π + normalisation decay channel. The result agrees withthe previous Belle measurement (8 . ± . ± . × − [21] and the BaBar measurement(8 . ± . ± . ± . ± . × − [22], obtained in a slightly larger phase-spaceregion. A multiplicative factor of 94.5% (96.2%) is required to scale the Belle (BaBar)results to the same phase-space region as in this analysis.24 able 8: The phase of the complex coefficients of the resonant contributions for the Isobarmodel and the K-matrix model. The first uncertainty is statistical, the second and the third areexperimental and model-dependent systematic uncertainties, respectively.Resonance Isobar (arg( c i ) ◦ ) K-matrix (arg( c i ) ◦ )Nonresonance 77 . ± . ± . ± . f (500) 38 . ± . ± . ± . f (980) 138 . ± . ± . ± . f (2020) 258 . ± . ± . ± . ρ (770) 0.0 (fixed) 0.0 (fixed) ω (782) 176 . ± . ± . ± . . ± . ± . ± . ρ (1450) 149 . ± . ± . ± . . ± . ± . ± . ρ (1700) 103 . ± . ± . ± . . ± . ± . ± . f (1270) 158 . ± . ± . ± . . ± . ± . ± . D π − P-wave 266 . ± . ± . ± . . ± . ± . ± . D ∗ (2400) − . ± . ± . ± . . ± . ± . ± . D ∗ (2460) − . ± . ± . ± . . ± . ± . ± . D ∗ (2760) − . ± . ± . ± . . ± . ± . ± . m ( D π ± ) > . /c . The first uncertainty is statistical, the second and the third areexperimental and model-dependent systematic uncertainties, respectively.Resonance Isobar ( F i %) K-matrix ( F i %)Nonresonance 2 . ± . ± . ± .
80 n/a f (500) 13 . ± . ± . ± .
45 n/a f (980) 1 . ± . ± . ± .
54 n/a f (2020) 1 . ± . ± . ± .
00 n/aS-wave 16 . ± . ± . ± .
46 16 . ± . ± . ± . ρ (770) 37 . ± . ± . ± .
98 36 . ± . ± . ± . ω (782) 0 . ± . ± . ± .
03 0 . ± . ± . ± . ρ (1450) 1 . ± . ± . ± .
22 2 . ± . ± . ± . ρ (1700) 0 . + 0 . − . ± . ± .
06 0 . ± . ± . ± . f (1270) 10 . ± . ± . ± .
10 9 . ± . ± . ± . D π − P-wave 9 . ± . ± . ± .
73 9 . ± . ± . ± . D ∗ (2400) − . ± . ± . ± .
35 9 . ± . ± . ± . D ∗ (2460) − . ± . ± . ± .
50 28 . ± . ± . ± . D ∗ (2760) − . ± . ± . ± .
09 1 . ± . ± . ± . The branching fraction of each quasi-two-body decay, B → r i h , with r i → h h , isgiven by B ( B → r i h ) × B ( r i → h h ) = B ( B → D π + π − ) × F i ε corr i , (34)where the resonant states ( h h ) = ( D π − ) , ( π + π − ). The fit fractions F i , defined in Eq. (3),25 able 10: Correction factors due to the D ∗ (2010) − veto. Resonance ε corr i % f (500) 99 . ± . f (980) 98 . ± . f (2020) 99 . ± . . ± . ρ (770) 98 . ± . ω (782) 99 . ± . ρ (1450) 95 . ± . ρ (1700) 96 . ± . f (1270) 91 . ± . D ∗ (2400) − . ± . D ∗ (2460) − .D ∗ (2760) − . are obtained from the Dalitz plot analysis and are listed in Table 9. The correction factors, ε corr i , account for the cut-off due to the D ∗ (2010) − veto. They are obtained by generatingpseudo experiment samples for each resonance over the Dalitz plot and applying the samerequirement ( m ( D π ± ) > . /c ). They are summarised in Table 10. The correctionfactors are the same for the Isobar model and the K-matrix model. The effects due to theuncertainties of the masses and widths of the resonances are included in the uncertaintiesgiven in the table.Using the overall B → D π + π − decay branching fraction, the fit fractions ( F i )and the correction factors ( ε corr i ), the branching fractions of quasi two-body decays arecalculated in Table 11. The first observation of the decays B → D f (500), B → D f (980), B → D ρ (1450), as well as B → D ∗ (2760) − π + , and the first evidenceof B → D f (2020) are reported. The present world averages [32] of the branchingfractions B ( B → D ρ (770)) × B ( ρ (770) → π + π − ), B ( B → D f (1270)) × B ( f (1270) → π + π − ), B ( B → D ∗ (2400) − π + ) × B ( D ∗ (2400) − → D π − ), and B ( B → D ∗ (2460) − π + ) ×B ( D ∗ (2460) − → D π − ) are improved considerably. When accounting for the branchingfractions of the ω (782) and f (1270) to π + π − , one obtains the following results for theIsobar model B ( B → D ω (782)) = (2 . ± . ± . ± . ± . +0 . − . ) × − (35)and B ( B → D f (1270)) = (16 . ± . ± . ± . ± . +0 . − . ) × − . (36)For the K-matrix model, one obtains B ( B → D ω (782)) = (2 . ± . ± . ± . ± . +0 . − . ) × − (37)and B ( B → D f (1270)) = (16 . ± . ± . ± . ± . +0 . − . ) × − . (38)26 able 11: Measured branching fractions of B ( B → rh ) × B ( r → h h ) for the Isobar andK-matrix models. The first uncertainty is statistical, the second the experimental systematic, thethird the model-dependent systematic, and the fourth the uncertainty from the normalisation B → D ∗ (2010) − π + channel.Resonance Isobar ( × − ) K-matrix ( × − ) f (500) 11 . ± . ± . ± . ± . f (980) 1 . ± . ± . ± . ± .
06 n/a f (2020) 1 . ± . ± . ± . ± .
06 n/aS-wave 14 . ± . ± . ± . ± . . ± . ± . ± . ± . ρ (770) 32 . ± . ± . ± . ± . . ± . ± . ± . ± . ω (782) 0 . ± . ± . ± . ± .
02 0 . ± . ± . ± . ± . ρ (1450) 1 . ± . ± . ± . ± .
06 1 . ± . ± . ± . ± . ρ (1700) 0 . ± . ± . ± . ± .
02 0 . ± . ± . ± . ± . f (1270) 9 . ± . ± . ± . ± . . ± . ± . ± . ± . D ∗ (2400) − . ± . ± . ± . ± . . ± . ± . ± . ± . D ∗ (2460) − . ± . ± . ± . ± . . ± . ± . ± . ± . D ∗ (2760) − . ± . ± . ± . ± .
05 1 . ± . ± . ± . ± . In both models, the fifth uncertainty is due to knowledge of the π + π − decay rates [32].The results are consistent with the measurement of the decay B → D ω (782), using thedominant ω (782) → π + π − π decay [32, 40]. f (980) and f (500) resonances In the Isobar model, significant contributions from both B → D f (500) and B → D f (980) decays are observed. The related branching fraction measurements can be usedto obtain information on the substructure of the f (980) and f (500) resonances within thefactorisation approximation. As discussed in Sec. 1, two models for the quark structure ofthose states are considered: q ¯ q or [ qq (cid:48) ][¯ q ¯ q (cid:48) ] (tetraquarks). In both models, mixing anglesbetween different quark states are determined using our measurements. In the q ¯ q model,the mixing between s ¯ s and u ¯ u or d ¯ d can be written as | f (980) (cid:105) = cos ϕ mix | s ¯ s (cid:105) + sin ϕ mix | n ¯ n (cid:105) , (39) | f (500) (cid:105) = − sin ϕ mix | s ¯ s (cid:105) + cos ϕ mix | n ¯ n (cid:105) , (40)where | n ¯ n (cid:105) ≡ ( | u ¯ u (cid:105) + | d ¯ d (cid:105) ) / √ ϕ mix is the mixing angle. In the [ qq (cid:48) ][¯ q ¯ q (cid:48) ] model, themixing angle, ω mix , is introduced and the mixing becomes | f (980) (cid:105) = cos ω mix | n ¯ ns ¯ s (cid:105) + sin ω mix | u ¯ ud ¯ d (cid:105) , (41) | f (500) (cid:105) = − sin ω mix | n ¯ ns ¯ s (cid:105) + cos ω mix | u ¯ ud ¯ d (cid:105) . (42)In both cases, the following variable is defined r f = B ( B → D f (980)) B ( B → D f (500)) × Φ(500)Φ(980) , (43)27 able 12: Systematic uncertainties on r f . The sum in quadrature of the uncertainties is alsoreported. Source r f PID 0 . . < . . . D ∗ (2010) − veto 0 . . . ππ res. mass, width 0 . f (500) model 0 . f (980) model 0 . . B rest frame. The value of their ratio is Φ(500) / Φ(980) = 1 . ± . . The value of the branching fraction B ( f (500) → π + π − ) = 2 / ππ final states. The ratio B ( f (980) → K + K − ) / B ( f (980) → π + π − ) = 0 . +0 . − . , obtainedfrom an average of the measurements by the BaBar [93] and BES [94] collaborations, isused to estimate the branching fraction B ( f (980) → π + π − ). Assuming that the ππ and KK decays are dominant in the f (980) decays, B ( f (980) → π + π − ) = 0 . ± .
06 isobtained. This gives r f = 0 . +0 . − . , taking into account the systematic uncertainties, as listed in Table 12.The parameter r f is related to the mixing angle by the equation r f = tan ϕ mix × (cid:12)(cid:12)(cid:12)(cid:12) F ( B → f (980)) F ( B → f (500)) (cid:12)(cid:12)(cid:12)(cid:12) (44)in the q ¯ q model and by r f = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − √ ω mix tan ω mix + √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12) F ( B → f (980)) F ( B → f (500)) (cid:12)(cid:12)(cid:12)(cid:12) (45)in the [ qq (cid:48) ][¯ q ¯ q (cid:48) ] tetraquark model [57, 58]. The form factors F ( B → f (980)) and F ( B → f (500)) are evaluated at the four-momentum transfer squared equal to the square ofthe D mass. Finally, values of the mixing angles as a function of form factor ratio are28 f fi (500))|/|F(B f fi |F(B ] (cid:176) M i x i ng a ng l e [ LHCb(a) (980))| f fi (500))|/|F(B f fi |F(B ] (cid:176) M i x i ng a ng l e [ LHCb(b)
Figure 13: Mixing angle as a function of form factor ratio for the (a) q ¯ q model and (b) [ qq (cid:48) ][¯ q ¯ q (cid:48) ]tetraquark model. Green band gives 1 σ interval around central values (black solid line). obtained in Fig. 13 for the q ¯ q model and the [ qq (cid:48) ][¯ q ¯ q (cid:48) ] tetraquark model. Such angles havealso been computed by LHCb for the decays B s ) → J/ψπ + π − [47–49].The expectation is that the ratio of form factors should be close to unity. However,LHCb has recently performed a search for the decay B s → D f (980) [95]. The limit seton this decay is below the value expected in a simple model based on our measured valueof B ( B → D f (500)) and assuming equal form factors. More complicated models maybe needed in order to explain all results.The above discussion is one possible interpretation of the results. Another possi-ble mechanism [91, 96] involves the generation of pseudo-scalar resonances through theinteractions of π + π − mesons. B → Dρ system The measured branching fraction of the B → D ρ (770) decay, presented in Table 11, canbe used to perform an isospin analysis of the B → Dρ system. Isospin symmetry relatesthe amplitudes of the decays B + → D ρ (770) + , B → D − ρ (770) + , and B → D ρ (770) ,which can be written as linear combinations of the isospin eigenstates A I with I = 1 / / A ( D ρ + ) = √ A / , (46) A ( D − ρ + ) = (cid:112) / A / + (cid:112) / A / ,A ( D ρ ) = (cid:112) / A / − (cid:112) / A / , leading to A ( D ρ + ) = A ( D − ρ + ) + √ A ( D ρ ) . (47)The strong phase difference between the amplitudes A / and A / is denoted by δ Dρ .Final-state interactions between the states D ρ and D − ρ + may lead to a value of δ Dρ different from zero and through constructive interference, to a larger value of B ( B → D ρ )29 able 13: Results of R Dρ and cos δ Dρ . Model R Dρ cos δ Dρ Isobar 0 . ± .
15 0 . +0 . − . K-matrix 0 . ± .
15 0 . +0 . − . than the prediction obtained within the factorisation approximation. In the heavy-quarklimit, the factorisation model predicts [97, 98] δ Dρ = O (Λ QCD /m b ) and the amplitude ratio R Dρ ≡ | A / |√ | A / | = 1 + O (Λ QCD /m b ), where m b represents the b quark mass and Λ QCD theQCD scale.Using our measurement of B ( B → D ρ ) together with the world average values of B ( B → D − ρ + ), B ( B + → D ρ + ), and the ratio of lifetimes τ ( B + ) /τ ( B ) [32], we obtain R Dρ = (cid:114) (cid:18) B ( D − ρ + ) + B ( D ρ )) B ( D ρ + ) × τ B + τ B − (cid:19) / (48)and cos δ Dρ = 14 R Dρ × (cid:18) τ B + τ B × B ( D − ρ + ) − B ( D ρ )) B ( D ρ + ) + 1 (cid:19) . (49)With a frequentist statistical approach [99], R Dρ and cos δ Dρ are calculated for the Isobarand K-matrix models in Table 13. These results are not significantly different from thepredictions of factorisation models. As opposed to the theoretical expectations [37, 41] andin contrast to the B → D ( ∗ ) π system [40], non-factorisable final-state interaction effects donot introduce a sizeable phase difference between the isospin amplitudes in the B → Dρ system . The precision on R Dρ and cos δ Dρ is dominated by that of the branching fractionsof the decays B + → D ρ (770) + (14%) and B → D − ρ (770) + (17%) [32]. The precisionof the branching fraction of the B → D ρ (770) decay is 7.3% (9.2%) for the Isobar(K-matrix) model (see Table 11). A Dalitz plot analysis of the B → D π + π − decay is presented. The decay model containsfour components from D π − resonances, four P-wave π + π − resonances and one D-wave π + π − resonance. Two models are used to describe the S-wave π + π − resonances. TheIsobar model uses four components, including the f (500), f (980), f (2020) resonancesand a nonresonant contribution. The K-matrix approach describes the π + π − S-wave usinga 5 × B → D π + π − and quasi-two-body decays are measured. Significant contributions fromthe f (500), f (980), ρ (1450) and D ∗ (2760) − mesons are observed for the first time. Forthe latter, this is a confirmation of the observation from previous inclusive measurements,and the spin-parity of this resonance is determined for the first time to be J P = 3 − .30his suggests a spectroscopic assignment of D , and shows that the 1 D family of charmresonances can be explored in Dalitz plot analysis of B -meson decays in the same way asrecently seen for the charm-strange resonances [25, 26]. Evidence for the f (2020) mesonis also seen for the first time. The measured branching fractions of two-body decays aremore precise than the existing world averages and there is good agreement between valuesfrom the Isobar and K-matrix models.The masses and widths of the D π − resonances are also determined. The measuredmasses and widths of the D ∗ (2400) − and D ∗ (2760) − states are consistent with the previousmeasurements. The precision on the D ∗ (2400) − meson is much improved. For themeasurement on the mass and width of the D ∗ (2460) − meson, the broad D ∗ (2400) − component was excluded from the fit model in the former LHCb inclusive analysis [29], dueto a high correlation with the background lineshape parameters, while here it is included.The present result therefore supersedes the former measurement.The significant contributions found for both the f (500) and f (980) allow us toconstrain on the mixing angle between the f (500) and f (980) resonances. An isospinanalysis in the B → Dρ decays using our improved measurement of the branching fractionof the decay B → D ρ is performed, indicating that non-factorisable effects fromfinal-state interactions are limited in the Dρ system. Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments forthe excellent performance of the LHC. We thank the technical and administrative staffat the LHCb institutes. We acknowledge support from CERN and from the nationalagencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3(France); BMBF, DFG, HGF and MPG (Germany); INFN (Italy); FOM and NWO (TheNetherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FANO(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (UnitedKingdom); NSF (USA). The Tier1 computing centres are supported by IN2P3 (France),KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC(Spain), GridPP (United Kingdom). We are indebted to the communities behind themultiple open source software packages on which we depend. We are also thankful forthe computing resources and the access to software R&D tools provided by Yandex LLC(Russia). Individual groups or members have received support from EPLANET, MarieSk(cid:32)lodowska-Curie Actions and ERC (European Union), Conseil g´en´eral de Haute-Savoie,Labex ENIGMASS and OCEVU, R´egion Auvergne (France), RFBR (Russia), XuntaGaland GENCAT (Spain), Royal Society and Royal Commission for the Exhibition of 1851(United Kingdom). 31 ppendicesA Unnormalised Legendre polynomial weighted mo-ments
Figures 14 and 15 show the distributions of the unnormalised Legendre polynomial weightedmoments < p UL > which display the contributions of resonances with spin larger than L/
2. The ρ (770) resonance can clearly be seen in the distributions with L ≤ D ∗ (2460) − resonance in the distributions with L ≤
4. Figures 16 and 17 display anexpanded version in low mass regions. The distributions from the Isobar and the K-matrixmodels are compatible with those from data.32 /c ) [GeV - p D( m / c / G e V æ U p Æ DataIsobar modelK-matrix modelLHCb(a) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - LHCb(b) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - LHCb(c) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - - LHCb(d) ] /c ) [GeV - p D( m / c / G e V æ U p Æ LHCb(e) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - LHCb(f) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - - LHCb(g) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - - LHCb(h)
Figure 14: The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspondto (a) to (h)) for background-subtracted and efficiency-corrected B → D π + π − data and theDalitz plot fit results as a function of m ( D π − ). /c ) [GeV - p + p ( m / c / G e V æ U p Æ DataIsobar modelK-matrix modelLHCb(a) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - LHCb(b) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ LHCb(c) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - - - - - LHCb(d) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - - LHCb(e) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - - LHCb(f) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - - - LHCb(g) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - - LHCb(h)
Figure 15: The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspondto (a) to (h)) for background-subtracted and efficiency-corrected B → D π + π − data and theDalitz plot fit results as a function of m ( π + π − ). /c ) [GeV - p D( m / c / G e V æ U p Æ DataIsobar modelK-matrix modelLHCb(a) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - LHCb(b) ] /c ) [GeV - p D( m / c / G e V æ U p Æ LHCb(c) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - LHCb(d) ] /c ) [GeV - p D( m / c / G e V æ U p Æ LHCb(e) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - LHCb(f) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - LHCb(g) ] /c ) [GeV - p D( m / c / G e V æ U p Æ - - LHCb(h)
Figure 16: The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspondto (a) to (h)) for background-subtracted and efficiency-corrected B → D π + π − data and theDalitz plot fit results as a function of m ( D π − ). Only results in the region m ( D π − ) < / c are shown. /c ) [GeV - p + p ( m / c / G e V æ U p Æ DataIsobar modelK-matrix modelLHCb(a) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - LHCb(b) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ LHCb(c) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - LHCb(d) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - - - LHCb(e) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - - LHCb(f) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - - - LHCb(g) ] /c ) [GeV - p + p ( m / c / G e V æ U p Æ - - LHCb(h)
Figure 17: The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspondto (a) to (h)) for background-subtracted and efficiency-corrected B → D π + π − data and theDalitz plot fit results as a function of m ( π + π − ). Only results in the region m ( π + π − ) < / c are shown. Systematic uncertainties on the parameters in theDalitz plot analysis
B.1 Systematic uncertainties for the Isobar model
Table 14: Systematic uncertainties on the D π − resonant masses (MeV/ c ) and widths (MeV)for the Isobar model. Source D ∗ (2400) − D ∗ (2460) − D ∗ (2760) − Γ m Γ m Γ m PID 1 . . < . < . . . . . . < . . . . . < . < . . . . . . < . . . . . . < . . . D ∗ (2010) − veto 4 . . < . < . . . . . . < . . . . . . . . . . . . < . . . ππ res. mass, width 3 . . . . . . B , D mass 0 . . . . . . f (500) model 2 . . . < . . . f (980) model 2 . . . . . . . . . . . . . . . . . . able 15: Systematic uncertainties on the moduli of the complex coefficients of the resonantcontributions for the Isobar model. The moduli are normalised to that of ρ (770). Source Nonres. f (500) ω (782) f (980) f (1270) ρ (1450)PID 0 .
02 0 . < .
01 0 . < .
001 0 . .
02 0 . < .
01 0 . < . < . < .
01 0 . < . < . < . < . < .
01 0 . < .
01 0 . < . < . < .
01 0 . < .
01 0 . < . < . D ∗ (2010) − veto 0 .
03 0 . < .
01 0 .
08 0 .
002 0 . .
04 0 . < .
01 0 . < .
001 0 . .
34 0 . < .
01 0 .
03 0 . < . .
11 0 .
10 0 .
01 0 .
04 0 .
001 0 . ππ res. mass, width 0 .
06 0 . < .
01 0 .
03 0 . < . f (500) model 0 .
36 n/a < .
01 0 .
46 0 .
004 0 . f (980) model 0 .
01 0 . < .
01 n/a < .
001 0 . .
51 0 .
80 0 .
01 0 .
46 0 .
005 0 . .
51 0 .
85 0 .
01 0 .
47 0 .
006 0 . ρ (1700) f (2020) D π − P-wave D ∗ (2400) − D ∗ (2460) − D ∗ (2760) − PID < .
001 0 .
06 0 .
14 0 . < . < . .
001 0 .
02 0 .
17 0 .
08 0 . < . < . < .
01 0 .
01 0 . < . < . .
001 0 .
02 0 .
10 0 .
03 0 . < . .
002 0 .
01 0 .
03 0 . < .
01 0 . D ∗ (2010) − veto 0 .
006 0 .
20 n/a 0 . < .
01 0 . .
006 0 .
21 0 .
25 0 .
31 0 .
02 0 . .
005 0 .
37 1 .
74 0 .
47 0 .
01 0 . .
002 0 .
27 0 .
05 0 .
12 0 . < . ππ res. mass, width 0 .
002 0 .
73 0 .
16 0 . < .
01 0 . f (500) model 0 .
004 1 .
56 0 .
62 0 .
20 0 .
01 0 . f (980) model 0 .
003 0 .
06 0 .
06 0 .
18 0 .
01 0 . .
008 1 .
78 1 .
86 0 .
59 0 .
02 0 . .
010 1 .
80 1 .
87 0 .
66 0 .
03 0 . able 16: Systematic uncertainties on the phases ( ◦ ) of the complex coefficients of the resonantcontributions for the Isobar model. The phase of ρ (700) is set to 0 ◦ as the reference. Source Nonres. f (500) ω (782) f (980) f (1270) ρ (1450)PID 0 . . . . . . . . . . . . < . < . < . < . . . . . . . . . . . < . . . . D ∗ (2010) − veto 1 . . . . . . . . . . . . . . < . . . . . . . . < . . ππ res. mass, width 2 . . . . . . f (500) model 3 . . . . . f (980) model 1 . . . . . . . . . . . . . . . . . ρ (1700) f (2020) D π − P-wave D ∗ (2400) − D ∗ (2460) − D ∗ (2760) − PID 0 . < . . . . . . . . . < . . < . . . . . . . . . . . . . . . < . . . D ∗ (2010) − veto 4 . . . . . . . . . . . . . . . . . . . . . . . ππ res. mass, width 0 . . . . . . f (500) model 0 . . . . . . f (980) model 2 . . . . . . . . . . . . . . . . . . able 17: Systematic uncertainties on the fit fractions (%) of the resonant contributions for theIsobar model. Source Nonres. f (500) ρ (770) ω (782) f (980) f (1270) ρ (1450)PID 0 .
02 0 .
23 0 .
37 0 .
01 0 .
04 0 .
07 0 . .
01 0 .
03 0 . < . < .
01 0 .
05 0 . .
01 0 .
09 0 .
39 0 .
01 0 .
01 0 . < . < .
01 0 .
09 0 . < .
01 0 .
01 0 . < . .
01 0 .
08 0 . < .
01 0 .
01 0 .
05 0 . D ∗ (2010) − veto 0 .
06 0 .
15 0 . < .
01 0 .
10 0 .
29 0 . .
07 0 .
31 0 .
61 0 .
01 0 .
11 0 .
31 0 . .
51 0 .
57 0 .
81 0 . < .
01 0 .
53 0 . .
21 0 .
03 0 .
50 0 .
01 0 .
07 0 .
10 0 . ππ res. mass, width 0 .
09 0 .
60 0 . < .
01 0 .
03 0 . < . f (500) model 0 .
57 2 .
25 0 .
04 0 .
01 0 .
50 0 .
94 0 . f (980) model 0 .
03 0 .
48 0 .
19 0 .
01 0 .
20 0 .
11 0 . .
80 2 .
45 0 .
98 0 .
03 0 .
54 1 .
10 0 . .
80 2 .
47 1 .
15 0 .
03 0 .
56 1 .
14 0 . ρ (1700) f (2020) S-wave D π − P-wave D ∗ (2400) − D ∗ (2460) − D ∗ (2760) − PID 0 .
01 0 .
01 0 .
31 0 .
19 0 .
12 0 .
43 0 . < .
01 0 .
03 0 .
04 0 .
02 0 .
06 0 .
28 0 . < .
01 0 .
01 0 .
12 0 .
13 0 .
14 0 .
18 0 . < .
01 0 .
01 0 .
05 0 .
06 0 .
04 0 .
09 0 . .
02 0 .
01 0 .
04 0 .
05 0 .
03 0 .
01 0 . D ∗ (2010) − veto 0 .
07 0 .
14 0 .
26 n/a 0 .
01 0 .
50 0 . .
07 0 .
15 0 .
43 0 .
24 0 .
20 0 .
74 0 . .
04 0 .
24 1 .
43 1 .
60 0 .
01 0 .
01 0 . .
02 0 .
21 0 .
07 0 .
17 0 .
28 0 .
44 0 . ππ res. mass, width 0 .
02 0 .
22 0 .
05 0 .
17 0 .
13 0 .
02 0 . f (500) model 0 .
03 0 .
92 0 .
25 0 .
60 0 .
13 0 .
22 0 . f (980) model 0 .
02 0 .
04 0 .
14 0 .
01 0 .
10 0 .
09 0 . .
06 1 .
00 1 .
46 1 .
73 0 .
35 0 .
50 0 . .
10 1 .
01 1 .
52 1 .
74 0 .
40 0 .
90 0 . .2 Systematic uncertainties for the K-matrix model Table 18: Systematic uncertainties on the D π − resonant masses (MeV/ c ) and widths (MeV)for the K-matrix model. Source D ∗ (2400) − D ∗ (2460) − D ∗ (2760) − Γ m Γ m Γ m PID 8 . . . . . . . . . < . . . < . < . < . < . < . < . . . . . . . . . . < . . . D ∗ (2010) − veto 14 . . . . . . . . . . . . . . . . . . . . < . . . . ππ res. mass, width 1 . . . . . . B , D mass 0 . . . . . . . . . . . . . . . . . . able 19: Systematic uncertainties on the moduli of the complex coefficients of the resonantcontributions for the K-matrix model. The moduli are normalised to that of ρ (770). Source ω (782) f (1270) ρ (1450) ρ (1700)PID < .
01 0 .
001 0 .
04 0 . < . < . < .
01 0 . < . < . < . < . < .
01 0 .
003 0 .
02 0 . < . < . < .
01 0 . D ∗ (2010) − veto 0 .
01 0 .
005 0 .
07 0 . .
01 0 .
006 0 .
08 0 . < .
01 0 . < .
01 0 . . < .
001 0 .
01 0 . ππ res. mass, width < . < . < .
01 0 . .
01 0 .
003 0 .
01 0 . .
01 0 .
007 0 .
08 0 . D π − P-wave D ∗ (2400) − D ∗ (2460) − D ∗ (2760) − PID 0 .
21 0 . < .
01 0 . .
12 0 .
01 0 . < . < . < . < .
01 0 . .
62 0 .
72 0 .
03 0 . .
13 0 . < .
01 0 . D ∗ (2010) − veto < .
01 0 . < .
01 0 . .
68 0 .
84 0 .
04 0 . .
52 0 . < .
01 0 . .
30 0 . < .
01 0 . ππ res. mass, width 0 .
05 0 . < .
01 0 . .
60 0 . < .
01 0 . .
91 0 .
98 0 .
04 0 . able 20: Systematic uncertainties on the phases ( ◦ ) of the complex coefficients of the resonantcontributions for the K-matrix model. The phase of ρ (700) is set to 0 ◦ as reference. Source ω (782) f (1270) ρ (1450) ρ (1700)PID 0 . . . . < . . < . . . < . < . . . . . . . . . . D ∗ (2010) − veto 1 . . . . . . . . . . . . . . . . ππ res. mass, width 0 . . . . . . . . . . . . D π − P-wave D ∗ (2400) − D ∗ (2460) − D ∗ (2760) − PID 3 . . . . . . . . < . < . < . . . . . . . . . . D ∗ (2010) − veto n/a 9 . . . . . . . . . . . . . . . ππ res. mass, width 0 . . . . . . . . . . . . able 21: Systematic uncertainties on the fit fractions (%) of the resonant contributions for theK-matrix model. Source ρ (770) ω (782) f (1270) ρ (1450) ρ (1700)PID 1 . < .
01 0 .
06 0 .
50 0 . .
33 0 .
01 0 .
04 0 .
01 0 . . < . < . < . < . .
17 0 .
01 0 .
45 0 .
22 0 . . < .
01 0 .
15 0 .
04 0 . D ∗ (2010) − veto 1 . < .
01 0 .
68 0 .
61 0 . .
13 0 .
01 0 .
83 0 .
82 0 . .
61 0 .
02 0 .
56 0 .
01 0 . .
49 0 .
01 0 .
15 0 .
21 0 . ππ res. mass, width 0 .
12 0 .
01 0 .
04 0 .
04 0 . .
79 0 .
02 0 .
58 0 .
21 0 . .
28 0 .
03 1 .
02 0 .
85 0 . D π − P-wave D ∗ (2400) − D ∗ (2460) − D ∗ (2760) − PID 0 .
77 0 .
59 0 .
13 0 .
28 0 . .
09 0 .
03 0 .
01 0 .
39 0 . . < . < . < . < . .
31 0 .
28 0 .
49 0 .
38 0 . .
23 0 .
13 0 .
08 0 .
03 0 . D ∗ (2010) − veto 1 .
44 n/a 0 .
69 0 .
86 0 . .
68 0 .
67 0 .
86 1 .
06 0 . .
08 0 .
63 0 .
23 0 .
28 0 . .
18 0 .
40 0 .
47 0 .
46 0 . ππ res. mass, width 0 .
07 0 .
07 0 .
02 0 .
01 0 . .
10 0 .
75 0 .
52 0 .
54 0 . .
01 1 .
00 1 .
01 1 .
19 0 . Results for the interference fit fractions
The central values of the interference fit fractions for the Isobar (K-matrix) model are givenin Table 22 (Table 23). The statistical, experimental systematic and model-dependentuncertainties on these quantities are given in Tables 24, 25 and 26 (Tables 27, 28 and 29).
Table 22: Interference fit fractions (%) of the resonant contributions for the Isobar model with m ( D π ± ) > . /c . The resonances are: ( A ) nonresonant S-wave, ( A ) f (500), ( A ) f (980), ( A ) f (2020), ( A ) ρ (770), ( A ) ω (782), ( A ) ρ (1450), ( A ) ρ (1700), ( A ) f (1270), ( A ) D π − P-wave, ( A ) D ∗ (2400) − , ( A ) D ∗ (2460) − , ( A ) D ∗ (2760) − . The diagonal elementscorrespond to the fit fractions given in Table 9. A A A A A A A A A A A A A A − . − .
25 0.00 0.00 0.00 0.00 0.13 0.79 − . − .
12 0.06 A − − . − .
53 0.00 0.00 0.00 0.00 0.14 3.37 0.97 3.81 0.57 A − − − . − .
60 0.63 − . − A − − − − . − . − . − . − . A − − − − − .
78 2.43 1.53 0.00 − . − . − . − . A − − − − − − .
01 0.00 0.00 0.00 0.01 − A − − − − − − − .
06 0.00 0.26 − .
74 0.94 0.04 A − − − − − − − − . − . − . − . A − − − − − − − − − . − . − . − . A − − − − − − − − − − .
01 0.00 A − − − − − − − − − − A − − − − − − − − − − − A − − − − − − − − − − − − Table 23: Interference fit fractions (%) of the resonant contributions for the K-matrix model with m ( D π ± ) > . /c . The resonances are: ( A ) K-matrix S-wave, ( A ) ρ (770), ( A ) ω (782),( A ) ρ (1450), ( A ) ρ (1700), ( A ) f (1270), ( A ) D π − P-wave, ( A ) D ∗ (2400) − , ( A ) D ∗ (2460) − ,( A ) D ∗ (2760) − The diagonal elements correspond to the fit fractions given in Table 9. A A A A A A A A A A A − .
06 2.37 − . − .
10 0.01 A − − .
84 4.20 2.10 0.00 − . − . − . − . A − − − .
01 0.00 0.00 0.00 0.01 − .
01 0.00 A − − − − .
43 0.00 − . − .
14 0.73 − . A − − − − − . − . − . − . A − − − − − − . − . − . − . A − − − − − − − A − − − − − − − A − − − − − − − − A − − − − − − − − − able 24: Statistical uncertainties on the interference fit fractions (%) of the resonant contribu-tions for the Isobar model with m ( D π ± ) > . /c . The resonances are: ( A ) nonresonantS-wave, ( A ) f (500), ( A ) f (980), ( A ) f (2020), ( A ) ρ (770), ( A ) ω (782), ( A ) ρ (1450),( A ) ρ (1700), ( A ) f (1270), ( A ) D π − P-wave, ( A ) D ∗ (2400) − , ( A ) D ∗ (2460) − , ( A ) D ∗ (2760) − . The diagonal elements correspond to the statistical uncertainties on the fit fractionsgiven in Table 9. A A A A A A A A A A A A A A A − A − − A − − − A − − − − A − − − − − A − − − − − − A − − − − − − − +0 . − . A − − − − − − − − A − − − − − − − − − A − − − − − − − − − − A − − − − − − − − − − − A − − − − − − − − − − − − Table 25: Experimental systematic uncertainties on the interference fit fractions (%) of theresonant contributions for the Isobar model with m ( D π ± ) > . /c . The resonancesare: ( A ) nonresonant S-wave, ( A ) f (500), ( A ) f (980), ( A ) f (2020), ( A ) ρ (770), ( A ) ω (782), ( A ) ρ (1450), ( A ) ρ (1700), ( A ) f (1270), ( A ) D π − P-wave, ( A ) D ∗ (2400) − , ( A ) D ∗ (2460) − , ( A ) D ∗ (2760) − . The diagonal elements correspond to the statistical uncertaintieson the fit fractions given in Table 9. A A A A A A A A A A A A A A A − A − − A − − − A − − − − A − − − − − A − − − − − − A − − − − − − − A − − − − − − − − A − − − − − − − − − A − − − − − − − − − − A − − − − − − − − − − − A − − − − − − − − − − − − able 26: Model-dependent systematic uncertainties on the interference fit fractions (%) ofthe resonant contributions for the Isobar model with m ( D π ± ) > . /c . The resonancesare: ( A ) nonresonant S-wave, ( A ) f (500), ( A ) f (980), ( A ) f (2020), ( A ) ρ (770), ( A ) ω (782), ( A ) ρ (1450), ( A ) ρ (1700), ( A ) f (1270), ( A ) D π − P-wave, ( A ) D ∗ (2400) − , ( A ) D ∗ (2460) − , ( A ) D ∗ (2760) − . The diagonal elements correspond to the statistical uncertaintieson the fit fractions given in Table 9. A A A A A A A A A A A A A A A − A − − A − − − A − − − − A − − − − − A − − − − − − A − − − − − − − A − − − − − − − − A − − − − − − − − − A − − − − − − − − − − A − − − − − − − − − − − A − − − − − − − − − − − − Table 27: Statistical uncertainties on the interference fit fractions (%) of the resonant contribu-tions for the K-matrix model with m ( D π ± ) > . /c . The resonances are: ( A )K-matrixS-wave, ( A ) ρ (770), ( A ) ω (782), ( A ) ρ (1450), ( A ) ρ (1700), ( A ) f (1270), ( A ) D π − P-wave,( A ) D ∗ (2400) − , ( A ) D ∗ (2460) − , ( A ) D ∗ (2760) − The diagonal elements correspond to thestatistical uncertainties on the fit fractions shown in Table 9. A A A A A A A A A A A A − A − − A − − − A − − − − A − − − − − A − − − − − − A − − − − − − − A − − − − − − − − A − − − − − − − − − able 28: Experimental systematic uncertainties on the interference fit fractions (%) of theresonant contributions for the K-matrix model with m ( D π ± ) > . /c . The resonancesare: ( A ) K-matrix S-wave, ( A ) ρ (770), ( A ) ω (782), ( A ) ρ (1450), ( A ) ρ (1700), ( A ) f (1270),( A ) D π − P-wave, ( A ) D ∗ (2400) − , ( A ) D ∗ (2460) − , ( A ) D ∗ (2760) − The diagonal elementscorrespond to the statistical uncertainties on the fit fractions shown in Table 9. A A A A A A A A A A A A − A − − A − − − A − − − − A − − − − − A − − − − − − A − − − − − − − A − − − − − − − − A − − − − − − − − − Table 29: Model-dependent systematic uncertainties on the interference fit fractions (%) of theresonant contributions for the K-matrix model with m ( D π ± ) > . /c . The resonancesare: ( A ) K-matrix S-wave, ( A ) ρ (770), ( A ) ω (782), ( A ) ρ (1450), ( A ) ρ (1700), ( A ) f (1270),( A ) D π − P-wave, ( A ) D ∗ (2400) − , ( A ) D ∗ (2460) − , ( A ) D ∗ (2760) − The diagonal elementscorrespond to the statistical uncertainties on the fit fractions shown in Table 9. A A A A A A A A A A A A − A − − A − − − A − − − − A − − − − − A − − − − − − A − − − − − − − A − − − − − − − − A − − − − − − − − − Results of the K-matrix parameters
The moduli and phases of the K-matrix parameters in Eq. (25) are listed in Table 30. Thebreak-down of systematic uncertainties are shown in Tables 31 and 32.
Table 30: The moduli and phases of the K-matrix parameters. The first uncertainty is statistical,the second the experimental systematic, and the third the model-dependent systematic. Themoduli are normalised to that of the ρ (770) contribution and the phase of ρ (770) is set to 0 ◦ .Parameter Modulus Phase ( ◦ ) f . ± . ± . ± . . ± . ± . ± . f . ± . ± . ± . . ± . ± . ± . f . ± . ± . ± . . ± . ± . ± . f . ± . ± . ± . . ± . ± . ± . f . ± . ± . ± . . ± . ± . ± . β . ± . ± . ± . . ± . ± . ± . β . ± . ± . ± . . ± . ± . ± . β . ± . ± . ± . . ± . ± . ± . β . ± . ± . ± . . ± . ± . ± . β . ± . ± . ± . . ± . ± . ± . ρ (770). Source f f f f f β β β β β PID 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . < . . . . . . . D ∗ (2010) − veto 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ππ res. mass, width 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . able 32: Systematic uncertainties on the phases ( ◦ ) of the K-matrix parameters. The phase of ρ (700) is set to 0 ◦ as the reference. Source f f f f f β β β β β PID 18 . . . . . . . . . . . . . . < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D ∗ (2010) − veto 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ππ res. mass, width 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eferences [1] N. Cabibbo, Unitary symmetry and leptonic decays , Phys. Rev. Lett. (1963) 531.[2] M. Kobayashi and T. Maskawa, CP -violation in the renormalizable theory of weakinteraction , Progress of Theoretical Physics (1973) 652.[3] BaBar collaboration, B. Aubert et al. , Measurement of time-dependent CP asymmetryin B → c ¯ cK ( ∗ )0 Decays , Phys. Rev.
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Sepp , N. Serra , J. Serrano , L. Sestini ,P. Seyfert , M. Shapkin , I. Shapoval , ,f , Y. Shcheglov , T. Shears , L. Shekhtman ,V. Shevchenko , A. Shires , R. Silva Coutinho , G. Simi , M. Sirendi , N. Skidmore ,I. Skillicorn , T. Skwarnicki , E. Smith , , E. Smith , J. Smith , M. Smith , H. Snoek ,M.D. Sokoloff , F.J.P. Soler , F. Soomro , D. Souza , B. Souza De Paula , B. Spaan ,P. Spradlin , S. Sridharan , F. Stagni , M. Stahl , S. Stahl , O. Steinkamp ,O. Stenyakin , F. Sterpka , S. Stevenson , S. Stoica , S. Stone , B. Storaci , S. Stracka ,t ,M. Straticiuc , U. Straumann , R. Stroili , L. Sun , W. Sutcliffe , K. Swientek ,S. Swientek , V. Syropoulos , M. Szczekowski , P. Szczypka , , T. Szumlak ,S. T’Jampens , T. Tekampe , M. Teklishyn , G. Tellarini ,f , F. Teubert , C. Thomas ,E. Thomas , J. van Tilburg , V. Tisserand , M. Tobin , J. Todd , S. Tolk ,L. Tomassetti ,f , D. Tonelli , S. Topp-Joergensen , N. Torr , E. Tournefier , S. Tourneur ,K. Trabelsi , M.T. Tran , M. 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Centro Brasileiro de Pesquisas F´ısicas (CBPF), Rio de Janeiro, Brazil Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil Center for High Energy Physics, Tsinghua University, Beijing, China LAPP, Universit´e Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France Clermont Universit´e, Universit´e Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France CPPM, Aix-Marseille Universit´e, CNRS/IN2P3, Marseille, France LAL, Universit´e Paris-Sud, CNRS/IN2P3, Orsay, France LPNHE, Universit´e Pierre et Marie Curie, Universit´e Paris Diderot, CNRS/IN2P3, Paris, France Fakult¨at Physik, Technische Universit¨at Dortmund, Dortmund, Germany Max-Planck-Institut f¨ur Kernphysik (MPIK), Heidelberg, Germany Physikalisches Institut, Ruprecht-Karls-Universit¨at Heidelberg, Heidelberg, Germany School of Physics, University College Dublin, Dublin, Ireland Sezione INFN di Bari, Bari, Italy Sezione INFN di Bologna, Bologna, Italy Sezione INFN di Cagliari, Cagliari, Italy Sezione INFN di Ferrara, Ferrara, Italy Sezione INFN di Firenze, Firenze, Italy Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy Sezione INFN di Genova, Genova, Italy Sezione INFN di Milano Bicocca, Milano, Italy Sezione INFN di Milano, Milano, Italy Sezione INFN di Padova, Padova, Italy Sezione INFN di Pisa, Pisa, Italy Sezione INFN di Roma Tor Vergata, Roma, Italy Sezione INFN di Roma La Sapienza, Roma, Italy Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Krak´ow, Poland AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science,Krak´ow, Poland National Center for Nuclear Research (NCBJ), Warsaw, Poland Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia Institute for High Energy Physics (IHEP), Protvino, Russia Universitat de Barcelona, Barcelona, Spain Universidad de Santiago de Compostela, Santiago de Compostela, Spain European Organization for Nuclear Research (CERN), Geneva, Switzerland Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland Physik-Institut, Universit¨at Z¨urich, Z¨urich, Switzerland Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, TheNetherlands NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine University of Birmingham, Birmingham, United Kingdom H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom Department of Physics, University of Warwick, Coventry, United Kingdom STFC Rutherford Appleton Laboratory, Didcot, United Kingdom School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom Imperial College London, London, United Kingdom School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom Department of Physics, University of Oxford, Oxford, United Kingdom Massachusetts Institute of Technology, Cambridge, MA, United States University of Cincinnati, Cincinnati, OH, United States University of Maryland, College Park, MD, United States Syracuse University, Syracuse, NY, United States Pontif´ıcia Universidade Cat´olica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to
Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China, associated to
Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to
Institut f¨ur Physik, Universit¨at Rostock, Rostock, Germany, associated to
National Research Centre Kurchatov Institute, Moscow, Russia, associated to
Yandex School of Data Analysis, Moscow, Russia, associated to
Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC, Valencia, Spain, associated to